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Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance. It is commonly used in the textile industry and often seen quoted on, for example, duvets and carpet underlay.The Shirley Institute in Manchester, England developed the tog as an easy-to-follow alternative to the SI unit of m2K/W. The name comes from the informal word "togs" for clothing which itself was probably derived from the word toga, a Roman garment.The basic unit of insulation coefficient is the RSI, (1 m2K/W). 1 tog = 0.1 RSI. There is also a clo clothing unit equivalent to 0.155 RSI or 1.55 tog...
The stone or stone weight (abbreviation: st.) is an English and imperial unit of mass now equal to 14 pounds (6.35029318 kg).England and other Germanic-speaking countries of northern Europe formerly used various standardised "stones" for trade, with their values ranging from about 5 to 40 local pounds (roughly 3 to 15 kg) depending on the location and objects weighed. The United Kingdom's imperial system adopted the wool stone of 14 pounds in 1835. With the advent of metrication, Europe's various "stones" were superseded by or adapted to the kilogram from the mid-19th century on. The stone continues...
Can you tell me why this question deserves to be negative?I tried to find faults and I couldn't: I did some research, I did all the calculations I could, and I think it is clear enough . I had deleted it and was going to abandon the site but then I decided to learn what is wrong and see if I ca...
I am a bit confused in classical physics's angular momentum. For a orbital motion of a point mass: if we pick a new coordinate (that doesn't move w.r.t. the old coordinate), angular momentum should be still conserved, right? (I calculated a quite absurd result - it is no longer conserved (an additional term that varies with time )
in new coordinnate: $\vec {L'}=\vec{r'} \times \vec{p'}$
$=(\vec{R}+\vec{r}) \times \vec{p}$
$=\vec{R} \times \vec{p} + \vec L$
where the 1st term varies with time. (where R is the shift of coordinate, since R is constant, and p sort of rotating.)
would anyone kind enough to shed some light on this for me?
From what we discussed, your literary taste seems to be classical/conventional in nature. That book is inherently unconventional in nature; it's not supposed to be read as a novel, it's supposed to be read as an encyclopedia
@BalarkaSen Dare I say it, my literary taste continues to change as I have kept on reading :-)
One book that I finished reading today, The Sense of An Ending (different from the movie with the same title) is far from anything I would've been able to read, even, two years ago, but I absolutely loved it.
I've just started watching the Fall - it seems good so far (after 1 episode)... I'm with @JohnRennie on the Sherlock Holmes books and would add that the most recent TV episodes were appalling. I've been told to read Agatha Christy but haven't got round to it yet
?Is it possible to make a time machine ever? Please give an easy answer,a simple one A simple answer, but a possibly wrong one, is to say that a time machine is not possible. Currently, we don't have either the technology to build one, nor a definite, proven (or generally accepted) idea of how we could build one. — Countto1047 secs ago
@vzn if it's a romantic novel, which it looks like, it's probably not for me - I'm getting to be more and more fussy about books and have a ridiculously long list to read as it is. I'm going to counter that one by suggesting Ann Leckie's Ancillary Justice series
Although if you like epic fantasy, Malazan book of the Fallen is fantastic
@Mithrandir24601 lol it has some love story but its written by a guy so cant be a romantic novel... besides what decent stories dont involve love interests anyway :P ... was just reading his blog, they are gonna do a movie of one of his books with kate winslet, cant beat that right? :P variety.com/2016/film/news/…
@vzn "he falls in love with Daley Cross, an angelic voice in need of a song." I think that counts :P It's not that I don't like it, it's just that authors very rarely do anywhere near a decent job of it. If it's a major part of the plot, it's often either eyeroll worthy and cringy or boring and predictable with OK writing. A notable exception is Stephen Erikson
@vzn depends exactly what you mean by 'love story component', but often yeah... It's not always so bad in sci-fi and fantasy where it's not in the focus so much and just evolves in a reasonable, if predictable way with the storyline, although it depends on what you read (e.g. Brent Weeks, Brandon Sanderson). Of course Patrick Rothfuss completely inverts this trope :) and Lev Grossman is a study on how to do character development and totally destroys typical romance plots
@Slereah The idea is to pick some spacelike hypersurface $\Sigma$ containing $p$. Now specifying $u(p)$ is trivial because the wave equation is invariant under constant perturbations. So that's whatever. But I can specify $\nabla u(p)|\Sigma$ by specifying $u(\cdot, 0)$ and differentiate along the surface. For the Cauchy theorems I can also specify $u_t(\cdot,0)$.
Now take the neigborhood to be $\approx (-\epsilon,\epsilon)\times\Sigma$ and then split the metric like $-dt^2+h$
Do forwards and backwards Cauchy solutions, then check that the derivatives match on the interface $\{0\}\times\Sigma$
Why is it that you can only cool down a substance so far before the energy goes into changing it's state? I assume it has something to do with the distance between molecules meaning that intermolecular interactions have less energy in them than making the distance between them even smaller, but why does it create these bonds instead of making the distance smaller / just reducing the temperature more?
Thanks @CooperCape but this leads me another question I forgot ages ago
If you have an electron cloud, is the electric field from that electron just some sort of averaged field from some centre of amplitude or is it a superposition of fields each coming from some point in the cloud? |
I've obtained some analytical results that I'd like to verify numerically by doing a double path integral. I haven't done this before with Mathematica and am unsure if it's possible, I'd like to do the following path integral;
$\int^{x_{a}}_{{x_{b}}} \int^{y_{a}}_{{y_{b}}}\exp \left[ \frac{i}{\hbar} \left( S[x(t)] - S[y(t)] \right) \right] \exp \left[ \frac{i}{\hbar} \int^{t}_{0}\left( F[x(\tau),y(\tau)] \right) d\tau \right] [dx(t)][dy(t)] $
where
$S[x(t)] = \int^{t}_{0} \left( \dot{x}(\tau) - v[x(\tau)] \right) d\tau$,
$v[x(\tau)] = x^{2}(\tau)$,
and
$F[x(\tau),y(\tau)] = \left(x(\tau) - y(\tau)\right)\left(\dot{x}(\tau) - \dot{y}(\tau)\right) - \left(x(\tau) - y(\tau)\right)^{2}$
(For context, it's quantum Brownian motion in which a particle is described within a bath of quantum harmonic oscillators)
Is it possible for Mathematica to do such calculations? If so, could someone point me in a direction to begin? |
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ALICE detector at the LHC. Transverse ...
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement is performed in the central pseudorapidity ...
Inclusive photon production at forward rapidities in proton-proton collisions at $\sqrt{s}$ = 0.9, 2.76 and 7 TeV
(Springer Berlin Heidelberg, 2015-04-09)
The multiplicity and pseudorapidity distributions of inclusive photons have been measured at forward rapidities ($2.3 < \eta < 3.9$) in proton-proton collisions at three center-of-mass energies, $\sqrt{s}=0.9$, 2.76 and 7 ...
Rapidity and transverse-momentum dependence of the inclusive J/$\mathbf{\psi}$ nuclear modification factor in p-Pb collisions at $\mathbf{\sqrt{\textit{s}_{NN}}}=5.02$ TeV
(Springer, 2015-06)
We have studied the transverse-momentum ($p_{\rm T}$) dependence of the inclusive J/$\psi$ production in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV, in three center-of-mass rapidity ($y_{\rm cms}$) regions, down to ...
Measurement of pion, kaon and proton production in proton–proton collisions at √s = 7 TeV
(Springer, 2015-05-27)
The measurement of primary π±, K±, p and p¯¯¯ production at mid-rapidity (|y|< 0.5) in proton–proton collisions at s√ = 7 TeV performed with a large ion collider experiment at the large hadron collider (LHC) is reported. ...
Two-pion femtoscopy in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
(American Physical Society, 2015-03)
We report the results of the femtoscopic analysis of pairs of identical pions measured in p-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Femtoscopic radii are determined as a function of event multiplicity and pair ...
Measurement of charm and beauty production at central rapidity versus charged-particle multiplicity in proton-proton collisions at $\sqrt{s}$ = 7 TeV
(Springer, 2015-09)
Prompt D meson and non-prompt J/$\psi$ yields are studied as a function of the multiplicity of charged particles produced in inelastic proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV. The results ...
Charged jet cross sections and properties in proton-proton collisions at $\sqrt{s}=7$ TeV
(American Physical Society, 2015-06)
The differential charged jet cross sections, jet fragmentation distributions, and jet shapes are measured in minimum bias proton-proton collisions at centre-of-mass energy $\sqrt{s}=7$ TeV using the ALICE detector at the ...
Centrality dependence of high-$p_{\rm T}$ D meson suppression in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Springer, 2015-11)
The nuclear modification factor, $R_{\rm AA}$, of the prompt charmed mesons ${\rm D^0}$, ${\rm D^+}$ and ${\rm D^{*+}}$, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at a centre-of-mass ...
K*(892)$^0$ and $\Phi$(1020) production in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2015-02)
The yields of the K*(892)$^0$ and $\Phi$(1020) resonances are measured in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV through their hadronic decays using the ALICE detector. The measurements are performed in multiple ... |
We consider a wireless communication system in which $N$ transmitter-receiverpairs want to communicate with each other. Each transmitter transmits data at acertain rate using a power that depends on the channel gain to its receiver. Ifa receiver can successfully receive the message, it sends an acknowledgment(ACK), else it sends a negative ACK (NACK). Each user aims to maximize itsprobability of successful transmission. We formulate this problem as astochastic game and propose a fully distributed learning algorithm to find acorrelated equilibrium (CE). In addition, we use a no regret algorithm to finda coarse correlated equilibrium (CCE) for our power allocation game. We alsopropose a fully distributed learning algorithm to find a Pareto optimalsolution. In general Pareto points do not guarantee fairness among the users,therefore we also propose an algorithm to compute a Nash bargaining solutionwhich is Pareto optimal and provides fairness among users. Finally, under thesame game theoretic setup, we study these equilibria and Pareto points wheneach transmitter sends data at multiple rates rather than at a fixed rate. Wecompare the sum rate obtained at the CE, CCE, Nash bargaining solution and thePareto point and also via some other well known recent algorithms.
We consider a Gaussian interference channel with independent direct and crosslink channel gains, each of which is independent and identically distributedacross time. Each transmitter-receiver user pair aims to maximize its long-termaverage transmission rate subject to an average power constraint. We formulatea stochastic game for this system in three different scenarios. First, weassume that each user knows all direct and cross link channel gains. Later, weassume that each user knows channel gains of only the links that are incidenton its receiver. Lastly, we assume that each user knows only its own directlink channel gain. In all cases, we formulate the problem of finding a Nashequilibrium (NE) as a variational inequality (VI) problem. We present a novelheuristic for solving a VI. We use this heuristic to solve for a NE of powerallocation games with partial information. We also present a lower bound on theutility for each user at any NE in the case of the games with partialinformation. We obtain this lower bound using a water-filling like powerallocation that requires only knowledge of the distribution of a user's ownchannel gains and average power constraints of all the users. We also provide adistributed algorithm to compute Pareto optimal solutions for the proposedgames. Finally, we use Bayesian learning to obtain an algorithm that convergesto an $\epsilon$-Nash equilibrium for the incomplete information game withdirect link channel gain knowledge only without requiring the knowledge of thepower policies of the other users.
We consider a wireless channel shared by multiple transmitter-receiver pairs.Their transmissions interfere with each other. Each transmitter-receiver pairaims to maximize its long-term average transmission rate subject to an averagepower constraint. This scenario is modeled as a stochastic game. We providesufficient conditions for existence and uniqueness of a Nash equilibrium (NE).We then formulate the problem of finding NE as a variational inequality (VI)problem and present an algorithm to solve the VI using regularization. We alsoprovide distributed algorithms to compute Pareto optimal solutions for theproposed game.
The optimal tradeoff between average service cost rate, average utility rate,and average delay is addressed for a state dependent M/M/1 queueing model, withcontrollable queue length dependent service rates and arrival rates. For amodel with a constant arrival rate $\lambda$ for all queue lengths, we obtainan asymptotic characterization of the minimum average delay, when the averageservice cost rate is a small positive quantity, $V$, more than the minimumaverage service cost rate required for queue stability. We show that dependingon the value of the arrival rate $\lambda$, the assumed service cost ratefunction, and the possible values of the service rates, the minimum averagedelay either: a) increases only to a finite value, b) increases without boundas $\log\frac{1}{V}$, c) increases without bound as $\frac{1}{V}$, or d)increases without bound as $\frac{1}{\sqrt{V}}$, when $V \downarrow 0$. We thenextend our analysis to (i) a complementary problem, where the tradeoff ofaverage utility rate and average delay is analysed for a M/M/1 queueing model,with controllable queue length dependent arrival rates, but a constant servicerate $\mu$ for all queue lengths, and (ii) a M/M/1 queueing model, withcontrollable queue length dependent service rates and arrival rates, for whichwe obtain an asymptotic characterization of the minimum average delay underconstraints on both the average service cost rate as well as the averageutility rate. The results that we obtain are useful in obtaining intuition aswell guidance for the derivation of similar asymptotic lower bounds, such asthe Berry-Gallager asymptotic lower bound, for discrete time queueing models.
We consider a fading point-to-point link with packets arriving randomly atrate $\lambda$ per slot to the transmitter queue. We assume that thetransmitter can control the number of packets served in a slot by varying thetransmit power for the slot. We restrict to transmitter scheduling policiesthat are monotone and stationary, i.e., the number of packets served is anon-decreasing function of the queue length at the beginning of the slot forevery slot fade state. For such policies, we obtain asymptotic lower bounds forthe minimum average delay of the packets, when average transmitter power is asmall positive quantity $V$ more than the minimum average power required fortransmitter queue stability. We show that the minimum average delay growseither to a finite value or as $\Omega\brap{\log(1/V)}$ or $\Omega\brap{1/V}$when $V \downarrow 0$, for certain sets of values of $\lambda$. These sets aredetermined by the distribution of fading gain, the maximum number of packetswhich can be transmitted in a slot, and the transmit power function of thefading gain and the number of packets transmitted that is assumed. We identifya case where the above behaviour of the tradeoff differs from that obtainedfrom a previously considered approximate model, in which the random queuelength process is assumed to evolve on the non-negative real line, and thetransmit power function is strictly convex. We also consider a fadingpoint-to-point link, where the transmitter, in addition to controlling thenumber of packets served, can also control the number of packets admitted inevery slot. Our approach, which uses bounds on the stationary probabilitydistribution of the queue length, also leads to an intuitive explanation of theasymptotic behaviour of average delay in the regime where $V \downarrow 0$.
We study sensor networks with energy harvesting nodes. The generated energyat a node can be stored in a buffer. A sensor node periodically senses a randomfield and generates a packet. These packets are stored in a queue andtransmitted using the energy available at that time at the node. For suchnetworks we develop efficient energy management policies. First, for a singlenode, we obtain policies that are throughput optimal, i.e., the data queuestays stable for the largest possible data rate. Next we obtain energymanagement policies which minimize the mean delay in the queue. We also compareperformance of several easily implementable suboptimal policies. A greedypolicy is identified which, in low SNR regime, is throughput optimal and alsominimizes mean delay. Next using the results for a single node, we developefficient MAC policies.
We study a sensor node with an energy harvesting source. The generated energycan be stored in a buffer. The sensor node periodically senses a random fieldand generates a packet. These packets are stored in a queue and transmittedusing the energy available at that time. We obtain energy management policiesthat are throughput optimal, i.e., the data queue stays stable for the largestpossible data rate. Next we obtain energy management policies which minimizethe mean delay in the queue.We also compare performance of several easilyimplementable sub-optimal energy management policies. A greedy policy isidentified which, in low SNR regime, is throughput optimal and also minimizesmean delay.
We consider stability of scheduled multiaccess message communication withrandom coding and joint maximum-likehood decoding of messages. The framework weconsider here models both the random message arrivals and the subsequentreliable communication by suitably combining techniques from queueing theoryand information theory. The number of messages that may be scheduled forsimultaneous transmission is limited to a given maximum value, and the channelsfrom transmitters to receiver are quasi-static, flat, and have independentfades. Requests for message transmissions are assumed to arrive according to ani.i.d. arrival process. Then, (i) we derive an outer bound to the region ofmessage arrival rate vectors achievable by the class of stationary schedulingpolicies, (ii) we show for any message arrival rate vector that satisfies theouterbound, that there exists a stationary state-independent policy thatresults in a stable system for the corresponding message arrival process, and(iii) in the limit of large message lengths, we show that the stability regionof message nat arrival rate vectors has information-theoretic capacity regioninterpretation.
We consider scheduled message communication over a discrete memorylessdegraded broadcast channel. The framework we consider here models both therandom message arrivals and the subsequent reliable communication by suitablycombining techniques from queueing theory and information theory. The channelfrom the transmitter to each of the receivers is quasi-static, flat, and withindependent fades across the receivers. Requests for message transmissions areassumed to arrive according to an i.i.d. arrival process. Then, (i) we derivean outer bound to the region of message arrival vectors achievable by the classof stationary scheduling policies, (ii) we show for any message arrival vectorthat satisfies the outerbound, that there exists a stationary``state-independent'' policy that results in a stable system for thecorresponding message arrival process, and (iii) under two asymptotic regimes,we show that the stability region of nat arrival rate vectors hasinformation-theoretic capacity region interpretation.
The stability of scheduled multiaccess communication with random coding andindependent decoding of messages is investigated. The number of messages thatmay be scheduled for simultaneous transmission is limited to a given maximumvalue, and the channels from transmitters to receiver are quasi-static, flat,and have independent fades. Requests for message transmissions are assumed toarrive according to an i.i.d. arrival process. Then, we show the following: (1)in the limit of large message alphabet size, the stability region has aninterference limited information-theoretic capacity interpretation, (2)state-independent scheduling policies achieve this asymptotic stability region,and (3) in the asymptotic limit corresponding to immediate access, thestability region for non-idling scheduling policies is shown to be identicalirrespective of received signal powers. |
The main purpose of linear regression is to estimate a mean difference of outcomes comparing adjacent levels of a regressor. There are many types of means. We are most familiar with the arithmetic mean.
$$AM(X) = \frac{\left( X_1 + X_2 + \ldots + X_n \right)}{n}$$
The AM is what is estimated using OLS and untransformed variables. The geometric mean is different:
$$GM(X) = \sqrt[\LARGE{n}]{\left( X_1 \times X_2 \times \ldots \times X_n \right)} = \exp(AM(\log(X))$$
Practically a GM difference is a multiplicative difference: you pay X% of a premium in interest when assuming a loan, your hemoglobin levels decrease X% after starting metformin, the failure rate of springs increase X% as a fraction of the width. In all of these instances, a raw mean difference makes less sense.
Log transforming estimates a geometric mean difference. If you log transform an outcome and model it in a linear regression using the following formula specification:
log(y) ~ x, the coefficient $\beta_1$ is a mean difference of the
log outcome comparing adjacent units of $X$. This is practically useless, so we exponentiate the parameter $e^{\beta_1}$ and interpret this value as a geometric mean difference.
For instance, in a study of HIV viral load following 10 weeks administration of ART, we might estimate prepost geometric mean of $e^{\beta_1} = 0.40$. That means whatever the viral load was at baseline, it was on average
60% lower or had a 0.6 fold decrease at follow-up. If the load was 10,000 at baseline, my model would predict it to be 4,000 at follow-up, if it were 1,000 at baseline, my model would predict it to be 400 at follow-up (a smaller difference on the raw scale, but proportionally the same).
This is an important distinction from other answers:The convention of multiplying the log-scale coefficient by 100 comes from the approximation $\log(x) \approx 1-x$ when $X$ is small. If the coefficient (on the log scale) is say 0.05, then $\exp(0.05) \approx 1.05$ and the interpretation is: a 5% "increase" in the outcome for a 1 unit "increase" in $X$. However, if the coefficient is 0.5 then $\exp(0.5) = 1.65$ and we interpret this as a 65% "increase" in $Y$ for a 1 unit "increase" in $X$. It is NOT a 50% increase.
Suppose we log transform a predictor:
y ~ log(x, base=2). Here, I am interested in a multiplicative change in $x$ rather than a raw difference. I now am interested in comparing participants differing by 2 fold in $X$. Suppose for instance, I am interested in measuring infection (yes/no) following exposure to blood-borne pathogen at various concentrations using an additive risk model. The biologic model may suggest that risk increases proportionately for every doubling of concentration. Then, I do not transform my outcome, but the estimated $\beta_1$ coefficient is interpreted as a risk difference comparing groups exposed at two-fold concentration differences of infectious material.
Lastly, the
log(y) ~ log(x) simply applies both definitions to obtain a multiplicative difference comparing groups differing multiplicatively in exposure levels. |
Details
Use the graphics module, to write a program that can plot an arbitrary polynomial. To accomplish this, you should define a function called
polynomial(x), which takes a floating point value x, and returns the result of calculating the value of the polynomial with the value of x. You should define some polynomial inside of this function. As a starter, use the following polynomial:
$$f(x) = x ^ 3 - 5 \cdot x ^ 2 + 2 \cdot x + 5$$
Using the defined function above, plot the polynomial on the turtle window. In order to make the plotted function easier to view, you should scale the output of your function up. More on this later.
Once you have plotted the function, you are going to then compute and plot a series of rectangles to estimate the area under the curve for some range of values from the domain of the previously defined polynomial. Define a function called
estimate_area(start, end, size), which takes integers for the starting
x value of the range, the ending x value of the range, and the width of the rectangle used to estimate the area under the curve.
This function should compute a series of rectangles of the size specified, with a height determined by the value of the polynomial at one of the points specified in that range. Your estimate of area under the curve is just the sum of the area of all of the rectangles combined. Print the estimated value of the area, and draw the rectangles you computed on the Graphics window as well.
Notes
There are two issues we need to deal with, in order to make our plots look realistic. The first is that our origin is in the upper left corner of the graphics window, and that the plots will look really tiny in comparison to the size of the window. We need to scale the drawn shapes to fill a better portion of the window.
Define a global constant called
SCALE. The larger the value assigned to this variable, the more "zoomed" in on the origin your drawing will be. We can scale the function by dividing the input to the polynomial function by this variable, and multiplying the output of this function by the same value:
Pulling back the curtain a little, this is simply treating each x coordinate as a smaller fraction of the actual x coordinate. The result of the polynomial needs to be scaled up by the same amount, to keep the relative proportions of the function looking correct. This form of scaling will only work in certain scenarios (mostly mathematical functions) but will fit the bill just fine.
The above
scaled_output will be the y value associated with the passed in x value. These then need to be shifted so that they are relative to the origin being moved to the center of the screen.
These are the points you will use to plot the function, and will be used in positioning of the rectangles as well.
"Hacker" Prompt
Each week, additional exercises related to the assignment will be provided at the end. These exercises are typically more challenging than the regular assignment. Bonus points will be provided to students who complete
any of the "Hacker" level assignments. Even Better Estimates:You likely assumed that the value you computed from the polynomial was one of the corners of the rectangle. This provides a decent estimate, but ultimately will always be an over (or under, depending on which corner, etc) estimate of what the value really is. Alter your program so that shifts where you compute the height of the rectangle so that you use the center of the rectangle. Convergence:The area under the curve can be computed by taking the integral of the polynomial. This can be best approximated by computing the area where the size of the rectangles approaches 0. While this is "impossible" for you to currently compute in python, we can compute how much the integral changes as we decrease the width of the rectangles. Use a forloop to compute and print a table of ever decreasing widths of rectangles, and their associated estimate of the area. Square Roots:There are some functions that we can plot, but have some issues that cause Python to not be able to compute values for an area under a curve. For example, \(\sqrt{x}\) cannot be computed, because you cannot compute the square root of negative values, (and the fact that it is not, technically, a function (\(\pm y = \sqrt{x}\))). Create a second program that allows you to plot the square root function. Grading
The assignment will be graded on the following requirements according to the course’s programming assignment rubric.
Functionality (75%): A functional program will: Have at leasttwo functions,
estimate_areaand
polynomial
Plot the
polynomialto the turtle window.
Scale the drawing of the
polynomialfunction based on the value of some defined constant variable.
Use Riemann Sums to estimate and print the area under the curve for some region of the function defined in
polynomial.
Draw the rectangles used to estimate the area to the turtle window. Style (25%): A program with good style will: include a header comment signifying the authors of the file, avoid magic numbers (literal primitive numbers), use meaningful names for variables and functions, have statements that are small (80 characters or less including leading space) and do one thing, and have functions that are small (40 lines or less including comments) and do one thing have a comment above functions that includes the purpose, the pre-conditions, and the post-conditions of the function. have spaces after commas in argument lists and spaces on both sides of binary operators (=, +, -, *, etc.).
If you've ever created a flip book animation you know that you can create animations by quickly changing images. In this assignment you will create animation like a flip book by moving and changing the image that is displayed.
Details
Create a Python program that uses the graphics module to create an animation of a character running and jumping like the following:
The program should have a function for the run animation, with parameters that change the start and end locations of the animation. The run animation should alternate between two different images so it looks like the character's legs are moving. The program should have a function for the jump animation, with parameters that change the start and end locations of the animation. The jump animation path should not be linear. It should be circular, parabolic, or sinusoidal to produce a smooth curve. The run function should be called twice, before and after the jump. The run function should be called once, in between the run animations. The transitions between the animations should be seamless.
You can use the images below that were used to produce the above animation, but feel free to use any other images you want.
Extra
Create an animation that tells a story. Include text and additional images in the animation.
Grading
The assignment will be graded on the following requirements according to the course’s programming assignment rubric.
Functionality (75%): A functional program will: have a run animation function with parameters for the start and end. have a jump animation function with parameters for the start and end. the run animation should alternate between two images. the jump animation should be nonlinear. The functions should be called to produce a seamless animation. Style (25%): A program with good style will: include a header comment signifying the authors of the file, avoid magic numbers (literal primitive numbers), use meaningful names for variables and functions, have statements that are small (80 characters or less including leading space) and do one thing, and have a comment above functions that includes the purpose, the pre-conditions, and the post-conditions of the function. have spaces after commas in argument lists and spaces on both sides of binary operators (=, +, -, *, etc.).
Computers represent colors using red, green, and blue because different combinations of those three lights can be used to produce lots of different colors. But people have an easier time thinking about colors in terms of hue, saturation, and brightness. Fortuneately, it is not difficult to convert between the two representations.
Details
Read about voltage dividers and then create a circuit with an RGB LED and a trimpot. Create a Python program that uses the rotation of the trimpot to change the hue of the LED. The program should have a function that uses an HSL to RGB conversion equations, with constant values for the saturation and lightness, to convert to timpot's rotation to the RGB values. Turning the trimpot through its full range should change the LED though the full range of hues.
Grading
The assignment will be graded on the following requirements according to the course’s programming assignment rubric.
Functionality (75%): A functional program will: have a function or functions that convert hue to RGB. have a circuit with a trimpot that can be read gPIo. have a circuit with a RGB LED that can be conrolled by gPIo. control the color of the LED by turning the trimpot. normalize the trimpot's range and the LED' to enable displaying the full range of hues. Style (25%): A program with good style will: include a header comment signifying the authors of the file, avoid magic numbers (literal primitive numbers), use meaningful names for variables and functions, have statements that are small (80 characters or less including leading space) and do one thing, and have spaces after commas in argument lists and spaces on both sides of binary operators (=, +, -, *, etc.). Submission
Submit your program as a .py file on the course Inquire page before class on Wednesday October 5
th. |
The affine evaluation map is a surjective homomorphism from the quantumtoroidal ${\mathfrak {gl}}_n$ algebra ${\mathcal E}'_n(q_1,q_2,q_3)$ to thequantum affine algebra $U'_q\widehat{\mathfrak {gl}}_n$ at level $\kappa$completed with respect to the homogeneous grading, where $q_2=q^2$ and$q_3^n=\kappa^2$. We discuss ${\mathcal E}'_n(q_1,q_2,q_3)$ evaluation modules. We give highestweights of evaluation highest weight modules. We also obtain the decompositionof the evaluation Wakimoto module with respect to a Gelfand-Zeitlin typesubalgebra of a completion of ${\mathcal E}'_n(q_1,q_2,q_3)$, which describes adeformation of the coset theory $\widehat{\mathfrak {gl}}_n/\widehat{\mathfrak{gl}}_{n-1}$.
On a Fock space constructed from $mn$ free bosons and lattice ${\Bbb{Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra$\mathscr {E}_m$ associated to $\mathfrak{gl}_m$, together with a level $m$action of the quantum toroidal algebra ${\mathscr E}_n$ associated to${\mathfrak {gl}}_n$. We prove that the $\mathscr {E}_m$ transfer matricescommute with the $\mathscr {E}_n$ transfer matrices after an appropriateidentification of parameters.
We use the Whittaker vectors and the Drinfeld Casimir element to show thateigenfunctions of the difference Toda Hamiltonian can be expressed viafermionic formulas. Motivated by the combinatorics of the fermionic formulas weuse the representation theory of the quantum groups to prove a number ofidentities for the coefficients of the eigenfunctions.
We construct an analog of the subalgebra $Ugl(n)\otimes Ugl(m)$ of $Ugl(m+n)$in the setting of quantum toroidal algebras and study the restrictions ofvarious representations to this subalgebra.
We identify the Taylor coefficients of the transfer matrices corresponding toquantum toroidal algebras with the elliptic local and non-local integrals ofmotion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate LongWave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidalsetting and the implications for the quantum KdV model. In particular, weconjecture that the spectrum of non-local integrals of motion of Bazhanov,Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equationsassociated to affine sl(2).
Iorgov, Lisovyy, and Teschner established a connection between isomonodromicdeformation of linear differential equations and Liouville conformal fieldtheory at $c=1$. In this paper we present a $q$ analog of their construction.We show that the general solution of the $q$-Painlev\'e VI equation is a ratioof four tau functions, each of which is given by a combinatorial series arisingin the AGT correspondence. We also propose conjectural bilinear equations forthe tau functions.
We have found the possible region of parameters of the minimal supersymmetricstandard model (MSSM) within the bounds from the experimental results of theHiggs mass, the rare decay mode of $b$-quark, the muon $g-2$, the dark matterabundance, and the direct searches for the lighter stop (i.e., one of thesupersymmetric partners of top quark) at the LHC. We present numerical resultsof calculations for the one loop effects of supersymmetric particles in theprocesses of $\tau^+ \tau^-$, $b \overline{b}$, $t \overline{t}$, and $Z h$production at the ILC by using benchmark points within the possible region ofthe MSSM parameters.
We introduce and study a category $\text{Fin}$ of modules of the Borelsubalgebra of a quantum affine algebra $U_q\mathfrak{g}$, where the commutativealgebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, hasfinitely many characteristic values. This category is a natural extension ofthe category of finite-dimensional $U_q\mathfrak{g}$ modules. In particular, weclassify the irreducible objects, discuss their properties, and describe thecombinatorics of the q-characters. We study transfer matrices corresponding tomodules in $\text{Fin}$. Among them we find the Baxter $Q_i$ operators and$T_i$ operators satisfying relations of the form $T_iQ_i=\prod_j Q_j+ \prod_kQ_k$. We show that these operators are polynomials of the spectral parameterafter a suitable normalization. This allows us to prove the Bethe ansatzequations for the zeroes of the eigenvalues of the $Q_i$ operators acting in anarbitrary finite-dimensional representation of $U_q\mathfrak{g}$.
We study highest weight representations of the Borel subalgebra of thequantum toroidal gl(1) algebra with finite-dimensional weight spaces. Inparticular, we develop the q-character theory for such modules. We introduceand study the subcategory of `finite type' modules. By definition, a moduleover the Borel subalgebra is finite type if the Cartan like current \psi^+(z)has a finite number of eigenvalues, even though the module itself can beinfinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogousto those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensorproduct W of Fock spaces and V is a highest weight module over the Borelsubalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces.Namely we show that for a special choice of finite type modules $V$ thecorresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u andsatisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatzequation for the zeroes of the eigenvalues of Q(u;p). Then we show that theeigenvalues of T_{V,W}(u;p) are given by an appropriate substitution ofeigenvalues of Q(u;p) into the q-character of V.
We establish the method of Bethe ansatz for the XXZ type model obtained fromthe R-matrix associated to quantum toroidal gl(1). We do that by using shufflerealizations of the modules and by showing that the Hamiltonian of the model isobtained from a simple multiplication operator by taking an appropriatequotient. We expect this approach to be applicable to a wide variety of models.
In our previous works on the XXZ chain of spin one half, we have studied theproblem of constructing a basis of local operators whose members have simplevacuum expectation values. For this purpose a pair of fermionic creationoperators have been introduced. In this article we extend this construction tothe spin one case. We formulate the fusion procedure for the creationoperators, and find a triplet of bosonic as well as two pairs of fermioniccreation operators. We show that the resulting basis of local operatorssatisfies the dual reduced qKZ equation.
We study one-point functions of the sine-Gordon model on a cylinder. Ourapproach is based on a fermionic description of the space of descendent fields,developed in our previous works for conformal field theory and the sine-Gordonmodel on the plane. In the present paper we make an essential addition bygiving a connection between various primary fields in terms of yet another kindof fermions. The one-point functions of primary fields and descendants areexpressed in terms of a single function defined via the data from thethermodynamic Bethe Ansatz equations.
We define and study representations of quantum toroidal $gl_n$ with naturalbases labeled by plane partitions with various conditions. As an application,we give an explicit description of a family of highest weight representationsof quantum affine $gl_n$ with generic level.
We have been developing a program package called GRACE/SUSY-loop whichautomatically calculates the MSSM amplitudes in one-loop order. We presentnumerical results of calculations for pair-production and three-body decay ofthe lighter stop ($\widetilde{t}_1$) at the International Linear Collider (ILC)using GRACE/SUSY-loop. Since the distributions of missing transverse energy(MET) depend on mass spectrum of SUSY particles, we consider two scenarios onthree-body decay of $\widetilde{t}_1$. In these scenarios, both QCD and EWcorrections have positive sign for decay widths and cross sections.
In third paper of the series we construct a large family of representationsof the quantum toroidal $\gl_1$ algebra whose bases are parameterized by planepartitions with various boundary conditions and restrictions. We study thecorresponding formal characters. As an application we obtain a Gelfand-Zetlintype basis for a class of irreducible lowest weight $\gl_\infty$-modules.
Extending our previous construction in the sine-Gordon model, we show how tointroduce two kinds of fermionic screening operators, in close analogy withconformal field theory with c<1.
GRACE/SUSY-loop is a program package for the automatic calculation of theMSSM amplitudes in one-loop order. We present features of GRACE/SUSY-loop,processes calculated using GRACE/SUSY-loop and an extension of the non-lineargauge formalism applied to GRACE/SUSY-loop.
We apply the fermionic description of CFT obtained in our previous work tothe computation of the one-point functions of the descendant fields in thesine-Gordon model.
The Grassmann structure of the critical XXZ spin chain is studied in thelimit to conformal field theory. A new description of Virasoro Verma modules isproposed in terms of Zamolodchikov's integrals of motion and two families offermionic creation operators. The exact relation to the usual Virasorodescription is found up to level 6.
We begin a study of the representation theory of quantum continuous$\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra dependson two parameters and is a deformed version of the enveloping algebra of theLie algebra of difference operators acting on the space of Laurent polynomialsin one variable. Fundamental representations of $\mathcal E$ are labeled by acontinuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcalE$ has many properties familiar from the representation theory of$\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infiniteconstructions of modules. Using tensor products of vector representations, weconstruct surjective homomorphisms from $\mathcal E$ to spherical double affineHecke algebras $S\ddot H_N$ for all $N$. A key step in this construction is anidentification of a natural bases of the tensor products of vectorrepresentations with Macdonald polynomials. We also show that one of the Fockrepresentations is isomorphic to the module constructed earlier by means of the$K$-theory of Hilbert schemes.
We construct a family of irreducible representations of the quantumcontinuous $gl_\infty$ whose characters coincide with the characters ofrepresentations in the minimal models of the $W_n$ algebras of $gl_n$ type. Inparticular, we obtain a simple combinatorial model for all representations ofthe $W_n$-algebras appearing in the minimal models in terms of $n$interrelating partitions.
The 1-loop corrected decay widths of sparticles (charginos, neutralinos,gluino and sfermions) in the framework of the MSSM are calculatedsystematically using GRACE/SUSY-loop, which is the program package for theautomatic calculation of the MSSM amplitudes in the 1-loop order. We presentthe renormalization scheme used in our system and show some numerical resultsof decay widths of sfermions and gluino using the SPS1a' parameter set andother SUSY parameter sets.
We derive a bosonic formula for the character of the principal space in thelevel $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from aknown fermionic formula for it. In our previous work, the latter was written asa sum consisting of Shapovalov scalar products of the Whittaker vectors for$U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. In this paper we compute these scalarproducts in the bosonic form, using the decomposition of the Whittaker vectorsin the Gelfand-Zetlin basis. We show further that the bosonic formula obtainedin this way is the quasi-classical decomposition of the fermionic formula.
With the aid of the creation operators introduced in our previous works, weshow how to construct a basis of the space of quasi-local operators for thehomogeneous XXZ chain.
In this article we unveil a new structure in the space of operators of theXXZ chain. We consider the space of all quasi-local operators, which areproducts of the disorder field with arbitrary local operators. In analogy withCFT the disorder operator itself is considered as primary field. In ourprevious paper, we have introduced the annhilation operators which mutuallyanti-commute and kill the primary field. Here we construct the creationcounterpart and prove the canonical anti-commutation relations with theannihilation operators. We show that the ground state averages of quasi-localoperators created by the creation operators from the primary field are given bydeterminants. |
Both the set of rational numbers $\mathbb{Q}$ and its complement are dense in $\mathbb{R}$, but the relationship between them is very asymmetric. For instance, the rationals are countable and have Lebesgue measure 0, whereas the irrationals are uncountable and have infinite Lebesgue measure. Is it possibly to decompose the real numbers into dense subsets in a more symmetric way, so that $\mathbb{R}$ can be written as a union of finitely many disjoint sets which can be mapped into each other by translation or reflection (i.e. are congruent)?
$$A=\bigcup_{n\in\mathbb Z}[2n,2n+1)$$ $$B=\bigcup_{n\in\mathbb Z}[2n+1,2n+2)$$ $$\mathbb R= [(A\cap\mathbb Q)\cup(B\setminus\mathbb Q)] \cup [(B\cap\mathbb Q)\cup(A\setminus\mathbb Q)] $$ The translation $x\mapsto x+1$ maps $[(A\cap\mathbb Q)\cup(B\setminus\mathbb Q)]$ onto $[(B\cap\mathbb Q)\cup(A\setminus\mathbb Q)].$
Yes, this is doable, via a construction like that of the Vitali set but for integers. Indeed, with Choice we can get an
uncountably dense example! (Note that bof's answer solves the problem as stated, without using Choice at all.)
For $x, y\in\mathbb{R}$, let $x\sim y$ if $x-y\in\mathbb{Z}$. Now via Choice we can get an uncountably dense transversal $T$ for $\sim$ - that is, $T$ is uncountably dense and contains exactly one real from each $\sim$-class. (Note that $[0, 1)$ is a non-dense transversal - the existence of a transversal, full stop, does not require choice.)
Now let $$A=\{t+2k: t\in T, k\in\mathbb{Z}\},\quad B=\{t+2k+1: t\in T, k\in\mathbb{Z}\}.$$ It's not hard to see that $B$ is gotten by shifting $A$ one unit (in either direction!), and that $A$ and $B$ are disjoint and cover $\mathbb{R}$. |
Given the sequence $a_1 = 0$ and $a_{n+1} = \dfrac{1}{2 \cdot\lfloor{a_n}\rfloor-a_n+1}$ and $p,q\in \mathbb N$ and coprime find $x$ so that $a_x = \dfrac{p}{q}$. I do not even know where would you start with a problem like this.
Observation: $a_k<1$ iff $k$ is odd.
Lemma: If $a_{2n}$ = $a_n$+1.
Proof: By induction. $a_2 = 1 = 1+a_1$. Further suppose $a_{2(n-1)}=a_{n-1}+1$. Denote $x=2\lfloor a_{n-1}\rfloor-a_{n-1}+1$. Then $$a_n=\frac 1x,$$ $$a_{2n-1} = \frac 1{x+1},$$ $$a_{2n} = \frac 1{2\cdot0-\frac1{x+1}+1} = \frac1{\frac{x}{x+1}}=\frac{x+1}{x}=1+\frac 1x = a_n+1.$$
Lemma proved.
Now consider a rational number and the following process with it. While it is greater than or equal to one, subtract one from the number. Otherwise apply the recurrent formula $x\to\frac1{1-x}$. In every application of the formula, the denominator decreases, so we will get eventually to the number 0.
We can follow the process backwards and assign elements $a_k$ to it. We start with $0=a_1$. When we add one to the value, we just jump from $a_k$ to $a_{2k}$. In the other case (after application of $\frac1{1-x}$), we are on an even index $a_k$. So $a_{k-1}$ is odd, so $a_k = \frac1{1-a_{k-1}}$. Since the function $\frac1{1-x}$ is injective, $a_{k-1}$ is the next value in the reverted sequence.
At the end, we reach the original rational number together with its position in the sequence. |
I'm calculating molar changes in thermodynamic properties due to reactions between gasses (assumed to be ideal gases). I can calculate $\Delta H$ easily enough, because it's just $\sum_i \nu_i\Delta_f H^\circ_i$, with $\nu_i$ the stoichiometric coefficients. $\Delta G$ (at standard pressure) can be calculated from $\sum_i \nu_i\mu_i$, with $\mu_i = \Delta_f G^\circ_i + RT\log[i]$. Since $\Delta G = \Delta H - T\Delta S$, I can calculate $\Delta S$ as $\frac{1}{T}(\Delta H - \Delta G)$.
So far so good. But I'm inexperienced in working with tabulated quantities (I'm a more of a theoretical physicist than a chemist) and keep making mistakes with signs and units. So I figured that as a sanity check I would calculate $\Delta S$ directly, using tabulated values of $S^\circ$. But then I realised I don't know how to do this, and it seems that none of the textbooks on my desk explain it either.
Based mostly on intuition, it seems like it should be $\Delta S = \sum_i\nu_i(S^\circ_i + R\log[i])$, with the $R\log[i]$ term having something to do with the entropy of mixing. For the example reaction I chose, this gave something with the approximately correct magnitude but the wrong sign (-15.6 instead of 14.2, which is what I get by calculating it from the Gibbs energy and the enthalpy).
So my question is, what is the correct way to calculate the entropy change due to an ideal gas reaction if, for some reason, you only have access to the concentrations and the standard entropies of the reactants? |
I am working on the problem
Consider the steady-state of the heat equation in a ball of radius a centred at the origin. In spherical coordinates, the ball occupied the region $0 \le r \le a$, $0 \le \theta \le \pi$ and $0 \le \phi < 2\pi$. It has a given temperature $g(\theta)$ imposed along its boundary, which is the sphere of radius $a$. Since the boundary condition is independent of $\phi$, we can assume that the temperature at the point $(r, \theta, \phi)$ in the ball is given as $u(r, \theta)$, which is given by the solution of the following boundary value problem,
$$\dfrac{1}{r^2} \dfrac{\partial}{\partial{r}} \left( r^2 \dfrac{\partial{u}}{\partial{r}} \right) + \dfrac{1}{r^2 \sin(\theta)} \dfrac{\partial}{\partial{\theta}} \left( \sin(\theta) \dfrac{\partial{u}}{\partial{\theta}} \right) = 0,$$
subject to boundary conditions
$u(a, \theta) = g(\theta)$ for $0 ≤ \theta ≤ \pi$.
(i) Show that the separation of variables $u(r, \theta) = R(r)S(\theta)$ leads to the equations
$$\dfrac{1}{\sin(\theta)} \dfrac{d}{d \theta} \left( \sin(\theta) \dfrac{dS}{d \theta} \right) + \lambda S = 0$$
and
$$(r^2 R')' - \lambda R = 0$$
(ii) Now let $\lambda = n(n + 1)$ for $n = 0,1,2,3, \dots$ and let $\mu = \cos(\theta)$, transform the ODE for $S(\theta)$ to the following Legendre’s equation:
$$(1 - \mu^2) \dfrac{\partial^2{S}}{\partial{\mu}^2} - 2\mu \dfrac{dS}{d \mu} + n(n + 1)S = 0$$
(iii) Solve the differential equation for $R$ for each eigenvalue $\lambda n = n(n + 1)$. (Hint: Try $R = Ar^m$.)
(iv) Given the solution of the Legendre’s equations are the Legendre polynomials $P_n(\mu) = P_n(\cos(θ))$, write the general solution for $u(r, \theta)$ as an infinite series.
I'm stuck on (iv) and just don't understand how to do this. I don't have very much experience with Legendre polynomials, so this is probably why. My textbook also doesn't have any solutions, so I am totally stuck. I would be very thankful if someone could please take the time to explain what (iv) is asking and show how (iv) is done. Thank you very much for your help! |
Spring 2018, Math 171 Week 8 Exponential Distribution
Let \(X \sim \mathrm{exp}(\lambda)\).
Find the distribution of \(Y = \lceil X \rceil\)
(Answer) \(\mathrm{geometric}(1-e^{-\lambda})\)
Show that \(X\) and \(Y\) are both memoryless
Find the distribution of \(\beta X\)
(Answer) \(\mathrm{exponential}(\lambda/\beta)\)
Find the distribution of \(e^{-X}\)
(Solution) Let \(Y = e^{-X}\). Note since \(X \in [0, \infty)\) we have \(Y \in (0, 1]\). \[\begin{aligned}F_Y(y) &= P(Y \le y)\\ &= P(e^{-X}\le y)\\&=P(-X \le \log(y))\\&= P(X \ge -\log(y))\\&=1-F_X(-\log(y))\end{aligned}\] \[\begin{aligned}f_Y(y) &= \frac{d}{dy}F_Y(y) \\ &= \frac{d}{dy}(1 - F_X(-\log(y))) \\ &= -f_X(-\log(y))\cdot \frac{-1}{y} \\ &= \frac{\lambda e^{-\lambda (-\log(y))}}{y} \\ &= \lambda y^{\lambda - 1}\end{aligned}\]
Let \(U \sim \mathrm{uniform}[0,1]\). Find the distribution of \(-\alpha\log{U}\)
(Answer) \(\mathrm{exponential}(1/\alpha)\)
Let \(X_1, X_2, \dots\overset{\mathrm{i.i.d}}{\sim} \mathrm{exp}(\lambda)\)
(Discussed) Suppose \(N \sim \mathrm{geo}(p)\). Find the distribution of \(Z = \sum_{i=1}^N X_i\).
Find the distribution of \(Q = \min(X_1, X_2, \dots X_n)\)
(Answer) \(\mathrm{exponential}(n\lambda)\)
Find the cumulative distribution of \(V = \max(X_1, X_2, \dots X_n)\)
(Answer) \((1-e^{-v\lambda})^n\) Poisson Process Basics
Let \(N(t)\) be a poisson process with rate \(\lambda\)
(Discussed) Find the probability of no arrivals in \((3,5]\)
(Discussed) Find the probability that there is exactly one arrival in each of the intervals: \((0,1], (1,2], (2,3], (3,4]\)
(Discussed) Find the probability that there are two arrivals in \((0,2]\) and three arrivals in \((1,4]\)
(Discussed) Find the covariance of \(N(t_1)\) and \(N(t_2)\) for \(0 < t_1 < t_2\) |
For the experiment of drawing two cards from the specified deck, define the random variable, $A$ as the count of aces drawn, and the event $A_1, A_2$ as "an ace is drawn first" and "... second" respectively.
In the event of $A=1$, the order is
not important. It does not matter whether the single ace is the first or second card drawn. The order is not specified by the event of "exactly one ace is drawn".
One solution is that $\mathsf P(A=1)={\tbinom 41\tbinom {36}1}\bigm/{\tbinom {40}2}$, which is the probability for selecting 1 from 4 aces and 1 from 36 non-aces given that 2 from 40 cards are selected. This solution does not concern itself with the order the cards are selected –in either numerator nor denominator–.
Another solution is that $\mathsf P(A=1)= \mathsf P(A_1\cap A_2^\complement)+\mathsf P(A_1^\complement\cap A_2) = (\dfrac 4{40}\cdot\dfrac{36}{39})+(\dfrac {36}{40}\cdot\dfrac 4{39})$
This solution partitions the event into two subsets, each an event in which the order the cards are drawn
is important. For each part, it matters that the ace is drawn first or second. $A_1\cap A_2^\complement$ specifies the order of the cards, as does the other part, $A_1^\complement\cap A_2$.
The solution to each part clearly concerns itself with the order of the draw. The relevant terms are obtained by evaluating, $\mathsf P(A_1\cap A_2^\complement)=\mathsf P(A_1)\mathsf P(A_2^\complement\mid A_1)$ and $\mathsf P(A_1^\complement\cap A_2)=\mathsf P(A_1^\complement)\mathsf P(A_2\mid A_1^\complement)$
However, as the event is the union of the parts — either part satisfies the event — then, as a whole, the order is
not important. That is, because the order is not important, we may measure the probability for each ordering, then add these results to obtain the measure for the event. |
Yes, list $E\cap C = \{x_n:n=1,2..\}$. For each $x$ let $N_x=\{n:x_n<x\}$. Define $f(x)=\sum_{n\in N_x}2^{-n}$.
I do not quite see what is the role of $E$ in this question, we could define $f$ as above, initially disregarding $E$ (and using $C = \{x_n:n=1,2..\}$) and then later restricting this function to $E$.
Edit. This may not be "discontinuous enough" at points of $C$, as discussed in comments below. So define also $M_x=\{n:x_n\le x\}$, define $g(x)=\sum_{n\in M_x}2^{-n}$, and define $h(x)=f(x)+g(x)$. I hope $h$ works, if not then I do not understand the question and may need to read all over again.
Edit. $h$ is continuous at points of $E\setminus C$ (verify:). But $h$ (and any function) would be continuous at those points of $C$ that are isolated in $E$, that is at any $c\in C$ which has a neighborhood which misses all other points in $E$. It is ok if $c$ is isolated in $C$,but not in $E$, in that case $h$ would be discontinuous at $c$. Indeed $c=x_m$ for some $m$ (according to the above definition of $f$). So, $2^{-m}$ is not one of the members of the sum that defines $f(c)=f(x_m)=\sum_{n\in N_{x_m}}2^{-n}$ (this is since $m\not\in N_{x_m}$, since in the definition of $N_x$ we have strict inequality, but it is not true that $x_m<x_m$). On the other hand, if $x>x_m$, then $2^{-m}$ is one of the members of the sum that defines $f(x)=\sum_{n\in N_{x}}2^{-n}$, this is because $m\in N_x$, since $x_m<x$. It follows that if $x>x_m$ then $f(x)\ge f(x_m)+2^{-m}$, so $\lim_{x\to x_m^+} f(x)\ge f(x_m)+2^{-m}$, here we are assuming the there are points in $E$ to the right of $x_m$, arbitrarily close to $x_m$ (that is, we are assuming that $x_m$ is not isolated from the right in $E$). So $f$ has a jump at $x_m$ (to the right of $x_m$) so $f$ is discontinuous (from the right) at $x_m$. Similarly, if $x_m$ is not isolated from the left in $E$ then one may show that $\lim_{x\to x_m^-}f(x)\le f(x_m)-2^{-m}$, so $g$ has a jump at $x_m$ (at the left of $x_m$). So, if $x_m$ is not isolated in $E$ then it is either not isolated from the left, or not isolated from the right, or both, so either $g$ of $f$ or both are discontinuous at $x_m$, so $h$ is discontinuous at $x_m$. You should be able to verify all these details yourself, once you know what you are trying to prove. |
Analysis --> Errors --> Error analysis
The Errors Analysis initiates a computational procedure that provides a statistical evaluation of the effect that errors in the layer thicknesses and refractive indices in a design will have on the spectral response of a designated spectral characteristic.
As the computations proceed, a sequence of curves corresponding to the statistical tests being performed is plotted. During the computations, the progress of the statistical procedure is indicated in the percentage bar at the bottom of the window. This tool allows
estimating the influence of errors in layer thicknesses and refractive indices on spectral characteristics of the current design.
Fig. 1. Error analysis setup.
The Error Analysis computation can be performed only for one spectral characteristic at a time. You should specify the
characteristic (Transmittance, Reflectance, phase characteristics etc.), the state of polarization and the angle of incidence prior to the calculations and the number er of tests. These specifications can be done at the Characteristic tab of the Error Analysis window (see right pane). OptiLayer performs calculations very fast even for a large number of tests.
Fig. 2. Example. For a beamsplitter BS_7, absolute errors of 0.5 nm and relative errors of 1% are specified, i.e. \(\delta=0.5\) nm and \(\Delta=1\%\).
In the
Error Analysis Setup window, there are two more tabs ( Thickness and Refractive index) that allow you to specify different kinds of expected (estimated) deposition errors.
In
Thickness tab there are three sub-tabs: Coating, Materials, and Thicknesses (see left pane). Important: Settings in all three tabs are independent. In the course of calculations, the errors specified in the last opened tab are taken into account.
In the
Coating sub-tab, you can specify absolute (Abs. RMS) or/and relative (Rel. RMS) level of errors in all coating layers. In the course of the calculations, instead of layer thicknesses \(d_1,...,d_m\), disturbed thicknesses are used:
\[ d_1^{(j)}=d_1+\delta_1^{(j)},...,\;d_m^{(j)}=d_m+\delta_m^{(j)}\; \mbox{(absolute errors)} \;\; d_1^{(j)}=d_1+\Delta_1^{(j)}\cdot d_1,...,d_m^{(j)}=d_m+\Delta_m^{(j)}\cdot d_m\; \mbox{(relative errors),} \]
where \(\{\delta_i^{(j)}\}\) [nm] and \(\{\Delta_i^{(j)}\}\) [%] are absolute and relative errors in layer thicknesses, respectively. The errors are normally distributed numbers with zero average and specified rms deviations \(\delta\) and \(\Delta\).
\(j=1,...,N\) where \(N\) is the number of statistical tests.
Fig. 3. Statistical error analysis: Calculation process with settings from Fig. 2.
Fig. 4. Statistical error analysis with settings from Fig. 2: expected and averaged (Exp) spectral characteristics, two curves, Exp-D and Exp+D, indicate the probability corridor for a given error level. The width of probability corridor corresponds to the level of the errors that were set in Error Analysis Setup window (Fig. 1).
In the course of the calculations, instead of layer thicknesses \(d_1,...,d_m\), disturbed thicknesses will be used. In the case of a two-material coating:
\[ d_1^{(j)}=d_1+\delta_{H,1}^{(j)},...,\;d_m^{(j)}=d_m+\delta_{L,m}^{(j)} \; \mbox{(absolute errors)} \]and
\[ d_1^{(j)}=d_1+\Delta_{H,1}^{(j)}\cdot d_1,...,d_m^{(j)}=d_m+\Delta_{L,m}^{(j)}\cdot d_m, \; \mbox{(relative errors)} \]
where \(\{\delta_{i,H}^{(j)}\}\), \(\{\delta_{i,L}^{(j)}\}\) [nm] and \(\{\Delta_{i,H}^{(j)}\}\), \(\{\Delta_{i,L}^{(j)}\}\), [%] are absolute and relative errors in layer thicknesses, respectively. The errors ar)e normally distributed numbers with zero average and specified rms deviations \(\delta_{H,L}\) and \(\Delta_{H,L}\).
This option is very convenient if you have not two but multiple thin-film materials or if you would like to estimate effect of errors on the spectral characteristics of stacks.
In the
Materials sub-tab, you can specify absolute (Abs. RMS) or/and relative (Rel. RMS) level of errors in layers. The errors in layers of different materials can be specified separately.
Fig. 5. Example. For a beamsplitter BS_7, relative errors of 0.5% in H-layers and absolute errors of 1nm are specified, i.e. \(\Delta_H=0.5\)% and \(\delta_L=1\) nm.
Fig. 6. Statistical error analysis: Calculation process with setting from Fig. 5.
Fig. 7. Statistical error analysis with settings from Fig. 5: Result.
In the
Thicknesses sub-tab, you can specify absolute (Abs. RMS) or/and relative (Rel. RMS) level of errors in all layers separately.
Fig. 8. Example. For a beamsplitter BS_7, relative errors of 1% in H-layers and relative errors of 0.5% in L-layers are specified.
In the course of the calculations, instead of layer thicknesses \(d_1,...,d_m\), disturbed thicknesses will be used. In the case of a two-material coating:
\[ d_1^{(j)}=d_1+\delta_{H,1}^{(j)},...,\;d_m^{(j)}=d_m+\delta_{L,m}^{(j)}\]in the case of absolute errors and
\[ d_1^{(j)}=d_1+\Delta_{H,1}^{(j)}\cdot d_1,...,d_m^{(j)}=d_m+\Delta_{L,m}^{(j)}\cdot d_m,\]
where \(\{\delta_{i,H}^{(j)}\}\), \(\{\delta_{i,L}^{(j)}\}\) [nm] and \(\{\Delta_{i,H}^{(j)}\}\), \(\{\Delta_{i,L}^{(j)}\}\), [%] are absolute and relative errors in layer thicknesses, respectively. The errors are normally distributed numbers with zero average and rms deviations specified in corresponding columns.
Using this sub-tab is recommended for
sophisticated error analysis.
Fig. 9. Statistical error analysis: Calculation process with setting from Fig. 8.
Fig. 10. Statistical error analysis with settings from Fig. 8: Result.
Fig. 11. Example. For a beamsplitter BS_7, a systematic offset of 0.04 in high-index layers and a systematic offset of 0.005 in low-index layers are specified.
In
Refractive index tab, you can specify absolute (RMS) or/and relative (Rel. RMS(%)) offsets in optical constants of layers (refractive indices and extinction coefficients), Fig. 11.
Important: if the is checked, then equal offsets for all layers of the same material are specified. Per Material Errors box
In the course of the calculations, instead of nominal refractive indices \(n_H, n_L\), disturbed refractive indices are used:
\[ n_{H,i}^{(j)}=n_H+\Sigma_{H}, \; n_{L,i}^{(j)}=n_L+\Sigma_{L}\]in the case of absolute errors and
\[ n_{H,i}^{(j)}=n_H\cdot(1+\Sigma_{H}), \; n_{L,i}^{(j)}=n_L\cdot (1+\Sigma_{L})\]
where \(\Sigma_H, \Sigma_L\) are
systematic offsets refractive indices of high- and low-index materials, respectively. The errors are normally distributed numbers with zero average and specified rms deviations \(\Sigma_H, \Sigma_L\).
The option allows you to specify errors in
substrate refractive indices and refractive index of the incident medium.
Fig. 12. Statistical error analysis: Calculation process with setting from Fig. 11.
Fig. 13. Statistical error analysis with settings from Fig. 11: expected and averaged (Exp) spectral characteristics, two curves, Exp-D and Exp+D, indicate the probability corridor for a given error level. The width of probability corridor corresponds to the level of the errors that were set in Error Analysis Setup window (Fig. 1).
If the
is Per Material Errors box unchecked, then different errors for all layers of the same material are specified. From the practical point of view, it can be addressed to instability from layer to layer of optical constants in the course of the deposition.
In the course of the calculations, instead of nominal refractive indices \(n_H, n_L\), disturbed refractive indices are used:
\[ n_{H,i}^{(j)}=n_H+\Sigma_{i,H}^{(j)}, \; n_{L,i}^{(j)}=n_L+\Sigma_{i,L}^{(j)}\]in the case of absolute errors and
\[ n_{H,i}^{(j)}=n_H\cdot(1+\Sigma_{i,H}^{(j)}), \; n_{L,i}^{(j)}=n_L\cdot (1+\Sigma_{i,L}^{(j)})\]
where \(\{\Sigma_{i,H}^{(j)}\}\) and \(\{\Sigma_{i,L}^{(j)}\}\) are
random offsets in layer refractive indices, respectively. The errors are normally distributed numbers with zero average and specified rms deviations \(\Sigma_H\) and \(\Sigma_L\).
Fig. 14. Example. For a beamsplitter BS_7, a random offset of 0.04 in high-index layers and a random offset of 0.0.005 in low-index layers are specified.
Fig. 15. Statistical error analysis: Calculation process with setting from Fig. 14.
Fig. 16. Statistical error analysis with settings from Fig. 14: expected and averaged (Exp) spectral characteristics, two curves, Exp-D and Exp+D, indicate the probability corridor for a given error level. The width of probability corridor corresponds to the level of the errors that were set in Error Analysis Setup window (Fig. 1).
Fig. 17. In the process of Error Analysis, a corridor is displayed that corresponds to the deviations of the spectral characteristics from their mathematical expectations. The width of the corridor depends on the probability for the selected characteristic value to fall within such a corridor. This probability can be set in the edit box Corridor Probability. By default, a reasonable value corresponding to one standard deviation is assigned to corridor. probability.
Fig. 18. Statistical error analysis with a standard probability corridor of one sigma specified in Fig. 17.
Fig. 19. Statistical error analysis with the probability corridor of two-sigma specified in Fig. 20. Obviously, expected disturbed spectral characteristics will exhibit larger deviations from the theoretical ones.
Fig. 20. The corridor of 95% corresponds to two-sigma.
Fig. 21. The corridor of 99.7% corresponds to three-sigma.
Pause before computing summary and Pause after each plot options allow you to slow down the Error Analysis procedure in order to save intermediate results in graphics form.
Fig. 22. Expected disturbed spectral characteristics with three-sigma settings will exhibit even larger deviations from the theoretical ones.
Fig. 23. Statistical error analysis in EFI (electric field intensity). Using Wavelength slider you can vary the current wavelength.
OptiLayer provides the statistical error analysis even of electric field intensity distribution. For this purpose, a check box
EFI Error Analysis is to be checked in Error Analysis Setup window (Fig. 1).
The calculations are performed for the angle of incidence and polarization state specified on the bottom panel of
EFI Error Analysis window. By changing the angle of incidence/polarization, calculations are performed on-the-fly. |
In (quasi) stable atoms the positive and negative charges form "charged" clouds, which can be represented as fractionally charged sub-clouds. In order to observe them as such, we have to have the same atomic state in the in- and out-states, i.e., we have to deal with elastic scattering in the first Born approximation. Then no atomic "polarization" effects are present. Let us consider for simplicity a large-angle scattering, i.e., scattering from the positive-charge sub-clouds (see formula (3) in this paper and here):
$$\text{scattering amplitude}\propto\int{|\psi_{nlm}(\vec{r}_a)|^2\text{e}^{\text{i}\frac{m_e}{M_A}\vec{q}\sum \vec{r}_a}d\tau}$$
This integral is just a sum of integrations over different sub-clouds. If one manages to prepare the target atoms in a certain polarized state $|n,l,m\rangle$ (in order not to average over different $l_z$ projections), the resulting cross section will be an elastic cross section of scattering from fractionally charged atomic sub-clouds. However, and let us not forget it, this will be an
inclusive (or deep inelastic) scattering with respect to the soft photon emissions, which carry away a tiny portion of the total transferred energy/momentum $|\vec{q}|$.
Such fractionally charged sub-clouds are confined in atoms and are never observed independently of atoms, just like quarks. I wonder whether this analogy between quarks and the "usual" sub-clouds is deep or superficial, to your opinion. |
Ultrapower
The intuitive idea behind ultrapower constructions (and ultraproduct constructions in general) is to take a sequence of already existing models and construct new ones from some combination of the already existing models. Ultrapower constructions are used in many major results involving elementary embeddings. A famous example is Scott's proof that the existence of a measurable cardinal implies $V\neq L$. Ultrapower embeddings are also used to characterize various large cardinal notions such as measurable, supercompact and certain formulations of rank into rank embeddings. Ultrapowers have a more concrete structure than general embeddings and are often easier to work with in proofs. Most of the results in this article can be found in [1].
Contents General construction
The general construction of an ultrapower supposes given an index set $X$ set for a collection of (non-empty) models $M_i$ with $i\in X$ and an ultrafilter $U$ over $X$. The ultrafilter $U$ is used to define equivalence classes over the structure $\prod_{i\in X} M_i$, the collection of all functions $f$ with domain $X$ such that $f(i)\in M_i$ for each $i\in X$. When the $M_i$ are identical to one another, we form an
ultrapower by "modding out" over the equivalence classes defined by $U$. In the general case where $M_i$ differs from $M_j$, we form a structure called the ultraproduct of $\langle M_i : i\in X\rangle$.
Two functions $f$ and $g$ are $U$-equivalent, denoted $f=_U g$, when the set of indices in $X$ where $f$ and $g$ agree is an element of the ultrafilter $U$ (intuitively, we think of $f$ and $g$ as disagreeing on a "small" subset of $X$). The $U$-equivalence class of $f$ is usually denoted $[f]$ and is the class of all functions $g\in \prod_{i\in X} M_i$ which are $U$-equivalent to $f$. When each $M_i$ happens to be the entire universe $V$, each $[f]$ is a proper class. To remedy this, we can use Scott's trick and only consider the members of $[f_U]$ of minimal rank to insure that $[f]$ is a set. The ultrapower (ultraproduct) is then denoted by $\text{Ult}_U(M) = M/U =\{[f]: f\in \prod_{i\in X} M_i\}$ with the membership relation defined by setting $[f]\in_U [g]$ when the set of all $i\in X$ such that $f(i)\in g(i)$ is in $U$.
Note that $U$ could be a principal ultrafilter over $X$ and in this case the ultraproduct is isomorphic to almost every $M_i$, so in this case nothing new or interesting is gained by considering the ultraproduct. However, even in the case where each $M_i$ is identical and $U$ is non-principal, we have the ultrapower isomorphic to each $M_i$ but the construction technique nevertheless yields interesting results. Typical ultrapower constructions concern the case $M_i=V$.
Formal definition
Given a collection of nonempty models $\langle M_i : i\in X \rangle$, we define the product of the collection $\langle M_i : i\in X \rangle$ as $$\prod_{i\in X}M_i = \{f:\text{dom}(f)=X \land (\forall i\in X)(f(i)\in M_i)\}$$
Given an ultrafilter $U$ on $X$, we then define the following relations on $\prod_{i\in X} M_i$: Let $f,g\in\prod_{i\in X} M_i$, then $$f =_U g \iff \{i\in X : f(i)=g(i)\}\in U$$ $$f \in_U g \iff \{i\in X : f(i)\in g(i)\}\in U$$
For each $f\in\prod_{i\in X} M_i$, we then define the
equivalence class of $f$ in $=_U$ as follows: $$[f]=\{g: f=_U g \land \forall h(h=_U f \Rightarrow \text{rank}(g)\leq \text{rank}(h) \}$$
Every member of the equivalence class of $f$ has the same rank, therefore the equivalence class is always a set, even if $M_i = V$. We now define the
ultraproduct of $\langle M_i : i\in X \rangle$ to be the model $\text{Ult}=(\text{Ult}_U\langle M_i : i\in X \rangle, \in_U)$ where: $$\text{Ult}_U\langle M_i : i\in X \rangle = \prod_{i\in X}M_i / U = \{[f]:f\in\prod_{i\in X}M_i\}$$
If there exists a model $M$ such that $M_i=M$ for all $i\in X$, then the ultraproduct is called the
ultrapower of $M$, and is denoted $\text{Ult}_U(M)$. Los' theorem Los' theorem is the following statement: let $U$ be an ultrafilter on $X$ and $Ult$ be the ultraproduct model of some family of nonempty models $\langle M_i : i\in X \rangle$. Then, for every formula $\varphi(x_1,...,x_n)$ of set theory and $f_1,...,f_n \in \prod_{i\in X}M_i$, $$Ult\models\varphi([f_1],...,[f_n]) \iff \{i\in X : \varphi(f_1(i),...,f_n(i))\}\in U$$
In particular, an ultrapower $\text{Ult}=(\text{Ult}_U(M), \in_U)$ of a model $M$ is elementarily equivalent to $M$. This is a very important result: to see why, let $f_x(i)=x$ for all $x\in M$ and $i\in X$, and now let $j_U(x)=[f_x]$ for every $x\in M$. Then $j_U$ is an elementary embedding by Los' theorem, and is called the
canonical ultrapower embedding $j_U:M\to\text{Ult}_U(M)$. Properties of ultrapowers of the universe of sets
Let $U$ be a nonprincipal $\kappa$-complete ultrafilter on some measurable cardinal $\kappa$ and $j_U:V\to\text{Ult}_U(V)$ be the canonical ultrapower embedding of the universe. Let $\text{Ult}=\text{Ult}_U(V)$ to simplify the notation. Then:
$U\not\in\text{Ult}$ $\text{Ult}^\kappa\subseteq\text{Ult}$ $2^\kappa\leq(2^\kappa)^{\text{Ult}}<j_U(\kappa)<(2^\kappa)^+$ If $\lambda>\kappa$ is a strong limit cardinal of cofinality $\neq\kappa$ then $j(\lambda)=\lambda$. If $\lambda$ is a limit ordinal of cofinality $\kappa$ then $j_U(\lambda)>lim_{\alpha\to\lambda}$ $j_U(\alpha)$, but if $\lambda$ has cofinality $\neq\kappa$, then $j_U(\lambda)=lim_{\alpha\to\lambda}$ $j_U(\alpha)$.
Also, the following statements are equivalent:
$U$ is a normal measure For every $X\subseteq\kappa$, $X\in U$ if and only if $\kappa\in j_U(X)$. In $(\text{Ult}_U(V),\in_U)$, $\kappa=[d]$ where $d:\kappa\to\kappa$ is defined by $d(\alpha)=\alpha$ for every $\alpha<\kappa$.
Let $j:V\to M$ be a nontrivial elementary embedding of $V$ into some transitive model $M$ with critical point $\kappa$ (which is a measurable cardinal), also let $D=\{X\subseteq\kappa:\kappa\in j(X)\}$ be the canonical normal fine measure on $\kappa$. Then:
There exists an elementary embedding $k:\text{Ult}\to M$ such that $k(j_D(x))=j(x)$ for every $x\in V$. Iterated ultrapowers
Given a nonprincipal $\kappa$-complete ultrafilter $U$ on some measurable cardinal $\kappa$, we define the
iterated ultrapowers the following way:$$(\text{Ult}^{(0)},E^{(0)})=(V,\in)$$$$(\text{Ult}^{(\alpha+1)},E^{(\alpha+1)})=\text{Ult}_{U^{(\alpha)}}(\text{Ult}^{(\alpha)},E^{(\alpha)})$$$$(\text{Ult}^{(\lambda)},E^{(\lambda)})=lim dir_{\alpha\to\lambda}\{(\text{Ult}^{(\alpha)},E^{(\alpha)}),i_{\alpha,\beta})$$where $\lambda$ is a limit ordinal, $limdir$ denotes direct limit, $i_{\alpha,\beta} : \text{Ult}^{(\alpha)}\to \text{Ult}^{(\beta)}$ is an elementary embedding defined as follows:$$i_{\alpha,\alpha}(x)=j^{(\alpha)}(x)$$$$i_{\alpha,\alpha+n}(x)=j^{(\alpha)}(j^{(\alpha+1)}(...(j^{(\alpha+n)}(x))...))$$$$i_{\alpha,\lambda}(x)=lim_{\beta\to\lambda}i_{\alpha,\beta}(x)$$and $j^{(\alpha)}:\text{Ult}^{(\alpha)}\to \text{Ult}^{(\alpha+1)}$ is the canonical ultrapower embedding from $\text{Ult}^{(\alpha)}$ to $\text{Ult}^{(\alpha+1)}$. Also, $U^{(\alpha)}=i_{0,\alpha}(U)$ and $\kappa^{(\alpha+1)}=i_{0,\alpha}(\kappa)$ where $\kappa=\kappa^{(0)}=crit(j^{(0)})$.
If $M$ is a transitive model of set theory and $U$ is (in $M$) a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, we can construct, within $M$, the iterated ultrapowers. Let us denote by $\text{Ult}^{(\alpha)}_U(M)$ the $\alpha$th iterated ultrapower, constructed in $M$.
Properties For every $\alpha$ the $\alpha$th iterated ultrapower $(\text{Ult}^{(\alpha)},E^{(\alpha)})$ is well-founded. This is due to $U$ being nonprincipal and $\kappa$-complete. The Factor Lemma: for every $\beta$, the iterated ultrapower $\text{Ult}^{(\beta)}_{U^{(\alpha)}}(\text{Ult}^{(\alpha)})$ is isomorphic to the iterated ultrapower $\text{Ult}^{(\alpha+\beta)}$. For every limit ordinal $\lambda$, $\text{Ult}^{(\lambda)}\subseteq \text{Ult}^{(\alpha)}$ for every $\alpha<\lambda$. Also, $\kappa^{(\lambda)}=lim_{\alpha\to\lambda}$ $\kappa^{(\alpha)}$. For every $\alpha$, $\beta$ such that $\alpha>\beta$, one has $\kappa^{(\alpha)}>\kappa^{(\beta)}$. If $\gamma<\kappa^{(\alpha)}$ then $i_{\alpha,\beta}(\gamma)=\gamma$ for all $\beta\geq\alpha$. If $X\subseteq\kappa^{(\alpha)}$ and $X\in \text{Ult}^{(\alpha)}$ then for all $\beta\geq\alpha$, one has $X=\kappa^{(\alpha)}\cap i_{\alpha,\beta}(X)$. The representation lemma
References Jech, Thomas J. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex Set Theory. |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance. It is commonly used in the textile industry and often seen quoted on, for example, duvets and carpet underlay.The Shirley Institute in Manchester, England developed the tog as an easy-to-follow alternative to the SI unit of m2K/W. The name comes from the informal word "togs" for clothing which itself was probably derived from the word toga, a Roman garment.The basic unit of insulation coefficient is the RSI, (1 m2K/W). 1 tog = 0.1 RSI. There is also a clo clothing unit equivalent to 0.155 RSI or 1.55 tog...
The stone or stone weight (abbreviation: st.) is an English and imperial unit of mass now equal to 14 pounds (6.35029318 kg).England and other Germanic-speaking countries of northern Europe formerly used various standardised "stones" for trade, with their values ranging from about 5 to 40 local pounds (roughly 3 to 15 kg) depending on the location and objects weighed. The United Kingdom's imperial system adopted the wool stone of 14 pounds in 1835. With the advent of metrication, Europe's various "stones" were superseded by or adapted to the kilogram from the mid-19th century on. The stone continues...
Can you tell me why this question deserves to be negative?I tried to find faults and I couldn't: I did some research, I did all the calculations I could, and I think it is clear enough . I had deleted it and was going to abandon the site but then I decided to learn what is wrong and see if I ca...
I am a bit confused in classical physics's angular momentum. For a orbital motion of a point mass: if we pick a new coordinate (that doesn't move w.r.t. the old coordinate), angular momentum should be still conserved, right? (I calculated a quite absurd result - it is no longer conserved (an additional term that varies with time )
in new coordinnate: $\vec {L'}=\vec{r'} \times \vec{p'}$
$=(\vec{R}+\vec{r}) \times \vec{p}$
$=\vec{R} \times \vec{p} + \vec L$
where the 1st term varies with time. (where R is the shift of coordinate, since R is constant, and p sort of rotating.)
would anyone kind enough to shed some light on this for me?
From what we discussed, your literary taste seems to be classical/conventional in nature. That book is inherently unconventional in nature; it's not supposed to be read as a novel, it's supposed to be read as an encyclopedia
@BalarkaSen Dare I say it, my literary taste continues to change as I have kept on reading :-)
One book that I finished reading today, The Sense of An Ending (different from the movie with the same title) is far from anything I would've been able to read, even, two years ago, but I absolutely loved it.
I've just started watching the Fall - it seems good so far (after 1 episode)... I'm with @JohnRennie on the Sherlock Holmes books and would add that the most recent TV episodes were appalling. I've been told to read Agatha Christy but haven't got round to it yet
?Is it possible to make a time machine ever? Please give an easy answer,a simple one A simple answer, but a possibly wrong one, is to say that a time machine is not possible. Currently, we don't have either the technology to build one, nor a definite, proven (or generally accepted) idea of how we could build one. — Countto1047 secs ago
@vzn if it's a romantic novel, which it looks like, it's probably not for me - I'm getting to be more and more fussy about books and have a ridiculously long list to read as it is. I'm going to counter that one by suggesting Ann Leckie's Ancillary Justice series
Although if you like epic fantasy, Malazan book of the Fallen is fantastic
@Mithrandir24601 lol it has some love story but its written by a guy so cant be a romantic novel... besides what decent stories dont involve love interests anyway :P ... was just reading his blog, they are gonna do a movie of one of his books with kate winslet, cant beat that right? :P variety.com/2016/film/news/…
@vzn "he falls in love with Daley Cross, an angelic voice in need of a song." I think that counts :P It's not that I don't like it, it's just that authors very rarely do anywhere near a decent job of it. If it's a major part of the plot, it's often either eyeroll worthy and cringy or boring and predictable with OK writing. A notable exception is Stephen Erikson
@vzn depends exactly what you mean by 'love story component', but often yeah... It's not always so bad in sci-fi and fantasy where it's not in the focus so much and just evolves in a reasonable, if predictable way with the storyline, although it depends on what you read (e.g. Brent Weeks, Brandon Sanderson). Of course Patrick Rothfuss completely inverts this trope :) and Lev Grossman is a study on how to do character development and totally destroys typical romance plots
@Slereah The idea is to pick some spacelike hypersurface $\Sigma$ containing $p$. Now specifying $u(p)$ is trivial because the wave equation is invariant under constant perturbations. So that's whatever. But I can specify $\nabla u(p)|\Sigma$ by specifying $u(\cdot, 0)$ and differentiate along the surface. For the Cauchy theorems I can also specify $u_t(\cdot,0)$.
Now take the neigborhood to be $\approx (-\epsilon,\epsilon)\times\Sigma$ and then split the metric like $-dt^2+h$
Do forwards and backwards Cauchy solutions, then check that the derivatives match on the interface $\{0\}\times\Sigma$
Why is it that you can only cool down a substance so far before the energy goes into changing it's state? I assume it has something to do with the distance between molecules meaning that intermolecular interactions have less energy in them than making the distance between them even smaller, but why does it create these bonds instead of making the distance smaller / just reducing the temperature more?
Thanks @CooperCape but this leads me another question I forgot ages ago
If you have an electron cloud, is the electric field from that electron just some sort of averaged field from some centre of amplitude or is it a superposition of fields each coming from some point in the cloud? |
K C Mittal
Articles written in Pramana – Journal of Physics
Volume 71 Issue 6 December 2008 pp 1279-1289 Research Articles
BARC is developing a technology for the accelerator-driven subcritical system (ADSS) that will be mainly utilized for the transmutation of nuclear waste and enrichment of U
233. Design and development of superconducting medium velocity cavity has been taken up as a part of the accelerator-driven subcritical system project. We have studied RF properties of 700 MHz, $\beta = 0.42$ single cell elliptical cavity for possible use in high current proton acceleration. The cavity shape optimization studies have been done using SUPERFISH code. A calculation has been done to find out the velocity range over which this cavity can accelerate protons efficiently and to select the number of cells/cavity. The cavity's peak electric and magnetic fields, power dissipation $P_{c}$, quality factor 𝑄 and effective shunt impedance $ZT^{2}$ were calculated for various cavity dimensions using these codes. Based on these analyses a list of design parameters for the inner cell of the cavity has been suggested for possible use in high current proton accelerator.
Volume 74 Issue 1 January 2010 pp 123-133 Research Articles
KALI-1000 pulse power system has been used to generate single pulse nanosecond duration high-power microwaves (HPM) from a virtual cathode oscillator (VIRCATOR) device. HPM power measurements were carried out using a transmitting–receiving system in the presence of intense high frequency (a few MHz) electromagnetic noise. Initially, the diode detector output signal could not be recorded due to the high noise level persisting in the ambiance. It was found that the HPM pulse can be successfully detected using wide band antenna, RF cable and diode detector set-up in the presence of significant electromagnetic noise. Estimated microwave peak power was $\sim 59.8$ dBm ($\sim 1$ kW) at 7 m distance from the VIRCATOR window. Peak amplitude of the HPM signal varies on shot-to-shot basis. Duration of the HPM pulse (FWHM) also varies from 52 ns to 94 ns for different shots.
Volume 76 Issue 3 March 2011 pp 501-511
Performance of the backward wave oscillator (BWO) is greatly enhanced with the introduction of plasma. Linear theory of the dispersion relation and the growth rate have been derived and analysed numerically for plasma-filled rippled wall rectangular waveguide driven by sheet electron beam. To see the effect of plasma on the TM
01 cold wave structure mode and on the generated frequency, the parameters used are: relativistic factor $\gamma = 1.5$ (i.e. $v/c = 0.741$), average waveguide height $y_0 = 1.445$ cm, axial corrugation period $z_0 = 1.67$ cm, and corrugation amplitude $\epsilon = 0.225$ cm. The plasma density is varied from zero to $2\times 10^{12}$ cm -3. The presence of plasma tends to raise the TM 01 mode cut-off frequency (14 GH$_z$ at $2 \times 10^{12}$ cm -3 plasma density) relative to the vacuum cut-off frequency (5 GH$_z$) which also causes a decrease in the group velocity everywhere, resulting in a flattening of the dispersion relation. With the introduction of plasma, an enhancement in absolute instability was observed.
Volume 78 Issue 4 April 2012 pp 635-649 Research Articles
Experimental investigation of the thermal conductivity of large grain and its dependence on the trapped vortices in parallel magnetic field with respect to the temperature gradient $\nabla T$ was carried out on four large-grain niobium samples from four different ingots. The zero-field thermal conductivity measurements are in good agreement with the measurements based on the theory of Bardeen–Rickayzen–Tewordt (BRT). The change in thermal conductivity with trapped vortices is analysed with the field dependence of the conductivity results of Vinen et al for low inductions and low-temperature situation. Finally, the dependence of thermal conductivity on the applied magnetic field in the vicinity of the upper critical field $H_{c2}$ is fitted with the theory of pure type-II superconductor of Houghton and Maki. Initial remnant magnetization in the sample shows a departure from the Houghton–Maki curve whereas the sample with zero trapped flux qualitatively agrees with the theory. A qualitative discussion is presented explaining the reason for such deviation from the theory. It has also been observed that if the sample with the trapped vortices is cycled through $T_c$, the subsequent measurement of the thermal conductivity coincides with the zero trapped flux results.
Volume 80 Issue 2 February 2013 pp 277-282 Research Articles
Transport of high current (∼kA range with particle energy $\sim 1$ MeV) planar electron beams is a topic of increasing interest for applications in high-power (1–10 GW) and high-frequency (10–20 GHz) microwave devices such as backward wave oscillator (
-1 , relativistic factor $\gamma = 1.16$, and beam voltage = ∼80 kV in notched wiggler magnet array. The calculation includes self-consistent effects of beam-generated fields. Our results show that the notched wiggler configuration with ∼6.97 kG magnetic field strength can provide vertical and horizontal confinements for a sheet electron beam with 0.3 cm thickness and 2 cm width. The feasibility calculation addresses to a system expected to drive for 13–20 GHz
Current Issue
Volume 93 | Issue 5 November 2019
Click here for Editorial Note on CAP Mode |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphenation near box question, I guess that makes sense, basically can't just rely on lccode anymore. I suppose you don't want the hyphenation code in my last answer by default?
@JosephWright anway if we rip out all the auto-testing (since mac/windows/linux come out the same anyway) but leave in the .cfg possibility, there is no actual loss of functionality if someone is still using a vms tex or whatever
I want to change the tracking (space between the characters) for a sans serif font. I found that I can use the microtype package to change the tracking of the smallcaps font (\textsc{foo}), but I can't figure out how to make \textsc{} a sans serif font.
@DavidCarlisle -- if you write it as "4 May 2016" you don't need a comma (or, in the u.s., want a comma).
@egreg (even if you're not here at the moment) -- tomorrow is international archaeology day: twitter.com/ArchaeologyDay , so there must be someplace near you that you could visit to demonstrate your firsthand knowledge.
@barbarabeeton I prefer May 4, 2016, for some reason (don't know why actually)
@barbarabeeton but I have another question maybe better suited for you please: If a member of a conference scientific committee writes a preface for the special issue, can the signature say John Doe \\ for the scientific committee or is there a better wording?
@barbarabeeton overrightarrow answer will have to wait, need time to debug \ialign :-) (it's not the \smash wat did it) on the other hand if we mention \ialign enough it may interest @egreg enough to debug it for us.
@DavidCarlisle -- okay. are you sure the \smash isn't involved? i thought it might also be the reason that the arrow is too close to the "M". (\smash[t] might have been more appropriate.) i haven't yet had a chance to try it out at "normal" size; after all, \Huge is magnified from a larger base for the alphabet, but always from 10pt for symbols, and that's bound to have an effect, not necessarily positive. (and yes, that is the sort of thing that seems to fascinate @egreg.)
@barbarabeeton yes I edited the arrow macros not to have relbar (ie just omit the extender entirely and just have a single arrowhead but it still overprinted when in the \ialign construct but I'd already spent too long on it at work so stopped, may try to look this weekend (but it's uktug tomorrow)
if the expression is put into an \fbox, it is clear all around. even with the \smash. so something else is going on. put it into a text block, with \newline after the preceding text, and directly following before another text line. i think the intention is to treat the "M" as a large operator (like \sum or \prod, but the submitter wasn't very specific about the intent.)
@egreg -- okay. i'll double check that with plain tex. but that doesn't explain why there's also an overlap of the arrow with the "M", at least in the output i got. personally, i think that that arrow is horrendously too large in that context, which is why i'd like to know what is intended.
@barbarabeeton the overlap below is much smaller, see the righthand box with the arrow in egreg's image, it just extends below and catches the serifs on the M, but th eoverlap above is pretty bad really
@DavidCarlisle -- i think other possible/probable contexts for the \over*arrows have to be looked at also. this example is way outside the contexts i would expect. and any change should work without adverse effect in the "normal" contexts.
@DavidCarlisle -- maybe better take a look at the latin modern math arrowheads ...
@DavidCarlisle I see no real way out. The CM arrows extend above the x-height, but the advertised height is 1ex (actually a bit less). If you add the strut, you end up with too big a space when using other fonts.
MagSafe is a series of proprietary magnetically attached power connectors, originally introduced by Apple Inc. on January 10, 2006, in conjunction with the MacBook Pro at the Macworld Expo in San Francisco, California. The connector is held in place magnetically so that if it is tugged — for example, by someone tripping over the cord — it will pull out of the socket without damaging the connector or the computer power socket, and without pulling the computer off the surface on which it is located.The concept of MagSafe is copied from the magnetic power connectors that are part of many deep fryers...
has anyone converted from LaTeX -> Word before? I have seen questions on the site but I'm wondering what the result is like... and whether the document is still completely editable etc after the conversion? I mean, if the doc is written in LaTeX, then converted to Word, is the word editable?
I'm not familiar with word, so I'm not sure if there are things there that would just get goofed up or something.
@baxx never use word (have a copy just because but I don't use it;-) but have helped enough people with things over the years, these days I'd probably convert to html latexml or tex4ht then import the html into word and see what come out
You should be able to cut and paste mathematics from your web browser to Word (or any of the Micorsoft Office suite). Unfortunately at present you have to make a small edit but any text editor will do for that.Givenx=\frac{-b\pm\sqrt{b^2-4ac}}{2a}Make a small html file that looks like<!...
@baxx all the convertors that I mention can deal with document \newcommand to a certain extent. if it is just \newcommand\z{\mathbb{Z}} that is no problem in any of them, if it's half a million lines of tex commands implementing tikz then it gets trickier.
@baxx yes but they are extremes but the thing is you just never know, you may see a simple article class document that uses no hard looking packages then get half way through and find \makeatletter several hundred lines of trick tex macros copied from this site that are over-writing latex format internals. |
Hello, I've never ventured into char before but cfr suggested that I ask in here about a better name for the quiz package that I am getting ready to submit to ctan (tex.stackexchange.com/questions/393309/…). Is something like latex2quiz too audacious?
Also, is anyone able to answer my questions about submitting to ctan, in particular about the format of the zip file and putting a configuration file in $TEXMFLOCAL/scripts/mathquiz/mathquizrc
Thanks. I'll email first but it sounds like a flat file with a TDS included in the right approach. (There are about 10 files for the package proper and the rest are for the documentation -- all of the images in the manual are auto-generated from "example" source files. The zip file is also auto generated so there's no packaging overhead...)
@Bubaya I think luatex has a command to force “cramped style”, which might solve the problem. Alternatively, you can lower the exponent a bit with f^{\raisebox{-1pt}{$\scriptstyle(m)$}} (modify the -1pt if need be).
@Bubaya (gotta go now, no time for followups on this one …)
@egreg @DavidCarlisle I already tried to avoid ascenders. Consider this MWE:
\documentclass[10pt]{scrartcl}\usepackage{lmodern}\usepackage{amsfonts}\begin{document}\noindentIf all indices are even, then all $\gamma_{i,i\pm1}=1$.In this case the $\partial$-elementary symmetric polynomialsspecialise to those from at $\gamma_{i,i\pm1}=1$,which we recognise at the ordinary elementary symmetric polynomials $\varepsilon^{(n)}_m$.The induction formula from indeed gives\end{document}
@PauloCereda -- okay. poke away. (by the way, do you know anything about glossaries? i'm having trouble forcing a "glossary" that is really an index, and should have been entered that way, into the required series style.)
@JosephWright I'd forgotten all about it but every couple of months it sends me an email saying I'm missing out. Oddly enough facebook and linked in do the same, as did research gate before I spam filtered RG:-)
@DavidCarlisle Regarding github.com/ho-tex/hyperref/issues/37, do you think that \textNFSSnoboundary would be okay as name? I don't want to use the suggested \textPUnoboundary as there is a similar definition in pdfx/l8uenc.def. And textnoboundary isn't imho good either, as it is more or less only an internal definition and not meant for users.
@UlrikeFischer I think it should be OK to use @, I just looked at puenc.def and for example \DeclareTextCompositeCommand{\b}{PU}{\@empty}{\textmacronbelow}% so @ needs to be safe
@UlrikeFischer that said I'm not sure it needs to be an encoding specific command, if it is only used as \let\noboundary\zzznoboundary when you know the PU encoding is going to be in force, it could just be \def\zzznoboundary{..} couldn't it?
@DavidCarlisle But puarenc.def is actually only an extension of puenc.def, so it is quite possible to do \usepackage[unicode]{hyperref}\input{puarenc.def}. And while I used a lot @ in the chess encodings, since I saw you do \input{tuenc.def} in an example I'm not sure if it was a good idea ...
@JosephWright it seems to be the day for merge commits in pull requests. Does github's "squash and merge" make it all into a single commit anyway so the multiple commits in the PR don't matter or should I be doing the cherry picking stuff (not that the git history is so important here) github.com/ho-tex/hyperref/pull/45 (@UlrikeFischer)
@JosephWright I really think I should drop all the generation of README and ChangeLog in html and pdf versions it failed there as the xslt is version 1 and I've just upgraded to a version 3 engine, an dit's dropped 1.0 compatibility:-) |
We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the topology on $\mathcal{D}_K(\Omega):=\lbrace \varphi \in C^{\infty}(\Omega) : \mathrm{supp}(\varphi) \subset K \rbrace$, defined by seminorm $p_{K_N}(\varphi):=\sup_{|\alpha| \leq N ; x \in K} |D^\alpha \varphi(x)|$. It can be shown that $\mathcal{U}:=\lbrace \varphi + U : \varphi \in \mathcal{D}(\Omega), U \in \mathcal{U}_0 \rbrace$ is a base for the vectorial topology on $\mathcal{D}(\Omega)$. As in the book Functional Analysis by Rudin, page 152-153. In particular the topology $\mathcal{T}$ on $\mathcal{D}(\Omega)$ it is Hausdorff topology, and the question I have is on this step:
If $\varphi_1 \neq \varphi_2$ are function test, we define:
$\displaystyle U:= \lbrace \varphi \in \mathcal{D}(\Omega) : \sup_{x \in \Omega} |\varphi(x)| < \sup_{x \in \Omega} |\varphi_1(x) - \varphi_2(x)| \rbrace$
we have $U \in \mathcal{U}_0$ and $\varphi_1 \notin \varphi_2 + U$. It follows that the singleton $\lbrace \varphi_1 \rbrace$ is a closed for topology $\mathcal{T}$ (why?). Then, since $\varphi_1 \neq \varphi_2$, there is $U' \in \mathcal{U}_0$ such that $\varphi_1 + U' \cap \varphi_2 + U = \emptyset$, so $\mathcal{T}$ is Hausdorff topology. (Is it correct this conclusion?) |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance. It is commonly used in the textile industry and often seen quoted on, for example, duvets and carpet underlay.The Shirley Institute in Manchester, England developed the tog as an easy-to-follow alternative to the SI unit of m2K/W. The name comes from the informal word "togs" for clothing which itself was probably derived from the word toga, a Roman garment.The basic unit of insulation coefficient is the RSI, (1 m2K/W). 1 tog = 0.1 RSI. There is also a clo clothing unit equivalent to 0.155 RSI or 1.55 tog...
The stone or stone weight (abbreviation: st.) is an English and imperial unit of mass now equal to 14 pounds (6.35029318 kg).England and other Germanic-speaking countries of northern Europe formerly used various standardised "stones" for trade, with their values ranging from about 5 to 40 local pounds (roughly 3 to 15 kg) depending on the location and objects weighed. The United Kingdom's imperial system adopted the wool stone of 14 pounds in 1835. With the advent of metrication, Europe's various "stones" were superseded by or adapted to the kilogram from the mid-19th century on. The stone continues...
Can you tell me why this question deserves to be negative?I tried to find faults and I couldn't: I did some research, I did all the calculations I could, and I think it is clear enough . I had deleted it and was going to abandon the site but then I decided to learn what is wrong and see if I ca...
I am a bit confused in classical physics's angular momentum. For a orbital motion of a point mass: if we pick a new coordinate (that doesn't move w.r.t. the old coordinate), angular momentum should be still conserved, right? (I calculated a quite absurd result - it is no longer conserved (an additional term that varies with time )
in new coordinnate: $\vec {L'}=\vec{r'} \times \vec{p'}$
$=(\vec{R}+\vec{r}) \times \vec{p}$
$=\vec{R} \times \vec{p} + \vec L$
where the 1st term varies with time. (where R is the shift of coordinate, since R is constant, and p sort of rotating.)
would anyone kind enough to shed some light on this for me?
From what we discussed, your literary taste seems to be classical/conventional in nature. That book is inherently unconventional in nature; it's not supposed to be read as a novel, it's supposed to be read as an encyclopedia
@BalarkaSen Dare I say it, my literary taste continues to change as I have kept on reading :-)
One book that I finished reading today, The Sense of An Ending (different from the movie with the same title) is far from anything I would've been able to read, even, two years ago, but I absolutely loved it.
I've just started watching the Fall - it seems good so far (after 1 episode)... I'm with @JohnRennie on the Sherlock Holmes books and would add that the most recent TV episodes were appalling. I've been told to read Agatha Christy but haven't got round to it yet
?Is it possible to make a time machine ever? Please give an easy answer,a simple one A simple answer, but a possibly wrong one, is to say that a time machine is not possible. Currently, we don't have either the technology to build one, nor a definite, proven (or generally accepted) idea of how we could build one. — Countto1047 secs ago
@vzn if it's a romantic novel, which it looks like, it's probably not for me - I'm getting to be more and more fussy about books and have a ridiculously long list to read as it is. I'm going to counter that one by suggesting Ann Leckie's Ancillary Justice series
Although if you like epic fantasy, Malazan book of the Fallen is fantastic
@Mithrandir24601 lol it has some love story but its written by a guy so cant be a romantic novel... besides what decent stories dont involve love interests anyway :P ... was just reading his blog, they are gonna do a movie of one of his books with kate winslet, cant beat that right? :P variety.com/2016/film/news/…
@vzn "he falls in love with Daley Cross, an angelic voice in need of a song." I think that counts :P It's not that I don't like it, it's just that authors very rarely do anywhere near a decent job of it. If it's a major part of the plot, it's often either eyeroll worthy and cringy or boring and predictable with OK writing. A notable exception is Stephen Erikson
@vzn depends exactly what you mean by 'love story component', but often yeah... It's not always so bad in sci-fi and fantasy where it's not in the focus so much and just evolves in a reasonable, if predictable way with the storyline, although it depends on what you read (e.g. Brent Weeks, Brandon Sanderson). Of course Patrick Rothfuss completely inverts this trope :) and Lev Grossman is a study on how to do character development and totally destroys typical romance plots
@Slereah The idea is to pick some spacelike hypersurface $\Sigma$ containing $p$. Now specifying $u(p)$ is trivial because the wave equation is invariant under constant perturbations. So that's whatever. But I can specify $\nabla u(p)|\Sigma$ by specifying $u(\cdot, 0)$ and differentiate along the surface. For the Cauchy theorems I can also specify $u_t(\cdot,0)$.
Now take the neigborhood to be $\approx (-\epsilon,\epsilon)\times\Sigma$ and then split the metric like $-dt^2+h$
Do forwards and backwards Cauchy solutions, then check that the derivatives match on the interface $\{0\}\times\Sigma$
Why is it that you can only cool down a substance so far before the energy goes into changing it's state? I assume it has something to do with the distance between molecules meaning that intermolecular interactions have less energy in them than making the distance between them even smaller, but why does it create these bonds instead of making the distance smaller / just reducing the temperature more?
Thanks @CooperCape but this leads me another question I forgot ages ago
If you have an electron cloud, is the electric field from that electron just some sort of averaged field from some centre of amplitude or is it a superposition of fields each coming from some point in the cloud? |
Misiurewicz points in the Mandelbrot set are strictly preperiodic. Defining the quadratic polynomial \(F_c(z) = z^2 + c\), then a Misiurewicz point with preperiod \(q > 0\) and period \(p > 0\) satisfies:
\[\begin{aligned} {F_c}^{q + p}(c) &= {F_c}^{q}(c) \\ {F_c}^{q' + p}(c) &\ne {F_c}^{q'}(c)\text{ for all } 0 \le q' < q \\ {F_c}^{q + p'}(c) &\ne {F_c}^{q}(c)\text{ for all } 1 \le p' < p \end{aligned}\]
where the first line says it is preperiodic, the second line says that the preperiod is exactly \(q\), and the third line says that the period is exactly \(p\). A naive solution of the first equation would be to use Newton's method for finding a root of \(f_1(c) = 0\) where \(f_1(c) = {F_c}^{q + p}(c) - {F_c}^{q}(c)\), and it does work: but the root found might have lower preperiod or lower period, so it requires checking to see if it's really the Misiurewicz point we want.
This need for checking felt unsatisfactory, so I tried to figure out a way to reject wrong solutions during the Newton's method iterations. The second line equation for exact preperiod gives \({F_c}^{q' + p}(c) - {F_c}^{q'}(c) \ne 0\), so I tried dividing \(f_1(c)\) by all those non-zero values to give an \(f_2(c)\) and applying Newton's method for finding a root of \(f_2(c) = 0\). So:
\[\begin{aligned} f_2(c) &= \frac{g_2(c)}{h_2(c)} \\ g_2(c) &= {F_c}^{q + p}(c) - {F_c}^{q}(c) \\ h_2(c) &= \prod_{q'=0}^{q-1}\left( {F_c}^{q' + p}(c) - {F_c}^{q'}(c) \right) \\ f_2'(c) &= \frac{g_2'(c) h_2(c) - g_2(c) h_2'(c)}{h_2(c)^2} \\ g_2'(c) &= ({F_c}^{q + p})'(c) - ({F_c}^{q})'(c) \\ h_2'(c) &= h_2(c) \sum_{q'=0}^{q-1} \frac{({F_c}^{q' + p})'(c) - ({F_c}^{q'})'(c)}{{F_c}^{q' + p}(c) - {F_c}^{q'}(c)} \end{aligned}\]
with the Newton step \(c_{n+1} = c_{n} - \frac{f_2(c_n)}{f_2'(c_n)}\) where \(c_0\) is an initial guess. Here are some image comparisons, where each pixel is coloured according the the root found (grey for wrong (pre)period, saturated circles surround each root within the basin of attraction, edges between basins coloured black, with the Mandelbrot set overlayed in white). The top half of each image uses the naive method, the bottom half the method detailed in this post (which seems better, because the saturated circles are larger):
The images are labelled with preperiod and period, but there might be an off-by-one error with respect to standard terminology: here I iterate \(F_c\) starting from \(c\), while some iterate \(F_c\) starting from \(0\). So my preperiods are one less than they would be if I'd started from \(0\). I tried extending the method to reject lower periods as well as lower preperiods, but it didn't work very well.
The C99 source code for this post is available: newton-misiurewicz.c, using code from my new (work-in-progress) mandelbrot-numerics library mandelbrot-numerics library (git HEAD at 7fe3b89465390a712c7427093b8fc5377d2e65b6 when this post was written). Compiled using OpenMP for parallelism, it takes a little over 25mins to run on my quad-core machine.
References: |
Generally, if the frequency of a signal or a particular band of signals is high, the bandwidth utilization is high as the signal provides more space for other signals to get accumulated. However, high frequency signals can't travel longer distances without getting attenuated. We have studied that transmission lines help the signals to travel longer distances.
Microwaves propagate through microwave circuits, components and devices, which act as a part of Microwave transmission lines, broadly called as Waveguides.
A hollow metallic tube of uniform cross-section for transmitting electromagnetic waves by successive reflections from the inner walls of the tube is called as a
Waveguide.
The following figure shows an example of a waveguide.
A waveguide is generally preferred in microwave communications. Waveguide is a special form of transmission line, which is a hollow metal tube. Unlike a transmission line, a waveguide has no center conductor.
The main characteristics of a Waveguide are −
The tube wall provides distributed inductance.
The empty space between the tube walls provide distributed capacitance.
These are bulky and expensive.
Following are few advantages of Waveguides.
Waveguides are easy to manufacture.
They can handle very large power (in kilo watts).
Power loss is very negligible in waveguides.
They offer very low loss (low value of alpha-attenuation).
When microwave energy travels through waveguide, it experiences lower losses than a coaxial cable.
There are five types of waveguides.
The following figures show the types of waveguides.
The types of waveguides shown above are hollow in the center and made up of copper walls. These have a thin lining of Au or Ag on the inner surface.
Let us now compare the transmission lines and waveguides.
The main difference between a transmission line and a wave guide is −
A
two conductor structure that can support a TEM wave is a transmission line.
A
one conductor structure that can support a TE wave or a TM wave but not a TEM wave is called as a waveguide.
The following table brings out the differences between transmission lines and waveguides.
Transmission Lines Waveguides Supports TEM wave Cannot support TEM wave All frequencies can pass through Only the frequencies that are greater than cut-off frequency can pass through One conductor transmission Two conductor transmission Reflections are less Wave travels through reflections from the walls of waveguide It has characteristic impedance It has wave impedance Propagation of waves is according to "Circuit theory" Propagation of waves is according to "Field theory" It has a return conductor to earth Return conductor is not required as the body of the waveguide acts as earth Bandwidth is not limited Bandwidth is limited Waves do not disperse Waves get dispersed
Phase Velocity is the rate at which the wave changes its phase in order to undergo a phase shift of
2π radians. It can be understood as the change in velocity of the wave components of a sine wave, when modulated.
Let us derive an equation for the Phase velocity.
According to the definition, the rate of phase change at
2π radians is to be considered.
Which means,
$λ$ / $T$ hence,
$$V = \frac{\lambda }{T}$$
Where,
$λ$ = wavelength and $T$ = time
$$V = \frac{\lambda }{T} = \lambda f$$
Since $f = \frac{1}{T}$
If we multiply the numerator and denominator by
2π then, we have
$$V = \lambda f = \frac{2\pi \lambda f}{2\pi }$$
We know that $\omega = 2\pi f$
and $\beta = \frac{2\pi }{f}$
The above equation can be written as,
$$V = \frac{2\pi f}{\frac{2\pi }{\lambda }} = \frac{\omega }{\beta }$$
Hence, the equation for Phase velocity is represented as
$$V_p = \frac{\omega }{\beta }$$
Group Velocity can be defined as the rate at which the wave propagates through the waveguide. This can be understood as the rate at which a modulated envelope travels compared to the carrier alone. This modulated wave travels through the waveguide.
The equation of Group Velocity is represented as
$$V_g = \frac{d\omega }{d\beta }$$
The velocity of modulated envelope is usually slower than the carrier signal. |
Epimorphism Preserves Commutativity Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.
Let $\circ$ be a commutative operation.
Then $*$ is also a commutative operation. Proof
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.
As an epimorphism is surjective, it follows that:
$\forall u, v \in T: \exists x, y \in S: \phi \left({x}\right) = u, \phi \left({y}\right) = v$
So:
\(\displaystyle u * v\) \(=\) \(\displaystyle \phi \left({x}\right) * \phi \left({y}\right)\) $\phi$ is a surjection \(\displaystyle \) \(=\) \(\displaystyle \phi \left({x \circ y}\right)\) Morphism Property \(\displaystyle \) \(=\) \(\displaystyle \phi \left({y \circ x}\right)\) Commutativity of $\circ$ \(\displaystyle \) \(=\) \(\displaystyle \phi \left({y}\right) * \phi {\left({x}\right)}\) Morphism Property \(\displaystyle \) \(=\) \(\displaystyle v * u\) by definition as above
$\blacksquare$
Note that this result is applied to epimorphisms.
Also see |
Yes, they're representations of $SO(8)$, more precisely $Spin(8)$ which is an "improvement" of $SO(8)$ that allows the rotation by 360 degrees to be represented by a matrix different from the unit matrix, namely minus unit matrix.
${\bf 8}_v$ transforms normally as $$ v\mapsto M v$$where $MM^T=1$ is the $8\times 8$ real orthogonal $SO(8)$ matrix. The spinor reps ${\bf 8}_s\oplus {\bf 8}_c$ label the left-handed and right-handed spinor, respectively. People usually learn spinors well before they study RNS string theory.
The spinor representation transforms under $SO(8)$ in a way that is fully encoded by the transformation rules under infinitesimal $SO(8)$ transformations, $1+i\omega_{ij} J^{ij}$ where $\omega$ are the angle parameters and $J$ are the generators.
In the Dirac spinor representation, $J_{ij}$ is written as$$ J_{ij} = \frac{\gamma_i \gamma_j - \gamma_j\gamma_i}{4}$$where $\gamma$ are the Dirac matrices that may be written as tensor products of Pauli matrices and the unit matrix and that obey$$\gamma_i\gamma_j+\gamma_j\gamma_i = 2\delta_{ij}\cdot {\bf 1}$$Each pair of added dimensions doubles the size of the Dirac matrices so the dimension of the total "Dirac" representation for $SO(2n)$ is $2^n$. For $n=4$ we get $2^4=16$.
This 16-dimensional spinor representation is real and may be split, according to the eigenvalue of the $\Gamma_9$ chirality matrix, to the 8-dimensional chiral (=Weyl) spinor representations labeled by the indices s,c.
For $SO(8)$, there are 3 real 8-dimensional irreducible representations that are "equally good" and may actually be permuted by an operation called "triality". This operation may be seen as the $S_3$ permutation symmetry of the 3 legs of the Mercedes-logo-like $SO(8)$ Dynkin diagram. I just wrote a text about it last night:
http://motls.blogspot.cz/2013/04/complex-real-and-pseudoreal.html?m=1
If you really need to explain what a representation of a group is, you should interrupt your studies of string theory and focus on group theory – keywords Lie groups, Lie algebras, and representation theory. Without this background, you would face similar confusion too often.This post imported from StackExchange Physics at 2014-03-07 16:30 (UCT), posted by SE-user Luboš Motl |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ... |
Definition:Fermat Number Contents Definition
A
Fermat number is a natural number of the form $2^{\paren {2^n} } + 1$, where $n = 0, 1, 2, \ldots$.
The number $2^{\paren {2^n} } + 1$ is, in this context, often denoted $F_n$.
\(\displaystyle 2^{\paren {2^0} } + 1\) \(=\) \(\displaystyle 3\) \(\displaystyle 2^{\paren {2^1} } + 1\) \(=\) \(\displaystyle 5\) \(\displaystyle 2^{\paren {2^2} } + 1\) \(=\) \(\displaystyle 17\) \(\displaystyle 2^{\paren {2^3} } + 1\) \(=\) \(\displaystyle 257\) \(\displaystyle 2^{\paren {2^4} } + 1\) \(=\) \(\displaystyle 65 \, 537\) \(\displaystyle 2^{\paren {2^5} } + 1\) \(=\) \(\displaystyle 4 \, 294 \, 967 \, 297\) \(\displaystyle 2^{\paren {2^6} } + 1\) \(=\) \(\displaystyle 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617\)
Both conventions are in place, sometimes in the same work.
However, in Section $257$ he defines $F_3 = 2^{2^3} + 1 = 257$ as the
$3$rd Fermat number.
Similarly, in Section $65,537$ he defines $F_4 = 2^{2^4} + 1 = 65 \, 537$ as the
$4$th Fermat number, and so on. Both of these naming conventions is more or less clumsy.
$\mathsf{Pr} \infty \mathsf{fWiki}$ takes the position that the cat has to jump one way or the other, and so uses the second of these conventions:
$F_n$ is the $n$th Fermat number. Also see Results about Fermat Numberscan be found here. Source of Name
This entry was named for Pierre de Fermat.
In $1640$, Pierre de Fermat wrote to Bernard Frénicle de Bessy that $2^n + 1$ is composite if $n$ is divisible by an odd prime.
On being unable to prove it, he sent the problem to Blaise Pascal, with the note:
I wouldn't ask you to work at it if I had been successful.
Pascal unfortunately did not take up the challenge.
In $1878$, he similarly found that $5 \times 2^{25} + 1$ is a divisor of $F_{23}$.
In $1909$, James C. Morehead and Alfred E. Western reported in
Bulletin of the American Mathematical Society that they had proved that $F_7$ and $F_8$ are not prime, but without having established what the prime factors are. $F_7 = \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {11 \, 141 \, 971 \, 095 \, 088 \, 142 \, 685 \times 2^9 + 1}$ $1 \, 238 \, 926 \, 361 \, 552 \, 897$
For example:
a divisor of $F_{1945}$ is known $19 \times 2^{9450} + 1$ is a divisor of $F_{9448}$ $5 \times 2^{23 \, 473} + 1$ is a divisor of $F_{23 \, 471}$ Sources 1937: Eric Temple Bell: Men of Mathematics... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs 1982: P.M. Cohn: Algebra Volume 1(2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $8$ 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $5$ 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $127$ 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $257$ 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $65,537$ 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $4,294,967,297$ 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $5$ 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $127$ 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $257$ 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $65,537$ 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $4,294,967,297$ 2008: David Nelson: The Penguin Dictionary of Mathematics(4th ed.) ... (previous) ... (next): Entry: Fermat number(P. de Fermat, 1640) |
Definition:Permutation on Polynomial Definition
Let $\map f {x_1, x_2, \ldots, x_n}$ denote a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\pi, \rho \in S_n$.
Then $\pi * f$ is the polynomial obtained by applying the permutation $\pi$ to the subscripts on the variables of $f$. This is called the permutation on the polynomial $f$ by $\pi$, or the $f$-permutation by $\pi$. Also known as
This is also called the
permutation of the polynomial.
Consider the polynomial on $3$ variables:
$\map f {x_1, x_2, x_3} = {x_1}^2 + 2 x_1 x_2 = 4 x_1 x_2 {x_3}^2$ Then: $\rho \circ f = {x_2}^2 + 2 x_2 x_3 = 4 x_2 x_3 {x_1}^2$ Also see Results about permutations on polynomialscan be found here. |
How to Model Moisture Flow in COMSOL Multiphysics®
Computing laminar and turbulent moisture flows in air is both flexible and user friendly with the
Moisture Flow multiphysics interfaces and coupling in the COMSOL Multiphysics® software. Available as of version 5.3a, this comprehensive set of functionality can be used to model coupled heat and moisture transport in air and building materials. Let’s learn how the Moisture Flow interface complements existing functionality, while highlighting its benefits.
Modeling Heat and Moisture Transport
Modeling the transport of heat and moisture through porous materials, or from the surface of a fluid, often involves including the surrounding media in the model in order to get accurate estimates of the conditions at the material surfaces. In the investigations of hygrothermal behavior of building envelopes, food packaging, and other common engineering problems, the surrounding medium is probably moist air (air with water vapor).
Moist air is the environing medium for applications such as building envelopes (illustration, left) and solar food drying (right). Right image by ArianeCCM — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
When considering porous media, the moisture transport process, which includes capillary flow, bulk flow, and binary diffusion of water vapor in air, depends on the nature of the material. In moist air, moisture is transported by diffusion and advection, where the advecting flow field in most cases is turbulent.
Computing heat and moisture transport in moist air requires the resolution of three sets of equations:
The Navier-Stokes equations, to compute the airflow velocity field \mathbf{u} and pressure p The energy equation, to compute the temperature T The moisture transport equation, to compute the relative humidity \phi
These equations are coupled together through the pressure, temperature, and relative humidity, which are used to evaluate the properties of air (density \rho(p,T,\phi); viscosity \mu(T,\phi); thermal conductivity k(T,\phi); and heat capacity C_p(T,\phi)); molecular diffusivity D(T) and through the velocity field used for convective transport.
With the addition of the
Moisture Flow multiphysics interface in version 5.3a, COMSOL Multiphysics defines all three of these equations in a few steps, as shown in the figure below. Using the Moisture Flow Multiphysics Interface
Whenever studying the flow of moist air, two questions should be asked:
Does the flow depend on moisture distribution? Does the nature of the flow require the use of a turbulence model?
If the answer is “yes” for at least one of these questions, then you should consider using the
Moisture Flow multiphysics interfaces, found under the Chemical Species Transport branch. The Moisture Flow group under the Chemical Species Transport branch of the Physics Wizard , with the single-physics interfaces and coupling node added with each version of the Moisture Flow predefined multiphysics interface.
The
Laminar Flow version of the multiphysics interface combines the Moisture Transport in Air interface with the Laminar Flow interface and adds the Moisture Flow coupling. Similarly, each version under Turbulent Flow combines the Moisture Transport in Air interface and the corresponding Turbulent Flow interface and adds the Moisture Flow coupling.
Besides providing a user-friendly way to define the coupled set of equations of the moisture flow problem, the multiphysics interfaces for turbulent flow handle the moisture-related turbulence variables required for the fluid flow computation.
Automatic Coupling Between Single-Physics Interfaces
One advantage of using the
Moisture Flow multiphysics interface is its usability. When adding the Moisture Flow node through the predefined interface, an automatic coupling of the Navier-Stokes equations is defined for the fluid flow and the moisture transport equations by the software (center screenshot in the image below) by using the following variables: The density and dynamic viscosity in the Navier-Stokes equations, which depend on the relative humidity variable from the Moisture Transportinterface through a mixture formula based on dry air and pure steam properties (left screenshot below) The velocity field and absolute pressure variables from the Single-Phase Flowinterface, which are used in the moisture transport equation (right screenshot below) Support for Turbulent Fluid Flow
The performance of the
Moisture Flow multiphysics interface is especially attractive when dealing with a turbulent moisture flow.
For turbulent flows, the turbulent mixing caused by the eddy diffusivity in the moisture convection is automatically accounted for by the COMSOL® software by enhancing the moisture diffusivity with a correction term based on the turbulent Schmidt number . The Kays-Crawford model is the default choice for the evaluation of the turbulent Schmidt number, but a user-defined value or expression can also be entered directly in the graphical user interface.
Selection of the model for the computation of the turbulent Schmidt number in the user interface of the Moisture Flow coupling.
In addition, for coarse meshes that may not be suitable for resolving the thin boundary layer close to walls,
Wall functions can be selected or automatically applied by the software. The wall functions are such that the computational domain is assumed to be located at a distance from the wall, the so-called lift-off position, corresponding to the distance from the wall where the logarithmic layer meets the viscous sublayer (or would meet it if there was no buffer layer in between). The moisture flux at the lift-off position, g_{wf}, which accounts for the flux to and from the wall, is automatically defined by the Moisture Flow interface, based on the relative humidity. Approximation of the flow field and the moisture flux close to walls when using wall functions in the turbulence model for fluid flow.
Note that the
Low-Reynolds and Automatic options for Wall Treatment are also available for some of the RANS models.
For more information, read this blog post on choosing a turbulence model.
Mass Conservation Across Boundaries
By using the
Moisture Flow interface, an appropriate mass conservation is granted in the fluid flow problem by the Screen and Interior Fan boundary conditions. A continuity condition is also applied on vapor concentration at the boundaries where the Screen feature is applied. For the Interior Fan condition, the mass flow rate is conserved in an averaged way and the vapor concentration is homogenized at the fan outlet, as shown in the figure below. Average mass flow rate conservation across a boundary with the Interior Fan condition. Example: Modeling Evaporative Cooling with the Moisture Flow Interface
Let’s consider evaporative cooling at the water surface of a glass of water placed in a turbulent airflow. The
Turbulent Flow, Low Reynolds k-ε interface, the Moisture Transport in Air interface, and the Heat Transfer in Moist Air interface are coupled through the Nonisothermal Flow, Moisture Flow, and Heat and Moisture coupling nodes. These couplings compute the nonisothermal airflow passing over the glass, the evaporation from the water surface with the associated latent heat effect, and the transport of both heat and moisture away from this surface.
By using the
Automatic option for Wall treatment in the Turbulent Flow, Low Reynolds k-ε interface, wall functions are used if the mesh resolution is not fine enough to fully resolve the velocity boundary layer close to the walls. Convective heat and moisture fluxes at lift-off position are added by the Nonisothermal Flow and Moisture Flow couplings. The temperature and relative humidity solutions after 20 minutes are shown below, along with the streamlines of the airflow velocity field. Temperature (left) and relative humidity (right) solutions with the streamlines of the velocity field after 20 minutes.
The temperature and relative humidity fields have a strong resemblance here, which is quite natural since the fields are strongly coupled and since both transport processes have similar boundary conditions, in this case. In addition, heat transfer is given by conduction and advection while mass transfer is described by diffusion and advection. The two transport processes originate from the same physical phenomena: conduction and diffusion from molecular interactions in the gas phase while advection is given by the total motion of the bulk of the fluid. Also, the contribution of the eddy diffusivity to the turbulent thermal conductivity and the turbulent diffusivity originate from the same physical phenomenon, which adds further to the similarity of the temperature and moisture field.
Next Steps
Learn more about the key features and functionality included with the Heat Transfer Module, and add-on to COMSOL Multiphysics:
Read the following blog posts to learn more about heat and moisture transport modeling:
How to Model Heat and Moisture Transport in Porous Media with COMSOL® How to Model Heat and Moisture Transport in Air with COMSOL®
Get a demonstration of the
Nonisothermal Flow and Heat and Moisture couplings in these tutorial models: Comments (2) CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science TAGS CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science |
Solving linear Inequalities means that comparison of two values or expressions. A linear inequality means that a relationship between two quantities that are not equal. In equations, one side is equal to the other side. To solving the linear inequalities, multiply, divide, or subtract the both side of the inequality equation to simplify the equation. Divide or multiplying inequalities by a negative number, must flip the inequality sign.
An linear inequalities which consists of more than one terms are raised only to the first power in terms, having the following forms, bx + c > 0 bx + c < 0 bx + c $\geq$ 0 bx + c $\leq$ 0 Where "b" represents the numerical coefficient of x, and "c" represents the constant term.
Properties Used on Solving Linear Inequalities:
Let a, b and c be real numbers. 1. Transitive Property If a < b and b < c then a < c 2. Addition Property If a < b then a + c < b + c 3. Subtraction Property If a < b then a - c < b - c 4. Multiplication Property i. If a < b and c is positive then c $\times$ a < c $\times$ b ii. If a < b and c is negative c $\times$ a > c $\times$ b Example 1: Solving the linear inequalities 6x - 7 > 2x + 1 Solution: Given 6x - 7 > 2x + 1 Add 7 to both sides and simplifying the equation 6x > 2x + 8 Subtract 2x to both sides and simplifying the equation 4x > 8 Multiply both sides by $\frac{1}{4}$ and simplifying the equation x > 2 Conclusion: The solution the interval (2, + infinity). Example 2: Solve the linear inequality 6x - 4 > 2x + 4 Solution: Add 4 to both sides and simplifying the equation 6x > 2x + 8 Subtracting 2x on both sides and simplifying the equation 4x > 8 Multiply both sides by $\frac{1}{4}$ and simplifying the equation x > 8 Example 3: Solve the linear inequalities 7x + 12 > 0 Solution: From the given problem 7x + 12 > 0 Subtract 12 on both sides, we get 7x + 12 - 12 > 0 - 12 7x > - 12 Divide by 7 on both sides,
$\frac{(7x)}{7}$ > - $\frac{12}{7}$
x > -1.71
Example 4: Solve the linear inequalities 3x + 20 > 0 Solution: From the given problem 3x + 20 > 0 Subtract 20 on both sides, we get 3x + 20 - 20 > 0 - 20 3x > - 20 Divide by 3 on both sides,
$\frac{(3x)}{3}$ > - $\frac{20}{3}$
x > - 6.33
Example 5: Solve the linear inequalities 5x - 25 < 0 Solution: From the given problem 5x - 25 < 0 Add 25 on both sides, we get 5x + 25 - 25 < 0 + 25 5x < 25 Divide by 5 on both sides, $\frac{(5x)}{5}$ < $\frac{25}{5}$ x < 5 Example 6: Solve the linear inequalities - 4x - 24 < 0 Solution: From the given problem - 4x - 24 < 0 Add 24 on both sides, we get - 4x + 24 - 24 < 0 + 24 - 4x < 24 Divide by -4 on both sides, $\frac{(-4x)}{-4}$ < $\frac{24}{-4}$ When we divide the negative term the inequality sign ‘<’ changes to’ >’ x > 6
Related Calculators Solve Linear Inequalities Calculator Graph Linear Inequalities Calculator Graph Linear Inequality Calculator Calculator for Solving Inequalities |
In general, you won't be able to replicate the option by a portfolio of the form $\Delta_t S_t + B_t$, though it is possible to do so with a portfolio of the form $\Delta_t^1 S_t + \Delta_t^2B_t$; see Chapter 3 of this book. Here, $B_t=e^{rt}$ is the value of the money-market account, and $r$ is the risk-free interest rate.On the other hand, you can create ...
This is a basic fact about futures trading and the storage of commodities.The phrase that was used by futures traders in the old days (and probably still today) was "the contango is limited by the carrying cost, there is no limit to the backwardation". This means that for example if spot gold is at 1200, gold dated one year from now cannot possibly sell ...
There is certainly much more to quantitative finance than technical analysis, and a previous question does a decent job of outlining the different areas, as does the wikipedia on "quantitative analyst".Even for what wikipedia terms an "algorithmic trading quant" or what Mark Joshi terms a "statistical arbitrage quant", technical analysis is just one tool ...
I had read some of them; actually, it does not exist an on-line library that collected them (or, better, it existed here, but it seems the website does not work anymore).I reported here below some of them that you did not find:More Than You Ever Wanted To Know* About Volatility SwapsModel RiskThe Volatility Smile And Its implied TreeEnhanced Numerical ...
The best overview I have seen so far is this paper which lists 214 (!) factors (or anomalies if you like) on over one hundred (!) pages:Harvey, Campbell R. and Liu, Yan and Zhu, Caroline, …and the Cross-Section of Expected Returns (February 3, 2015). Available at SSRN: https://ssrn.com/abstract=2249314 or http://dx.doi.org/10.2139/ssrn.2249314Abstract: ...
I think you might find this answer in The future language of quant programming? useful.People get this problem wrong because they always end up discussing the theoretical advantages of these languages rather than the practical uses of these languages.Theoretically speaking:Haskell is elegant and has many of the theoretical advantages (language ...
A hurst exponent, H, between 0 to 0.5 is said to correspond to a mean reverting process (anti-persistent), H=0.5 corresponds to Geometric Brownian Motion (Random Walk), while H >= 0.5 corresponds to a process which is trending (persistent).The hurst exponent is limited to a value between 0 to 1, as it corresponds to a fractal dimension between 1 and 2 (D=2-...
Hi Quantitative Finance has in my opinion two main streams.The first is about of valuation of some derivative contracts in a consistent way. This is a theory and once paradigms accepted it is coherent, it can considered as science at the same level as economy can pretend to this kind of terminology.The second is about making (or trying to) prediction(s) ...
C++Think in C++ can be a starting point. This is free. And, you might study Beginning Visual C++ 2010 by Ivan HortonQuantitative finance and C++ (if you are derivatives-oriented)You might find Mark Joshi as well as Daniel Duffy's writings of (great) interest.It is easy to find the references of both their books on a website such as Amazon.You can also ...
For a basic introduction, the three chapters in Hull's Options, Futures, and Other Derivatives on Binomial Trees, Wiener Processes and Ito's Lemma, and The Black-Scholes-Merton Model helped me start to understand the basic concepts within a broader context.After that, Shreve's two books seems to be pretty popular (see here and here). He explains things ...
Quantopian provides both the fundamental data (from Morningstar), as well as the backtest platform to reproduce results from the books you mentioned. Here's the introduction to our fundamentals offering: https://www.quantopian.com/posts/fundamental-data-from-morningstar-now-available-for-backtesting(disclosure: I'm the ceo of quantopian)
Quant in trading creates system that can be backtested, has a certain risk valuation. It is more like playing chess when you need to calculate multistep strategy.Let say certain instrument moves 1% a day. Our goal is to create strategy for one year (250 step strategy). If we use stock + options we get 50 or more entries a day into our system for analysis. ...
A detailed description of the Hurst Exponent can be found here. A further (rather short search of Google) turned up this site claiming to provide an Excel Workbook with, among other things, Hurst Exponent estimation.
Unfortunately, there is no correct answer for this question, it's like what car you should drive on your weekend.C++ is a popular language in quantitative finance, but it's usually (but not always!) only used to build the application backbone, such as derivative pricing. Why C++? C++ is a good choice because C++ is platform independent, we can natively ...
This thread will inevitably close because it doesn't meet community guidelines, but I respect your passion in this field and my best suggestion for you is that if you're trying to emulate a MFE education, go look up the course listings of any reputable MFE program, and then look into the sites for those (past) classes and see the recommended readings and ...
Yes, you can say they are traded on listed options, but only for a few limited markets, and not that liquid relative to options on a single asset.For instance, the commodity futures space, there are options on commodity spreads listed, and a strike of 0 would be the same as an exchange option.These options have some liquidity in energy and grain markets,...
I found these nice lecture note by Karl Sigman on the web. On page three you see if $X\sim N(\mu,\sigma)$ then the moment generating function (mgf) of $X$ is given by$$M_X(s) = E(exp(sX)) = \exp( \mu s + \sigma^2 s^2 /2)$$Thus for Brownian motion with drift $X_t$ you get$$M_{X_t}(s) = E(exp(s X_t)) = \exp( \mu t s + \sigma^2 s^2 t /2).$$Finally for $...
If you have a fairly good model of regime separation (of course requiring a good quantitative measure of regime state classifications -- momentum and reverting) and predictive likelihood (using something like a markov state transition matrix)-- one could weight contributions corresponding to next state probabilities. Of course, you will rarely get a ...
It depends the kind of information you look for.Questions and answer.This web site is really the best I know on quant finance. You can browse "tags" and go the the associated wiki pages to have summarized information.Wilmott Forum is not that bad;Nuclear Phynance is good too.Generic knowledge.It depends on what area on finance you are interested in....
The general ideaFor equity securities, a simple backtest will typically consist of two steps:Computation of the portfolio return resulting from your portfolio formation rule (or trading strategy)Risk-adjustment of portfolio returns using an asset pricing modelStep 2 is simply a regression and computationally very simple in Matlab. What's trickier is ...
There is no guarantee you can improve the Sharpe in this case, depending on the correlation of the returns streams. For the two asset case (you can model your strategies as assets and take a linear combination of them), if the correlation of the two assets is equal to the ratio of Sharpes (smaller to larger), there is zero diversification benefit.For ...
There is not a single 'interest-rate' to reduce, there are various interest rates in play.The central bank mandate is usually to control CPI or a similar measure of inflation (e.g. Bank of England's 2% inflation target for GBP). There are various tools for them to do this, including QE and setting the central bank rate.However, at the moment, the central ...
Just an update on my playlist, It has 33 videos now, roughly 3x more vids. I have included some more general economics and machine learning and programming vids, which have relevant applications in Q finance.https://www.youtube.com/watch?v=jXFNpDcYOxM&list=PLqMiStH7exaXmQqV7y-tg68f2ZYZK3Yur
Yes. Mark Joshi's book is a good preparation.For this question you are given some function random() yielding a uniform random number and what we want is a function next() which yields realizations of a random $X$ variable with values $v_j$ such that $P(X=v_j)=p_j$.From standard textbooks we know the following transformation: If $u_i$ are uniform random ...
1) In an academic sense could it be enough to use ML to create a new factor portfolio?The original FF papers (92,93) said something deep because they contradicted the dominant theory of the day. When you say in an academic sense, you may not get much respect from serious academics if you data mine a factor these days. However, as a statistical exercise, ...
Another way of staying "time-varying risk-premium", is saying that the risk-premium is predictable. However, that the fact that the risk-premium is predictable does not means that you can make money out of this.The best two references to understand this are:Cochrane (2008) - The dog that did not barkGoyal and Welch (2007)The first tells you what ...
I'm assuming you're talking about a European option. I did a similar problem for my homework recently, I used the in-out parity for pricing the up and in barrier option.Basically European Option = Knock up and in Option + Knock up and out optionYou can price the up and out easily using Binomial and use BS formula for pricing the European Option, then ... |
Idempotent idempotent element
An element $e$ of a ring, semi-group or groupoid equal to its own square: $e^2=e$. An idempotent $e$ is said to contain an idempotent $f$ (denoted by $e\geq f$) if $ef=e=fe$. For associative rings and semi-groups, the relation $\geq$ is a partial order on the set $E$ of idempotent elements, called the natural partial order on $E$. Two idempotents $u$ and $v$ of a ring are said to be orthogonal if $uv=0=vu$. With every idempotent of a ring (and also with every system of orthogonal idempotents) there is associated the so-called Peirce decomposition of the ring. For an $n$-ary algebraic relation $\omega$, an element $e$ is said to be an idempotent if $(e\ldots e)\omega=e$, where $e$ occurs $n$ times between the brackets.
Comments
An algebraic operation $\omega$ is sometimes said to be idempotent if every element of the set on which it acts is idempotent in the sense defined above. Such operations are also called affine operations; the latter name is preferable because an affine unary operation is not the same thing as an idempotent element of the semi-group of unary operations. In the theory of $R$-modules, the affine operations are those of the form
$$(x_1,\ldots,x_n)\mapsto\sum_{i=1}^nr_ix_i$$
with $\sum_{i=1}^nr_i=1$.
How to Cite This Entry:
Idempotent.
Encyclopedia of Mathematics.URL: http://www.encyclopediaofmath.org/index.php?title=Idempotent&oldid=39746 |
Integral Targets
OptiLayer enables you to specify
In OptiLayer, integral targets are represented as finite sums and can be considered as approximations of an integral expression with the help of rectangle rule.
As in conventional targets it is possible to specify
In the example (Fig. 1), transmittance (TA, no polarization) is integrated with three commonly used spectral weight functions. It is required that the integral values are to be more than 60%.
In order to specify the Integral type (
There are three predefined commonly used spectral weights distributions:
Spectral distributions of the integral weights can be plotted using Preview panel of the Integral Weights tab.
Switching between different distributions on the left panel, you can observe different spectral weights distributions.
Along with the predefined spectral distribution of the integral weight, it is possible to specify and load your own spectral distributions. The corresponding weight function \(W(\lambda)\) is to be specified at some wavelength grid. It can done:
If you would like to specify not integral but an averaged value (averaged over a spectral range), you can add new Integral Weights spreadsheets (Fig. 5). By default a spreadsheet containing unit values only will be created. You can set the number of spectral points and the spectral range.
Default names are F1, F2, etc. Of course, you can rename the distributions for your convenience.
Specifying this unit function in Integral Target window (Fig. 1), you will get averaged values in the specified spectral range over the specified number of spectral points.
Final expression for the term corresponding to the integral target in the merit function looks as follows:
\[ MF_{int}^2=\frac 1M\sum\limits_{i=1}M\left[\frac{F^{(i)}-\hat{F}^{(i)}}{\Delta F^{(i)}}\right]^2, \]
where \(M\) is the number of different integrals in the resulting target (Fig. 1), \(F^{(i)}\) are target values, \(\Delta F^{(i)}\) are corresponding tolerances (Fig. 1).
If
\[ F=\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) C(\lambda)d\lambda,\]
where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(C(\lambda)\) is a spectral characteristic of the coating.
If your integral characteristic should be normalized with respect to the spectral distribution of integral weights, then Normalized check box should be checked. In this case:
\[ F=\frac{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda)C(\lambda)d\lambda}{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) d\lambda},\]
where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(C(\lambda)\) is a spectral characteristic of a coating.
If your integral characteristic should take spectral distribution of source/detector into account, then
\[ F=\int\limits_{\lambda_d}^{\lambda_u} W(\lambda)D(\lambda) S(\lambda) C(\lambda)d\lambda,\]
where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(C(\lambda)\) is a spectral characteristic of a coating, \(D(\lambda)\) and \(S(\lambda)\) are spectral distributions of the detector and light source, respectively.
If your integral characteristic should take spectral distribution of source/detector into account and If your integral characteristic should
\[ F=\frac{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda)D(\lambda) S(\lambda) C(\lambda)d\lambda}{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) S(\lambda) D(\lambda)d\lambda},\]
where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(S(\lambda)\) is a spectral characteristic of a coating, \(D(\lambda)\) and \(S(\lambda)\) are spectral distributions of the detector and light source, respectively. |
We prove that square-tiled surfaces having fixed combinatorics of horizontalcylinder decomposition and tiled with smaller and smaller squares becomeasymptotically equidistributed in any ambient linear $GL(\mathbb R)$-invariantsuborbifold defined over $\mathbb Q$ in the moduli space of Abeliandifferentials. Moreover, we prove that the combinatorics of the horizontal andof the vertical decompositions are asymptotically uncorrelated. As aconsequence, we prove the existence of an asymptotic distribution for thecombinatorics of a "random" interval exchange transformation with integerlengths. We compute explicitly the absolute contribution of square-tiled surfaceshaving a single horizontal cylinder to the Masur-Veech volume of any ambientstratum of Abelian differentials. The resulting count is particularly simpleand efficient in the large genus asymptotics. We conjecture that thecorresponding relative contribution is asymptotically of the order $1/d$, where$d$ is the dimension of the stratum, and prove that this conjecture isequivalent to the long-standing conjecture on the large genus asymptotics ofthe Masur-Veech volumes. We prove, in particular, that the recent results ofChen, M\"oller and Zagier imply that the conjecture holds for the principalstratum of Abelian differentials as the genus tends to infinity. Our result on random interval exchanges with integer lengths allows to makeempirical computation of the probability to get a $1$-cylinder pillowcase covertaking a "random" one in a given stratum. We use this technique to derive theapproximate values of the Masur-Veech volumes of strata of quadraticdifferentials of all small dimensions.
A meander is a topological configuration of a line and a simple closed curvein the plane (or a pair of simple closed curves on the 2-sphere) intersectingtransversally. Meanders can be traced back to H. Poincar\'e and naturallyappear in various areas of mathematics, theoretical physics and computationalbiology (in particular, they provide a model of polymer folding). Enumerationof meanders is an important open problem. The number of meanders with 2Ncrossings grows exponentially when N grows, but the longstanding problem on theprecise asymptotics is still out of reach. We show that the situation becomesmore tractable if one additionally fixes the topological type (or the totalnumber of minimal arcs) of a meander. Then we are able to derive simpleasymptotic formulas for the numbers of meanders as N tends to infinity. We alsocompute the asymptotic probability of getting a simple closed curve on a sphereby identifying the endpoints of two arc systems (one on each of the twohemispheres) along the common equator. The new tools we bring to bear are basedon interpretation of meanders as square-tiled surfaces with one horizontal andone vertical cylinders. The proofs combine recent results on Masur-Veechvolumes of the moduli spaces of meromorphic quadratic differentials in genuszero with our new observation that horizontal and vertical separatrix diagramsof integer quadratic differentials are asymptotically uncorrelated. Theadditional combinatorial constraints we impose in this article yield explicitpolynomial asymptotics.
We study the rational Picard group of the projectivized moduli space ofholomorphic n-differentials on complex genus g stable curves. We define (n - 1)natural classes in this Picard group that we call Prym-Tyurin classes. Weexpress these classes as linear combinations of boundary divisors and thedivisor of n-differentials with a double zero. We give two different proofs ofthis result, using two alternative approaches: an analytic approach thatinvolves the Bergman tau function and its vanishing divisor and analgebro-geometric approach that involves cohomological computations on theuniversal curve.
We derive a local index theorem in Quillen's form for families ofCauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces)that are quotients of the hyperbolic plane by the action of cofinite finitelygenerated Fuchsian groups. Each conical point (or a conjugacy class ofprimitive elliptic elements in the Fuchsian group) gives rise to an extra termin the local index theorem that is proportional to the symplectic form of a newK\"{a}hler metric on the moduli space of Riemann orbisurfaces. We find a simpleformula for a local K\"{a}hler potential of the elliptic metric and show thatwhen the order of elliptic element becomes large, the elliptic metric convergesto the cuspidal one corresponding to a puncture on the orbisurface (or aconjugacy class of primitive parabolic elements). We also give a simple exampleof a relation between the elliptic metric and special values of Selberg's zetafunction.
We compute the number of coverings of ${\mathbb{C}}P^1\setminus\{0, 1,\infty\}$ with a given monodromy type over $\infty$ and given numbers ofpreimages of 0 and 1. We show that the generating function for these numbersenjoys several remarkable integrability properties: it obeys the Virasoroconstraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchyand satisfies a topological recursion in the sense of Eynard-Orantin.
Branched covers of the complex projective line ramified over $0,1$ and$\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree areeffectively enumerated. More precisely, branched covers of a given ramificationprofile over $\infty$ and given numbers of preimages of $0$ and $1$ areconsidered. The generating function for the numbers of such covers is shown tosatisfy a PDE that determines it uniquely modulo a simple initial condition.Moreover, this generating function satisfies an infinite system of PDE's calledthe KP (Kadomtsev-Petviashvili) hierarchy. A specification of this generatingfunction for certain values of parameters generates the numbers of {\emdessins} of given genus and degree, thus providing a fast algorithm forcomputing these numbers.
We explicitly compute the diverging factor in the large genus asymptotics ofthe Weil-Petersson volumes of the moduli spaces of $n$-pointed complexalgebraic curves. Modulo a universal multiplicative constant we prove theexistence of a complete asymptotic expansion of the Weil-Petersson volumes inthe inverse powers of the genus with coefficients that are polynomials in $n$.This is done by analyzing various recursions for the more general intersectionnumbers of tautological classes, whose large genus asymptotic behavior is alsoextensively studied.
The Hultman numbers enumerate permutations whose cycle graph has a givennumber of alternating cycles (they are relevant to the Bafna-Pevzner approachto genome comparison and genome rearrangements). We give two newinterpretations of the Hultman numbers: in terms of polygon gluings and asintegrals over the space of complex matrices, and derive some properties oftheir generating functions.
The tau function on the moduli space of generic holomorphic 1-differentialson complex algebraic curves is interpreted as a section of a line bundle on theprojectivized Hodge bundle over the moduli space of stable curves. Theasymptotics of the tau function near the boundary of the moduli space of1-differentials is computed, and an explicit expression for the pullback of theHodge class on the projectivized Hodge bundle in terms of the tautologicalclass and the classes of boundary divisors is derived. This expression is usedto clarify the geometric meaning of the Kontsevich-Zorich formula for the sumof the Lyapunov exponents associated with the Teichm\"uller flow on the Hodgebundle.
A relatively fast algorithm for evaluating Weil-Petersson volumes of modulispaces of complex algebraic curves is proposed. On the basis of numerical data,a conjectural large genus asymptotics of the Weil-Petersson volumes iscomputed. Asymptotic formulas for the intersection numbers involving$\psi$-classes are conjectured as well. The precision of the formulas is highenough to believe that they are exact.
We propose a universal approach to a range of enumeration problems in graphs.The key point is in contracting suitably chosen symmetric tensors placed at thevertices of a graph along the edges. In particular, this leads to an algorithmthat counts the number of d-regular subgraphs of an arbitrary graph includingthe number of d-factors (previously we considered the case d=2 with a specialemphasis on the enumeration of Hamiltonian cycles; cf. math.CO/0403339). Webriefly discuss the problem of the computational complexity of this algorithm.
Following Penrose, we introduce a family of graph functions defined in termsof contractions of certain products of symmetric tensors along the edges of agraph. Special cases of these functions enumerate edge colorings and cycles ofarbitrary length in graphs (in particular, Hamiltonian cycles).
We show that the real-valued function $S_\alpha$ on the moduli space$\mathcal{M}_{0,n}$ of pointed rational curves, defined as the critical valueof the Liouville action functional on a hyperbolic 2-sphere with $n\geq 3$conical singularities of arbitrary orders $\alpha=\{\alpha_1,...,\alpha_n\}$,generates accessory parameters of the associated Fuchsian differential equationas their common antiderivative. We introduce a family of Kaehler metrics on$\mathcal{M}_{0,n}$ parameterized by the set of orders $\alpha$, explictlyrelate accessory parameters to these metrics, and prove that the functions$S_\alpha$ are their Kaehler potentials.
A formula for the generating function of the Weil-Petersson volumes of modulispaces of pointed curves that is identical to the genus expansion of the freeenergy in two dimensional gravity is obtained. The contribution of arbitrarygenus is expressed in terms of the Bessel function $J_0$. |
Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production
1.
School of Mathematical Sciences, Peking University, Beijing, 100871, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities
inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: $ u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v) $, $ \tau_1v_t = \Delta v-a_1v+b_1w $, $ \tau_2w_t = \Delta w-a_2w+b_2u $, under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\Bbb{R}^{n} $ ($ n\geq 1 $), where $ \tau_i,a_i,b_i>0 $ ($ i = 1,2 $) are constants, and the diffusivity $ D $ and the density-dependent sensitivity $ S $ satisfy $ D(s)\geq a_0(s+1)^{-\alpha} $ and $ 0\leq S(s)\leq b_0(s+1)^{\beta} $ for all $ s\geq 0 $ with $ a_0,b_0>0 $ and $ \alpha,\beta\in\Bbb R $. It is proved that if $ \alpha+\beta<3 $ and $ n = 1 $, or $ \alpha+\beta<4/n $ with $ n\geq 2 $, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: $ u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w) $, $ z_t = \Delta z-\rho z+\mu u $, $ w_t = \Delta w-\delta w+\gamma u $, where $ \chi,\mu,\xi,\gamma,\rho,\delta>0 $, and the diffusivity $ D $ fulfills $ D(s)\geq c_0(s+1)^{M-1} $ for any $ s\geq 0 $ with $ c_0>0 $ and $ M\in\Bbb R $. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. $ \chi\mu = \xi\gamma $), the solution is globally bounded if $ M>-1 $ and $ n = 1 $, or $ M>2-4/n $ with $ n\geq 2 $. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion. Mathematics Subject Classification:Primary: 35B35, 35B40, 35K55; Secondary: 92C17. Citation:Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328
References:
[1]
R. A. Adams,
Sobolev Spaces, Academic Press, New York, 1975.
Google Scholar
[2] [3]
J. Bergh and J. Löfström,
[4] [5] [6]
A. Friedman,
[7]
K. Fujie and T. Senba,
Application of an Adams type inequality to a two-chemical substances chemotaxis system,
[8]
H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,
[9]
D. Henry,
[10] [11]
M. Hieber and J. Prüss,
Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations,
[12] [13]
B. Hu and Y. Tao,
To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production,
[14]
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,
[15] [16] [17]
H. Jin and T. Xiang,
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions,
[18] [19]
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva,
[20]
J. Lankeit,
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion,
[21] [22] [23] [24] [25]
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signalling, microglia, and Alzheimer's disease senile plague: Is there a connection?,
[26]
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.Google Scholar
[27] [28] [29]
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations,
[30] [31] [32]
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,
[33]
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production,
[34] [35]
H. Triebel,
[36] [37] [38] [39]
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,
[40] [41]
show all references
References:
[1]
R. A. Adams,
Sobolev Spaces, Academic Press, New York, 1975.
Google Scholar
[2] [3]
J. Bergh and J. Löfström,
[4] [5] [6]
A. Friedman,
[7]
K. Fujie and T. Senba,
Application of an Adams type inequality to a two-chemical substances chemotaxis system,
[8]
H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,
[9]
D. Henry,
[10] [11]
M. Hieber and J. Prüss,
Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations,
[12] [13]
B. Hu and Y. Tao,
To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production,
[14]
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,
[15] [16] [17]
H. Jin and T. Xiang,
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions,
[18] [19]
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva,
[20]
J. Lankeit,
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion,
[21] [22] [23] [24] [25]
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signalling, microglia, and Alzheimer's disease senile plague: Is there a connection?,
[26]
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.Google Scholar
[27] [28] [29]
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations,
[30] [31] [32]
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,
[33]
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production,
[34] [35]
H. Triebel,
[36] [37] [38] [39]
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,
[40] [41]
[1]
Philippe Laurençot.
Global bounded and unbounded solutions to a chemotaxis system with indirect signal production.
[2]
Wei Wang, Yan Li, Hao Yu.
Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity.
[3]
Youshan Tao, Michael Winkler.
A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production.
[4]
Liangchen Wang, Yuhuan Li, Chunlai Mu.
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source.
[5]
Miaoqing Tian, Sining Zheng.
Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species.
[6] [7]
Sachiko Ishida, Tomomi Yokota.
Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity.
[8]
Yilong Wang, Zhaoyin Xiang.
Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system.
[9]
Ling Liu, Jiashan Zheng.
Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source.
[10]
Tomasz Cieślak, Kentarou Fujie.
Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity.
[11]
Masaaki Mizukami.
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity.
[12]
Hao Yu, Wei Wang, Sining Zheng.
Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity.
[13] [14] [15]
Johannes Lankeit, Yulan Wang.
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption.
[16]
Hua Zhong, Chunlai Mu, Ke Lin.
Global weak solution and boundedness in a three-dimensional competing chemotaxis.
[17]
Chunhua Jin.
Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion.
[18]
Jiashan Zheng.
Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion.
[19]
Hai-Yang Jin, Tian Xiang.
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions.
[20]
Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota.
Boundedness in a quasilinear parabolic-parabolic
Keller-Segel system with the sensitivity $v^{-1}S(u)$.
2018 Impact Factor: 1.008
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Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China
$ \left\{ \begin{array}{ll} -\triangle u+V(x)u+\phi u = f(x,u), \ \ \ \ x\in { \mathbb{R} }^{2},\\ \triangle \phi = u^2, \ \ \ \ x\in { \mathbb{R} }^{2}, \end{array} \right. $
$ V(x) $
$ f(x, u) $
$ x $
$ f(x, u) $
$ u $
$ L^s( \mathbb{R} ^N) $ Keywords:Schrödinger-Poisson system, logarithmic convolution potential, ground state solution, axially symmetric. Mathematics Subject Classification:Primary: 35J20; Secondary: 35Q55. Citation:Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
References:
[1] [2] [3] [4] [5] [6]
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in hartree and thomas-fermi type theories,
[7] [8]
J. Chen, S. T. Chen and X. H. Tang, Ground state solutions for the planar asymptotically periodic Schrödinger-Poisson system,
[9]
J. Chen, X. H. Tang and S. T. Chen, Existence of ground states for fractional Kirchhoff equations with general potentials via Nehari-Pohozaev manifold,
[10]
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb R^3$,
[11] [12] [13]
S. T. Chen and X. H. Tang, Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities,
[14] [15] [16]
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,
[17] [18] [19] [20]
P. Markowich, C. Ringhofer and C. Schmeiser,
The concentration-compactness principle in the calculus of variations. the locally compact case. Ⅰ & Ⅱ,
[21] [22] [23]
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth,
[24] [25]
J. J. Sun and S. W. Ma,
Ground state solutions for some Schrödinger-Poisson systems with periodic potentials,
[26] [27]
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains,
[28]
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential,
[29]
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials,
[30]
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions,
[31]
Z. P. Wang and H. S. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb R^3$,
[32] [33]
show all references
References:
[1] [2] [3] [4] [5] [6]
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in hartree and thomas-fermi type theories,
[7] [8]
J. Chen, S. T. Chen and X. H. Tang, Ground state solutions for the planar asymptotically periodic Schrödinger-Poisson system,
[9]
J. Chen, X. H. Tang and S. T. Chen, Existence of ground states for fractional Kirchhoff equations with general potentials via Nehari-Pohozaev manifold,
[10]
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb R^3$,
[11] [12] [13]
S. T. Chen and X. H. Tang, Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities,
[14] [15] [16]
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,
[17] [18] [19] [20]
P. Markowich, C. Ringhofer and C. Schmeiser,
The concentration-compactness principle in the calculus of variations. the locally compact case. Ⅰ & Ⅱ,
[21] [22] [23]
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth,
[24] [25]
J. J. Sun and S. W. Ma,
Ground state solutions for some Schrödinger-Poisson systems with periodic potentials,
[26] [27]
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains,
[28]
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential,
[29]
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials,
[30]
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions,
[31]
Z. P. Wang and H. S. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb R^3$,
[32] [33]
[1]
Sitong Chen, Junping Shi, Xianhua Tang.
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity.
[2]
Xu Zhang, Shiwang Ma, Qilin Xie.
Bound state solutions of Schrödinger-Poisson system with critical exponent.
[3]
Xianhua Tang, Sitong Chen.
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials.
[4]
Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang.
Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent.
[5]
Margherita Nolasco.
Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential.
[6]
Lun Guo, Wentao Huang, Huifang Jia.
Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $.
[7]
Zhengping Wang, Huan-Song Zhou.
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$.
[8] [9]
Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang.
Standing waves for Schrödinger-Poisson system with general nonlinearity.
[10] [11]
Yi He, Lu Lu, Wei Shuai.
Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents.
[12]
Zhanping Liang, Yuanmin Song, Fuyi Li.
Positive ground state solutions of a quadratically coupled schrödinger system.
[13]
Mingzheng Sun, Jiabao Su, Leiga Zhao.
Infinitely many solutions for a Schrödinger-Poisson system with
concave and convex nonlinearities.
[14]
Claudianor O. Alves, Minbo Yang.
Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$.
[15] [16] [17]
Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng.
Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$.
[18]
Xiaoyan Lin, Yubo He, Xianhua Tang.
Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential.
[19] [20]
2018 Impact Factor: 1.008
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Spring 2018, Math 171 Week 6 Exit Distributions
A person is terminally ill. On a day when the person is awake, there is an 0.2 chance they will die overnight, and they are equally likely to be awake or unconscious the next day. On a day when the person is unconscious, there is an 0.2 chance they will be awake the next day, and they are equally likely to stay unconscious or die.
Let \(X_n\) be the person’s state on day \(n\) (awake, unconscious, or dead). Show that \((X_n)_{n\ge 0}\) is a Markov chain. Find its transition matrix.
(Answer) \[\begin{matrix}& \mathbf A & \mathbf U & \mathbf D \cr \mathbf A & 0.4 & 0.4 & 0.2 \cr \mathbf U & 0.2 & 0.4 & 0.4 \cr \mathbf D & 0 & 0 & 1\end{matrix}\]
Compute the probability that the person spends at least one day awake before dying given that they are initially unconscious.
(Answer) \(\frac{0.2}{1-0.4} = \frac 1 3\)
Compute the expected number of days the person will spend awake before dying given that they are initially unconscious.
(Answer) \(\frac 5 7\)
In the Markov chain corresponding to the following transition matrix \[\begin{matrix} & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 \cr \mathbf 1 & 0.1 & 0.5 & 0.2 & 0.2 \cr \mathbf 2 & 0.2 & 0.4 & 0.4 & 0 \cr \mathbf 3 & 0 & 0.5 & 0.3 & 0.2 \cr \mathbf 4 & 0.5 & 0.5 & 0 & 0 \end{matrix}\] compute the probability that the chain reaches state 1 before it reaches state 4 for each starting state.
(Answer) Define \(h(x)=P_x(T_1 < T_4)\). Then \(h(1)=1\), \(h(4)=0\), and \[\begin{bmatrix}h(2) \cr h(3)\end{bmatrix} = \left(I - \begin{bmatrix}0.4 & 0.4 \cr 0.5 & 0.3\end{bmatrix}\right)^{-1}\begin{bmatrix}0.2 \cr 0\end{bmatrix}\]
(Discussed) In the Markov chain corresponding to the following transition matrix \[\begin{matrix} & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 \cr \mathbf 1 & 0.1 & 0.5 & 0.2 & 0.2 & 0 \cr \mathbf 2 & 0.4 & 0.2 & 0.3 & 0 & 0.1\cr \mathbf 3 & 0 & 0.5 & 0.3 & 0.2 & 0\cr \mathbf 4 & 0.2 & 0 & 0.5 & 0.1 & 0.2 \cr \mathbf 5 & 0.1 & 0.1 & 0.2 & 0.1 & 0.5 \end{matrix}\] Compute the probability that the chain reaches states 1 or 2 before it reaches state 5 for each starting state.
Exit Times
In the Markov chain from 1.2 compute the expected time taken to reach state 4 from each of the other states.
(Answer) \(g(x)=\mathbb E_x[T_4]\). Then \(g(4)=0\) and \[\begin{bmatrix}g(1) \cr g(2) \cr g(3)\end{bmatrix} = \left(I - \begin{bmatrix}0.1 & 0.5 & 0.2 \cr 0.2 & 0.4 & 0.4 \cr 0 & 0.5 & 0.3 \end{bmatrix}\right)^{-1} \begin{bmatrix}1 \cr 1 \cr 1\end{bmatrix}\]
(Discussed) In the Markov chain from 1.3 compute the expected time taken to reach either of states 2 or 5 from each of the other states. That is, \(\mathbb E_x[T]\) where \(T = \min \{n \ge 0 \mid X_n \in \{2, 5\}\}\) |
To be precise, that code generates draws from a
shifted lognormal distribution.
Define $R_t = \frac{P_t + D_t}{P_{t-1}} - 1$ as the
return from $t-1$ to $t$.
Define $r_t = \log \left( 1 + R_t \right)$ as the
log return from $t-1$ to $t$. (Note if $D_t=0$ then $r_t = \log P_t - \log P_{t-1}$.)
Your code above generates
returns where the corresponding log return follows the normal distribution.
As a practical matter, the linear approximation of $f(x) = \log(1+x)$ around $x=0$ is given by $x$, hence for $r_t$ near zero, we have $r_t \approx R_t$. If a return is .02, the log return is .0198. (Of course, this breaks down the farther one is from zero.) |
Orthogonal group An orthogonal group is a group of all linear transformations of an $n$-dimensional vector space $V$ over a field $k$ which preserve a fixed non-singular quadratic form $Q$ on $V$ (i.e. linear transformations $\def\phi{\varphi}\phi$ such that $Q(\phi(v))=Q(v)$ for all $v\in V$). An orthogonal group is a classical group. The elements of an orthogonal group are called orthogonal transformations of $V$ (with respect to $Q$), or also automorphisms of the form $Q$. Furthermore, let ${\rm char\;} k\ne 2$ (for orthogonal groups over fields with characteristic 2 see [Di], [2]) and let $f$ be the non-singular symmetric bilinear form on $V$ related to $Q$ by the formula
$$f(u,v)=\frac{1}{2}(Q(u+v) - Q(u) - Q(v)).$$ The orthogonal group then consists of those linear transformations of $V$ that preserve $f$, and is denoted by $\def\O{ {\rm O} }\O_n(k,f)$, or (when one is talking of a specific field $k$ and a specific form $f$) simply by $\O_n$. If $B$ is the matrix of $f$ with respect to some basis of $V$, then the orthogonal group can be identified with the group of all $(n\times n)$-matrices $A$ with coefficients in $k$ such that $A^TBA = B$ (${}^T$ is transposition).
The description of the algebraic structure of an orthogonal group is a classical problem. The determinant of any element from $\O_n$ is equal to 1 or $-1$. Elements with determinant 1 are called rotations; they form a normal subgroup $\O_n^+(k,f)$ (or simply $\O_n^+$) of index 2 in the orthogonal group, called the rotation group. Elements from $\O_n\setminus \O_n^+$ are called inversions. Every rotation (inversion) is the product of an even (odd) number of reflections from $\O_n$.
Let $Z_n$ be the group of all homotheties $\def\a{\alpha}\phi_\a : v\mapsto \a v$, $\a\in k$, $\a\ne 0$, of the space $V$. Then $\O_n\cap Z_n$ is the centre of $\O_n$; it consists of two elements: $\phi_1$ and $\phi_{-1}$. If $n$ is odd, then $\O_n$ is the direct product of its centre and $\O_n^+$. If $n\ge 3$, the centre of $\O_n^+$ is trivial if $n$ is odd, and coincides with the centre of $\O_n$ if $n$ is even. If $n=2$, the group $\O_n^+$ is commutative and is isomorphic either to the multiplicative group $k^*$ of $k$ (when the Witt index $\nu$ of $f$ is equal to 1), or to the group of elements with norm 1 in $k(\sqrt-\Delta)$, where $\Delta$ is the discriminant of $f$ (when $\nu=0$). The commutator subgroup of $\O_n(k,f)$ is denoted by $\def\Om{\Omega}\Om_n(k,f)$, or simply by $\Om_n$; it is generated by the squares of the elements from $\O_n$. When $n\ge 3$, the commutator subgroup of $\O_n^+$ coincides with $\Om_n$. The centre of $\Om_n$ is $\Om_n\cap Z_n$.
Other classical groups related to orthogonal groups include the canonical images of $\O_n^+$ and $\Om_n$ in the projective group; they are denoted by ${\rm P}\O_n^+(k,f)$ and ${\rm P}\Om_n(k,f)$ (or simply by ${\rm P}\O_n^+$ and ${\rm P}\Om_n$) and are isomorphic to $\O_n^+/(\O_n^+\cap Z_n)$ and $\Om_n/(\Om_n\cap Z_n)$, respectively.
The basic classical facts about the algebraic structure describe the successive factors of the following series of normal subgroups of an orthogonal group:
$$\O_n\supset \O_n^+\supset \Om_n\supset \Om_n\cap Z_n \supset \{e\}.$$ The group $\O_n/\O_n^+$ has order 2. Every element in $\O_n/\Om_n$ has order 2, thus this group is defined completely by its cardinal number, and this number can be either infinite or finite of the form $2^\a$ where $\a$ is an integer. The description of the remaining factors depends essentially on the Witt index $\nu$ of the form $f$.
First, let $\nu\ge 1$. Then $\O_n^+/\Om_n \simeq k^*/{k^*}^2$ when $n>2$. This isomorphism is defined by the spinor norm, which defines an epimorphism from $\O_n^+$ on $k^*/{k^*}^2$ with kernel $\Om_n$. The group $\Om_n\cap Z_n$ is non-trivial (and consists of the transformations $\phi_1$ and $\phi_{-1}$) if and only if $n$ is even and $\Delta\in {k^*}^2$. If $n\ge 5$, then the group ${\rm P}\Om_n = \Om_n/(\Om_n\cap Z_n)$ is simple. The cases where $n=3,4$ are studied separately. Namely, ${\rm P}\Om_3 = \Om_3$ is isomorphic to $\def\PSL{ {\rm PSL}}\PSL_2(k)$ (see Special linear group) and is also simple if $k$ has at least 4 elements (the group $\O_3^+$ is isomorphic to the projective group $\def\PGL{ {\rm PGL}}\PGL_2(k)$). When $\nu=1$, the group ${\rm P}\Om_4 = \Om_4$ is isomorphic to the group $\PSL_2(k(\sqrt{\Delta}))$ and is simple (in this case $\Delta\notin k^2$), while when $\nu=2$, the group ${\rm P}\Om_4$ is isomorphic to $\PSL_2(k)\times \PSL_2(k)$ and is not simple. In the particular case when $k = \R$ and $Q$ is a form of signature $(3,1)$, the group ${\rm P}\Om_4 = \Om_4\simeq \PSL_2(\C)$ is called the Lorentz group.
When $\nu = 0$ (i.e. $Q$ is an anisotropic form), these results are not generally true. For example, if $k=\R$ and $Q$ is a positive-definite form, then $\Om_n = O_n^+$, although $\R^*/{\R^*}^2$ consists of two elements; when $k=\Q$, $n=4$, one can have $\Delta\in k^2$, but $\phi_{-1}\notin \Om_4$. When $\nu=0$, the structures of an orthogonal group and its related groups essentially depend on $k$. For example, if $k=\R$, then ${\rm P}\O_n^+$, $n\ge 3$, $n\ne 4$, $\nu=0$, is simple (and ${\rm P}\O_4^+$ is isomorphic to the direct product $\O_3^+ \times \O_3^+$ of two simple groups); if $k$ is the field of $p$-adic numbers and $\nu=0$, there exists in $\O_3$ (and $\O_4$) an infinite normal series with Abelian quotients. Important special cases are when $k$ is a locally compact field or an algebraic number field. If $k$ is the field of $p$-adic numbers, then $n=0$ is impossible when $\nu\ge 5$. If $k$ is an algebraic number field, then there is no such restriction and one of the basic results is that ${\rm P}\Om_n$, when $\nu=0$ and $n\ge 5$, is simple. In this case, the study of orthogonal groups is closely connected with the theory of equivalence of quadratic forms, where one needs the forms obtained from $Q$ by extension of coefficients to the local fields defined by valuations of $k$ (the Hasse principle).
If $k$ is the finite field $\F_q$ of $q$ elements, then an orthogonal group is finite. The order of $\O_n^+$ for $n$ odd is equal to
$$(q^{n-1}-1)q^{n-2}(q^{n-3}-1)q^{n-4}\cdots (q^2-1)q,$$ while when $n=2m$ it is equal to
$$\def\e{\epsilon}(q^{2m-1}-\e q^{m-1})(q^{2m-2}-1)q^{2m-3}\cdots(q^2-1)q,$$ where $\e=1$ if $(-1)^m\Delta\in \F_q^2$ and $\e=-1$ otherwise. These formulas and general facts about orthogonal groups when $\nu\ge 1$ also allow one to calculate the orders of $\Om_n$ and ${\rm P}\Om_n$, since $\nu\ge 1$ when $n\ge 3$, while the order of $k^*/{k^*}^2$ is equal to 2. The group ${\rm P}\Om_n$, $n\ge 5$, is one of the classical simple finite groups (see also Chevalley group).
One of the basic results on automorphisms of orthogonal groups is the following: If $n\ge 3$, then every automorphism $\phi$ of $\O_n$ has the form $\phi(u)=\chi(u)gug^{-1}$, $u\in \O_n$, where $\chi$ is a fixed homomorphism of $\O_n$ into its centre and $g$ is a fixed bijective semi-linear mapping of $V$ onto itself satisfying $Q(g(v))=r_gQ^\sigma(v)$ for all $v\in V$, where $r_g\in k^*$ while $\sigma$ is an automorphism of $k$. If $\nu\ge 1$ and $n\ge 6$, then every automorphism of $\O_n^+$ is induced by an automorphism of $\O_n$ (see [Di], ).
Like the other classical groups, an orthogonal group has a geometric characterization (under certain hypotheses). Indeed, let $ Q$ be an anisotropic form such that $Q(v)\in k^2$ for all $v\in V$. In this case $k$ is a Pythagorean orderable field. For a fixed order of the field $k$, any sequence $((H_s)_{1\le s\le n}$ constructed from a linearly independent basis $((h_s)_{1\le s\le n}$, where $H_s$ is the set of all linear combinations of the form $\def\l{\lambda}\sum_{j=1}^sl_jh_j$, $\l_s\ge 0$, is called an $n$-dimensional chain of incident half-spaces in $V$. The group $\O_n$ has the property of free mobility, i.e. for any two $n$-dimensional chains of half-spaces there exists a unique transformation from $\O_n$ which transforms the first chain into the second. This property characterizes an orthogonal group: If $L$ is any ordered skew-field and $G$ is a subgroup in ${\rm GL_n(L)}$, $n\ge 3$, having the property of free mobility, then $L$ is a Pythagorean field, while $G=\O_n(L,f)$, where $f$ is an anisotropic symmetric bilinear form such that $f(v,v)\in L_1^2$ for any vector $v$.
Let $\def\bk{ {\bar k}}\bk$ be a fixed algebraic closure of the field $k$. The form $f$ extends naturally to a non-singular symmetric bilinear form ${\bar f}$ on $V\otimes_k \bk$, and the orthogonal group $\O_n(\bk,f)$ is a linear algebraic group defined over $k$ with $\O_n(k,f)$ as group of $k$-points. The linear algebraic groups thus defined (for various $f$) are isomorphic over $\bk$ (but in general not over $k$); the corresponding linear algebraic group over $\bk$ is called the orthogonal algebraic group $\O_n(\bk)$. Its subgroup $\O_n^+(\bk,{\bar f})$ is also a linear algebraic group over $\bk$, and is called a properly orthogonal, or special orthogonal algebraic group (notation: $\def\SO{ {\rm SO}}\SO_n(\bk)$); it is the connected component of the identity of $\O_n(\bk)$. The group $\SO_n(\bk)$ is an almost-simple algebraic group (i.e. does not contain infinite algebraic normal subgroups) of type $B_s$ when $n=2s+1$, $s\ge 1$, and of type $D_s$ when $n=2s$, $s\ge 3$. The universal covering group of $\SO_n$ is a spinor group.
If $ k=\R,\C$ or a $p$-adic field, then $\O_n(k,f)$ has a canonical structure of a real, complex or $p$-adic analytic group. The Lie group $\O_n(\R,f)$ is defined up to isomorphism by the signature of the form $f$; if this signature is $(p,q)$, $p+q=n$, then $\O_n(\R,f)$ is denoted by $\O_(p,q)$ and is called a pseudo-orthogonal group. It can be identified with the Lie group of all real $(n\times n)$-matrices $A$ which satisfy
$$A^TI_{p,q}A = I_{p,q}\qquad\textrm{ where }I_{p,q} = \begin{pmatrix}1_p & 0 \\ 0 & -1_q\end{pmatrix}$$ ($1_s$ denotes the unit $(s\times s)$-matrix). The Lie algebra of this group is the Lie algebra of all real $(n\times n)$-matrices $X$ that satisfy the condition $X^TI_{p,q} = -I_{p.q}X$. In the particular case $q=0$, the group $\O(p,q)$ is denoted by $\O(n)$ and is called a real orthogonal group; its Lie algebra consists of all skew-symmetric real $(n\times n)$-matrices. The Lie group $\O(p,q)$ has four connected components when $q\ne 0$, and two connected components when $q=0$. The connected component of the identity is its commutator subgroup, which, when $q=0$, coincides with the subgroup $\def\SO{ {\rm SO}}\SO(n)$ in $\O(n)$ consisting of all transformations with determinant 1. The group $\O(p,q)$ is compact only when $q=0$. The topological invariants of $\SO(n)$ have been studied. One of the classical results is the calculation of the Betti numbers of the manifold $\SO(n)$: Its Poincaré polynomial has the form
$$\prod_{s=1}^m(1+t^{4s-1})$$ when $n=2m+1$, and the form
$$(1+t^{2m-1})\prod_{s=1}^{m-1}(1+t^{4s-1})$$ when $n=2m$. The fundamental group of the manifold $\SO(n)$ is $\Z_2$. The calculation of the higher homotopy groups $\pi_l(\SO(n))$ is directly related to the classification of locally trivial principal $\SO(n)$-fibrations over spheres. An important part in topological $K$-theory is played by the periodicity theorem, according to which, when $N\gg n$, there are the isomorphisms
$$\pi_{n+8}(\O(N)) \simeq \pi_{n}(\O(N));$$ further,
$$\pi_n(\O(N)) \simeq \Z_2$$ if $n=0,1$;
$$\pi_n(\O(N)) \simeq \Z$$ if $n=3,7$; and
$$\pi_n(\O(N)) = 0$$ if $n=2,4,5,6$. The study of the topology of the group $\O(p,q)$ reduces in essence to the previous case, since the connected component of the identity of $\O(p,q)$ is diffeomorphic to the product $\SO(p)\times \SO(q)$ on a Euclidean space.
Comments
A Pythagorean field is a field in which the sum of two squares is again a square.
References
[Ar] E. Artin, "Geometric algebra", Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101 [Bo] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", 2, Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210 [Di] J.A. Dieudonné, "La géométrie des groups classiques", Springer (1955) Zbl 0221.20056 [Di2] J. Dieudonné, "On the automorphisms of the classical groups", Mem. Amer. Math. Soc., 2, Amer. Math. Soc. (1951) MR0045125 Zbl 0042.25603 [Hu] D. Husemoller, "Fibre bundles", McGraw-Hill (1966) MR0229247 Zbl 0144.44804 [OM] O.T. O'Meara, "Introduction to quadratic forms", Springer (1973) Zbl 0259.10018 [We] H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502 [Zh] D.P. Zhelobenko, "Compact Lie groups and their representations", Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013 How to Cite This Entry:
Orthogonal group.
Encyclopedia of Mathematics.URL: http://www.encyclopediaofmath.org/index.php?title=Orthogonal_group&oldid=34320 |
Difference between revisions of "Inaccessible"
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Therefore $2$-inaccessibility is weaker than $3$-inaccessibility, which is weaker than $4$-inaccessibility... all of which are weaker than $\omega$-inaccessibility, which is weaker than $\omega+1$-inaccessibility, which is weaker than $\omega+2$-inaccessibility...... all of which are weaker than hyperinaccessibility, etc.
Therefore $2$-inaccessibility is weaker than $3$-inaccessibility, which is weaker than $4$-inaccessibility... all of which are weaker than $\omega$-inaccessibility, which is weaker than $\omega+1$-inaccessibility, which is weaker than $\omega+2$-inaccessibility...... all of which are weaker than hyperinaccessibility, etc.
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==Hyper-inaccessible==
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==Hyper-inaccessible ==
A cardinal $\kappa$ is ''hyperinaccessible'' if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-hyperinaccessible cardinals. Continuing, $\kappa$ is ''hyperhyperinaccessible'' if $\kappa$ is $\kappa$-hyperinaccessible.
A cardinal $\kappa$ is ''hyperinaccessible'' if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-hyperinaccessible cardinals. Continuing, $\kappa$ is ''hyperhyperinaccessible'' if $\kappa$ is $\kappa$-hyperinaccessible.
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More generally, $\kappa$ is ''hyper${}^\alpha$-inaccessible'' if it is hyperinaccessible and for every $\beta\lt\alpha$ it is $\kappa$-hyper${}^\beta$-inaccessible, where $\kappa$ is ''$\alpha$-hyper${}^\beta$-inaccessible'' if it is hyper${}^\beta$-inaccessible and for every $\gamma<\alpha$, it is a limit of $\gamma$-hyper${}^\beta$-inaccessible cardinals.
More generally, $\kappa$ is ''hyper${}^\alpha$-inaccessible'' if it is hyperinaccessible and for every $\beta\lt\alpha$ it is $\kappa$-hyper${}^\beta$-inaccessible, where $\kappa$ is ''$\alpha$-hyper${}^\beta$-inaccessible'' if it is hyper${}^\beta$-inaccessible and for every $\gamma<\alpha$, it is a limit of $\gamma$-hyper${}^\beta$-inaccessible cardinals.
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Every [[Mahlo]] cardinal $\kappa$ is
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Every [[Mahlo]] cardinal $\kappa$ is $\$-inaccessible .
Revision as of 14:05, 29 April 2019 Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy, although weaker notions such as the worldly cardinals can still be viewed as large cardinals.
A cardinal $\kappa$ being inaccessible implies the following:
$V_\kappa$ is a model of ZFC and so inaccessible cardinals are worldly. The worldly cardinals are unbounded in $\kappa$, so $V_\kappa$ satisfies the existence of a proper class of worldly cardinals. $\kappa$ is an aleph fixed point and a beth fixed point, and consequently $V_\kappa=H_\kappa$. (Solovay)there is an inner model of a forcing extension satisfying ZF+DC in which every set of reals is Lebesgue measurable; in fact, this is equiconsistent to the existence of an inaccessible cardinal. For any $A\subseteq V_\kappa$, the set of all $\alpha<\kappa$ such that $\langle V_\alpha;\in,A\cap V_\alpha\rangle\prec\langle V_\kappa;\in,A\rangle$ is club in $\kappa$.
An ordinal $\alpha$ being inaccessible is equivalent to the following:
$V_{\alpha+1}$ satisfies $\mathrm{KM}$. $\alpha>\omega$ and $V_\alpha$ is a Grothendiek universe. $\alpha$ is $\Pi_0^1$-Indescribable. $\alpha$ is $\Sigma_1^1$-Indescribable. $\alpha$ is $\Pi_2^0$-Indescribable. $\alpha$ is $0$-Indescribable. $\alpha$ is a nonzero limit ordinal and $\beth_\alpha=R_\alpha$ where $R_\beta$ is the $\beta$-th regular cardinal, i.e. the least regular $\gamma$ such that $\{\kappa\in\gamma:\mathrm{cf}(\kappa)=\kappa\}$ has order-type $\beta$. $\alpha = \beth_{R_\alpha}$. $\alpha = R_{\beth_\alpha}$. $\alpha$ is a weakly inaccessible strong limit cardinal (see weakly inaccessible below). Contents Weakly inaccessible cardinal
A cardinal $\kappa$ is
weakly inaccessible if it is an uncountable regular limit cardinal. Under GCH, this is equivalent to inaccessibility, since under GCH every limit cardinal is a strong limit cardinal. So the difference between weak and strong inaccessibility only arises when GCH fails badly. Every inaccessible cardinal is weakly inaccessible, but forcing arguments show that any inaccessible cardinal can become a non-inaccessible weakly inaccessible cardinal in a forcing extension, such as after adding an enormous number of Cohen reals (this forcing is c.c.c. and hence preserves all cardinals and cofinalities and hence also all regular limit cardinals). Meanwhile, every weakly inaccessible cardinal is fully inaccessible in any inner model of GCH, since it will remain a regular limit cardinal in that model and hence also be a strong limit there. In particular, every weakly inaccessible cardinal is inaccessible in the constructible universe $L$. Consequently, although the two large cardinal notions are not provably equivalent, they are equiconsistent.
There are a few equivalent definitions of weakly inaccessible cardinals. In particular:
Letting $R$ be the transfinite enumeration of regular cardinals, a limit ordinal $\alpha$ is weakly inaccessible if and only if $R_\alpha=\aleph_\alpha$ A nonzero cardinal $\kappa$ is weakly inaccessible if and only if $\kappa$ is regular and there are $\kappa$-many regular cardinals below $\kappa$; that is, $\kappa=R_\kappa$. A regular cardinal $\kappa$ is weakly inaccessible if and only if $\mathrm{REG}$ is unbounded in $\kappa$ (showing the correlation between weakly Mahlo cardinals and weakly inaccessible cardinals, as stationary in $\kappa$ is replaced with unbounded in $\kappa$) Levy collapse
The Levy collapse of an inaccessible cardinal $\kappa$ is the $\lt\kappa$-support product of $\text{Coll}(\omega,\gamma)$ for all $\gamma\lt\kappa$. This forcing collapses all cardinals below $\kappa$ to $\omega$, but since it is $\kappa$-c.c., it preserves $\kappa$ itself, and hence ensures $\kappa=\omega_1$ in the forcing extension.
Inaccessible to reals
A cardinal $\kappa$ is
inaccessible to reals if it is inaccessible in $L[x]$ for every real $x$. For example, after the Levy collapse of an inaccessible cardinal $\kappa$, which forces $\kappa=\omega_1$ in the extension, the cardinal $\kappa$ is of course no longer inaccessible, but it remains inaccessible to reals. Universes
When $\kappa$ is inaccessible, then $V_\kappa$ provides a highly natural transitive model of set theory, a universe in which one can view a large part of classical mathematics as taking place. In what appears to be an instance of convergent evolution, the same universe concept arose in category theory out of the desire to provide a hierarchy of notions of smallness, so that one may form such categories as the category of all small groups, or small rings or small categories, without running into the difficulties of Russell's paradox. Namely, a
Grothendieck universe is a transitive set $W$ that is closed under pairing, power set and unions. That is, (transitivity) If $b\in a\in W$, then $b\in W$. (pairing) If $a,b\in W$, then $\{a,b\}\in W$. (power set) If $a\in W$, then $P(a)\in W$. (union) If $a\in W$, then $\cup a\in W$.
The
Grothendieck universe axiom is the assertion that every set is an element of a Grothendieck universe. This is equivalent to the assertion that the inaccessible cardinals form a proper class. Degrees of inaccessibility
A cardinal $\kappa$ is
$1$-inaccessible if it is inaccessible and a limit of inaccessible cardinals. In other words, $\kappa$ is $1$-inaccessible if $\kappa$ is the $\kappa^{\rm th}$ inaccessible cardinal, that is, if $\kappa$ is a fixed point in the enumeration of all inaccessible cardinals. Equivalently, $\kappa$ is $1$-inaccessible if $V_\kappa$ is a universe and satisfies the universe axiom.
More generally, $\kappa$ is $\alpha$-inaccessible if it is inaccessible and for every $\beta\lt\alpha$ it is a limit of $\beta$-inaccessible cardinals.
$1$-inaccessibility is already consistency-wise stronger than the existence of a proper class of inaccessible cardinals, and $2$-inaccessibility is stronger than the existence of a proper class of $1$-inaccessible cardinals. More specifically, a cardinal $\kappa$ is $\alpha$-inaccessible if and only if for every $\beta<\alpha$: $$V_{\kappa+1}\models\mathrm{KM}+\text{There is a proper class of }\beta\text{-inaccessible cardinals}$$
As a result, if $\kappa$ is $\alpha$-inaccessible then for every $\beta<\alpha$: $$V_\kappa\models\mathrm{ZFC}+\text{There exists a }\beta\text{-inaccessible cardinal}$$
Therefore $2$-inaccessibility is weaker than $3$-inaccessibility, which is weaker than $4$-inaccessibility... all of which are weaker than $\omega$-inaccessibility, which is weaker than $\omega+1$-inaccessibility, which is weaker than $\omega+2$-inaccessibility...... all of which are weaker than hyperinaccessibility, etc.
Hyper-inaccessible and more
A cardinal $\kappa$ is
hyperinaccessible if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-hyperinaccessible cardinals. Continuing, $\kappa$ is hyperhyperinaccessible if $\kappa$ is $\kappa$-hyperinaccessible.
More generally, $\kappa$ is
hyper${}^\alpha$-inaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$ it is $\kappa$-hyper${}^\beta$-inaccessible, where $\kappa$ is $\alpha$-hyper${}^\beta$-inaccessible if it is hyper${}^\beta$-inaccessible and for every $\gamma<\alpha$, it is a limit of $\gamma$-hyper${}^\beta$-inaccessible cardinals.
Meta-ordinal terms are terms like $Ω^α · β + Ω^γ · δ +· · ·+Ω^\epsilon · \zeta + \theta$ where $α, β...$ are ordinals. They are ordered as if $Ω$ were an ordinal greater then all the others. $(Ω · α + β)$-inaccessible denotes $β$-hyper${}^α$-inaccessible, $Ω^2$-inaccessible denotes hyper${}^\kappa$-inaccessible $\kappa$ etc. Every Mahlo cardinal $\kappa$ is $\Omega^α$-inaccessible for all $α<\kappa$. Similar hierarchy exists for Mahlo cardinals below weakly compact. All such properties can be killed softly be forcing to make them any weaker properties from this family.[1] |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphenation near box question, I guess that makes sense, basically can't just rely on lccode anymore. I suppose you don't want the hyphenation code in my last answer by default?
@JosephWright anway if we rip out all the auto-testing (since mac/windows/linux come out the same anyway) but leave in the .cfg possibility, there is no actual loss of functionality if someone is still using a vms tex or whatever
I want to change the tracking (space between the characters) for a sans serif font. I found that I can use the microtype package to change the tracking of the smallcaps font (\textsc{foo}), but I can't figure out how to make \textsc{} a sans serif font.
@DavidCarlisle -- if you write it as "4 May 2016" you don't need a comma (or, in the u.s., want a comma).
@egreg (even if you're not here at the moment) -- tomorrow is international archaeology day: twitter.com/ArchaeologyDay , so there must be someplace near you that you could visit to demonstrate your firsthand knowledge.
@barbarabeeton I prefer May 4, 2016, for some reason (don't know why actually)
@barbarabeeton but I have another question maybe better suited for you please: If a member of a conference scientific committee writes a preface for the special issue, can the signature say John Doe \\ for the scientific committee or is there a better wording?
@barbarabeeton overrightarrow answer will have to wait, need time to debug \ialign :-) (it's not the \smash wat did it) on the other hand if we mention \ialign enough it may interest @egreg enough to debug it for us.
@DavidCarlisle -- okay. are you sure the \smash isn't involved? i thought it might also be the reason that the arrow is too close to the "M". (\smash[t] might have been more appropriate.) i haven't yet had a chance to try it out at "normal" size; after all, \Huge is magnified from a larger base for the alphabet, but always from 10pt for symbols, and that's bound to have an effect, not necessarily positive. (and yes, that is the sort of thing that seems to fascinate @egreg.)
@barbarabeeton yes I edited the arrow macros not to have relbar (ie just omit the extender entirely and just have a single arrowhead but it still overprinted when in the \ialign construct but I'd already spent too long on it at work so stopped, may try to look this weekend (but it's uktug tomorrow)
if the expression is put into an \fbox, it is clear all around. even with the \smash. so something else is going on. put it into a text block, with \newline after the preceding text, and directly following before another text line. i think the intention is to treat the "M" as a large operator (like \sum or \prod, but the submitter wasn't very specific about the intent.)
@egreg -- okay. i'll double check that with plain tex. but that doesn't explain why there's also an overlap of the arrow with the "M", at least in the output i got. personally, i think that that arrow is horrendously too large in that context, which is why i'd like to know what is intended.
@barbarabeeton the overlap below is much smaller, see the righthand box with the arrow in egreg's image, it just extends below and catches the serifs on the M, but th eoverlap above is pretty bad really
@DavidCarlisle -- i think other possible/probable contexts for the \over*arrows have to be looked at also. this example is way outside the contexts i would expect. and any change should work without adverse effect in the "normal" contexts.
@DavidCarlisle -- maybe better take a look at the latin modern math arrowheads ...
@DavidCarlisle I see no real way out. The CM arrows extend above the x-height, but the advertised height is 1ex (actually a bit less). If you add the strut, you end up with too big a space when using other fonts.
MagSafe is a series of proprietary magnetically attached power connectors, originally introduced by Apple Inc. on January 10, 2006, in conjunction with the MacBook Pro at the Macworld Expo in San Francisco, California. The connector is held in place magnetically so that if it is tugged — for example, by someone tripping over the cord — it will pull out of the socket without damaging the connector or the computer power socket, and without pulling the computer off the surface on which it is located.The concept of MagSafe is copied from the magnetic power connectors that are part of many deep fryers...
has anyone converted from LaTeX -> Word before? I have seen questions on the site but I'm wondering what the result is like... and whether the document is still completely editable etc after the conversion? I mean, if the doc is written in LaTeX, then converted to Word, is the word editable?
I'm not familiar with word, so I'm not sure if there are things there that would just get goofed up or something.
@baxx never use word (have a copy just because but I don't use it;-) but have helped enough people with things over the years, these days I'd probably convert to html latexml or tex4ht then import the html into word and see what come out
You should be able to cut and paste mathematics from your web browser to Word (or any of the Micorsoft Office suite). Unfortunately at present you have to make a small edit but any text editor will do for that.Givenx=\frac{-b\pm\sqrt{b^2-4ac}}{2a}Make a small html file that looks like<!...
@baxx all the convertors that I mention can deal with document \newcommand to a certain extent. if it is just \newcommand\z{\mathbb{Z}} that is no problem in any of them, if it's half a million lines of tex commands implementing tikz then it gets trickier.
@baxx yes but they are extremes but the thing is you just never know, you may see a simple article class document that uses no hard looking packages then get half way through and find \makeatletter several hundred lines of trick tex macros copied from this site that are over-writing latex format internals. |
-
Physics
(
http://mymathforum.com/physics/
)
SenatorArmstrong October 5th, 2017 11:44 AM
Finding frequency of oscillation
I am a little stuck and would love a hint or two if anyone has got any for me.
There's a particle with mass m that is trapped in a potential.
$$U(x) = \frac{-U_o}{(\frac{a}{a})^2 + 1}$$
where $$U_o > 1$$ and $$a>0$$. Assuming the amplitude is small, what would be the frequency of oscillation?
My plan of attack was to use Taylor Expansion for $U(x)$ considering the particle remains close to equilibrium (x=a).
$$U(x) = U(a) + U'(a)(x-a)+ \frac{1}{2}U''(a)(x-a)^2 + ... \approx U_o + \frac{1}{2}k(x-a)^2$$
My problem now is trying to figure out frequency from this information. Perhaps there is some relation with Simple Harmonic Oscillation and a potential function.
Thanks for any tips!
greg1313 October 5th, 2017 11:56 AM
Looks like there is a typo. You've got
a over a squared plus one in the denominator of the right-hand side.
SenatorArmstrong October 5th, 2017 04:09 PM
Quote:
Originally Posted by
greg1313
(Post 581601)
Looks like there is a typo. You've got
a over a squared plus one in the denominator of the right-hand side.
Embarrassing... Thanks for telling me!
Should be...
$$U(x) = \frac{-U_o}{(\frac{x}{a})^2 + 1}$$
romsek October 5th, 2017 05:16 PM
Quote:
Originally Posted by
SenatorArmstrong
(Post 581613)
Embarrassing... Thanks for telling me!
Should be...
$$U(x) = \frac{-U_o}{(\frac{x}{a})^2 + 1}$$
$U(x) \approx -U_0+\dfrac{U_0 x^2}{a^2}$
$\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$
and thus the restoring force is given by
$\kappa = \dfrac{2U_0}{a^2}$
should be cookie cutter from here
SenatorArmstrong October 6th, 2017 09:27 AM
Quote:
Originally Posted by
romsek
(Post 581617)
$U(x) \approx -U_0+\dfrac{U_0 x^2}{a^2}$
$\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$
and thus the restoring force is given by
$\kappa = \dfrac{2U_0}{a^2}$
should be cookie cutter from here
Thanks for the help romsek.
Wouldn't the restoring force be equal to the derivative of the potential function? What happened to the x in the in $\kappa = \dfrac{2U_0}{a^2}$ ? compared to $\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$
where there is the x. Do we remove the x since we are assuming the amplitude is small?
romsek October 6th, 2017 09:36 AM
Quote:
Originally Posted by
SenatorArmstrong
(Post 581654)
Thanks for the help romsek.
Wouldn't the restoring force be equal to the derivative of the potential function? What happened to the x in the in $\kappa = \dfrac{2U_0}{a^2}$ ? compared to $\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$
where there is the x. Do we remove the x since we are assuming the amplitude is small?
I'm sorry I guess I wasn't clear.
Yes, the force is the derivative of the potential.
The "spring constant", $\kappa$ is what I gave you.
With that constant you can determine the period of oscillation from the commonly known formula for simple harmonic oscillators with no damping.
SenatorArmstrong October 6th, 2017 09:48 AM
Quote:
Originally Posted by
romsek
(Post 581617)
$U(x) \approx -U_0+\dfrac{U_0 x^2}{a^2}$
$\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$
and thus the restoring force is given by
$\kappa = \dfrac{2U_0}{a^2}$
should be cookie cutter from here
Do you think this is an acceptable way to relate frequency and restoring force?
$F = \frac{2U_o}{a^2} = -kx = ma$
Then considering... $\sqrt{\frac{k}{m}} = 2 \pi f$ $\Rightarrow \frac{k}{4\pi^2 f} = m$
I then multiply both sides by $a$
$\frac{ak}{4\pi^2 f} = ma = F$
Then plug back in and solve for f
$ f = \sqrt{\frac{a^3 k}{8U_o \pi^2}}$
However, I have a constant k that is now in the formula that I did not have originally.
romsek October 6th, 2017 10:05 AM
Quote:
Originally Posted by
SenatorArmstrong
(Post 581656)
Do you think this is an acceptable way to relate frequency and restoring force?
$F = \frac{2U_o}{a^2} = -kx = ma$ Then considering... $\sqrt{\frac{k}{m}} = 2 \pi f$ $\Rightarrow \frac{k}{4\pi^2 f} = m$ I then multiply both sides by $a$ $\frac{ak}{4\pi^2 f} = ma = F$ Then plug back in and solve for f $ f = \sqrt{\frac{a^3 k}{8U_o \pi^2}}$ However, I have a constant k that is now in the formula that I did not have originally.
...
and $k = \dfrac{2U_o}{a^2}$
so
$f = \sqrt{\dfrac{a}{4\pi^2m}}$
now you are back to your original constants.... (you dropped the $m$ in the original)
SenatorArmstrong October 7th, 2017 10:30 AM
Quote:
Originally Posted by
romsek
(Post 581658)
...
and $k = \dfrac{2U_o}{a^2}$
so
$f = \sqrt{\dfrac{a}{4\pi^2m}}$
now you are back to your original constants.... (you dropped the $m$ in the original)
Crystal clear. Thank you for your help as always!
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1 2006-Spring 1.2 Problem 1
Define an indicator variable \(I_i\) as:
\[ I_i = \begin{cases} 1 & \text{if $i^{th}$ and $(i+1)^{th}$ cards are different (H,T) or (T,H)},\\ 0 & \text{otherwise} \end{cases} \]
Now the number of runs in a sequence of \(n\) coin tosses is given by: \[ R_n = 1+ \sum_{i=2}^{n-1} I_i \forall n\geq 3 \]
Thus,
\[ \begin{align} ER_n &= 1 + \sum_{i=2}^{n-1} EI_i \\ &= 1 + \sum_{i=2}^{n-1} P(I_i=1) \end{align} \]
\(P(I_i=1)\) is given by : \(P(I_i=1)=p\times q + q \times p\) (heads followed by tails or tails followed by heads)
And hence, \[ ER_n = 1+(n-2) \times (2pq) \]
Check: - \(ER_1 = 1\) and that is \(ER_1=1\) - \(P(R_2=1)=p^2 + q^2\) and \(P(R_2=2)=pq+qp=2pq\) thus \(E(R_2)=(p^2+q^2)+4pq = (p+q)^2+2pq = 1+2pq\) which is what \(ER_n\) formula gives us for n=2
1.2.1 Variance Calculation:
To calculate: \(\sigma^2=Var(R_n)\)
\(Var(R_n) = E(R_n^2)-(ER_n)^2\)
So we focus on calculating \(ER_n^2\):
\[\begin{align} ER_n^2 &= E((1+\sum_{i=2}^{n-1}I_i)^2)\\ &= E(1+(\sum_{i=1}^{n-1}I_i)^2 + 2\sum_{i=2}^{n-1}I_i))\\ &= E(1+(\sum{i=2}^{n-1}I_i^2 + 2\sum_{2\leq i < j}^{n-1} I_iI_j) + 2\sum_{i=2}^{n-1}I_i)\\ &= 1 + (n-2)\times(2pq) + 2\sum_{2\leq i < j}^{n-1} I_iI_j + 2(n-2)(2pq) \\ &= 1+3(n-2)\times(2pq) + 2\sum_{2\leq i < j}^{n-1} I_iI_j\\ \end{align}\]
In order to calculate \(EI_iI_j\), we consider following 3 cases:
Case 1: \(j-i=1\), then \(P(I_i=1, I_j=1)\) = Probaility that \(i^{th}, (i+1)^{th} \mathrm{and} (i+2)^{th}\) cards are different. For \(i=2\) to \(i={n-1}\) there are \((n-2-1)=n-3\) such terms and \(P(I_i,I_j=1)= pqp+qpq=pq\)
Case 2: \(j-i>=2\), then \(P(I_i=1,I_j=1) = P(I_i=1)P(I_j=1)\) , that is these events are independent unless they occur next to each other as in Case 1. and hence \(P(I_i=1,I_j=1)=P(I_i=1)P(I_j=1) = (pq)^2\) and there are \((n-4)\) such ways to choose
Thus, \(ER_n^2 = 1+6pq(n-2)+2\times((n-3)\times (pq)+(n-4)\times(pq)^2)\) Substitute in (skipping/TODO) \(Var(R_n) = ER_n^2-(ER_n)^2\)
1.3 Problem 2
\(X = \{X_1,X_2 \dots, X_n \}\) and \(Y=\sum_{i=1}^{N}c_iX_i\)
To determine : - Distribution of Y
We use characteristic functions
Using the characteristic function of a normal RV:
\(\phi_X(t) = E[e^{itX}] = e^{-it\mu- \frac{1}{2}\sigma^2t^2}\)
For multivariate case:
\(\phi_X(\bf{t}) = E[e^{i\bf{t^T}X}] = e^{-i\bf{t^T}\mu- \frac{1}{2}\bf{t^T}\sum^2 \bf{t}}\)
Now, \(Y=c^TX\) where \(c=[c_1,c_2 \dots c_n]\) (\(Y=\sum_{i=1}^{N}c_iX_i\))
Thus,
\[ \begin{align} \phi_Y(\bf{t}) &= E[e^{i\bf{t^T}Y}]\\ &= E[e^{i\bf{t}c^TX}]\\ &= \phi_X(tc^T)\\ &= e^{-ic^T\bf{t}\mu - \frac{1}{2}\bf{t}^Tc\sum^2 \bf{t}c^T} \end{align} \]
and thus comparing with the characteristic function we started with:
\(Y \sim N(c^T\mu, a\sum a^T)\)
TODO: Check again the transposes
1.4 Problem 3
TODO |
OptiChar Module of OptiLayer Thin Film Software
In OptiChar, a thin film is represented by a model including spectral dependencies of refractive index and extinction coefficient, film thickness, dependence of optical parameters on the thickness of a thin film (bulk inhomogeneity), thickness of surface overlayer, and porosity.
The thin-film model can be described by a vector of model parameters \(X\). The coordinates of the vector \(X\) can be the parameters describing wavelength dependencies of the refractive indices/extinction coefficients, film thickness, degree of bulk inhomogeneity, porosity, thickness of the surface overlayer etc. The model parameters are determined by the minimization of the discrepancy function \(DF\) estimating the closeness between experimental and model data:
\[ DF^2(X)=\sum\limits_{j=1}^N \left[\frac{S(X;\lambda_j)-\hat{S}(\lambda_j)}{\Delta_j}\right]^2, \]
where \(S\) is the spectral characteristic of the model film, \(\hat{S}\) is the corresponding experimental spectral characteristic, \(\{\lambda_j\}\) is the wavelength grid in the experimental spectral range, \(\Delta_j\) are measurement tolerances.
If spectral characteristics (transmittance, reflectance) are measured in %, and then the default \(\Delta_j\) values for OptiLayer, OptiChar, OptiRE is 1%. In this case, the value of \(DF\) around one means that RMS deviation of model data from the experiment one is about 1%. This value should be compared with expected accuracy of your measurements. This expected accuracy should include not only random noise, but also systematic errors (drifts, offsets).
A special interface provides an opportunity for a flexible and well-grounded choice of the specific thin film model depending on the available experimental data, its accuracy, and
Sophisticated mathematical algorithms enable you to reliably study even the most fine effects in thin films caused by a small absorption, small bulk and surface inhomogeneities.
Using
There are multiple factor affecting accuracy of the characterization results even in simplest characterization problems. Among them are:
If you specify Tolerances as real expected accuracy at each wavelength (Woollam Ellipsometers provide this information, for example), then you should not try to minimize \(DF\) below 1. When you reach this threshold value, you should stop and try to analyze the results.
All our characterization models and methodology have been verified in the frame of collaboration with scientists from world leading research groups.
References: |
Epsilon naught, $\epsilon_0$ The ordinal $\epsilon_0$, commonly given the British pronunciation "epsilon naught," is the smallest ordinal $\alpha$ for which $\alpha=\omega^\alpha$ and can be equivalently characterized as the supremum
$$\epsilon_0=\sup\{\omega,\omega^\omega,\omega^{\omega^\omega},\ldots\}$$
The ordinals below $\epsilon_0$ exhibit an attractive finitistic normal form of representation, arising from an iterated Cantor normal form involving only finite numbers and the expression $\omega$ in finitely iterated exponentials, products and sums.
The ordinal $\epsilon_0$ arises in diverse proof-theoretic contexts. For example, it is the proof-theoretic ordinal of the first-order Peano axioms.
Epsilon numbers
The ordinal $\epsilon_0$ is the first ordinal in the hierarchy of $\epsilon$-numbers, where $\epsilon_\alpha$ is the $\alpha^{\rm th}$ fixed point of the exponential function $\beta\mapsto\omega^\beta$. These can also be defined inductively, as $\epsilon_{\alpha+1}=\sup\{\epsilon_\alpha+1,\omega^{\epsilon_\alpha+1},\omega^{\omega^{\epsilon_\alpha+1}},\ldots\}$, and $\epsilon_\lambda=\sup_{\alpha\lt\lambda}\epsilon_\alpha$ for limit ordinals $\lambda$. The epsilon numbers therefore form an increasing continuous sequence of ordinals. Every uncountable infinite cardinal $\kappa$ is an epsilon number fixed point $\kappa=\epsilon_\kappa$. |
Neurons (Activation Functions)¶
Neurons can be attached to any layer. The neuron of each layer will affect the output in the forward pass and the gradient in the backward pass automatically unless it is an identity neuron. Layers have an identity neuron by default [1].
class
Neurons.
Identity¶
An activation function that does not change its input.
class
Neurons.
ReLU¶
Rectified Linear Unit. During the forward pass, it inhibits all inhibitions below some threshold \(\epsilon\), typically 0. In other words, it computes point-wise \(y=\max(\epsilon, x)\). The point-wise derivative for ReLU is\[\begin{split}\frac{dy}{dx} = \begin{cases}1 & x > \epsilon \\ 0 & x \leq \epsilon\end{cases}\end{split}\]
epsilon¶
Specifies the minimum threshold at which the neuron will truncate. Default
0.
Note
ReLU is actually not differentiable at \(\epsilon\). But it has
subdifferential\([0,1]\). Any value in that interval can be taken as a subderivative, and can be used in SGD if we generalize from gradient descent to subgradientdescent. In the implementation, we choose the subgradient at \(x==0\) to be 0. class
Neurons.
LReLU¶
Leaky Rectified Linear Unit. A Leaky ReLU can help fix the “dying ReLU” problem. ReLU’s can “die” if a large enough gradient changes the weights such that the neuron never activates on new data.\[\begin{split}\frac{dy}{dx} = \begin{cases}1 & x > 0 \\ 0.01 & x \leq 0\end{cases}\end{split}\]
class
Neurons.
Sigmoid¶
Sigmoid is a smoothed step function that produces approximate 0 for negative input with large absolute values and approximate 1 for large positive inputs. The point-wise formula is \(y = 1/(1+e^{-x})\). The point-wise derivative is\[\frac{dy}{dx} = \frac{-e^{-x}}{\left(1+e^{-x}\right)^2} = (1-y)y\]
class
Neurons.
Tanh¶
Tanh is a transformed version of Sigmoid, that takes values in \(\pm 1\) instead of the unit interval. input with large absolute values and approximate 1 for large positive inputs. The point-wise formula is \(y = (1-e^{-2x})/(1+e^{-2x})\). The point-wise derivative is\[\frac{dy}{dx} = 4e^{2x}/(e^{2x} + 1)^2 = (1-y^2)\]
class
Neurons.
Exponential¶
The exponential function.\[y = exp(x)\]
[1] This is actually not true: not all layers in Mocha support neurons. For example, data layers currently does not have neurons, but this feature could be added by simply adding a neuron property to the data layer type. However, for some layer types like loss layers or accuracy layers, it does not make much sense to have neurons. |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance. It is commonly used in the textile industry and often seen quoted on, for example, duvets and carpet underlay.The Shirley Institute in Manchester, England developed the tog as an easy-to-follow alternative to the SI unit of m2K/W. The name comes from the informal word "togs" for clothing which itself was probably derived from the word toga, a Roman garment.The basic unit of insulation coefficient is the RSI, (1 m2K/W). 1 tog = 0.1 RSI. There is also a clo clothing unit equivalent to 0.155 RSI or 1.55 tog...
The stone or stone weight (abbreviation: st.) is an English and imperial unit of mass now equal to 14 pounds (6.35029318 kg).England and other Germanic-speaking countries of northern Europe formerly used various standardised "stones" for trade, with their values ranging from about 5 to 40 local pounds (roughly 3 to 15 kg) depending on the location and objects weighed. The United Kingdom's imperial system adopted the wool stone of 14 pounds in 1835. With the advent of metrication, Europe's various "stones" were superseded by or adapted to the kilogram from the mid-19th century on. The stone continues...
Can you tell me why this question deserves to be negative?I tried to find faults and I couldn't: I did some research, I did all the calculations I could, and I think it is clear enough . I had deleted it and was going to abandon the site but then I decided to learn what is wrong and see if I ca...
I am a bit confused in classical physics's angular momentum. For a orbital motion of a point mass: if we pick a new coordinate (that doesn't move w.r.t. the old coordinate), angular momentum should be still conserved, right? (I calculated a quite absurd result - it is no longer conserved (an additional term that varies with time )
in new coordinnate: $\vec {L'}=\vec{r'} \times \vec{p'}$
$=(\vec{R}+\vec{r}) \times \vec{p}$
$=\vec{R} \times \vec{p} + \vec L$
where the 1st term varies with time. (where R is the shift of coordinate, since R is constant, and p sort of rotating.)
would anyone kind enough to shed some light on this for me?
From what we discussed, your literary taste seems to be classical/conventional in nature. That book is inherently unconventional in nature; it's not supposed to be read as a novel, it's supposed to be read as an encyclopedia
@BalarkaSen Dare I say it, my literary taste continues to change as I have kept on reading :-)
One book that I finished reading today, The Sense of An Ending (different from the movie with the same title) is far from anything I would've been able to read, even, two years ago, but I absolutely loved it.
I've just started watching the Fall - it seems good so far (after 1 episode)... I'm with @JohnRennie on the Sherlock Holmes books and would add that the most recent TV episodes were appalling. I've been told to read Agatha Christy but haven't got round to it yet
?Is it possible to make a time machine ever? Please give an easy answer,a simple one A simple answer, but a possibly wrong one, is to say that a time machine is not possible. Currently, we don't have either the technology to build one, nor a definite, proven (or generally accepted) idea of how we could build one. — Countto1047 secs ago
@vzn if it's a romantic novel, which it looks like, it's probably not for me - I'm getting to be more and more fussy about books and have a ridiculously long list to read as it is. I'm going to counter that one by suggesting Ann Leckie's Ancillary Justice series
Although if you like epic fantasy, Malazan book of the Fallen is fantastic
@Mithrandir24601 lol it has some love story but its written by a guy so cant be a romantic novel... besides what decent stories dont involve love interests anyway :P ... was just reading his blog, they are gonna do a movie of one of his books with kate winslet, cant beat that right? :P variety.com/2016/film/news/…
@vzn "he falls in love with Daley Cross, an angelic voice in need of a song." I think that counts :P It's not that I don't like it, it's just that authors very rarely do anywhere near a decent job of it. If it's a major part of the plot, it's often either eyeroll worthy and cringy or boring and predictable with OK writing. A notable exception is Stephen Erikson
@vzn depends exactly what you mean by 'love story component', but often yeah... It's not always so bad in sci-fi and fantasy where it's not in the focus so much and just evolves in a reasonable, if predictable way with the storyline, although it depends on what you read (e.g. Brent Weeks, Brandon Sanderson). Of course Patrick Rothfuss completely inverts this trope :) and Lev Grossman is a study on how to do character development and totally destroys typical romance plots
@Slereah The idea is to pick some spacelike hypersurface $\Sigma$ containing $p$. Now specifying $u(p)$ is trivial because the wave equation is invariant under constant perturbations. So that's whatever. But I can specify $\nabla u(p)|\Sigma$ by specifying $u(\cdot, 0)$ and differentiate along the surface. For the Cauchy theorems I can also specify $u_t(\cdot,0)$.
Now take the neigborhood to be $\approx (-\epsilon,\epsilon)\times\Sigma$ and then split the metric like $-dt^2+h$
Do forwards and backwards Cauchy solutions, then check that the derivatives match on the interface $\{0\}\times\Sigma$
Why is it that you can only cool down a substance so far before the energy goes into changing it's state? I assume it has something to do with the distance between molecules meaning that intermolecular interactions have less energy in them than making the distance between them even smaller, but why does it create these bonds instead of making the distance smaller / just reducing the temperature more?
Thanks @CooperCape but this leads me another question I forgot ages ago
If you have an electron cloud, is the electric field from that electron just some sort of averaged field from some centre of amplitude or is it a superposition of fields each coming from some point in the cloud? |
fixed in 10.0.2 Update
I have tried like these. I think there is a bug.
Plot[1/Sqrt[-1 + 2^2 Sech[x]^2], {x, 0, ArcCosh[2]}, Ticks -> {{ArcCosh[2]}, Automatic}]
This is the antiderivative.
primitive = Integrate[1/Sqrt[-1 + 2^2*Sech[x]^2], x]; Plot[primitive, {x, 0, ArcCosh[2]}, Ticks -> {{ArcCosh[2]}, {π/4, π/2}}]
Limit[primitive, x -> 0]
0
So far, that's right. Look at this any situation.Limits of
primitive are same regardless of the direction. And the limit value is minus. Is this right?
(version 10)
Limit[primitive, x -> ArcCosh[2], Direction -> -1] // FullSimplify
-π/2
Limit[primitive, x -> ArcCosh[2], Direction -> 1] // FullSimplify
-π/2
But this computation is right at version 9
(version 9)
Limit[primitive, x -> ArcCosh[2], Direction -> 1] // FullSimplify
π/2
And as mentioned earlier
origin, the definite integral is an erroneous conclusion at version 9 also.
===============================================================
Edit
The definite integral is solved using substitution method as Dr. Wolfgang Hintze says like this.
$u$ =$\frac{\cosh ^2(x)-1}{a^2-1}$
$dx$ = $\frac{\left(a^2-1\right)}{2 \sinh (x) \cosh (x)}du$
$\int_0^1 \frac{a^2-1}{2 \sinh (x) \cosh (x) \sqrt{a^2 \text{sech}^2(x)-1}} \, du$
$\frac{1}{2} \int_0^1 \frac{a^2-1}{\sqrt{a^2 \sinh ^2(x)-\sinh ^2(x) \cosh ^2(x)}} \, du$
$\frac{1}{2} \int_0^1 \frac{1}{\sqrt{\frac{\left(\cosh ^2(x)-1\right) \left(a^2-\cosh ^2(x)+1-1\right)}{\left(a^2-1\right) \left(a^2-1\right)}}} \, du$
$\frac{1}{2} \int_0^1 \frac{1}{\sqrt{u (1-u)}} \, du=\frac{\pi }{2}$
It is solved in the real number region. And
ArcCos[2]is also real number. But I don't konw why mathematica make $\int_0^{\cosh ^{-1}(2)} \frac{1}{\sqrt{2^2 \text{sech}^2(x)-1}}\, dx$ appear a imaginary term like $\left(\frac{1}{2}-i\right) \pi$ .
===============================================================
Origin
I have tried two expressions.(version 10)
(1)
Integrate[1/Sqrt[-1 + 2^2*Sech[x]^2], {x, 0, ArcCosh[2]}]
$\left(\frac{1}{2}-i\right) \pi$
(2)
$Assumptions = {a > 1}; Integrate[1/Sqrt[-1 + a^2*Sech[x]^2], {x, 0, ArcCosh[a]}]
$\frac{\pi }{2}$
What difference does it make it?
The first computing (1) makes the imaginary term.
- I π. I don't know why it did such result?
(version 9)
Integrate[1/Sqrt[-1 + 2^2*Sech[x]^2], {x, 0, ArcCosh[2]}]
$\frac{3 \pi }{2}$
If possible, I want to know mathematica's detail process. |
Symbols:Greek/Sigma Contents Sigma
The $18$th letter of the Greek alphabet.
Minuscules: $\sigma$ and $\varsigma$ Majuscule: $\Sigma$
The $\LaTeX$ code for \(\sigma\) is
\sigma .
The $\LaTeX$ code for \(\varsigma\) is
\varsigma .
The $\LaTeX$ code for \(\Sigma\) is
\Sigma .
$\Sigma$ Let $\mathcal E$ be an experiment. The event space of $\mathcal E$ is usually denoted $\Sigma$ (Greek capital sigma), and is the set of all outcomes of $\mathcal E$ which are interesting.
Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
The composite is called the summation of $\tuple {a_1, a_2, \ldots, a_n}$, and is written: $\displaystyle \sum_{j \mathop = 1}^n a_j = \tuple {a_1 + a_2 + \cdots + a_n}$ The $\LaTeX$ code for \(\displaystyle \sum_{j \mathop = 1}^n a_j\) is
\displaystyle \sum_{j \mathop = 1}^n a_j .
The $\LaTeX$ code for \(\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j\) is
\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j .
The $\LaTeX$ code for \(\displaystyle \sum_{\map \Phi j} a_j\) is
\displaystyle \sum_{\map \Phi j} a_j .
$\map \sigma n$ Let $n$ be an integer such that $n \ge 1$.
That is:
$\displaystyle \map \sigma n = \sum_{d \mathop \divides n} d$
where $\displaystyle \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
The $\LaTeX$ code for \(\map \sigma n\) is
\map \sigma n .
$\sigma$ $\sigma = \dfrac q A$
where:
$\sigma$ $\sigma = \dfrac m A$
where:
</onlyinclude>
The $\LaTeX$ code for \(\sigma\) is
\sigma . |
Linear ordering isotonic regression can be understood as approximating given series of 1-dimensional observations with non-decreasing function. It is similar to inexact smoothing splines, with the difference that we use monotonicity, rather than smoothness, to remove noise from the data. General isotonic regression is approximating given series of values with values satisfying a given partial ordering. Problem
Given values $ a_1 $, ..., $ a_n $ and their positive weights $ w_1 $, ..., $ w_n $, approximate them by $ y_1 $, ... $ y_n $ as close as possible, subject to a set of constraints of kind $ y_i \ge y_j $ :
Given values $ \mathbf{a} \in \Bbb{R}^n $, weights $ \mathbf{w} \in \Bbb{R}^n $ such that $ w_i>0 $ for all i, set of constraints $ E \subset \{1,...,n\}^2 $, minimize $ \sum_i w_i \left( y_i-a_i \right)^2 $ with respect to $ \mathbf{y} \in \Bbb{R}^n $ subject to $ y_i \ge y_j $ for all $ (i,j)\in E $
If all weights equal to 1, the problem is called
unweighted isotonic regression, otherwise it is called weighted isotonic regression.
The graph $ G=(V=\{1,2,...n\},E=E) $ is supposed to be acyclic. For example, it cannot be that, among others, there are constraints $ x_2 \ge x_5 $ and $ x_5 \ge x_2 $.
Example
Minimize $ 3(y_1-2)^2+(y_2-1)^2+(y_3-4)^2+2y_4^2 $ subject to
$ y_1\le y_2 $ $ y_2\le y_3 $ $ y_2\le y_4 $ Linear ordering
Of a particular interest is
linear ordering isotonic regression: Given values $ \mathbf{a} \in \Bbb{R}^n $, weights $ \mathbf{w} \in \Bbb{R}^n $ such that $ w_i>0 $ for all i, minimize $ \sum_i w_i \left( y_i-a_i \right)^2 $ with respect to $ \mathbf{y} \in \Bbb{R}^n $ subject to $ y_1 \le y_2\le\;...\;\le y_n $
For linear ordering isotonic regression, there is a simple linear algorithm, called Pool Adjacent Violators Algorithm (PAVA).
If all weights are equal 1, the problem is called
unweighted linear ordering isotonic regression. Example Isotonic regression for $ 10^6 $ points. Left: points $ (x_i, a_i) $, where $ a_i=0 \text{ or }1 $. The probability that $ a_i=1 $ is determined by logistic function. Only 1000 points shown. Right: outcome of isotonic regression (black graph) versus logistic function (red graph). The logistic function had been restored with high precision. Linear ordering isotonic regression as a model
Sometimes, linear ordering isotonic regression is applied to a set of observations $ (x_i, y_i) $, $ 1 \le i \le n $, where x is the explanatory variable, y is the dependent variable. The observations are sorted by their x's, then the isotonic regression is applied to y's with additional restriction $ x_i=x_{i+1}\Rightarrow y_i=y_{i+1} $ for all $ i $.
Other variants Non-euclidean metrics
Sometimes other metrics are used instead of the Euclidean metric, for instance $ L_1 $ metrics:
$ \sum_i w_i |y_i-a_i|\! $
or unweighted $ L_\infty $ metrics:
$ \max_i |y_i-a_i| $ Points at a grid
Sometimes, values are placed at 2d or higher-dimensional grid. The fit value must increase at each dimension, i. e.
Minimize $ \sum_{ij} w_{ij}\left(y_{ij}-x_{ij}\right)^2 $ with respect to $ y $ subject to $ y_{ij} \le y_{kl} \text{ if } i \le j,\ k \le l $ Algorithms Pool Adjacent Violators Algorithm
Pool Adjacent Violators Algorithm (PAVA) is a linear time (and linear memory) algorithm for linear ordering isotonic regression.
Preliminary considerations
The algorithm is based on the following theorem:
Theorem: For an optimal solution, if $ a_i \ge a_{i+1} $, then $ y_i = y_{i+1} $ Proof: suppose the opposite, i. e. $ y_i < y_{i+1} $. Then for sufficiently small $ \varepsilon $, we can set $ y_i^\mathrm{new} = y_i + w_{i+1} \varepsilon $ $ y_{i+1}^\mathrm{new} = y_{i+1} - w_i \varepsilon $
which decreases the sum $ \sum_i w_i(y_i-a_i)^2\! $ without violating the constraints. Therefore, our original solution was not optimal. Contradiction.
Since $ y_i = y_{i+1} $, we can combine both points $ (w_i, a_i) $ and $ (w_{i+1}, a_{i+1}) $ to a new point $ \left(w_i + w_{i+1}, {w_i a_i + w_{i+1} a_{i+1} \over w_i + w_{i+1}}\right) $.
However, after we combine two points $ (w_i, a_i) $ and $ (w_{i+1},a_{i+1}) $ to the new point $ \left(w_i^\prime, a_i^\prime\right) $, this new point can violate the constraint $ a_{i-1}\le a_i^\prime $. In this case it should be combined with the (i-1)-th point. If the combined point violates its constraint, it should be combined with the previous point, etc.
Algorithm Input: values $ a_1,...,a_n\! $ weights $ w_1,...,w_n\! $ Output: non-decreasing values $ y_1,...,y_n\! $ minimizing $ \sum_i w_i (y_i-a_i)^2 \! $ The Algorithm $ a'_1 \leftarrow a_1 $ $ w'_1 \leftarrow w_1 $ $ j \leftarrow 1 $ $ S_0 \leftarrow 0 $ $ S_1 \leftarrow 1 $ for $ i=2,3,...,n\! $ do $ j \leftarrow j+1 $ $ a'_j \leftarrow a_i $ $ w'_j \leftarrow w_i $ while $ j>1\! $ and $ a'_j < a'_{j-1} \! $ do $ a'_{j-1} \leftarrow {w'_j a'_j + w'_{j-1} a'_{j-1} \over w'_j + w'_{j-1}} $ $ w'_{j-1} \leftarrow w'_j + w'_{j-1} $ $ j \leftarrow j-1 $ $ S_j \leftarrow i $ for $ k=1,2,...,j\! $ do for $ l=S_{k-1}+1,...,S_k\! $ do $ y_l=a'_k\! $ for $ l=S_{k-1}+1,...,S_k\! $ do
Here S defines to which old points each new point corresponds.
Arbitrary case algorithms
In the arbitrary case, this can be solved as a quadratic problem. The best algorithm takes $ \Theta(n^4) $ time, see:
Maxwell, WL and Muckstadt, JA (1985), "Establishing consistent and realistic reorder intervals in production-distribution systems", Operations Research33, pp. 1316-1341. Spouge J, Wan H, and Wilbur WJ (2003), "Least squares isotonic regression in two dimensions", J. Optimization Theory and Apps. 117, pp. 585-605. Implementations R isoreg
The function
isoreg performs unweighted linear ordering isotonic regression. It does not require any packages. For many simple cases, it is enough.
Example of usage:
x=sort(rnorm(10000)) y=x+rnorm(10000) y.iso=isoreg(y)$yf plot(x,y,cex=0.2); lines(x,y.iso,lwd=2,col=2)
The
isoreg function also implemenets linear ordering isotonic regression as a model: x=rnorm(10000) y=x+rnorm(10000) y.iso=isoreg(x,y)$yf plot(x,y,cex=0.2); lines(sort(x),y.iso,lwd=2,col=2) Iso
The package
Iso contains three functions: pava - linear ordering isotonic regression, weighted or unweighted. biviso - 2-d isotonic regression ufit - unimodal order (increasing then decreasing)
Example of usage:
install.packages("Iso") # should be done only once library("Iso") # should be done once per session x=sort(rnorm(10000)) y=x+rnorm(10000) y.iso=pava(y) plot(x,y,cex=0.2); lines(x,y.iso,lwd=2,col=2) isotone isotone is the most advanced package. gpava - linear ordering isotonic regression, weighted or unweighted, for any metrics. Similarly to isoreg, gpava can implement linear ordering isotonic regression as a model. activeSet - general isotonic regression for any metrics.
Example of usage:
install.packages("isotone") # should be done only once library("isotone") # should be done once per session x=sort(rnorm(10000)) y=x+rnorm(10000) y.iso=gpava(y)$x plot(x,y,cex=0.2); lines(x,y.iso,lwd=2,col=2) Comparison of speed
Since all three libraries somehow implement the PAVA algorithm, we can compare their speed.
As we can see on the graph below, for unweighted linear order isotonic regression (LOIR) with $ n>200 $,
isoreg should be used. For weighted LOIR and unweighted LOIR with $ n \le 200 $, pava should be used. gpava should be used only for non-Euclidean metrics.
Also, the implementations of weighted simple ordering isotonic regression on R are far from perfect.
Java
Weka, a free software collection of machine learning algorithms for data mining tasks written in the University of Waikato, contains, among others, an isotonic regression classifier. The classifier is deeply engrained into the Weka's hierarchy of classes and cannot be used without Weka.
Python
While scikit-learn implements the isotonic regression, Andrew Tulloch made the Cython implementation of the algorithm for linear ordering isotnic regression, which is 14-5000 times faster, depending on the data size. See Speeding up isotonic regression in scikit-learn by 5,000x. If you just want the code, click here.
Usage Calibration of output for categorical probabilities For more information, see "Predicting Good Probabilities With Supervised Learning", by Alexandru Niculescu-Mizil and Rich Caruana.
Most of the supervised learning algorithms, for example Random Forests [1], boosted trees, SVM etc. are good in predicting the most probable category,
but not the probability. Some of them tend to overestimate high probabilities and underestimate low probabilities. (One notable exception is neural networks, which themselves produce a well calibrated prediction.)
In order to obtain the correct probability, unweighted linear order isotonic regression can be used:
$ \hat y^\mathrm{final} = f(\hat y) $
where
$ \hat y^\mathrm{final} $ is a final prediction for probability $ \hat y $ is a prediction given by the model $ f $ is a non-decreasing function
In order to find f, model's predictions on the validation set are matched with output variable, and the isotonic regression is applied to the pairs.
Alternative approach is to pass $ \hat p $ via a sigmoid function:
$ \hat y^\mathrm{final} = {1\over 1 + e^{a-b\hat y}} $
This is called Platt calibration.
At small dataset sizes, Platt calibration is better than isotonic regression. Starting at 200-5000 observations, the isotonic regression surpasses Platt calibration slightly. Note also that this kind of isotonic regression is easier and faster than the logistic regression needed by Platt calibration.
Calibration of recommendation models The problem
We have a model M which predicts residuals r of another model, trained on the training set. For cases with small number of users or items, the model is overfitted, and predictions need relaxation:
$ \hat r_{ui}^\mathrm{final} = f(\hat S_{ui})\hat r_{ui} $
where
$ \hat r_{ui}^\mathrm{final} $ is a final prediction for user u, item i $ \hat r_{ui} $ is a prediction given by the model $ f $ is a non-decreasing function $ S_{ui} $ may be either $ S_u $ (number of observations with user u in the training set), or $ S_i $ (number of observations with item i in the training set), or $ \min(S_u, S_i) $, depending on the nature of the model. The letter S stands for Support. Solution
Given set of triplets $ (\hat r_k, r_k, S_k) $ from validation database, we have:
$ \min_f \sum_k \left(r_k - \hat r_k f(S_k)\right)^2 = \min_f \sum_k \hat r_k^2 \left( f(S_k) - \frac{r_k}{\hat r_k} \right)^2 + \operatorname{const} $
Therefore, for k-th observation we should set weight $ w_k=\hat y_k^2 $ and value $ y_k = r_k / \hat r_k $. The triplets $ (S_k, w_k, y_k) $ then sorted with respect to S, and the isotonic regression applied to them.
If a function directly predicts the rating or acceptance rate, rather than predicting the residual of another model, we can define $ r_{ui} $ as the difference between this model and some simpler model (i. e. average rating for the item plus average rating correction for the user). This model is called
basic model. Analysis and calibration of input variables
Often, we know an output variable depends on an input variable monotonically, but other than this we have very little prior knowledge about the dependence. In this case, isotonic regression can provide important hints about the dependence.
Below is a real-world example of bankrupcy flag vs revolving utilization of unsecured lines, from the Give Me Some Credit contest.
The dependence can be used in several ways:
for calibration of the input variable, maybe after smoothing. In the case given by the example, uncalibrated indicator produced the correlation of -0.002, due to outliers. The standard calibration, log(x+1), produced the correlation of 0.18. The isotonic regression calibration produced the correlation of 0.30. by the person who makes the final decision, in the context of human-machine cooperation. for filtering outliers. In this case, we see that the outlier filter should be set very low, probably around 1.1 or lower. The horizontal line starts at x=1.029. Nonmetric multidimensional scaling
A multidimensional scaling (MDS) algorithm, given the matrix of item–item similarities, puts items into N-dimensional space such that similar items would be closer to each other than dissimilar items. This is particulary useful for visualization purposes.[2]
There are several kinds of MDS, depending on the relationship between similarity and distance, as well as the definition of distance. For nonmetric MDS, the distance may be an arbitrary monotonic function of similarity. The function is typically found using isotonic regression. [3]. Nonmetric multidimensional scaling is implemented as isoMDS in the MASS library. [4]
Advantages, disadvantages, and post-processing
Advantages:
Non-parametric Fast (linear time) Simple
Disadvantages:
one or several points at the ends of the interval are sometimes noisy it compares favorably to other approaches only if there is big enough statistics ($ n\gtrapprox 10^3 $ and ideally $ n>10^4\! $)
These disadvantages can be improved by smoothing the outcome of isotonic regression. This way, we can get the best of both worlds (smoothness and monotonicity).
Isotonic Regression Algorithms, by Quentin F. Stout - review of the best current existing algorithms Articles Predicting Good Probabilities With Supervised Learning, by Alexandru Niculescu-Mizil, Rich Coruana Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods, by John C. Platt |
I'd like to plot the function $f$ in this answer https://math.stackexchange.com/a/788818/66096
Let $h(x) = \begin{cases} e^{ -\frac{1}{1 - (1-2x)^2}} & \mbox{ for } 0<x < 1\\ 0 & \mbox{ otherwise} \end{cases}$ Then let $$g(x)=\sum_{n=1}^\infty h(n^2(x-n))=\begin{cases}h(n^2(x-n))&\text{if }n\le x<n+1, n\in\mathbb N\\0&\text{otherwise}\end{cases}$$ And let $f(x)=\int_0^xg(t)\,\mathrm dt$.
I only know the basics of
Mathematica, so I've been trying to plot piecewise functions without success.
Here is what I got
h[x_] = Piecewise[{{E^(-(1 - (1 - 2 x)^2)^(-1)), 0 < x < 1}, {0, x > 1}, {0, x < 0}}]
But how do I deal with $g$ ? I could input partial sums with the series expression, but at the same time, I could save computations by just defining it piecewise. |
Alright, I have this group $\langle x_i, i\in\mathbb{Z}\mid x_i^2=x_{i-1}x_{i+1}\rangle$ and I'm trying to determine whether $x_ix_j=x_jx_i$ or not. I'm unsure there is enough information to decide this, to be honest.
Nah, I have a pretty garbage question. Let me spell it out.
I have a fiber bundle $p : E \to M$ where $\dim M = m$ and $\dim E = m+k$. Usually a normal person defines $J^r E$ as follows: for any point $x \in M$ look at local sections of $p$ over $x$.
For two local sections $s_1, s_2$ defined on some nbhd of $x$ with $s_1(x) = s_2(x) = y$, say $J^r_p s_1 = J^r_p s_2$ if with respect to some choice of coordinates $(x_1, \cdots, x_m)$ near $x$ and $(x_1, \cdots, x_{m+k})$ near $y$ such that $p$ is projection to first $m$ variables in these coordinates, $D^I s_1(0) = D^I s_2(0)$ for all $|I| \leq r$.
This is a coordinate-independent (chain rule) equivalence relation on local sections of $p$ defined near $x$. So let the set of equivalence classes be $J^r_x E$ which inherits a natural topology after identifying it with $J^r_0(\Bbb R^m, \Bbb R^k)$ which is space of $r$-order Taylor expansions at $0$ of functions $\Bbb R^m \to \Bbb R^k$ preserving origin.
Then declare $J^r p : J^r E \to M$ is the bundle whose fiber over $x$ is $J^r_x E$, and you can set up the transition functions etc no problem so all topology is set. This becomes an affine bundle.
Define the $r$-jet sheaf $\mathscr{J}^r_E$ to be the sheaf which assigns to every open set $U \subset M$ an $(r+1)$-tuple $(s = s_0, s_1, s_2, \cdots, s_r)$ where $s$ is a section of $p : E \to M$ over $U$, $s_1$ is a section of $dp : TE \to TU$ over $U$, $\cdots$, $s_r$ is a section of $d^r p : T^r E \to T^r U$ where $T^k X$ is the iterated $k$-fold tangent bundle of $X$, and the tuple satisfies the following commutation relation for all $0 \leq k < r$
$$\require{AMScd}\begin{CD} T^{k+1} E @>>> T^k E\\ @AAA @AAA \\ T^{k+1} U @>>> T^k U \end{CD}$$
@user193319 It converges uniformly on $[0,r]$ for any $r\in(0,1)$, but not on $[0,1)$, cause deleting a measure zero set won't prevent you from getting arbitrarily close to $1$ (for a non-degenerate interval has positive measure).
The top and bottom maps are tangent bundle projections, and the left and right maps are $s_{k+1}$ and $s_k$.
@RyanUnger Well I am going to dispense with the bundle altogether and work with the sheaf, is the idea.
The presheaf is $U \mapsto \mathscr{J}^r_E(U)$ where $\mathscr{J}^r_E(U) \subset \prod_{k = 0}^r \Gamma_{T^k E}(T^k U)$ consists of all the $(r+1)$-tuples of the sort I described
It's easy to check that this is a sheaf, because basically sections of a bundle form a sheaf, and when you glue two of those $(r+1)$-tuples of the sort I describe, you still get an $(r+1)$-tuple that preserves the commutation relation
The stalk of $\mathscr{J}^r_E$ over a point $x \in M$ is clearly the same as $J^r_x E$, consisting of all possible $r$-order Taylor series expansions of sections of $E$ defined near $x$ possible.
Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; that is for every point $x \in M$ there exists a $p \in P$ such that $\| x - p\|_{2}<\varepsilon$....
The same result should be true for abstract Riemannian manifolds. Do you know how to prove it in that case?
I think there you really do need some kind of PDEs to construct good charts.
I might be way overcomplicating this.
If we define $\tilde{\mathcal H}^k_\delta$ to be the $\delta$-Hausdorff "measure" but instead of $diam(U_i)\le\delta$ we set $diam(U_i)=\delta$, does this converge to the usual Hausdorff measure as $\delta\searrow 0$?
I think so by the squeeze theorem or something.
this is a larger "measure" than $\mathcal H^k_\delta$ and that increases to $\mathcal H^k$
but then we can replace all of those $U_i$'s with balls, incurring some fixed error
In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in a metric space (X, d). It is named after the German mathematician Hermann Minkowski and the French mathematician Georges Bouligand.To calculate this dimension for a fractal S, imagine this fractal lying on an evenly spaced grid, and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid...
@BalarkaSen what is this
ok but this does confirm that what I'm trying to do is wrong haha
In mathematics, Hausdorff dimension (a.k.a. fractal dimension) is a measure of roughness and/or chaos that was first introduced in 1918 by mathematician Felix Hausdorff. Applying the mathematical formula, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas...
Let $a,b \in \Bbb{R}$ be fixed, and let $n \in \Bbb{Z}$. If $[\cdot]$ denotes the greatest integer function, is it possible to bound $|[abn] - [a[bn]|$ by a constant that is independent of $n$? Are there any nice inequalities with the greatest integer function?
I am trying to show that $n \mapsto [abn]$ and $n \mapsto [a[bn]]$ are equivalent quasi-isometries of $\Bbb{Z}$...that's the motivation. |
Compounds of boron with hydrogen are called boranes. One of these boranes has the empirical formula BH_3 and a molecular mass of 28 amu. What is its molecular formula?
Solution: We will first find the formula mass of the empirical formula, BH_3.
mass = 10.81\,\text{amu} + (3\times 1.008\,\text{amu})=13.83 amu We now know that one molecular unit has a mass of 13.83 amu. To figure out how many units are in 28 amu, we simply dividing 28 amu by 13.83 amu. \frac{28\,\text{amu}}{13.83\,\text{amu}}=2.02\approx 2
This means the molecular formula is (BH_3)_2. Another way to write this would be, B_2H_6 |
Spring 2018, Math 171 Week 9 Miscillaneous Poisson Process Problems
Let \(X_1, X_2, \dots\overset{\mathrm{i.i.d}}{\sim} \mathrm{exp}(\lambda)\), and let \(N(t)\) be a poisson process with rate \(\lambda\)
Show the equality \(P(\sum_{i=1}^n X_i \le t) = P(N(t) \ge n)\)
Find an analogous equality for \(P(s \le \sum_{i=1}^n X_i \le t)\)
(Answer) \(P(N(s) < n, N(t) \ge n)\)
\(n+m\) cars approach \(n\) toll booths \((m < n)\). The time taken for a car to pay its toll is exponentially distributed with rate \(\lambda\). When a toll booth becomes available, the next car in line fills it instantly if there are any cars waiting.
What is the expected time before the first car exits the tolls?
(Answer) \(\frac{1}{n\lambda}\)
What is the expected time before the \(m^\mathrm{th}\) car exits the tolls?
(Answer) \(\frac{m}{n\lambda}\)
What is the expected time before the last car exits the tollbooths?
(Solution)
Let \(\tau_1, \tau_2, \dots \tau_{n+m}\) be the times between successive cars exiting any of the tollbooths.
Then \(\tau_1\), the time of the first car to exit any of the tolls, is the minimum of \(n\) exponential waiting times (for each of the tolls), and is therefore distributed exponential with parameter \(n\lambda\).
Likewise, \(\tau_2, \dots \tau_{m+1}\) are each the minimum of \(n\) exponential waiting times each (since the tolls are all occupied by cars until \(m+1\) cars have passed) and are therefore exponential with parameter \(n\lambda\).
After \(m+1\) cars have passed, there are more tollbooths than cars left, so there are no cars occupying some of the tollbooths. For this reason, \(\tau_{m+2}\) is distributed exponentially with parameter \((n-1)\lambda\), \(\tau_{m+3}\) is distributed exponentially with parameter \((n-2)\lambda\), and so on.
The quantity of interest is therefore \[\begin{aligned}\mathbb{E}[\tau_1 + \dots + \tau_{m+n}] &= \mathbb{E}[\tau_1] + \dots + \mathbb{E}[\tau_{n+m}]\\&=\frac{m+1}{n\lambda} + \frac{1}{(n-1)\lambda} + \frac{1}{(n-2)\lambda} + \dots \frac{1}{\lambda}\end{aligned}\]
What is the probability that the \((n+1)\)st car to enter a toll exits before the first car to enter a toll?
(Answer) \(\frac{n-1}{n} \frac{1}{2}\) Compound Poisson Process
Let \(X_1, X_2, \dots\) be a sequence of i.i.d. random variables with mean \(\mu\) and variance \(\sigma^2\), and let \(N(t)\) be a poisson process with rate \(\lambda\) independent of all the \(X_k\). Define \(S(t) = \sum_{k=1}^{N(t)}X_k\).
Compute \(\mathbb{E}[S(t)]\)
(Answer) \(t \lambda \mu\)
Compute \(\mathrm{Cov}(S(t_1), S(t_2))\) for \(t_1 < t_2\)
(Solution)\[\begin{aligned}\mathrm{Cov}(S(t_1), S(t_2)) &= \mathrm{Cov}(S(t_1), S(t_2) - S(t_1) + S(t_1))\\ &= \mathrm{Cov}(S(t_1), S(t_2) - S(t_1)) + \mathrm{Var}(S(t_1))\\&=\mathrm{Cov}(\sum_{k=1}^{N(t_1)}X_k, \sum_{k=N(t_1)+1}^{N(t_2)}X_k) + \mathrm{Var}(S(t_1))\\&=\mathrm{Var}(S(t_1))\end{aligned}\] Where the last equality comes from the independence of \(X_i\) and \(X_j\) for \(i \neq j\) and the fact that the sums \(\sum_{k=1}^{N(t_1)}X_k\) and \(\sum_{k=N(t_1)+1}^{N(t_2)}X_k\) do not overlap in indices.
Compute \(\mathrm{Var}(S(t))\)
(Answer) \(t\lambda(\sigma^2 + \mu^2)\)
Suppose the \(X_k\) have MGF \(M_X(u)\). Compute the MGF of \(S(t)\)
(Answer) \(e^{t\lambda(M_X(u)-1)}\) Thinning and Superposition
Let \(N_1(t)\) and \(N_2(t)\) be independent poisson processes with rates \(\lambda_1\) and \(\lambda_2\). Find the probability that the \(m_1^{\mathrm{th}}\) arrival of \(N_1(t)\) occurs before the \(m_2^{\mathrm{th}}\) arrival of \(N_2(t)\).
Let \(N(t)\) be a poisson process with rate \(\lambda\) and let each arrival of the process be identified as either type 1 with probability \(p\) or type 2 with probability \(1-p\). Find the probability that the \(m_1^{\mathrm{th}}\) arrival of type 1 occurs before the \(m_2^{\mathrm{th}}\) arrival of type 2.
\(n+m\) cars approach \(n\) toll booths \((m < n)\). The time taken for a car to pass through a toll booth is exponentially distributed with rate \(\lambda\). When a toll booth becomes available, a car fills it instantly if there are any cars waiting.
(Discussed) Describe the exits of first \(m\) cars from the toll booths as a superposition of poisson processes
(Discussed) What is the probability that the first car comes through the leftmost tollbooth?
(Discussed) What is the probability that all of the first \(m\) cars comes through the leftmost tollbooth?
What is the probability that each of the first \(m\) cars go exit through a different tollbooth?
(Answer) \(\frac{n-1}{n}\frac{n-2}{n} \dots \frac{n-m+1}{n}\) |
Every vector space over a field of positive characteristic $p$ is in particular a vector space over $\mathbb{F}_p$. Any subgroup of such a vector space is a subspace (exercise), and conversely. Assuming the axiom of choice, any such subspace is a direct sum of copies of $\mathbb{F}_p$.
Every vector space over a field of characteristic zero is in particular a vector space over $\mathbb{Q}$. Assuming the axiom of choice, it is in fact a direct sum of copies of $\mathbb{Q}$.
The subgroups of a direct sum must be direct sums of subgroups, so your question reduces to the classification of subgroups of $\mathbb{Q}$.
Let $H$ be a subgroup of a $\mathbb{Q}$-vector space $V$. By restricting our attention to the subspace spanned by $H$, we may WLOG assume that $H$ spans $V$ as a vector space. Take a basis $e_i, i \in I$ of $V$ consisting of elements of $H$ and consider the short exact sequence$$0 \to \bigoplus_i \mathbb{Z} e_i \to V \cong \bigoplus_i \mathbb{Q} e_i \to \bigoplus_i \mathbb{Q}/\mathbb{Z} \to 0.$$
Since $e_i \in H$ for all $i$, it's not hard to see that $H$ is necessarily the preimage of its image in $\bigoplus \mathbb{Q}/\mathbb{Z}$, so our problem reduces to the problem of classifying subgroups of $M = \bigoplus_i \mathbb{Q}/\mathbb{Z}$. By partial fraction decomposition (which makes sense for rational numbers just as well as rational functions even if nobody's ever told you this!), $M$ is the direct sum of its Sylow $p$-subgroups$$M_p \cong \bigoplus_i \mathbb{Z}(p^{\infty})$$
where $\mathbb{Z}(p^{\infty})$ is the Prüfer $p$-group (I really don't like this notation but it appears to be standard).
Proposition: Let $H$ be a subgroup of $M$, and let $H_p$ be its image in $M_p$ (regarded as a subgroup of $M$). Then $H = \bigoplus_p H_p$.
Proof. Let $h \in H$ be an element. By multiplying $h$ by appropriate powers of every prime not equal to $p$ which occurs in the denominators of the components of $h$, we can find an element which has the same image as $h$ in $M_p$ but which has zero image in $M_q, q \neq p$. Applying this algorithm to preimages in $H$ of every element in $H_p$, we conclude that every $H_p$ is a subgroup of $H$, and the conclusion follows.
So our problem reduces to the problem of classifying the subgroups of $M_p$. First we make the following observation about subgroups of $\mathbb{Z}(p^{\infty})$. Every element of $\mathbb{Z}(p^{\infty})$ has the form $\frac{a}{p^n}$ where $(a, p) = 1$ and $n$ is unique; call $n$ the
valuation $\nu_p \left( \frac{a}{p^n} \right)$. Every element of valuation $n$ generates the subgroup$$P_n = \left\{ \frac{a}{p^n} : a \in \mathbb{Z} \right\} \cong \mathbb{Z}/p^n\mathbb{Z}$$
of $\mathbb{Z}(p^{\infty})$, and these subgroups are totally ordered by inclusion. It follows that in fact they are the only subgroups of $\mathbb{Z}(p^{\infty})$.
Edit: Hum! So according to the comments this question is hopeless in full generality. Well, at least the constructions so far provide a large class of examples (obtained by taking direct sums of subgroups of $\mathbb{Z}(p^{\infty})$). |
The question shows you already found the correct numeric result.I would just suggest working on the presentation so that people readingyour calculations do not misunderstand them. (Eventually you may alsobe solving problems that are complicated enough that even
you willnot be able to remember what you're doing unless you explain it carefully.)
I would assign a different number or name to the time of each event. It often helps to choose a convenient event and measure all timesrelative to that event. So if the event is when the first drop starts falling,the first drop starts falling at $0$ seconds and the second dropstarts falling at $1$ second.When you are first setting up the problem you have not yet calculatedwhen the first drop hits the ground, so give that a name:you might say it happens at $t_1$ seconds. And you can say the second drop hits the ground at $t_2$ seconds.
So the first drop falls for a total of $t_1$ seconds, which givesa vertical distance of$$ 0=1000−5t_1^2.$$From this is follows that $t_1 = 10\sqrt2$; this is really the same thingyou already did, just giving a unique name to this time so it's clear which time we're talking about.
At time $t_1$, when the first drop hits the ground, the second drop hasonly been falling for $t_1 - 1$ seconds. So its height above the groundat that instant is $$1000 - 5(t_1 - 1)^2 = 1000 - 5(10\sqrt2 - 1)^2 \approx 136.$$
And of course at $t_2$ seconds, when the second drop hits the ground,the drop has been falling for $t_2 - 1$ seconds, and so$$ 0=1000−5(t_2 - 1)^2.$$From this we can figure out that $t_2 = 10\sqrt2 + 1$;personally, however, I would just observe that since the two drops travelthe same distance with the same initial velocity and acceleration,they take the same elapsed time, so $t_2 - 1 = t_1 = 10\sqrt2$and we can immediately solve this to find that $t_2 = 10\sqrt2 + 1 = t_1 + 1$without dealing with a quadratic.
If we wanted a more long-winded solution I suppose we could say thatif the second drop has been falling for $\Delta t_{21}$ seconds when thefirst drop hits the ground, then $t_1 = \Delta t_{21} + 1$(because the first drop started one second earlier)and therefore $$ 0=1000−5(\Delta t_{21} + 1)^2.$$That's the equation $0=1000−5(t + 1)^2$ that you wrote,but writing $\Delta t_{21}$ instead of $t$ is a reminder that thisequation was true with regard to a very specific amount of elapsed time.we can solve that equation to find out that $\Delta t_{21} = 10\sqrt2 - 1$, and then find the height of thesecond drop above the ground at time $t_1$ by evaluating$1000 - 5(\Delta t_{21} - 1)^2,$but I'm already having more trouble remembering which symbol means whatthan I did in the previous solution, so I think maybe this isn't the easiestway to go. |
This question already has an answer here:
how to prove :
an odd perfect number cannot be a prime number or a product of two prime numbers or power of prime number.
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This question already has an answer here:
how to prove :
an odd perfect number cannot be a prime number or a product of two prime numbers or power of prime number.
Let $\sigma$ denote the sum-of-divisors function. A positive integer $N$ is said to be perfect if $\frac{\sigma(N)}{N}=2$.
Let $p^\alpha$ be an odd prime power and suppose that $N=p^\alpha$ is an odd perfect number. Since $\sigma(p^\alpha)=1+p+p^2+\dots+p^\alpha=\frac{p^{\alpha+1}-1}{p-1}$, we can observe that
$$\sigma(p^\alpha) = \frac{p^{\alpha+1}-1}{p-1} < \frac{p^\alpha p}{p-1} \implies \frac{\sigma(p^\alpha)}{p^\alpha} < \frac{p}{p-1} \le \frac{3}{2} < 2$$
since $p \ge 3$, so $N$ must be deficient.
Now, let $p^\alpha$ and $q^\beta$ be two odd prime powers with $p<q$ and suppose that $N=p^\alpha q^\beta$ is an odd perfect number. Noting that $\sigma$ is multiplicative, we obtain
$$\sigma(p^\alpha q^\beta) = \frac{p^{\alpha+1}-1}{p-1} \cdot \frac{q^{\beta+1}-1}{q-1} < \frac{p^\alpha p}{p-1} \cdot \frac{q^\beta q}{q-1} \implies \frac{\sigma(p^\alpha q^\beta)}{p^\alpha q^\beta} < \frac{p}{p-1} \cdot \frac{q}{q-1} \le \frac{3}{2} \cdot \frac{5}{4} = \frac{15}{8} < 2$$
since $p \ge 3$ and $q \ge 5$, so again $N$ must be deficient.
Mostly you push symbols around with the intent of getting a nice factorization:
$$(p+1)(q+1)=2pq \iff 2=(p-1)(q-1),$$
$$\frac{p^{a+1}-1}{p-1}=2p^a \iff p^{a+1}-2p+1=0 \iff p^2-2p+1=(p-1)^2\le0.$$
We also assume $a\ge1$ so $p^{a+1}\ge p^2$ for the prime-power case (since the $p^1$ case is easy).
You should be familiar with $\sigma_1$'s explicit formula $\displaystyle\sigma_1\left(\prod_p p^{v_p}\right)=\prod_p\frac{p^{v_p+1}-1}{p-1}$. |
(Sorry was asleep at that time but forgot to log out, hence the apparent lack of response)
Yes you can (since $k=\frac{2\pi}{\lambda}$). To convert from path difference to phase difference, divide by k, see this PSE for details http://physics.stackexchange.com/questions/75882/what-is-the-difference-between-phase-difference-and-path-difference
Yes you can (since $k=\frac{2\pi}{\lambda}$). To convert from path difference to phase difference, divide by k, see this PSE for details
http://physics.stackexchange.com/questions/75882/what-is-the-difference-between-phase-difference-and-path-difference |
Loss Layers¶ class
HingeLossLayer¶
Compute the hinge loss for binary classification problems:\[\frac{1}{N}\sum_{i=1}^N \max(1 - \mathbf{y}_i \cdot \hat{\mathbf{y}}_i, 0)\]
Here \(N\) is the batch-size, \(\mathbf{y}_i \in \{-1,1\}\) is the ground-truth label of the \(i\)-th sample, and \(\hat{\mathbf{y}}_i\) is the corresponding prediction.
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
Should be a vector containing two symbols. The first one specifies the name for the prediction \(\hat{\mathbf{y}}\), and the second one specifies the name for the ground-truth \(\mathbf{y}\).
class
MultinomialLogisticLossLayer¶
The multinomial logistic loss is defined as \(\ell = -w_g\log(x_g)\), where \(x_1,\ldots,x_C\) are probabilities for each of the \(C\) classes conditioned on the input data, \(g\) is the corresponding ground-truth category, and \(w_g\) is the
weightfor the \(g\)-th class (default 1, see bellow).
If the conditional probability blob is of the shape
(dim1, dim2, ..., dim_channel, ..., dimN), then the ground-truth blob should be of the shape
(dim1, dim2, ..., 1, ..., dimN). Here
dim_channel, historically called the “channel” dimension, is the user specified tensor dimension to compute loss on. This general case allows to produce multiple labels for each sample. For the typical case where only one (multi-class) label is produced for one sample, the conditional probability blob is the shape
(dim_channel, dim_num)and the ground-truth blob should be of the shape
(1, dim_num).
The ground-truth should be a
zero-basedindex in the range of \(0,\ldots,C-1\).
Should be a vector containing two symbols. The first one specifies the name for the conditional probability input blob, and the second one specifies the name for the ground-truth input blob.
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
weights¶
This can be used to specify weights for different classes. The following values are allowed
Empty array (default). This means each category should be equally weighted. A 1D vector of length
channels. This defines weights for each category.
An (N-1)D tensor of the shape of a data point. In other words, the sameshape as the prediction except that the last mini-batch dimension isremoved. This is equivalent to the above case if the prediction is a 2Dtensor of the shape
channels-by-
mini-batch.
An ND tensor of the same shape as the prediction blob. This allows us tofully specify different weights for different data points ina mini-batch. See
SoftlabelSoftmaxLossLayer.
dim¶
Default
-2(penultimate). Specify the dimension to operate on.
normalize¶
Indicating how weights should be normalized if given. The following values are allowed
:local(default): Normalize the weights locally at each location (w,h), across the channels.
:global: Normalize the weights globally.
:no: Do not normalize the weights.
The weights normalization are done in a way that you get the same objective function when specifying
equal weightsfor each class as when you do not specify any weights. In other words, the total sum of the weights are scaled to be equal to weights x height x channels. If you specify
:no, it is your responsibility to properly normalize the weights.
class
SoftlabelSoftmaxLossLayer¶
Like the
SoftmaxLossLayer, except that this deals with
soft labels. For multiclass classification with \(K\) categories, we call an integer value \(y\in\{0,\ldots,K-1\}\) a hard label. In contrast, a soft label is a vector on the \(K\)-dimensional simplex. In other words, a soft label specifies a probability distribution over all the \(K\) categories, while a hard label is a special case where all the probability masses concentrates on one single category. In this case, this loss is basically computing the KL-divergence D(p||q), where p is the ground-truth softlabel, and q is the predicted distribution.
dim¶
Default
-2(penultimate). Specify the dimension to operate on.
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
Should be a vector containing two symbols. The first one specifies the name for the conditional probability input blob, and the second one specifies the name for the ground-truth (soft labels) input blob.
class
SoftmaxLossLayer¶
This is essentially a combination of
MultinomialLogisticLossLayerand
SoftmaxLayer. The given predictions \(x_1,\ldots,x_C\) for the \(C\) classes are transformed with a softmax function\[\sigma(x_1,\ldots,x_C) = (\sigma_1,\ldots,\sigma_C) = \left(\frac{e^{x_1}}{\sum_j e^{x_j}},\ldots,\frac{e^{x_C}}{\sum_je^{x_j}}\right)\]
which essentially turn the predictions into non-negative values with exponential function and then re-normalize to make them look like probabilties. Then the transformed values are used to compute the multinomial logsitic loss as\[\ell = -w_g \log(\sigma_g)\]
Here \(g\) is the ground-truth label, and \(w_g\) is the weight for the \(g\)-th category. See the document of
MultinomialLogisticLossLayerfor more details on what the weights mean and how to specify them.
The shapes of the inputs are the same as for the
MultinomialLogisticLossLayer: the multi-class predictions are assumed to be along the channel dimension.
The reason we provide a combined softmax loss layer instead of using one softmax layer and one multinomial logistic layer is that the combined layer produces the back-propagation error in a more numerically robust way.\[\frac{\partial \ell}{\partial x_i} = w_g\left(\frac{e^{x_i}}{\sum_j e^{x_j}} - \delta_{ig}\right) = w_g\left(\sigma_i - \delta_{ig}\right)\]
Here \(\delta_{ig}\) is 1 if \(i=g\), and 0 otherwise.
Should be a vector containing two symbols. The first one specifies the name for the conditional probability input blob, and the second one specifies the name for the ground-truth input blob.
dim¶
Default
-2(penultimate). Specify the dimension to operate on. For a 4D vision tensor blob, the default value (penultimate) translates to the 3rd tensor dimension, usually called the “channel” dimension.
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
class
SquareLossLayer¶
Compute the square loss for real-valued regression problems:\[\frac{1}{2N}\sum_{i=1}^N \|\mathbf{y}_i - \hat{\mathbf{y}}_i\|^2\]
Here \(N\) is the batch-size, \(\mathbf{y}_i\) is the real-valued (vector or scalar) ground-truth label of the \(i\)-th sample, and \(\hat{\mathbf{y}}_i\) is the corresponding prediction.
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
Should be a vector containing two symbols. The first one specifies the name for the prediction \(\hat{\mathbf{y}}\), and the second one specifies the name for the ground-truth \(\mathbf{y}\).
class
BinaryCrossEntropyLossLayer¶
A simpler alternative to
MultinomialLogisticLossLayerfor the special case of binary classification.\[-\frac{1}{N}\sum_{i=1}^N \log(p_i)y_i + \log(1-p_i)(1-y_i)\]
Here \(N\) is the batch-size, \(\mathbf{y}_i\) is the ground-truth label of the \(i\)-th sample, and :math:
p_iis the corresponding prediction.
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
Should be a vector containing two symbols. The first one specifies the name for the prediction \(\hat{\mathbf{y}}\), and the second one specifies the name for the binary ground-truth labels \(\mathbf{p}\).
class
GaussianKLLossLayer¶
Given two inputs
muand sigmaof the same size representing the means and standard deviations of a diagonal multivariate Gaussian distribution, the loss is the Kullback-Leibler divergence from that to the standard Gaussian of the same dimension.
Used in variational autoencoders, as in Kingma & Welling 2013, as a form of regularization.\[D_{KL}(\mathcal{N}(\mathbf{\mu}, \mathrm{diag}(\mathbf{\sigma})) \Vert \mathcal{N}(\mathbf{0}, \mathbf{I}) ) = -\frac{1}{2}\left(\sum_{i=1}^N (\mu_i^2 + \sigma_i^2 - 2\log\sigma_i) - N\right)\]
weight¶
Default
1.0. Weight of this loss function. Could be useful when combining multiple loss functions in a network.
Should be a vector containing two symbols. The first one specifies the name for the mean vector \(\mathbf{\mu}\), and the second one the vector of standard deviations \(\mathbf{\sigma}\). |
Today i ran onto this simple problem, which seemed to be interesting to me. Given the illustration bellow, the problem is states as:
Two identical cylinders roll in between two identical planks. If the velocity of each cylinder is $\vec{v}$ and the velocity of the bottom plank is $\vec{u}$ ($|\vec{v}| > |\vec{u}|$), find the velocity the man standing on the top plank needs to obtain with respect to it[the top plank], for which he covers distance $s$ in $t$ seconds in the stationary reference frame ( observer's frame ). Will the velocity vector point right or left? ( $\vec{v} $ and $\vec{u} $ are both given with respect to the stationary reference frame ). All friction is to be neglected and the cylinders are assumed to be rotating without slipping.
Interestingly, a naive approach would be to immediately write $$\vec{v_m} = \frac{\vec{s}}{t} - \vec{v}$$ since one might think that the velocity of the upper plank with respect to the stationary observer is$\vec{v}$.
However, after inspecting the problem more thoroughly, I've come to the following conclusion:
The velocity of the upper plank comes solely from the rotation of the cylinders. Angular velocity of each of the cylinders is$\ $ $\omega = \frac{v-u}{R}$
From here it seems that the velocity of the upper plank with respect to the stationary reference frame might be $$\vec{u}' =\vec{v} - \vec{u}$$
And the sought velocity is: $$\vec{v_m} = \frac{\vec{s}}{t} - \vec{u'}$$
Which seems to be ok intuitvely ( the greater the magnitude of $\vec{v}$, the smaller the relative speed of the man needs to be ). However, it's still bugging me and making me believe the motion of the bottom plank affects the motion of the upper plank ( other than in the elaborated way ). Is there something wrong with this reasoning?
Note: Although this is not a school problem, I'll tag it as homework because this seems the kind of a problem that would appear in homework problems. EDIT: By "stationary reference frame" I mean the frame on which the observer is standing ( e.g. Earth ), observing motion of the system" |
The title is a little ambiguous, but I didn't know how else to put it. What I'm trying to do is solve a system of equations
system1 = {-I*ω*a1 == -I*ω1*a1 - I*J12*a2 - I*J13*a3 - κ1[ω]/2*a1 - γ1[ω]/2*a1 + Sqrt[κ1[ω]]*ain, -I*ω*a2 == -I*ω2*a2 - I*J12*a1 - I*J23*a3 - γ2/2*a2, -I*ω*a3 == -I*ω3*a3 - I*J13*a1 - I*J23*a2 - γ3/2*a3};
Which I then solve for a1, a2 and a3:
s = Solve[system1, {a1, a2, a3}];
And then, finally, I am interested in a function $R(\omega)$ which is given by
I*Sqrt[κ1]*(a1 /. s)/ain - 1
If I simply use all the above and evaluate that last term, it works. I get a (rather ugly) expression of the type
$\left\{-1+\frac{2 i \text{$\kappa $1} \left(4 \text{J23}^2+(\text{$\gamma $2}-2 i (\omega -\text{$\omega $2})) (\text{$\gamma $3}-2 i (\omega -\text{$\omega $3}))\right)}{4 \text{J12}^2 (\text{$\gamma $3}-2 i (\omega -\text{$\omega $3}))-16 i \text{J12} \text{J13} \text{J23}+4 \text{J13}^2 (\text{$\gamma $2}-2 i (\omega -\text{$\omega $2}))+(\text{$\gamma $1}+\text{$\kappa $1}-2 i \omega +2 i \text{$\omega $1}) \left(4 \text{J23}^2+(\text{$\gamma $2}-2 i \omega +2 i \text{$\omega $2}) (\text{$\gamma $3}-2 i \omega +2 i \text{$\omega $3})\right)}\right\}$
(I apologize for the format, I don't know how to give variables and parameters subscripts in mathematica so for example $J_{12}$ comes out as $J12$)
However, I want to define this as a function, and I am not sure if
R[ω_] := I*Sqrt[κ1]*(a1 /. s)/ain - 1
does the trick.
Now, the real problem comes from the fact that I don't want just this analytic expression, I want to plot $R(\omega)$. The real and imaginary parts, as well as the argument. For this I need values for my constants of course, which is fine. But the problems start appearing when I consider for example $\kappa_1$, which is not a constant but a function. To be clear, every term in these equations, apart from $a_1$, $a_2$, $a_3$,$\kappa_1$ and $\gamma_1$, are constants. These other five depend on the only variable in the equations, $\omega$.
The main issue is that I don't know how I should do this. I can define for example $\kappa_1$ the way I want to without any problems using
κ1[ω_] := 1/(ω*Zc*(C1 + CJ1 + CJ3));
This gives me a function $\kappa_1$ that I can evaluate. But how do I put this into my system of equations, system1? Do I have to replace
κ1 with κ1[ω]?
If I do that, then I run into trouble when solving the system and defining an equation. If anyone coudld help, I'd be very grateful. For completeness, below is the full list of parameters:
C1 = 190*10^-15;C2 = 240*10^-15;C3 = 270*10^-15;L1 = 1.74*10^ -9;L2 = L1;L3 = L1;CJ1 = 30*10^-15;CJ2 = CJ1;CJ3 = CJ1;R1 = 2*10^5;R2 = R1;R3 = R1;Zc = 50;Cmatrix = {{C1 + Cκ, CJ1, CJ3}, {CJ1, C2, CJ2}, {CJ3, CJ2, C3}};Lmatrix = {{L1, 0, 0}, {0, L1, 0}, {0, 0, L1}};Linv = Inverse[Lmatrix];Cinv = Inverse[Cmatrix];ω1 = Sqrt[Cinv[[1, 1]]*Linv[[1, 1]]];ω2 = Sqrt[Cinv[[2, 2]]*Linv[[2, 2]]];ω3 = Sqrt[Cinv[[3, 3]]*Linv[[3, 3]]];J12 = 0.5*Cinv[[1, 2]]/(Sqrt[Cinv[[1, 1]]]*Cinv[[2, 2]])*Sqrt[ω1*ω2];J13 = 0.5*Cinv[[1, 3]]/(Sqrt[Cinv[[1, 1]]]*Cinv[[3, 3]])*Sqrt[ω1*ω3];J23 = 0.5*Cinv[[2, 3]]/(Sqrt[Cinv[[2, 2]]]*Cinv[[3, 3]])*Sqrt[ω3*ω2];Cκ = 80*10^-15;γ2 = 1/(R2*(C2 + CJ1 + CJ2));γ3 = 1/(R3*(C3 + CJ2 + CJ3));κtilde = 1/(Zc*Cκ);BigK[ω_] := 1 + (κtilde/ω)^2;Cκtilde[ω_] := Cκ*(1 - 1/BigK[ω]);κ1[ω_] := 1/(BigK[ω]*Zc*(C1 + CJ1 + Cκtilde[ω] + CJ3));γ1[ω_] := 1/(R1*(C1 + CJ1 + CJ3 + Cκtilde[ω])); |
Inaccessible
Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy (although there are some weaker large cardinal notions, such as universe cardinals).
If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC, but this is not an equivalence, since the weaker notion of universe cardinal also have this feature, and are not all regular when they exist. Every inaccessible cardinal $\kappa$ is a beth fixed point, and consequently $V_\kappa=H_\kappa$. (Zermelo) The models of second-order ZFC are precisely the models $\langle V_\kappa,\in\rangle$ for an inaccessible cardinal $\kappa$. The uncountable Grothedieck universes are precisely the sets of the form $V_\kappa$ for an inaccessible cardinal $\kappa$. Weakly inaccessible
A cardinal $\kappa$ is
weakly inaccessible if it is an uncountable regular limit cardinal. Under the GCH, this is equivalent to inaccessibility, since under GCH every limit cardinal is a strong limit cardinal. So the difference between weak and strong inaccessibility only arises when GCH fails badly. Every inaccessible cardinal is weakly inaccessible, but forcing arguments show that any inaccessible cardinal can become a non-inaccessible weakly inaccessible cardinal in a forcing extension, such as after adding an enormous number of Cohen reals (this forcing is c.c.c. and hence preserves all cardinals and cofinalities and hence also all regular limit cardinals). Meanwhile, every weakly inaccessible cardinal is fully inaccessible in any inner model of GCH, since it will remain a regular limit cardinal in that model and hence also be a strong limit there. In particular, every weakly inaccessible cardinal is inaccessible in the constructible universe $L$. Consequently, although the two large cardinal notions are not provably equivalent, they are equiconsistent. Grothendieck universe
The concept of Grothendieck universes arose in category theory out of the desire to create a hierarchy of notions of smallness, so that one may form such categories as the category of all small groups, or small rings or small categories, without running into the difficulties of Russell's paradox.
A
Grothendieck universe is a transitive set $W$ that is closed under pairing, power set and unions. That is, (transitivity) If $b\in a\in W$, then $b\in W$. (pairing) If $a,b\in W$, then $\{a,b\}\in W$. (power set) If $a\in W$, then $P(a)\in W$. (union) If $a\in W$, then $\cup a\in W$. Universe axiom
The
Grothendieck universe axiom is the assertion that every set is an element of a Grothendieck universe. This is equivalent to the assertion that the inaccessible cardinals form a proper class. |
Remember that we have supposed two hypothesis $latex {\left\{ f_{0},f_{1}\right\} }&fg=000000$ elements of $latex {\mathcal{F}}&fg=000000$. Denote $latex {P_{0}}&fg=000000$ and $latex {P_{1}}&fg=000000$ two probability measures under $latex {(\mathcal{X},\mathcal{A})}&fg=000000$ under $latex {f_{0}}&fg=000000$ and $latex {f_{1}}&fg=000000$ respectively. If $latex {P_{0}}&fg=000000$ and $latex {P_{1}}&fg=000000$ are very “close”, then it is hard to distinguish $latex {f_{0}}&fg=000000$ and $latex {f_{1}}&fg=000000$ and […]
Photos of Johann Radon and Otto Nikodym. Sources: Apprendre les Mathématiques and Wikipedia. Consider the simplest case, $latex {M=1}&fg=000000$ with two hypothesis $latex {\{f_{1},f_{2}\}}&fg=000000$ belonging to $latex {\mathcal{F}}&fg=000000$. According to the last post, we need only to find lower bounds for the minimax probability of error $latex {p_{e,1}}&fg=000000$. Today, we will find a bound using […]
In the last publication, we defined a minimax lower bound as $latex \displaystyle \mathcal{R}^{*}\geq cs_{n} &fg=000000$ where $latex {\mathcal{R}^{*}\triangleq\inf_{\hat{f}}\sup_{f\in\mathcal{F}}\mathbb E\left[d^{2}(\hat{f}_{n},f)\right]}&fg=000000$ and $latex {s_{n}\rightarrow0}&fg=000000$. The big issue with this definition is to take the supremum over a massive set $latex {\mathcal{F}}&fg=000000$ and then the infimum over all the possible estimators of $latex {f}&fg=000000$.
In my most recent research, I’m working on finding “Minimax Lower Bounds” for some kind of estimators. Therefore, to learn a little more and get my ideas clear, I’ll going to start a series of posts about the topic. I pretend to make some review in the general method and introduce some bounds depending on […]
This week I am going to present three applications of the Delta method theorem. The first is a direct one and it is about the behavior in distribution of the sample variance. The second one is an hypothesis test in the variance when the sample is normal. Finally, the third is an interesting application in […] |
I would like to start this blog with some basic ideas about density estimation and nonparametric regression.
The study of the probability density function (pdf) is called nonparametric estimation. This kind of estimation can serve as a block building in nonparametric regression.
The typical regression problem is setting as follows. Assume that we have a set of explanatory variables $X_1,\ldots,X_d$ and an explained variable $Y$ related in the following way:
\begin{equation} Y=X^{\top}\beta+\varepsilon\label{eq:linear_model} \end{equation} where $\varepsilon$ is independent of $X$ and $\E[\varepsilon]=0$.
Taking expectations on both sides, we can see model \eqref{eq:linear_model} as,
\begin{equation} \label{eq:cond_model} \E[X\vert Y]=X_{1}\beta_{1}+\cdots+X_{d}\beta_{d}=\mathbf{X}^{\top}\beta. \end{equation}
where $\E[X\vert Y]$ is the conditional expectation of $Y$ given $X$.
Unfortunately, for many real problems, equation \eqref{eq:cond_model} is not sufficient.
To tackle this situation, we can transform it into,
\begin{equation} \E[X\vert Y]=m(\mathbf{X}), \end{equation} Here $m(\mathbf{X})$ is the true, unknown regression function.
Just to get the things in perspective. Suppose that $\mathbf{X}=(X_1,X_2)$ and the real model for the conditional expectation is
\begin{equation} \label{eq:example} \E[X\vert Y]=\beta_1 X_1 + \beta_2 X_2 +\beta_3 X_2^2. \end{equation}
Now given a data sample, you have to estimate $\E[X\vert Y]$ as accurately as possible in
one single trial. That means, you can not change the model if the data does not fit well.
In parametric models, the task is relative easy: Estimate the $\beta$’s producing the minimum error given the formula \eqref{eq:example}. The advantage is that you have to deal only with a finite number of parameters give a parametric structure of your model.
However, if you relax the latter condition, and the only assumption is that there exist a function $m(\cdot)$ (which could be differentiable).
The question here: How could we perform the regression under this new rules? This type of regression is called nonparametric and we will return to it later.
To start, we will build the nonparametric density as a preamble in the next posts. |
Difference between revisions of "Superstrong"
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Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for the [[axiom of determinacy]]. However, Shelah had then discovered that [[Shelah]] cardinals were a weaker bound that still sufficed to imply the consistency strength of (ZF+)AD. After this, it was found that the existence of
+
Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for the [[axiom of determinacy]]. However, Shelah had then discovered that [[Shelah]] cardinals were a weaker bound that still sufficed to imply the consistency strength of (ZF+)AD. After this, it was found that the existence of [[Woodin]] cardinals was equiconsistent to AD. Woodin-ness is a significant weakening of superstrongness.
''Most results in this article can be found in <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''
''Most results in this article can be found in <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''
Revision as of 11:21, 14 October 2018
Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for the axiom of determinacy. However, Shelah had then discovered that Shelah cardinals were a weaker bound that still sufficed to imply the consistency strength of $\text{(ZF+)AD}$. After this, it was found that the existence of infinitely many Woodin cardinals was equiconsistent to $\text{AD}$. Woodin-ness is a significant weakening of superstrongness.
Most results in this article can be found in [1] unless indicated otherwise. Contents Definitions
There are, like most critical point variations on measurable cardinals, multiple equivalent definitions of superstrongness. In particular, there is an elementary embedding definition and an extender definition.
Elementary Embedding Definition
A cardinal $\kappa$ is
$n$-superstrong (or $n$-fold superstrong when referring to the $n$-fold variants) iff it is the critical point of some elementary embedding $j:V\rightarrow M$ such that $M$ is a transitive class and $V_{j^n(\kappa)}\subset M$ (in this case, $j^{n+1}(\kappa):=j(j^n(\kappa))$ and $j^0(\kappa):=\kappa$).
A cardinal is
superstrong iff it is $1$-superstrong.
The definition quite clearly shows that $\kappa$ is $j^n(\kappa)$-strong. However, the least superstrong cardinal is never strong.
Extender Definition
A cardinal $\kappa$ is
$n$-superstrong (or $n$-fold superstrong) iff there is a $(\kappa,\beta)$-extender $\mathcal{E}$ for a $\beta>\kappa$ with $V_{j^n_{\mathcal{E}}(\kappa)}\subseteq$ $Ult_{\mathcal{E}}(V)$ (where $j_{\mathcal{E}}$ is the canonical ultrapower embedding from $V$ into $Ult_{\mathcal{E}}(V)$).
A cardinal is
superstrong iff it is $1$-superstrong. Relation to other large cardinal notions measurable = $0$-superstrong = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge $n$-superstrong $n$-fold supercompact $(n+1)$-fold strong, $n$-fold extendible $(n+1)$-fold Woodin, $n$-fold Vopěnka $(n+1)$-fold Shelah almost $n$-huge super almost $n$-huge $n$-huge super $n$-huge $(n+1)$-superstrong
Let $M$ be a transitive class $M$ such that there exists an elementary embedding $j:V\to M$ with $V_{j(\kappa)}\subseteq M$, and let $\kappa$ be its superstrong critical point. While $j(\kappa)$ need not be an inaccessible cardinal in $V$, it is always worldly and the rank model $V_{j(\kappa)}$ satisfies $\text{ZFC+}$"$\kappa$ is strong" (although $\kappa$ may not be strong in $V$).
Superstrong cardinals have strong upward reflection properties, in particular there are many measurable cardinals
above a superstrong cardinal. Every $n$-huge cardinal is $n$-superstrong, and so $n$-huge cardinals also have strong reflection properties. Remark however that if $\kappa$ is strong or supercompact, then it is consistent that there is no inaccessible cardinals larger than $\kappa$: this is because if $\lambda>\kappa$ is inaccessible, then $V_\lambda$ satisfies $\kappa$'s strongness/supercompactness. Thus it is clear that supercompact cardinals need not be superstrong, even though they have higher consistency strength. In fact, because of the downward reflection properties of strong/supercompact cardinals, if there is a superstrong above a strong/supercompact $\kappa$, then there are $\kappa$-many superstrong cardinals below $\kappa$; same with hugeness instead of superstrongness. In particular, the least superstrong is strictly smaller than the least strong (and thus smaller than the least supercompact).
Every $1$-extendible cardinal is superstrong and has a normal measure containing all of the superstrongs less than said $1$-extendible. This means that the set of all superstronges less than it is stationary. Similarly, every cardinal $\kappa$ which is $2^\kappa$-supercompact is larger than the least superstrong cardinal and has a normal measure containing all of the superstrongs less than it.
Every superstrong cardinal is Woodin and has a normal measure containing all of the Woodin cardinals less than it. Thus the set of all Woodin cardinals below it is stationary, and so is the set of all measurables smaller than it. Superstrongness is consistency-wise stronger than Hyper-Woodinness.
Letting $\kappa$ be superstrong, $\kappa$ can be forced to $\aleph_2$ with an $\omega$-distributive, $\kappa$-c.c. notion of forcing, and in this forcing extension there is a normal $\omega_2$-saturated ideal on $\omega_1$. [3]
Superstrongness is not Laver indestructible. [4]
References Kanamori, Akihiro. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex The higher infinite. Kentaro, Sato. Double helix in large large cardinals and iteration ofelementary embeddings., 2007. www bibtex Jech, Thomas J. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex Set Theory. Bagaria, Joan and Hamkins, Joel David and Tsaprounis, Konstantinos and Usuba, Toshimichi. Superstrong and other large cardinals are never Laver indestructible.Archive for Mathematical Logic 55(1-2):19--35, 2013. www arχiv DOI bibtex |
Transform to $y_i=\sin^2 x_i$. Then $\sum_iy_i=1$, and with $\sin x_i=\sqrt{y_i}$ and $\cos x_i=\sqrt{1-y_i}$ the inequality becomes
$$\sum_i\left(\sqrt{1-y_i}-3\sqrt{y_i}\right)\ge0\;.$$
The left-hand side is $10$ times the average value of $f(y)=\sqrt{1-y}-3\sqrt y$ at the $y_i$. The graph of $f$ has an inflection point at $y=1/\left(1+3^{-2/3}\right)\approx0.675$ and is convex to its left. Either none or one of the $y_i$ can be to its right. If none are, the average value of $f$ is greater or equal to the value at the average, $f(1/10)=0$. If one is, say, $y_i$, then we can bound the average value of the remaining ones by the value at their average, so in this case
$$\sum_i\left(\sqrt{1-y_i}-3\sqrt{y_i}\right)\ge\sqrt{1-y_1}-3\sqrt{y_1}+9\sqrt{1-(1-y_1)/9}-27\sqrt{(1-y_1)/9}\;.$$
This is non-negative with a single root at $1/10$, so the inequality holds in both cases. |
I've never actually solved a problem like this before, but it looks pretty trivial so I'll give it a shot. My apologies if this is wrong.
i. Let $\Phi$ denote the formulae of propositional logic that can be formed from the connectives in $C$ and the variables in $P$. More precisely, lets us defined that $\Phi$ is the smallest collection of formulae such that
If $X \in P$, then $X \in \Phi$ If $\phi,\psi \in \Phi$ then $\phi \wedge \psi \in \Phi$. If $\phi,\psi \in \Phi$ then $\phi \vee \psi \in \Phi$.
Furthermore, let $I$ denote a valuation such that $I \models X$ for every $X \in P$, and let $\Phi'$ denote the set of all formulae $\phi$ of propositional logic such that $I \models \phi$. The problem becomes:
Show that $\Phi \subseteq \Phi'$.
Now for the important realization:
Since $\Phi$ is the least set satisfying 1,2 and 3, thus it suffices to show that $\Phi'$ also satisfies 1,2 and 3.
That's it, the rest is easy. We continue:
In other words, it suffices to show the following.
If $X \in P$, then $I \models X$. If $I \models \phi,\psi$, then $I \models \phi \wedge \psi$. If $I \models \phi,\psi$, then $I \models \phi \vee \psi$.
But this is trivial.
True, because we assumed that $I \models X$ for every $X \in P.$ True, by the definition of $\wedge$. True, by the definition of $\vee$.
ii. I'm not sure what the definition of "adequate" is, but I'm guessing this is even easier. If it just means: "can be used to express all functions of the form $\mathbb{B}^n \rightarrow \mathbb{B},$" well just take any function returning "FALSE" whenever all arguments are true and you'll have your counterexample. |
As the title says:
Suppose $a,b,c > 0$ are constants. Consider $ay'' + by' + cy = 0$. Suppose that $y(x)$ is a solution. Prove that $\lim_{x\to\infty}y(x) = 0$. This is problem 34 from p.1160 in section 17.1 of Stewart Single Variable Calculus, 8th edition (2015).
The characteristic equation is of course $ar^2 + br + c = 0$, for which $r = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}$.
For $b^2 - 4ac > 0$ and $r_1 \neq r_2$ then $y(x) = C_1e^{r_1x} + C_2e^{r_2x}$ for some constants $C_1, C_2$. For $b^2 - 4ac = 0$ and $r = r_1 = r_2$ then $y(x) = C_1e^{rx} + C_2xe^{rx}$ for some constants $C_1, C_2$. For $b^2 - 4ac < 0$ and $r = \alpha + \beta i$ then $y(x) = e^{\alpha x}(C_1cos(\beta x) + C_2isin(\beta x))$ for some constants $C_1, C_2$.
I can’t see any obvious reason that the limit of any of these is $0$. How do I approach this? |
We are going to introduce the histogram as a simple nonparametric density estimator. I will divide this presentation in several posts for simplicity reasons.
Select and origin $latex {x_0}&fg=000000$ and divide the real line into bins of binwidth
$latex \displaystyle B_j = \left[x_0 – (j-1)h, x_0 + (j-1)h\right) \quad j\in {\mathbb Z}. &fg=000000$
Let $latex {n_j}&fg=000000$ be how many observations falls into $latex {B_j}&fg=000000$. Let $latex {\hat{f}_j=\frac{n_j}{n}}&fg=000000$ and let $latex {f_j=\int_{B_j} f(u)du}&fg=000000$. Finally plot the histogram erecting a bar over each bin with height $latex {f_j}&fg=000000$ and width $latex {h}&fg=000000$
More formally, the histogram is given by
$latex \displaystyle \hat{f}_h(x)=\frac{1}{nh}\sum_{i=1}^n \sum_j I(X_i\in B_j) I(x\in B_j) &fg=000000$
where $latex {I}&fg=000000$ is the indicator function.
If $latex {m_j}&fg=000000$ is the center of $latex {B_j}&fg=000000$, it is clear that the histogram assigns the same estimate $latex {\hat{f}_n(m_j)}&fg=000000$ for each $latex {x\in \left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right) }&fg=000000$. This is rather restrictive but later we will see better alternatives.
Derivation
The probability of that an observation of $latex {X}&fg=000000$ will fall into the bin $latex {x\in \left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right) }&fg=000000$ is given by
$latex \displaystyle \mathop{\mathbb P}\left( X\in \left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right) \right) = \int_{m_j-\frac{h}{2}}^{m_j+\frac{h}{2}} f(u)du, &fg=000000$
which is just the area under $latex {f}&fg=000000$ in the interval $latex {\left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right)}&fg=000000$. We can approximate this area by the area of a rectangle with height $latex {f(m_j)}&fg=000000$ and width $latex {h}&fg=000000$,
$latex \displaystyle \int_{m_j-\frac{h}{2}}^{m_j+\frac{h}{2}} f(u)du \approx f(m_j)h \ \ \ \ \ (1)&fg=000000$
A natural estimate for this probability is the relative frequency of observations in this interval. That means:
$latex \displaystyle \mathop{\mathbb P}\left( X\in x\in \left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right) \right) \approx \frac{1}{n} \sharp \left\lbrace X_i\in \left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right) \right\rbrace. \ \ \ \ \ (2)&fg=000000$
$latex \displaystyle \hat{f}_h (m_j) = \frac{1}{n} \sharp \left\lbrace X_i\in \left[ m_j -\frac{h}{2},m_j +\frac{h}{2} \right) \right\rbrace. &fg=000000$
In the next post we will study the statistical properties of this estimator and we will make some practical examples. |
$\pu{3 g}$ of $\ce{Mg}$ are placed in $\pu{500 mL}$ of $\pu{0.625 M}$ $\ce{AgNO3}$. When the reaction is complete, what is the molarity of the $\ce{AgNO3}$ solution?
I'm thinking you need to use $M_1V_1=M_2V_2$, but I'm stuck on how to use it.
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$\pu{3 g}$ of $\ce{Mg}$ are placed in $\pu{500 mL}$ of $\pu{0.625 M}$ $\ce{AgNO3}$. When the reaction is complete, what is the molarity of the $\ce{AgNO3}$ solution?
I'm thinking you need to use $M_1V_1=M_2V_2$, but I'm stuck on how to use it.
This question appears to be off-topic. The users who voted to close gave this specific reason:
Whenever possible, start with writing down equation(s) for the chemical reactions taking place in the system and don't use random formulas. Magnesium as a more active metal is going to reduce silver in the solution, thus lowering its molarity:
$$\ce{\overset{0}{Mg} (s) + 2\overset{+1}{Ag}NO3 (aq) -> \overset{+2}{Mg}(NO3)2 (aq) + \overset{0}{Ag} (s)}$$
Final concentration $c_2(\ce{AgNO3})$ is pretty much defined by the remaining silver(I) nitrate when magnesium is depleted in the reaction:
$$c_2(\ce{AgNO3}) = c_1(\ce{AgNO3}) - \Delta c(\ce{AgNO3})\label{eq:1}\tag{1}$$
$c_1(\ce{AgNO3})$ is the known initial concentration; $V$ is the volume; $\Delta c(\ce{AgNO3})$ is the change in concentration:
$$\Delta c(\ce{AgNO3}) = \frac{\Delta n(\ce{AgNO3})}{V}\label{eq:2}\tag{2}$$
where the unknown amount of silver nitrate $\Delta n(\ce{AgNO3})$ can be found knowing stoichiometry of the reaction (assuming the reaction is complete and is irreversible), mass and molar mass of magnesium $m(\ce{Mg})$ and $M(\ce{Mg})$, respectively:
$$\Delta n(\ce{AgNO3}) = 2n(\ce{Mg}) = \frac{2m(\ce{Mg})}{M(\ce{Mg})}\label{eq:3}\tag{3}$$
At this point we can rewrite \eqref{eq:1} using \eqref{eq:2} and \eqref{eq:3} since all the variables are known:
$$c_2(\ce{AgNO3}) = c_1(\ce{AgNO3}) - \frac{2m(\ce{Mg})}{VM(\ce{Mg})} = \pu{0.625 M} - \frac{2\cdot\pu{3 g}}{\pu{0.500 L}\cdot\pu{24 g mol-1}} = \pu{0.125 M}$$ |
I'm not sure where I could pose a challenge to find best $f(n)$ so people will join in. $n\ge 5$ will never probably be proven optimal, but some lucky computations or out of the box analysis might give nice results.
(Given $n$ fixed digits and operations $(+,-,\times,\div)$, whats the highest $N\in\mathbb N$, such that all numbers $1\dots N$ can be built? $f(n)=N$)
@TheSimpliFire You mentioned base, is it true that using digits $\lt b$ means we can represent some number $N$ using $\le (b+1)\log_b N$ digits, if only $+,\times$ are allowed?
If $b=2$, $3\log_2 N$ bound is given: https://arxiv.org/pdf/1310.2894.pdf and explained: " The upper bound can be obtained by writing $N$ in binary and finding a representation using Horner’s algorithm."
So if we actually allow $\le b$ digits, we have $log_b N$ digits and that many bases, so the bound would be $2\log_b N$? https://en.wikipedia.org/wiki/Horner%27s_method @TheSimpliFire
The problem is inverting the bound which is not trivial if $b\ne 2$.
For example, we can build $1=2-1$ using $1,2$ digits but adding onto $5$ and having now a set $1,2,5$ does NOT allow to rebuild $1$ since all digits must be used.
So keeping consecutive integers from $n-1$ digit case is not guaranteed. This is the issue.
The $d$ is fixed at $n$ digits and all need to be used.
Thats why I took $d_i=2^{i-1}$ digit sets since we can divide two largest to get the $n-1$ case and this allows to obtain bound $f(n)\ge2^n-1$ eventually.
Inductively.
$i=1,\dots,n$
This is not the issue if all digits are $1$'s also, on which they give bound $3\log_2 N\ge a(N)$ which can be translated to $f(n)\ge 2^{N/3}$ since multiplying two $1$'s reduces the case to $n-1$ and allows induction.
We need to inductively build digits $d_i$ so next set can achieve at least what previous one did.
Otherwise, it is hard to prove the next step is better when adding more digits.
For example we can add $d_0,d_0/2,d_0/2$ where $d_0$ can be anything since $d_0-d_0/2-d_0/2$ reduces us to case $n-3$. The comments discuss setting better bounds using similar construction (on my last question)
I'm not sure if you have the full context of the question or if this makes sense so sorry for clogging up the chat :P |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance. It is commonly used in the textile industry and often seen quoted on, for example, duvets and carpet underlay.The Shirley Institute in Manchester, England developed the tog as an easy-to-follow alternative to the SI unit of m2K/W. The name comes from the informal word "togs" for clothing which itself was probably derived from the word toga, a Roman garment.The basic unit of insulation coefficient is the RSI, (1 m2K/W). 1 tog = 0.1 RSI. There is also a clo clothing unit equivalent to 0.155 RSI or 1.55 tog...
The stone or stone weight (abbreviation: st.) is an English and imperial unit of mass now equal to 14 pounds (6.35029318 kg).England and other Germanic-speaking countries of northern Europe formerly used various standardised "stones" for trade, with their values ranging from about 5 to 40 local pounds (roughly 3 to 15 kg) depending on the location and objects weighed. The United Kingdom's imperial system adopted the wool stone of 14 pounds in 1835. With the advent of metrication, Europe's various "stones" were superseded by or adapted to the kilogram from the mid-19th century on. The stone continues...
Can you tell me why this question deserves to be negative?I tried to find faults and I couldn't: I did some research, I did all the calculations I could, and I think it is clear enough . I had deleted it and was going to abandon the site but then I decided to learn what is wrong and see if I ca...
I am a bit confused in classical physics's angular momentum. For a orbital motion of a point mass: if we pick a new coordinate (that doesn't move w.r.t. the old coordinate), angular momentum should be still conserved, right? (I calculated a quite absurd result - it is no longer conserved (an additional term that varies with time )
in new coordinnate: $\vec {L'}=\vec{r'} \times \vec{p'}$
$=(\vec{R}+\vec{r}) \times \vec{p}$
$=\vec{R} \times \vec{p} + \vec L$
where the 1st term varies with time. (where R is the shift of coordinate, since R is constant, and p sort of rotating.)
would anyone kind enough to shed some light on this for me?
From what we discussed, your literary taste seems to be classical/conventional in nature. That book is inherently unconventional in nature; it's not supposed to be read as a novel, it's supposed to be read as an encyclopedia
@BalarkaSen Dare I say it, my literary taste continues to change as I have kept on reading :-)
One book that I finished reading today, The Sense of An Ending (different from the movie with the same title) is far from anything I would've been able to read, even, two years ago, but I absolutely loved it.
I've just started watching the Fall - it seems good so far (after 1 episode)... I'm with @JohnRennie on the Sherlock Holmes books and would add that the most recent TV episodes were appalling. I've been told to read Agatha Christy but haven't got round to it yet
?Is it possible to make a time machine ever? Please give an easy answer,a simple one A simple answer, but a possibly wrong one, is to say that a time machine is not possible. Currently, we don't have either the technology to build one, nor a definite, proven (or generally accepted) idea of how we could build one. — Countto1047 secs ago
@vzn if it's a romantic novel, which it looks like, it's probably not for me - I'm getting to be more and more fussy about books and have a ridiculously long list to read as it is. I'm going to counter that one by suggesting Ann Leckie's Ancillary Justice series
Although if you like epic fantasy, Malazan book of the Fallen is fantastic
@Mithrandir24601 lol it has some love story but its written by a guy so cant be a romantic novel... besides what decent stories dont involve love interests anyway :P ... was just reading his blog, they are gonna do a movie of one of his books with kate winslet, cant beat that right? :P variety.com/2016/film/news/…
@vzn "he falls in love with Daley Cross, an angelic voice in need of a song." I think that counts :P It's not that I don't like it, it's just that authors very rarely do anywhere near a decent job of it. If it's a major part of the plot, it's often either eyeroll worthy and cringy or boring and predictable with OK writing. A notable exception is Stephen Erikson
@vzn depends exactly what you mean by 'love story component', but often yeah... It's not always so bad in sci-fi and fantasy where it's not in the focus so much and just evolves in a reasonable, if predictable way with the storyline, although it depends on what you read (e.g. Brent Weeks, Brandon Sanderson). Of course Patrick Rothfuss completely inverts this trope :) and Lev Grossman is a study on how to do character development and totally destroys typical romance plots
@Slereah The idea is to pick some spacelike hypersurface $\Sigma$ containing $p$. Now specifying $u(p)$ is trivial because the wave equation is invariant under constant perturbations. So that's whatever. But I can specify $\nabla u(p)|\Sigma$ by specifying $u(\cdot, 0)$ and differentiate along the surface. For the Cauchy theorems I can also specify $u_t(\cdot,0)$.
Now take the neigborhood to be $\approx (-\epsilon,\epsilon)\times\Sigma$ and then split the metric like $-dt^2+h$
Do forwards and backwards Cauchy solutions, then check that the derivatives match on the interface $\{0\}\times\Sigma$
Why is it that you can only cool down a substance so far before the energy goes into changing it's state? I assume it has something to do with the distance between molecules meaning that intermolecular interactions have less energy in them than making the distance between them even smaller, but why does it create these bonds instead of making the distance smaller / just reducing the temperature more?
Thanks @CooperCape but this leads me another question I forgot ages ago
If you have an electron cloud, is the electric field from that electron just some sort of averaged field from some centre of amplitude or is it a superposition of fields each coming from some point in the cloud? |
Spring 2018, Math 171 Week 7 Martingales
Let \((X_n)_{n \ge 0}\) be i.i.d. uniform on \([-1, 0) \cup (0, 1]\)
Show that \((M_n)_{n \ge 0}\) with \(M_n = X_0 + \dots + X_n\) is a Martingale.
Show that \((M_n)_{n \ge 0}\) with \(M_n = \frac{1}{X_0} + \dots + \frac{1}{X_n}\) is not a Martingale.
(Answer) \(\mathbb{E}|M_n| = \infty\)
Let \((X_n)_{n \ge 1}\) be i.i.d. uniform on \(\{-1, 1\}\). AKA Rademacher distributed. Let \(S_n = X_1 + \dots + X_n\), \(S_0 = 0\).
Compute the moment generating function of \(S_n\). That is, \(\mathbb{E}[e^{tS_n}]\)
(Answer) \(\left(\frac{e^t + e^{-t}}{2}\right)^n\)
Find the odd moments of \(S_n\). That is, \(\mathbb{E}[S_n^{2m+1}]\). Explain briefly why the result makes sense.
(Answer) 0. Makes sense because \(S_n\) is symmetric about 0.
Find a formula for the even moments of \(S_n\). That is, \(\mathbb{E}[S_n^{2m}]\).
(Answer) \(\sum_{k=0}^n \binom{n}{k}\left(\frac{1}{2}\right)^n (2k-n)^{2m}\)
For what values of \(c_n\) is \(M_n = S_n^2 - c_n\) a martingale?
(Answer) \(c_n = n\)
Find a formula for \(\mathbb{E}[S_{n+1}^{2m} \mid S_n=t]\) which depends only on \(m\) and \(t\)
(Answer) \(\sum_{k=0}^m {2m \choose 2k} t^{2k}\) Optional Stopping
Problem 5.16 from the textbook (2nd edition)
Problem 5.17 from the textbook (2nd edition)
Problem 5.8 from the textbook (2nd edition) |
Type:Improvement Status:Closed Priority:Major Resolution:Fixed Affects Version/s:3.7 Fix Version/s:3.7 Component/s:Forum Testing Instructions: Log in as admin Create a site with 2 users (ensure both users have profile images set) Create a course with a forum and discussion Enrol the 2 users in the course Enable portfolios on the site (Site administration > advanced features > enable portfolios) Enable and make visible the file download portolio plugin (site administration > plugins > portfolio > manage porfolios) Log in as user 1 and post in the discussion and add a text file as an attachment to the post Log in as user 2 and post the following text in the discussion: When $a \ne 0$, there are two solutions to (ax^2 + bx + c = 0) and they are $$x = {-b \pm \sqrt {b^2-4ac} \over 2a}.$$ Log in as admin View the discussion CONFIRM that the posts have the author's profile image rendered CONFIRM that you can export the attachment posted by user 1 (click the plus icon next to the attachment) and/or the post CONFIRM that the post from user 2 has had the mathjax text filtered to show the equations (easiest way to confirm this would be to post the same text in a different Moodle install forum without these changes and confirm that it looks the same) Log in as admin Create a site with 2 users (ensure both users have profile images set) Create a course with a forum and discussion Enrol the 2 users in the course Enable portfolios on the site (Site administration > advanced features > enable portfolios) Enable and make visible the file download portolio plugin (site administration > plugins > portfolio > manage porfolios) Log in as user 1 and post in the discussion and add a text file as an attachment to the post Log in as user 2 and post the following text in the discussion: When $a \ne 0$, there are two solutions to (ax^2 + bx + c = 0) and they are $$x = {-b \pm \sqrt{b^2-4ac}
\over 2a}.$$
When $a \ne 0$, there are two solutions to (ax^2 + bx + c = 0) and they are $$x = {-b \pm \sqrt {b^2-4ac} Log in as admin View the discussion CONFIRMthat the posts have the author's profile image rendered CONFIRMthat you can export the attachment posted by user 1 (click the plus icon next to the attachment) and/or the post CONFIRMthat the post from user 2 has had the mathjax text filtered to show the equations (easiest way to confirm this would be to post the same text in a different Moodle install forum without these changes and confirm that it looks the same) Affected Branches:MOODLE_37_STABLE Fixed Branches:MOODLE_37_STABLE Epic Link: Pull from Repository: Pull Master Branch: MDL-65394-master Pull Master Diff URL:
Improve the rendering speed of the new forum rendering. At the moment it's marginally faster than the old rendering but there should be some things we can improve. |
OptiRE module of OptiLayer Thin Film Software
OptiRE is intended for the post-production characterization (reverse engineering) of optical coatings based on spectral photometric or/and ellipsometric data. Reverse engineering provides a feedback for the design-production chain. Its main purpose is to discover errors in parameters of produced coatings, calibrate monitoring device and thus to help raise the quality of an optical coating production.
There is a variety of auxiliary options making your work with experimental data and obtained results very flexible:
Post-production characterization (Reverse Engineering) algorithms are based on the optimization of the
\[ DF^2(X)=\left(\frac 1L\sum\limits_{j=1}^L \frac{S(X;\lambda_j)-\hat{S}(\lambda_j)}{\Delta_j}\right)^2 \rightarrow \min,\]
where \(S\) is the model spectral characteristic of the coating, \(\hat{S}\) is the measurement characteristic, \(X\) is the vector of model parameters, \(\{\lambda_j\}, j=1,...,L\) is the wavelength grid, \(\Delta_j\) are measurement tolerances.
OptiLayer provides a wide set of optical coating models and very powerful algorithms as well as special mathematical tools such as Tikhonov's regularization. Choosing of he model adequately describing your optical coating and verification of the results are not straightforward tasks. OptiLayer proposes various tools for reliable post-production characterization.
All our reverse engineering models and approaches have been carefully tested and verified in the frame of collaboration with researches from world leading laboratories and institutes. You can find many useful advises in our publications on characterization and on reverse engineering.
References: |
Consider the famous work equation due to a continuous charge distribution:
$W=\frac{\varepsilon_{0}}{2}\left ( \int_{volume \space space}\left \| \vec{E} \right \|^{2}.d \tau+\oint_{S}V.\vec{E}.d \vec{a} \right )$
Note:
$V\left(r \right)=\frac{1}{4 \pi \varepsilon _{0}}\frac{q}{r}$
$\vec{E}= \frac{1}{4\pi\varepsilon _{0}}\frac{q}{r^{2}}$
The author of the text
Introduction to Electrodynamics claims that
..well the integral of $E^{2}$ can only increase; evidently the surface integral must decrease correspondingly to leave the sum intact.
In fact, at large distance from the charge, E goes like $\frac{1}{r^{2}}$ and V like $\frac{1}{r}$ while the surface area grows like $r^{2}$; roughly speaking, then, the surface integral goes down like $\frac{1}{r}$
Could someone explain further on the bold?
Thanks in advance. |
In finance, many stochastic processes $X(t)$ are defined via \begin{equation} dX = \text{(some drift term)} dt + \sigma X^\gamma dW_t \end{equation} with $\gamma = 1/2$ (for instance the Heston model or the CIR process). Generally, this is called a square-root process. My question is: How does one justify the choice of $\gamma = 1/2$. I am aware that it is convenient to chose $0 < \gamma < 1$ since for $\gamma > 1$, no unique Martingale measure exists. But why exactly $\gamma = 1/2$ and not, say $\gamma = 6/7$. (I have found one related question here Why square root of volatility in Heston model? but no satisfying answer has been given.)
C.I.R Process belongs the class of affine diffusion processes.For processes within this class, a closed form solution of the characteristic functionexists(Duffie,et al). For more details, Suppose we have given a scalar SDEs, i.e., $$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ this process ($\{X_t\}_{0\leq t\leq T}$) is said to be of the affine form if \begin{align} &&\mu(X_t,t)=\alpha_0+\alpha_1X_t\\ &&\sigma^2(X_t,t)=\beta_0+\beta_1X_t \end{align} where $\alpha_j,\beta_j\in R$. We claim C.I.R Process belongs the class of affine diffusion processes,because
\begin{align} & \mu (t,{{r}_{t}})\,\,=\kappa (\theta -{{r}_{t}})=\underbrace{\kappa \theta }_{{{\alpha }_{0}}}+\underbrace{(-\kappa )}_{{{\alpha }_{1}}}\,{{r}_{t}} \\ & {{\sigma }^{2}}(t,{{r}_{t}})=\sigma^2r_t=\underbrace{0}_{\beta_0}+\underbrace{{{\sigma }^{2}}\,}_{\beta {{}_{1}}}\,{{r}_{t}} \\ \end{align}Now, if $\gamma\ne\frac{1}{2}$,then C.I.R Process doesn't belong the class of affine diffusion processes(please check yourself) and the discounted characteristic function
is not of the following form $$\Phi(\phi,r_t,t,T)=e^{A(\phi,\tau)+B(\phi,\tau)r_t}$$Consequently, conditional distribution of on $r_t$ doesn't follow a non-central chi-square distribution. If $\gamma>\frac{1}{2}$ e.g $\gamma=\frac{6}{7}$ then Feller's Condition holds for any value of $\kappa$ and $\theta$ (we know $\kappa,\theta>0$)
$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)=\kappa \theta>0 $$ In other words, $r_t$ is always positive and this is inconsistent with financial Modeling. Also,if $\gamma<\frac{1}{2}$ then
$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)\rightarrow-\infty $$ In other words, $r_t$ is always negative and this is inconsistent with reality. |
06/04/2011, 09:43 PM (This post was last modified: 06/04/2011, 10:26 PM by tommy1729.)
(06/04/2011, 01:13 PM)Gottfried Wrote: Sometimes we find easter-eggs even after easter...
For the alternating iteration-series
(definitions as copied and extended from previous post, see below)
we find a rational polynomial for p=4. That means
(maybe this is trivial and a telescoping sum only, didn't check this thorough)
<hr>
Another one:
<hr>
Code:
\\ define function f(x) for forward iteration and g(x) for backward iteration (=negative height)
\\(additional parameter h for positive integer heights is possible)
f(x,h=1) = for(k=1,h,x = x^2 - 0.5 ); return (x) ;
g(x,h=1) = for(k=1,h,x = sqrt(0.5 + x) ); return (x) ;
\\ do analysis at central value for alternating sums x0=1
x = 1.0
sp(x) = sumalt(h=0,(-1)^h * f(x , h))
sn(x) = sumalt(h=0,(-1)^h * g(x , h))
y(x) = sp(x) + sn(x) - x
this is not my expertise ... yet.
but i think i have seen those before in some far past.
for starters , i related your sums to equations of type f(x) = f(g(x)).
also , ergodic theory studies averages of type
F(x) = lim n-> oo 1/n (f^[0](x) + f^[1](x) + ... f^[n](x).)
hidden telescoping can indeed occur.
and sometimes we can rewrite to an integral.
but again , this is not my expertise yet.
you gave me extra question instead of an answer :p
in particular i do not understand your matrix idea in this thread.
my guess is that when you start at 1.0 , you use carleman matrices to compute the sum and one carleman matrix will not converge ( lies outside the radius ) for 1.0 ; so one is wrong and the other is not.
talking about alternating series 1/2 -1/3 + 1/5 -1/7 + 1/11 - ...
i believe this has a closed form/name and if i recall correctly its called the first mertens constant ...
there was something else i wanted to say ... forgot :s
edit : i do not know how to rewrite an average as a sum or superfunction ( do know integral and perhaps infinite product )... i say that because it might be usefull to see the link with the " ergodic average " ( or whatever its called ).
it bothers me , i wanna get rid of this " lim **/n " term for averages. ( might also be of benefit for number theory and statistics )
(06/04/2011, 09:43 PM)tommy1729 Wrote: in particular i do not understand your matrix idea in this thread.
You may look at alternating sum of iterates (here: of exponential function)
There I describe the method first time however with another function as basis: the exponential function.
The problem of convergence of series of matrices surfaces, and the question of convergence of the shortcutformula for the geometric series especially.
Nearly everything was completely new for me, so this article should be rewritten; anyway in its naivety it might be a good introductory impulse to understand the key idea for that matrix-method and possibly engage in the area which I call now "iteration-series" in resemblance to "powerseries" and "dirichletseries".
Gottfried
Gottfried Helms, Kassel
06/05/2011, 11:40 AM (This post was last modified: 06/05/2011, 12:35 PM by Gottfried.)
Looking back at the article on the alternating iteration-series of exponential there was some confirmation for the matrix-based method missing. While I could use the serial summation (Abel- or Eulersummation of the explicite iterates) for the crosscheck of the matrix-method for the bases, where the powertower of infinite height converges, I could not do that for the other bases due to too fast growth of terms/iterated exponentials.
But well, if I take the (complex) fixpoint t
as initial value, then the alternating series
reduces to
, which should be meaningful for each base, whether its exponential fixpoint is real or not.
With this I have now (at least) one check-value by serial summation for the comparision with the matrix-method.
The matrix-method, dimension 32x32, for instance for base e
, which has a divergent iteration-series, comes out near the expected result
to three/four digits and the same was true for the conjugate of t
. If the convergence could be accelerated, then this gives another confirmation for the applicability of this method for the iteration-series.
Gottfried Helms, Kassel
(03/03/2009, 12:15 PM)Gottfried Wrote: serial summation
0.709801988103 towards 2'nd fixpoint:
0.419756033790 towards 1'st fixpoint:
Matrix-method:
0.580243966210 towards 2'nd fixpoint // incorrect, doesn't match serial summation
0.419756033790 towards 1'st fixpoint // matches serial summation
a reason might be this : the vandermonde matrix must have a determinant <> 1 for almost all functions.
hence the determinant of f^h(x) and f^-h(x) cannot both satisfy to be in the radius ( determinant < 1 = within radius 1 ) for (1 - A)^-1.
basicly just taylor series radius argument for matrices.
have you considered this ?
if i am correct about that , the question becomes : what if the determinant of f(x) is 1 ? will the matrix method agree on both fixpoints ?
(06/05/2011, 01:45 PM)tommy1729 Wrote: if i am correct about that , the question becomes : what if the determinant of f(x) is 1 ? will the matrix method agree on both fixpoints ?
How do you compute or at least estimate the determinant of an (infinite sized) Carleman-matrix (as simply transposed of "matrix-operators")?
Gottfried
Gottfried Helms, Kassel
ive noticed we used both the terms vandermonde and carleman matrix.
ofcourse its carleman matrix and not vandermonde ! also note that the 2 matrix-method number must sum to 1 !! 0.580243966210 + 0.41975603379 =0.9999999999 = 1 simply because 1/(1+x) + 1/(1+(1/x)) = 1. - which also shows the importance of the determinant !! - because of this sum = 1 , the matrix methods cannot match the serial summation.(*) this is similar to my determinant argument made before , just an equivalent restatement. * the sum of both serials is related to the equation f(g(x)) = f(x) , whereas the sum of matrix methods just gives 1 for all x.
(06/06/2011, 11:01 AM)tommy1729 Wrote: 0.580243966210
+
0.41975603379
=0.9999999999 = 1
simply because 1/(1+x) + 1/(1+(1/x)) = 1.
Yes, that observation was exactly what I was discussing when I presented these considerations here since 2007; especially I had a conversation with Andy on this. The next step which is obviously to do, is to search for the reason
why powerseries-based methods disagree with the serial summation - and always only one of the results.
And then possibly for some adaption/cure, so that the results can be made matching. For instance, Ramanujan-summation for divergent series includes one integral term to correct for the change-of-order-of-summation which is an internal detail in that summation method, possibly we should find something analoguous here.
Quote:also note that the 2 matrix-method number must sum to 1 !!
- which also shows the importance of the determinant !! -
Thank you for the double exclamation. They don't introduce a determinant of an infinite sized matrix but make much noise, which I do not like as you know from earlier conversations of mine in sci.math. So I'll stop that small conversation on your postings here as I don't have to say much more relevant at the moment for the other occasional and interested reader.
Gottfried
Gottfried Helms, Kassel
10/19/2017, 10:38 AM (This post was last modified: 10/19/2017, 10:40 AM by Gottfried.)
(06/06/2011, 12:47 PM)Gottfried Wrote: (06/06/2011, 11:01 AM)tommy1729 Wrote: 0.580243966210
+
0.41975603379
=0.9999999999 = 1
simply because 1/(1+x) + 1/(1+(1/x)) = 1.
Yes, that observation was exactly what I was discussing when I presented these considerations here since 2007; especially I had a conversation with Andy on this. The next step which is obviously to do, is to search for the reason why powerseries-based methods disagree with the serial summation - and always only one of the results.
(...)
It should be mentioned also in this thread, that the reason for this problem of matching the Carleman-based and the simple serial summation based results is simple and simple correctable.
1) The Carleman-matrix is always based on the power series of a function f(x) and more specifically of a function g(x+t_0)-t_0 where t_0 is the attracting fixpoint for the function f(x). For that option the Carleman-matrix-based and the serial summation approach evaluate to the same value.
2) But for the other direction of the iteration series, with iterates of the inverse function f^[-1] () we need the Carleman matrix developed at that fixpoint t_1 which is attracting for f^[-1](x) and do the Neumann-series then of this Carlemanmatrix. This evaluates then again correctly and in concordance with the series summation. (Of course, "serial summation" means always to possibly include Cesaro or Euler summation or the like).
So with the correct adapation of the required two Carleman-matrices and their Neumann-series we reproduce correctly the iteration-series in question in both directions.
Gottfried
Gottfried Helms, Kassel
10/19/2017, 04:50 PM (This post was last modified: 10/19/2017, 05:21 PM by sheldonison.)
(10/19/2017, 10:38 AM)Gottfried Wrote: ...
1) The Carleman-matrix is always based on the power series of a function f(x) and more specifically of a function g(x+t_0)-t_0 where t_0 is the attracting fixpoint for the function f(x). For that option the Carleman-matrix-based and the serial summation approach evaluate to the same value.
2) But for the other direction of the iteration series, with iterates of the inverse function f^[-1] () we need the Carleman matrix developed at that fixpoint t_1 which is attracting for f^[-1](x) ...
So with the correct adapation of the required two Carleman-matrices and their Neumann-series we reproduce correctly the iteration-series in question in both directions.
Gottfried
Is there a connection between the Carlemann-matrix and the Schröder's equation,
? Here lambda is the derivative at the fixed point;
, and then the iterated function g(x+1)= f(g(x)) can be generated from the inverse Schröder's equation:
Does the solution to the Carlemann Matrix give you the power series for
?
I would like a Matrix solution for the Schröder's equation. I have a pari-gp program for the formal power series for both
, iterating using Pari-gp's polynomials, but a Matrix solution would be easier to port over to a more accessible programming language and I thought maybe your Carlemann solution might be what I'm looking for
- Sheldon
10/19/2017, 09:33 PM (This post was last modified: 10/23/2017, 11:56 PM by Gottfried.)
(10/19/2017, 04:50 PM)sheldonison Wrote: (10/19/2017, 10:38 AM)Gottfried Wrote: ...
1) The Carleman-matrix is always based on the power series of a function f(x) and more specifically of a function g(x+t_0)-t_0 where t_0 is the attracting fixpoint for the function f(x). For that option the Carleman-matrix-based and the serial summation approach evaluate to the same value.
2) But for the other direction of the iteration series, with iterates of the inverse function f^[-1] () we need the Carleman matrix developed at that fixpoint t_1 which is attracting for f^[-1](x) ...
So with the correct adapation of the required two Carleman-matrices and their Neumann-series we reproduce correctly the iteration-series in question in both directions.
Gottfried
Is there a connection between the Carlemann-matrix and the Schröder's equation, ? Here lambda is the derivative at the fixed point; , and then the iterated function g(x+1)= f(g(x)) can be generated from the inverse Schröder's equation:
Does the solution to the Carlemann Matrix give you the power series for ?
I would like a Matrix solution for the Schröder's equation. I have a pari-gp program for the formal power series for both , iterating using Pari-gp's polynomials, but a Matrix solution would be easier to port over to a more accessible programming language and I thought maybe your Carlemann solution might be what I'm looking for
Hi Sheldon - yes that connection is exceptionally simple. The Schröder-function is simply expressed by the eigenvector-matrices which occur by diagonalization of the Carleman-matrix for function f(x).
In my notation, with a Carlemanmatrix F for your function f(x) we have with a vector V(x) = [1,x,x^2,x^3,...]
Then by diagonalization we find a solution in M and D such that
The software must take care, that the eigenvectors in M are correctly scaled, for instance in the triangular case, (where f(x) has no constant term) the diagonal in M is the diagonal unit matrix I such that indeed M is in the Carleman-form. (Using M=mateigen(F) in Pari/GP does not suffice, you must scale the columns in M appropriately - I've built my own eigen-solver for triangular matrices which I can provide to you).
Then we have
We need here only to take attention for the problem, that non-triangular Carlemanmatrices of finite size - as they are only available to our software packages - give not the correct eigenvectors for the true power series of f(x). To learn about this it is best to use functions which have triangular Carleman-matrices, so for instance $f(x)=ax+b$ $f(x) = qx/(1+qx) $ or $f(x) = t^x-1 $ or the like where also the coefficient at the linear term is not zero and not 1.
For the non-triangular matrices, for instance for $f(x)=b^x$ the diagonalization gives only rough approximations to an -in some sense- "best-possible" solution for fractional iterations and its eigenvector-matrices are in general not Carleman or truncated Carleman. But they give nonetheless real-to-real solutions also for $b > \eta $ and seem to approximate the Kneser-solution when the size of the matrices increase.
You can have my Pari/GP-toolbox for the adequate handling of that type of matrices and especially for calculating the diagonalization for $t^x-1$ such that the eigenvectormatrices are of Carleman-type and true truncations of the \psi-powerseries for the Schröder-function (for which the builtin-eigensolver in Pari/GP does not take care). If you are interested it is perhaps better to contact me via email because the set of routines should have also some explanations with them and I expect some need for diadactical hints.
<hr>
For a "preview" of that toolbox see perhaps page 21 ff in http://go.helms-net.de/math/tetdocs/Cont...ration.pdf
which discusses the diagonalization for $t^x -1$ with its schroeder-function (and the "matrix-logarithm" method for the $ e^x - 1$ and $ \sin(x)$ functions which have no diagonalization in the case of finite size).
Gottfried Helms, Kassel |
When we measure position for example, how does the system "know" that we're measuring position in order to collapse to a position eigenvector? Does the wave function always evolve from the state that it collapsed to? For example, if we measure the position (whatever that means) does the wave evolve from a delta function?
The system doesn't "know" anything.
The only uncontroversial statement one can make about the (strong) measurement of a quantum system is that you will make the correct predictions if you assume that the state after the measurement was the eigenstate corresponding to the measured value of the observable (so, for position, indeed a $\delta$-function, if we ignore issues with that not being a real function, which would be a distraction here). But what we mean by "state" in the first place - i.e. what ontology, if any, corresponds to the statement "the system is in the quantum state $\lvert \psi\rangle$" - is ambiguous to begin with:
Whether the original state "collapsed" to this new state, whether the "state" is just an imperfect representation of our knowledge and the "collapse" is just updating our information (cf. "$\psi$-ontic" vs "$\psi$-epistemic", see e.g. this answer by Emilio Pisanty) instead of an actual physical process, or something else entirely, is a matter of quantum interpretation. In some interpretations, there is collapse, in others there isn't, but in any case, the formalism of quantum mechanics itself does not provide a single "correct" interpretation.
That is, your question is essentially unanswerable unless you specify the interpretation within which it is to be answered. But none of the predictions of quantum mechanics depends on it anyway - you do not need to have a concept of "how" collapse works to compute the outcome of measurements.
The collapse happens in all bases. What I mean by that is that the wavefunction can be expressed in any basis you want to. It's just that the easiest basis to look at right after measurement is the one corresponding to what you measured, since the state is the eigenstate corresponding to your measurement.
Always remember the wavefunction isn't physical. It's an abstract thing that we can only describe and "look at" as shadows from their projections. We can choose any projection we want to, but that choice doesn't change the wavefunction
This question is about what is called the "preferred basis problem" and it is a well-studied aspect of quantum measurement theory.
There are two aspects to the measurement problem:
If we adopt the collapse postulate, then for any given measurement-like interaction, how is the measurement basis determined?
What is the nature of the evolution of the system such that it finally arrives in one state of that basis?
The question here is mainly concerned with 1. This is the part of the measurement problem which can be resolved by the study of decoherence, which goes as follows.
It can happen that for one basis an off-diagonal density matrix element such as $\langle \phi_i |\psi\rangle \langle \psi | \phi_j \rangle$ (where $\phi_i$ are states of the basis) will either evolve very quickly or can be sensitive to very small disturbances, whereas for another basis this may not be so. In this case the off-diagonal elements of the density matrix average to zero over any practical timescale, so we have decoherence between states of such a basis. It is called a pointer basis. It is a basis in which the density matrix of the sytem is diagonal. In this case the future evolution of the system is indistinguishable from that of a system which is in one and only one of these basis states, drawn randomly with a probability obtained from the density matrix in the standard way.
One can also get a diagonal density matrix by taking an average over parts of the environment which have become entangled with the system. In either case the resulting decoherence solves the preferred basis problem, but it does not address the wider issue of exactly how to interpret the physical implications of the mathematics of quantum theory. That is, you can still take your pick from single-world or many-world interpretations, and the ontological status of the wavefunction or state-vector is not settled by this type of study.
As far as I know, the "collapse" (or the environmental decoherence that imitates a collapse) is always to the position basis.
I think that a lot of the confusion surrounding this issue comes from the fact that there is a symmetry between position and momentum in the Hamiltonian formalism of QM, so it looks as though wavefunction collapse should have no reason to prefer one over the other. However, in every realistic physical theory, the position-momentum symmetry is explicitly broken by the actual Hamiltonian, which is local in the position basis and not local in any other basis for the position-momentum space. We don't know why this is the case, and we might eventually discover that it is itself the result of some dynamical symmetry-breaking process, but for now the "preferred" status of the position basis is just a brute fact of the laws of physics.
This doesn't mean that the collapse/decoherence is to a Platonic basis of Dirac delta functions. There is presumably something more subtle going on at quantum gravitational scales that we (or at least I) don't yet understand.
This "collapse"language is completely navel gazing as far as measurements go.
In this answer I show a position measurement, which I copy:
This event was "measured" a few decades ago, by being immortalized in a picture. The data accompanying the picture, the magnetic field ,allow to measure the momentum of the electron, (again and again if one wants) and also the vertex where it appeared. What is collapsing? a balloon? One could remeasure a number of such interactions and derive the probability of a $K^-$ hitting an atom and giving a distribution of the electron momenta.
That distribution is connected with the supposedly collapsing wave function!!!
In quantum mechanics , one event means just a sampling from a probability distribution. When you throw dice and get a six, is anything collapsing?
There is one wavefunction describing the $K^-$ riding along towards an electron in an atom, and another wavefunction after the $K^-$ has interacted with an atom . From then on a different wave function will describe the freed electron and the $K^-$ will obey a different wavefunction because its momentum has changed. This means that an accumulation of events will describe these two different wavefunction/probability distributions. Not a single instance.
Thats all. |
Learning Objectives
Apply the work-energy theorem to find information about the motion of a particle, given the forces acting on it Use the work-energy theorem to find information about the forces acting on a particle, given information about its motion
We have discussed how to find the work done on a particle by the forces that act on it, but how is that work manifested in the motion of the particle? According to Newton’s second law of motion, the sum of all the forces acting on a particle, or the net force, determines the rate of change in the momentum of the particle, or its motion. Therefore, we should consider the work done by all the forces acting on a particle, or the
net work, to see what effect it has on the particle’s motion.
Let’s start by looking at the net work done on a particle as it moves over an infinitesimal displacement, which is the dot product of the net force and the displacement:
\[dW_{net} = \vec{F}_{net} \cdotp d \vec{r}. \nonumber\]
Newton’s second law tells us that
\[\vec{F}_{net} = m \left(\dfrac{d \vec{v}}{dt}\right) \nonumber\]
so
\[dW_{net} = m \left(\dfrac{d \vec{v}}{dt}\right) \cdotp d \vec{r}. \nonumber\]
For the mathematical functions describing the motion of a physical particle, we can rearrange the differentials dt, etc., as algebraic quantities in this expression, that is,
\[\begin{align*} dW_{net} &= m \left(\dfrac{d \vec{v}}{dt}\right) \cdotp d \vec{r} \\[5pt] &= m\, d \vec{v}\; \cdotp \left(\dfrac{d \vec{r}}{dt}\right) \\[5pt] &= m \vec{v}\; \cdotp d \vec{v}, \end{align*}\]
where we substituted the velocity for the time derivative of the displacement and used the commutative property of the dot product. Since derivatives and integrals of scalars are probably more familiar to you at this point, we express the dot product in terms of Cartesian coordinates before we integrate between any two points A and B on the particle’s trajectory. This gives us the net work done on the particle:
\[\begin{align} W_{net,\; AB} & = \int_{A}^{B} (mv_{x} dv_{x} + mv_{y}dv_{y} + mv_{z}dv_{z} \\[4pt] & = \frac{1}{2} m \left| v_{x}^{2} + v_{y}^{2} + v_{z}^{2} \right|_{A}^{B} = \left|\frac{1}{2} mv^{2} \right|_{A}^{B} = K_{B} - K_{A} \ldotp \end{align} \label{7.8}\]
In the middle step, we used the fact that the square of the velocity is the sum of the squares of its Cartesian components, and in the last step, we used the definition of the particle’s kinetic energy. This important result is called the work-energy theorem.
Work-Energy Theorem
The net work done on a particle equals the change in the particle’s kinetic energy:
\[W_{net} = K_{B} - K_{A} \ldotp \label{7.9}\]
According to this theorem, when an object slows down, its final kinetic energy is less than its initial kinetic energy, the change in its kinetic energy is negative, and so is the net work done on it. If an object speeds up, the net work done on it is positive. When calculating the net work, you must include all the forces that act on an object. If you leave out any forces that act on an object, or if you include any forces that don’t act on it, you will get a wrong result.
The importance of the work-energy theorem, and the further generalizations to which it leads, is that it makes some types of calculations much simpler to accomplish than they would be by trying to solve Newton’s second law. For example, in Newton’s Laws of Motion, we found the speed of an object sliding down a frictionless plane by solving Newton’s second law for the acceleration and using kinematic equations for constant acceleration, obtaining
$$v_{f}^{2} = v_{i}^{2} + 2g(s_{f} - s_{i}) \sin \theta,$$
where \(s\) is the displacement down the plane.
We can also get this result from the work-energy theorem (Equation \ref{7.9}). Since only two forces are acting on the object—gravity and the normal force—and the normal force doesn’t do any work, the net work is just the work done by gravity. This only depends on the object’s weight and the difference in height, so
$$W_{net} = W_{grav} = -mg (y_{f} - y_{i}),$$
where y is positive up. The work-energy theorem says that this equals the change in kinetic energy:
$$-mg (y_{f} - y_{i}) = \frac{1}{2} (v_{f}^{2} - v_{i}^{2}) \ldotp$$
Using a right triangle, we can see that (y
f − y i) = (s f − s i)sin \(\theta\), so the result for the final speed is the same.
What is gained by using the work-energy theorem? The answer is that for a frictionless plane surface, not much. However, Newton’s second law is easy to solve only for this particular case, whereas the work-energy theorem gives the final speed for any shaped frictionless surface. For an arbitrary curved surface, the normal force is not constant, and Newton’s second law may be difficult or impossible to solve analytically. Constant or not, for motion along a surface, the normal force never does any work, because it’s perpendicular to the displacement. A calculation using the work-energy theorem avoids this difficulty and applies to more general situations.
Problem-Solving Strategy: Work-Energy Theorem
Draw a free-body diagram for each force on the object. Determine whether or not each force does work over the displacement in the diagram. Be sure to keep any positive or negative signs in the work done. Add up the total amount of work done by each force. Set this total work equal to the change in kinetic energy and solve for any unknown parameter. Check your answers. If the object is traveling at a constant speed or zero acceleration, the total work done should be zero and match the change in kinetic energy. If the total work is positive, the object must have sped up or increased kinetic energy. If the total work is negative, the object must have slowed down or decreased kinetic energy
Example \(\PageIndex{1}\): Loop-the-Loop
The frictionless track for a toy car includes a loop-the-loop of radius \(R\). How high, measured from the bottom of the loop, must the car be placed to start from rest on the approaching section of track and go all the way around the loop?
Strategy
The free-body diagram at the final position of the object is drawn in Figure \(\PageIndex{2}\). The gravitational work is the only work done over the displacement that is not zero. Since the weight points in the same direction as the net vertical displacement, the total work done by the gravitational force is positive. From the work-energy theorem, the starting height determines the speed of the car at the top of the loop,
$$mg(y_{2} - y_{1}) = \frac{1}{2} mv_{2}^{2}, \nonumber$$
where the notation is shown in the accompanying figure. At the top of the loop, the normal force and gravity are both down and the acceleration is centripetal, so
$$a_{top} = \frac{F}{m} = \frac{N + mg}{m} = \frac{v_{2}^{2}}{R} \ldotp \nonumber$$
The condition for maintaining contact with the track is that there must be some normal force, however slight; that is, \(N > 0\). Substituting for \(v_{2}^{2}\) and N, we can find the condition for y
1. Solution
Implement the steps in the strategy to arrive at the desired result:
$$N = \frac{-mg + mv_{2}^{2}}{R} = \frac{-mg + 2mg(y_{1} - 2R)}{R} > 0\; or\; y_{1} > \frac{5R}{2} \ldotp \nonumber$$
Significance
On the surface of the loop, the normal component of gravity and the normal contact force must provide the centripetal acceleration of the car going around the loop. The tangential component of gravity slows down or speeds up the car. A child would find out how high to start the car by trial and error, but now that you know the work-energy theorem, you can predict the minimum height (as well as other more useful results) from physical principles. By using the work-energy theorem, you did not have to solve a differential equation to determine the height.
Exercise \(\PageIndex{1}\)
Suppose the radius of the loop-the-loop in Example \(\PageIndex{1}\) is 15 cm and the toy car starts from rest at a height of 45 cm above the bottom. What is its speed at the top of the loop?
In situations where the motion of an object is known, but the values of one or more of the forces acting on it are not known, you may be able to use the work-energy theorem to get some information about the forces. Work depends on the force and the distance over which it acts, so the information is provided via their product.
Example \(\PageIndex{2}\): Determining a Stopping Force
A bullet from a 0.22 LR-caliber cartridge has a mass of 40 grains (2.60 g) and a muzzle velocity of 1100 ft./s (335 m/s). It can penetrate eight 1-inch pine boards, each with thickness 0.75 inches. What is the average stopping force exerted by the wood, as shown in Figure \(\PageIndex{3}\)?
Strategy
We can assume that under the general conditions stated, the bullet loses all its kinetic energy penetrating the boards, so the work-energy theorem says its initial kinetic energy is equal to the average stopping force times the distance penetrated. The change in the bullet’s kinetic energy and the net work done stopping it are both negative, so when you write out the work-energy theorem, with the net work equal to the average force times the stopping distance, that’s what you get. The total thickness of eight 1-inch pine boards that the bullet penetrates is 8 x \(\frac{3}{4}\) in. = 6 in. = 15.2 cm.
Solution
Applying the work-energy theorem, we get
$$W_{net} = - F_{ave} \Delta s_{stop} = - K_{initial} , \nonumber$$
so
$$F_{ave} = \frac{\frac{1}{2} mv^{2}}{\Delta s_{stop}} = \frac{\frac{1}{2} (2.66 \times 10^{-3}\; kg)(335\; m/s)^{2}}{0.152\; m} = 960\; N \ldotp \nonumber$$
Significance
We could have used Newton’s second law and kinematics in this example, but the work-energy theorem also supplies an answer to less simple situations. The penetration of a bullet, fired vertically upward into a block of wood, is discussed in one section of Asif Shakur’s recent article [“Bullet-Block Science Video Puzzle.”
The Physics Teacher (January 2015) 53(1): 15-16]. If the bullet is fired dead center into the block, it loses all its kinetic energy and penetrates slightly farther than if fired off-center. The reason is that if the bullet hits off-center, it has a little kinetic energy after it stops penetrating, because the block rotates. The work-energy theorem implies that a smaller change in kinetic energy results in a smaller penetration. You will understand more of the physics in this interesting article after you finish reading Angular Momentum.
Learn more about work and energy in this PhET simulation (https://phet.colorado.edu/en/simulation/the-ramp) called “the ramp.” Try changing the force pushing the box and the frictional force along the incline. The work and energy plots can be examined to note the total work done and change in kinetic energy of the box.
Contributors
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0). |
The last post I forget to say that we use Mikownski classes of densities because the MISE is a risk corresponding to the $latex {\mathbb L^2({\mathbb R})}&fg=000000$ norm. Thus, it is natural to assume that $latex {p}&fg=000000$ is smooth with respect to this norm. Another way to describe smoothness in $latex {\mathbb L^{2}({\mathbb R})}&fg=000000$ are Sobolev classes:
Definition 1 Let $latex {\beta\geq1}&fg=000000$ and an integer $latex {L>0}&fg=000000$. The Sobolev class$latex {\mathcal{S}(\beta,L)}&fg=000000$ is the set of all $latex {\beta-1}&fg=000000$ times differentiable functions $latex {f:{\mathbb R}\rightarrow{\mathbb R}}&fg=000000$ having absolutely continuous derivate $latex {f^{(\beta-1)}}&fg=000000$ and satisfying
$latex \displaystyle \int(f^{(\beta)}(x))^{2}dx\leq L^{2}. &fg=000000$
It is also interesting notice that, for an integer $latex {\beta}&fg=000000$, we have the inclusion $latex {\mathcal{S}(\beta,L)\subset\mathcal{H}(\beta,L)}&fg=000000$ by the generalized Minkowski inequality. Also, for these classes, we attain the
same rate of convergence as Nikol’ski classes.
Theorem 2:Suppose that, for an integer $latex {\beta\geq1}&fg=000000$: the function $latex {K}&fg=000000$ is a kernel of order $latex {\beta-1}&fg=000000$ satisfying the conditions
$latex \displaystyle {\displaystyle \int K^{2}(u)du}<\infty,\qquad\int|u|^{\beta}|K(u)|du<\infty; &fg=000000$
the density $latex {p}&fg=000000$ is $latex {\beta-1}&fg=000000$ times differentiable, its derivative $latex {p^{(\beta-1)}}&fg=000000$ is absolutely continuous on $latex {{\mathbb R}}&fg=000000$ and
$latex \displaystyle \int(p^{(\beta)}(x))^{2}dx<\infty. &fg=000000$
Then for all $latex {n\geq1}&fg=000000$ and all $latex {h>0}&fg=000000$ the mean integrated squared error of the kernel estimator $latex {\hat{p} _{n}}&fg=000000$ satisfies
$latex \displaystyle \begin{array}{rl} \mathrm{MISE}\triangleq & \mathbb E_{p}\int(\hat{p} _{n}(x)-p(x))^{2}dx\\ \leq & \frac{1}{nh}\int K^{2}(u)du+\frac{h^{2\beta}}{(l!)^{2}}\left(\int|u|^{\beta}|K(u)|du\right)^{2}\int(p^{(\beta)}(x))^{2}dx. \end{array} &fg=000000$
Proof: Due decomposition of MISE and the Proposition to bound the variance we can bound the variance term. For the bias term, we apply this inequality with $latex {l=\left\lfloor \beta\right\rfloor =\beta-1}&fg=000000$, replacing $latex {L}&fg=000000$ by $latex {\left(\int(p^{(\beta)}(x))^{2}dx\right)^{1/2}}&fg=000000$. By generalized Minkowski inequality we have for all $latex {t\in{\mathbb R}}&fg=000000$ and $latex {0\leq\theta\leq1}&fg=000000$ that
$latex \displaystyle \begin{array}{rl} \displaystyle\int\left(p^{(l)}(x+t)-p^{(l)}(x)\right)^{2}dx & =\displaystyle\int\left(p^{(l)}(x)+t\int_{0}^{1}p^{(l)}(x+\theta t)d\theta-p^{(l)}(x)\right)^{2}dx\\ & =\displaystyle\int\left(t\int_{0}^{1}p^{(l+1)}(x+\theta t)d\theta\right)^{2}dx\\ &\leq \displaystyle t^{2}\left(\int_{0}^{1}\left[\int\left(p^{(\beta)}(x+\theta t)\right)^{2}dx\right]^{1/2}d\theta\right)^{2}\\ & =\displaystyle t^{2}\left(\int_{0}^{1}\left[\int\left(p^{(\beta)}(u)\right)^{2}du\right]^{1/2}d\theta\right)^{2}\\ & = \displaystyle t^{2}\int\left(p^{(\beta)}(u)\right)^{2}du. \end{array} &fg=000000$
$latex \Box&fg=000000$
The natural questions that arises now are:
How to choose a kernel $latex {K}&fg=000000$ and the bandwidth $latex {h}&fg=000000$ in an optimal way? And, this optimal is in a some meaning consistent?
Let’s answer these questions the next week. As always, please use the comment section for questions, suggestion or improvements.
See you next time.
Source: Tsybakov, A. (2009). Introduction to nonparametric estimation. Springer. |
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
# Exercise 1: Derive the Equations of Motion
---
The Lagrange points are the Solar System's favorite parking spots. Named for Italian mathematician Joseph Louis Lagrange, Lagrange points are locations where a small object can rotate synchronously with a moon or planet as that object rotates around another object like a planet or the Sun. In the Earth-Sun system, for example, the Lagrange points are locations where NASA can put a telescope like the [James Web Space Telescope](https://en.wikipedia.org/wiki/James_Webb_Space_Telescope) and it will orbit the Sun but at exactly the same rate as the Earth orbits the Sun.
For simplicity we will assume that the Earth orbits the Sun in a circular orbit with exactly 1 AU between their centers of mass. It turns out that this assumption is not actually necessary, but it does make things conceptually simpler.
To find the Lagrange points we need to define a coordinate system with the origin at the center of mass of the Earth-Sun system. Place the Sun along the $-\hat{x}$ axis at $x = - R_\odot$ and the Earth along the $+\hat{x}$ axis at $x = R_E$. Using Newton's Law of Gravitation, show that the acceleration experienced by a small mass located at $(x, y)$ is given by
$$ a_x (x, y) = - \frac{G M_\odot (x + R_\odot)}{((x + R_\odot)^2 + y^2 )^{3/2}} - \frac{G M_E (x - R_E)}{((x - R_E)^2 + y^2 )^{3/2}} $$
$$ a_y (x, y) = - \frac{G M_\odot y}{((x + R_\odot)^2 + y^2 )^{3/2}} - \frac{G M_E y}{((x - R_E)^2 + y^2 )^{3/2}} $$
To rotate synchronously with the Earth an object would need to have an acceleration at the Lagrange point $\left[ x_L, y_L \right]$ such that
$$\vec{a} (x_L, y_L) = \left[ - \omega^2 x, - \omega^2 y \right] $$
Combine these to show that the Lagrange points are found when
$$ 0 = - \frac{G M_\odot (x + R_\odot)}{((x + R_\odot)^2 + y^2 )^{3/2}} - \frac{G M_E (x - R_E)}{((x - R_E)^2 + y^2 )^{3/2}} + \omega^2 x $$
$$ 0 = - \frac{G M_\odot y}{((x + R_\odot)^2 + y^2 )^{3/2}} - \frac{G M_E y}{((x - R_E)^2 + y^2 )^{3/2}} + \omega^2 y $$
are satisfied.
#Exercise 2: Plot the Right-Hand Side of the Equations of Motion
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It is often useful when looking for solutions of nonlinear equations to move all the terms to one side (usually the right-hand side) and then look for places where the right-hand side equals zero. Finding zeros of nonlinear equations can be done using iterative methods such as Newton's method, but these require an initial guess.
To get some idea of where to guess it is often useful to plot the expressions which we wish would equal zero - the right-hand sides. This gives us "the lay of the land". Using an appropriate 2D plotting routine create plots of the right hand sides for interesting regions where you might go looking for Lagrange points.
#Exercise 3: Find the First Three Lagrange Points
---
The first three Lagrange points can be found by making an intelligent guess, namely that a good place to look is along the x-axis where $y = 0$. S
how that this satisfies the y equation and simplifies the x equation. With this simplified single equation, use Newton's method along with some good guesses from Exercise 2 to find L1, L2, and L3.
#Exercise 4: Find L4 and L5
---
The fourth and fifth Lagrange points do not lie along the x-axis, so we need to bring back the y equation we threw out in the Exercise 3. This will require the use of the generalized version of Newton's method for systems of equations. Use your new version of Newton's method to find L4 and L5.
#Exercise 5: Use an Optimized Library to Find the Lagrange Points Faster
---
Newton's method is roughly 400 years old, so it should come as no surprise to you that we can do better. For example, Newton's method requires us to be able to take analytical derivatives, which is not always possible and/or easy. Newton's method fails if the Jacobian matrix becomes singular, and it has a tendency to be very picky about the initial conditions it is given. It also shouldn't surprise you that your implementation of Newton's method isn't using your computer as efficiently as it could.
To solve this find an improved root-finding algorithm in a suitable computational library for your programming language. For examples, look at SciPy.Optimize for Python, the Optimization package in Matlab, MINPACK in Fortran, the GNU Scientific Library in C/C++, or the Intel Math Kernel Library for both C/C++ and Fortran. Using the online documentation find an appropriate routine, figure out how to feed in your problem, and interpret the output. Measure how much faster the new routine is than your implementation of Newton's method.
#Exercise 6: Are the Lagrange Points Stable or Unstable Equilibria?
---
Once you have found the Lagrange points, you know that if an object is placed exactly at each of those points the object will stay there relative to the Sun and Earth. But what if they aren't perfectly at the Lagrange point? Are those locations stable equilbria such that an object will oscillate back and forth around the Lagrange point, or are they unstable such that the object will accelerate away from the Lagrange point over time? To check this we need to compute the eigenvalues of the Jacobian at each Lagrange point. Take the eigenvalues of the Jacobian and characterize each of the Lagrange points as stable or unstable. |
Naive Bayes methods are a set of supervised learning algorithms based on applying Bayes’ theorem with the “naive” assumption of conditional independence between every pair of features given the value of the class variable. Bayes’ theorem states the following relationship, given class variable \(y\) and dependent feature vector \(x_1\) through \(x_n\), :
Using the naive conditional independence assumption that
for all \(i\), this relationship is simplified to
Since \(P(x_1, \dots, x_n)\) is constant given the input, we can use the following classification rule:
and we can use Maximum A Posteriori (MAP) estimation to estimate \(P(y)\) and \(P(x_i \mid y)\); the former is then the relative frequency of class \(y\) in the training set.
The different naive Bayes classifiers differ mainly by the assumptions they make regarding the distribution of \(P(x_i \mid y)\).
In spite of their apparently over-simplified assumptions, naive Bayes classifiers have worked quite well in many real-world situations, famously document classification and spam filtering. They require a small amount of training data to estimate the necessary parameters. (For theoretical reasons why naive Bayes works well, and on which types of data it does, see the references below.)
Naive Bayes learners and classifiers can be extremely fast compared to more sophisticated methods. The decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a one dimensional distribution. This in turn helps to alleviate problems stemming from the curse of dimensionality.
On the flip side, although naive Bayes is known as a decent classifier, it is known to be a bad estimator, so the probability outputs from
predict_proba are not to be taken too seriously.
References:
GaussianNB implements the Gaussian Naive Bayes algorithm for classification. The likelihood of the features is assumed to be Gaussian:
The parameters \(\sigma_y\) and \(\mu_y\) are estimated using maximum likelihood.
>>> from sklearn import datasets >>> iris = datasets.load_iris() >>> from sklearn.naive_bayes import GaussianNB >>> gnb = GaussianNB() >>> y_pred = gnb.fit(iris.data, iris.target).predict(iris.data) >>> print("Number of mislabeled points out of a total %d points : %d" ... % (iris.data.shape[0],(iris.target != y_pred).sum())) Number of mislabeled points out of a total 150 points : 6
MultinomialNB implements the naive Bayes algorithm for multinomially distributed data, and is one of the two classic naive Bayes variants used in text classification (where the data are typically represented as word vector counts, although tf-idf vectors are also known to work well in practice). The distribution is parametrized by vectors \(\theta_y = (\theta_{y1},\ldots,\theta_{yn})\) for each class \(y\), where \(n\) is the number of features (in text classification, the size of the vocabulary) and \(\theta_{yi}\) is the probability \(P(x_i \mid y)\) of feature \(i\) appearing in a sample belonging to class \(y\).
The parameters \(\theta_y\) is estimated by a smoothed version of maximum likelihood, i.e. relative frequency counting:
where \(N_{yi} = \sum_{x \in T} x_i\) is the number of times feature \(i\) appears in a sample of class \(y\) in the training set \(T\), and \(N_{y} = \sum_{i=1}^{n} N_{yi}\) is the total count of all features for class \(y\).
The smoothing priors \(\alpha \ge 0\) accounts for features not present in the learning samples and prevents zero probabilities in further computations. Setting \(\alpha = 1\) is called Laplace smoothing, while \(\alpha < 1\) is called Lidstone smoothing.
ComplementNB implements the complement naive Bayes (CNB) algorithm. CNB is an adaptation of the standard multinomial naive Bayes (MNB) algorithm that is particularly suited for imbalanced data sets. Specifically, CNB uses statistics from the
complement of each class to compute the model’s weights. The inventors of CNB show empirically that the parameter estimates for CNB are more stable than those for MNB. Further, CNB regularly outperforms MNB (often by a considerable margin) on text classification tasks. The procedure for calculating the weights is as follows:
where the summations are over all documents \(j\) not in class \(c\), \(d_{ij}\) is either the count or tf-idf value of term \(i\) in document \(j\), \(\alpha_i\) is a smoothing hyperparameter like that found in MNB, and \(\alpha = \sum_{i} \alpha_i\). The second normalization addresses the tendency for longer documents to dominate parameter estimates in MNB. The classification rule is:
i.e., a document is assigned to the class that is the
poorest complement match.
References:
BernoulliNB implements the naive Bayes training and classification algorithms for data that is distributed according to multivariate Bernoulli distributions; i.e., there may be multiple features but each one is assumed to be a binary-valued (Bernoulli, boolean) variable. Therefore, this class requires samples to be represented as binary-valued feature vectors; if handed any other kind of data, a
BernoulliNB instance may binarize its input (depending on the
binarize parameter).
The decision rule for Bernoulli naive Bayes is based on
which differs from multinomial NB’s rule in that it explicitly penalizes the non-occurrence of a feature \(i\) that is an indicator for class \(y\), where the multinomial variant would simply ignore a non-occurring feature.
In the case of text classification, word occurrence vectors (rather than word count vectors) may be used to train and use this classifier.
BernoulliNB might perform better on some datasets, especially those with shorter documents. It is advisable to evaluate both models, if time permits.
References:
Naive Bayes models can be used to tackle large scale classification problems for which the full training set might not fit in memory. To handle this case,
MultinomialNB,
BernoulliNB, and
GaussianNB expose a
partial_fit method that can be used incrementally as done with other classifiers as demonstrated in Out-of-core classification of text documents. All naive Bayes classifiers support sample weighting.
Contrary to the
fit method, the first call to
partial_fit needs to be passed the list of all the expected class labels.
For an overview of available strategies in scikit-learn, see also the out-of-core learning documentation.
Note
The
partial_fit method call of naive Bayes models introduces some computational overhead. It is recommended to use data chunk sizes that are as large as possible, that is as the available RAM allows.
© 2007–2018 The scikit-learn developers
Licensed under the 3-clause BSD License. http://scikit-learn.org/stable/modules/naive_bayes.html |
For physics research, there is a differential equation that I simulate again and again. It would be wonderful to speed it up. Each time I run it, part of the input function $I(t)$ changes. It takes a few minutes each time, and after running a few hundred iterations per day, it is eating up a bunch of time. Each time I run it, the input function $I(t)$ only changes a little bit, and the output $\textbf{m(t)}$ also only changes a little bit.
The equation is the Landau -Lifshitz-Gilbert-Slonczewski Equation (LLGS) :
$$ \frac{d\textbf{m}}{dt}=-|\gamma|\textbf{m} \times \textbf{B}_\textrm{eff}-|\gamma|\alpha_\textrm{g}\textbf{m}\times[\textbf{m}\times\textbf{B}_\textrm{eff}]+|\gamma|\alpha_\textrm{j}~I(t)~\textbf{m}\times[\textbf{m}\times\textbf{p}] $$
In this equation, I am solving for $\bf{m}=\bf{m}(t)$, which is magnetization (a 3-D vector). $I(t)$ is current, an input parameter that is a function of time, which changes each time the LLGS is solved. $t$ for time, $\gamma$, $\alpha_\textrm{g}$, $\alpha_\textrm{j}$ are scalar constants, and $\textbf{B}_\textrm{eff}$ and $\textbf{p}$ are constant vectors.
Here is my code for solving this equation:
gamma = 176;alphag = 0.01;alphajConstant = 0.00603;p = {Cos[Pi/6], 0, Sin[Pi/6]};current[t_] := Piecewise[{{3, t <= 50}, {(5-3)/(150-50) (t - 50) + 3, 50 < t < 150}, {5, t >= 150}}] + .01*Sin[2*Pi*30*t];Beff[t_] := {0, 0, 1.5 - 0.8*(m[t].{0, 0, 1})};cons[t_] := -gamma*Cross[m[t], Beff[t]];tGilbdamp[t_] := alphag*Cross[m[t], cons[t]];tSlondamp[t_] := current[t]*alphajConstant*gamma*Cross[m[t], Cross[m[t], p]];LLGS = {m'[t] == cons[t] + tGilbdamp[t] + tSlondamp[t], m[0] == {0, 0, 1}};sol1 = NDSolve[LLGS, {m}, {t, 0, 200}, MaxSteps -> \[Infinity]]; mm[t_] = m[t] /. sol1[[1]] ;
To reiterate, this equation is solving for $\textbf{m(t)}$=
m[t] with a chaging input parameter $I(t)$=
current[t] . What might changes? In the perhaps a second current & simulation would look like this:
current[t_] := Piecewise[{{3.1, t <= 60}, {(5.1-3.1)/(200-60) (t - 60) + 3.1, 60< t < 200}, {5.1, t >= 200}}] + .05*Sin[2*Pi*31*t];LLGS = {m'[t] == cons[t] + tGilbdamp[t] + tSlondamp[t], m[0] == {0, 0, 1}};sol1 = NDSolve[LLGS, {m}, {t, 0, 200}, MaxSteps -> \[Infinity]];
What I am considering:
One thought is to use the functionality for NDSolve Reinitialize . Will this require me to change how my
LLGS equation is formatted? I assume that to use this I will need to change how
current[t] appears in my equation, and have it instead be a set of initial conditions. Is this the best option to speed things up?
Perhaps I should define a method? Perhaps there a way to use previous solutions as a starting point?
Is there a better way to make this go faster? Both individually, and in repetition? |
Hello guys! I was wondering if you knew some books/articles that have a good introduction to convexity in the context of variational calculus (functional analysis). I was reading Young's "calculus of variations and optimal control theory" but I'm not that far into the book and I don't know if skipping chapters is a good idea.
I don't know of a good reference, but I'm pretty sure that just means that second derivatives have consistent signs over the region of interest. (That is certainly a sufficient condition for Legendre transforms.)
@dm__ yes have studied bells thm at length ~2 decades now. it might seem airtight and has stood the test of time over ½ century, but yet there is some fineprint/ loopholes that even phd physicists/ experts/ specialists are not all aware of. those who fervently believe like Bohm that no new physics will ever supercede QM are likely to be disappointed/ dashed, now or later...
oops lol typo bohm bohr
btw what is not widely appreciated either is that nonlocality can be an emergent property of a fairly simple classical system, it seems almost nobody has expanded this at length/ pushed it to its deepest extent. hint: harmonic oscillators + wave medium + coupling etc
But I have seen that the convexity is associated to minimizers/maximizers of the functional, whereas the sign second variation is not a sufficient condition for that. That kind of makes me think that those concepts are not equivalent in the case of functionals...
@dm__ generally think sampling "bias" is not completely ruled out by existing experiments. some of this goes back to CHSH 1969. there is unquestioned reliance on this papers formulation by most subsequent experiments. am not saying its wrong, think only that theres very subtle loophole(s) in it that havent yet been widely discovered. there are many other refs to look into for someone extremely motivated/ ambitious (such individuals are rare). en.wikipedia.org/wiki/CHSH_inequality
@dm__ it stands as a math proof ("based on certain assumptions"), have no objections. but its a thm aimed at physical reality. the translation into experiment requires extraordinary finesse, and the complex analysis starts with CHSH 1969. etc
While it's not something usual, I've noticed that sometimes people edit my question or answer with a more complex notation or incorrect information/formulas. While I don't think this is done with malicious intent, it has sometimes confused people when I'm either asking or explaining something, as...
@vzn what do you make of the most recent (2015) experiments? "In 2015 the first three significant-loophole-free Bell-tests were published within three months by independent groups in Delft, Vienna and Boulder.
All three tests simultaneously addressed the detection loophole, the locality loophole, and the memory loophole. This makes them “loophole-free” in the sense that all remaining conceivable loopholes like superdeterminism require truly exotic hypotheses that might never get closed experimentally."
@dm__ yes blogged on those. they are more airtight than previous experiments. but still seem based on CHSH. urge you to think deeply about CHSH in a way that physicists are not paying attention. ah, voila even wikipedia spells it out! amazing
> The CHSH paper lists many preconditions (or "reasonable and/or presumable assumptions") to derive the simplified theorem and formula. For example, for the method to be valid, it has to be assumed that the detected pairs are a fair sample of those emitted. In actual experiments, detectors are never 100% efficient, so that only a sample of the emitted pairs are detected. > A subtle, related requirement is that the hidden variables do not influence or determine detection probability in a way that would lead to different samples at each arm of the experiment.
↑ suspect entire general LHV theory of QM lurks in these loophole(s)! there has been very little attn focused in this area... :o
how about this for a radical idea? the hidden variables determine the probability of detection...! :o o_O
@vzn honest question, would there ever be an experiment that would fundamentally rule out nonlocality to you? and if so, what would that be? what would fundamentally show, in your opinion, that the universe is inherently local?
@dm__ my feeling is that something more can be milked out of bell experiments that has not been revealed so far. suppose that one could experimentally control the degree of violation, wouldnt that be extraordinary? and theoretically problematic? my feeling/ suspicion is that must be the case. it seems to relate to detector efficiency maybe. but anyway, do believe that nonlocality can be found in classical systems as an emergent property as stated...
if we go into detector efficiency, there is no end to that hole. and my beliefs have no weight. my suspicion is screaming absolutely not, as the classical is emergent from the quantum, not the other way around
@vzn have remained civil, but you are being quite immature and condescending. I'd urge you to put aside the human perspective and not insist that physical reality align with what you expect it to be. all the best
@dm__ ?!? no condescension intended...? am striving to be accurate with my words... you say your "beliefs have no weight," but your beliefs are essentially perfectly aligned with the establishment view...
Last night dream, introduced a strange reference frame based disease called Forced motion blindness. It is a strange eye disease where the lens is such that to the patient, anything stationary wrt the floor is moving forward in a certain direction, causing them have to keep walking to catch up with them. At the same time, the normal person think they are stationary wrt to floor. The result of this discrepancy is the patient kept bumping to the normal person. In order to not bump, the person has to walk at the apparent velocity as seen by the patient. The only known way to cure it is to remo…
And to make things even more confusing:
Such disease is never possible in real life, for it involves two incompatible realities to coexist and coinfluence in a pluralistic fashion. In particular, as seen by those not having the disease, the patient kept ran into the back of the normal person, but to the patient, he never ran into him and is walking normally
It seems my mind has gone f88888 up enough to envision two realities that with fundamentally incompatible observations, influencing each other in a consistent fashion
It seems my mind is getting more and more comfortable with dialetheia now
@vzn There's blatant nonlocality in Newtonian mechanics: gravity acts instantaneously. Eg, the force vector attracting the Earth to the Sun points to where the Sun is now, not where it was 500 seconds ago.
@Blue ASCII is a 7 bit encoding, so it can encode a maximum of 128 characters, but 32 of those codes are control codes, like line feed, carriage return, tab, etc. OTOH, there are various 8 bit encodings known as "extended ASCII", that have more characters. There are quite a few 8 bit encodings that are supersets of ASCII, so I'm wary of any encoding touted to be "the" extended ASCII.
If we have a system and we know all the degrees of freedom, we can find the Lagrangian of the dynamical system. What happens if we apply some non-conservative forces in the system? I mean how to deal with the Lagrangian, if we get any external non-conservative forces perturbs the system?Exampl...
@Blue I think now I probably know what you mean. Encoding is the way to store information in digital form; I think I have heard the professor talking about that in my undergraduate computer course, but I thought that is not very important in actually using a computer, so I didn't study that much. What I meant by use above is what you need to know to be able to use a computer, like you need to know LaTeX commands to type them.
@AvnishKabaj I have never had any of these symptoms after studying too much. When I have intensive studies, like preparing for an exam, after the exam, I feel a great wish to relax and don't want to study at all and just want to go somehwere to play crazily.
@bolbteppa the (quanta) article summary is nearly popsci writing by a nonexpert. specialists will understand the link to LHV theory re quoted section. havent read the scientific articles yet but think its likely they have further ref.
@PM2Ring yes so called "instantaneous action/ force at a distance" pondered as highly questionable bordering on suspicious by deep thinkers at the time. newtonian mechanics was/ is not entirely wrong. btw re gravity there are a lot of new ideas circulating wrt emergent theories that also seem to tie into GR + QM unification.
@Slereah No idea. I've never done Lagrangian mechanics for a living. When I've seen it used to describe nonconservative dynamics I have indeed generally thought that it looked pretty silly, but I can see how it could be useful. I don't know enough about the possible alternatives to tell whether there are "good" ways to do it. And I'm not sure there's a reasonable definition of "non-stupid way" out there.
← lol went to metaphysical fair sat, spent $20 for palm reading, enthusiastic response on my leadership + teaching + public speaking abilities, brought small tear to my eye... or maybe was just fighting infection o_O :P
How can I move a chat back to comments?In complying to the automated admonition to move comments to chat, I discovered that MathJax is was no longer rendered. This is unacceptable in this particular discussion. I therefore need to undo my action and move the chat back to comments.
hmmm... actually the reduced mass comes out of using the transformation to the center of mass and relative coordinates, which have nothing to do with Lagrangian... but I'll try to find a Newtonian reference.
One example is a spring of initial length $r_0$ with two masses $m_1$ and $m_2$ on the ends such that $r = r_2 - r_1$ is it's length at a given time $t$ - the force laws for the two ends are $m_1 \ddot{r}_1 = k (r - r_0)$ and $m_2 \ddot{r}_2 = - k (r - r_0)$ but since $r = r_2 - r_1$ it's more natural to subtract one from the other to get $\ddot{r} = - k (\frac{1}{m_1} + \frac{1}{m_2})(r - r_0)$ which makes it natural to define $\frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2}$ as a mass
since $\mu$ has the dimensions of mass and since then $\mu \ddot{r} = - k (r - r_0)$ is just like $F = ma$ for a single variable $r$ i.e. an spring with just one mass
@vzn It will be interesting if a de-scarring followed by a re scarring can be done in some way in a small region. Imagine being able to shift the wavefunction of a lab setup from one state to another thus undo the measurement, it could potentially give interesting results. Perhaps, more radically, the shifting between quantum universes may then become possible
You can still use Fermi to compute transition probabilities for the perturbation (if you can actually solve for the eigenstates of the interacting system, which I don't know if you can), but there's no simple human-readable interpretation of these states anymore
@Secret when you say that, it reminds me of the no cloning thm, which have always been somewhat dubious/ suspicious of. it seems like theyve already experimentally disproved the no cloning thm in some sense. |
Definition:Factorial/Definition 1 Definition
Let $n \in \Z_{\ge 0}$ be a positive integer.
The factorial of $n$ is defined inductively as: $n! = \begin{cases} 1 & : n = 0 \\ n \left({n - 1}\right)! & : n > 0 \end{cases}$
$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \, 040 \\ 8 & 40 \, 320 \\ 9 & 362 \, 880 \\ 10 & 3 \, 628 \, 800 \\ \end{array}$
Also see Results about factorialscan be found here.
Before that, various symbols were used whose existence is now of less importance.
Notations for $n!$ in history include the following:
$\sqbrk n$ as used by Euler $\mathop{\Pi} n$ as used by Gauss $\left\lvert {\kern-1pt \underline n} \right.$ and $\left. {\underline n \kern-1pt} \right\rvert$, once popular in England and Italy. Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration that $2$, $3$, $4$, etc., should be found in mathematical results. Sources 1965: Seth Warner: Modern Algebra... (previous) ... (next): $\S 19$ 1971: Robert H. Kasriel: Undergraduate Topology... (previous) ... (next): $\S 1.18$: Sequences Defined Inductively: Exercise $2$ 1980: David M. Burton: Elementary Number Theory(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms(3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(6)$ |
ISSN:
1078-0947
eISSN:
1553-5231
All Issues
Discrete & Continuous Dynamical Systems - A
January 2013 , Volume 33 , Issue 1
Special Issue
Tribute to Jean Mawhin
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Abstract:
Jean Mawhin will celebrate his seventieth birthday on December 11, 2012, most likely in Heusy (Verviers), a lovely city in the foothills of the Ardennes, in Belgium, where he was born and resided all his life. He received his pre-university education there and then continued with his studies of mathematics at the Université de Liège, where he received the degree of
Docteur en Sciences Mathématiques, avec la plus grande distinctionon February 10, 1969. He continued as Maítre de Conférencesat Liège from 1969 until 1973 and simultaneously was appointed as Chargé de Coursat the (newly established) Universitè Catholique de Louvain at Louvain-la-Neuve, from 1970 to 1974, where he then served as Professeurfrom 1974 to 1977 and was promoted to Professeur Ordinairein 1977.
For more information please click the “Full Text” above.
Abstract:
We consider Kolmogorov-type systems which are not necessarily competitive or cooperative. Our main result shows that such systems cannot have nontrivial periodic solutions whose orbits are orbitally stable. We obtain our results under two assumptions that we consider to be natural assumptions.
Abstract:
We are concerned with the existence of single- and multi-bump solutions of the equation $-\Delta u+(\lambda a(x)+a_0(x))u=|u|^{p-2}u$, $x\in{\mathbb R}^N$; here $p>2$, and $p<\frac{2N}{N-2}$ if $N\geq 3$. We require that $a\geq 0$ is in $L^\infty_{loc}({\mathbb R}^N)$ and has a bounded potential well $\Omega$, i.e. $a(x)=0$ for $x\in\Omega$ and $a(x)>0$ for $x\in{\mathbb R}^N$\$\bar{\Omega}$. Unlike most other papers on this problem we allow that $a_0\in L^\infty({\mathbb R}^N)$ changes sign. Using variational methods we prove the existence of multibump solutions $u_\lambda$ which localize, as $\lambda\to\infty$, near prescribed isolated open subsets $\Omega_1,\dots,\Omega_k\subset\Omega$. The operator $L_0:=-\Delta+a_0$ may have negative eigenvalues in $\Omega_j$, each bump of $u_\lambda$ may be sign-changing.
Abstract:
We investigate general properties, such as existence and uniqueness, continuous dependence on data and continuation, of solutions to retarded functional differential equations with infinite delay on a differentiable manifold.
Abstract:
We deal with a class of functionals $I$ on a Banach space $X,$ having the structure $I=\Psi+\mathcal G,$ with $\Psi : X \to (- \infty , + \infty ]$ proper, convex, lower semicontinuous and $\mathcal G: X \to \mathbb{R} $ of class $C^1.$ Also, $I$ is $G$-invariant with respect to a discrete subgroup $G\subset X$ with $\mbox{dim (span}\ G)=N$. Under some appropriate additional assumptions we prove that $I$ has at least $N+1$ critical orbits. As a consequence, we obtain that the periodically perturbed $N$-dimensional relativistic pendulum equation has at least $N+1$ geometrically distinct periodic solutions.
Abstract:
We extend the study [1] of gradient systems with equal depth multiple-well potentials to the case when some of the wells are degenerate, in the sense that the Hessian is non positive at those wells. The exponentially small speed, in terms of distances between fronts, typical of non degenerate potentials is replaced by an algebraic upper bound, whose degree depends on the degeneracy of the wells.
Abstract:
We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.
Abstract:
This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \ 1 \leq p \leq +\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\Bbb{R}^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
Abstract:
sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish
a prioribounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties. Abstract:
In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$
Abstract:
We prove results on when nonlinear elliptic equations have infinitely many bifurcations if the nonlinearities grow rapidly.
Abstract:
In this paper we study critical points of the functional \begin{eqnarray*} J_{\epsilon}(u):= \frac{\epsilon^p}{p}\int_0^1|u_x|^pdx+\int_0^1F(u)dx, \; u∈w^{1,p}(0,1), \end{eqnarray*} where F:$\mathbb{R}$→$\mathbb{R}$ is assumed to be a double-well potential. This functional represents the total free energy in phase transition models. We consider a non-classical choice for $F$ modeled on $F(u)=|1-u^2|^{\alpha}$ where $1< \alpha < p$. This choice leads to the existence of multiple continua of critical points that are not present in the classical case $\alpha= p = 2$. We prove that the interior of these continua are local minimizers. The energy of these local minimizers is strictly greater than the global minimum of $J_{\epsilon}$. In particular, the existence of these continua suggests an alternative explanation for the
slow dynamicsobserved in phase transition models. Abstract:
We study positive solutions $y(u)$ for the first order differential equation $$y'=q(c\,{y}^{\frac{1}{p}}-f(u))$$ where $c>0$ is a parameter, $p>1$ and $q>1$ are conjugate numbers and $f$ is a continuous function in $[0,1]$ such that $f(0)=0=f(1)$. We shall be particularly concerned with positive solutions $y(u)$ such that $y(0)=0=y(1)$. Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the $p$-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for $p=2$.
Abstract:
Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.
Abstract:
By using differential inequalities we improve some estimates of W.S. LOUD for the ultimate bound and asymptotic stability of the solutions to the Duffing equation $ u''+ c{u'} + g(u)= f(t)$ where $c>0$, $f $ is measurable and essentially bounded, and $g$ is continuously differentiable with $g'\ge b>0$.
Abstract:
This paper is concerned with non-selfadjoint elliptic problems having a principal part in divergence form and involving an indefinite weight. We study the asymptotic behavior of the principal eigenvalues when the first order term (drift term) becomes larger and larger. Several of our results also apply to elliptic operators in general form.
Abstract:
We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations $L(p)x=f(x)+b(t),\;p=d/dt$ with $2\pi$-periodic forcing $b$ and periodic $f$ we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of $2\pi$-periodic solutions.
Abstract:
This paper is about systems of variational inequalities of the form: $$ \left\{ \begin{array}{l} ‹A_k U_k+ F_k (u) , v_k -u_k› ≥ 0,\; ∀ v_k ∈ K_k \\ u_k ∈ K_k , \end{array} \right. $$ $(k=1,\dots , m)$, where $A_k$ and $F_k$ are multivalued mappings with possibly non-power growths and $K_k$ is a closed, convex set. We concentrate on the noncoercive case and follow a sub-supersolution approach to obtain the existence and enclosure of solutions to the above system between sub- and supersolutions.
Abstract:
In this work we study the periodic solutions, their stability and bifurcation for the class of Duffing differential equation $x''+ \epsilon C x'+ \epsilon^2 A(t) x +b(t) x^3 = \epsilon^3 \Lambda h(t)$, where $C>0$, $\epsilon>0$ and $\Lambda$ are real parameter, $A(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions and $\epsilon$ is sufficiently small. Our results are proved using the averaging method of first order.
Abstract:
This paper deals with integral equations of the form \begin{eqnarray*} x(t)=\tilde{x}+∫_a^td[A]x+f(t)-f(a), t∈[a,b], \end{eqnarray*} in a Banach space $X,$ where $-\infty\ < a < b < \infty$, $\tilde{x}∈ X,$ $f:[a,b]→X$ is regulated on [a,b] and $A(t)$ is for each $t∈[a,b], $ a linear bounded operator on $X,$ while the mapping $A:[a,b]→L(X)$ has a bounded variation on [a,b] Such equations are called generalized linear differential equations. Our aim is to present new results on the continuous dependence of solutions of such equations on a parameter. Furthermore, an application of these results to dynamic equations on time scales is given.
Abstract:
We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation \begin{equation*} \Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = f(t,u) \quad \hbox{ in } {]-r,r[}, \qquad u(-r)=a, \, u(r) = b. \end{equation*} We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side $f$ of the equation.
Abstract:
We prove the existence of an unbounded sequence of critical points of the functional \begin{equation*} J_{\lambda}(u) =\frac{1} {p} ∫_{\mathbb{R} ^N}{||x|^{-α\nabla^k} u|} ^p - λ h(x){||x|^{-α+k}u|} ^p - \frac{1} {q} ∫_{\mathbb{R} ^N}Q(x){||x|^{-b}u|} ^q \end{equation*} related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. We assume $Q\le 0$ at 0 and infinity and consider two essentially different cases: $h\equiv 1$ and $h$ in a certain weighted Lebesgue space.
Abstract:
Consider the following nonlinear elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u - u=\mu_1u^3+\beta uv^2,\ & \hbox{in}\ \Omega\\ -\Delta v - v= \mu_2v^3+\beta vu^2,\ & \hbox{in}\ \Omega\\ u,v>0\ \hbox{in}\ \Omega, \ u=v=0,\ & \hbox{on}\ \partial\Omega, \end{array} \right. \end{equation*}where $\mu_1,\mu_2>0$ are constants and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\leq3$. We study the existence and non-existence of positive solutions and give bifurcation results in terms of the coupling constant $\beta$.
Abstract:
We consider second-order Euler-Lagrange systems which are periodic in time. Their periodic solutions may be characterized as the stationary points of an associated action functional, and we study the dynamical implications of minimizing the action. Examples are well-known of stable periodic minimizers, but instability always holds for periodic solutions which are minimal in the sense of Aubry-Mather.
Abstract:
In this paper we discuss some general results on families of symmetric and doubly-symmetric solutions in reversible Hamiltonian systems having several independent first integrals. We describe a set-up for such solutions which allows the application of classical continuation and bifurcation results.
Abstract:
In 1806 Poisson published one of the first papers on functional differential equations. Among others he studied an example with a state-dependent delay, which is motivated by a geometric problem. This example is not covered by recent results on initial value problems for differential equations with state-dependent delay. We show that the example generates a semiflow of differentiable solution operators, on a manifold of differentiable functions and away from a singular set. Initial data in the singular set produce multiple solutions.
Abstract:
Let $f\in C(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m},% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m})$ and $p\in C([0,T],% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m})$ be continuous functions. We consider the $T$ periodic boundary value problem (*) $u^{\prime}(t)=f(u(t))+p(t),$ $u(0)=u(T).$ It is shown that when $f$ is a coercive gradient function, or the bounded perturbation of a coercive gradient function, and the Brouwer degree $d_{B}(f,B(0,r),0)\neq0$ for large $r$, there is a solution for all $p.$ A result for bounded $f$ is also obtained.
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Definition:Quotient (Algebra) Definition
Let $a, b \in \Z$ be integers such that $b \ne 0$.
From the Division Theorem:
$\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$ The value $q$ is defined as the quotient of $a$ on division by $b$, or the quotient of $\dfrac a b$.
When $x, y \in \R$ the quotient is still defined:
The quotient of $x$ on division by $y$ is defined as the value of $q$ in the expression: $\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \left|{y}\right|$ From the definition of the Modulo Operation: $x \bmod y := x - y \left \lfloor {\dfrac x y}\right \rfloor = r$
it can be seen that the
quotient of $x$ on division by $y$ is defined as: $q = \left \lfloor {\dfrac x y}\right \rfloor$ |
import "gonum.org/v1/gonum/mathext"
Package mathext implements special math functions not implemented by the Go standard library.
AiryAi returns the value of the Airy function at z. The Airy function here, Ai(z), is one of the two linearly independent solutions to
y'' - y*z = 0.
See http://mathworld.wolfram.com/AiryFunctions.html for more detailed information.
AiryAiDeriv returns the value of the derivative of the Airy function at z. The Airy function here, Ai(z), is one of the two linearly independent solutions to
y'' - y*z = 0.
See http://mathworld.wolfram.com/AiryFunctions.html for more detailed information.
Beta returns the value of the complete beta function B(a, b). It is defined as
Γ(a)Γ(b) / Γ(a+b)
Special cases are:
B(a,b) returns NaN if a or b is Inf B(a,b) returns NaN if a and b are 0 B(a,b) returns NaN if a or b is NaN B(a,b) returns NaN if a or b is < 0 B(a,b) returns +Inf if a xor b is 0.
See http://mathworld.wolfram.com/BetaFunction.html for more detailed informations.
CompleteB computes an associate complete elliptic integral of the 2nd kind, 0≤m≤1. It returns math.NaN() if m is not in [0,1].
B(m) = \int_{0}^{π/2} {\cos^2θ} / {\sqrt{1-m{\sin^2θ}}} dθ
CompleteD computes an associate complete elliptic integral of the 2nd kind, 0≤m≤1. It returns math.NaN() if m is not in [0,1].
D(m) = \int_{0}^{π/2} {\sin^2θ} / {\sqrt{1-m{\sin^2θ}}} dθ
CompleteE computes the complete elliptic integral of the 2nd kind, 0≤m≤1. It returns math.NaN() if m is not in [0,1].
E(m) = \int_{0}^{π/2} {\sqrt{1-m{\sin^2θ}}} dθ
CompleteK computes the complete elliptic integral of the 1st kind, 0≤m≤1. It returns math.NaN() if m is not in [0,1].
K(m) = \int_{0}^{π/2} 1/{\sqrt{1-m{\sin^2θ}}} dθ
Digamma returns the logorithmic derivative of the gamma function at x.
ψ(x) = d/dx (Ln (Γ(x)).
EllipticE computes the Legendre's elliptic integral of the 2nd kind E(phi,m), 0≤m<1:
E(\phi,m) = \int_{0}^{\phi} \sqrt{1-m\sin^2(\theta)} d\theta
Legendre's elliptic integrals can be expressed as symmetric elliptic integrals, in this case:
E(\phi,m) = \sin\phi R_F(\cos^2\phi,1-m\sin^2\phi,1)-(m/3)\sin^3\phi R_D(\cos^2\phi,1-m\sin^2\phi,1)
The definition of E(phi,k) where k=sqrt(m) can be found in NIST Digital Library of Mathematical Functions (http://dlmf.nist.gov/19.2.E5).
EllipticF computes the Legendre's elliptic integral of the 1st kind F(phi,m), 0≤m<1:
F(\phi,m) = \int_{0}^{\phi} 1 / \sqrt{1-m\sin^2(\theta)} d\theta
Legendre's elliptic integrals can be expressed as symmetric elliptic integrals, in this case:
F(\phi,m) = \sin\phi R_F(\cos^2\phi,1-m\sin^2\phi,1)
The definition of F(phi,k) where k=sqrt(m) can be found in NIST Digital Library of Mathematical Functions (http://dlmf.nist.gov/19.2.E4).
EllipticRD computes the symmetric elliptic integral R_D(x,y,z):
R_D(x,y,z) = (1/2)\int_{0}^{\infty}{1/(s(t)(t+z))} dt, s(t) = \sqrt{(t+x)(t+y)(t+z)}.
The arguments x, y, z must satisfy the following conditions, otherwise the function returns math.NaN():
0 ≤ x,y ≤ upper, lower ≤ z ≤ upper, lower ≤ x+y,
where:
lower = (5/(2^1022))^(1/3) = 4.809554074311679e-103, upper = ((2^1022)/5)^(1/3) = 2.079194837087086e+102.
The definition of the symmetric elliptic integral R_D can be found in NIST Digital Library of Mathematical Functions (http://dlmf.nist.gov/19.16.E5).
EllipticRF computes the symmetric elliptic integral R_F(x,y,z):
R_F(x,y,z) = (1/2)\int_{0}^{\infty}{1/s(t)} dt, s(t) = \sqrt{(t+x)(t+y)(t+z)}.
The arguments x, y, z must satisfy the following conditions, otherwise the function returns math.NaN():
0 ≤ x,y,z ≤ upper, lower ≤ x+y,y+z,z+x,
where:
lower = 5/(2^1022) = 1.112536929253601e-307, upper = (2^1022)/5 = 8.988465674311580e+306.
The definition of the symmetric elliptic integral R_F can be found in NIST Digital Library of Mathematical Functions (http://dlmf.nist.gov/19.16.E1).
GammaIncReg computes the regularized incomplete Gamma integral.
GammaIncReg(a,x) = (1/ Γ(a)) \int_0^x e^{-t} t^{a-1} dt
The input argument a must be positive and x must be non-negative or GammaIncReg will panic.
See http://mathworld.wolfram.com/IncompleteGammaFunction.html or https://en.wikipedia.org/wiki/Incomplete_gamma_function for more detailed information.
GammaIncRegComp computes the complemented regularized incomplete Gamma integral.
GammaIncRegComp(a,x) = 1 - GammaIncReg(a,x) = (1/ Γ(a)) \int_0^\infty e^{-t} t^{a-1} dt
The input argument a must be positive and x must be non-negative or GammaIncRegComp will panic.
GammaIncRegCompInv computes the inverse of the complemented regularized incomplete Gamma integral. That is, it returns the x such that:
GammaIncRegComp(a, x) = y
The input argument a must be positive and y must be between 0 and 1 inclusive or GammaIncRegCompInv will panic. GammaIncRegCompInv should return a positive number, but can return 0 even with non-zero y due to underflow.
GammaIncRegInv computes the inverse of the regularized incomplete Gamma integral. That is, it returns the x such that:
GammaIncReg(a, x) = y
The input argument a must be positive and y must be between 0 and 1 inclusive or GammaIncRegInv will panic. GammaIncRegInv should return a positive number, but can return NaN if there is a failure to converge.
InvRegIncBeta computes the inverse of the regularized incomplete beta function. It returns the x for which
y = I(x;a,b)
The domain of definition is 0 <= y <= 1, and the parameters a and b must be positive. For other values of x, a, and b InvRegIncBeta will panic.
Lbeta returns the natural logarithm of the complete beta function B(a,b). Lbeta is defined as:
Ln(Γ(a)Γ(b)/Γ(a+b))
Special cases are:
Lbeta(a,b) returns NaN if a or b is Inf Lbeta(a,b) returns NaN if a and b are 0 Lbeta(a,b) returns NaN if a or b is NaN Lbeta(a,b) returns NaN if a or b is < 0 Lbeta(a,b) returns +Inf if a xor b is 0.
MvLgamma returns the log of the multivariate Gamma function. Dim must be greater than zero, and MvLgamma will return NaN if v < (dim-1)/2.
See https://en.wikipedia.org/wiki/Multivariate_gamma_function for more information.
NormalQuantile computes the quantile function (inverse CDF) of the standard normal. NormalQuantile panics if the input p is less than 0 or greater than 1.
RegIncBeta returns the value of the regularized incomplete beta function I(x;a,b). It is defined as
I(x;a,b) = B(x;a,b) / B(a,b) = Γ(a+b) / (Γ(a)*Γ(b)) * int_0^x u^(a-1) * (1-u)^(b-1) du.
The domain of definition is 0 <= x <= 1, and the parameters a and b must be positive. For other values of x, a, and b RegIncBeta will panic.
Zeta computes the Riemann zeta function of two arguments.
Zeta(x,q) = \sum_{k=0}^{\infty} (k+q)^{-x}
Note that Zeta returns +Inf if x is 1 and will panic if x is less than 1, q is either zero or a negative integer, or q is negative and x is not an integer.
See http://mathworld.wolfram.com/HurwitzZetaFunction.html or https://en.wikipedia.org/wiki/Multiple_zeta_function#Two_parameters_case for more detailed information.
Path Synopsis internal/amos Package amos implements functions originally in the Netlib code by Donald Amos. internal/cephes Package cephes implements functions originally in the Netlib code by Stephen Mosher. internal/gonum Package gonum contains functions implemented by the gonum team. |
We show that sheet closures appear as associated varieties of affine vertexalgebras. Further, we give new examples of non-admissible affine vertexalgebras whose associated variety is contained in the nilpotent cone. We alsoprove some conjectures from our previous paper and give new examples of lisseaffine W-algebras.
We prove the long-standing conjecture on the coset construction of theminimal series principal $W$-algebras of ADE types in full generality. We dothis by first establishing Feigin's conjecture on the coset realization of theuniversal principal $W$-algebras, which are not necessarily simple. Asconsequences, the unitarity of the "discrete series" of principal $W$-algebrasis established, a second coset realization of rational and unitary W-algebrasof type $A$ and $D$ are given and the rationality of Kazama-Suzuki coset vertexsuperalgebras is derived.
Attached to a vertex algebra $\mathcal{V}$ are two geometric objects. The{\it associated scheme} of $\mathcal{V}$ is the spectrum of Zhu's Poissonalgebra $R_{\mathcal{V}}$. The {\it singular support} of $\mathcal{V}$ is thespectrum of the associated graded algebra $\text{gr}(\mathcal{V})$ with respectto Li's canonical decreasing filtration. There is a closed embedding from thesingular support to the arc space of the associated scheme, which is anisomorphism in many interesting cases. In this note we give an example of anon-quasi-lisse vertex algebra for which the isomorphism is not true as schemesbut true as varieties.
We introduce the notion of chiral symplectic cores in a vertex Poissonvariety, which can be viewed as analogs of symplectic leaves in Poissonvarieties. As an application we show that any quasi-lisse vertex algebra is aquantization of the arc space of its associated variety, in the sense that itsreduced singular support coincides with the arc space of its associatedvariety. We also show that the coordinate ring of the arc space of Slodowyslices is free over its vertex Poisson center, and the latter coincides withthe vertex Poisson center of the coordinate ring of the arc space of the dualof the corresponding simple Lie algebra.
We introduce a notion of quasi-lisse vertex algebras, which generalizesadmissible affine vertex algebras. We show that the normalized character of anordinary module over a quasi-lisse vertex operator algebra has a modularinvariance property, in the sense that it satisfies a modular lineardifferential equation. As an application we obtain the explicit characterformulas of simple affine vertex algebras associated with the Deligneexceptional series at level $-h^{\vee}/6-1$, which express the homogeneousSchur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelliand van Rees, as quasi-modular forms.
The Bershadsky-Polyakov algebra is the $\mathcal{W}$-algebra associated to$\mathfrak{s}\mathfrak{l}_3$ with its minimal nilpotent element $f_{\theta}$.For notational convenience we define $\mathcal{W}^{\ell} = \mathcal{W}^{\ell -3/2} (\mathfrak{s}\mathfrak{l}_3, f_{\theta})$. The simple quotient of$\mathcal{W}^{\ell}$ is denoted by $\mathcal{W}_{\ell}$, and for $\ell$ apositive integer, $\mathcal{W}_{\ell}$ is known to be $C_2$-cofinite andrational. We prove that for all positive integers $\ell$, $\mathcal{W}_{\ell}$contains a rank one lattice vertex algebra $V_L$, and that the coset$\mathcal{C}_{\ell} = \text{Com}(V_L, \mathcal{W}_{\ell})$ is isomorphic to theprincipal, rational $\mathcal{W}(\mathfrak{s}\mathfrak{l}_{2\ell})$-algebra atlevel $(2\ell +3)/(2\ell +1) -2\ell$. This was conjectured in the physicsliterature over 20 years ago. As a byproduct, we construct a new family ofrational, $C_2$-cofinite vertex superalgebras from $\mathcal{W}_{\ell}$
Let $\mathfrak{g}$ be a simple, finite-dimensional Lie (super)algebraequipped with an embedding of $\mathfrak{s} \mathfrak{l}_2$ inducing theminimal gradation on $\mathfrak{g}$. The corresponding minimal$\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{g}, e_{-\theta})$ introduced byKac and Wakimoto has strong generators in weights $1,2,3/2$, and all operatorproduct expansions are known explicitly. The weight one subspace generates anaffine vertex (super)algebra $V^{k'}(\mathfrak{g}^{\natural})$ where$\mathfrak{g}^{\natural} \subset \mathfrak{g}$ denotes the centralizer of$\mathfrak{s} \mathfrak{l}_2$. Therefore $\mathcal{W}^k(\mathfrak{g},e_{-\theta})$ has an action of a connected Lie group $G^{\natural}_0$ with Liealgebra $\mathfrak{g}^{\natural}_0$, where $\mathfrak{g}^{\natural}_0$ denotesthe even part of $\mathfrak{g}^{\natural}$. We show that for any reductivesubgroup $G \subset G^{\natural}_0$, and for any reductive Lie algebra$\mathfrak{g}' \subset \mathfrak{g}^{\natural}$, the orbifold $\mathcal{O}^k =\mathcal{W}^k(\mathfrak{g}, e_{-\theta})^{G}$ and the coset $\mathcal{C}^k =\text{Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-\theta}))$ arestrongly finitely generated for generic values of $k$. Here $V(\mathfrak{g}')$denotes the affine vertex algebra associated to $\mathfrak{g}'$. We findexplicit minimal strong generating sets for $\mathcal{C}^k$ when $\mathfrak{g}'= \mathfrak{g}^{\natural}$ and $\mathfrak{g}$ is either $\mathfrak{s}\mathfrak{l}_n$, $\mathfrak{s}\mathfrak{p}_{2n}$,$\mathfrak{s}\mathfrak{l}(2|n)$ for $n\neq 2$,$\mathfrak{p}\mathfrak{s}\mathfrak{l}(2|2)$, or$\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|4)$. Finally, we conjecture somesurprising coincidences among families of cosets $\mathcal{C}_k$ which are thesimple quotients of $\mathcal{C}^k$, and we prove several cases of ourconjecture.
We use affine W-algebras to quantize Mishchenko-Fomenko subalgebras forcentralizers of nilpotent elements in simple Lie algebras under certainassumptions that are satisfied for all cases in type A and all minimalnilpotent cases outside type $E_8$.
For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, wedescribe a new family of relaxed highest weight representations of$V_k(\mathfrak{g})$. They are simple quotients of representations of the affineKac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following$\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principalnilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$;2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which areinduced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over$\mathfrak{sl}_3$. This in particular gives the classification of all simplepositive energy weight representations of $V_k(\mathfrak{g})$ with finitedimensional weight spaces for $\mathfrak{g}=\mathfrak{sl}_3$.
These are lecture notes from author's mini-course during Session 1: "Vertexalgebras, W-algebras, and application" of INdAM Intensive research period"Perspectives in Lie Theory", at the Centro di Ricerca Matematica Ennio DeGiorgi, Pisa, Italy. December 9, 2014 -- February 28, 2015.
We give an explicit description for the weight three generator of the cosetvertex operator algebra $C_{L_{\widehat{\sl_{n}}}(l,0)\otimesL_{\widehat{\sl_{n}}}(1,0)}(L_{\widehat{\sl_{n}}}(l+1,0))$, for $n\geq 2, l\geq1$. Furthermore, we prove that the commutant$C_{L_{\widehat{\sl_{3}}}(l,0)\otimesL_{\widehat{\sl_{3}}}(1,0)}(L_{\widehat{\sl_{3}}}(l+1,0))$ is isomorphic to the$\W$-algebra $\W_{-3+\frac{l+3}{l+4}}(\sl_3)$, which confirms the conjecturefor the $\sl_3$ case that $C_{L_{\widehat{\frak g}}(l,0)\otimesL_{\widehat{\frak g}}(1,0)}(L_{\widehat{\frak g}}(l+1,0))$ is isomorphic to$\W_{-h+\frac{l+h}{l+h+1}}(\frak g)$ for simply-laced Lie algebras ${\frak g}$with its Coxeter number $h$ for a positive integer $l$.
We prove the conjectual isomorphism between the level $k$$\widehat{sl}_2$-parafermion vertex operator algebra and the $(k+1,k+2)$minimal series $W_k$-algebra for all integers $k \ge 2$. As a consequence, weobtain the conjectural isomorphism between the $(k+1,k+2)$ minimal series$W_k$-algebra and the coset vertex operator algebra $SU(k)_1 \otimesSU(k)_1/SU(k)_2$.
We study modularity of the characters of a vertex (super)algebra equippedwith a family of conformal structures. Along the way we introduce the notionsof rationality and cofiniteness relative to such a family. We apply the resultsto determine modular transformations of trace functions on admissible modulesover affine Kac-Moody algebras and, via BRST reduction, trace functions onminimal series representations of principal affine W-algebras.
We produce in an explicit form free generators of the affine W-algebra oftype A associated with a nilpotent matrix whose Jordan blocks are of the samesize. This includes the principal nilpotent case and we thus recover thequantum Miura transformation of Fateev and Lukyanov.
We investigate the irreducibility of the nilpotent Slodowy slices that appearas the associated variety of W-algebras. Furthermore, we provide new examplesof vertex algebras whose associated variety has finitely many symplecticleaves.
We consider a lifting of Joseph ideals for the minimal nilpotent orbitclosure to the setting of affine Kac-Moody algebras and find new examples ofaffine vertex algebras whose associated varieties are minimal nilpotent orbitclosures. As an application we obtain a new family of lisse ($C_2$-cofinite)W-algebras that are not coming from admissible representations of affineKac-Moody algebras.
We develop some basic properties such as $p$-centers of affine vertexalgebras and free field vertex algebras in prime characteristic. We show thatthe Wakimoto-Feigin-Frenkel homomorphism preserves the $p$-centers by providingexplicit formulas. This allows us to formulate the notion of baby Wakimotomodules, which in particular provides an interpretation in the context ofmodular vertex algebras for Mathieu's irreducible character formula of modularaffine Lie algebras at the critical level.
We prove the rationality of all the minimal series principal W-algebrasdiscovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family ofrational and C_2-cofinite vertex operator algebras. A key ingredient in ourproof is the study of Zhu's algebra of simple W-algebras via the quantizedDrinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebracommutes with the reduction functor. Using this general fact we determine themaximal spectrums of the associated graded of Zhu's algebra of all theadmissible affine vertex algebras as well.
First, we establish the relation between the associated varieties of modulesover Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, weprove the Feigin-Frenkel conjecture on the singular supports of G-integrableadmissible representations. In fact we show that the associated variates ofG-integrable admissible representations are irreducible G-invariantsubvarieties of the nullcone of g, by determining them explicitly. Third, weprove the C_2-cofiniteness of a large number of simple W-algebras, includingall minimal series principal W-algebras and the exceptional W-algebras recentlydiscovered by Kac-Wakimoto.
We study the vertex algebras associated with modular invariantrepresentations of affine Kac-Moody algebras at fractional levels, whose simplehighest weight modules are classified by Joseph's characteristic varieties. Weshow that an irreducible highest weight representation of a non-twisted affineKac-Moody algebra at an admissible level k is a module over the associatedsimple affine vertex algebra if and only if it is an admissible representationwhose integral root system is isomorphic to that of the vertex algebra itself.This in particular proves the conjecture of Adamovic and Milas on therationality of admissible affine vertex algebras in the category O.
We prove the conjecture of Frenkel, Kac and Wakimoto on the existence oftwo-sided BGG resolutions of G-integrable admissible representations of affineKac-Moody algebras at fractional levels. As an application we establish thesemi-infintie analogue of the generalized Borel-Weil theorem for mimimalparabolic subalgebras which enables an inductive study of admissiblerepresentations.
We prove the conjecture of Kac-Wakimoto on the rationality of exceptionalW-algebras for the first non-trivial series, namely, for theBershadsky-Polyakov vertex algebras $W_3^{(2)}$ at level $k=p/2-3$ with$p=3,5,7,...$. This gives new examples of rational conformal field theories.
We determine Zhu's algebra and C_2-algebra of parafermion vertex operatoralgebras for sl_2. Moreover, we prove the C_2-cofiniteness of parafermionvertex operator algebras for any finite dimensional simple Lie algebras.
Let g be a complex simple Lie algebra, f a nilpotent element of g. We showthat (1) the center of the W-algebra $W^{cri}(g,f)$ associated with (g,f) atthe critical level coincides with the Feigin-Frenkel center of the affine Liealgebra associated with g, (2) the centerless quotient $W_{\chi}(g,f)$ of$W^{cri}(g,f)$ corresponding to an oper $\chi$ on the disc is simple, (3) thesimple quotient $W_{\chi}(g,f)$ is a quantization of the jet scheme of theintersection of the Slodowy slice at f with the nilpotent cone of g.
We study the restricted category O for an affine Kac--Moody algebra at thecritical level. In particular, we prove the first part of the Feigin-Frenkelconjecture: the linkage principle for restricted Verma modules. Moreover, weprove a version of the BGGH-reciprocity principle and we determine the blockdecomposition of the restricted category O. For the proofs we need a deformedversion of the classical structures, so we mostly work in a relative setting. |
Learning Objectives
Calculate the acceleration vector given the velocity function in unit vector notation. Describe the motion of a particle with a constant acceleration in three dimensions. Use the one-dimensional motion equations along perpendicular axes to solve a problem in two or three dimensions with a constant acceleration. Express the acceleration in unit vector notation. Instantaneous Acceleration
In addition to obtaining the displacement and velocity vectors of an object in motion, we often want to know its
acceleration vector at any point in time along its trajectory. This acceleration vector is the instantaneous acceleration and it can be obtained from the derivative with respect to time of the velocity function, as we have seen in a previous chapter. The only difference in two or three dimensions is that these are now vector quantities. Taking the derivative with respect to time \(\vec{v}\)(t), we find
$$\vec{a} (t) = \lim_{t \rightarrow 0} \frac{\vec{v} (t + \Delta t) - \vec{v} (t)}{\Delta t} = \frac{d\vec{v} (t)}{dt} \ldotp \label{4.8}$$
The acceleration in terms of components is
$$\vec{a} (t) = \frac{dv_{x} (t)}{dt}\; \hat{i} + \frac{dv_{y} (t)}{dt}\; \hat{j} + \frac{dv_{z} (t)}{dt}\; \hat{k} \ldotp \label{4.9}$$
Also, since the velocity is the derivative of the position function, we can write the acceleration in terms of the second derivative of the position function:
$$\vec{a} (t) = \frac{d^{2} x(t)}{dt^{2}}\; \hat{i} + \frac{d^{2} y(t)}{dt^{2}}\; \hat{j} + \frac{d^{2} z(t)}{dt^{2}}\; \hat{k} \ldotp \label{4.10}$$
Example 4.4
Finding an Acceleration Vector
A particle has a velocity of \(\vec{v}\)(t) = 5.0t \(\hat{i}\) + t
2 \(\hat{j}\) − 2.0t 3 \(\hat{k}\) m/s. (a) What is the acceleration function? (b) What is the acceleration vector at t = 2.0 s? Find its magnitude and direction. Solution We take the first derivative with respect to time of the velocity function to find the acceleration. The derivative is taken component by component: $$\vec{a} (t) = 5.0\; \hat{i} + 2.0t\; \hat{j} - 6.0t^{2}\; \hat{k}\; m/s^{2} \ldotp$$ Evaluating \(\vec{a}\) (2.0\; s) = 5.0 \(\hat{i}\) + 4.0 \(\hat{j}\) - 24.0 \(\hat{k}\) m/s 2gives us the direction in unit vector notation. The magnitude of the acceleration is \(|\vec{a} (2.20\; s)| = \sqrt{5.0^{2} + 4.0^{2} + (-24.0)^{2}} = 24.8\; m/s^{2} \ldotp\) Significance
In this example we find that acceleration has a time dependence and is changing throughout the motion. Let’s consider a different velocity function for the particle.
Example 4.5
Finding a Particle Acceleration
A particle has a position function \(\vec{r}\) (t) = (10t − t
2) \(\hat{i}\) + 5t \(\hat{j}\) + 5t \(\hat{k}\) m. (a) What is the velocity? (b) What is the acceleration? (c) Describe the motion from t = 0 s. Strategy
We can gain some insight into the problem by looking at the position function. It is linear in y and z, so we know the acceleration in these directions is zero when we take the second derivative. Also, note that the position in the x direction is zero for t = 0 s and t = 10 s.
Solution Taking the derivative with respect to time of the position function, we find \(\vec{v}\)(t) = (10 − 2t) \(\hat{i}\) + 5 \(\hat{j}\) + 5 \(\hat{k}\) m/s. The velocity function is linear in time in the x direction and is constant in the y and z directions. Taking the derivative of the velocity function, we find $$\vec{a}(t) = −2\; \hat{i} m/s^{2} \ldotp$$The acceleration vector is a constant in the negative x-direction. The trajectory of the particle can be seen in Figure 4.9. Let’s look in the y and z directions first. The particle’s position increases steadily as a function of time with a constant velocity in these directions. In the x direction, however, the particle follows a path in positive x until t = 5 s, when it reverses direction. We know this from looking at the velocity function, which becomes zero at this time and negative thereafter. We also know this because the acceleration is negative and constant—meaning, the particle is decelerating, or accelerating in the negative direction. The particle’s position reaches 25 m, where it then reverses direction and begins to accelerate in the negative x direction. The position reaches zero at t = 10 s.
Exercise 4.2
Suppose the acceleration function has the form \(\vec{a}\)(t) = a \(\hat{i}\) + b \(\hat{j}\) + c \(\hat{k}\) m/s
2, where a, b, and c are constants. What can be said about the functional form of the velocity function? Constant Acceleration
Multidimensional motion with constant acceleration can be treated the same way as shown in the previous chapter for one-dimensional motion. Earlier we showed that three-dimensional motion is equivalent to three one-dimensional motions, each along an axis perpendicular to the others. To develop the relevant equations in each direction, let’s consider the two-dimensional problem of a particle moving in the xy plane with constant acceleration, ignoring the z-component for the moment. The acceleration vector is
$$\vec{a} = a_{0x}\; \hat{i} + a_{0y}\; \hat{j} \ldotp$$
Each component of the motion has a separate set of equations similar to Equation 3.10–Equation 3.14 of the previous chapter on one-dimensional motion. We show only the equations for position and velocity in the x- and y-directions. A similar set of kinematic equations could be written for motion in the z-direction:
$$x(t) = x_{0} + (v_{x})_{avg} t \label{4.11}$$
$$v_{x}(t) = v_{0x} + a_{x}t \label{4.12}$$
$$x(t) = x_{0} + v_{0x} t + \frac{1}{2} a_{x} t^{2} \label{4.13}$$
$$v_{x}^{2} (t) = v_{0x}^{2} + 2a_{x}(x − x_{0}) \label{4.14}$$
$$y(t) = y_{0} + (v_{y})_{avg} t \label{4.15}$$
$$v_{y}(t) = v_{0y} + a_{y} t \label{4.16}$$
$$y(t) = y_{0} + v_{0y} t + \frac{1}{2} a_{y} t^{2} \label{4.17}$$
$$v_{y}^{2} (t) = v_{0y}^{2} + 2a_{y}(y − y_{0}) \ldotp \label{4.18}$$
Here the subscript 0 denotes the initial position or velocity. Equation 4.11 to Equation 4.18 can be substituted into Equation 4.2 and Equation 4.5 without the z-component to obtain the position vector and velocity vector as a function of time in two dimensions:
$$\vec{r} (t) = x(t)\; \hat{i} + y(t)\; \hat{j}\; and\; \vec{v} (t) = v_{x} (t)\; \hat{i} + v_{y} (t)\; \hat{j} \ldotp$$
The following example illustrates a practical use of the kinematic equations in two dimensions.
Example 4.6
A Skier
Figure 4.10 shows a skier moving with an acceleration of 2.1 m/s
2 down a slope of 15° at t = 0. With the origin of the coordinate system at the front of the lodge, her initial position and velocity are
$$\vec{r} (0) = (7.50\; \hat{i} - 50.0\; \hat{j}) m$$
and
$$\vec{v} (0) = (4.1\; \hat{i} - 1.1\; \hat{j}) m/s$$
(a) What are the x- and y-components of the skier’s position and velocity as functions of time? (b) What are her position and velocity at t = 10.0 s?
Strategy
Since we are evaluating the components of the motion equations in the x and y directions, we need to find the components of the acceleration and put them into the kinematic equations. The components of the acceleration are found by referring to the coordinate system in Figure 4.10. Then, by inserting the components of the initial position and velocity into the motion equations, we can solve for her position and velocity at a later time t.
Solution The origin of the coordinate system is at the top of the hill with y-axis vertically upward and the x-axis horizontal. By looking at the trajectory of the skier, the x-component of the acceleration is positive and the y-component is negative. Since the angle is 15° down the slope, we find $$a_{x} = (2.1\; m/s^{2}) \cos(15^{o}) = 2.0\; m/s^{2}$$$$a_{y} = (−2.1\; m/s^{2}) \sin (15^{o}) = −0.54\; m/s^{2} \ldotp$$Inserting the initial position and velocity into Equation 4.12 and Equation 4.13 for x, we have $$x(t) = 75.0\; m + (4.1\; m/s)t + \frac{1}{2} (2.0\; m/s^{2})t^{2}$$$$v_{x}(t) = 4.1\; m/s + (2.0\; m/s^{2})t \ldotp$$For y, we have $$y(t) = -50.0.0\; m + (-1.1\; m/s)t + \frac{1}{2} (-0.54\; m/s^{2})t^{2}$$$$v_{y}(t) = -1.1\; m/s + (-0.54\; m/s^{2})t \ldotp$$ Now that we have the equations of motion for x and y as functions of time, we can evaluate them at t = 10.0 s: $$x(10.0\; s) = 75.0\; m + (4.1\; m/s) (10.0\; s) + \frac{1}{2} (2.0\; m/s^{2})(10.0\; s)^{2} = 216.0\; m$$$$v_{x}(10.0\; s) = 4.1\; m/s + (2.0\; m/s^{2})(10.0\; s) = 24.1\; m/s$$ $$y(10.0) = -50.0.0\; m + (-1.1\; m/s)(10.0\; s) + \frac{1}{2} (-0.54\; m/s^{2})(10.0\; s)^{2}$$$$v_{y}(10.0\; s) = -1.1\; m/s + (-0.54\; m/s^{2})(10.0\; s) \ldotp$$The position and velocity at t = 10.0 s are, finally $$\vec{r} (10.0\; s) = (216.0\; \hat{i} - 88.0\; \hat{j}) m$$$$\vec{v} (10.0\;s ) = (24.1\; \hat{i} - 6.5\; \hat{j}) m/s \ldotp$$The magnitude of the velocity of the skier at 10.0 s is 25 m/s, which is 60 mi/h. Significance
It is useful to know that, given the initial conditions of position, velocity, and acceleration of an object, we can find the position, velocity, and acceleration at any later time.
With Equation 4.8 through Equation 4.10 we have completed the set of expressions for the position, velocity, and acceleration of an object moving in two or three dimensions. If the trajectories of the objects look something like the “Red Arrows” in the opening picture for the chapter, then the expressions for the position, velocity, and acceleration can be quite complicated. In the sections to follow we examine two special cases of motion in two and three dimensions by looking at projectile motion and circular motion.
Simulation
At this University of Colorado Boulder website, you can explore the position velocity and acceleration of a ladybug with an interactive simulation that allows you to change these parameters.
Contributors
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0). |
Pythagoras's Theorem Contents Theorem
Then:
$a^2 + b^2 = c^2$
Consider the triangle shown below.
This new figure is shown below.
Now to calculate the area of this figure.
Thus we have the calculate of the large square to be:
$4 \left({\dfrac 1 2 a b}\right) + c^2 = 2 a b + c^2$
On the other hand, we can calculate the area of the large square to be:
$\left({a + b}\right)^2 = a^2 + 2 a b + b^2$ Thus: $a^2 + 2 a b + b^2 = 2 a b + c^2 \iff a^2 + b^2 = c^2$
$\blacksquare$
We have:
$\dfrac b c = \dfrac d b$
and:
$\dfrac a c = \dfrac e a$
That is:
$b^2 = c d$ $a^2 = c e$
Adding, we now get:
$a^2 + b^2 = c d + c e = c \left({d + e}\right) = c^2$
$\blacksquare$
The area of the big square is $c^2$.
It is also equal to $4 \dfrac {a b} 2 + \left({a - b}\right)^2$.
So:
\(\displaystyle c^2\) \(=\) \(\displaystyle 4 \frac {a b} 2 + \left({a - b}\right)^2\) \(\displaystyle \) \(=\) \(\displaystyle 2 a b + a^2 - 2 a b + b^2\) \(\displaystyle \) \(=\) \(\displaystyle a^2 + b^2\)
$\blacksquare$
Rearrange the pieces to make the two squares on the right, with areas $a^2$ and $b^2$.
$\blacksquare$
The two squares both have the same area, that is, $\left({a + b}\right)^2$.
The one on the left has four triangles of area $\dfrac {a b} 2$ and a square of area $c^2$.
The one on the right has four triangles of area $\dfrac {a b} 2$ and two squares: one of area $a^2$ and one of area $b^2$.
Take away the triangles from both of the big squares and you are left with $c^2 = a^2 + b^2$.
$\blacksquare$
We have that $CH = BS = AB = AJ$.
Hence the result follows directly from Pythagoras's Theorem for Parallelograms.
$\blacksquare$
We have:
$\angle CAB \cong \angle DCB$ $\angle ABC \cong \angle ACD$
Then we have:
$\triangle ADC \sim \triangle ACB \sim \triangle CDB$
Use the fact that if $\triangle XYZ \sim \triangle X'Y'Z'$ then:
$\dfrac {\left({XYZ}\right)} {\left({X' Y' Z'}\right)} = \dfrac {XY^2} {X'Y'^2} = \dfrac {h_z^2} {h_{z'}^2} = \dfrac {t_z^2} {t_{z'}^2} = \ldots$
where $\left({XYZ}\right)$ represents the area of $\triangle XYZ$.
This gives us: $\dfrac {\left({ADC}\right)} {\left({ACB}\right)} = \dfrac {AC^2} {AB^2}$
and
$\dfrac {\left({CDB}\right)} {\left({ACB}\right)} = \dfrac {BC^2} {AB^2}$
Taking the sum of these two equalities we obtain:
$\dfrac {\left({ADC}\right)} {\left({ACB}\right)} + \dfrac{\left({CDB}\right)} {\left({ACB}\right)} = \dfrac {BC^2} {AB^2} + \dfrac {AC^2} {AB^2}$
Thus:
$\dfrac{\overbrace{\left({ADC}\right) + \left({CDB}\right)}^{\left({ACB}\right)}} {\left({ACB}\right)} = \dfrac {BC^2 + AC^2} {AB^2}$
This gives us $\therefore AB^2 = BC^2 + AC^2$ as desired.
$\blacksquare$
$\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n + 1} } {\left({2 n + 1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$ $\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n} } {\left({2 n}\right)!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$ From these, we derive the proof that $\cos^2 x + \sin^2 x = 1$. $\sin \theta = \dfrac {\text{Opposite}} {\text{Hypotenuse}}$ $\cos \theta = \dfrac {\text{Adjacent}} {\text{Hypotenuse}}$
Let $\text{Adjacent} = a, \text{Opposite} = b, \text{Hypotenuse} = c$, as in the diagram at the top of the page.
Thus:
\(\displaystyle \cos^2 x + \sin^2 x\) \(=\) \(\displaystyle 1\) Sum of Squares of Sine and Cosine \(\displaystyle \implies \ \ \) \(\displaystyle \left({\frac a c}\right)^2 + \left({\frac b c}\right)^2\) \(=\) \(\displaystyle 1\) \(\displaystyle \implies \ \ \) \(\displaystyle a^2 + b^2\) \(=\) \(\displaystyle c^2\) multiplying both sides by $c^2$
$\blacksquare$
Construct squares $BDEC$ on $BC$, $ABFG$ on $AB$ and $ACKH$ on $AC$.
Construct $AL$ parallel to $BD$ (or $CE$).
Since $\angle BAC$ and $\angle BAG$ are both right angles, from Two Angles making Two Right Angles make Straight Line it follows that $CA$ is in a straight line with $AG$.
We have that $\angle DBC = \angle FBA$, because both are right angles.
We add $\angle ABC$ to each one to make $\angle FBC$ and $\angle DBA$.
By common notion 2, $\angle FBC = \angle DBA$.
By Triangle Side-Angle-Side Equality, $\triangle ABD = \triangle FBC$.
So, by Parallelogram on Same Base as Triangle has Twice its Area, the parallelogram $BDLM$ is twice the area of $\triangle ABD$.
So, by Parallelogram on Same Base as Triangle has Twice its Area, the parallelogram $ABFG$ is twice the area of $\triangle FBC$.
So $BDLM = 2 \triangle ABD = 2 \triangle FBC = ABFG$.
By the same construction, we have that $CELM = 2 \triangle ACE = 2 \triangle KBC = ACKH$.
But $BDLM + CELM$ is the whole of the square $BDEC$.
$\blacksquare$
Also known as Pythagoras's theorem was known to the Pythagoreans as the Theorem of the Bride, from its numerological significance. Also see Source of Name
This entry was named for Pythagoras of Samos.
Pythagoras's Theorem, or at least certain specific instances of it, was known to the ancient Egyptians, and it may date back further than that.
Little is actually known about Pythagoras himself, and it is uncertain whether he actually proved it or not.
Some traditional sources suggest that when he discovered this theorem, he was so delighted he sacrificed an ox to the gods in thanksgiving. However, given his philosophical outlook, this is vanishingly unlikely.
Once upon a time there were three ladies of the First Peoples of America sitting around the campfire.
On a reindeer skin sat a lady who was the mother of a fine young warrior who weighed $140$ pounds.
On a buffalo skin sat a lady who was the mother of a fine young warrior who weighed $160$ pounds.
The third lady, as well she might, was sitting on the skin of a hippopotamus, as she herself weighed a mighty $300$ pounds.
As you can see: The squaw on the hippopotamus is equal to the sons of the squaws on the other two hides. Sources Pythagorean Theorem and its many proofs - $96$ proofs of Pythagoras's Theorem 1967: George McCarty: Topology: An Introduction with Application to Topological Groups... (previous) ... (next): $\text{III}$: Pythagoras' Theorem 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $2$ 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $5$ 1992: George F. Simmons: Calculus Gems... (previous) ... (next): Chapter $\text {B}.1$: The Pythagorean Theorem 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $2$ 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $5$ 2008: David Nelson: The Penguin Dictionary of Mathematics(4th ed.) ... (previous) ... (next): Entry: Pythagoras' theorem 2008: Ian Stewart: Taming the Infinite... (previous) ... (next): Chapter $2$: The Logic Of Shape 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics(5th ed.) ... (previous) ... (next): Entry: Pythagoras' Theorem |
Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou, 450001, China
The paper investigates the upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with structural damping: $ u_{tt}-M(\|\nabla u\|^2)\Delta u+(-\Delta)^\alpha u_t+f(u) = g(x,t) $, where $ \alpha\in(1/2, 1) $ is said to be a dissipative index. It shows that when the nonlinearity $ f(u) $ is of supercritical growth $ p: 1 \leq p< p_{\alpha}\equiv\frac{N+4\alpha}{(N-4\alpha)^+} $, the related evolution process has a pullback attractor for each $ \alpha\in(1/2, 1) $, and the family of pullback attractors is upper semicontinuous with respect to $ \alpha $. These results extend those in [
Keywords:Non-autonomous Kirchhoff wave equation, structural damping, dissipative index, pullback attractor, upper semicontinuity. Mathematics Subject Classification:Primary: 37L15, 37B55, 35B41; Secondary: 35B33, 35B65, 35L05. Citation:Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036
References:
[1]
A. V. Babin and M. I. Vishik,
[2] [3]
F. D. M. Bezerra, A. N. Carvalho, J. W. Cholewa and M. J. D. Nascimento,
Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of dynamics,
[4]
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero,
A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor,
[5]
T. Caraballo, G. Łukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems,
[6]
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Surez,
Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system,
[7]
A. N. Carvalho and J. A. Langa,
Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds,
[8] [9]
A. N. Carvalho, J. A. Langa and J. C. Robinson,
[10] [11] [12] [13]
X. Fan and S. Zhou,
Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type,
[14]
M. M. Freitas, P. Kalita and J. A. Langa,
Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations,
[15] [16]
P. G. Geredeli and I. Lasiecka,
Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity,
[17]
J. K. Hale and G. Raugel,
Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,
[18]
V. Kalantarov and S. Zelik,
Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,
[19]
G. Kirchhoff,
[20] [21] [22]
K. Ono,
Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings,
[23]
K. Ono,
On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,
[24]
C. Y. Sun, D. M. Cao and J. Q. Duan,
Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,
[25]
Y. H. Wang and C. K. Zhong,
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models,
[26] [27]
Z. J. Yang, P. Y. Ding and L. Li,
Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity,
[28] [29]
Z. J. Yang and Y. N. Li,
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models,
[30]
Z. J. Yang and Z. M. Liu,
Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation,
[31]
Z. J. Yang and Z. M. Liu,
Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation,
show all references
References:
[1]
A. V. Babin and M. I. Vishik,
[2] [3]
F. D. M. Bezerra, A. N. Carvalho, J. W. Cholewa and M. J. D. Nascimento,
Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of dynamics,
[4]
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero,
A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor,
[5]
T. Caraballo, G. Łukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems,
[6]
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Surez,
Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system,
[7]
A. N. Carvalho and J. A. Langa,
Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds,
[8] [9]
A. N. Carvalho, J. A. Langa and J. C. Robinson,
[10] [11] [12] [13]
X. Fan and S. Zhou,
Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type,
[14]
M. M. Freitas, P. Kalita and J. A. Langa,
Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations,
[15] [16]
P. G. Geredeli and I. Lasiecka,
Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity,
[17]
J. K. Hale and G. Raugel,
Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,
[18]
V. Kalantarov and S. Zelik,
Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,
[19]
G. Kirchhoff,
[20] [21] [22]
K. Ono,
Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings,
[23]
K. Ono,
On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,
[24]
C. Y. Sun, D. M. Cao and J. Q. Duan,
Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,
[25]
Y. H. Wang and C. K. Zhong,
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models,
[26] [27]
Z. J. Yang, P. Y. Ding and L. Li,
Longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity,
[28] [29]
Z. J. Yang and Y. N. Li,
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models,
[30]
Z. J. Yang and Z. M. Liu,
Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation,
[31]
Z. J. Yang and Z. M. Liu,
Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation,
[1]
Zhaojuan Wang, Shengfan Zhou.
Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise.
[2]
Yonghai Wang, Chengkui Zhong.
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models.
[3]
Ling Xu, Jianhua Huang, Qiaozhen Ma.
Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory.
[4]
Zhijian Yang, Yanan Li.
Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models.
[5]
Wen Tan.
The regularity of pullback attractor for a non-autonomous
[6]
Rodrigo Samprogna, Tomás Caraballo.
Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay.
[7]
T. Caraballo, J. A. Langa, J. Valero.
Structure of the pullback attractor
for a non-autonomous scalar differential inclusion.
[8]
Shengfan Zhou, Caidi Zhao, Yejuan Wang.
Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems.
[9]
Zhaojuan Wang, Shengfan Zhou.
Existence and upper semicontinuity of attractors for
non-autonomous stochastic lattice systems with random coupled
coefficients.
[10]
Zhaojuan Wang, Shengfan Zhou.
Random attractor for stochastic non-autonomous damped wave equation with critical exponent.
[11]
Shengfan Zhou, Min Zhao.
Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise.
[12]
Chunyou Sun, Daomin Cao, Jinqiao Duan.
Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor.
[13]
Zhaojuan Wang, Shengfan Zhou.
Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise.
[14] [15] [16]
Fang Li, Bo You.
Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping.
[17]
Xue-Li Song, Yan-Ren Hou.
Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain.
[18]
Bo You, Yanren Hou, Fang Li, Jinping Jiang.
Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian.
[19]
Suping Wang, Qiaozhen Ma.
Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay.
[20]
Shengfan Zhou, Jinwu Huang, Xiaoying Han.
Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices.
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Hey guys! I built the voltage multiplier with alternating square wave from a 555 timer as a source (which is measured 4.5V by my multimeter) but the voltage multiplier doesn't seem to work. I tried first making a voltage doubler and it showed 9V (which is correct I suppose) but when I try a quadrupler for example and the voltage starts from like 6V and starts to go down around 0.1V per second.
Oh! I found a mistake in my wiring and fixed it. Now it seems to show 12V and instantly starts to go down by 0.1V per sec.
But you really should ask the people in Electrical Engineering. I just had a quick peek, and there was a recent conversation about voltage multipliers. I assume there are people there who've made high voltage stuff, like rail guns, which need a lot of current, so a low current circuit like yours should be simple for them.
So what did the guys in the EE chat say...
The voltage multiplier should be ok on a capacitive load. It will drop the voltage on a resistive load, as mentioned in various Electrical Engineering links on the topic. I assume you have thoroughly explored the links I have been posting for you...
A multimeter is basically an ammeter. To measure voltage, it puts a stable resistor into the circuit and measures the current running through it.
Hi all! There is theorem that links the imaginary and the real part in a time dependent analytic function. I forgot its name. Its named after some dutch(?) scientist and is used in solid state physics, who can help?
The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. These relations are often used to calculate the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the analyticity condition, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics these relations are known under the names...
I have a weird question: The output on an astable multivibrator will be shown on a multimeter as half the input voltage (for example we have 9V-0V-9V-0V...and the multimeter averages it out and displays 4.5V). But then if I put that output to a voltage doubler, the voltage should be 18V, not 9V right? Since the voltage doubler will output in DC.
I've tried hooking up a transformer (9V to 230V, 0.5A) to an astable multivibrator (which operates at 671Hz) but something starts to smell burnt and the components of the astable multivibrator get hot. How do I fix this? I check it after that and the astable multivibrator works.
I searched the whole god damn internet, asked every god damn forum and I can't find a single schematic that converts 9V DC to 1500V DC without using giant transformers and power stage devices that weight 1 billion tons....
something so "simple" turns out to be hard as duck
In peskin book of QFT the sum over zero point energy modes is an infinite c-number, fortunately, it's experimental evidence doesn't appear, since experimentalists measure the difference in energy from the ground state. According to my understanding the zro pt energy is the same as the ground state, isn't it?
If so, always it is possible to substract a finite number (higher exited state for e.g.) from this zero point enrgy (which is infinte), it follows that, experimentally we always obtain infinte spectrum.
@AaronStevens Yeah, I had a good laugh to myself when he responded back with "Yeah, maybe they considered it and it was just too complicated". I can't even be mad at people like that. They are clearly fairly new to physics and don't quite grasp yet that most "novel" ideas have been thought of to death by someone; likely 100+ years ago if it's classical physics
I have recently come up with a design of a conceptual electromagntic field propulsion system which should not violate any conservation laws, particularly the Law of Conservation of Momentum and the Law of Conservation of Energy. In fact, this system should work in conjunction with these two laws ...
I rememeber that Gordon Freeman's thesis was "Observation of Einstein-Podolsky-Rosen Entanglement on Supraquantum Structures by Induction Through Nonlinear Transuranic Crystal of Extremely Long Wavelength (ELW) Pulse from Mode-Locked Source Array "
In peskin book of QFT the sum over zero point energy modes is an infinite c-number, fortunately, it's experimental evidence doesn't appear, since experimentalists measure the difference in energy from the ground state. According to my understanding the zro pt energy is the same as the ground state, isn't it? If so, always it is possible to substract a finite number (higher exited state for e.g.) from this zero point enrgy (which is infinte), it follows that, experimentally we always obtain infinte spectrum.
@ACuriousMind What confuses me is the interpretation of Peskin to this infinite c-number and the experimental fact
He said, the second term is the sum over zero point energy modes which is infnite as you mentioned. He added," fortunately, this energy cannot be detected experm., since the experiments measure only the difference between from the ground state of H".
@ACuriousMind Thank you, I understood your explanations clearly. However, regarding what Peskin mentioned in his book, there is a contradiction between what he said about the infinity of the zero point energy/ground state energy, and the fact that this energy is not detectable experimentally because the measurable quantity is the difference in energy between the ground state (which is infinite and this is the confusion) and a higher level.
It's just the first encounter with something that needs to be renormalized. Renormalizable theories are not "incomplete", even though you can take the Wilsonian standpoint that renormalized QFTs are effective theories cut off at a scale.
according to the author, the energy differenc is always infinite according to two fact. the first is, the ground state energy is infnite, secondly, the energy differenc is defined by substituting a higher level energy from the ground state one.
@enumaris That is an unfairly pithy way of putting it. There are finite, rigorous frameworks for renormalized perturbation theories following the work of Epstein and Glaser (buzzword: Causal perturbation theory). Just like in many other areas, the physicist's math sweeps a lot of subtlety under the rug, but that is far from unique to QFT or renormalization
The classical electrostatics formula $H = \int \frac{\mathbf{E}^2}{8 \pi} dV = \frac{1}{2} \sum_a e_a \phi(\mathbf{r}_a)$ with $\phi_a = \sum_b \frac{e_b}{R_{ab}}$ allows for $R_{aa} = 0$ terms i.e. dividing by zero to get infinities also, the problem stems from the fact that $R_{aa}$ can be zero due to using point particles, overall it's an infinite constant added to the particle that we throw away just as in QFT
@bolbteppa I understand the idea that we need to drop such terms to be in consistency with experiments. But i cannot understand why the experiment didn't predict such infinities that arose in the theory?
These $e_a/R_{aa}$ terms in the big sum are called self-energy terms, and are infinite, which means a relativistic electron would also have to have infinite mass if taken seriously, and relativity forbids the notion of a rigid body so we have to model them as point particles and can't avoid these $R_{aa} = 0$ values. |
You have already got "practical" answers, so I intend to answer form another point of view.
There is a quite famous theorem due to Stone and von Neumann, later improved by Mackay, and finally by Dixmier and Nelson, roughly speaking establishing the following result within the most elementary version. (Another version of the theorem focuses on the unitary groups generated by $X$ and $P$ avoiding problems with domains, however I stick here to the self-adjoint operator version.)
THEOREM. (rough statement "for physicists")
If you have a couple of self-adjoint operators $X$ and $P$ defined on a Hilbert space $H$ such that are conjugated to each other:
\begin{equation}[X,P] = i \hbar I \quad\quad\quad (1)\end{equation}
and there is a cyclic vector for $X$ and $P$, then there exists a unitary operator $U : L^2(R, dx)\to H$ such that:
$$(U^{-1} X U )\psi (x)= x\psi(x)\quad \mbox{and}\quad (U^{-1} P U )\psi (x)= -i\hbar \frac{d\psi(x)}{dx}\:.\quad (2)$$
(The rigorous statement, in this Nelson-like version is reads as follows
THEOREM.
Let $X$ and $P$ be a pair of self-adjoint operators on a complex Hilbert space $H$ such that (a) they veryfy (1) on a common invariant dense subspace $S\subset H$, (b) $X^2+P^2$ is essentially self-adjoint on $S$ and (c) there is a cyclic vector in $S$ for $X$ and $P$. Then there exists a unitary operator $U : L^2(R, dx)\to H$ such that (2) are valid for $\psi \in C_0^{\infty}(R)$.
Notice that the operators defined in the right-hand sides of (2) admits unique self-adjoint extensions so they completely fix the operators representing respective observables. We can equally replace $C_0^\infty(R)$for the Schwartz space ${\cal S}(R)$ in the last statement.)
Barring technicalities, all that means that
commutation relations actually fix position and momentum observables as well as the Hilbert space. For instance, referring to Murod Abdukhakimov's answer, if the addition of $\partial f$ to the standard expressions of $X$ and $P$ gives rise to truly self-adjoint operators, then a unitary transformation (just that connecting $\psi$ to $\psi'$ in Murod Abdukhakimov's answer) gets rid of the deformation restoring the standard expression. Remember that unitary transformations do not alter all physical objects of QM.
The result extends to $R^n$, i.e., concerning particles in space for $n=3$.Dropping the irreducibility requirement (there is a cyclic vector for $X$ and $P$) the thesis holds anyway but $H$ decompose into a direct sum (not direct integral!) of closed subspaces where the strong statement is valid.
There are important consequences of this fundamental theorem. First of all $H$ must be saparable as $L^2(R,dx)$ is. Moreover no time operator $T$ (conjugated with the Hamiltonian operator $H$) exists if the Hamiltonian operator id bounded below as physics requires. The latter statement is due to the fact that the theorem fixes the spectra of $X$ and $P$ as the whole real axes in both cases, so that the spectrum of $H$ would not be bounded below if $T,H$ were a conjugated pair of operators.A similar no-go theorem arises concerning quantization of a particle on a circle when one tries to define position and impulse self-adjoint operators.The attempt to solve these no-go results gave rise to more general formulation of quantum mechanics based on the notion of POVM and eventually turned out to be very useful in other contexts as quantum information theory.
An important observation is that Stone-von Neumann - MacKay - Dixmier -Nelson's result fails when dealing with infinite dimensional systems.That is, roughly speaking, passing from the (symplectic space) of a finite number of particle to the (symplectic space) of a field. In that case the canonical commutation relations of $X_i$ and $P_j$ are replaced by those of the quantum fields. E.g:,
$$[\phi(t, x), \pi(t, y)] = i \hbar \delta(x,y) I$$
or more sophisticated versions of them. In this juncture, there exist infinitely many representations of the algebra of observables that cannot be connected by unitary operators. This is a well-known phenomenon in QFT or quantum statistical mechanics (in the thermodynamic limit).For instance the free theory and the interacting theory of a given quantum field cannot be represented in the same Hilbert space once one assumes standard requirements on states and observables (the so called Haag's theorem and this is the deep reason why LSZ formalism uses the weak topology instead of the strong one as in standard quantum theory of the scattering).
If one includes superselections charges in the algebra of observables, non unitarily equivalent representations of the algebra arise automatically giving rise to sectors.
In QFT in curved spacetime the appearance of inequivalent representations of the algebra of observables is a quite common phenomenon due to the presence of curvature of the spacetime.This post imported from StackExchange Physics at 2014-04-12 19:04 (UCT), posted by SE-user V. Moretti |
@JosephWright Well, we still need table notes etc. But just being able to selectably switch off parts of the parsing one does not need... For example, if a user specifies format 2.4, does the parser even need to look for e syntax, or ()'s?
@daleif What I am doing to speed things up is to store the data in a dedicated format rather than a property list. The latter makes sense for units (open ended) but not so much for numbers (rigid format).
@JosephWright I want to know about either the bibliography environment or \DeclareFieldFormat. From the documentation I see no reason not to treat these commands as usual, though they seem to behave in a slightly different way than I anticipated it. I have an example here which globally sets a box, which is typeset outside of the bibliography environment afterwards. This doesn't seem to typeset anything. :-( So I'm confused about the inner workings of biblatex (even though the source seems....
well, the source seems to reinforce my thought that biblatex simply doesn't do anything fancy). Judging from the source the package just has a lot of options, and that's about the only reason for the large amount of lines in biblatex1.sty...
Consider the following MWE to be previewed in the build in PDF previewer in Firefox\documentclass[handout]{beamer}\usepackage{pgfpages}\pgfpagesuselayout{8 on 1}[a4paper,border shrink=4mm]\begin{document}\begin{frame}\[\bigcup_n \sum_n\]\[\underbrace{aaaaaa}_{bbb}\]\end{frame}\end{d...
@Paulo Finally there's a good synth/keyboard that knows what organ stops are! youtube.com/watch?v=jv9JLTMsOCE Now I only need to see if I stay here or move elsewhere. If I move, I'll buy this there almost for sure.
@JosephWright most likely that I'm for a full str module ... but I need a little more reading and backlog clearing first ... and have my last day at HP tomorrow so need to clean out a lot of stuff today .. and that does have a deadline now
@yo' that's not the issue. with the laptop I lose access to the company network and anythign I need from there during the next two months, such as email address of payroll etc etc needs to be 100% collected first
@yo' I'm sorry I explain too bad in english :) I mean, if the rule was use \tl_use:N to retrieve the content's of a token list (so it's not optional, which is actually seen in many places). And then we wouldn't have to \noexpand them in such contexts.
@JosephWright \foo:V \l_some_tl or \exp_args:NV \foo \l_some_tl isn't that confusing.
@Manuel As I say, you'd still have a difference between say \exp_after:wN \foo \dim_use:N \l_my_dim and \exp_after:wN \foo \tl_use:N \l_my_tl: only the first case would work
@Manuel I've wondered if one would use registers at all if you were starting today: with \numexpr, etc., you could do everything with macros and avoid any need for \<thing>_new:N (i.e. soft typing). There are then performance questions, termination issues and primitive cases to worry about, but I suspect in principle it's doable.
@Manuel Like I say, one can speculate for a long time on these things. @FrankMittelbach and @DavidCarlisle can I am sure tell you lots of other good/interesting ideas that have been explored/mentioned/imagined over time.
@Manuel The big issue for me is delivery: we have to make some decisions and go forward even if we therefore cut off interesting other things
@Manuel Perhaps I should knock up a set of data structures using just macros, for a bit of fun [and a set that are all protected :-)]
@JosephWright I'm just exploring things myself “for fun”. I don't mean as serious suggestions, and as you say you already thought of everything. It's just that I'm getting at those points myself so I ask for opinions :)
@Manuel I guess I'd favour (slightly) the current set up even if starting today as it's normally \exp_not:V that applies in an expansion context when using tl data. That would be true whether they are protected or not. Certainly there is no big technical reason either way in my mind: it's primarily historical (expl3 pre-dates LaTeX2e and so e-TeX!)
@JosephWright tex being a macro language means macros expand without being prefixed by \tl_use. \protected would affect expansion contexts but not use "in the wild" I don't see any way of having a macro that by default doesn't expand.
@JosephWright it has series of footnotes for different types of footnotey thing, quick eye over the code I think by default it has 10 of them but duplicates for minipages as latex footnotes do the mpfoot... ones don't need to be real inserts but it probably simplifies the code if they are. So that's 20 inserts and more if the user declares a new footnote series
@JosephWright I was thinking while writing the mail so not tried it yet that given that the new \newinsert takes from the float list I could define \reserveinserts to add that number of "classic" insert registers to the float list where later \newinsert will find them, would need a few checks but should only be a line or two of code.
@PauloCereda But what about the for loop from the command line? I guess that's more what I was asking about. Say that I wanted to call arara from inside of a for loop on the command line and pass the index of the for loop to arara as the jobname. Is there a way of doing that? |
The following problem is from CLRS (31.1-13, Page 933, 3rd edition):
Give an efficient algorithm to convert a given $\beta$-bit (binary) integer to a decimal representation. Argue that if multiplication or division of integers whose length is at most $\beta$ takes time $M(\beta)$, then we can convert binary to decimal in time $Θ(M(\beta)\log \beta)$. (Hint: Use a divide-and-conquer approach, obtaining the top and bottom halves of the result with separate recursions.)
By a simple divide and conquer (by dividing $\beta$-bit integer into two $\beta/2$-bit integers), I obtain the recurrence $T(\beta) = 2T(\beta/2) + M(\beta)$. However, how to argue that it is $O(M(\beta) \log \beta)$?
My attempt: I think the result relies on $M(\beta)$. If $M(\beta) = \Theta(\beta)$, then $T(\beta) = \Theta(M(\beta)\log \beta)$. But, if, for example, $M(\beta) = o(\beta)$ or $M(\beta) = \omega(\beta^2)$, you may not obtain $T(\beta) = \Theta(M(\beta)\log \beta)$.
According to the hint, it seems that we should use different recursions to obtain $O(M(\beta)\log \beta)$ and $\Omega(M(\beta) \log \beta)$, respectively. But how? |
The initial interest in the zeros is their connection with the distribution of primes, which is often done via asymptotic statements about the prime counting function. In analytic number theory, it is standard fare to have an arithmetic function defined by a summation formula, and then modify it into a form that is easier to manipulate and obtain results for, in such a way that the asymptotic results about the modified function can be translated into results about the original function very easily. This is certainly the case with $\pi(x)$, which is why I bring this up. Most of the information that is relevant here can be found on the Explicit formula article at Wikipedia, for explicit formulas for the $\pi(x)$ function using the zeros of the Riemann zeta function. Two key highlights:
$(1)$
"This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their 'expected' positions."
$(2)$
"Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors."
With the very basics of complex numbers we see that $x^\rho$, as a function of $x$, has a magnitude given by $x^{\Re (\rho)}$ and its argument by $\Im(\rho)\cdot\log x$. The imaginary parts thus contribute oscillatory behavior to the explicit formulas, while the real parts say which imaginary parts dominate over others and by how much - this is some meaning behind the 'Fourier transform' description. Indeed, given the dominant term in an asymptote for $\pi$ we have roughly the primes' "expected positions" (we are taking some license in referring to positioning when we are actually speaking of distribution in the limit), and the outside terms will speak to how much $\pi$
deviates from the expected dominant term as we take $x$ higher and higher in value. If one of the real parts differed from the others, it would privilege some deviation over others, changing our view of the regularity in the primes' distribution.
Eventually it also became clear that more and more results in number theory - even very accessible results that belie how deep the Riemann Hypothesis really has come to be - were equivalent to or could only be proven on the assumption of RH. See for example here or here or here. I'm not sure if any truly comprehensive list of the consequences or equivalences actually exists!
Moreover it is clear now that RH is not an isolated phenomenon, and instead exists as a piece in a much bigger puzzle (at least as I see it). The $\zeta$ function is a trivial case of a Dirichlet $L$-function as well a case of a trivial case of a Dedekind $\zeta$ function, and there is respectively a Generalized Riemann Hypothesis (GRH) and Extended Riemann Hypothesis for these two more general classes of functions. There are numerous analogues to the zeta function and RH too - many of these have already gained more ground or already had the analogous RH proven!
It is now wondered what the appropriate definition of an $L$-function "should" be, that is, morally speaking - specifically it must have some analytic features and of course a functional equation involving a reflection, gamma function, weight, conductor etc. but the precise recipe we need to create a slick theory is not yet known. (Disclaimer: this paragraph comes from memory of reading something a long time ago that I cannot figure out how to find again to check. Derp.)
Finally, there is the spectral interpretation of the zeta zeros that has arisen. There is the Hilbert-Pólya conjecture. As the Wikipedia entry describes it,
In a letter to Andrew Odlyzko, dated January 3, 1982, George Pólya said that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts of the zeros of the Riemann zeta function corresponded to eigenvalues of an unbounded self adjoint operator.
This has spurred quantum-mechanical approaches to the Riemann Hypothesis. Moreover, we now have serious empirical evidence of a connection between the zeta zeros and random matrix theory, specifically that their pair-correlation matches that of Gaussian Unitary Ensembles (GUEs)...
The year: 1972. The scene: Afternoon tea in Fuld Hall at the Institute for Advanced Study. The camera pans around the Common Room, passing by several Princetonians in tweeds and corduroys, then zooms in on Hugh Montgomery, boyish Midwestern number theorist with sideburns. He has just been introduced to Freeman Dyson, dapper British physicist.
Dyson: So tell me, Montgomery, what have you been up to?
Montgomery: Well, lately I've been looking into the distribution of the zeros of the Riemann zeta function.
Dyson: Yes? And?
Montgomery: It seems the two-point correlations go as... (turning to write on a nearby blackboard): $$1-\left(\frac{\sin\pi x}{\pi x}\right)^2$$
Dyson: Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix?
(Source:
The Spectrum of Riemannium.)
If so inclined one can see the empirical evidence in pretty pictures e.g. here. |
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