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On phaseless compressed sensing with partially known support
School of Mathematics, Tianjin University, Tianjin 300072, China
We establish a theoretical framework for the problem of phaseless compressed sensing with partially known signal support, which aims at generalizing the Null Space Property and the Strong Restricted Isometry Property from phase retrieval to partially sparse phase retrieval. We first introduce the concepts of the Partial Null Space Property (P-NSP) and the Partial Strong Restricted Isometry Property (P-SRIP); and then show that both the P-NSP and the P-SRIP are exact recovery conditions for the problem of partially sparse phase retrieval. We also prove that a random Gaussian matrix $ A\in \mathbb{R}^{m\times n} $ satisfies the P-SRIP with high probability when $ m = O(t(k-r)\log(\frac{n-r}{t(k-r)})). $
Keywords:Phase retrieval, compressed sensing, phaseless compressed sensing, partial null space property, partial strong restricted isometry property. Mathematics Subject Classification:Primary: 90C90; Secondary: 94A12. Citation:Ying Zhang, Ling Ma, Zheng-Hai Huang. On phaseless compressed sensing with partially known support. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019014
References:
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R. Balan, B. Bodmann, P. G. Casazza and D. Edidin,
Saving phase: injectivity and stability for phase retrieval,
[3]
A.S. Bandeira, J. Cahill, D. Mixon and A. Nelson,
Painless reconstruction from magnitudes of frame coefficients,
[4] [5] [6]
O. Bunk, A. Diza, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy and J. F. van der Veen,
Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,
[7]
T. Cai and A. Zhang,
Sparse representation of a polytope and recovery of sparse signals and low-rank matrices,
[8] [9] [10]
E. J. Candès, T. Strohmer and V. Voroninski,
Exact and stable signal recovery from magnitude measurements via convex programming,
[11]
A. Conca, D. Edidin, M. Hering and C. Vinzant,
An algebraic characterization of injectivity in phase retrieval,
[12] [13] [14]
M. P. Friedlander, H. Mansour, R. Saab and O. Yilmaz,
Recovering compressively sampled signals using partial support information,
[15] [16] [17] [18] [19]
J. Miao, T. Ishikawa, Q. Shen and T. Earnest,
Extending X-ray crystallography to allow the imagine of non-crystalline materials, cells and single protein complexes,
[20] [21]
D. T. Peng, N. H. Xiu and J. Yu,
$S_{1/2}$ regularization methods and fixed point algorithms for affine rank minimization problems,
[22]
H. Qiu, X. Chen, W. Liu, G. Zhou, Y. J. Wang and J. Lai,
A fast $l_1$-solver and its applications to robust face recognition,
[23]
N. Vaswani and W. Lu,
Modified-CS: Modifying compressive sensing for problems with partially known support,
[24]
V. Voroninski and Z. Q. Xu,
A strong restricted isometry property, with an application to phaseless compressed sensing,
[25] [26] [27]
Y. Wang, W. Liu, L. Caccetta and G. Zhou,
Parameter selection for nonnegative $l_1$ matrix/tensor sparse decomposition,
[28]
Y. Wang, G. Zhou, L. Caccetta and W. Liu,
An alternative Lagrange-dual based algorithm for sparse signal reconstruction,
[29] [30]
G. W. You, Z. H. Huang and Y. Wang,
A theoretical perspective of solving phaseless compressive sensing via its nonconvex relaxation,
[31]
L. J. Zhang, L. C. Kong, Y. Li and S. L. Zhou,
A smoothing iterative method for quantile regression with nonconvex $l_p$ penalty,
show all references
References:
[1] [2]
R. Balan, B. Bodmann, P. G. Casazza and D. Edidin,
Saving phase: injectivity and stability for phase retrieval,
[3]
A.S. Bandeira, J. Cahill, D. Mixon and A. Nelson,
Painless reconstruction from magnitudes of frame coefficients,
[4] [5] [6]
O. Bunk, A. Diza, F. Pfeiffer, C. David, B. Schmitt, D. K. Satapathy and J. F. van der Veen,
Diffractive imaging for periodic samples: Retrieving one-dimensional concentration profiles across microfluidic channels,
[7]
T. Cai and A. Zhang,
Sparse representation of a polytope and recovery of sparse signals and low-rank matrices,
[8] [9] [10]
E. J. Candès, T. Strohmer and V. Voroninski,
Exact and stable signal recovery from magnitude measurements via convex programming,
[11]
A. Conca, D. Edidin, M. Hering and C. Vinzant,
An algebraic characterization of injectivity in phase retrieval,
[12] [13] [14]
M. P. Friedlander, H. Mansour, R. Saab and O. Yilmaz,
Recovering compressively sampled signals using partial support information,
[15] [16] [17] [18] [19]
J. Miao, T. Ishikawa, Q. Shen and T. Earnest,
Extending X-ray crystallography to allow the imagine of non-crystalline materials, cells and single protein complexes,
[20] [21]
D. T. Peng, N. H. Xiu and J. Yu,
$S_{1/2}$ regularization methods and fixed point algorithms for affine rank minimization problems,
[22]
H. Qiu, X. Chen, W. Liu, G. Zhou, Y. J. Wang and J. Lai,
A fast $l_1$-solver and its applications to robust face recognition,
[23]
N. Vaswani and W. Lu,
Modified-CS: Modifying compressive sensing for problems with partially known support,
[24]
V. Voroninski and Z. Q. Xu,
A strong restricted isometry property, with an application to phaseless compressed sensing,
[25] [26] [27]
Y. Wang, W. Liu, L. Caccetta and G. Zhou,
Parameter selection for nonnegative $l_1$ matrix/tensor sparse decomposition,
[28]
Y. Wang, G. Zhou, L. Caccetta and W. Liu,
An alternative Lagrange-dual based algorithm for sparse signal reconstruction,
[29] [30]
G. W. You, Z. H. Huang and Y. Wang,
A theoretical perspective of solving phaseless compressive sensing via its nonconvex relaxation,
[31]
L. J. Zhang, L. C. Kong, Y. Li and S. L. Zhou,
A smoothing iterative method for quantile regression with nonconvex $l_p$ penalty,
[1]
Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz.
Compressed sensing and dynamic mode decomposition.
[2]
Yingying Li, Stanley Osher.
Coordinate descent optimization for
[3] [4] [5]
Mikhail Krastanov, Michael Malisoff, Peter Wolenski.
On the strong invariance property for non-Lipschitz dynamics.
[6] [7] [8]
Zdzisław Brzeźniak, Paul André Razafimandimby.
Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces.
[9] [10]
Michael Röckner, Jiyong Shin, Gerald Trutnau.
Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and
non-explosion results.
[11]
Masahito Ohta.
Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement.
[12] [13] [14]
Vianney Perchet, Marc Quincampoix.
A differential game on Wasserstein space. Application to weak approachability with partial monitoring.
[15] [16] [17]
Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto.
Regular maps with the specification property.
[18] [19] [20]
2018 Impact Factor: 1.025
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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? |
Using the solution from the_candyman
the_candyman (https://math.stackexchange.com/users/51370/the-candyman), Calculate the coordinates of the third vertex of triangle given the other two and the length of edges in the cheapest computational way, URL (version: 2017-02-22): https://math.stackexchange.com/q/2156910
Let $A = (x_A, y_A)$ and $B = (x_B,y_B)$ the known vertices of your triangle. Let's call $d_{AB}$, $d_{BC}$ and $d_{CA}$ the lengths of each side.
Translate your points subtracting $x_A$ and $y_A$ so that $A$ corresponds with the origin. That is:
$$A' = (0, 0), B' = (x_B-x_A, y_B-y_A ) = (x_B', y_B').$$
Rotate $B'$ so that it lies on the $x$-axis. This can be done without knowing the angle, indeed:
$$A'' = (0,0), B'' = (d_{AB}, 0).$$
Anyway, the value of the rotation angle is important for the next steps. In particular it is $$\theta = \arctan2\left(y_B-y_A,x_B-x_A\right),$$
where $\arctan2(\cdot, \cdot)$ is defined in details here.
At this point, it is easy to find $C''$. Notice that there are two solutions, since the point $C''$ can be placed above or below the side $AB$.
$$x_C'' = \frac{d_{AB}^2+d_{AC}^2-d_{BC}^2}{2d_{AB}},$$
and
$$y_C'' = \pm\frac{\sqrt{(d_{AB}+d_{AC}+d_{BC})(d_{AB}+d_{AC}-d_{BC})(d_{AB}-d_{AC}+d_{BC})(-d_{AB}+d_{AC}+d_{BC})}}{2d_{AB}}.$$
Now, rotate back your point $C''$ using $-\theta$ (see step 2), thus obtaining $C'$. Finally, translate $C'$ by adding $x_A$ and $y_A$ to the components in order to obtain $C$.
My question is:
How can I perform step 4 and five? |
Calculate Pearson Correlation Confidence Interval in Python
import numpy as npfrom scipy import stats
Recently, many studies have been arguing that we should report effect sizes along with confidence intervals, as opposed to simply reporting p values (e.g., see this paper).
In Python, however, there is no functions to directly obtain confidence intervals (CIs) of Pearson correlations. I therefore decided to do a quick ssearch and come up with a wrapper function to produce the correlation coefficients, p values, and CIs based on
scipy.stats and
numpy.
There are many tutorials on the detailed steps and I mainly followed this one.
Detailed steps
Let’s use a random dataset for an example.
> x = np.random.randint(1, 10, 10)> y = np.random.randint(1, 10, 10)> xarray([6, 4, 3, 3, 2, 5, 8, 2, 6, 1])> yarray([3, 3, 9, 4, 9, 4, 6, 9, 7, 9])
The first step involves transformation of the correlation coefficient into a Fishers’ Z-score.
> r, p = stats.pearsonr(x,y)> r,p(-0.5356559002279192, 0.11053303487716389)> r_z = np.arctanh(r)> r_z-0.5980434968020534
The corresponding standard deviation is $se = \dfrac{1}{\sqrt{N-3}}$:
> se = 1/np.sqrt(x.size-3)> se0.3779644730092272
CI under the transformation can be calculated as $r_z \pm z_{\alpha/2}\times se$, where $z_{\alpha/2}$ is can be calculated using
scipy.stats.norm.ppf function:
> alpha = 0.05> z = stats.norm.ppf(1-alpha/2)> lo_z, hi_z = r_z-z*se, r_z+z*se> lo_z, hi_z(-1.3388402513358, 0.14275325773169323)
Finally, we can reverse the transformation by
np.tanh:
> lo, hi = np.tanh((lo_z, hi_z))> lo, hiarray([-0.87139341, 0.1417914 ])
We can validate this in R:
> x=c(6, 4, 3, 3, 2, 5, 8, 2, 6, 1)> y=c(3, 3, 9, 4, 9, 4, 6, 9, 7, 9)> cor.test(x,y) Pearson's product-moment correlationdata: x and yt = -1.7942, df = 8, p-value = 0.1105alternative hypothesis: true correlation is not equal to 095 percent confidence interval: -0.8713934 0.1417914sample estimates: cor-0.5356559
A wrapper function
I wrapped all these steps up within a single function on Github gist: |
eISSN:
2163-2480 Evolution Equations & Control Theory
March 2016 , Volume 5 , Issue 1
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Abstract:
We discuss the notion of the well productivity index (PI) for the generalized Forchheimer flow of fluid through porous media. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, [5,3,4]. Here we generalize our results from [4] in case of arbitrary order of the nonlinearity of the flow. In case of the compressible gas flow the mathematical model of the PI is studied for the first time. In contrast to slightly compressible fluid the PI stays ``almost'' constant for a long period of time, but then it blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The ``greater'' are the initial reserves, the larger is this critical value. We present numerical and theoretical results for the time asymptotic of the PI and its stability with respect to the initial data.
Abstract:
In this paper we study the behavior of the energy and the $L^{2}$ norm of solutions of the wave equation with localized linear damping in exterior domain. Let $u$ be a solution of the wave system with initial data $\left( u_{0},u_{1}\right) $. We assume that the damper is positive at infinity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that:
1. If $(u_{0},u_{1}) $ belong to $H_{0}^{1}( \Omega) \times L^{2}( \Omega ) ,$ then the total energy $ E_{u}(t) \leq C_{0}(1+t) ^{-1}I_{0}$ and $\Vert u(t) \Vert _{L^{2}}^{2}\leq C_{0}I_{0},$ where \begin{equation*} I_{0}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}. \end{equation*} 2. If the initial data $\left( u_{0},u_{1}\right) $ belong to $ H_{0}^{1}\left( \Omega \right) \times L^{2}\left( \Omega \right) $ and verifies \begin{equation*} \left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}<+\infty , \end{equation*} then the total energy $E_{u}\left( t\right) \leq C_{2}\left( 1+t\right) ^{-2}I_{1}$ and $\left\Vert u\left( t\right) \right\Vert _{L^{2}}^{2} \leq C_{2} \left( 1+t\right) ^{-1}I_{1},$ where \begin{equation*} I_{1}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}+\left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}^{2} \end{equation*} and \begin{equation*} d\left( x\right) =\left\{ \begin{array}{lc} \left\vert x\right\vert & d\geq 3, \\ \left\vert x\right\vert \ln \left( B\left\vert x\right\vert \right) & d=2, \end{array} \right. . \end{equation*} with $B$ $\underset{x\in \Omega }{\inf } \left\vert x\right\vert \geq 2$.
Abstract:
We investigate a class of semilinear parabolic and elliptic problems with fractional dynamic boundary conditions. We introduce two new operators, the so-called fractional Wentzell Laplacian and the fractional Steklov operator, which become essential in our study of these nonlinear problems. Besides giving a complete characterization of well-posedness and regularity of bounded solutions, we also establish the existence of finite-dimensional global attractors and also derive basic conditions for blow-up.
Abstract:
We consider the stochastic linear quadratic optimal control problem for state equations of the Itô-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.
Abstract:
We study the Cauchy problem of the relativistic Nordström-Vlasov system. Under some additional conditions, total energy for weak solutions with BV scalar field are shown to be conserved.
Abstract:
In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations.
Abstract:
We study the free dynamic operator $\mathcal{A}$ which arises in the study of a heat-viscoelastic structure model with highly coupled boundary conditions at the interface between the heat domain and the contiguous structure domain. We use Baiocchi's characterization on the interpolation of subspaces defined by a constrained map [1], [16,p 96] to identify a relevant subspace $V_0$ of both $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^∗)^{\frac{1}{2}})$, which is sufficient to determine the optimal regularity of the interface (boundary) $\to$ interior map $\mathcal{A}^{-1} \mathcal{B}_N$ from the interface to the energy space. Here, $\mathcal{B}_N$ is the (boundary) control operator acting at the interface in the Neumann boundary conditions.
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ok, suppose we have the set $U_1=[a,\frac{a+b}{2}) \cup (\frac{a+2}{2},b]$ where $a,b$ are rational. It is easy to see that there exists a countable cover which consists of intervals that converges towards, a,b and $\frac{a+b}{2}$. Therefore $U_1$ is not compact. Now we can construct $U_2$ by taking the midpoint of each half open interval of $U_1$ and we can similarly construct a countable cover that has no finite subcover.
By induction on the naturals, we eventually end up with the set $\Bbb{I} \cap [a,b]$. Thus this set is not compact
I am currently working under the Lebesgue outer measure, though I did not know we cannot define any measure where subsets of rationals have nonzero measure
The above workings is basically trying to compute $\lambda^*(\Bbb{I}\cap[a,b])$ more directly without using the fact $(\Bbb{I}\cap[a,b]) \cup (\Bbb{I}\cap[a,b]) = [a,b]$ where $\lambda^*$ is the Lebesgue outer measure
that is, trying to compute the Lebesgue outer measure of the irrationals using only the notions of covers, topology and the definition of the measure
What I hope from such more direct computation is to get deeper rigorous and intuitve insight on what exactly controls the value of the measure of some given uncountable set, because MSE and Asaf taught me it has nothing to do with connectedness or the topology of the set
Problem: Let $X$ be some measurable space and $f,g : X \to [-\infty, \infty]$ measurable functions. Prove that the set $\{x \mid f(x) < g(x) \}$ is a measurable set. Question: In a solution I am reading, the author just asserts that $g-f$ is measurable and the rest of the proof essentially follows from that. My problem is, how can $g-f$ make sense if either function could possibly take on an infinite value?
@AkivaWeinberger For $\lambda^*$ I can think of simple examples like: If $\frac{a}{2} < \frac{b}{2} < a, b$, then I can always add some $\frac{c}{2}$ to $\frac{a}{2},\frac{b}{2}$ to generate the interval $[\frac{a+c}{2},\frac{b+c}{2}]$ which will fullfill the criteria. But if you are interested in some $X$ that are not intervals, I am not very sure
We then manipulate the $c_n$ for the Fourier series of $h$ to obtain a new $c_n$, but expressed w.r.t. $g$.
Now, I am still not understanding why by doing what we have done we're logically showing that this new $c_n$ is the $d_n$ which we need. Why would this $c_n$ be the $d_n$ associated with the Fourier series of $g$?
$\lambda^*(\Bbb{I}\cap [a,b]) = \lambda^*(C) = \lim_{i\to \aleph_0}\lambda^*(C_i) = \lim_{i\to \aleph_0} (b-q_i) + \sum_{k=1}^i (q_{n(i)}-q_{m(i)}) + (q_{i+1}-a)$. Therefore, computing the Lebesgue outer measure of the irrationals directly amounts to computing the value of this series. Therefore, we first need to check it is convergent, and then compute its value
The above workings is basically trying to compute $\lambda^*(\Bbb{I}\cap[a,b])$ more directly without using the fact $(\Bbb{I}\cap[a,b]) \cup (\Bbb{I}\cap[a,b]) = [a,b]$ where $\lambda^*$ is the Lebesgue outer measure
What I hope from such more direct computation is to get deeper rigorous and intuitve insight on what exactly controls the value of the measure of some given uncountable set, because MSE and Asaf taught me it has nothing to do with connectedness or the topology of the set
Alessandro: and typo for the third $\Bbb{I}$ in the quote, which should be $\Bbb{Q}$
(cont.) We first observed that the above countable sum is an alternating series. Therefore, we can use some machinery in checking the convergence of an alternating series
Next, we observed the terms in the alternating series is monotonically increasing and bounded from above and below by b and a respectively
Each term in brackets are also nonegative by the Lebesgue outer measure of open intervals, and together, let the differences be $c_i = q_{n(i)-q_{m(i)}}$. These form a series that is bounded from above and below
Hence (also typo in the subscript just above): $$\lambda^*(\Bbb{I}\cap [a,b])=\sum_{i=1}^{\aleph_0}c_i$$
Consider the partial sums of the above series. Note every partial sum is telescoping since in finite series, addition associates and thus we are free to cancel out. By the construction of the cover $C$ every rational $q_i$ that is enumerated is ordered such that they form expressions $-q_i+q_i$. Hence for any partial sum by moving through the stages of the constructions of $C$ i.e. $C_0,C_1,C_2,...$, the only surviving term is $b-a$. Therefore, the countable sequence is also telescoping and:
@AkivaWeinberger Never mind. I think I figured it out alone. Basically, the value of the definite integral for $c_n$ is actually the value of the define integral of $d_n$. So they are the same thing but re-expressed differently.
If you have a function $f : X \to Y$ between two topological spaces $X$ and $Y$ you can't conclude anything about the topologies, if however the function is continuous, then you can say stuff about the topologies
@Overflow2341313 Could you send a picture or a screenshot of the problem?
nvm I overlooked something important. Each interval contains a rational, and there are only countably many rationals. This means at the $\omega_1$ limit stage, thre are uncountably many intervals that contains neither rationals nor irrationals, thus they are empty and does not contribute to the sum
So there are only countably many disjoint intervals in the cover $C$
@Perturbative Okay similar problem if you don't mind guiding me in the right direction. If a function f exists, with the same setup (X, t) -> (Y,S), that is 1-1, open, and continous but not onto construct a topological space which is homeomorphic to the space (X, t).
Simply restrict the codomain so that it is onto? Making it bijective and hence invertible.
hmm, I don't understand. While I do start with an uncountable cover and using axiom of choice to well order the irrationals, the fact that the rationals are countable means I eventually end up with a countable cover of the rationals. However the telescoping countable sum clearly does not vanish, so this is weird...
In a schematic, we have the following, I will try to figure this out tomorrow before moving on to computing the Lebesgue outer measure of the cantor set:
@Perturbative Okay, kast question. Think I'm starting to get this stuff now.... I want to find a topology t on R such that f: R, U -> R, t defined by f(x) = x^2 is an open map where U is the "usual" topology defined by U = {x in U | x in U implies that x in (a,b) \subseteq U}.
To do this... the smallest t can be is the trivial topology on R - {\emptyset, R}
But, we required that everything in U be in t under f?
@Overflow2341313 Also for the previous example, I think it may not be as simple (contrary to what I initially thought), because there do exist functions which are continuous, bijective but do not have continuous inverse
I'm not sure if adding the additional condition that $f$ is an open map will make an difference
For those who are not very familiar about this interest of mine, besides the maths, I am also interested in the notion of a "proof space", that is the set or class of all possible proofs of a given proposition and their relationship
Elements in a proof space is a proof, which consists of steps and forming a path in this space For that I have a postulate that given two paths A and B in proof space with the same starting point and a proposition $\phi$. If $A \vdash \phi$ but $B \not\vdash \phi$, then there must exists some condition that make the path $B$ unable to reach $\phi$, or that $B$ is unprovable under the current formal system
Hi. I believe I have numerically discovered that $\sum_{n=0}^{K-c}\binom{K}{n}\binom{K}{n+c}z_K^{n+c/2} \sim \sum_{n=0}^K \binom{K}{n}^2 z_K^n$ as $K\to\infty$, where $c=0,\dots,K$ is fixed and $z_K=K^{-\alpha}$ for some $\alpha\in(0,2)$. Any ideas how to prove that? |
So far in my education career I have only met differential equations as small parts of courses on other stuff. Solving special cases as part of calculus, solving simple systems as a part of linear algebra. This coming semester I'm going to have two courses devoted entirely to differential equations, so I thought I would try to gain some understanding that isn't purely mechanical.
Here's an example I have some questions about.
This is example 10.2.1 from 'KALKULUS' (3rd edition) by Lindstrøm. The translation is mine. The part where the differential equation is solved has been removed.
An animal population consists today of $P$ animals and has a growth rate $r$. How big is the population in $t$ years?
Let $y(t)$ be the population size after $t$ years. In the time between $t$ and $t+\Delta t$ the population increases from $y(t)$ to $y(t+\Delta t)$, i.e. an increase of $y(t+\Delta t)-y(t)$. We can also derive this increase in another way: The growth rate is $r$, which means that the population increase per time unit is $ry(t)$. During a small time interval from $t$ to $t+\Delta t$ the population increase is approximately $ry(t)\Delta t$, an approximation that gets better with smaller $\Delta t$'s. If we equate these expressions, we get
$$y(t+\Delta t)-y(t)\approx ry(t)\Delta t$$
Dividing by $\Delta t$, we get
$$\frac{y(t+\Delta t)-y(t)}{\Delta t}\approx ry(t)$$
Letting $\Delta t$ go towards zero, this gives
$$y'(t)=ry(t)$$
Thus we have a differential equation that $y$ has to satisfy:
$$y'(t)-ry(t)=0$$
NOTE: We could have gotten this differential equation faster by using the fact that the growth rate $r$ by definition means that $y'(t)=ry(t).$ We have chosen the more elaborate approach because it shows a general thought process which can be used in several situations.
Why is the population increase approximately $ry(t)Δt$ during a small time interval from $t$ to $t+Δt$?
How is this thought process different than using the definition? In the real world it has been observed that the growth of a population tend to depend on the size of the population, so you want the change in population to depend on its size. In the language of calculus one way to write this out is $y'(t)=ry(t)$. What's the advantage of going through the more elaborate reasoning? |
Integrate over the region in the first octant above the parabolic cylinder and below the paraboloid
I could not get the limits right even that I tried many one but I still could not get it
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We can set up the integral as follows...
$$ \iiint F(x,y,z)dzdydx$$
For the z integration or bounds it is simply from the lower surface ($z=y^2$) to the upper surface $(z=8-2x^2-y^2)$ So now we have...
$$ \iint \!\!\int_{y^2}^{8-2x^2-y^2}(8xz) dzdydx $$
Now, we can think of the integral as being resolved onto the xy-plane with z=0. Setting the two functions of x and y equal to each other and simplifying, we get $x^2+y^2=4$. Now, trying to find the y bounds we solve to $y$ in terms of $x$. Thus, $y=\sqrt{4-x^2}$. Next, we need to integrate along the x axis where $y=0$. This is from $0$ to $2$. And our integral ends up being:
$$\int_0^2\!\!\!\int_0^{\sqrt{4-x^2}}\!\!\int_{y^2}^{8-2x^2-y^2}(8xz)dzdydx$$
It has been a while since I have done these so please correct me if there is an error or a better way to do it. |
Symbols:Greek/Pi Contents Pi
The $16$th letter of the Greek alphabet.
Minuscules: $\pi$ and $\varpi$ Majuscule: $\Pi$
The $\LaTeX$ code for \(\pi\) is
\pi .
The $\LaTeX$ code for \(\varpi\) is
\varpi .
The $\LaTeX$ code for \(\Pi\) is
\Pi .
$\pi$ $\map \pi x$ That is: $\displaystyle \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$ The $\LaTeX$ code for \(\map \pi x\) is
\map \pi x .
$\pi_i$
The notation $\pi_i$ is often used for the $i$th projection.
The $\LaTeX$ code for \(\pi_i\) is
\pi_i .
$\map {\Pi_X} s$
Let $p_X$ be the probability mass function for $X$.
The probability generating function for $X$, denoted $\map {\Pi_X} s$, is the formal power series defined by: $\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$ The $\LaTeX$ code for \(\map {\Pi_X} s\) is
\map {\Pi_X} s .
$\displaystyle \prod_{j \mathop = 1}^n a_j$
Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
The composite is called the product of $\tuple {a_1, a_2, \ldots, a_n}$, and is written: $\displaystyle \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$ The $\LaTeX$ code for \(\displaystyle \prod_{j \mathop = 1}^n a_j\) is
\displaystyle \prod_{j \mathop = 1}^n a_j .
The $\LaTeX$ code for \(\displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j\) is
\displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j .
The $\LaTeX$ code for \(\displaystyle \prod_{\map \Phi j} a_j\) is
\displaystyle \prod_{\map \Phi j} a_j . |
April 15th, 2016, 09:42 PM
# 1
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Joined: Mar 2016
From: Nepal
Posts: 37
Thanks: 4
Explain Fourier transform please
I bumped into Fourier transform and from its applications I found it was very important for scientists. But I don't know why. I tried going through Wikipedia, but it didn't help. I really want to understand this.
Please help. In layman's terms as far as possible.
Last edited by skipjack; April 16th, 2016 at 05:10 AM.
April 16th, 2016, 12:52 AM
# 2
Senior Member
Joined: Jun 2015
From: England
Posts: 915
Thanks: 271
Quote:
You need to be careful which one you mean.
There are many repetitive phenomena in Nature and in Man's experience.
The tides in the oceans
Soundwaves in the air
Music
Electrical waves in electrical apparatus
Mechanical vibrations in buildings and machinery.
The list goes on and on.
The Fast Fourier Transform (often abbreviated to FFT or sometimes FT) is a numerical mathematical method for extracting Fourier Series coefficients from tables of measurements on the phenomena.
These coefficients are used by engineers and scientists for many design, repair and inspection processes.
The second mathematical process is an analytical mathematics technique for solving some difficult mathematical equations by replacing the variables with different ones that offer easier equations.
This is what is meant by 'transform'.
There are many examples of transform methods in mathematics. Do you know the equation of a circle and of a parabola?
Last edited by skipjack; April 16th, 2016 at 05:13 AM.
April 16th, 2016, 08:51 AM
# 4
Senior Member
Joined: Jun 2015
From: England
Posts: 915
Thanks: 271
Quote:
$\displaystyle {x^2} + {y^2} + 2gx + = {R^2}$
which has two variables, x and y
we can transform it into a pair of equations in one variable
$\displaystyle x = R\cos t$
$\displaystyle y = R\sin t$
Similarly with the parabola
$\displaystyle {y^2} = 4ax$
we can transform it into a pair of equations in one variable.
$\displaystyle x = a{t^2}$
$\displaystyle y = 2at$
First some more introduction.
M = 5.73204 x 1597.235676
Find M
In the past this was a difficult multiplication.
It was made easier by the logarithmic transformation into an addition.
So a multiplication equation is transformed into and addition
logM = log(5.73204) + log(1597.235676)
M = antilog (logM)
The General Fourier Transform includes the Laplace transform, amongst others.
These types of transformation do the same thing for differential equations,
That is they transform a difficult differential equation into an addition.
Once you have perfome the addition you have to perform the inverse transformation, just as taking the antilog above.
Do you understand differential equations?
Last edited by studiot; April 16th, 2016 at 08:53 AM.
Tags explain, fourier, transform
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Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
No. If $f:(0,1) \to [0,1]$ were continuous and bijective, there would be a unique point $x \in (0,1)$ such that $f(x) = 1$. However, since $f$ is continuous, the intervals $[x - \varepsilon, x]$ and $[x, x + \varepsilon]$ would be mapped to intervals $[a,1]$ and $[b,1]$, say. By bijectivity we'd have $a, b \lt 1$. Thus every value strictly between $\max{\{a,b\}}$ and $1$ would be assumed at least twice, contradicting bijectivity.
Let $f:(0,1) \rightarrow [0,1]$ be continuous and surjective. (Actually, we just need to suppose that $0$ and $1$ are in the image of $f$.) Let $a,b \in (0,1)$ such that $f(a)=0$ and $f(b)=1$. Let $I=[a,b]$ if $a<b$ or $I=[b,a]$ if $b<a$. Then, by the intermediate value theorem, $f(I)$ is an interval that contains $0$ and $1$ and so $f(I)$ contains $[0,1]$, which implies $f(I)=[0,1]$. But then $f$ cannot be injective because there are lots of points in $(0,1)\setminus I$.
Suppose that $f:(0,1) \rightarrow [0,1]$ is 1-1 and continuous. By the intermediate value theorem, the image of any interval under $f$ is an interval. Since $f$ is 1-1, it is either (strictly) monotone increasing or decreasing. Hence, $f(0,1)$ is an interval. Without loss of generality, assume $f$ is increasing; were it not this analysis would apply to $1 - f$.
Suppose now that $f$ is onto; then we must have some $t\in(0,1)$ with $f(t) = 1$. Because $f$ is strictly monotone increasing, we would have to have $f(s) > 1$, for $t \le s < 1$. This violates the premise that $f(0,1) \subseteq [0,1]$. Hence, $f$ cannot be onto.
Since Theo gave an answer I am going to be nitpicking and add one remark. When speaking about continuity (especially when tagging under [topology]) it is best to mention the topology you are working with. In this case, you mean in the
standard topology.
Otherwise, consider the discrete topology, i.e. every set is open:
Let $f\colon [0,1]\to (0,1)$ be any bijection, it is continuous since all sets are open, the preimage of an open set is an open set, thus $f$ is continuous.
There does not exist a continuous bijection from (0,1) to [0,1]. Indeed, let $f$ be such a function. Let consider a sequence $x_n=1-1/n$. Then from the sequence $(f(x_n))$ we can choose a subsequence $(f(x_{n_k}))$ which is convergent. Let denote this limit by $y$. Obviously, $y \in [0,1]$. Since $f^{-1}$ also is continuous, we get $f^{-1}(y)=\lim_{k \to +\infty}f^{-1}(f(x_{n_k}))=\lim_{k \to \infty}x_{n_k}=1$. But $1 \notin (0,1)$.
Remark(Why $f^{-1}$ must be continuous under our assumption?) By our assumption $f:(0,1)\to [0,1]$ is continuous bijection. Then $f:(0,1)\to [0,1]$ must be injective and continuous which following invariance of domain (see, http://en.wikipedia.org/wiki/Invariance_of_domain is homeomorphism. Hence $f^{-1}: [0,1]\to (0,1)$ is continuous.
protected by Asaf Karagila♦ Jan 3 '16 at 19:20
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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 \(\mathbf{\sigma}\). |
Two popular methods to find the bandwidth $latex {h}&fg=000000$ for the nonparametric density estimator are the plug-in method and the method cross-validation. The first one we will focus in the “quick and dirty” plug-in method introduced by Silverman (1986). In cross-validation we will minimize a modified version of the quadratic risk of $latex {\hat{f}_{h}}&fg=000000$.
The normal reference rule
This method works well only if the true density is very smooth. Assume that $latex {f}&fg=000000$ is normal distributed. Then we have
$latex \displaystyle h_{plug}=1.06\sigma n^{-1/5}. &fg=000000$
Usually $latex {\sigma}&fg=000000$ is estimated by $latex {\min\{s,Q/1.34\}}&fg=000000$ where $latex {s}&fg=000000$ is the sample standard deviation and $latex {Q}&fg=000000$ is the interquartile range. Recall that the interquartile range is the $latex {75^{\text{th}}}&fg=000000$ percentile minus the $latex {25^{\text{th}}}&fg=000000$ percentile. Here, $latex {Q/1.34}&fg=000000$ gives a consistent estimate of $latex {\sigma}&fg=000000$ if the data comes from a $latex {N(\mu,\sigma^{2})}&fg=000000$.
We can summarize this method in the following way
$latex \displaystyle h_{plug}=1.06\min\left\{ \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}},\frac{Q}{1.34}\right\} n^{-1/5}. &fg=000000$
Define the
integrated squared error as
$latex \displaystyle \begin{array}{rl} \displaystyle {\rm ISE}(\hat{f}_{h}) & =\displaystyle \int\left(\hat{f}_{h}(x)-f(x)\right)^{2}dx\nonumber \\ & =\displaystyle \int\hat{f}_{h}^{2}(x)dx-2\int\hat{f}_{h}(x)f(x)dx+\int f^{2}(x)dx. \end{array} &fg=000000$
Notice that the MISE is indeed the expected value of ISE. Our goal is minimize the ISE as small as possible. Remark that the last term in (0) does not depends on $latex {h}&fg=000000$, so minimize this risk the is equivalent to minimizing the expected value of
$latex \displaystyle {\rm ISE}(\hat{f}_{h})-\int f^{2}(x)dx=\int\hat{f}_{h}^{2}(x)dx-2\int\hat{f}_{h}(x)f(x)dx &fg=000000$
If we look closer the term $latex {\int\hat{f}_{h}(x)f(x)dx}&fg=000000$ we notice that is the expected value of $latex {\mathbb E(\hat{f}_{h}(X))}&fg=000000$. The straight estimate for this expected value is
$latex \displaystyle \frac{1}{n}\sum_{i=1}^{n}\hat{f}_{h}(X_{i})=\frac{1}{n^{2}h}\sum_{i=1}^{n}\sum_{j=1}^{n}K\left(\frac{X_{j}-X_{i}}{h}\right). \ \ \ \ \ (1)&fg=000000$
The problem with it is that the observations to estimate the expectation are dependent of the observations to estimate $latex {\hat{f}_{h}}&fg=000000$. The solution to solve this, it is remove the $latex {i^{\text{th}}}&fg=000000$ observation for $latex {\hat{f}_{h}}&fg=000000$. Then, we define the leave-one-out cross-validation estimator of $latex {\int\hat{f}_{h}(x)f(x)dx}&fg=000000$ as
$latex \displaystyle \frac{1}{n}\sum_{i=1}^{n}\hat{f}_{h,-i}(X_{i}), &fg=000000$
where
$latex \displaystyle \hat{f}_{h,-i}(x)=\frac{1}{n-1}\mathop{\sum_{j=1}^{n}}_{j\neq i}K_{h}(x-X_{j}). &fg=000000$
The following figure illustrates the idea behind the leave-one cross validation. The idea is to take one data point as your test data and the rest as your training data for each iteration.
Following with the $latex {\int\hat{f}_{h}^{2}(x)dx}&fg=000000$ term we have
$latex \displaystyle \begin{array}{rl} \displaystyle \int\hat{f}_{h}^{2}(x)dx & =\displaystyle \int\left(\frac{1}{n}\sum_{i=1}^{n}K_{h}(x-X_{i})\right)^{2}dx\\ & =\displaystyle \frac{1}{n^{2}h^{2}}\sum_{i=1}^{n}\sum_{i=1}^{n}\int K\left(\frac{x-X_{i}}{h}\right)K\left(\frac{x-X_{j}}{h}\right)dx\\ & =\displaystyle \frac{1}{n^{2}h}\sum_{i=1}^{n}\sum_{i=1}^{n}\int K\left(u\right)K\left(\frac{X_{i}-X_{j}}{h}-u\right)du\\ & =\displaystyle \frac{1}{n^{2}h}\sum_{i=1}^{n}\sum_{i=1}^{n}K*K\left(\frac{X_{i}-X_{j}}{h}\right). \end{array} &fg=000000$
where $latex {K*K}&fg=000000$ means the convolution of $latex {K}&fg=000000$ with itself.
Finally it is possible define a reasonable criterion to choose the bandwidth,
$latex \displaystyle CV(h)=\frac{1}{n^{2}h}\sum_{i=1}^{n}\sum_{j=1}^{n}K*K\left(\frac{X_{i}-X_{j}}{h}\right)-\frac{2}{n(n-1)}\sum_{i=1}^{n}\mathop{\sum_{j=1}^{n}}_{j\neq i}K_{h}(X_{i}-X_{j}). &fg=000000$
Note: An alternative way to implement the leave-one-out ,
$latex \displaystyle CV(h)=\int\hat{f}_{h}^{2}(x)dx-\frac{2}{n(n-1)}\sum_{i=1}^{n}\mathop{\sum_{j=1}^{n}}_{j\neq i}K_{h}(X_{i}-X_{j}) &fg=000000$
and then calculate numerically the integral.
Sources: Hardle, W. (2004). Nonparametric and Semiparametric Models. Springer Series in Statistics. Springer. Tsybakov, A. (2009). Introduction to nonparametric estimation. Springer. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, volume 26. Chapman & Hall/CRC. Related articles Check your missing-data imputations using cross-validation (andrewgelman.com) |
We study separability problem using general symmetric informationallycomplete measurements and propose separability criteria in$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ and$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\cdots\otimes\mathbb{C}^{d_{n}}$.Our criteria just require less local measurements and provide experimentalimplementation in detecting entanglement of unknown quantum states.
In this paper we introduce the notion of the stability of a sequence ofmodules over Hecke algebras. We prove that a finitely generated consistentsequence associated with Hecke algebras is representation stable.
In this paper, we present some generalized monogamy inequalities based onnegativity and convex-roof extended negativity (CREN). These monogamy relationsare satisfied by the negativity of $N$-qubit quantum systems $ABC_1\cdotsC_{N-2}$, under the partitions $AB|C_1\cdots C_{N-2}$ and $ABC_1|C_2\cdotsC_{N-2}$. Furthermore, the $W$-class states are used to test these generalizedmonogamy inequalities.
In recent papers, the theory of representations of finite groups has beenproposed to analyzing the violation of Bell inequalities. In this paper, weapply this method to more complicated cases. For two partite system, Alice andBob each make one of $d$ possible measurements, each measurement has $n$outcomes. The Bell inequalities based on the choice of two orbits are derived. Theclassical bound is only dependent on the number of measurements $d$, but thequantum bound is dependent both on $n$ and $d$. Even so, when $d$ is largeenough, the quantum bound is only dependent on $d$. The subset of probabilitiesfor four parties based on the choice of six orbits under group action isderived and its violation is described. Restricting the six orbits to threeparties by forgetting the last party, and guaranteeing the classical boundinvariant, the Bell inequality based on the choice of four orbits is derived.Moreover, all the corresponding nonlocal games are analyzed.
We study the problem of distinguishing maximally entangled quantum states byusing local operations and classical communication (LOCC). A question offundamental interest is whether any three maximally entangled states in$\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are distinguishable by LOCC. Inthis paper, we restrict ourselves to consider the generalized Bell states. Andwe prove that any three generalized Bell states in$\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are locally distinguishable.
We studied the construction problem of the unextendible product basis (UPB).We mainly give a method to construct a UPB of a quantum system through the UPBsof its subsystem. Using this method and the UPBs which are known for us, weconstruct different kinds of UPBs in general bipartite quantum system. Then weuse these UPBs to construct a family of UPBs in multipartite quantum system.The UPBs can be used to construct the bound entangled states with differentranks.
In this paper, we give a BLM realization of the positive part of the quantumgroup of $U_v(gl_n)$ with respect to RTT relations.
In \cite{fl}, the authors get a new presentation of two-parameter quantumalgebra $U_{v,t}(\mathfrak{g})$. Their presentation can cover all Kac-Moodycases. In this paper, we construct a suitable Hopf pairing such that$U_{v,t}(sl_{n})$ can be realized as Drinfeld double of certain Hopfsubalgebras with respect to the Hopf pairing. Using Hopf pairing, we constructa $R$-matrix for $U_{v,t}(sl_{n})$ which will be used to give the Schur-Weyldual between $U_{v,t}(sl_{n})$ and Hecke algebra $H_{k}(v,t)$. Furthermore,using the Fusion procedure we construct the primitive orthogonal idempotents of$H_{k}(v,t)$. As a corollary, we give the explicit construction of irreducible$U_{v,t}(sl_{n})$-representations of $V^{\otimes k}$.
In this paper, we give an geometric description of the Schur-Weyl duality fortwo-parameter quantum algebras $U_{v, t}(gl_n)$, where $U_{v, t}(gl_n)$ is thedeformation of $U_v(I, \cdot)$, the classic Shur-Weyl duality $(U_{r, s}(gl_n),V^{\otimes d}, H_d(r, s))$ can be seen as a corollary of the Shur-Weyl duality$(U_{v, t}(gl_n), V^{\otimes d}, H_d(v, t))$ by using the galois descendapproach. we also establish the Shur-Weyl duality between the algebras$\widetilde{U_{v, t}(gl_N)^m}$, $\widehat{U_{v, t}(gl_N)^m}$ and Heck algebra$H_k(v, t)$.
In this paper, we introduce and study the quantum deformations of the clustersuperalgebra. Then we prove the quantum version of the Laurent phenomenon forthe super-case.
In this paper, we study the concurrence of arbitrary dimensional tripartitequantum systems. An explicit operational lower bound of concurrence is obtainedin terms of the concurrence of sub-states. A given example show that our lowerbound may improve the well known existing lower bounds of concurrence. Thesignificance of our result is to get a lower bound when we study theconcurrence of arbitrary dimensional multipartite quantum systems.
We study the concurrence of arbitrary dimensional multipartite quantumsystems. An explicit analytical lower bound of concurrence is obtained in termsof the concurrences of sub-quantum systems. Detailed examples are given to showthat our lower bounds improve the existing lower bounds of concurrence.
We study the unextendible maximally entangled bases (UMEB) in$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect it with the partialHadamard matrix. Firstly, we show that for a given special UMEB in$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrixcan not extend to a complete Hadamard matrix in $\mathbb{C}^{d}$. As acorollary, any $(d-1)\times d$ partial Hadamard matrix can extend to a completeHadamard matrix. Then we obtain that for any $d$ there is an UMEB except $d=p\\text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. Finally, we arguethat there exist different kinds of constructions of UMEB in$\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and$d=3\times5 \times7$.
In this paper, we study the one-way local operations and classicalcommunication (LOCC) problem. In $\mathbb{C}^d\otimes\mathbb{C}^d$ with$d\geq4$, we construct a set of $3\lceil\sqrt{d}\rceil-1$ one-way LOCCindistinguishable maximally entangled states which are generalized Bell states.Moreover, we show that there are four maximally entangled states which cannotbe perfectly distinguished by one-way LOCC measurements for any dimension$d\geq 4$.
In this paper, we mainly study the local indistinguishability of multipartiteproduct states. Firstly, we follow the method of Z.-C. Zhang \emph{et al}[Phys.Rev. A 93, 012314(2016)] to give another more concise set of $2n-1$ orthogonalproduct states in $\mathbb{C}^m\otimes\mathbb{C}^n(4\leq m\leq n)$ which cannot be distinguished by local operations and classical communication(LOCC).Then we use the 3 dimension cubes to present some product states which give usan intuitive view how to construct locally indistinguishable product states intripartite quantum system. At last, we give an explicit construction of locallyindistinguishable orthogonal product states for general multipartite system.
V.S.Guba had proved that the R.Thompson group $T$ satisfies polynomialisoperimetric inequality and $\Phi_T(n)\preceq n^7$, where $\Phi_T$ is the Dehnfunction of group $T$. In this paper, we show that $\Phi_T(n)\preceq n^5$.
We study the local indistinguishability of mutually orthogonal product basisquantum states in the high-dimensional quantum system. In the quantum system of$\mathbb{C}^d\otimes\mathbb{C}^d$, where $d$ is odd, Zhang \emph{et al} haveconstructed $d^2$ orthogonal product basis quantum states which are locallyindistinguishable in [Phys. Rev. A. {\bf 90}, 022313(2014)]. We find a subsetcontains with $6d-9$ orthogonal product states which are still locallyindistinguishable. Then we generalize our method to arbitrary bipartite quantumsystem $\mathbb{C}^m\otimes\mathbb{C}^n$. We present a small set with only$3(m+n)-9$ orthogonal product states and prove these states are LOCCindistinguishable. Even though these $3(m+n)-9$ product states are LOCCindistinguishable, they can be distinguished by separable measurements. Thisshows that separable operations are strictly stronger than the local operationsand classical communication.
We give a explicit construction of $d$ locally indistinguishable orthogonalmaximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d$ for any $d\geq 4$. This gives an answer tothe conjecture proposed by S. Bandyopadhyay in 2009. Thus it reflects the nonlocality of the fundamentalfeature of quantum mechanics.
In this paper, we give a method for the local unitary equivalent problemwhich is more efficient than that was proposed by Bin Liu $et \ al$\cite{bliu}.
We solved the unextendible maximally entangled basis (UMEB) problem in$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d'}(d\neq d')$,the results turn out to bethat there always exist a UMEB.In addition,there might be two sets of UMEB withdifferent numbers.The main difficult is to prove the unextendibility of the setof states.We give an explicit construction of UMEB by considering the Schmidtnumber of the complementary space of the states we construct.
In this paper, we give a quantum cluster algebra structure on the deformedGrothendieck ring of $\CC_{n}$, where $\CC_{n}$ is a full subcategory of finitedimensional representations of $U_q(\widehat{sl_{2}})$ defined in section II.
In this paper, we prove one case of the conjecture given by Hernandez andLeclerc\cite{HL0}. Specifically, we give a cluster algebra structure on theGrothendieck ring of a full subcategory of the finite dimensionalrepresentations of a simply-laced quantum affine algebra $U_q(\widehat{\g})$.In the procedure, we also give a specific description of compatible subsets oftype $E_{6}$. As a conclusion, for every exchange relation of cluster algebrathere exists a exact sequence of the full subcategory corresponding to it.
This paper studies the Kazhdan-Lusztig coefficients $\mu(u,w)$ of theKazhdan-Lusztig polynomials $P_{u,w}$ for the lowest cell ${c_{0}}$ of anaffine Weyl group of type $\widetilde{G_{2}}$ and gives an estimation$\mu(u,w)\leqslant 3$ for $u,w\in c_{0}$.
In this paper, we develop 2-dimensional algebraic theory which closelyfollows the classical theory of modules. The main results are givingdefinitions of 2-module and the representation of 2-ring. Moreover, for a2-ring $\cR$, we prove that its modules form a 2-Abelian category.
In this paper, we will construct the injective resolution of any$\cR$-2-module, define the right derived 2-functor, and give some relatedproperties of the derived 2-functor in ($\cR$-2-Mod). |
B P Singh
Articles written in Pramana – Journal of Physics
Volume 3 Issue 2 August 1974 pp 61-73 Nuclear Physics
The structure of the low-lying states of
58Ni has been calculated in shell model by assuming an inert 56Ni core plus two valence nucleons in the p 3/2, f 5/2 and p 1/2 orbitals. The two-body matrix elements are first expressed in terms of seven radial matrix elements and these are then parametrized to give best fit between the computed and the observed energies of the levels below 4 MeV. The wave-functions obtained using these two-body matrix elements are used to study the concept of effective charges. It is found that a single effective charge is not sufficient to predict the
Volume 8 Issue 1 January 1977 pp 91-97 Nuclear And Particle Physics
max=0.12 MeV, max=0.21 MeV, 2(cos 4(cos max 0.12 MeV→557→53 keV cascade and 2(cos 4(cos max=0.21 MeV→444 keV→53 keV cascade.
Spins and parities of the 650, 537 and 93 keV levels of
103Rh are deduced by triple angular correlation and the internal conversion coefficient studies. Multipolarities of the transitions are also determined.
Volume 12 Issue 3 March 1979 pp 243-250 Nuclear And Particle Physics
Some of the low-lying states in many isotopes
144Nd, 148Sm, 152Gd and 156Gd show a similar typical behaviour. The first 2 + is regarded as a single quadrupole phonon state and 3 − as a single octupole phonon state. The levels with the spins and parities 1 −, 5 −, 3 −, 4 −, etc. are considered due to the simultaneous excitation of quadrupole and octupole phonons. If this consideration is correct, then the transition from − to 2 + states must contain an appreciable max 800 keV→ 144Nd.
Volume 30 Issue 3 March 1988 pp 245-249 Condensed Matter Physics
The method of generalized least squares has been used to deconvolute the Compton profile measurements in nickel. The method depends on two arbitrary parameters namely the cut-off parameter
Volume 47 Issue 5 November 1996 pp 401-410
The excitation functions for the reactions
127I( 129Cs, 127I( 127Cs, 133Cs( 135La and 133Cs( 133La have been measured up to ≈50 MeV 0=4 (2
Volume 87 Issue 4 October 2016 Article ID 0056 Regular
We have synthesized, characterized and studied the third-order nonlinear optical properties of two different nanostructures of polydiacetylene (PDA), PDA nanocrystals and PDA nanovesicles, along with silver nanoparticles-decorated PDA nanovesicles. The second molecular hyperpolarizability $\gamma (−\omega; \omega,−\omega,\omega$) of the samples has been investigated by antiresonant ring interferometric nonlinear spectroscopic (ARINS) technique using femtosecond mode-locked Ti:sapphire laser in the spectral range of 720–820 nm. The observed spectral dispersion of $\gamma$ has been explained in the framework of three-essential states model and a correlation between the electronic structure and optical nonlinearity of the samples has been established. The energy of two-photon state, transition dipole moments and linewidth of the transitions have been estimated. We have observed that the nonlinear optical properties of PDA nanocrystals and nanovesicles are different because of the influence of chain coupling effects facilitated by the chain packing geometry of the monomers. On the other hand, our investigation reveals that the spectral dispersion characteristic of $\gamma$ for silver nanoparticles-coated PDA nanovesicles is qualitatively similar to that observed for the uncoated PDA nanovesicles but bears no resemblance to that observed in silver nanoparticles. The presence of silver nanoparticles increases the $\gamma$ values of the coated nanovesicles slightly as compared to that of the uncoated nanovesicles, suggesting a definite but weak coupling between the free electrons of the metal nanoparticles and $\pi$ electrons of the polymer in the composite system. Our comparative studies show that the arrangement of polymer chains in polydiacetylene nanocrystals is more favourable for higher nonlinearity.
Current Issue
Volume 93 | Issue 5 November 2019
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Spring 2018, Math 171 Week 3 Stopping/Non-Stopping times
Let \(T_1, T_2\) be stopping times for some Markov Chain \(\{X_n:n \ge 0\}\). Which of the following will also necessarily be stopping times? Prove your claims.
(Discussed) \(T_3=5\)
\(T_4=T_1 + T_2 + 1\)
(Discussed) \(T_5=T_1 + T_2 - 1\)
(Solution) \(T_5\) will not necessarily be a stopping time. Suppose \(\{X_n:n \ge 0\}\) is the Markov Chain corresponding to the transition matrix \[P = \begin{matrix} & \mathbf 0 & \mathbf 1 \cr \mathbf 0 & 1/2 & 1/2 \cr \mathbf 1 & 0 & 1 \end{matrix}\] Suppose further that \(T_1 = \min\{n \ge 0: X_n = \mathbf 0\}\) and \(T_2 = \min\{n \ge 0: X_n = \mathbf 1\}\). If \(T_5\) were a stopping time we would have \(P(T_5 = n | X_n = x_n, \dots, X_0=x_0)\in \{0, 1\} \; \forall n\). However, by definition of \(T_5\)\[P(T_5 = 0 | X_0=\mathbf 0) = P(T_1 + T_2 = 1 | X_0=\mathbf 0)\] by directly enumerating the possibilities we see \[= P(T_1 = 1, T_2 = 0 | X_0=\mathbf 0)\] \[+ P(T_1 = 0, T_2 = 1 | X_0=\mathbf 0)\] and now using the Multiplication Rule \[= P(T_1 = 1 | T_2 = 0, X_0=\mathbf 0)P(T_2 = 0 | X_0=\mathbf 0)\] \[+ P(T_2 = 1 | T_1 = 0, X_0=\mathbf 0)P(T_1 = 0 | X_0=\mathbf 0)\] since \(T_1\) and \(T_2\) are stopping times this simplifies to \[= P(T_1 = 1 | X_0=\mathbf 0)P(T_2 = 0 | X_0=\mathbf 0)\]\[ + P(T_2 = 1 | X_0=\mathbf 0)P(T_1 = 0 | X_0=\mathbf 0)\] each term of which can be computed \[=0 \cdot 0 + \frac 1 2 \cdot 1\]\[= \frac 1 2 \notin \{0, 1\}\]
(Discussed) Solve problem 4(ii) on HW 2
(Discussed) Show Lemma 1.3 from the textbook Classification of States
Consider the Markov chain defined by the following transition matrix: \[P = \begin{matrix} & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 & \mathbf 6 & \mathbf 7 & \mathbf 8 \cr \mathbf 1 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 \cr \mathbf 2 & 0.5 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 \cr \mathbf 3 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0.5 \cr \mathbf 4 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 \cr \mathbf 5 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \cr \mathbf 6 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \cr \mathbf 7 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 \cr \mathbf 8 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}\] Identify the transient and recurrent states, and the irreducible closed sets in the Markov chain. Give reasons for your answers.
Consider the Markov chain defined by the following transition matrix: \[P = \begin{matrix} & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 & \mathbf 6 \cr \mathbf 1 & 0 & 0 & 1 & 0 & 0 & 0 \cr \mathbf 2 & 0 & 0 & 0 & 0 & 0 & 1 \cr \mathbf 3 & 0 & 0 & 0 & 0 & 1 & 0 \cr \mathbf 4 & 0.25 & 0.25 & 0 & 0.5 & 0 & 0 \cr \mathbf 5 & 1 & 0 & 0 & 0 & 0 & 0 \cr \mathbf 6 & 0 & 0.5 & 0 & 0 & 0 & 0.5 \end{matrix}\] Identify the transient and recurrent states, and the irreducible closed sets in the Markov chain. Give reasons for your answers.
Stationary Distributions
Recall a stationary distribution is a vector \(\pi\) satisfying: \[\sum _i \pi(i)=1\] \[\pi(i)\ge 0, \quad \forall i\] \[\pi P = \pi\]
Compute any and all stationary distributions of \[P = \begin{bmatrix} 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 \end{bmatrix}\] If you claim \(P\) has a unique stationary distribution, please justify.
(Partially Discussed) Under what circumstances is the stationary distribution of \[P = \begin{bmatrix} 1-r & 0 & r \cr p & 1-p & 0 \cr 0 & q & 1-q \end{bmatrix}\] unique? Justify your answer. Compute the stationary distribution in this case.
Compute any and all stationary distributions of \[P = \begin{bmatrix} 1 & 0 \cr 0 & 1 \end{bmatrix}\] If you claim \(P\) has a unique stationary distribution, please justify.
Compute any and all stationary distributions of \[P = \begin{bmatrix} P_1 & 0 \cr 0 & P_2 \end{bmatrix}\] where \(P_1\) has a unique stationary distribution \(\pi_1\) and \(P_2\) has a unique stationary distribution \(\pi_2\). If you claim \(P\) has a unique stationary distribution, please justify.
Compute any and all stationary distributions of \[P = \begin{bmatrix} 0 & p & 0 & 1-p \cr q & 0 & 1-q & 0 \cr 0 & 1-r & 0 & r \cr 1-s & 0 & s & 0 \end{bmatrix}\] If you claim \(P\) has a unique stationary distribution, please justify. |
What is critical velocity?
Critical velocity is defined as the speed at which a falling object reaches when both gravity and air resistance are equalised on the object.
The other way of defining critical velocity is the speed and direction at which the fluid can flow through a conduit without becoming turbulent. Turbulent flow is defined as the irregular flow of the fluid with continuous change in magnitude and direction. It is the opposite of laminar flow which is defined as the flow of fluid in parallel layers without disruptions of the layers.
Critical velocity formula
Following is the mathematical representation of critical velocity with the dimensional formula:
\(V_{C}=\frac{R_{e}\eta }{\rho r}\)
Where,
Vc: critical velocity
Re: Reynolds number (ratio of inertial forces to viscous forces)
𝜂: coefficient of viscosity
r: radius of the tube
⍴: density of the fluid
Dimensional formula of: Reynolds number (Re): M 0L 0T 0 Coefficient of viscosity (𝜂): M 1L -1T -1 Radius (r) : M 0L 1T 0 Density of fluid (⍴): M 1L -3T 0 Critical velocity: \(V_{c}=\frac{\left [ M^{0}L^{0}T^{0} \right ]\left [ M^{1}L^{-1}T^{-1} \right ]}{\left [ M^{1}L^{-3}T^{0} \right ]\left [ M^{0}L^{1}T^{0} \right ]}\)
∴\( V_{c}=M^{0}L^{1}T^{-1}\)
SI unit of critical velocity is ms
-1 Reynolds number
Reynolds number is defined as the ratio of inertial forces to viscous forces. Mathematical representation is as follows:
\(R_{e}=\frac{\rho uL}{\mu }=\frac{uL}{\nu }\)
Where,
⍴: density of the fluid in kg.m
-3
𝜇: dynamic viscosity of the fluid in m
2s
u: velocity of the fluid in ms
-1
L: characteristic linear dimension in m
𝜈: kinematic viscosity of the fluid in m
2s -1
Depending upon the value of Reynolds number, flow type can be decided as follows:
If Re is between 0 to 2000, the flow is streamlined or laminar If Re is between 2000 to 3000, the flow is unstable or turbulent If Re is above 3000, the flow is highly turbulent
Reynolds number with respect to laminar and turbulent flow regimes are as follows:
When the Reynolds number is low that is the viscous forces are dominant, laminar flow occurs and are characterised as a smooth, constant fluid motion When the Reynolds number is high that is the inertial forces are dominant, turbulent flow occurs and tends to produce vortices, flow instabilities and chaotic eddies.
Following is the derivation of Reynolds number:
\(R_{e}=\frac{ma}{\tau A}=\frac{\rho V.\frac{du}{dt}}{\mu \frac{du}{dy}.A}\propto \frac{\rho L^{3}\frac{du}{dt}}{\mu \frac{du}{dy}L^{2}}=\frac{\rho L\frac{dy}{dt}}{\mu }=\frac{\rho u_{0}L}{\mu }=\frac{u_{0}L}{\nu }\)
Where,
t: time
y: cross-sectional position
\(u=\frac{dx}{dt}\) : flow speed
τ: shear stress in Pa
A: cross-sectional area of the flow
V: volume of the fluid element
u
0: maximum speed of the object relative to the fluid in ms -1
L: a characteristic linear dimension
𝜇: dynamic viscosity of the fluid in Pa.s
𝜈: kinematic viscosity in m
2s
⍴: density of the fluid in kg.m
-3
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Definition:Generating Function Contents Definition
Let $A = \left \langle {a_n}\right \rangle$ be a sequence in $\R$.
Then $\displaystyle G_A \left({z}\right) = \sum_{n \mathop \ge 0} a_n z^n$ is called the generating function for the sequence $A$. The definition can be modified so that the lower limit of the summation is $b$ where $b > 0$ by assigning $a_k = 0$ where $0 \le k < b$.
The variable $z$ is a dummy variable, known as the
parameter of $G_A \left({z}\right)$. The coefficient of $z^n$ extracted from $G \left({z}\right)$ is the $n$th term of $S$, and can be denoted: $\left[{z^n}\right] G \left({z}\right) := a_n$
Let $A = \left \langle {a_{m, n} }\right \rangle$ be a doubly subscripted sequence in $\R$ for $m, n \in \Z_{\ge 0}$.
Then $\displaystyle G_A \left({w, z}\right) = \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ is called the generating function for the sequence $A$. Also denoted as
When the sequence is understood, $G \left({z}\right)$ can be used.
Different authors may use different symbols.
$x$ is often used instead.
In the field of probability theory $s$ tends to be the symbol of choice.
Some authors use $\zeta$.
Also see Results about generating functionscan be found here.
Many others since have developed the technique further.
A generating function is a clothesline on which we hang up a sequence of numbers for display. Sources 1956: G. Pólya: On Picture-Writing( Amer. Math. Monthly Vol. 63: 689 – 697) www.jstor.org/stable/2309555 1971: George E. Andrews: Number Theory... (previous) ... (next): $\text {3-4}$ Generating Functions: Definition $\text {3-3}$ 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction... (previous) ... (next): $\S 4.1$: Generating functions 1990: Herbert S. Wilf: generatingfunctionology 1994: Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science(2nd ed.) $\S 1.2$ 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms(3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(1)$ |
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The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron identification and triggering, the TRD ...
Constraining the magnitude of the Chiral Magnetic Effect with Event Shape Engineering in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Elsevier, 2018-02)
In ultrarelativistic heavy-ion collisions, the event-by-event variation of the elliptic flow $v_2$ reflects fluctuations in the shape of the initial state of the system. This allows to select events with the same centrality ...
First measurement of jet mass in Pb–Pb and p–Pb collisions at the LHC
(Elsevier, 2018-01)
This letter presents the first measurement of jet mass in Pb-Pb and p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV and 5.02 TeV, respectively. Both the jet energy and the jet mass are expected to be sensitive to jet ...
First measurement of $\Xi_{\rm c}^0$ production in pp collisions at $\mathbf{\sqrt{s}}$ = 7 TeV
(Elsevier, 2018-06)
The production of the charm-strange baryon $\Xi_{\rm c}^0$ is measured for the first time at the LHC via its semileptonic decay into e$^+\Xi^-\nu_{\rm e}$ in pp collisions at $\sqrt{s}=7$ TeV with the ALICE detector. The ...
D-meson azimuthal anisotropy in mid-central Pb-Pb collisions at $\mathbf{\sqrt{s_{\rm NN}}=5.02}$ TeV
(American Physical Society, 2018-03)
The azimuthal anisotropy coefficient $v_2$ of prompt D$^0$, D$^+$, D$^{*+}$ and D$_s^+$ mesons was measured in mid-central (30-50% centrality class) Pb-Pb collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_{\rm ...
Search for collectivity with azimuthal J/$\psi$-hadron correlations in high multiplicity p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 and 8.16 TeV
(Elsevier, 2018-05)
We present a measurement of azimuthal correlations between inclusive J/$\psi$ and charged hadrons in p-Pb collisions recorded with the ALICE detector at the CERN LHC. The J/$\psi$ are reconstructed at forward (p-going, ...
Systematic studies of correlations between different order flow harmonics in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(American Physical Society, 2018-02)
The correlations between event-by-event fluctuations of anisotropic flow harmonic amplitudes have been measured in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE detector at the LHC. The results are ...
$\pi^0$ and $\eta$ meson production in proton-proton collisions at $\sqrt{s}=8$ TeV
(Springer, 2018-03)
An invariant differential cross section measurement of inclusive $\pi^{0}$ and $\eta$ meson production at mid-rapidity in pp collisions at $\sqrt{s}=8$ TeV was carried out by the ALICE experiment at the LHC. The spectra ...
J/$\psi$ production as a function of charged-particle pseudorapidity density in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV
(Elsevier, 2018-01)
We report measurements of the inclusive J/$\psi$ yield and average transverse momentum as a function of charged-particle pseudorapidity density ${\rm d}N_{\rm ch}/{\rm d}\eta$ in p-Pb collisions at $\sqrt{s_{\rm NN}}= 5.02$ ...
Energy dependence and fluctuations of anisotropic flow in Pb-Pb collisions at √sNN=5.02 and 2.76 TeV
(Springer Berlin Heidelberg, 2018-07-16)
Measurements of anisotropic flow coefficients with two- and multi-particle cumulants for inclusive charged particles in Pb-Pb collisions at 𝑠NN‾‾‾‾√=5.02 and 2.76 TeV are reported in the pseudorapidity range |η| < 0.8 ... |
RTD From HwB RTD = Resistive Temperature Device
Common type of RTD is Pt-100, which is a temperature sensor made from platinum. Resistance varies with temperature. 100Ω at 0 °C.
Contents Pt-100
Temperature Resistance Pt-100
α=0.003750 °C
-1 Pt-100
α=0.003850 °C
-1 °C Ω Ω -200 19.9 18.5 -100 61.2 60.3 0 100 100 100 138 139 200 174 176 300 209 212 400 243 247 500 275 281 600 307 314 700 337 345 800 366 376 850 380 390 Callendar-Van Dusen Equation Given from RTD manufacturer:
Alpha, α
Delta, δ Beta, β Calculated from α, δ & β:
A = <math>\alpha + \frac{\alpha \times \delta}{100}</math>
B = <math>-\frac{\alpha \times \delta}{100^2}</math>
C = <math>-\frac{\alpha \times \beta}{100^4}</math>
Given:
R
T = Resistance (Ω) at T°C R 0 = Resistance (Ω) at 0°C (100 Ω for Pt-100) T = Temperature in °C Temperature according to ITS-90 (International Temperature Scale of 1990. Calculated
R T = R 0 x (1 + A x T + B x T² + (T-100 °C) x C x T³) for T < 0 °C R T = R 0 x (1 + A x T + B x T²) for T > 0 °C Callendar-Van Dusen Constants Constants
The purity of the metal will determine the constants.
Given Calculated Alpha, α Delta, δ Beta, β A B C °C -1 °C °C °C -1 °C -2 °C -4 0.003750 1.605 0.16 0.381 x 10 -3 -6.02 x 10 -7 -6.0 x 10 -12 0.003850 1.4999 0.10863 3.908 x 10 -3 -5.775 x 10 -7 -4.183 x 10 -12 0.003902 1.52 0.11 3.96 x 10 -3 -5.93 x 10 -7 -4.3 x 10 -12 0.003911 3.9692 × 10 -3 –5.8495 × 10 -7 –4.233 × 10 -12 0.003916 3.9739 × 10 -3 –5.870 × 10 -7 –4.4 × 10 -12 0.003920 3.9787 × 10 -3 –5.8686 × 10 -7 –4.167 × 10 -12 0.003928 3.9888 × 10 -3 –5.915 × 10 -7 –3.85 × 10 -12 Alpha, α, is sometimes known as TCR 0.003850 is according to IEC 751-2 standard. Made from 99.99% pure platinum. Typical data
Standard Typical data α Tolerance R 0 BS EN 60751 1996 0.003850 °C -1 ±0.05%, ±0.03%, ±0.02%, ±0.01% 100 Ω DIN 43760 1980 0.003850 °C -1 ±0.05% (1/2 DIN B), ±0.03% (1/3 DIN B), ±0.02% (1/5 DIN B), ±0.01% (1/10 DIN B) 100 Ω IEC 60751:1995 0.003850 °C -1 ±0.05%, ±0.03%, ±0.02%, ±0.01% 100 Ω JIS C1604 - 1981 (Japanese Industrial Standard) 0.003916 °C -1 ±0.15ºC, ±0.2ºC, ±0.5ºC 100 Ω or 50 Ω US Standard Curve 0.003916 °C -1 ±0.1 ohms 100 Ω or 50 Ω BS 2G 148 (British Aircraft Industry) 0.003900 °C -1 ±0.1% (at 0ºC) 130 Ω Graph Temperature vs Resistance Non linearity
This graph shows the affect of B & C in the Callendar-Van Dusen equation. Percentage error if the RTD would have been assumed to be linear.
Standards IEC 60751:1995 BS EN 60751 1996 DIN 43760 1980 BS 2G 148 Sources Thermal Developments International Ltd.: Hand made platinum resistance temperature detectors Honeywell - Sensing and Control: Reference and Application Data - Temperature Sensors Platinum RTDs Pentronic: Platinum resistance thermometers |
Computation Layers¶ class
ArgmaxLayer¶
Compute the arg-max along the channel dimension. This layer is only used in the test network to produce predicted classes. It has no ability to do back propagation.
tops¶
Blob names for output and input.
class
ChannelPoolingLayer¶
1D pooling over the channel dimension.
kernel¶
Default 1, pooling kernel size.
stride¶
Default 1, stride for pooling.
pad¶
Default (0,0), a 2-tuple specifying padding in the front and the end.
pooling¶
Default
Pooling.Max(). Specify the pooling function to use.
tops¶
Blob names for output and input.
class
ConvolutionLayer¶
Convolution in the spatial dimensions.
kernel¶
Default (1,1), a 2-tuple specifying the width and height of the convolution filters.
stride¶
Default (1,1), a 2-tuple specifying the stride in the width and height dimensions, respectively.
pad¶
Default (0,0), a 2-tuple specifying the two-sided padding in the width and height dimensions, respectively.
n_filter¶
Default 1. Number of filters.
n_group¶
Default 1. Number of groups. This number should divide both
n_filterand the number of channels in the input blob. This parameter will divide the input blob along the channel dimension into
n_groupgroups. Each group will operate independently. Each group is assigned with
n_filter/
n_groupfilters.
neuron¶
Default
Neurons.Identity(), can be used to specify an activation function for the convolution outputs.
filter_regu¶
Default
L2Regu(1), the regularizer for the filters.
bias_regu¶
Default
NoRegu(), the regularizer for the bias.
filter_lr¶
Default 1.0. The local learning rate for the filters.
bias_lr¶
Default 2.0. The local learning rate for the bias.
class
CropLayer¶
Do image cropping. This layer is primarily used only on top of data layer so backpropagation is currently not implemented.
crop_size¶
A (width, height) tuple of the size of the cropped image.
random_crop¶
Default
false. When enabled, randomly place the cropping box instead of putting at the center. This is useful to produce random perturbation of the input images during training.
random_mirror¶
Default
faulse. When enabled, randomly (with probability 0.5) mirror the input images (flip the width dimension).
tops¶
Blob names for output and input.
class
DropoutLayer¶
Dropout is typically used during training, and it has been demonstrated to be effective as regularizers for large scale networks. Dropout operates by randomly “turn off” some responses. Specifically, the forward computation is\[\begin{split}y = \begin{cases}\frac{x}{1-p} & u > p \\ 0 & u <= p\end{cases}\end{split}\]
where \(u\) is a random number uniformly distributed in [0,1], and \(p\) is the
ratiohyper-parameter. Note the output is scaled by \(1-p\) such that \(\mathbb{E}[y] = x\).
ratio¶
The probability \(p\) of turning off a response. Or could also be interpreted as the ratio of all the responses that are turned off.
The names of the input blobs dropout operates on. Note this is a
in-place layer, so there is no
topsproperty. The output blobs will be the same as the input blobs.
class
ElementWiseLayer¶
Element-wise layer implements basic element-wise operations on inputs.
operation¶
Element-wise operation. Built-in operations are in module
ElementWiseFunctors, including
Add,
Subtract,
Multiplyand
Divide.
tops¶
Output blob names, only one output blob is allowed.
Input blob names, count must match the number of inputs
operationtakes.
class
InnerProductLayer¶
Densely connected linear layer. The output is computed as\[y_i = \sum_j w_{ij}x_j + b_i\]
where \(w_{ij}\) are the weights and \(b_i\) are bias.
output_dim¶
Output dimension of the linear map. The input dimension is automatically decided via the inputs.
weight_init¶
Default
XavierInitializer(). Specify how the weights \(w_{ij}\) should be initialized.
bias_init¶
Default
ConstantInitializer(0), initializing the bias \(b_i\) to 0.
bias_regu¶
Default
NoRegu(). Regularizer for the bias. Typically no regularization should be applied to the bias.
weight_lr¶
Default 1.0. The local learning rate for the weights.
bias_lr¶
Default 2.0. The local learning rate for the bias.
tops¶
Blob names for output and input.
class
LRNLayer¶
Local Response Normalization Layer. It performs normalization over local input regions via the following mapping\[x \rightarrow y = \frac{x}{\left( \beta + (\alpha/n)\sum_{x_j\in N(x)}x_j^2 \right)^p}\]
Here \(\beta\) is the shift, \(\alpha\) is the scale, \(p\) is the power, and \(n\) is the size of the local neighborhood. \(N(x)\) denotes the local neighborhood of \(x\) of size \(n\) (including \(x\) itself). There are two types of local neighborhood:
LRNMode.AcrossChannel(): The local neighborhood is a region of shape (1, 1, \(k\), 1) centered at \(x\). In other words, the region extends across nearby channels (with zero padding if needed), but has no spatial extent. Here \(k\) is the kernel size, and \(n=k\) in this case.
LRNMode.WithinChannel(): The local neighborhood is a region of shape (\(k\), \(k\), 1, 1) centered at \(x\). In other words, the region extends spatially (in
boththe width and the channel dimension), again with zero padding when needed. But it does not extend across different channels. In this case \(n=k^2\).
kernel¶
Default 5, an integer indicating the kernel size. See \(k\) in the descriptions above.
scale¶
Default 1.
shift¶
Default 1 (yes, 1, not 0).
power¶
Default 0.75.
mode¶
Default
LRNMode.AcrossChannel().
tops¶
Names for output and input blobs. Only one input and one output blob are allowed.
class
PoolingLayer¶
2D pooling over the 2 image dimensions (width and height).
kernel¶
Default (1,1), a 2-tuple of integers specifying pooling kernel width and height, respectively.
stride¶
Default (1,1), a 2-tuple of integers specifying pooling stride in the width and height dimensions respectively.
pad¶
Default (0,0), a 2-tuple of integers specifying the padding in the width and height dimensions respectively. Paddings are two-sided, so a pad of (1,0) will pad one pixel in both the left and the right boundary of an image.
pooling¶
Default
Pooling.Max(). Specify the pooling operation to use.
tops¶
Blob names for output and input.
class
PowerLayer¶
Power layer performs element-wise operations as\[y = (ax + b)^p\]
where \(a\) is
scale, \(b\) is
shift, and \(p\) is
power. During back propagation, the following element-wise derivatives are computed:\[\frac{\partial y}{\partial x} = pa(ax + b)^{p-1}\]
Power layer is implemented separately instead of as an Element-wise layer for better performance because there are some many special cases of Power layer that could be computed more efficiently.
power¶
Default 1
scale¶
Default 1
shift¶
Default 0
tops¶
Blob names for output and input.
class
ReshapeLayer¶
Reshape a blob. Can be useful if, for example, you want to make the
flatoutput from an
InnerProductLayer
meaningfulby assigning each dimension spatial information.
Internally there is no data copying going on. The total number of elements in the blob tensor after reshaping should be the same as the original blob tensor.
width¶
Default 1. The new width after reshaping.
height¶
Default 1. The new height after reshaping.
channels¶
Default 1. The new channels after reshaping.
tops¶
Blob names for output and input.
class
SoftmaxLayer¶
Compute softmax over the channel dimension. The inputs \(x_1,\ldots,x_C\) are mapped as\[\sigma(x_1,\ldots,x_C) = (\sigma_1,\ldots,\sigma_C) = \left(\frac{e^{x_1}}{\sum_j e^{x_j}},\ldots,\frac{e^{x_C}}{\sum_je^{x_j}}\right)\]
class
SplitLayer¶
Split layer produces identical
copies[1] of the input. The number of copies is determined by the length of the
topsproperty. During back propagation, derivatives from all the output copies are added together and propagated down.
This layer is typically used as a helper to implement some more complicated layers.
Input blob names, only one input blob is allowed.
tops¶
Output blob names, should be more than one output blobs.
[1] All the data is shared, so there is no actually data copying. |
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? |
Here is the BCS state:
$$ \left|\Psi_\mathrm{BCS}\right\rangle = \prod_k \left( u_k - v_ke^{i \phi} c_{k\uparrow}^{\dagger} c_{-k\downarrow}^{\dagger}\right) \left|0\right\rangle.$$
When I develop the BCS state to understand what it means, I will have a state like this : \begin{align} \left|\Psi_\mathrm{BCS}\right\rangle & = \prod_k u_k |0\rangle + \sum_{k_0}\prod_{k \neq k_0} u_k(-v_{k_0}e^{i \phi})|k_0\uparrow, -k_0\downarrow\rangle \\ & \quad+ \sum_{k_0<k_1}\prod_{k \neq k_0,k_1}u_k(-1)^2e^{2i \phi}v_{k_0}v_{k_1}|k_1\uparrow, -k_1\downarrow, k_0\uparrow, -k_0\downarrow\rangle+ \cdots \end{align} So we have an infinite superposition of cooper pairs (one cooper pair + 2 pairs + 3 pairs + etc). Is this understanding correct?
In my course they say that $|v_k|^2$ is the probability to have a Fermi quasiparticle with wavevector $k$. But I don't understand this. |
4 2017-Spring 4.1 Paper 4.2 Problem 1
Let the three points be \((x_1,y_1); (x_2,y_2); (x_3,y_3)\) Expected area of the rectangle with sides parallel to the coordinate axes is: \(E[\big(\max{(x_1,x_2,x_3)}-\min{(x_1,x_2,x_3)}\big)\big(\max{(y_1,y_2,y_3)}-\min{(y_1,y_2,y_3)}\big)]\) where \(x_i ~ U(0,1)\), \(y_i ~ U(0,1)\) and \(x_i, y_i\) are independent.
Thus, expected area can be simplified to \(E[\big(\max{(x_1,x_2,x_3)}-\min{(x_1,x_2,x_3)}\big)]^2\)
Let \(\max{(x_1,x_2,x_3)\) be represents as \(x_{(1)}\) and \(\min{(x_1,x_2,x_3)\) as \(x_{(3)}\) Then \(P_{X_{(1)}}(X_{(1)} < x) = P(X_{1}<x)P(X_{2}<x)P(X_{3}<x) = x^3 \implies E[X_{(1)}]=\frac{3}{4}\)
Similarly,
\(P_{X_{(3)}}(X_{(3)} < x) = 1-(1-x)^3 \implies P_{X_{(3)}}(X_{(3)} = x) = 3(1-x)^2 \implies E[X_{(3)}]=\frac{1}{4}\)
Thus, expected area = \((3/4-1/4)^2= 1/16\)
4.3 Problem 2
\(f(x,y) g(\sqrt{x^2+y^2})\) Consider the transformation:
\[ \begin{align*} x &= r \cos(\theta)\\ y &= r \sin(\theta) \end{align*} \]
Then \(f(r \cos(\theta), r \sin(\theta)) = g(r)\)
Reverse tranformation gives: \[ \begin{align*} Z &= X/Y = \tan(\theta)\\ r^2 &= y^2\sec^2(\theta) \end{align*} \]
Since \(f(r, \theta) = g(r)\) is independnet of \(\theta\), \(\theta ~ U\). Let \(\theta ~ U(0,2\pi)\)
Now, \(\theta = \tan^{-1}((Z)\)
\(f_Z(z) = f_\theta(\tan^{-1}(z)) |\frac{\partial \theta}{\partial z}|+ f_\theta(\pi+\tan^{-1}(z)) |\frac{\partial \theta}{\partial z}| = \frac{1}{2\pi}\)
\(\frac{\partial \theta}{\partial z} = 1/sec^2(\theta) = \frac{1}{z^2+1}\)
Thus, $f_Z(z) = =
4.4 Problem 3 4.5 Part a
\(E[X_{n+1}^r|X_n] = \int_0^{cx_n} x^r \frac{1}{cx_n} dx = \frac{(cx_n)^r}{r+1}\)
4.6 Part b
For \(r=1\):
\(E[X_{n+1}|X_n] = \frac{cx_n}{2}\)
$EX_{n+1}= E[E[X_{n+1}|X_n] ]= $ Thus $ = ^n 0 $ as \(n \rightarrow \infty\) as < 1$
For \(r=2\):
$EX_{n+1}^2 = $
$ = ^n $ as \(n \rightarrow \infty\)
4.7 Part c 4.7.1 ToDO 4.8 Problem 4 4.8.1 Part a
All \(n\) boys in single block \(\implies\) rreat them as one unit. Now for \(m\) girls there are \(m+1\) spots where we can put this one ‘unit’ in \(m+1\) ways while the girls can be arranged in \(m!\) ways and among the unit the boys can be arranged in \(n!\) ways.
Hence, the required probability: \(\frac{(m+1)n!m!}{(n+m)!}\)
4.8.2 Part b
For \(n>m\) the probability is zero,
If \(n\leq m\) we arrange the girls first leaving \(m+1\) spaces for \(n\) boys which can be filled in \({m+1 \choose n} \times n!\) and the girls can be arranged in \(m!\) ways
Required probability: \(\frac{{m+1 \choose n}m!n!}{(n+m)!}\)
4.8.3 Part c
Define: \(I_i=1\) if \((i-1,i,i+1)=(g,b,g)\) and \(0\) otherwise.
Then \(EW = E\sum_{i=1}^{n+m-2} I_i = (n+m-2) EI_1 = (n+m-2)P(I_1=1) = (n+m-2) \frac{n(m)(m-1)}{(n+m)!}\) |
I heard that friction depends only on the normal force but not on the contact area. Let's take a cube and a sphere which are of same weight (then normal force will also same ) but the force needed to move these two objects is different, why?
Let's look at this problem from the point of view of equations of motion, see diagram below:
Firstly let's make a few assumptions.
Ball and cube are of same weight ($mg$) and same size. Simple friction model $F_f=\mu F_n$ holds and $\mu$ is independent of speed. Both objects are completely stationary (no sliding, rolling or tumbling) at $t=0$, at which point a purely horizontal force $F$ starts acting on the centre of gravity of the objects. A) Case of the Ball:
If we assume there is enough friction the ball will start translating
and rolling (but without sliding or slipping) and at each instant:
$$v=\omega R,$$
with $v$ the translational speed, $\omega$ the angular speed and $R$ the radius of the ball.
For the translational speed we can set up the Newtonian equation:
$$ma=F-F_f,$$
$$ma=F-\mu mg,$$
$$m\frac{dv}{dt}=F-\mu mg.$$
Integrated between $0,0$ and $t,v$ we get:
$$v=\bigg (\frac{F}{m}-\mu g \bigg )t.$$
And:
$$\omega=\frac{v}{R}.$$
The kinetic energy $K$ of the object is now the sum of translational energy and rotational energy and also the work $W$ done on the object:
$$K=\frac{mv^2}{2}+\frac{I \omega^2}{2}=W.$$
With $I=\frac{2mR^2}{5}$ the inertial moment of the ball and inserting we get:
$$K=\frac{7mv^2}{10}=W_A.$$
B) Case of the Cube:
Although $F$ causes a tumbling moment $FR$ around the forward contact point, we'll assume no tumbling actually occurs (we can also prevent it by lowering the line of the force $F$ closer to the floor, thereby reducing the tumbling couple).
The translational equation of motion is the same as in case A:
$$ma=F-F_f,$$
so we also obtain:
$$v=\bigg (\frac{F}{m}-\mu g \bigg )t.$$
The kinetic energy is now limited to translational energy:
$$K=\frac{mv^2}{2}.$$
But this is not equal to the total work done on the cube, as it doesn't take into account friction work.
The friction work is given by:
$$W_f=F_fx,$$
or:
$$W_f=\mu mgx.$$
Where $x$ is the displacement over the time $t$.
We know from above that the acceleration $a$ is:
$$a=\frac{F}{m}-\mu g,$$
and the displacement $x$ is:
$$x=\frac{at^2}{2}=\frac{1}{2}\bigg (\frac{F}{m}-\mu g \bigg )t^2=\frac{1}{2}vt.$$
So for $W_f$ we get:
$$W_f=\frac{1}{2}\mu mgvt.$$
The total work $W_B$ now done becomes:
$$W_B=\frac{mv^2}{2}+\frac{\mu mgvt}{2}.$$
We can now also compare $W_A$ and $W_B$ but it's rather a long derivation, so I'll only post the conclusion here. For:
$$F>\frac{7\mu mg}{2},$$
then:
$$W_A>W_B$$.
However, in this earlier answer I showed that for rolling without slipping the critical coefficient of friction $\mu_c$ is given by:
$$\mu_c=\frac{FI}{mg(I+mR^2)}$$
Extracting $F$ from that expression and with $I=\frac{2mv^2}{5}$ we then find that:
$$F=\frac{7\mu_c mg}{2},$$
which proves that:
$$\large{W_A=W_B}.$$
Note also that where $\mu>\mu_c$ then $\mu_c$ needs to be used in the equations of motion as otherwise we would be over-estimating the friction force and torque.
C) Conclusions:
All things being equal and in the presence of sufficient friction, the velocity (translation) of a sphere and a cube are the same. Both cases require the same amount of work but in the case of a (non-tumbling) cube part of that work is lost as friction work and thus
not conserved.
Whenever one applies a sideways force trough the center of gravity of an object, that force has two components: 1) a direct force that tries to overcome friction and slide the object, and 2) a torque that uses friction to produce a rotation of the object by lifting its center of gravity over the leading edge.
A short, flat object will tend to slide because the torque needed to raise the center of gravity over the long edge is large compared to the force needed to overcome friction; the same object place on its short edge will tumble, because the torque needed to raise the center of gravity over that short edge is comparatively small.
Thus rolling friction must always be less than sliding friction, by definition, because for an object to roll the counter-force of friction that leads to torque must be smaller than the counter-force of friction that must be overcome for a slide.
A wheel, then, is simply a special case in which (ideally) the center of gravity of the object never needs to be lifted, and so the friction needed to induce a roll (ideally) is vanishingly small.
therefore it is easier to move sphere than cube of same weight.
This is a very common question asked by students like me in mechanical engineering. The friction is irrespective of area means the friction generated, the vector F (The letter with which we denote) will form whether area in contact is less or more
But the ability to stop is determined by the Number of friction vectors developed per area of contact. On the surface of a cube the magnitude of the friction vector will be the same as the one developed on the point of contact of the sphere(Irrespective of the surface of contact).
But the number of friction vectors produced per unit area will be different. This is one of the explanation given to me by our professor. Hence the cube requires more force than a sphere.
protected by Qmechanic♦ Nov 19 '15 at 16:33
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead? |
Flow Completion Time (FCT) is probably the most important user-perceived metric. I wrote this post is to crystallize several ingredients of the line of related work.
Why FCT matters?
One has to differentiate between user-perceived metrics and metrics cared by the network operators. Users are mainly concerned about the time (i.e., FCT) to finish the current network transaction (web browsing, file transfer), unlike operators that are interested in network utilization, throughput, fairness and so forth. Optimizing operator defined metrics such as the network throughput does not necessarily imply the improvement of FCT. Why Flow-Completion Time is the Right Metric for Congestion Control provides details on such gap for canonical congestion control mechanisms.
What is the optimal policy to minimize FCT theoretically?
Shortest Remaining Processing Time (SRPT) is the optimal policy in a single link setting in terms of optimizing FCT,
regardless of the flow size distribution. Analysis of SRPT Scheduling: Investigating Unfairness investigates the objections to SRTP (not knowing the flow size a priori, unclear performance improvements over PS, starvation for large flows) with the metrics of job response time and job slowdown in an M/G/1 system. The paper mainly looks at a truncated heavy-tailed distribution, i.e., the job size is sampled i.i.d. from the Bounded Pareto distribution. It proves that SRPT outperforms PS scheduling w.r.t. mean slow down for all job distributions, besides the well-known facts about its FCT optimality. It also derives formulas for M/G/1/SRPT under overload.
Unfortunately, there is no universally optimal policy in a multi-link setting. pFabric: Minimal Near-Optimal Datacenter Transport summarizes the issue: even under the simplified network fabric, minimizing the average FCT corresponds to the NP-hard sum-multicoloring problem. However, the greedy flow scheduling algorithm with an ideal big switch view promises to provide at least a 2-approximation to the optimal.
How to minimize FCT with a limited number of service queues?
Even with a single link, there is a practical problem: commodity switches typically support an only limited number of $K$ FIFO queues (e.g., $K=4\sim8$), which prevents us from differentiating flows by the key value of flow size in a fine-grained manner (ps, there are active works to enhance the switch hardware to support e.g., priority queue heap with O(log n) operation complexity).
With this, one could only use $K-1$ thresholds to roughly differentiate the flows w.r.t. the flow size. pFabric: Minimal Near-Optimal Datacenter Transport provides a derivation of a 2-queue example, as summarized below (I included the omitted intermediate steps). The authors also generalize it to an arbitrary finite number of queues in the technical report.
Objective: theoretically justify that the average FCT optimization depends on the threshold and the flow size distribution Assumption: assume the flow size is known or accurately measured a priori Model representation: link capacity 1; flow arrival rate (NOT packet arrival rate) $\lambda$ following Poisson process; flow size CDF $S\sim F_{S}(\cdot)$; the link is NOT overloaded, i.e., total load $\rho=\lambda E(s) < 1$; a single threshold $t$ that classifies flows into high priority queue 0 and low priority queue 1 Derivation: the aggregate Poisson process could be split into 2 independent poison processes at each service queue, with the rate $\lambda_{0}=\lambda F_{S}(t)$ and $\lambda_{1}=\lambda(1-F_{S}(t))$. The drain rate of queue 0 is $\mu_{0}=1$ and the drain rate of queue 1 is the remaining bandwidth after serving queue 0, which is $\mu_{1}=1-\rho B_{S}(t)$, where $B_{S}(\cdot)$ denotes the traffic fraction in bytes (which means when setting the threshold $t$, the fraction of total bytes of flows less than the size $t$ compared with the total bytes of all traffic). we have Pollaczek-Khinchin (P-K) Mean Formula states that the mean waiting time $W$ of a M/G/1 queue follows where $\rho=\lambda/\mu=\lambda E(S)$ refers to the workload/utilization. Here a flow is treated as a job in the theoretical model, instead of diving into the microscopic view of packet streams (though it feels to deviate from the actual case). Here $E(S^{2})$ refers to the second moment of the service time distribution, notthe flow size distribution. Also, here $\mu$ refers to the service rate for each flow, NOT the drain rate. Note that the $W$ here is different from the mean spent time $W^{‘}$ (waiting time + service time) in the Little’s Law. For each queue $i (i=0, 1)$ we have $\rho_{i}=\frac{\rho B_{S}^{i}}{\mu_{i}}$, where $\mu_{i}$ is the service/drain rate for queue $i$. Hence, we could obtain the mean waiting time for each queue,
We could now compute the average normalized flow completion time (by flow size):
With $F^{‘}_{S}(t)=f_{S}(t)$, we could obtain the final result, as stated in the pFabric paper:
It is clear from the result that $FCT_{n}$ depends on the threshold, flow size distribution and the workload. The paper also attaches a numerical plot visualizing the function $FCT_{n}(t)$ for various workload 10–80% and web search flow size distribution.
What if we are agnostic of the flow size oracle?
Information-Agnostic Flow Scheduling for Commodity Data Centers eliminates the assumption on the availability of the flow size information and borrows the Multi-Level-Feedback-Queue (MLFQ) in OS to emulate SJF without the oracle. The key to the proposal is how to set the optimal demotion thresholds. The paper gives derivation to the optimal thresholds given the fixed workload distribution and load. Though Sum-of-Linear-Ratios (SoLR) problem is generally NP-hard, the paper provides a closed-form analytical solution assuming M/M/1 queue. The paper also identifies that due to the traffic variations, the demotion thresholds mismatch could lead to performance degradation severely. The follow-up work AuTO: Scaling Deep Reinforcement Learning for Datacenter-Scale Automatic Traic Optimization leverages novel Deep Reinforcement Learning to automatically set the configurations adaptive to the traffic pattern. |
What is the difference? I know there is the (almost) same question What's the difference between helicity and chirality? but when a particle is given as left-handed. Is it helicity or chirality?
When we consider spinors of the Lorentz group $SO(3,1)$, recall that the universal covering of $SO(3,1)^+$ (the component of the Lorentz group connected to the identity) is isomorphic to $SL(2,\mathbb C)$.
Two-component spinors are elements of two-dimensional irreducible modules of $SL(2,\mathbb C)$. However, noting that the complexification of the Lie algebra of $SL(2,\mathbb C)$ is $A_1 \oplus A_1$, there are two inequivalent such modules.
These modules have weights $(1,0)$ and $(0,1)$, or in physics language, spins $(\frac12,0)$ and $(0,\frac12)$ respectively. Objects with indices corresponding to each have different transformation properties, namely, for the former,
$$\psi_\alpha \to M^\beta_\alpha \psi_\beta$$
for some $M\in SL(2,\mathbb C)$ whereas for the latter,
$$\psi_{\dot\alpha} \to \overline M^{\dot\beta}_{\dot\alpha} \psi_{\dot\beta}.$$
Typically, we refer to the undotted indices as
left-handed and the dotted indices as right-handed. Note that in some cases in lower dimension, they are not distinct (which is simpler as one does not need Van der Waerden notation to distinguish them.)
It should be noted they transform in the same way under rotations, but they transform oppositely under boosts, motivating the nomenclature. Normally you may have been introduced to spinors first through the Dirac spinor, which lies in the $(\frac12,0) \oplus (0,\frac12)$ representation, being comprised of two chiral spinors.
Helicity is the projection of spin onto momentum of a particle:
$$ h = \frac{\vec s\cdot \vec p}{|\vec p|} $$
If a particle with spin-1 moves exactly in the same direction as its spin points (let's say the spin point in $z$-direction and it also moves in $z$-direction), then the helicity is $h=+1$. If it moves in the exact opposite direction, towards $-z$, the helicity is $h=-1$.
As for the terminology, a particle for which $h=-|\vec s|$ is called left-handed, and $h=-|\vec s|$ is called right-handed.
Chirality is a property of a particle. As Wikipedia puts it, " it is determined by whether the particle transforms in a right- or left-handed representation of the Poincaré group."
Though terms like "left-chiral" and "right-chiral" might be more suitable, people usually also use the terms left-handed and right-handed when talking about chirality. Another thing to watch out for is that for massless particles, its helicity is the same as its chirality. |
The second is defined as the time it takes for 9,192,631,770 wavelengths of a certain transition of the cesium-133 atom to pass a fixed point. What is the frequency of this electromagnetic radiation? What is the wavelength?
Solution:
The frequency of this electromagnetic radiation is already given to us in the question. It is simply 9,192,631,770 per second.
To calculate the wavelength, we will use this formula:\lambda=\dfrac{c}{v}
(Where \lambda is the wavelength, c is the speed of light, and v is frequency)
Speed of light = 2.99792458\times 10^8\dfrac{m}{s}
Substituting the values we know and solving for \lambda gives us:
\lambda=0.0326122557 m
This question can be found in General Chemistry, 9th edition, chapter 7, question 7.42 |
ISSN:
2155-3289
eISSN:
2155-3297 Numerical Algebra, Control & Optimization
2014 , Volume 4 , Issue 1
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Abstract:
Inspired by the results in [S. S. Dragomir and I. Gomm, Num. Alg. Cont. $\&$ Opt., 2 (2012), 271--278], we give some new bounds for two mappings related to the Hermite--Hadamard inequality for convex functions of two variables, and apply them to special functions to get some results for the $p$-logarithmic mean. We also apply the Hermite--Hadamard inequality to matrix functions in this paper.
Abstract:
In this paper, by virtue of the epigraph technique, we construct a new kind of closedness-type constraint qualification, which is the sufficient and necessary condition to guarantee the strong duality between a cone constraint composite optimization problem and its dual problem holds. Under this closedness-type constraint qualification condition, we obtain a formula of subdifferential for composite functions and study a cone constraint composite DC optimization problem in locally convex Hausdorff topological vector spaces.
Abstract:
In this paper, we will consider the problem of partially sparse signal recovery (PSSR), which is the signal recovery from a certain number of linear measurements when its part is known to be sparse. We establish and characterize partial $s$-goodness for a sensing matrix in PSSR. We show that the partial $s$-goodness condition is equivalent to the partial null space property (NSP), and is weaker than partial restricted isometry property. Moreover, this provides a verifiable approach for partial NSP via partial $s$-goodness constants. We also give exact and stable partially $s$-sparse recovery via the partial $l_1$-norm minimization under mild assumptions.
Abstract:
In this paper, we establish convexity of some functions associated with symmetric cones, called SC trace functions. As illustrated in the paper, these functions play a key role in the development of penalty and barrier function methods for symmetric cone programs. With trace function method we offer much simpler proofs to these useful inequalities.
Abstract:
Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing edge-coloring of $G$ if $F_{\sigma}(u)\not= F_{\sigma}(v)$ for any $uv\in E(G)$, where $F_{\sigma}(u)$ denotes the set of colors of edges incident with $u$. A total-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing total-coloring of $G$ if $S_{\sigma}(u)\not= S_{\sigma}(v)$ for any $uv\in E(G)$, where $S_{\sigma}(u)$ denotes the set of colors of edges incident with $u$ together with the color assigned to $u$. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of $G$ is denoted by $\chi_a^{'}(G)$ (resp. $\chi^{''}_{a}(G)$). In this paper, we provide upper bounds for these parameters of the Cartesian product $G$ □ $H$ of two graphs $G$ and $H$. We also determine exact value of these parameters for the Cartesian product of a bipartite graph and a complete graph or a cycle, the Cartesian product of a complete graph and a cycle, the Cartesian product of two trees and the Cartesian product of regular graphs.
Abstract:
Multifingered hand-arm robots play an important role in dynamic manipulation tasks. They can grasp and move various shaped objects. It is important to plan the motion of the arm and appropriately control the grasping forces for the multifingered hand-arm robots. In this paper, we perform the grasping force based manipulation of the multifingered hand-arm robot by using neural networks. The motion parameters are analyzed and planned with the constraint of the multi-arms kinematics. The optimal grasping force problem is recast as the second-order cone program. The semismooth Newton method with the Fischer-Burmeister function is then used to efficiently solve the second-order cone program. The neural network manipulation system is obtained via the fitting of the data that are generated from the optimal manipulation simulations. The simulations of optimal grasping manipulation are performed to demonstrate the effectiveness of the proposed approach.
Abstract:
In this paper, we propose an iterative method for calculating the largest eigenvalue of nonhomogeneous nonnegative polynomials. This method is a generalization of the method in [19]. We also prove this method is convergent for irreducible nonhomogeneous nonnegative polynomials.
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ISSN:
2155-3289
eISSN:
2155-3297 Numerical Algebra, Control & Optimization
2014 , Volume 4 , Issue 3
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Abstract:
In this paper, we provide a robust control approach for controlling the autonomous bicycle kinematics with the objective of stabilizing the bicycle steer $\delta$ and roll $\phi$ angles. The dynamical model is the so-called 'Whipples Bicycle Model', where the roll (lean) angle and the steer angle of the bicycle are the two outputs of the model and the torques across the roll and steer angle as the two control variables. Two control design methods are developed based on $H_\infty$ and $H_2$-norm optimization using dynamic output feedback. The ensuing results are compared with an adaptive control scheme. The autonomous bicycle was tested for varying velocities.
Abstract:
Recurrent neural networks (RNNs) have emerged as a promising tool in modeling nonlinear dynamical systems. The convergence is one of the most important issues of concern among the dynamical properties for the RNNs in practical applications. The reason is that the viability of many applications of RNNs depends on their convergence properties. We study in this paper the convergence properties of the weighted state space search algorithm (WSSSA) -- a derivative-free and non-random learning algorithm which searches the neighborhood of the target trajectory in the state space instead of the parameter space. Because there is no computation of partial derivatives involved, the WSSSA has a couple of salient features such as simple, fast and cost effective. In this study we provide a necessary and sufficient condition that required for the convergence of the WSSSA. Restrictions are offered that may help assure convergence of the of the WSSSA to the desired solution. The asymptotic rate of convergence is also analyzed. Our study gives insights into the problem and provides useful information for the actual design of the RNNs. A numerical example is given to support the theoretical analysis and to demonstrate that it is applicable to many applications.
Abstract:
In this paper we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle's dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. The second method is based on the TV-Stokes model. By smoothing the tangential vector and imposing the divergence free condition, we propose an image zooming method based on the TV-Stokes model using the dual formulation. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.
Abstract:
Schilders' factorization can be used as a basis for preconditioning indefinite linear systems which arise in many problems like least-squares, saddle-point and electronic circuit simulations. Here we consider its application to resistor network modeling. In that case the sparsity of the matrix blocks in Schilders' factorization depends on the sparsity of the inverse of a permuted incidence matrix. We introduce three different possible permutations and determine which permutation leads to the sparsest inverse of the incidence matrix. Permutation techniques are based on types of sub-digraphs of the network of an incidence matrix.
Abstract:
Microgrids are smaller, self-contained electricity grids featuring distributed generation (e.g., solar photovoltaic panels, wind turbines, biomass), energy storage technologies, and power system control devices that enable self-coordinated operations. Microgrids can be seen as a key technology for greater integration of renewable energy resources. However, the uncertain nature in power generated by these resources poses challenges to its integration into the electric grid. In this paper, we present a demand-side management stochastic optimization model to operate an isolated microgrid under uncertain power generation and demand.
Abstract:
The timetabling problem is to find a schedule of activities in space/time that satisfies a prescribed set of operational and resource constraints and which maximizes an objective function that reflects the value of the schedule. Constructing an effective timetable is always a challenging task for any scheduler. Most literature research focuses on specific applications and the resulting models are not easily applied to problems other than those for which they were designed for. In this paper, we construct a general model for university course timetabling. Our model incorporates a total of 17 different types of requirements identified in the literature as well as three new constraint types that we think should be part of the restrictions in a general university based timetabling model. An integer programming (IP) model is presented which incorporates restrictions that need to be satisfied and requests that are included in the objective function. We implement and test our models using the AIMMS mathematical software package. Computational results on a number of case studies are favorable and demonstrate the value of our approach.
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I cracked KCl and made two samples. One is a fine powder, another is a coarse powder. Using these samples, diffraction intensity was measured. The results were as followed.
[peak number, relative intensity (fine), relative intensity (coarse)]1: 100, 1002: 52, 353: 16, 104: 19, 145: 27, 126: 19, 87: 6, 38: 13, 6
By the way, relative intensity can be theoretically calculated, using
$I = |F(h,k,l,\vec{K})|^2 \cdot p \cdot L(\theta) $ (eq.1) .
I wrote a simple program and evaluated its values.
[peak number, relative intensity]1: 1002: 67.13: 21.94: 9.75: 25.76: 18.77: 6.18: 11.0
Now I have two questions.
What is the difference between the fine powder and the coarse one? (eq.1) doesn't include how the powder was cracked, so it's seemed that there is no difference. But the relative intensities are in fact different. I've heard the size of the powder is related to extinction effect, but don't know how to add the size factor to (eq.1).
Why are the theoretical data and the experimental data different? When thinking about relative-intensity(fine) and that of theoretical, the values of the peak of 1,5,6,7,8 are similar, and 2,3,4 are far from similar. It's seemed so strange to me.
Any hints will be appreciated. |
While this does not answer the question asked (and will therefore not be accepted), I provide this response in hopes it will help others and promote worthwhile discussion.
David Baird provides a simple explanation for propagating the error through a linear least squares fit inIn his book Experimentation: An Introduction to Measurement Theory and Experiment Design, David Baird provides a simple explanation for doing a linear least squares fit using the diffences between the measured and fit values to estimate the error of the fit parameters. The best fit for the parameters $m$ and $b$ in $$y=mx+b$$ will is determined using his eqn. (6.3):
Essentially, weight the fit by dividing $x_i$ and $y_i$ values by the corresponding $dy_i$. As commented below, when reducing this to a matrix problem as described in the OP's wiki-link, make sure to also weight the $1$, i.e., $X=[x, 1]/dy$. Perform the fit to get the parameters $m$ and $b$ in $$y=mx+b$$ Calculate the differences of each $y_i$ value from the fit, $\delta y_i$ Calculate the standard deviation of the fit parameters using$$\sigma_y = \sqrt{\frac{\sum(\delta y_i)^2}{n-2}}$$$$\sigma_m = \sigma_y \sqrt{\frac{n}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$$$\sigma_b = \sigma_y \sqrt{\frac{\sum{x_i^2}}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$
$$m = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum(x^2_i) - (\sum x_i)^2} $$
I am not certain this is$$b = \frac{\sum(x_i^2)\sum y_i - \sum x_i \sum x_iy_i}{n\sum(x^2_i) - (\sum x_i)^2} $$
After obtaining $m$ and $b$, a complete propagationstandard deviation for the fit parameters can be obtained by calculating the differences of each $y_i$ value from the fit, $\delta y_i = y_i - (m x_i +b)$. From these differences, one calculates the standard deviation of the data from the fit line using:
$$\sigma_y = \sqrt{\frac{\sum(\delta y_i)^2}{n-2}}$$
and then the standard deviation of the parameters using$$\sigma_m = \sigma_y \sqrt{\frac{n}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$and$$\sigma_b = \sigma_y \sqrt{\frac{\sum{x_i^2}}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$
To include the measurement error but, $dy_i$, in the fit one would divide the initial system of equations by $dy_i$ giving $$\frac{y_i}{dy_i} = m\frac{x_i}{y_i} + b \frac{1}{dy_i}$$ then repeat Baird's derivation to get the weighted fit parameters
$$m = \frac{\sum \frac{1}{dy_i}\sum \frac{x_i y_i}{dy_i^2} - \sum \frac{x_i}{dy_i^2} \sum \frac{y_i}{dy_i}}{\sum \frac{1}{dy_i}\sum \frac{x^2_i}{dy_i^2} - \sum \frac{x_i}{dy_i}\sum \frac{x_i}{dy_i^2}} $$
$$b = \frac{\sum \frac{x_i^2}{dy_i^2}\sum \frac{y_i}{dy_i} - \sum \frac{x_i}{dy_i} \sum \frac{x_iy_i}{dy_i^2}}{\sum \frac{1}{dy_i}\sum \frac{x^2_i}{dy_i^2} - \sum \frac{x_i}{dy_i}\sum \frac{x_i}{dy_i^2}} $$
Notice that the $b\frac{1}{dy_i}$ term makes it seems reasonableso you cannot simply divide $x_i$ and $y_i$ by $dy_i$ (as pointed out in the comments below).
Unfortunately, this does not propagate the measurement error into an error in the fit parameters though.
While this does not answer the question asked (and will therefore not be accepted), I provide this response in hopes it will help others and promote worthwhile discussion.
David Baird provides a simple explanation for propagating the error through a linear least squares fit in his book Experimentation: An Introduction to Measurement Theory and Experiment Design.
Essentially, weight the fit by dividing $x_i$ and $y_i$ values by the corresponding $dy_i$. As commented below, when reducing this to a matrix problem as described in the OP's wiki-link, make sure to also weight the $1$, i.e., $X=[x, 1]/dy$. Perform the fit to get the parameters $m$ and $b$ in $$y=mx+b$$ Calculate the differences of each $y_i$ value from the fit, $\delta y_i$ Calculate the standard deviation of the fit parameters using$$\sigma_y = \sqrt{\frac{\sum(\delta y_i)^2}{n-2}}$$$$\sigma_m = \sigma_y \sqrt{\frac{n}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$$$\sigma_b = \sigma_y \sqrt{\frac{\sum{x_i^2}}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$
I am not certain this is a complete propagation of the error but it seems reasonable.
While this does not answer the question asked (and will therefore not be accepted), I provide this response in hopes it will help others and promote worthwhile discussion.
In his book Experimentation: An Introduction to Measurement Theory and Experiment Design, David Baird provides a simple explanation for doing a linear least squares fit using the diffences between the measured and fit values to estimate the error of the fit parameters. The best fit for the parameters $m$ and $b$ in $$y=mx+b$$ will is determined using his eqn. (6.3):
$$m = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum(x^2_i) - (\sum x_i)^2} $$
$$b = \frac{\sum(x_i^2)\sum y_i - \sum x_i \sum x_iy_i}{n\sum(x^2_i) - (\sum x_i)^2} $$
After obtaining $m$ and $b$, a standard deviation for the fit parameters can be obtained by calculating the differences of each $y_i$ value from the fit, $\delta y_i = y_i - (m x_i +b)$. From these differences, one calculates the standard deviation of the data from the fit line using:
$$\sigma_y = \sqrt{\frac{\sum(\delta y_i)^2}{n-2}}$$
and then the standard deviation of the parameters using$$\sigma_m = \sigma_y \sqrt{\frac{n}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$and$$\sigma_b = \sigma_y \sqrt{\frac{\sum{x_i^2}}{n\sum{x_i^2}-\left(\sum{x_i}\right)^2}}$$
To include the measurement error, $dy_i$, in the fit one would divide the initial system of equations by $dy_i$ giving $$\frac{y_i}{dy_i} = m\frac{x_i}{y_i} + b \frac{1}{dy_i}$$ then repeat Baird's derivation to get the weighted fit parameters
$$m = \frac{\sum \frac{1}{dy_i}\sum \frac{x_i y_i}{dy_i^2} - \sum \frac{x_i}{dy_i^2} \sum \frac{y_i}{dy_i}}{\sum \frac{1}{dy_i}\sum \frac{x^2_i}{dy_i^2} - \sum \frac{x_i}{dy_i}\sum \frac{x_i}{dy_i^2}} $$
$$b = \frac{\sum \frac{x_i^2}{dy_i^2}\sum \frac{y_i}{dy_i} - \sum \frac{x_i}{dy_i} \sum \frac{x_iy_i}{dy_i^2}}{\sum \frac{1}{dy_i}\sum \frac{x^2_i}{dy_i^2} - \sum \frac{x_i}{dy_i}\sum \frac{x_i}{dy_i^2}} $$
Notice that the $b\frac{1}{dy_i}$ term makes it so you cannot simply divide $x_i$ and $y_i$ by $dy_i$ (as pointed out in the comments below).
Unfortunately, this does not propagate the measurement error into an error in the fit parameters though. |
Quadratic Expressions
A quadratic expression is a polynomial with degree two. Some examples of quadratic expressions:
\[1 - x + {x^2},\,\,\,\,\,\,{y^2} + 1,\,\,\,\,\,\, - 3{z^2}\]
Some examples of expressions which are not quadratic:
\[ - x + 2,\,\,\,\,3 + {y^3} - {y^2} + 1,\,\,\,\,{z^{10}} - 3\]
Note that for a polynomial expression to classify as a quadratic, the powers of the coefficients of the variables are irrelevant. Thus, the following are also quadratic expressions:
\[ - \sqrt 2 {x^2} + \sqrt 3 x - \frac{1}{2},\,\,\,\,{\pi ^3}{y^2} - {2^{ - \frac{1}{3}}}y + 1,\,\,\,\,\frac{{{z^2}}}{{{3^{10}}}} - \frac{z}{{{3^5}}}\,\]
We can use any letter to represent the variable in a quadratic expression. For example, the expression \( - 3{t^2} + t - 2\) is a quadratic in the variable \(t\). The expression \(x - a - {a^2}\) is
not a quadratic in \(x\), but it is a quadratic in \(a\).
If the variable is \(x\), then the simplest quadratic expression is \({x^2}\), whereas the general quadratic expression is of the form \(a{x^2} + bx + c\). There can be at the most three terms in a quadratic expression:
the square term (the term containing \({x^2}\))
the linear term (the term containing \(x\))
the constant term (the term which is independent of the variable)
We will use the notation
\[\begin{align}&{Q_1}\left( x \right)\,:\;2{x^2} - 5x + 2\,\\&{Q_2}\left( y \right)\,:\;1 - 7{y^2}\end{align}\] |
A fibonacci number is a number in the sequence of numbers 1, 1, 3, 5, 8, 13, 21, 34, 55, ...... Each number in the sequence except the first two 1's, is got by adding the previous two terms.
Fibonacci sequence can therefore be defined as $a_{1}$ = $a_{2}$ = 1
$a_{n} = a_{n-1} + a_{n-2}$
, where n is a positive integer which is greater than 2.
This is a recursive formula to find a term in the Fibonacci Sequence.
An important property of Fibonacci Numbers is that the ratio between two consecutive terms approximates very close to $\varphi$, the golden ratio after initial few terms. $\varphi$ is an irrational number which is equal to $\frac{\sqrt{5}+1}{2}$.
This property is used in finding a general term in fibonacci sequence.
The formula to be used to find a general term in Fibonacci Sequence is as follows:
$a_{n}$ = $\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\sqrt{5}}$.
The value of the expression after substituting the value of n can be found using the calculator.
Verification Test for Fibonacci Number:
Suppose a number N is given. If the computed value of 5 $N^{2}$ + 4 or 5 $N^{2}$ - 4 is a perfect square, then N is a Fibonacci number.
Solved Examples
Question 1:
Find the 14
th
term in the fibonacci sequence.
Solution:
We need to find $a_{14}$
here. $a_{14}$ = $\frac{(1+\sqrt{5})^{14}-(1-\sqrt{5})^{14}}{2^{14}\sqrt{5}}$ The value returned by the calculator for the above expression is 377 Hence, the 14 th term in fibonacci sequence is 377. The ratio between any two consecutive terms is $\varphi$. That is, $\frac{a_{n}}{a_{n-1}}$ = $\frac{\sqrt{5}+1}{2}$ This property can be used to find the preceding or succeeding term of a given fibonacci number.
Question 2:
987 is a Fibonacci Number. Find the preceding and succeeding numbers in the sequence.
Solution:
Let us find the preceding number using the above relation. We have $a_{n}$
= 987 and let $a_{n-1}$ = x. $\frac{987}{x}$ = $\frac{\sqrt{5}+1}{2}$ 987 x 2 = x $(\sqrt{5}+1)$ x = $\frac{987\times 2}{\sqrt{5}+1}$ = 609.99954689....... Rounding to the nearest integer, x = 610. The number succeeding 987 is obtained by adding the number preceding to it. Hence, the number that comes after 987 in the sequence is 987 + 610 = 1597. We can also find the next number using the following proportion: $\frac{y}{987}$ = $\frac{\sqrt{5}+1}{2}$ y = 987 x $\frac{\sqrt{5}+1}{2}$ y = $\frac{987(\sqrt{5}+1)}{2}$ = 1596.9995468...... Rounding this, we get the number 1597.
Question 3:
Check whether the number 4181 is a Fibonacci number or not.
Solution:
Let us compute the two expressions given for the test.
5$N^{2}$ + 4 = $5(4181)^{2}$ + 4 = 87,403,809 $\sqrt{87,403,809}$ = 9349.000428 The computed value for 5$N^{2}$ + 4 is not a perfect square. Similarly, 5$N^{2}$ - 4 = 87,403,801. $\sqrt{87,403,801}$ = 9349 The computed value of 5$N^{2}$ - 4 is a perfect square. Hence, 4181 is a Fibonacci number. |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane polars $(r , \theta)$, we have the defining relations $$r\sin \theta=x=\pm R^2−b^2 \sin\sigma, r\cos\theta=z=R\cos\sigma$$ I am having a tough time visualising what this is?
Consider the function $f(z) = Sin\left(\frac{1}{cos(1/z)}\right)$, the point $z = 0$a removale singularitya polean essesntial singularitya non isolated singularitySince $Cos(\frac{1}{z})$ = $1- \frac{1}{2z^2}+\frac{1}{4!z^4} - ..........$$$ = (1-y), where\ \ y=\frac{1}{2z^2}+\frac{1}{4!...
I am having trouble understanding non-isolated singularity points. An isolated singularity point I do kind of understand, it is when: a point $z_0$ is said to be isolated if $z_0$ is a singular point and has a neighborhood throughout which $f$ is analytic except at $z_0$. For example, why would $...
No worries. There's currently some kind of technical problem affecting the Stack Exchange chat network. It's been pretty flaky for several hours. Hopefully, it will be back to normal in the next hour or two, when business hours commence on the east coast of the USA...
The absolute value of a complex number $z=x+iy$ is defined as $\sqrt{x^2+y^2}$. Hence, when evaluating the absolute value of $x+i$ I get the number $\sqrt{x^2 +1}$; but the answer to the problem says it's actually just $x^2 +1$. Why?
mmh, I probably should ask this on the forum. The full problem asks me to show that we can choose $log(x+i)$ to be $$log(x+i)=log(1+x^2)+i(\frac{pi}{2} - arctanx)$$ So I'm trying to find the polar coordinates (absolute value and an argument $\theta$) of $x+i$ to then apply the $log$ function on it
Let $X$ be any nonempty set and $\sim$ be any equivalence relation on $X$. Then are the following true:
(1) If $x=y$ then $x\sim y$.
(2) If $x=y$ then $y\sim x$.
(3) If $x=y$ and $y=z$ then $x\sim z$.
Basically, I think that all the three properties follows if we can prove (1) because if $x=y$ then since $y=x$, by (1) we would have $y\sim x$ proving (2). (3) will follow similarly.
This question arised from an attempt to characterize equality on a set $X$ as the intersection of all equivalence relations on $X$.
I don't know whether this question is too much trivial. But I have yet not seen any formal proof of the following statement : "Let $X$ be any nonempty set and $∼$ be any equivalence relation on $X$. If $x=y$ then $x\sim y$."
That is definitely a new person, not going to classify as RHV yet as other users have already put the situation under control it seems...
(comment on many many posts above)
In other news:
> C -2.5353672500000002 -1.9143250000000003 -0.5807385400000000 C -3.4331741299999998 -1.3244286800000000 -1.4594762299999999 C -3.6485676800000002 0.0734728100000000 -1.4738058999999999 C -2.9689624299999999 0.9078326800000001 -0.5942069900000000 C -2.0858929200000000 0.3286240400000000 0.3378783500000000 C -1.8445799400000003 -1.0963522200000000 0.3417561400000000 C -0.8438543100000000 -1.3752198200000001 1.3561451400000000 C -0.5670178500000000 -0.1418068400000000 2.0628359299999999
probably the weirdness bunch of data I ever seen with so many 000000 and 999999s
But I think that to prove the implication for transitivity the inference rule an use of MP seems to be necessary. But that would mean that for logics for which MP fails we wouldn't be able to prove the result. Also in set theories without Axiom of Extensionality the desired result will not hold. Am I right @AlessandroCodenotti?
@AlessandroCodenotti A precise formulation would help in this case because I am trying to understand whether a proof of the statement which I mentioned at the outset depends really on the equality axioms or the FOL axioms (without equality axioms).
This would allow in some cases to define an "equality like" relation for set theories for which we don't have the Axiom of Extensionality.
Can someone give an intuitive explanation why $\mathcal{O}(x^2)-\mathcal{O}(x^2)=\mathcal{O}(x^2)$. The context is Taylor polynomials, so when $x\to 0$. I've seen a proof of this, but intuitively I don't understand it.
@schn: The minus is irrelevant (for example, the thing you are subtracting could be negative). When you add two things that are of the order of $x^2$, of course the sum is the same (or possibly smaller). For example, $3x^2-x^2=2x^2$. You could have $x^2+(x^3-x^2)=x^3$, which is still $\mathscr O(x^2)$.
@GFauxPas: You only know $|f(x)|\le K_1 x^2$ and $|g(x)|\le K_2 x^2$, so that won't be a valid proof, of course.
Let $f(z)=z^{n}+a_{n-1}z^{n-1}+\cdot\cdot\cdot+a_{0}$ be a complex polynomial such that $|f(z)|\leq 1$ for $|z|\leq 1.$ I have to prove that $f(z)=z^{n}.$I tried it asAs $|f(z)|\leq 1$ for $|z|\leq 1$ we must have coefficient $a_{0},a_{1}\cdot\cdot\cdot a_{n}$ to be zero because by triangul...
@GFauxPas @TedShifrin Thanks for the replies. Now, why is it we're only interested when $x\to 0$? When we do a taylor approximation cantered at x=0, aren't we interested in all the values of our approximation, even those not near 0?
Indeed, one thing a lot of texts don't emphasize is this: if $P$ is a polynomial of degree $\le n$ and $f(x)-P(x)=\mathscr O(x^{n+1})$, then $P$ is the (unique) Taylor polynomial of degree $n$ of $f$ at $0$. |
If you are expecting someone to solve the Kerr metric equations, you probably need to hire a professional mathematician; but if you want an approximation, we can make that happen. Lets start with some simple results and eventually work our way to advanced results.
Objective
Our objective is to calculate the maximum one dimensional stress tensor acting in the direction along the axis tangential to the assumed spherical surface of the black hole. Since all our forces come from the gravity of the black hole, they will all be acting in this direction, so I'm not going to use any vectors.
Assumptions $\text{M}_{b}$ is the mass of the black hole and it equals$1.99\times10^{31} \text{ kg}$ (10 times the mass of the sun). The object in question is a 1 km long, 100m radius cylindrical rod, with the rod aligned in the direction of a radial line from the center of the black hole outwards. The object has constant density $\rho = 1$, its just not that important right now. The object is 'suspended' with it's midpoint centered at 3/4 the Schwarzschild radius of the black hole. Schwarzschild radius
Given by $$r_s = \frac{2GM}{c^2}$$ where G is the universal gravitation constant ($6.67\times10^{-11}\frac{\text{N}\cdot\text{m}^2}{\text{kg}^2}$), M is the mass of the black hole, and c is the speed of light ($3.00\times10^{8}\frac{\text{m}}{\text{s}}$). Therefore $$r_s = \frac{2\cdot 6.67\times10^{-11} \cdot 1.99\times10^{31}}{(3.00\times10^{8})^2} = 29500 \text{m}. $$ Therefore, with some rounding, the near-hole point of our rod is at 22 km, the far-hole point is at 23km.
Force of gravity as a function of distance from near-hole point
Let us define a coordinate system in one dimension with $l = 0$ as the near-hole point, and $l = 1000$ (in meters) as the far-hole point of our rod. We will calculate the force of gravity on each infinitesimally small slice of the rod as a function of it's $l$ coordinate.
The force of gravity on a mass is $$F = G\frac{m_1m_2}{r^2}.$$ The mass of a slice of the rod (equivalent to the distance derivative of the mass of the rod) is equal to the mass of a circle $\frac{dm}{dl} = \rho \pi (100)^2$. Therefore the distance derivative of the force of gravity on a slice is $$\frac{dF_{slice}}{dl} = 6.67\times10^{-11} \frac{1.99\times10^{31}\cdot \rho \pi (100)^2}{(23000 + l)^2} = \frac{4.17\times10^{25}}{(23000 + l)^2} $$
Integrate the distance derivative of the force of gravity
To find the net force between points $l = a$ and $l = b$, we integrate the distance derivative of the force of gravity with respect to distance from the near-hole point. $$\int_a^b \frac{4.17\times10^{25}}{(23000 + l)^2} dl = \left.\frac{-4.17\times10^{25}}{23000 + l}\right|^b_a = -4.17\times10^{25}\left(\frac{1}{23000+b}-\frac{1}{23000+a}\right)$$
Solving this for the net force on the entire rod, we get $$-4.17\times10^{25}\left(\frac{1}{23000+1000}-\frac{1}{23000+0}\right) = 7.55\times10^{19} \text{N}.$$
Now that force has to be counteracted by a 'lift' force keeping the rod out of the black hole. For simplification let us assume that the counteracting force acts equally on each slice of the rod, so each a slice of the rod from a to b is pulled out of the black hole with force $$F_{lift} = -7.55\times10^{19}\cdot\frac{b-a}{1000}.$$ Note the force is negative because it is acting in the direction out of the hole.
Solve for stress at any point in the rod
In this simplification, the highest gravity force will be at the lowest point closest to $l = 0$. Therefore, the stress causing force at any distance $x$ in this rod is going to be the net force of gravity and lift for all slices below it minus the net force of gravity and lift for all points above it. $$\begin{align}F_{net} =&\left.\frac{-4.17\times10^{25}}{23000 + l}\right|^x_0 - 7.55\times10^{19}\cdot\frac{x-0}{1000}- \left.\frac{-4.17\times10^{25}}{23000 + l}\right|^{1000}_x + 7.55\times10^{19}\cdot\frac{1000-x}{1000} \\=& 3.55\times10^{21}-\frac{8.34\times10^{25}}{23000+x} + 7.55\times10^{16}\cdot (1000-2x)\end{align}$$
The net force graph looks like this:
Maximum force is $1.61\times10^{18} \text{ N}$ at $l=500$.
Stress defined as $\sigma = \frac{\text{F}}{\text{A}}$. The cross sectional area is $\pi(100)^2 = 31415 \text{m}^2$, so maximum stress is $$\sigma = \frac{1.61\times10^{18} \text{ N}}{31415 \text{m}^2} = 5.12\times10^{13} \text{Pa}.$$
Conclusion
The calculation works and produces logical results. Stress should be zero at the ends of the rod (there is nothing to pull away from) and should be maximum in the center. The stress produced is very high, as would be expected 23km from the center of a black hole.
Required yield strength is about 51 TPa. The required material strength is probably not achievable with any known or theoretical material. I can't find anything with a yield strength over 1 TPa, much less 51. |
Category:Definitions/Integral Domains An integral domain $\struct {D, +, \circ}$ is: a commutative ring which is non-null with a unity in which there are no (proper) zero divisors, that is: $\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$
that is (from the Cancellation Law of Ring Product of Integral Domain) in which all non-zero elements are cancellable.
Subcategories
This category has the following 10 subcategories, out of 10 total.
A ► Definitions/Associates (9 P) D ► Definitions/Divisibility (1 C, 7 P) G I ► Definitions/Integers (5 C, 45 P) L ► Definitions/Lowest Common Multiple (5 P) O ► Definitions/Ordered Integral Domains (1 C, 11 P) Pages in category "Definitions/Integral Domains"
The following 25 pages are in this category, out of 25 total. |
Errors in layer thicknesses and instability of the refractive indices of thin film materials are the main reasons of the deviations of experimental spectral characteristics of produced optical coatings from theoretical performances of corresponding multilayer designs.
Errors in layer thicknesses are inevitable even in modern deposition plants equipped with high precision monitoring devices.
It is known also that
optical constants of layer materials may deviate from the specified in the corresponding theoretical designs. These deviations are explained by various factors inside the deposition chamber, for example, change of materials in melting pot, unstable substrate temperature, cleaning conditions, etc. nominal constants
It is known that
different solutions of the same design problem exhibit different sensitivities to deposition errors. Some designs may be sensitive with respect to even small errors. Other solutions are stable to errors of large magnitudes.
Stability of the design solutions is tightly interconnected with a monitoring technique used to control the deposition process. For example, the same design solution might be stable in the case of time-monitoring control and sensitive to errors in the cases of broad band or monochromatic monitoring.
Three ways to find a stable design
1) To obtain a set of solutions of a design problem, provide computational manufacturing experiments simulating real deposition runs and choose solutions providing highest production yields estimations (see for example, article on yield analysis and our paper)
2) To use special algorithms
(Robust Synthesis) taking into account stability requirement already in the course of the synthesis process. In Fig. 1 you can see results of the error analysis performed for a conventional and a robust designs of a two-line filter. The robust design (right panel) exhibit much higher stability to thickness errors in index offsets than a conventional design (left panel).
3) To use a deterministic approach taking errors in optical constants into account. In OptiLayer, this approach is realized in the form of Environments manager.
Fig. 1. Illustrating example: Stability of the spectral characteristics of conventional (left panel) and robust (right panel) two-line filters.
With
Robust synthesis, you can obtain designs which are typically more stable with respect to production errors than standard (conventional) designs.
Robust design can be activated by checking
Robust Synthesis Enabled check box available through Synthesis --> Options --> Robust tab (Fig. 2).
The new algorithm is based on a simultaneous optimization of a sets of design (
design cloud) located in a vicinity of a basic, so called pivotal design.
Robust Synthesis options require more computations than analogous standard OptiLayer synthesis options (
about M times more, if M is the number of samples of the design cloud).
Reasonable
values are 100-200. M
The number of the designs in the design cloud
can be specified in M The number of samples field on Robust tab.
The standard merit function is generalized in the following way:
\(GMF=\left\{\sum\limits_{j=0}^M MF^2_j/(M+1)\right\}^{1/2}, \) (Eq. 1)
\(MF_j=\left\{\frac 1L \sum\limits_{l=0}^L \left[\frac{S(X^{(j)},\lambda_l)-\hat{S}_l}{\Delta S_l}\right]^2\right\}^{1/2} \) (Eq. 2)
\(S(X^{(j)},\lambda)\) and \(\hat{S}_l\) are actual and target values of spectral characteristics, \(\Delta S_l\) are target tolerances, \(\{\lambda_l\}, l=1,...,L\) is the wavelength grid.
\(X^{(0)}={d_1,...,d_m, n_H,n_L}\) is the pivotal design and \(X^{(j)}={d_1^{(j)},...,d_m^{(j)}, n_H^{(j)},n_L^{(j)}}\) are disturbed designs from the cloud.
\(MF_0\) is the standard merit function, \(MF_j\) are merit functions corresponding to designs from the cloud.
Fig. 2. Activating Robust synthesis option and describing characteristics of the design cloud.
OptiLayer allows you to specify various expected production errors. In the simplest case, there are
errors in layer thicknesses only and no offsets of the optical constants.
Errors in layer thicknesses can be specify in the absolute and/or relative scales (
Tolerance size panel, Absolute and Relative fields on the Robust tab, see Fig. 2). In the example in Fig. 2, relative errors of 1% are specified.
No Drift in Type column specifies absence of offsets in layer refractive indices.
Absolute errors in layer thicknesses:
\(d_i^{(j)}=d_i+\delta_i^{(j)} \;\;\) (Eq. 3)
Relative errors in layer thicknesses:
\(d_i^{(j)}=d_i+\Delta_i^{(j)}d_i \;\; \) (Eq. 4)
Please, note that these parameters and the values of the refractive index offsets are not directly corresponding to the deposition process accuracy, yet they are connected with it. These parameters should not be considered literally, they are merely control parameters of the algorithm.
Fig. 3. Systematic offsets of H and L materials from [-0.01; 0.01] and [-0.005; 0.005] are specified.
If you would like to take into account offsets of the refractive indices as well, you need to specify
Index Drift level and Type of the offset.
If
Per Material type is chosen then systematic offsets will take the for all layers of the same material. same value
\(n_{H,L}^{(j)}=n_{H,L}+\Sigma_{H,L}^{(j)}, \;\;j=1,...,M\) (Eq. 5)
The algorithm with
systematic offset is applied when it is assumed that actual refractive indices can be shifted with respect to the nominal ones in the course of the deposition.
The systematic errors \(\Sigma_{H,L}^{(j)}\) are random normally distributed errors with zero means and standard deviations \(\Sigma_{H,L}\).
If
Per Layer type is chosen (Fig. 3) then random offsets will take for different layers of the same material. different values
\(n_{H,L}^{(j)}=n_{H,L}+\sigma_{H,L,i}^{(j)}, \;\;j=1,...,M\) (Eq. 6)
The random errors \(\sigma_{H,L,i}^{(j)}\) are random normally distributed errors with zero means and standard deviations \(\sigma_{H,L}\).
The algorithm with
random offset is applied when it is assumed that actual refractive indices are not stable in the course of the deposition process due to various reasons (for example, instabilities of substrate temperature).
Example. Two-line Filter: target transmittance is 100% in the ranges 598-602 nm and 698-702 nm, target transmittance is zero in the ranges 500-580 nm, 615-693 nm, and 720-800 nm (Fig. 4).
Layer materials are Nb
2O 5 and SiO 2, Suprasil substrate.
First, a set of conventional designs was obtained. The structure of one of the conventional designs and its spectral characteristics are shown in Fig. 4.
Merit function value (MF) is 2.9.
Fig. 4. Spectral transmittance and structure of a conventional 31-layer design solution.
Assume, that the expected level of errors in layer thicknesses is 1% and there are systematic offsets of refractive indices, 0.01 of Nb
2O 5 and 0.005 of SiO 2. A robust solution can be found with settings shown in Fig. 3 ( Per Material). It means that \(\Sigma_{H}=0.01\) and \(\Sigma_{L}=0.005.\)
As a starting design, a single layer can be used. Gradual Evolution can be used for synthesis.
As a result, a 27-layer solution (Fig. 5) is obtained.
The merit function value is 10.3 that is larger than in the case of the conventional design (Fig. 4).
It is a typical situation for the robust synthesis:
robust design solutions approximate target specifications a little bit worse than the conventional designs. The reason is fundamental: additional stability requirements are taken into account in the course of the merit function optimization, i.e. the generalized merit function containing multiple terms (Eq. 2) is optimized instead of the standard merit function.
Fig. 5. Spectral transmittance and structure of the robust 27-layer robust solution obtained assuming 1% errors in layer thicknesses and 0.01 and 0.005 systematic offsets in high- and low-refractive indices, respectively.
Fig. 6.
Spectral transmittance and profile of the robust 29-layer robust solution obtained assuming 1% errors in layer thicknesses and 0.01 and 0.005 random offsets in high- and low-refractive indices, respectively.
If the expected level of errors in layer thicknesses is 1% and there are random offsets of refractive indices, 0.01 of Nb
2O 5 and 0.005 of SiO 2. A robust solution can be found with settings shown in Fig. 3 ( Per Layer). It means that \(\sigma_{H}=0.01\) and \(\sigma_{L}=0.005.\)
As a result, a 29-layer solution (Fig. 6) is synthesized.
The merit function value is 6.5. As expected it is bigger than in the case of the conventional design (Fig. 4).
Important! Merit function (MF) displayed on the bottom of the evaluation window (Fig. 7 and 8) is calculated in two different ways. If the robust option is disabled, the merit function is calculated in the standard way. If the robust option is enabled, MF is calculated via Eqs. 1 and 2. In Fig. 7 and 8 we can see MF values of conventional and robust two-line filters, respectively. Standard MF is smaller at the conventional design. At the same time, the generalized merit function GMF is smaller in the case of the robust solution.
Fig. 7. Calculation of the merit function when robust option is disabled/enabled (conventional design).
Fig. 8. Calculation of the merit function when robust option is disabled/enabled (robust design).
After obtaining the robust design or a series of the robust designs, several important questions arise:
How to
evaluate stability of the obtained robust designs?
How to compare stability of the conventional and robust designs?
What design solution is the most stable to deposition errors?
There are no simple answers to these questions. However, there are recommendations, which are typically help you to evaluate the designs stability (of course, with respect to the monitoring technique in use).
Computational experiments simulating the deposition process can help you to evaluate stability of your design solution.
1) In the case of non-optical monitoring technique (quartz crystal or time monitoring), statistical error analysis it recommended (see below).
2) In the case of broadband monitoring (BBM), computational manufacturing with BBM are recommended (without witness chips or with witness chips).
3) In the case monochromatic monitoring, simulations without witness chips or indirect monitoring can be used.
In the course of the error analysis, it is reasonable to specify the same levels of the errors in layer thicknesses and offsets of refractive indices.
In the case of statistical error analysis, designs with imposed errors in layer parameters are generated and spectral characteristics of the disturbed designs are calculated.
Fig. 9. Activating statistical error analysis.
In the example above (Two-Line Filter), the level of errors can be equal to 1% (
Rel. RMS (%) column).
If in the course of the robust synthesis Tolerance size was specified in absolute values, then that value should be specified in
Rel.RMS(%) column.
If in the course of the robust synthesis refractive index offsets were specified
Per Material, then Per Material Errors box should be checked and the values of offsets should be specified in the RMS column on the Refractive Index tab (Fig. 10).
This is, however, just a general recommendation. Of course, other reasonable error levels/settings can be used in the course of the statistical analysis.
Fig. 10. Specifying errors in layer thicknesses and refractive index offsets in the course of the statistical error analysis.
Fig. 11. Results of the statistical error analysis of the conventional 31-layer conventional design (Fig. 4). Spectral characteristics of the disturbed design degrade significantly especially around the high transmission zone at 700 nm.
Fig. 12. Results of the statistical error analysis of the robust 27-layer design (Fig. 5). It is seen that the design is more stable.
The numerical measure
helps to evaluate the averaged stability of the design solution. This value is displayed on the bottom of the Error Analysis window. E(dMF)
The value E(dMF) is expected deviation of the spectral characteristics of the theoretical design averaged by the number of the wavelength and the number of the statistical tests (
The number of tests field on the Error Analysis Setup window, Fig. 9).
Fig. 13. Comparison of the E(dMF) values of the conventional 31-layer and 27-layer robust designs.
Important notes:
The robust algorithm takes not all sources of the deposition errors into account. Influence of some factors should be considered separately.
It is recommended to obtain and analyze a series of good design solutions using various design techniques.
It can happen that the levels of errors in layer parameters is to high to meet target requirements with the help of the robust algorithm.
It is reasonable to stop the computations if either the generalized merit function almost does not decrease or there are very insignificant changes in the pivotal design. It means that a state of dynamic equilibrium has been achieved and at the specified error level it is not reasonable to search for a more complicated design.
Almost all OptiLayer optimization algorithms support the robust synthesis.
You maybe also interested in the following articles: |
When we come across real situations like " If the incomes of two persons is in the ratio 3 : 6 and their expenditures are in the ratio 2 : 5, find their income if both of them save Rs. 400." It is very difficult to solve this by trial and error method. It is more important now than ever before for everyone, specially teachers and students to aware of latest changes in the Syllabus and the best new methods, programs, and devices in the field right away. We focus on simple ways of learning to inspire students. Unfortunately, many of us have treated algebra as a complicated subject and skip to solve related problems because of different reasons. We have solved linear equations and inequalities in the way that will help you to understand concept easily.
Definition What are Linear Equations?
An algebraic equations with degree one are known as linear equations.
Examples
: 2x + 1 = 0, 10y = 7 and 1/3x - 1 = 0
Some Important Points:For x, y variables; p,q coefficients and s is a constant value. 1. General form: Linear Equations of one variable is px + s = 0. Linear Equations of two variables is px + qy + s = 0 2. The only solution to the linear equation of one variable is x = - s/p. 3. The value which satisfy the equation is called the solution. 4. The solution to the linear equation of one variable can be represented on the number line. 5. When we plot a graph for an equation with 2 variables that gives a straight line. What are linear inequalities? A statement of inequality between two expressions involving a single variable with highest power 1, is called a linear inequality. Examples : 2x + 4 < 5, 6(x - 4) $\geq$ 5x -2 Some Important Points:
1. Domain of the variable: The set from which the values of the variable are replaced in an inequality, is called replacement set of the domain of the variable.
2. Solution Set: The set of all values of x from the domain (replacement set) which satisfy, the given inequality is called the solution set.
Solving Linear Equations and Inequalities in One Variable
A linear equation in 1 variable, say x, is an equation that can be written in the form: ax+b = 0 (a and b are real number and a $\neq$ 0). The properties used in solving linear equations and linear inequalities are very similar, except the multiplication property. Check for linear equations properties and linear inequalities properties for more understanding.
Linear Equations and Inequalities in Two Variables
The general form of linear equations of two variables is ax + by + c = 0.
Here the variables used are x and y. As the value of changes the value of x also changes the value of y correspondingly.
Variable, x can take infinitely many values on the real number line, there will be corresponding real value for y.
For Example : Find any two solutions of the linear equation of two variables 4x + y = 8. Represent graphically.
When 4x + y = 8 => y = 8 - 4x
Let us assume some values for x
when x = -2, y = 8 - 4 ( -2) = 8 + 8 = 16
Hence
one of the solution of the above equation is ( -2, 16 )
When x = 0, y = 8 - 4 ( 0 ) = 8 - 0 = 8
The
other solution is ( 0,8).
The pair of values (x,y) are called the set of solutions to the given equation.
When we plot the points on the two-dimensional graph, we get a straight line. The line will divide the graph into two regions.
The graph will be as shown below:
Linear Inequalities in two variables
The general form of linear inequalities of 2 variables will be of the form ax + by < c, ax + by $\leq$ c, ax + by > c, ax + by $\geq$ c.
We know that the graph of the equation ax + by = c, is a straight line which divides the xy-plane into two parts which are represented by ax + by $\geq$ c,
and ax + by $\leq$ c. These two parts are known as closed half spaces.
The region ax + by < c and ax + by > c are known as the open half spaces.
These half spaces are known as the solution sets of the corresponding inequalities.
Functions and Linear Equations and Inequalities
Functions are represented as y = f(x), where x is an independent variable and y is a dependent variable. Slope - intercept form is one of the examples of linear equations.
Example :
Linear equations:
y = f(x) = 2x+3, y = f(x) = 3x$^2$ - 2x + 6.
Whereas Linear inequalities: y = f(x) > 2x + 3, y = f(x) $\geq$, 3x$^2$ - 2x + 6
Few Important facts
:
1. The graph of the line is a straight line which can be extended indefinitely on both the direction.
3. The Domain of the function is the set of all real numbers,
x = { x : x belongs to Real numbers }
4. The range of the function is also the set of all real numbers which depend on x.
y = { y : y belongs to real numbers, y = f(x) }
5. A linear equation in one variable have one solution whereas a linear inequality can have many.
Examples Example 1
:
S
olve 2x + 5 = 9, and represent the solution on the number line
Solution
: We have 2x + 5 = 9
=> 2x + 5 - 5 = 9 - 5
=> 2x = 4
=> x = 4/2 = 2
Hence, We can see the solution being represented on the number line.
Example 2
: Represent the region which satisfy the two inequalities 4x + y $\leq$ 8 , and 15 x + 7 y > 105 in two dimensional graph.
Solution: Step 1: Write inequalities into equations as
4x + y = 8 ---------(1) and
15 x + 7y = 105 ----------(2)
Step 2: Find the x and y intercepts of each line.
4x + y = 8
15 x + 7y = 105
Step 3: Analyze the solution area.
Since the inequality is 4x + y $\leq$ 8
(1) We draw thick line in the graph.
(2) when we plug in (0,0) , we get 4(0) + 0 < 8 => 0 < 8, which is a true statement.
Hence we shade the region which does contain the origin
.
Since the inequality is 15 x + 7y > 105,
(1) We draw the dotted line in the graph.
(2) By substituting (0,0) we get 15(0)+7(0) > 105 => 0 > 105, which is a false statement.
Hence we shade the region which does not contain the origin
.
Step 4: Design the graph
.
The graphs of these two inequalities are shown below.
Step 5: Get your solution.
The shaded region represents the desired solution set.
Example 3: Solve 2x + 5 $\geq$ 13. Represent the solution on the number line, if x belongs to set of Real numbers. Solution:
We have 2x + 5 $\geq$ 13 and the replacement set is "Real Numbers"..
2x + 5 $\geq$ 13
=> 2x + 5 - 5 $\geq$ 13 - 5
=> 2x $\geq$ 8
=> x $\geq$ 8/2
=> x $\geq$ 4
In the above number line, the darkened portion x $\geq$ 4, is the solution set for the inequality.
Practice Questions
1. Solve the equation for x. 4 ( x- 5) = 40.
2. Check if x = 4 is the solution of the equation 6x + 4 = 20
3. Find any four solutions of the equation 3x + y = 10.
4. Find the solution set of the inequality, 2x + 5 $\leq$ 15, where x belongs to Real numbers.
Express your answer in interval form and also represent on the real number line.
5. Graph the equation 4x + 7y = 28, by finding the x and y intercepts.
6. Shade the region which satisfy the inequality 3x + y $\geq$ - 2. |
This question already has an answer here:
I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is.
Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ $x$$y$$z$
find the maximum and minimum values subject to the constraint: $g$($x$,$y$,$z$) $=$ $x^2$+2$y^2$+3$z^2$
$$F=x y z +\lambda \left(x^2+2 y^2+3 z^2-a\right)$$ Computing derivatives $$F'_x=y z+2 \lambda x=0\tag 1$$ $$F'_y=x z+4 \lambda y=0\tag 2$$ $$F'_z=x y+6 \lambda z=0\tag 3$$ $$F'_\lambda=x^2+2 y^2+3 z^2-a=0\tag 4$$ Now, I should consider equations $(1,2,3)$ and solve them for $x,y,z$ in terms of $\lambda$.
Multiplying equations 1,2,3 by $x,y,z$ we obtain that $2x^2=4y^2=6z^2$. From here I found the corresponding multiples that $x^2$ and $y^2$ are in terms of $z$ and plugged into equation 4 to solve for z. I found that $x$ Was a multiple of $z$ by 3 and $y$ was a multiple of $z$ by $\frac{3}{2}$ I found this by setting $4y^2$=6$z^2$ and got $\frac{6}{4}$ = $\frac{3}{2}$ Now plugging these into equation 4 I obtained 3+$\frac{3}{2}$+$3z^2-6$=0 My algebra lead me to $z=$ $\pm\frac{1}{\sqrt{2}}$ but if done right z should equal $\pm\frac{\sqrt{2}}{\sqrt{3}}$
I got this by adding 6 over to the right then subtracted 3 leaving me: $\frac{3}{2}$+$3z^2$=$3$ then subtracting $\frac{3}{2}$ lead me to : $3z^2$= $\frac{3}{2}$ and dividing by 3 gives $\frac{3}{2} \div \frac{3}{1}$ which is equivalent to $\frac{3}{2} \times \frac{1}{3}$ = $\frac{3}{6}$ and you can see that really leaves me with $z^2$= $\frac{1}{3}$ which is equivalent to $z=$ $\pm$ $\frac{1}{\sqrt{3}}$
What did I do incorrectly and how do I proceed from here once I have found z. Also how is it that you can write the two functions as two functions added together giving you $F$. |
i have the following exercice:
prouve the existence of $\psi_1 \in \mathcal{D}(\mathbb{R})$ such as $\psi_1(0)=0$ and $\psi_1'(0)=0$.
Let $\psi_0 \in \mathcal{D}(\mathbb{R})$ such as $\psi_0=1$ in $V(0)$, and let $\varphi \in \mathcal{D}(\mathbb{R})$ We note $f(x)= \varphi(x)-\varphi(0)\psi_0(x)- \varphi'(0)\psi_1(x)$ such as $f(0)=f'(0)=0$.
prouve that there exist $g \in \mathcal{D}(\mathbb{R})$ such $f(X)= x^2 g(x)$.
resolve the equation $x^2 T=0$.
resolve the equation $x^2 T=\delta$.
I try to do this. for question 1, it's OK. for question 2. i try to methods, but i have questions
Method 1. for $x \neq 0$, we have $f(x)=\dfrac{g(x)}{x^2}$. My difficultie is how we define $g(0)$?
b. With the Taylor developement integral remainder of $f$, in the neighbourhood of zero, in order 2
$$ f(x)= f(0)+ xf'(0)+ \dfrac{x^2}{2!} \displaystyle\int_0^1 (1-t)^2 f''(tx) dt. $$ beacause of $f(0)= f'(0)=0$, then we have $g(x)= \dfrac{1}{2!} \displaystyle\int_0^1(1-t)^2 f''(tx) dt.$ it's clear that $g \in C^\infty(\Bbb R)$ and $\mathrm{supp\,} g $ is compact beacause $\mathrm{supp\,} f'' \subset \mathrm{supp\,} f' \subset \mathrm{supp\,} f$ et $\mathrm{supp\,} f$ est compact car $\mathrm{supp\,} f \subset \{\mathrm{supp\,} \varphi\} \cup \{\mathrm{supp\,} \psi_0\} \cup \{\mathrm{supp\,} \psi_1\}$.
the method 2 is it true?
for question 3 and 4 i don't understand how we can resolve the two equations in $\mathcal{D}'(\mathbb{R})$.
Thank's to help me. |
I'm trying to solve the following differential equation by using the method of Frobenius. I'm however, having some trouble in doing so, I was hoping someone could help me out.
$2ty''+(1+t)y'-2y=0$
ATTEMPT:First of all, we need to control if we can actually use Frobenius' method. By looking at $2ty''$ we can easily conclude that $t=0$ is the only singular point of the differential equation. Then we need to control whether $t \cdot (1+t)$ and $ t^2 \cdot -2 $ are both analytic at $t=0$, which is true. Therefore, we can indeed use the method of Frobenius.
We let $y$ be of the form $y=\sum_{n=0}^{\infty}a_n \cdot t^{n+r}$, then we can get the following by differentiating:
$y=\sum_{n=0}^{\infty}a_n \cdot t^{n+r}$
$y'=\sum_{n=0}^{\infty}(n+r)a_n \cdot t^{n+r-1}$
$y''=\sum_{n=0}^{\infty}(n+r)(n+r-1)a_n \cdot t^{n+r-2}$
We now plug these back in our differential equation:
$\begin{align} 2\sum_{n=0}^{\infty}(n+r)(n+r-1)a_n \cdot t^{n+r-1}+\sum_{n=0}^{\infty}(n+r)a_n \cdot t^{n+r-1} + \sum_{n=0}^{\infty}(n+r)a_n \cdot t^{n+r} - 2\sum_{n=0}^{\infty}a_n \cdot t^{n+r} =\\ \sum_{n=0}^\infty a_n[2(n+r)(n+r-1)+n+r]t^{n+r-1}+\sum_{n=1}^\infty a_{n-1}[n+r-3]t^{n+r-1} =\\ [a_0(2r(r-1))+ra_0]t^r+[2a_1(r+1)(r)+a_1(r+1)+a_0(r-2)]t^{r+1}+\sum_{n=2}^\infty[a_n[2(n+r)(n+r-1)+n+r] + a_{n-1}(n+r-3)]t^{n+r-1} = 0 \end{align}$ So then, out of the first term of the sum (since it has to be 0, as the sum is equal to zero), we can solve for r, which gives us $r=0$ or $r=\frac{1}{2}$, since $a_0$ is assumed to be non-zero. Otherwise it would just change the index of r.
Then, we set $a_0$ to be $1$. and we can solve for $a_0$ by using the second term. $r=0$ gives us that $a_1$ has to be $2$. $r= \frac{1}{2}$ gives us that $a_1$ has to be equal to $\frac{1}{2}$. The last term of the equation above gives us the recurrent relationship. I believe that when we have the recurrent relationship, by plugging in the two different $a_1$ values, we will get two linearly independent solutions for the differential equation and therefore we'll get the general solution of the differential equation. I'm not able however to find this recurrent relationship.
QUESTIONS: I'm having quite a bit of trouble with using this method to solve differential equations, so here are a couple of questions I want to ask: Is it always necessary to write out the first and second term of the sum, as to get the values for $r$ and $a_1$? Is it okay to just set $a_0$ equal to 1? In the sum, I chose to change the powers of t equal to n+r-1 as to factor these powers out, I have also set these to n+r in a previous attempt, however, that I couldn't solve. Is there a general approach as to how to set these powers? Do you always set them at the lowest power of t in the total differential equation? Is this a correct and efficient manner to solve such a problem with Frebenius' method? Or is there some work that could be omitted?
Thanks for your time,
K. Kamal |
So I was trying to prove that the characteristic of an integral domain is either $0$ or prime. I got stuck, so I searched for a proof and I came across the following proof online
Now I almost want to accept this proof, except for the following (possibly silly and pedantic) issue. In the above proof we know that $n_0 \in \mathbb{N} \subseteq \mathbb{Z}$, so since $n$ is not prime, we factorize $n = m \cdot k$ (where $\cdot$ represents multiplication on the integers in the ring $(\mathbb{Z}, +, \cdot)$). Now in the above proof the following is asserted $$n(1_D) = \underbrace{1_D + \dots + 1_D}_{n \text{ times}} =0_D \implies m\cdot k(1_D) = \underbrace{\left(1_D + \dots + 1_D\right)}_{m \text{ times}} \ \bullet \underbrace{\left(1_D + \dots + 1_D\right)}_{k \text{ times}}$$
Now seemingly it seems that multiplication of $m$ and $k$ in $\mathbb{Z}$ is inducing (ring) multiplication of elements in $D$ (when I thought we'd only end up with addition of the $1_D$'s $mk$ times).
Is there a reason why this happens? |
I was wondering, what is the motivation behind the payoff of the cash swaptions being multiplied by the swap annuity? $$c(S_{\theta, T})=\sum_{i=\theta+1}^{T}\tau_i\frac{1}{{(1+S_{\theta,T}(\theta))}^{\tau_{\theta,i}}}$$ Why not using the classic one: $$A_{t} = \sum_{i=1}^{T} P_{t,T_i}\tau_i$$
Thank you in advance for your answer!
Cheers
S. |
I'm studying Classical Mechanics by Goldstein. I solved a problem but I have a question.
Pro 2.18
A point mass is constrained to move on a massless hoop of radius a fixed in a vertical plane that rotates about its vertical symmetry axis with constant angular speed ω. Obtain the Lagrange equations of motion assuming the only external forces arise from gravity. What are the constants of motion? Show that if ω is greater than a critical value ω0, there can be a solution in which the particle remains stationary on the hoop at a point other than at the bottom, but that if ω < ω0, the only stationary point for the particle is at the bottom of the hoop. What is the value of ω0?
So I proceeded like this solution here.
So here, when we choose only one generalized coordinate $\theta$(polar angle), energy function $h$ is not same as the energy. But in the text (chapter about Lagrangian) it says that if potential $V=V(q)$, $h=E$. For this problem $V=mga \cos \theta$, (or negative, according to how define $\theta$ or axis) so it satisfies the condition that potential only depends on generalized coordinate, not on generalized velocity. So $h$ should be $E$, but apparently not.
What is wrong here?
I know that if I set the azimuthal angle as an independent variable, such a contradiction doesn't appear. But I cannot see why I should do that ( the problem says that azimuthal angle is not independent variable, and derivation of $h=E$ says nothing about that.)
Surely something must be wrong with my reasoning, because if the Lagrangian of a system is $L=\frac{1}{2}my'^2+mgy$, we can insert the constant horizontal kinetic energy $\frac{1}{2}mx'^2$ (x is not generalized coordinate here), but that would destroy $h=E$. Can someone explain this to me? |
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1. Observation of a peaking structure in the J/psi phi mass spectrum from B-+/- -> J/psi phi K-+/- decays
PHYSICS LETTERS B, ISSN 0370-2693, 06/2014, Volume 734, Issue 370-2693 0370-2693, pp. 261 - 281
A peaking structure in the J/psi phi mass spectrum near threshold is observed in B-+/- -> J/psi phi K-+/- decays, produced in pp collisions at root s = 7 TeV...
PHYSICS, NUCLEAR | ASTRONOMY & ASTROPHYSICS | PHYSICS, PARTICLES & FIELDS | Physics - High Energy Physics - Experiment | Physics | High Energy Physics - Experiment | scattering [p p] | J/psi --> muon+ muon | experimental results | Particle Physics - Experiment | Nuclear and High Energy Physics | Phi --> K+ K | vertex [track data analysis] | CERN LHC Coll | B+ --> J/psi Phi K | Peaking structure | hadronic decay [B] | Integrated luminosity | info:eu-repo/classification/ddc/ddc:530 | Data sample | final state [dimuon] | mass enhancement | width [resonance] | (J/psi Phi) [mass spectrum] | Breit-Wigner [resonance] | 7000 GeV-cms | leptonic decay [J/psi] | PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
PHYSICS, NUCLEAR | ASTRONOMY & ASTROPHYSICS | PHYSICS, PARTICLES & FIELDS | Physics - High Energy Physics - Experiment | Physics | High Energy Physics - Experiment | scattering [p p] | J/psi --> muon+ muon | experimental results | Particle Physics - Experiment | Nuclear and High Energy Physics | Phi --> K+ K | vertex [track data analysis] | CERN LHC Coll | B+ --> J/psi Phi K | Peaking structure | hadronic decay [B] | Integrated luminosity | info:eu-repo/classification/ddc/ddc:530 | Data sample | final state [dimuon] | mass enhancement | width [resonance] | (J/psi Phi) [mass spectrum] | Breit-Wigner [resonance] | 7000 GeV-cms | leptonic decay [J/psi] | PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Journal Article
2. Measurement of the ratio of the production cross sections times branching fractions of B c ± → J/ψπ ± and B± → J/ψK ± and ℬ B c ± → J / ψ π ± π ± π ∓ / ℬ B c ± → J / ψ π ± $$ \mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm }{\pi}^{\pm }{\pi}^{\mp}\right)/\mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm}\right) $$ in pp collisions at s = 7 $$ \sqrt{s}=7 $$ TeV
Journal of High Energy Physics, ISSN 1029-8479, 1/2015, Volume 2015, Issue 1, pp. 1 - 30
The ratio of the production cross sections times branching fractions σ B c ± ℬ B c ± → J / ψ π ± / σ B ± ℬ B ± → J / ψ K ± $$ \left(\sigma...
B physics | Branching fraction | Hadron-Hadron Scattering | Quantum Physics | Quantum Field Theories, String Theory | Classical and Quantum Gravitation, Relativity Theory | Physics | Elementary Particles, Quantum Field Theory
B physics | Branching fraction | Hadron-Hadron Scattering | Quantum Physics | Quantum Field Theories, String Theory | Classical and Quantum Gravitation, Relativity Theory | Physics | Elementary Particles, Quantum Field Theory
Journal Article
2017, Critical Voices, ISBN 1845198646, xxiv, 298 pages
Book
2010, Classic thinkers., ISBN 9780745625836, viii, 252
Book
2013, ISBN 9781441115140, xxxiii, 139 pages
Book
2005, ISBN 9780748619917, viii, 454
Book
Physics Letters B, ISSN 0370-2693, 05/2016, Volume 756, Issue C, pp. 84 - 102
A measurement of the ratio of the branching fractions of the meson to and to is presented. The , , and are observed through their decays to , , and ,...
scattering [p p] | pair production [pi] | statistical | Phi --> K+ K | f0 --> pi+ pi | High Energy Physics - Experiment | Compact Muon Solenoid | pair production [K] | mass spectrum [K+ K-] | Ratio B | Large Hadron Collider (LHC) | 7000 GeV-cms | leptonic decay [J/psi] | (b)over-bar(s) | J/psi --> muon+ muon | experimental results | Nuclear and High Energy Physics | Physics and Astronomy | branching ratio [B/s0] | CERN LHC Coll | Violating Phase Phi(s) | B/s0 --> J/psi Phi | CMS collaboration ; proton-proton collisions ; CMS ; B physics | Physics | Física | hadronic decay [f0] | Decay | colliding beams [p p] | hadronic decay [Phi] | mass spectrum [pi+ pi-] | B/s0 --> J/psi f0
scattering [p p] | pair production [pi] | statistical | Phi --> K+ K | f0 --> pi+ pi | High Energy Physics - Experiment | Compact Muon Solenoid | pair production [K] | mass spectrum [K+ K-] | Ratio B | Large Hadron Collider (LHC) | 7000 GeV-cms | leptonic decay [J/psi] | (b)over-bar(s) | J/psi --> muon+ muon | experimental results | Nuclear and High Energy Physics | Physics and Astronomy | branching ratio [B/s0] | CERN LHC Coll | Violating Phase Phi(s) | B/s0 --> J/psi Phi | CMS collaboration ; proton-proton collisions ; CMS ; B physics | Physics | Física | hadronic decay [f0] | Decay | colliding beams [p p] | hadronic decay [Phi] | mass spectrum [pi+ pi-] | B/s0 --> J/psi f0
Journal Article
9. Evidence for a Narrow Near-Threshold Structure in the J/psi phi Mass Spectrum in B+ -> J/psi phi K+ Decays
PHYSICAL REVIEW LETTERS, ISSN 0031-9007, 06/2009, Volume 102, Issue 24
Journal Article
Journal of Immunology, ISSN 0022-1767, 06/2016, Volume 196, Issue 12, pp. 4977 - 4986
Increased osteoclastogenesis is responsible for osteolysis, which is a severe consequence of inflammatory diseases associated with bone destruction, such as...
RHEUMATOID-ARTHRITIS | IMMUNE-SYSTEM | METASTASIS | OSTEOBLAST PROLIFERATION | DIFFERENTIATION | IMMUNOLOGY | TRANSCRIPTION FACTOR | EXPRESSION | GENOME-WIDE ASSOCIATION | BONE-RESORPTION | MICRORNA | Tumor Necrosis Factor-alpha - metabolism | MicroRNAs - antagonists & inhibitors | Forkhead Box Protein O3 - genetics | Nuclear Proteins - genetics | Immunoglobulin J Recombination Signal Sequence-Binding Protein - genetics | Macrophages - immunology | Immunoglobulin J Recombination Signal Sequence-Binding Protein - metabolism | Bone Resorption | Macrophages - pathology | Down-Regulation | NFATC Transcription Factors - metabolism | Gene Expression Regulation | Nuclear Proteins - metabolism | Inflammation | Transcription Factors - genetics | Osteoclasts - metabolism | Forkhead Box Protein O3 - metabolism | Transcription Factors - metabolism | Animals | Sequence Analysis, RNA | Mice | MicroRNAs - genetics | Osteogenesis | Tumor Necrosis Factor-alpha - antagonists & inhibitors | NFATC Transcription Factors - genetics | Positive Regulatory Domain I-Binding Factor 1 | Index Medicus | Abridged Index Medicus
RHEUMATOID-ARTHRITIS | IMMUNE-SYSTEM | METASTASIS | OSTEOBLAST PROLIFERATION | DIFFERENTIATION | IMMUNOLOGY | TRANSCRIPTION FACTOR | EXPRESSION | GENOME-WIDE ASSOCIATION | BONE-RESORPTION | MICRORNA | Tumor Necrosis Factor-alpha - metabolism | MicroRNAs - antagonists & inhibitors | Forkhead Box Protein O3 - genetics | Nuclear Proteins - genetics | Immunoglobulin J Recombination Signal Sequence-Binding Protein - genetics | Macrophages - immunology | Immunoglobulin J Recombination Signal Sequence-Binding Protein - metabolism | Bone Resorption | Macrophages - pathology | Down-Regulation | NFATC Transcription Factors - metabolism | Gene Expression Regulation | Nuclear Proteins - metabolism | Inflammation | Transcription Factors - genetics | Osteoclasts - metabolism | Forkhead Box Protein O3 - metabolism | Transcription Factors - metabolism | Animals | Sequence Analysis, RNA | Mice | MicroRNAs - genetics | Osteogenesis | Tumor Necrosis Factor-alpha - antagonists & inhibitors | NFATC Transcription Factors - genetics | Positive Regulatory Domain I-Binding Factor 1 | Index Medicus | Abridged Index Medicus
Journal Article
12. Editorial Comment on: Age, Body Mass Index, and Gender Predict 24-Hour Urine Parameters in Recurrent Idiopathic Calcium Oxalate Stone Formers by Otto et al.(From: Otto BJ, Bozorgmehri S, Kuo J, et al. J Endourol 2017;31:1335–1341)
Journal of Endourology, ISSN 0892-7790, 01/2018, Volume 32, Issue 1, pp. 76 - 76
Journal Article
Physics Letters B, ISSN 0370-2693, 12/2015, Volume 751, Issue C, pp. 63 - 80
An observation of the decay and a comparison of its branching fraction with that of the decay has been made with the ATLAS detector in proton–proton collisions...
PARTICLE ACCELERATORS
PARTICLE ACCELERATORS
Journal Article
Journal of High Energy Physics, ISSN 1126-6708, 2012, Volume 2012, Issue 5
Journal Article
15. Search for rare decays of $$\mathrm {Z}$$ Z and Higgs bosons to $${\mathrm {J}/\psi } $$ J/ψ and a photon in proton-proton collisions at $$\sqrt{s}$$ s = 13$$\,\text {TeV}$$ TeV
The European Physical Journal C, ISSN 1434-6044, 2/2019, Volume 79, Issue 2, pp. 1 - 27
A search is presented for decays of $$\mathrm {Z}$$ Z and Higgs bosons to a $${\mathrm {J}/\psi } $$ J/ψ meson and a photon, with the subsequent decay of the...
Nuclear Physics, Heavy Ions, Hadrons | Measurement Science and Instrumentation | Nuclear Energy | Quantum Field Theories, String Theory | Physics | Elementary Particles, Quantum Field Theory | Astronomy, Astrophysics and Cosmology
Nuclear Physics, Heavy Ions, Hadrons | Measurement Science and Instrumentation | Nuclear Energy | Quantum Field Theories, String Theory | Physics | Elementary Particles, Quantum Field Theory | Astronomy, Astrophysics and Cosmology
Journal Article |
2 2013-Spring 2.2 Problem 1
Given: \(E[X|Y]=X\ and \ E[Y|X]=X\)
To Show:
2.2.1 Part (a): \(P(X=Y)=1\)
\[ E[Y]=E[E[Y|X]]=E[X]=\mu_x \]
Thus, \(\mu_y=\mu_x\)
Also, \[ \begin{align} Cov(X,Y) &=E[XY]-E[X]E[Y]\\ &= E[E[XY|X]]-\mu_x^2\\ &= E[XE[Y|X]]-\mu_x^2\\ &= E[X^2]-\mu_x^2\\ &= \sigma_x^2 \end{align} \]
Repeating the above with \(E[XY]= E[E[XY|Y]]\) would give \(Cov(X,Y)=\sigma_y^2\) and hence \(Cov(X,Y)=\sigma_x^2=\sigma_y^2=Var(X)\) which implies \(X=Y\) or \(P(X=Y)=1\)
Note, we implicitly used the requirement of the variance being finite[This is what is implied by the function being squared integrable: \(\int |f(X)|^2dx < \infty\) |
The pressure at the bottom of the Mariana Trench in the Pacific Ocean is $1090$ bar. What temperature will the two allotropes of tin be at equilibrium? Assume that the molar volume, energy, and entropy change does not vary with temperature.
Relevant data: Density of white tin: $7.287$ g/mL; Density of grey tin: $5.766$ g/mL
At $1$ bar: The enthalpy change from white tin to grey tin is $-2.016$ kJ/mol and the entropy change is $-7.04$ J/mol K.
My attempt: Since the two species are in equilibrium, I thought first to use the Clausius Clapeyron equation. Assuming that the molar volume is independent with temperature, I calculated the molar volume change to be $4.26\cdot 10^{-6} \frac{\pu{m^3}}{\pu{mol}}$ from white tin to grey tin. Now, I substituted these values into the equation to find that $$\frac{dP}{dT}=\frac{\Delta S}{T\Delta V} =-\frac{1.6\cdot 10^7}{T}$$upon integration, I find that $$P-P^\circ = 1.6\cdot 10^7 \ln(\frac {T^\circ}{T}).$$Substituting the temperature of equilibrium at one bar from the data given I find that the temperature is $286.4 \pu{ K},$ thus $T^\circ = 286.4\pu{K}$ and $P^\circ = 10^5\pu{Pa}.$ Thus at $P = 1090\cdot 10^5\pu{Pa},$ $T = 0.317\pu{K}.$ However, the temperature given in the answer is much higher ($220.35$ K). The given answer was derived from a completely different method not involving the Clausius Clapeyron equation.
Hence I have the following question:
Question: Why is it not valid to use the Clausius Clapeyron equation in this scenario? (Or did I make a mistake in my derivation that caused my values to be off?) |
This question already has an answer here:
Consider the following complex power series $$\sum_{n \geq 1} \frac{z^n}{n} \,\,\,\,\,\,\, z \in \mathbb{C}$$
It surely converges conditionally for $z=-1$ (for alternating series test) and for $z=1$ it diverges (it is the harmonic series).
My question is: how can one show that the power series converges conditionally for
any $z \in \mathbb{C}$ such that $|z|=1$ (except for $z=1$)? |
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. |
Contents 1 Math 181 Honors Calculus, Fall 2008, Prof. Bell 1.1 Did 'ya Know? 1.2 Getting started editing 1.3 Lecture Notes_MA181Fall2008bell 1.4 Extra Credit 1.5 Homework Help 1.6 Cross-subject Issues 1.7 Interesting Articles About Calculus and Math 1.8 Learn LaTeX_MA181Fall2008bell 1.9 Solution to Exams_MA181Fall2008bell 1.10 Useful Formulas_MA181Fall2008bell 1.11 Study Guides Math 181 Honors Calculus, Fall 2008, Prof. Bell Did 'ya Know?
This week I thought that I'd demonstrate how to add ellipses in a LaTeX equation.
$ \begin{align} \sum_n &= 1 + 2 + 3 + ... + n + ... + \infty \\ &= 0 + 1 + 2 + \cdots + n + \cdots + \infty \end{align} $
In the first example, three consecutive dots were typed, but the second example uses LaTeX's \cdots command. This stands for 'center dots'. There are also other 'dot' functions.
\ldots $ \ldots $ \therefore $ \therefore $ \cdots $ \cdots $ \dagger $ \dagger $ \vdots $ \vdots $ \clubsuit $ \clubsuit $ \ddots $ \ddots $ \doteq $ \doteq $ Extra Credit Homework Help
Homework Whatever_MA181Fall2008bell (the last one)
Cross-subject Issues
Help! The ECE students are struggling with complex numbers. Can you shed some light here?
Interesting Articles About Calculus and Math
Single Elimination Forever_MA181Fall2008bell - Simple question to make you think a little.
Common Latex Commands http://authors.aps.org/latex/ |
Definition:Square/Function Contents Definition
Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$.
The
square (function) on $\F$ is the mapping $f: \F \to \F$ defined as: $\forall x \in \F: \map f x = x \times x$
where $\times$ denotes multiplication.
The
square (function) on $\F$ is the mapping $f: \F \to \F$ defined as: $\forall x \in \F: \map f x = x^2$
where $x^2$ denotes the $2$nd power of $x$.
Square Function in Specific Number Systems
Specific contexts in which the square function is used include the following:
The
(real) square function is the real function $f: \R \to \R$ defined as: $\forall x \in \R: \map f x = x^2$
The
(integer) square function is the integer function $f: \Z \to \Z$ defined as: $\forall x \in \Z: \map f x = x^2$ Also see Results about the square functioncan be found here. |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane polars $(r , \theta)$, we have the defining relations $$r\sin \theta=x=\pm R^2−b^2 \sin\sigma, r\cos\theta=z=R\cos\sigma$$ I am having a tough time visualising what this is?
Consider the function $f(z) = Sin\left(\frac{1}{cos(1/z)}\right)$, the point $z = 0$a removale singularitya polean essesntial singularitya non isolated singularitySince $Cos(\frac{1}{z})$ = $1- \frac{1}{2z^2}+\frac{1}{4!z^4} - ..........$$$ = (1-y), where\ \ y=\frac{1}{2z^2}+\frac{1}{4!...
I am having trouble understanding non-isolated singularity points. An isolated singularity point I do kind of understand, it is when: a point $z_0$ is said to be isolated if $z_0$ is a singular point and has a neighborhood throughout which $f$ is analytic except at $z_0$. For example, why would $...
No worries. There's currently some kind of technical problem affecting the Stack Exchange chat network. It's been pretty flaky for several hours. Hopefully, it will be back to normal in the next hour or two, when business hours commence on the east coast of the USA...
The absolute value of a complex number $z=x+iy$ is defined as $\sqrt{x^2+y^2}$. Hence, when evaluating the absolute value of $x+i$ I get the number $\sqrt{x^2 +1}$; but the answer to the problem says it's actually just $x^2 +1$. Why?
mmh, I probably should ask this on the forum. The full problem asks me to show that we can choose $log(x+i)$ to be $$log(x+i)=log(1+x^2)+i(\frac{pi}{2} - arctanx)$$ So I'm trying to find the polar coordinates (absolute value and an argument $\theta$) of $x+i$ to then apply the $log$ function on it
Let $X$ be any nonempty set and $\sim$ be any equivalence relation on $X$. Then are the following true:
(1) If $x=y$ then $x\sim y$.
(2) If $x=y$ then $y\sim x$.
(3) If $x=y$ and $y=z$ then $x\sim z$.
Basically, I think that all the three properties follows if we can prove (1) because if $x=y$ then since $y=x$, by (1) we would have $y\sim x$ proving (2). (3) will follow similarly.
This question arised from an attempt to characterize equality on a set $X$ as the intersection of all equivalence relations on $X$.
I don't know whether this question is too much trivial. But I have yet not seen any formal proof of the following statement : "Let $X$ be any nonempty set and $∼$ be any equivalence relation on $X$. If $x=y$ then $x\sim y$."
That is definitely a new person, not going to classify as RHV yet as other users have already put the situation under control it seems...
(comment on many many posts above)
In other news:
> C -2.5353672500000002 -1.9143250000000003 -0.5807385400000000 C -3.4331741299999998 -1.3244286800000000 -1.4594762299999999 C -3.6485676800000002 0.0734728100000000 -1.4738058999999999 C -2.9689624299999999 0.9078326800000001 -0.5942069900000000 C -2.0858929200000000 0.3286240400000000 0.3378783500000000 C -1.8445799400000003 -1.0963522200000000 0.3417561400000000 C -0.8438543100000000 -1.3752198200000001 1.3561451400000000 C -0.5670178500000000 -0.1418068400000000 2.0628359299999999
probably the weirdness bunch of data I ever seen with so many 000000 and 999999s
But I think that to prove the implication for transitivity the inference rule an use of MP seems to be necessary. But that would mean that for logics for which MP fails we wouldn't be able to prove the result. Also in set theories without Axiom of Extensionality the desired result will not hold. Am I right @AlessandroCodenotti?
@AlessandroCodenotti A precise formulation would help in this case because I am trying to understand whether a proof of the statement which I mentioned at the outset depends really on the equality axioms or the FOL axioms (without equality axioms).
This would allow in some cases to define an "equality like" relation for set theories for which we don't have the Axiom of Extensionality.
Can someone give an intuitive explanation why $\mathcal{O}(x^2)-\mathcal{O}(x^2)=\mathcal{O}(x^2)$. The context is Taylor polynomials, so when $x\to 0$. I've seen a proof of this, but intuitively I don't understand it.
@schn: The minus is irrelevant (for example, the thing you are subtracting could be negative). When you add two things that are of the order of $x^2$, of course the sum is the same (or possibly smaller). For example, $3x^2-x^2=2x^2$. You could have $x^2+(x^3-x^2)=x^3$, which is still $\mathscr O(x^2)$.
@GFauxPas: You only know $|f(x)|\le K_1 x^2$ and $|g(x)|\le K_2 x^2$, so that won't be a valid proof, of course.
Let $f(z)=z^{n}+a_{n-1}z^{n-1}+\cdot\cdot\cdot+a_{0}$ be a complex polynomial such that $|f(z)|\leq 1$ for $|z|\leq 1.$ I have to prove that $f(z)=z^{n}.$I tried it asAs $|f(z)|\leq 1$ for $|z|\leq 1$ we must have coefficient $a_{0},a_{1}\cdot\cdot\cdot a_{n}$ to be zero because by triangul...
@GFauxPas @TedShifrin Thanks for the replies. Now, why is it we're only interested when $x\to 0$? When we do a taylor approximation cantered at x=0, aren't we interested in all the values of our approximation, even those not near 0?
Indeed, one thing a lot of texts don't emphasize is this: if $P$ is a polynomial of degree $\le n$ and $f(x)-P(x)=\mathscr O(x^{n+1})$, then $P$ is the (unique) Taylor polynomial of degree $n$ of $f$ at $0$. |
Momentum is the product of mass and the velocity of the object. Any object moving with mass possesses momentum. The only difference in angular momentum is that it deals with rotating or spinning objects. So is it the rotational equivalent of linear momentum?
What is Angular Momentum?
If you try to get on a bicycle and try to balance without a kickstand you are probably going to fall off. But once you start pedalling, these wheels pick up angular momentum. They are going to resist change, thereby balancing gets easier.
Angular momentum is defined as:
The property of any rotating object given by moment of inertia times angular velocity.
It is the property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object. It is a
vector quantity, which implies that here along with magnitude, direction is also considered.
Symbol The angular momentum is a vector quantity, denoted by \(\vec{L}\) Units It is measured using SI base units: Kg.m 2.s -1 Dimensional formula The dimensional formula is: [M][L] 2[T] -1 You may also want to check out these topics given below! Relation Between Torque And Moment Of Inertia Relation Between Torque And Speed Radial Acceleration Relation Between Kinetic Energy And Momentum Angular Momentum Formula
Angular momentum can be experienced by an object in two situations. They are:
Point object: The object accelerating around a fixed point. For example, Earth revolving around the sun. Here the angular momentum is given by: Where, \(\vec{L}\) is the angular velocity r is the radius (distance between the object and the fixed point about which it revolves) \( \vec{p}\) is the linear momentum. Extended object: The object, which is rotating about a fixed point. For example, Earth rotating about its axis. Here the angular momentum is given by: Where, \(\vec{L}\) is the angular momentum. I is the rotational inertia. \( \vec{\omega }\) is the angular velocity. Angular Momentum Quantum Number
Angular momentum quantum number is synonymous to Azimuthal quantum number or secondary quantum number. It is a quantum number of an atomic orbital which decides the angular momentum and describes the size and shape of the orbital. The typical value ranges from
0 to 1. Angular Momentum and Torque
Consider the same point mass attached to a string, the string is tied to a point, and now if we exert a torque on the point mass, it will start rotating around the centre,
The particle of mass m will travel with a perpendicular velocity V┴ which is the velocity that is perpendicular to the radius of the circle; r is the distance of the particle for the centre of its rotation.The magnitude of L→ is given by:
L = rmv sin ϕ Where, Φ is the angle between r→ and p→ p⊥ and v⊥ are the components of p→ and v→ perpendicular to r→ . r⊥ is the perpendicular distance between the fixed point and the extension of p→ .
Notice the equation L = r⊥mv the angular momentum of the body only changes when there is a net torque applied on it. So, when there is no torque applied, the perpendicular velocity of the body will depend upon the radius of the circle. I.e. the distance from the centre of mass of the body to the centre of the circle. Thus,
for shorter radius, velocity will be high. for higher radius, velocity will be low.
as to conserve the angular momentum of the body.
Right-Hand Rule
The direction of angular momentum is given by the right-hand rule, which states that:
If you position your right hand such that the fingers are in the direction of r. Then curl them around your palm such that they point towards the direction of Linear momentum(p). The outstretched thumb gives the direction of angular momentum(L). Examples of Angular Momentum
We knowingly or unknowingly come across this property in many instances. Some examples are explained below.
Ice-skater
When an ice-skater goes for a spin she starts off with her hands and legs far apart from the centre of her body. But when she needs more angular velocity to spin, she gets her hands and leg closer to her body. Hence, her angular momentum is conserved and she spins faster.
Gyroscope
A gyroscope uses the principle of angular momentum to maintain its orientation. It utilises a spinning wheel which has 3 degrees of freedom. When it is rotated at high speed it locks on to the orientation, and it won’t deviate from its orientation. This is useful in space applications where the attitude of a spacecraft is a really important factor to be controlled.
Angular Momentum Questions (FAQs) Q1: Calculate the angular momentum of a pully of 2 kg, radius 0.1 m, rotating at a constant angular velocity of 4 rad/sec. Ans: Substitute the given values like m=2kg and r=0.1 m in I=1/2mr² (formula of the moment of inertia) we get I= 0.01 kg.m 2
Angular momentum is given by L=Iω, thus, substituting the values we get L=0.04kg.m².s-¹.
Q2: Give the expression for Angular momentum. Ans: \(\vec{L}=I\times \vec{\omega }\) or \(\vec{L}=r\times \vec{p}\) Q3: For an isolated rotating body, how are angular velocity and radius related? Ans: For an isolated rotating body angular velocity is inversely proportional to the radius. Q5: Write the dimensional formula for Angular momentum. Ans: The dimensional formula is ML 2T -1 Q6: When an ice-skater goes for a spin, what happens to her spinning speed when she stretches her hands? Ans: Spinning speed reduces. Q7: How can an ice-skater increase his/her spinning speed? Ans: By bringing hands closer, thus reducing the radius increases the angular velocity. Q8: If the moment of inertia of an isolated system is halved. What happens to its angular velocity? Ans: Angular velocity will be doubled. Q9: Calculate the angular moment of the object. When an object with the moment of inertia I = 5 kg.m² is made to rotate 1 rad/sec speed. Ans: Substituting the given value in formula L=Iω we get L=5kg.m 2.s -1. Stay tuned with BYJU’S for more such interesting articles. Also, download BYJU’S – The Learning App for loads of interactive, engaging Physics videos with an unlimited academic assistance. |
I'm trying to find solutions for the Poisson equation under Neumann conditions, and have a couple of questions. More specifically, I'm interested in the gradient of the function $\phi(x)$ in a space $\Omega \subset \mathbb{R}^d$. (note that I'm only interested in the gradient. For my problem I do not care about $\phi(x)$ at all. I know two things about $\phi(x)$. First, I know the Laplacian on the entire set $\Omega$: $$ \nabla^2 \phi(x)=f(x)\quad \forall \quad x \in \Omega $$ Second, the following boundary condition: $$ \nabla \phi(x)n=0 \quad \forall \quad x \in \partial \Omega $$ where n is the outward unit normal to $\Omega$. As I understand it, the solution for $\phi(x)$ is given by: $$ \phi(x_0)=\int_\Omega f(x) G(x,x_0) dx + boundary terms+arbitrary constant $$ And my object of interest, the gradient of $\phi$ is given by: $$ \nabla_{x_0} \phi(x_0)=\int_\Omega f(x) \nabla_{x_0} G(x,x_0) dx +\nabla_{x_0} boundary terms $$ where $G$ is the Green function of my problem.
I have a couple of questions:
Does anybody know a text that works out this problem under Neumann conditions? I have seen many treatises where they look at Dirchilet conditions, but none with Neumann. I would particularly be interested in how to define the Green function exactly.
Are my boundary terms zero (because of the rather simple boundary condition) in the problem?
I'm trying to get a feel for the Green's function in different spaces. Is defining the Green's function in this problem somehow similar to determining the appropriate bounds for integration? For example, suppose that the problem occurs in only 1 dimension. In that case, the gradient of the Green's function, $\nabla_{x_0} G(x,x_0)$ should be a stepwise function that takes value 1 for all $x$ smaller than $x_0$ and value 0 thereafter right? To me this seems to be the only way to retrieve the standard solution for a one-dimensional problem.
Should it not be easier to retrieve the gradient of the Green's function (which I'm interested in), rather than the Green's function itself? Is there any text that treats this issue?
Many thanks for any help you can offer. |
trying to determine if the series is conditionally convergent or divergent. $$\sum_{n = 1}^\infty \frac{2^{n^{2}}}{n!}$$ with n! i tried the ratio test on the series $$\frac{2^{(n+1)^{2}}}{(n+1)!} * \frac{n!}{2^{n^{2}}} = \frac{2^{2n+1}}{(n+1)} $$ which is > 1 as $n\to \infty$ and is overall divergent ? not sure if I am on the right track.
Well, applying the ratio test:
$$\lim_{\text{n}\to\infty}\space\left|\frac{\frac{2^{\left(\text{n}+1\right)^2}}{\left(\text{n}+1\right)!}}{\frac{2^{\text{n}^2}}{\text{n}!}}\right|=\lim_{\text{n}\to\infty}\space\left|\frac{2^{2\text{n}+1}}{\text{n}+1}\right|\space\to\space\infty\tag1$$
Note that ${2^{n^2} \over n!} = { (2^n)^n \over n!} \ge { (2^n)^n \over n^n}= ({2^n \over n})^n \ge 1$. Hence $\sum_n {2^{n^2} \over n!} \ge N$ for all $N$. |
Suppose that a polynomial $p(x,y)$ defined on $\mathbb{R}^2$ is identically zero on some open ball (in the Euclidean topology). How does one go about proving that this must be the zero polynomial?
WLOG suppose that the center of ball is the origin and write
$$ p(x,y)=\sum _{i,j=0}^ma_{i,j}x^iy^j $$
Plug in $x=y=0$. You find that $a_{0,0}=0$. Take the partial derivative with respect to $x$ and set $x=y=0$. You find that $a_{1,0}=0$. You should be able to finish it from here by continuining similarly. . .
To make notation simpler, let $(a,b)$ be the center of the open ball. Let $g(x,y)=f(a+x,b+y)$. Then the polynomial $g$ is identically $0$ in an open ball containing the origin. We show that $g(x,y)$ is identically $0$.
Consider any line through the origin. We will show that $g(x,y)=0$ at all points on that line. The lines are given by $y=kx$ where $k$ is a constant, and, easily forgotten, $x=0$.
Let $P(t)=g(t,kt)$ (for the line $x=0$, let $P(t)=g(0,t)$).
Then $P(t)$ is a polynomial, and is identically $0$ in an interval. In particular, $P(t)=0$ for infinitely many $t$. Thus $P(t)$ must be identically $0$ (a non-zero polynomial has only finitely many roots).
We conclude that $g$ is identically $0$ on every line through the origin, and hence everywhere.
Note that essentially the same argument works for polynomials in $n$ variables.
This follows purely algebraically by induction on degree using the fact that a polynomial has no more roots than its degree over a domain - see my prior post.
Write $p(x,y) = \sum_{i=0}^n q_i(x)y^i$, where $q_i(x)$ are polynomials in $x$.
Pick a point $(x_0,y_0)$ interior to $U$, where $U$ is your open ball. Then there exists a $r>0$ so that $B_r(x_0,y_0) \subset U$.
Pick any $x_1 \in (x_0-r,x_0+r)$, and chose some $\delta>0$ so that $\{x_1\} \times (y_0-\delta ,y_0+\delta) \subset U$.
Then $p(x_1,y)= \sum_{i=0}^n q_i(x_1)y^i$ is a polynomial in $y$ with constant coefficients $q_i(x_1)$ which is identically zero on on the interval $(y_0-\delta, y_0+\delta)$. Thus $p(x_1,y)$ is the zero polynomial.
Hence $q_i(x_1)=0$. But since $x_1$ was arbitrary in $(x_0-r,x_0+r)$, each $q_i$ has infinitelly many roots, thus each $q_i=0$.
A stronger, but still reasonably easy to prove, statement follows from the combinatorial Nullstellensatz. It's actually enough to require that $p$ is identically zero on a lattice with sufficiently many points. |
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 \(\mathbf{\sigma}\). |
Wheel resistance to forward movement
According to wikipedia, rolling or wheel resistance to forward movement calculations can be simplified if the vehicle does not move fast, which is our case. Not having found the rolling resistance coefficient for a tire on grass, we take the one of sand, 0.3.
The rolling resistance is given by:
Rolling resistance
\[ F_{rr}=C_{rr}\cdot F_{g} \]
Gravitational force
\[ F_{g}=m \cdot g \]
where:
\( F_{rr} \) : rolling resistance force in N.
\( C_{rr} \) : rolling resistance coefficient.
\( m \) : mass of the vehicle (the mower for us) in Kg.
\( g \) : constant of gravitation, in \( m/s^{2} \)
With our specifications, \( F_{rr} \) is, for a flat vehicle, all wheels are identical and the weight is evenly distributed:
\[ F_{rr}=C_{rr}\cdot m\cdot g=20\cdot 9.81\cdot 0.3=58.8N\approx 60N \]
Total resistance to forward movement
Taking into account the slope, the calculation is a little more complex: we must add the components due to gravitation. In addition, the rolling or wheel resistance to forward movement decreases with increasing gradient.
The total resistance is due to the force of gravity and rolling resistance, that is to say:
\[ \overrightarrow{F_{rtot}}=\overrightarrow{F_{rr}}+\overrightarrow{F_{rg}} \]
where:
\( \overrightarrow{F_{rr}} \) is the rolling resistance force.
\( \overrightarrow{F_{rg}} \) is the strength of resistance due to gravitation.
\( \overrightarrow{F_{rtot}} \) is the total resistance force: that which the motors must overcome.
From the diagram above, and the triangle of forces, we calculate:
Rolling resistance:
\[ F_{rr}=C_{rr}\cdot F_{p}= C_{rr}\cdot F_{g}\cdot cos(\alpha ) \]
Gravity resistance:
\[ F_{rg}=F_{g}\cdot sin(\alpha ) \]
Gravitational force:
\[ F_{g}=m\cdot g \]
by replacing the terms, you get:
\[ F_{rtot}=C_{rr}\cdot F_{g}\cdot cos(\alpha )+F_{g}\cdot sin{\alpha }=F_{g}\cdot (C_{rr}\cdot cos(\alpha )+sin(\alpha ))=m\cdot g\cdot (C_{rr}\cdot cos(\alpha )+sin(\alpha )) \]
Based on the general specifications, it is easy to calculate the total resistance force in the grass for a maximum gradient of 45°.
\[ F_{rtot}=20\cdot 9.81\cdot (0.3\cdot 0.7+0.7)=177.5N\approx 180N \]
Note: the values calculated above correspond to the maximum forces at which the robot could be confronted. |
Denote by $f$ a monotonically decreasing, convex function defined on $[0,\infty)$ that has a derivative $f'$ on $(0,\infty)$.
I would like to show that if $f(0)$ exists and is finite (and $\lim_{x \to 0} f(x) = f(0)$), then the right hand limit $f_+'(x) = \lim_{h \searrow 0} \frac{f(x+h)-f(x)}{h}$ exists and is finite at $0$, and that $\lim_{x\to 0} f'(x) = f_+'(0)$ (in my setting it would be fine to assume that $f'$ is continuous (or even differentiable) on $(0,\infty)$).
I have so far tried to follow (and then modify) https://proofwiki.org/wiki/Convex_Real_Function_is_Left-Hand_and_Right-Hand_Differentiable and the cited reference (1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach). In this case it is shown (on the interior of an interval) that $F_x(h) = \frac{f(x+h)-f(x)}{h}$ is an increasing function in $h$ and hence $\lim_{h\to 0} F_x(h) = f_+'(x)$ has to exists. This existence is already unclear to me, and the closest answer to this I could find was mentioned in the question Proof that Right hand and Left hand derivatives always exist for convex functions. where an inequality involving limits based on $h' < 0$ is used. In my case I can not reflect around the boundary point and I am hence searching for another way to show the statement.
Any hints or comments are greatly appreciated. |
In CP2K, Restrained Electrostatic Potential (RESP) charges can be fitted for periodic and nonperiodic systems. It is automatically decided by the program whether a periodic or nonperiodic RESP fit is carried out. If the electrostatic (Hartree) potential is periodic (i.e. a periodic Poisson solver is used), a periodic RESP fit is performed. If the Hartree potential is computed using a nonperiodic Poisson solver, the nonperiodic fitting is employed. In any case, a least squares fitting procedure at defined grid points $\mathbf{r}_k$ is carried out,
\begin{equation} R_{\mathrm{esp}}=\frac{1}{N}\sum_k^N{(V_{\mathrm{QM}}(\mathbf{r}_k)-V_{\mathrm{RESP}}(\mathbf{r}_k))^2}, \end{equation}
where $V_{\mathrm{QM}}$ is the quantum mechanical (QM) potential and $V_{\mathrm{RESP}}$ the potential generated by the RESP charges. $N$ is the number of selected fit points.
The fitted potential is obtained from a set of point charges $\{q_a\}$ centered at atom $a$ according to \begin{equation} V_{\mathrm{RESP}}(\mathbf{r}_k) = \sum_a\frac{q_a}{|\mathbf{R}_a-\mathbf{r}_k|}. \end{equation} For more details see: J. Phys. Chem., 97 , 10269-10280 (1993).
The fitted potential is generated from the charge distribution $\rho$, \begin{equation} \rho(\mathbf{r})=\sum_a{q_a g_a(\mathbf{r},\mathbf{R}_a)}, \end{equation} where $g_a$ is a Gaussian function centered at atom $a$. The periodic fitting is embedded in a Gaussian and plane waves (GPW) framework and described in detail in Phys. Chem. Chem. Phys., 17 , 14307-14316 (2015). In the periodic case, CP2K offers also the possibility to fit the variance of the potential instead of the absolute values, see below.
The RESP fitting is a post-SCF step and included as a subsection of the PROPERTIES section.
&PROPERTIES &RESP &SPHERE_SAMPLING &END &END RESP &END PROPERTIES
With this basis setup, the following defaults are employed:
.TRUE. by default.
There are different options to sample the fit points. The fit points can be sampled in shells/spheres around the atoms or, for slab-like systems, in a certain range above the surface. In any case, the systems and the sampled fitting points can be printed as .xyz file and visualized with, e.g. VMD, by enabling:
&RESP .... &PRINT &COORD_FIT_POINTS &END &END &END RESP
For better visualization it is recommended to center the coordinates of the systems using CENTER_COORDINATES.
figure 1. The spherical shells are defined by a minimal radius r$_{\mathrm{min}}$ and a maximal radius r$_{\mathrm{max}}$. The paramters r$_{\mathrm{min}}$ and r$_{\mathrm{max}}$ can be defined specifically for each element and are, by default, based on the van der Waals (vdW) radii. For the vdW radii, the values from the Cambridge Structural Database
CAMBRIDGE or the Universal Force Field
UFF can be specified via AUTO_VDW_RADII_TABLE. Using the keywords
AUTO_RMIN_SCALE and
AUTO_RMAX_SCALE, r$_{\mathrm{min}}$ and r$_{\mathrm{max}}$ are then calculated as follows:
These settings can be overwritten for all atoms using the keywords
RMIN and
RMAX or only for specific atoms using
RMIN_KIND and
RMAX_KIND. In the following example, r$_{\mathrm{min}}$ is overwritten for all carbon atoms to 2.1$\,\mathring{\mathrm{A}}$.
&SPHERE_SAMPLING AUTO_VDW_RADII_TABLE CAMBRIDGE AUTO_RMIN_SCALE 1.0 AUTO_RMAX_SCALE 10.0 RMIN_KIND 2.1 C &END
RESP charges can be also fitted for slab-like systems. In this case, the potential should be well reproduced above the surface where, e.g., adsorption processes take place. The input for a flat monolayer, see figure 2, is for example:
&SLAB_SAMPLING RANGE 1.0 3.0 ATOM_LIST 1..32 SURF_DIRECTION Z &END &SLAB_SAMPLING RANGE 1.0 3.0 ATOM_LIST 1..32 SURF_DIRECTION -Z &END
SURF_DIRECTION. With the keyword
ATOM_LIST the atoms that constitute the surface are defined. This list should contain indexes of atoms of the first surface layer.
RANGE defines that the points are sampled between 1-3 $\mathring{\mathrm{A}}$ above the surface layer.
The sampling technique is flexible enough to follow a corrugation of the surface. The sampling technique works as follows: An orthogonal box with box length $abc$ is constructed over each surface atom, see figure 3. All fitting points within this box are included in the fitting. The height $c$ of the box is defined by the keyword
RANGE. The length $a=b$ are given by the keyword
LENGTH. For flat surface layers,
LENGTH is set to a sufficiently large values (3 $\mathring{\mathrm{A}}$ or more). For corrugated surface layers,
LENGTH should be in the range of the distance between the surface atoms.
An input example for a corrugated graphene layer on ruthenium, see figure 4, is given below.
ATOM_LIST lists in this case the indexes of the carbon atoms.
&SLAB_SAMPLING RANGE 2.0 4.0 LENGTH 2.0 ATOM_LIST 1..1250 SURF_DIRECTION Z &END
A constraint on the total charge of the system is introduced by the keyword INTEGER_TOTAL_CHARGE, which is set by default to
.TRUE.. Further explicit constraints can be given via the subsection CONSTRAINT. It is possible to enforce the same charges for chemically equivalent atoms, e.g. for the hydrogen atoms of a methyl group. The corresponding input is:
&CONSTRAINT EQUAL_CHARGES ATOM_LIST 1 2 3 &END
where
ATOM_LIST lists the indexes of the atoms that should have the same charge.
The definition of more elaborate constraints is also possible. The constraints are always linear following the formula $\sum_i^{n\_list}c_iq_i=t$. The sum is running over the atoms given in
ATOM_LIST and $t$ is the target value of the constraint given by TARGET. The coefficients $\{c_i\}$ are defined by the keyword ATOM_COEF. With the following input it is achieved that the (absolute value) of the charge on atom 3 is twice as large as the charge on atom 5, i.e. $q_3=2q_5$.
&CONSTRAINT ATOM_LIST 3 5 ATOM_COEF 1.0 -2.0 TARGET 0.0 &END
To avoid unphysical values for the fitted charges, restraints can be set. The restraints in CP2K are addressed by harmonic penalty functions, \begin{equation} R_{\mathrm{rest}} = \beta \sum_j (q_j-t_j)^2,\end{equation}where $t_j$ is the target value for charge $q_j$ and $\beta$ is the strength of the restraint. By default, all elements except hydrogen are weakly restrained to zero, i.e. the keyword RESTRAIN_HEAVIES_TO_ZERO is set to
.TRUE. by default. The strength of this restraint is controlled by RESTRAIN_HEAVIES_STRENGTH. Restraints can be also defined explicitly via the subsection RESTRAINT:
&RESP ... &RESTRAINT ATOM_LIST 1..3 TARGET -0.18 STRENGTH 0.0001 &END &RESTRAINT ATOM_LIST 4 TARGET 0.21 STRENGTH 0.0001 &END RESTRAIN_HEAVIES_TO_ZERO .FALSE. &END RESP
In this example, charges on atoms with indexes 1..3 are restrained to -0.18 and the charge on atom 4 to 0.21. The target values $t_j$ of the restraints can be, e.g., inspired from DDAPC, Mulliken charges etc.The strength $\beta$ of the restraint is defined by STRENGTH. Large values for $\beta$ will limit increasingly the flexibility of the charge fitting and decrease the quality of the fit. If only the explicitly given restraints should be used,
RESTRAIN_HEAVIES_TO_ZERO must be switched to
.FALSE..
CP2K offers also the possibility to fit the variance of the potential as proposed in J. Chem. Theory Comput., 5 , 2866–2878 (2009). This is only valid for periodic systems, since the reference state of the ESP is arbitrary in the periodic case. The modified residual reads:
\begin{equation} R_{\mathrm{repeat}}=\frac{1}{N}\sum_k^N{(V_{\mathrm{QM}}(\mathbf{r}_k)-V_{\mathrm{RESP}}(\mathbf{r}_k)-\delta)^2}, \end{equation} where \begin{equation} \delta = \frac{1}{N}\sum_k^N(V_{\mathrm{QM}}(\mathbf{r}_k)-V_{\mathrm{RESP}}(\mathbf{r}_k)). \end{equation}
When $V_{\mathrm{QM}}$ is obtained from, e.g., a plane wave code and the periodicity of $V_{\mathrm{RESP}}$ is later treated by, e.g., Ewald summation, both potentials will have different offsets. The modified residual $R_{\mathrm{repeat}}$ was introduced to overcome this problem. In CP2k, $V_{\mathrm{QM}}$ and $V_{\mathrm{RESP}}$ are both evaluated with the same method, the GPW approach, and have thus the same offset. However, fitting the variance of the potential is an easier task than fitting the absolute values and avoids a strong fluctuation of the charges. A stabilization of the fitting procedure is thus also expected in CP2K when $\delta\ne0$. The value of $\delta$ depends on the sampling of fitting points. If all points are included, it strictly holds that $\delta=0$, since we have $\sum_k^{N_{all}}V(r_k)=0$ (in CP2K). In this case, the original and modified residuals are identical, i.e. $R_{\mathrm{esp}}= R_{\mathrm{repeat}}$. If the fitting points are sampled in spheres around the atom, which is done for molecular periodic structures like MOFs, $\delta$ will be non-zero and fitting the variance is strongly recommended. When sampling the fitting points above a surface, we often find that $\delta\sim 0$. However, minimizing $R_{\mathrm{repeat}}$ will partly also yield improvements for such systems.
To enable this option, add the keyword
USE_REPEAT_METHOD:
&RESP ... USE_REPEAT_METHOD &END RESP
Note that
RESTRAIN_HEAVIES_TO_ZERO is then automatically switched to
.FALSE.. Furthermore, the definition of explicit restraints is usually not necessary.
To obtain REPEAT charges in a stricter sense, i.e. as computed by the ''REPEAT code'', sphere sampling has to be enabled, the van der Waals radii must be retrieved from the Universal Force Field and the total charge must be retained. The corresponding input is &RESP USE_REPEAT_METHOD &SPHERE_SAMPLING AUTO_VDW_RADII_TABLE UFF &END &END RESP
Use the keyword
AUTO_RMIN_SCALE and
AUTO_RMAX_SCALE to scale the van der Waals radii as described above. Note that small numerical deviations compared to the REPEAT code are possible since the fitting is embedded in a GPW framwork as described in Phys. Chem. Chem. Phys., 17 , 14307-14316 (2015) , whereas the REPEAT code uses Ewald summation.
A measure for the quality of the fit are the root-mean square (RMS) error
\begin{equation} \mathrm{RMS}=\sqrt{\frac{\sum_k^N~(V_{\mathrm{QM}}(\mathbf{r}_k)-V_{\mathrm{RESP}}(\mathbf{r}_k))^2}{N}} \end{equation}
and the relative root-mean square (RRMS) error \begin{equation} \mathrm{RRMS}=\sqrt{\frac{\sum_k^N~(V_{\mathrm{QM}}(\mathbf{r}_k)-V_{\mathrm{RESP}}(\mathbf{r}_k))^2}{\sum_k^N~V_{{\mathrm{QM}}}(\mathbf{r}_k)^2}}. \end{equation}
Both errors are printed to the output file. They should be as small as possible. Typical values can be found here:
When the variance is fitted, $V_{\mathrm{RESP}}$ is shifted by $\delta$ with respect to $V_{\mathrm{QM}}$. Thus, $V_{\mathrm{RESP}}$ is replaced by $\tilde{V}_{\mathrm{RESP}}=V_{\mathrm{RESP}}+\delta$ in the formulas for the RMS and RRMS values.
The RESP potential can be printed in cube file format using the following option: &RESP .... &PRINT &V_RESP_CUBE &END &END &END RESP
The QM potential can be as well printed as cube file using V_HARTREE_CUBE. The cube files can be visualized with, e.g. VMD, and the RESP and the QM potential can be directly compared. Note that $\tilde{V}_{\mathrm{RESP}}$ is printed instead of $V_{\mathrm{RESP}}$ when the variance is fitted. |
This may be overtly obvious or simple and I'm being very dense, but it's something that has been bothering me. I am confused about how correlation functions of generic spin operators work in 2-d CFTs. There is a formula that is quoted in many texts (e.g. diFrancesco's text, see section 5.1-5.2) for the four point function of operators with weights $(h_i, \bar{h}_i)$:
\(\langle \phi_1(z_1,\bar{z}_1) \phi_2(z_2,\bar{z}_2) \phi_3(z_3,\bar{z}_3) \phi_4(z_4,\bar{z}_4)\rangle = f(\eta,\bar{\eta}) \prod z_{ij}^{h/3-h_i-h_j} \bar{z}_{ij}^{\bar{h}/3-\bar{h}_i - \bar{h}_j}\)
where the product is supposed to be for $i<j$ up to 4 and $h = \sum_i h_i$. The function $f$ is allowed to depend arbitrarily on the conformally invariant cross ratio $\eta$ and its conjugate. Here lies the crux of my problem:
If we consider any tensor; for example, $D^{\mu \nu \sigma}$, which denotes the four point function of a spin-3 current and three scalars, then it is clear that under the change to the complex coordinates, we will have a tensor $D^{abc}$ where $a,b,c$ may denote either $z$ or $\bar{z}$. If the current is symmetric, then of course we may eliminate all but 4 degrees of freedom. The issue now is the following: how does the above four point function capture all of these independent components? Setting $(h_1,\bar{h}_1) = (h_1, h_1-3)$, and the rest scalars so that $\bar{h}=h$, we clearly only have an expression for one of the components. Which one is it? What happens to the other degrees of freedom? Must we assume tracelessness? |
Briefly, we shall see the definition of a kernel density estimator in the multivariate case.
Suppose that the data is d-dimensional so that $latex {X_{i}=(X_{i1},\ldots,X_{id})}&fg=000000$. We will use the product kernel
$latex \displaystyle \hat{f}_{h}(x)=\frac{1}{nh_{1}\cdots h_{d}}\left\{ \prod_{j=1}^{d}K\left(\frac{x_{j}-X_{ij}}{h_{j}}\right)\right\} . &fg=000000$
The risk is given by
$latex \displaystyle \mathrm{MISE}\approx\frac{\left(\mu_{2}(K)\right)^{4}}{4}\left[\sum_{j=1}^{d}h_{j}^{4}\int f_{jj}^{2}(x)dx+\sum_{j\neq k}h_{j}^{2}h_{k}^{2}\int f_{jj}f_{kk}dx\right]+\frac{\left(\int K^{2}(x)dx\right)^{d}}{nh_{1}\cdots h_{d}} &fg=000000$
where $latex {f_{jj}}&fg=000000$ is the second partial derivative of $latex {f}&fg=000000$. The optimal bandwidth satisfies $latex {h_{i}=O(n^{-1/(4+d)})}&fg=000000$ leading to a risk of order $latex {O(n^{-4/(4+d)})}&fg=000000$ (for further details see Hardle (2004)).
The interesting effect of $latex {O(n^{-4/(4+d)})}&fg=000000$ here is that the risk increase exponentially as the dimension grows. We call to this behavior the
curse of dimensionality. This phenomena says that the data is more sparse as we increase the dimensionality. This table from Silverman (1986) shows the sample size required to ensure a relative mean squared error less than 0.1 at 0 when the density is multivariate normal and the optimal bandwidth is selected.
Dimension Sample size 1 4 2 19 3 67 4 223 5 768 6 2790 7 10,700 8 43,700 9 187,000 10 842,000
For this reason it is important to search methods for dimension reduction. One of these methods was proposed by Li (1991) in its article Sliced Inverse Regression for Dimension Reduction. I used this method to find another efficient estimator based in a Taylor approximation (see Solís Chacón, M et al. (2012) ). In a next post I going to talk a little about the details of those articles.
Sources: Hardle, W. (2004). Nonparametric and Semiparametric Models. Springer Series in Statistics. Springer. Li, K.-C. (1991). Sliced Inverse Regression for Dimension Reduction. Journal of the American Statistical Association, 86(414), 316-327. Retrieved from http://www.jstor.org/stable/2290563 Tsybakov, A. (2009). Introduction to nonparametric estimation. Springer. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, volume 26. Chapman & Hall/CRC. Solís Chacón, M., Loubes, J.-M., Clement, M. & Da Veiga, S. (2012). Efficient estimation of conditional covariance matrices for dimension reduction. Arxiv preprint arXiv: Retrieved from http://arxiv.org/abs/1110.3238 |
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. |
Difference between revisions of "Inaccessible"
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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.
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.
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A cardinal $\kappa$ is ''inaccessible'', also
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A cardinal $\kappa$ is ''inaccessible'', also called ''strongly inaccessible'', if it is an [[uncountable]] [[regular]] [[strong limit]] cardinal.
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$\kappa$ inaccessible
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*
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$V_\kappa$ is a model of ZFC and so inaccessible cardinals are [[worldly]]
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* (
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, .
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* $\kappa$ is an [[aleph fixed point]] and a [[beth fixed point]], and consequently $V_\kappa=H_\kappa$.
− +
* ()there is an inner model of a forcing extension satisfying ZF+DC in which every set of reals is Lebesgue measurablein , is of an inaccessible cardinal.
−
*
+
* , the set of $\kappa$ $\kappa$ is [[]] $\$.
− + − + − + + + + + + + + + + + +
==Weakly inaccessible cardinal==
==Weakly inaccessible cardinal==
Line 23: Line 31:
*Letting $R$ be the transfinite enumeration of [[regular]] cardinals, a limit ordinal $\alpha$ is weakly inaccessible if and only if $R_\alpha=\aleph_\alpha$
*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 $\kappa$
+
*A nonzero cardinal $\kappa$ is weakly inaccessible if and only if $\kappa$ is regular and there are $\kappa$-many regular cardinals below $\kappa$
*A regular cardinal $\kappa$ is weakly inaccessible if and only if $\mathrm{REG}$ is unbounded in $\kappa$ (showing the correlation between [[Mahlo|weakly Mahlo]] cardinals and weakly inaccessible cardinals, as stationary in $\kappa$ is replaced with unbounded in $\kappa$)
*A regular cardinal $\kappa$ is weakly inaccessible if and only if $\mathrm{REG}$ is unbounded in $\kappa$ (showing the correlation between [[Mahlo|weakly Mahlo]] cardinals and weakly inaccessible cardinals, as stationary in $\kappa$ is replaced with unbounded in $\kappa$)
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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.
−
==Hyper-inaccessible==
+
==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.
−
Every [[Mahlo]] cardinal $\kappa$ is
+
Every [[Mahlo]] cardinal $\kappa$ is $\$-inaccessible .
+ + Latest revision as of 09:13, 4 May 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 $α<\kappa$ and probably more. Similar hierarchy exists for Mahlo cardinals below weakly compact. All such properties can be killed softly by forcing to make them any weaker properties from this family.[1]
ReferencesMain library |
Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity
1.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
2.
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
$\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$
$ f $
$ x $
$ f(\cdot,0)\equiv 0 $
$ y $
$ |x|\to\infty $
$ y $
$ f_u(x,0) $
$ f_{uu}(x,0) $
$ f(x,\cdot) $ Keywords:Elliptic equations on the entire space, quasiperiodic solutions, center manifold, Birkhoff normal form, KAM theorem. Mathematics Subject Classification:Primary: 35B08, 35B15, 35J61; Secondary: 37J40, 37K55. Citation:Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020077
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In mechanics, Poisson’s ratio is the negative of the ratio of transverse strain to lateral or axial strain. It is named after Siméon Poisson and denoted by the Greek letter ‘nu’, It is the ratio of the amount of transversal expansion to the amount of axial compression for small values of these changes.
What is Poisson’s Ratio?
Poisson’s ratio is
“the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here, Compressive deformation is considered negative Tensile deformation is considered positive.
Greek letter ‘nu’,ν
Poisson’s ratio = – Lateral strain / Longitudinal strain
-1.0 to +0.5
Unitless quantity
Scalar quantity
Poisson’s Ratio Formula
Imagine a piece of rubber, in the usual shape of a cuboid. Then imagine pulling it along the sides. What happens now?
It will compress in the middle. If the original length and breadth of the rubber are taken as L and B respectively, then when pulled longitudinally, it tends to get compressed laterally. In simple words, length has increased by an amount dL and the breadth has increased by an amount dB.
In this case,\(\varepsilon _{t}=-\frac{dB}{B}\) \(\varepsilon _{l}=-\frac{dL}{L}\)
The formula for Poisson’s ratio is,\(Poisson’s\;ratio=\frac{Transverse\;starin}{Longitudinal\;strain}\) \(\Rightarrow \nu =-\frac{\varepsilon _{t}}{\varepsilon _{l}}\)
where,
εt is the Lateral or Transverse Strain
εl is the Longitudinal or Axial Strain\(\nu \) is the Poisson’s Ratio
The strain on its own is defined as the change in dimension (length, breadth, area…) divided by the original dimension.
Poisson Effect
When a material is stretched in one direction, it tends to compress in the direction perpendicular to that of force application and vice versa. The measure of this phenomenon is given in terms of Poisson’s ratio. For example, a rubber band tends to become thinner when stretched.
Poisson’s ratio values for different material
It is the ratio of transverse contraction strain to longitudinal extension strain, in the direction of the stretching force. There can be a stress and strain relation which is generated with the application of force on a body.
For tensile deformation, Poisson’s ratio is positive. For compressive deformation, it is negative.
Here, the negative Poisson ratio suggests that the material will exhibit a positive strain in the transverse direction, even though the longitudinal strain is positive as well.
For most materials, the value of Poisson’s ratio lies in the range,
0 to 0.5.
A few examples of Poisson ratio is given below for different materials.
Concrete
0.1 – 0.2
Cast iron
0.21 – 0.26
Steel
0.27 – 0.30
Rubber
0.4999
Gold
0.42 – 0.44
Glass
0.18 – 0.3
Cork
0.0
Copper
0.33
Clay
0.30 – 0.45
Stainless steel
0.30 – 0.31
Foam
0.10 – 0.50
Practice Questions On Poisson’s Ratio Q1: Does Poisson’s ratio is dependent on temperature? Ans:In general, Colder temperature decreases both strains and high-temperature increases both horizontal and vertical strain. Thus, the net effect on Poisson’s Ratio is small since the change in both horizontal and vertical strain is by a similar amount. Q2: Is Poisson’s ratio constant? Ans: Poisson’s ratio for material remains approximately constant within elastic limits Q3: Define Poisson’s ratio Ans:The ratio of transverse strain to longitudinal strain in the direction of the stretching force. Q4: Write the Poisson’s ratio formula Ans:\(Poisson’s\;ratio=\frac{Transverse\;starin}{Longitudinal\;strain}\) Q5: State true or False. Poisson’s ratio is negative for compressive deformation. Ans: True Q6: State true or False. Poisson’s ratio is negative for Tensile deformation. Ans: False. Poisson’s ratio is Positive for Tensile deformation Q7: What is the Poisson’s ratio of concrete? Ans: The Poisson’s ratio of concrete is 0.1 to 0.2. Q8: What does the Poisson’s ratio 0.5 mean? Ans: Poisson’s ratio 0.5 means a perfectly incompressible material is deformed elastically at small strains. Q9: What is the units of Poisson’s ratio? Ans: Poisson’s ratio is the unitless scalar quantity. Q10: What is the Poisson’s ratio of cork? Ans: Poisson’s ratio of cork is 0.0.
Hope you have understood about Poisson’s ratio, How it is defined, its Symbol, Units, Formula, Terms and Values for various materials.
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H. O. Ibraheem
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Methods Funct. Anal. Topology
24 (2018), no. 3, 207-239
Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on $X$ that is concentrated on $\mathbb K(X)$. We consider the multiplicative (current) group $C_0(X\to\mathbb R_+)$ consisting of functions on $X$ that take values in $\mathbb R_+=(0,\infty)$ and are equal to 1 outside a compact set. Each element $\theta\in C_0(X\to\mathbb R_+)$ maps $\mathbb K(X)$ onto itself; more precisely, $\theta$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i \theta(s_i)s_i\delta_{x_i}$. Thus, elements of $C_0(X\to\mathbb R_+)$ transform the weights of discrete Radon measures. We study conditions under which the measure $\mu$ is quasi-invariant under the action of the current group $C_0(X\to\mathbb R_+)$ and consider several classes of examples. We further assume that $X=\mathbb R^d$ and consider the group of local diffeomorphisms $\operatorname{Diff}_0(X)$. Elements of this group also map $\mathbb K(X)$ onto itself. More precisely, a diffeomorphism $\varphi\in \operatorname{Diff}_0(X)$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i s_i\delta_{\varphi(x_i)}$. Thus, diffeomorphisms from $\operatorname{Diff}_0(X)$ transform the atoms of discrete Radon measures. We study quasi-invariance of $\mu$ under the action of $\operatorname{Diff}_0(X)$. We finally consider the semidirect product $\mathfrak G:=\operatorname{Diff}_0(X)\times C_0(X\to \mathbb R_+)$ and study conditions of quasi-invariance and partial quasi-invariance of $\mu$ under the action of $\mathfrak G$. |
We summarize here the main ideas of the Newtonian model/approach by listing the (1) constructs, i.e., the “things” or ideas that are get “used” in the model, (2) the relationships–in mathematical or sentence form–that connect the constructs in meaningful ways, and (3) the ways of representing the relationships. Developing a deep and rich understanding of the relationships in a model/approach comes slowly. It is absolutely not something you can simply memorize. This understanding comes only with repeated hard mental effort over a period of time. A good test you can use to see if you are “getting it” is whether you can tell a full story about each of the relationships. It is the meaning behind the equations, behind the simple sentence relationships, that is important for you to acquire. With this kind of understanding, you can apply a model/approach to the analysis of phenomena you have not thought about before. You can
reason with the model.
Example: Drag Force on a Barge
Suppose two tugboats push on a barge at different angles, as shown in Figure. The first tugboat exerts a force of \(2.7 \times 10^5 \space N \) in the
x-direction, and the second tugboat exerts a force of \(3.6 \times 10^5 \space N \) in the y-direction.
Figure 8.4.1. (a) A view from above of two tugboats pushing on a barge. (b) The free-body diagram for the ship contains only forces acting in the plane of the water. It omits the two vertical forces—the weight of the barge and the buoyant force of the water supporting it cancel and are not shown. Since the applied forces are perpendicular, the x- and y-axes are in the same direction as \(F_x\) and \(F_y\) The problem quickly becomes a one-dimensional problem along the direction of \(F_{app} \), since friction is in the direction opposite to \(F_{app} \).
If the mass of the barge is \(5.0 \times 10^6 \space kg \) and its acceleration is observed to be \(7.5 \times 10^{-2} \space m/s^2 \) in the direction shown, what is the drag force of the water on the barge resisting the motion? (Note: drag force is a frictional force exerted by fluids, such as air or water. The drag force opposes the motion of the object.)
Strategy
The directions and magnitudes of acceleration and the applied forces are given in Figure (a). We will define the total force of the tugboats on the barge as \(F_{app} \) so that:\[ F_{app} = F_x + F_y \]
Since the barge is flat bottomed, the drag of the water \(F_D\) will be in the direction opposite to \(F_{app} \) as shown in the free-body diagram in Figure (b). The system of interest here is the barge, since the forces on
it are given as well as its acceleration. Our strategy is to find the magnitude and direction of the net applied force \(F_{app} \), and then apply Newton’s second law to solve for the drag force \(F_D\). Solution
Since \(F_x\) and \(F_y\) are perpendicular, the magnitude and direction of \(F_{app}\) are easily found. First, the resultant magnitude is given by the Pythagorean theorem:
\[F_{app} = \sqrt{F_x^2 + F_y^2} \]
\[F_{app} = \sqrt{(2.7 \times 10^5 \space N)^2 + (3.6 \times 10^5 \space N)^2} = 4.5 \times 10^5 \space N. \]
The angle is given by \[ \theta = tan^{-1} \left(\dfrac{F_y}{F_x} \right) \]
\[ \theta = tan^{-1} \left( \dfrac{3.6 \times 10^5 \space N}{2.7 \times 10^5 \space N} \right) = 53^o, \]
which we know, because of Newton’s first law, is the same direction as the acceleration. \(F_D\) is in the opposite direction of \(F_{app} \), since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as \(F_{app} \), but its magnitude is slightly less than \(F_{app} \). The problem is now one-dimensional. From Figure (b) we can see that
\[F_{net} = F_{app} - F_D. \] But Newton’s second law states that \[ F_{net} = ma \]
Thus, \[ F_{app} - F_D = ma \]
This can be solved for the magnitude of the drag force of the water \(F_D\) in terms of known quantities: \[ F_D = F_{app} - ma \] Substituting known values gives \[ F_D = (4.5 \times 10^5 \space N) - (5.0 \times 10^6 \space kg)(7.5 \times 10^{-2} \space m/s^2) = 7.5 \times 10^4 \space N \]
The direction of \(F_D\) has already been determined to be in the direction opposite to \(F_{app} \) or at an angle of \(53^o\) south of west.
Discussion
The numbers used in this example are reasonable for a moderately large barge. It is certainly difficult to obtain larger accelerations with tugboats, and small speeds are desirable to avoid running the barge into the docks. Drag is relatively small for a well-designed hull at low speeds, consistent with the answer to this example, where \(F_D\) is less than 1/600th of the weight of the ship.
Listed here are the major, most important constructs, relationships, and representations of the Newtonian model. Constructs
\(F_{A\:on\:B}\)
net force \(\Sigma F\)(net Torque \(\Sigma\tau\) )
acceleration, \(a\) angular acceleration, \(\alpha\)
mass rotational inertia
components
Relationships Newton's 3 rd law:
\[F_{A\:on\:B}=-F_{B\:on\:A}\]
\[J_{A\:on\:B}=-J_{B\:on\:A}\]
Newton's 1 st law:
\[ If\: \Sigma F = 0,\: then\: \Delta v = 0 \]
Newton's 2 nd law:
\[ \Sigma F= dp/dt \enspace \Sigma F= ma \enspace \Sigma\tau = dL/dt \enspace \Sigma\tau = I\alpha \]
Special Case of Constant Acceleration
\[ x(t) = x_0 + v_{x0} t + \frac{1}{2} a_{x0} t^2 , \]
\[ v_x(t) = v_{x0} + a_{x0}t, \]
\[ a_x(t)= a_{x0} \]
Rotation
\[ \frac{ds}{dt} = v_{tangential}= r\frac{d\theta}{dt} \]
\[\omega = \frac{d\theta}{dt} ~~~~~~~~~~\enspace v_{tangential}=r\omega \]
\[\alpha= \frac{d\omega}{dt} ~~~~~~~~~~\enspace a_tangential=r\alpha \]
Special Case of circular motion as constant speed
\[ v_{tangential}=r\omega \]
\[ v_{radial}=0. \]
\[ a_{radial}=-r\omega^2 = -\frac{v^2}{r} \]
\[ a_{tangential} = 0\]
\[ \Sigma F_{radial}=ma_{radial}= -mr\omega^2=-\frac{mv^2}{r} \]
Representations
Force Diagrams
Vector(arrow) representation of Newton's laws |
In this paper we consider a variation of the Merton’s problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model.
We formulate a general Bayesian disorder detection problem, which generalizes models considered in the literature. We study properties of basic statistics, which allow us to reduce problems of quickest detection of disorder moments to optimal stopping problems. Using general results, we consider in detail a disorder problem for Brownian motion on a finite time segment.
We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not assume the knowledge of eigenfunctions of the correspondinge covariance operator. This makes it possible to generalize many previous results in this area. We also obtain ultimate results connected with exact small deviation of Brownian excursion and Brownian meander as well as for Bessel processes and their local times.
We consider two models of summation of independent identically distributed random variables with a parameter. The first is motivated by financial applications and the second by contact models for species migration. We characterize the limiting distributions and their bifurcations under different relationships between the parameter and the number of summands. We find that in the phase transition we may get limiting distributions that are quite different from those that come up in standard limit theorems. Our results suggest that these limiting distributions may provide better models, at least for certain aggregation levels. Moreover, we show how the parameter determines at which aggregation levels these models apply.\
We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We additionally suppose that ${{E}} |X_{11}|^{4 + \delta} =: \mu_{4+\delta} < \infty$ for some $\delta > 0$. Under these conditions we show that the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-1/2} X$ and Wigner's semicircle law is of order $(nv)^{-1}$, where $v$ is the distance in the complex plane to the real line. Furthermore, we outline applications such as the rate of convergence of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues, and eigenvector delocalization.
This paper introduces matrix-variate t-distributions for which degree of freedom is a multivariate parameter. A relation for a density function is obtained. © 2015 Society for Industrial and Applied Mathematics.
We consider the multichannel queueing system with nonidentical servers and regenerative input flow. The necessary and sufficient condition for ergodicity is established, and functional limit theorems for high and ultra-high load are proved. As a corollary, the ergodicity condition for queues with unreliable servers is obtained. Suggested approaches are used to prove the ergodic theorem for systems with limitations. We also consider the hierarchical networks of queueing systems
This paper contains detailed exposition of the results presented in the short communication [M. V. Zhitlukhin and A. A. Muravlev, Russian Math. Surveys, 66 (2011), pp. 1012–1013]. We consider Chernoff’s problem of sequential testing of two hypotheses about the sign of the drift of a Brownian motion under the assumption that it is normally distributed. We obtain an integral equation which characterizes the optimal decision rule and find its solution numerically.
We consider the problem of optimal estimation of the value of a vector parameter \thetavector=(\theta_0,...,\theta_n)^⊤ of the drift term in a fractional Brownian motion represented by the finite sum i=0^nii(t) over known functions \varphi_i(t), \alli. For the value of parameter \thetavector, we obtain a maximum likelihood estimate as well as Bayesian estimates for normal and uniform a priori distributions.
The main result of this paper is a counterpart of the theorem of Monroe [Ann. Probab., 6 (1978), pp. 42--56] for a geometric Brownian motion: A process is equivalent to a time change of a geometric Brownian motion if and only if it is a nonnegative supermartingale. We also provide a link between our main result and Monroe's [Ann. Math. Statist., 43 (1972), pp. 1293--1311]. This is based on the concept of a minimal stopping time, which is characterized in Monroe [Ann. Math. Statist., 43 (1972), pp. 1293--1311] and Cox and Hobson [Probab. Theory Related Fields, 135 (2006), pp. 395--414] in the Brownian case. Finally, we suggest a sufficient condition for minimality (for the processes other than a Brownian motion) complementing the discussion in the aforementioned papers.
We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock price appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein- Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor Y in an interval [0, T0] for fixed T0 > 0, and use sequential estimation. We show that the consumption and investment strategy calculated through this sequential procedure is δ-optimal.
Let (X1; Y1),..., (XN; YN) be independent identically distributed random vectors. We show, that if the statistics LX = a1*X1+... + aN*XN and LY = b1*Y1 + ... + bN*YN are epsilon - independent, then under some conditions X = (X1,..., XN), Y = (Y1,..., YN) are "epsilon in the power alpha"-independent for some alpha> 0.
We study the Markov exclusion process for a particle system with a local interaction in the integer strip. This process models the exchange of velocities and particle-hole exchange of the liquid molecules. It is shown that the mean velocity profile corresponds to the behavior which is characteristic for incompressible viscous liquid. We prove the existence of phase transition between laminar and turbulent profiles.
We consider a basic stochastic particle system consisting of
N identical particles with isotropic k-particle synchronization, ${k\ge 2}$. In the limit when both the number of particles N and the time $t=t(N)$ grow to infinity we study an asymptotic behavior of a coordinate spread of the particle system. We describe three time stages of $t(N)$ for which a qualitative behavior of the system is completely different. Moreover, we discuss the case when a spread of the initial configuration depends on N and increases to infinity as $N\rightarrow\infty$.
The formula for calculating the entropy and the Hausdorff dimension of an invariant Erdos measure for the pseudogolden ratio and all values Bernoulli parameter is obtained. This formula make possible calculating the entropy and the Hausdorff dimension with high accuracy.
We characterize the set $W$ of possible joint laws of terminal values of a nonnegative submartingale $X$ of class $(D)$, starting at 0, and the predictable increasing process (compensator) from its Doob--Meyer decomposition. The set of possible values remains the same under certain additional constraints on $X$, for example, under the condition that $X$ is an increasing process or a squared martingale. Special attention is paid to extremal (in a certain sense) elements of the set $W$ and to the corresponding processes. We relate also our results with Rogers's results on the characterization of possible joint values of a martingale and its maximum.
We study the structure of the functional of accumulation defined on the trajectories of semi-Markov process with a finite set of states. As t -> ∞ this functional increases linearly and the coefficient is linear-fractional functional relative to the probability measure, defining homogeneous Markov randomized control strategy.
We consider a model of an insurance company investing its reserve into a risky asset whose price follows a geometric Lévy process. We show that the nonruin probability is a viscosity solution of a second order integro-differential equation and prove a uniqueness theorem for the latter. |
Beer-Lambert Law derivation helps us to define the relationship of the intensity of visible UV radiation with the exact quantity of substance present. The Derivation of Beer-Lambert Law has many applications in modern day science. Used in modern-day labs for testing of medicines, organic chemistry and to test with quantification. These are some of the fields that this law finds its uses in. Beer-Lambert Law Statement
The Beer-Lambert law states that:
for a given material sample path length and concentration of the sample are directly proportional to the absorbance of the light.
The Beer-Lambert law is expressed as:
A = εLc
where,
A is the amount of light absorbed for a particular wavelength by the sample ε is the molar extinction coefficient L is the distance covered by the light through the solution c is the concentration of the absorbing species
Following is an equation to solve for molar extinction coefficient:
\(\epsilon =\frac{A}{Lc}\)
But Beer-Lambert law is a combination of two different laws: Beer’s law and Lambert law.
What is Beer’s Law?
Beer’s law was stated by August Beer which states that concentration and absorbance are directly proportional to each other.
What is Lambert Law?
Lambert law was stated by Johann Heinrich Lambert which states that absorbance and path length are directly proportional.
Beer-Lambert Law Formula
\(I=I_{0}e^{-\mu (x)}\)
Where,
I is the intensity I 0is the initial intensity x is the depth in meters 𝜇 is the coefficient of absorption Following is the table explaining concepts related to Physics laws Schematic Diagram of Beer-Lambert Law
Absorption of energy causes the absorption of light as well usually by electrons. Different forms of light such as visible light and ultraviolet light get absorbed in this process. Therefore, change in the intensity of light due to absorption, interference, and scattering leads to:
ΔI = I 0 – IT
The following equations are necessary for us to obtain our ultimate derivative equation. Transmittance is measured as the ratio of light passing through a substance. It can be calculated as
IT/ I 0. To calculate the of transmittance percentage we can do so by:
Percent TransmittanceAnother key metric is absorbance that is defined as the amount of light absorbed. This is usually calculated as the negative of transmittance and is given by:
Absorbance (A) The rate of decrease in the intensity of light with the thickness of the material the light is directly proportional to the intensity of the incident light. Mathematically, it can be expressed as:
As k’= Proportionality constant
Taking in the reciprocal of the equation we get,
Integrating the above equation we also get,
In the above equation, b and C is the constant of integration and
I T is the intensity being transmitted at the thickness
In order to solve the above equation with the constant of integration, we then get,
While solving for C in the equation will give us,
Converting to
log10 we get,
Rearranging the equation we get,
Lambert Derivation
Thus, Lambert’s law was formed and it states that the monochromatic radiation changes exponentially and decreases when it passes through a medium of uniform thickness.
Beer Derivation
Thus, this concludes the
derivation of Beer-Lambert law. This goes to show you that in order to derive a particular law, there are a lot of different equations that need to be found out first, to achieve the ultimate result.
Stay tuned with BYJU’S to learn more about other Physics related concepts. |
Spring 2018, Math 171 Week 2 Markov/Non-Markov Chains
(Discussed) Example 1.2 from the book (Ehrenfest Chain)
(Discussed) At \(t=0\) an urn contains \(N\) balls, \(M\) of which are red, \(N-M\) of which are green. Each day (\(t = 1, 2, \dots\)) a ball is drawn without replacement. Let \(X_n\) be the color of the ball drawn at \(t=n\). Is \(\{X_n:N \ge n \ge 1\}\) a Markov Chain? Prove your claim.
Example 1.6 from the book (Inventory Chain)
Let \(\{X_n:n \ge 0\}\) be a Markov Chain on the state space \(\mathcal{S}=\{0, 1, 2\}\). Define \[Y_n=I_{[X_n \ge 1]} = \begin{cases}1 &\text{if } X_n =1,2 \cr 0 &\text{if } X_n = 0\end{cases}\] Under what circumstances, if any, is \(\{Y_n:n \ge 0\}\) a Markov Chain?
Stopping/Non-Stopping Times
Let \(\{X_n:n \ge 0\}\) be a Markov Chain. Which of the following will necessarily be stopping times? Prove your claims.
(Discussed) \(T=\min\{n \ge 0: X_n = x\}\) (Discussed) \(T=\max\{n \ge 0: X_n = x\}\) \(T=\min\{n \ge 0: X_n = X_{n-1}\}\) (Discussed) \(T=\min\{n \ge 0: X_{n+1} = X_{n}\}\)
Let \(T_1, T_2\) be stopping times for some Markov Chain \(\{X_n:n \ge 0\}\). Which of the following will also necessarily be stopping times? Prove your claims.
\(T_1 + T_2\)
(Answer) Yes
\(T_1 - T_2\)
(Answer) No
\(\min(T_1, T_2)\)
(Answer) Yes
\(\max(T_1, T_2)\)
(Answer) Yes Linear Algebra
Compute or write down the inverse of the matrix \[A = \begin{bmatrix} a & b \cr c & d \end{bmatrix}\]
(Answer) \[\frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}\]
Under what circumstances is the following matrix invertible? Under these circumstances, compute its inverse. \[A = \begin{bmatrix} a & 0 & 0 \cr b & c & 0 \cr 0 & d & e \end{bmatrix}\]
(Answer) \(ace \neq 0\)\[\begin{bmatrix}1/a & 0 & 0 \\ -b/ac & 1/c & 0\\bd/ace & -d/ce & 1/e\end{bmatrix}\]
Compute a left eigenvector of \[P = \begin{bmatrix} 1-r & 0 & r \cr p & 1-p & 0 \cr 0 & q & 1-q \end{bmatrix}\] corresponding to eigenvalue \(1\)
(Answer) \(ace \neq 0\)\[\begin{bmatrix}1/a & 0 & 0 \\ -b/ac & 1/c & 0\\bd/ace & -d/ce & 1/e\end{bmatrix}\] |
The intersection of two groups $(U,\odot_U)$ and $(V,\odot_V)$ is first, and foremost, the
set $$ G = U \cap V \text{.}$$To turn this into a group, one would need to define a suitable operation $\odot_G$ on $G$. But where is that operation supposed to come from? Since $U$ and $V$ can be completely different groups, which just happen to be constructed over two non-disjoint sets $U$ and $V$, it's not at all obvious how $\odot_G$ is supposed to be defined. Thus, the intersection of two groups is merely a set, not a group. The same holds of course for the union of two sets - again, where would the operation come from that turns the union into a group?
In fact, since we generally consider groups only
up to isomorphisms, i.e we treat two groups $G_1,G_2$ as the same group $G$ if they only difference between the two is the names of the elements, the union or intersection of two groups isn't even well-defined. For any pair of groups $U,V$ we can find some set-theoretic representation of $U$ and $V$ such that $U \cap V = \emptyset$, and another such that $U \cap V \neq \emptyset$.
Now constrast this with the situation of two
subgroups $U,V$ of some group $(H,\odot_H)$. In this case, we know that $\odot_U$ and $\odot_V$ are simply the restrictions of $\odot_H$ to $U$ respectively $V$, and the two operations will therefore agree on the intersection of $U$ and $V$. So we can very naturally endow the set $$ G = U \cap V$$with the operation $$ \odot_G = \odot_H\big|_{U \cap V} = \odot_U\big|_{U \cap V} = \odot_V\big|_{U \cap V} \text{.}$$ |
Moving Charges Create Magnetic Fields
In the last section we learned that a magnetic field affects moving charges. By Newton’s third law, the moving charges must exert an equal and opposite force on whatever produced the \(\mathbf{B}\) field. In other words, the moving charge must create its own \(\mathbf{B}\) field! By using Newton’s third law we can complete the description of the indirect model we started earlier:
\[\text{Moving charges } \xrightarrow{\text{creates field}} \mathbf{ B } \xrightarrow{\text{exerts force on}} \text{Moving charges}\]
As we learned in Physics 7B moving charges constitute an electric current; a concept that is particular useful if we have a steady flow of charge. Considering a separate charge \(q\), the indirect model becomes:
\[\text{Current } \xrightarrow{\text{creates field}} \mathbf{ B } \xrightarrow{\text{exerts force on}} \text{Moving charge }q\]
Electromagnetism: A History
While we have worked this out, it is far from clear what currents and moving charges have to do with anything related to the fridge magnets or bar magnets that make magnetism familiar to us. In essence we have cheated: the ideas of how a magnetic field affects moving charges were not known until the mid-1800s. Before that, the only thing known about magnetism was that some materials can produce magnetic fields and these attract (or repel) certain kinds of other similar materials, and that the Earth had its own magnetic field which aligns these magnetic materials. These facts were known to the Greeks as early as 600 BC. The question of why certain materials where magnetic while others did not appear to be, and what phenomenon created these magnetic fields was not addressed until 1820.
In 1820, Dutch physicist Hans Christian Ørsted had set up an experiment to show that large electric currents could be used to heat a wire. While demonstrating this to a group of students in his house, he noticed that a compass on his bookshelf changed direction whenever his “kettle” was switched on. After months of investigation, Ørsted concluded that
an electric current could create a magnetic field. This was big news at the time, because prior to this only magnetic fields were known to affect other magnetic materials. This was a watershed moment in the history of science, as it was the first link between electric and magnetic phenomena. Originally we experience these as two distinct forces, two distinct fields. Ørsted’s finding was the first step on the road that led humankind to find that these apparently dissimilar phenomena were in fact linked. This unification of seemingly disconnected ideas is still at the core of fundamental research: much hope is placed on possibly unifying all forces in nature. What is a Magnet?
The finding that electricity and magnetism are linked caused a huge revolution in science, but we now want to return to our question of what makes a magnet a magnet. Ørsted showed us that electric currents created a magnetic field, but where are the currents in a magnetized piece of Iron? People could not answer this question until the late 1800s, and even then they were met with skepticism. The answer relied on the existence of atoms: in a nutshell, the origins of magnetism are found to be the electric currents produced by the electrons orbiting (i.e. making a current loop) atomic nuclei.
The magnetism of certain materials also depends on spin orientation.
Spin endows the electrons with an intrinsic angular momentum. This “intrinsic spin” can only have two possible values (you might have heard that an electron can be “spin up” or “spin down” in your physics or chemistry classes). Spin is a purely quantum mechanical phenomenon, but for the purposes of thinking about magnetism in our current discussion, consider spin an additional way to produce a loop of current (the smallest one you can imagine!)
The fact that atoms exist is something we now take for granted, but most scientists thought of it as ludicrous until 1905 (due in large part to a separate contribution by Einstein)! They reasoned that all things in they saw in motion lost energy due to friction, and the idea of electrons perpetually moving around in atoms seemed absurd to them.
Only Some Materials Are Magnetic
To summarize, all our experiments point to the following finding: to get a magnetic field, we need a net motion of charge. How does this idea explain magnetism in materials? Imagine helium, an atom with two electrons. Now if these electrons go around in opposite directions, the currents they produce will be opposite to each other. The magnetic field of one current loop will cancel the magnetic field of the other, leading to no net magnetic field. The spin also affects the magnetic field, and if the spins are pointed in the same direction (both up or both down) the field gets stronger, while if they are aligned in opposite directions the field gets weaker. In helium, the spins of the two electrons are paired up-down, so helium would not be very magnetic. As it turns out, helium is an “anti-magnet” and tries to stop any magnetic field going through it. This effect is called
diamagnetism and is a manifestation of Lenz’s law which we have not covered yet.
However, there are many materials whose atoms have an odd number of electrons but who don't exhibit magnetism, how can that be? So far we have discussed the effects of the magnetic field on individual atoms; we need to consider the possibilities that these atoms are also interacting strongly with
each other. Since a material is made up of \(\sim 10^{23}\) atoms, tiny effects, like the ones between atoms, can become large if each atom contributes to it. Highly magnetic materials are generally metals, because metals have many outer electrons that act as if they're "free," and can interact strongly with external fields and with each other. Ampère’s Law
We see that magnetism boils down to moving charges affecting other moving charges. It is no surprise that an explanation of the phenomena of bar magnets took so long; it required a serendipitous observation of how two superficially unlike phenomena (electricity and magnetism) affected one another
and the atomic model. We have already studied how a magnetic field affects a moving charge. Now we turn to the quantitative question of how, exactly, a current produces a magnetic field. Field Produced by a Long, Straight Wire
We will first study a simple test case: a long straight wire carrying a current. We want to understand the magnetic field produced by this wire, i.e. how strong it is in magnitude, where it points (recall it is a vector), and how does it vary with position. In other words, we want to map the \(\mathbf{B}\) field.
We will retrace some of Ørsted’s steps. He showed that a current-carrying conductor produces a magnetic field. A simple way to demonstrate this is to place several compass needles in a horizontal plane (for example, the surface of a table) near a long wire placed vertically (running up and down through the table surface). Let’s assume the current direction to be coming from the bottom and going toward the top of the table. When there is no current in the wire, all needles point in the same direction (magnetic north). As soon as current starts flowing in the wire, all needles will deflect. We have produced a magnetic field!
The first thing that one notices when doing this experiment, is that the needles orient themselves in a recognizable pattern; if we draw a circle on the table with the wire at the center and we place the compasses along the circle, we will notice that all the compass needles will orient themselves
tangential to the circle. In other words, the field lines for \(\mathbf{B}\) from a long straight wire at a distance \(r\) from the wire will have the shape of concentric circles of radius \(r\). For our experiment with the current coming out of the table, we find that the \(\mathbf{B}\) field direction is counterclockwise along the circles. If we flip the direction of the current, the compass needles will still point tangent to the circle, but now their north poles point in a clockwise direction. Right-Hand Rule
This observation for the direction of the \(\mathbf{B}\) field can be summarized with the following convenient rule,
right-hand-rule #1 (RHR #1): Point the thumb of your right hand along a wire in the direction of the conventional (positive) current. Your fingers will now curve naturally in the direction of the magnetic field.
What can we infer about the magnitude of the magnetic field? By symmetry, the magnetic field should have the same magnitude everywhere along the circle. Why is this? Every point along the circle is equidistant from the wire, and have the same distance \(r\) from the wire. Likewise, if the wire is infinitely long, we could chose to place a horizontal plane (with compasses on it) anywhere along the wire. Moving this horizontal plane up or down along the wire, should also have no effect in our results. To be mathematically precise, we can set the plane of the wire to be the x-y plane, and the direction along the wire to be the z-axis. We see that, by symmetry, the magnetic field does not depend on the z coordinate.
Mapping the Field
Let's mathematically relate every quantity that we've talked about. We expect that the magnitude of the field at any point will depend only on the perpendicular distance \(r\) between the wire and that point. All points at the same distance \(r\) will have the same magnitude (that is why the compass needles around a wire arranged in a circle with radius \(r\)). It is also not unexpected that the magnitude of the field will be larger if we have a larger current. It turns out that the magnitude is proportional to the current. Likewise, you probably expect that if we start moving away from the wire, the magnetic field will get weaker the farther we move. The equation that relates all these quantities to the magnetic field magnitude at a point \(\overrightarrow{r}\) is:
\[| \mathbf{B} (\overrightarrow{r})| = \dfrac{\mu_0 I}{2 \pi r}\]
where \(I\) is the current in the wire and \(r\) is the perpendicular distance from the wire to the point we are interested in. In our experiment with a wire along the z-axis, the distance \(r\) would be the length of a vector perpendicular to the wire that points directly to our position. The constant \(\mu_0\) is a proportionality constant called the
magnetic permeability of the vacuum, which has the value \(\mu_0 = 4 \times 10^{−7} \text{T }· \text{ m/A}\)
Example #1
a) A long straight wire carrying a current \(I\) produces a magnetic field of \(\mathbf{B} = 1.0 \times 10^{−4} \text{ T}\) at a distance of \(2 \text{ cm}\) away from the wire. Find the current \(I\) carried by the wire. How close to the wire is the field a magnitude of \(1 \text{ T}\)? b) A proton is moving at \(\mathbf{v} = 1.5 \times 10^3 \text{ m/s}\) parallel to the wire, in the same direction as the current, a distance of \(1.0 \text{ cm}\) away from the wire. Find the magnitude and direction of the magnetic force exerted on the proton by the magnetic field produced by the wire.
Solution
a) We use our equation \(| \mathbf{B} (\overrightarrow{r})| = \frac{\mu_0 I}{2 \pi r}\) to solve for the current \(I\). Once we find the current, we set the field equal to \(1 \text{ T}\) and solve for the distance \(r\) from the wire. Solve for \(I\): Rearranging our equation we find \[I = 2 \pi r |\mathbf{B}|/ \mu_0\] \[I = 2 \pi (0.02\text{ m})(1.0 \times 10^{-4} \text{ T})/ 4 \pi \times 10^{-7} \text{ T m/A}\] \[I = 10 \text{ A}\] Solve for \(r\) with new \(\mathbf{B}\): \[r = \mu_0 I/ 2 \pi |\mathbf{B}| = 2 \mathrm{\mu m}\] b) To find the force on the proton, we must know the direction of \(\mathbf{B}\) at the proton's location and from there we can determine the direction of the force and its magnitude. Direction of \(\mathbf{B}\): From the Right-Hand Rule above, we find that the magnetic field produced by the wire at the proton’s location is going into the page. Direction of \(\mathbf{F}\): By Right-Hand Rule #2, the magnetic force on the proton is directed towards the wire. Magnitude of \(\mathbf{F}\): We know that \(|\mathbf{B}| = 0.1 \text{ T}\) at 2 cm from the wire. At 1 cm from the wire, because \(|\mathbf{B}| \propto 1/r\), we must have \(|\mathbf{B}| = 0.2 \text{ T}\). We now insert \(\mathbf{B}\) into the equation for the magnetic force \[\mathbf{F} = q|\mathbf{v}||\mathbf{B}| \sin \theta\] \[\mathbf{F} = (1.6 \times 10^{−19}\text{ C})(1.5 \times 10^3 \text{ m/s})(0.2 \text{ T})(\sin 90°)\] \[\mathbf{F} = 4.8 \times 10^{-17} \text{ N}\] Field Produced by a Circular Current
We would now like to describe the magnetic field from another simple configuration of electrical current. Consider a coil of radius \(r_0\), made up of \(N\) loops of wire, all carrying a current \(I\). For ease, we adopt a convention where the axis of the coil is the z-axis. This configuration is displayed below from two views.
Suppose we are interested in the magnetic field along the coil’s axis, at point \(P\), a distance \(z\) along the axis from the coil. We can treat the coil as \(N\) copies of a single loop of current \(I\), or as one loop with current \(NI\). If we apply the right-hand-rule to a small piece of these loops, we find that no matter which part of the loop we choose, the resulting \(\mathbf{B}\)-field points upward at our position \(P\), or indeed any point on the axis of the coil (the z-axis).
It makes sense that the magnitude of \(\mathbf{B}\) will decrease as we get further away from the coil. It also makes sense that the magnitude will be directly proportional to the number of wire loops and to the current flowing through the loop. Indeed, it turns out the magnetic field at distance \(z\) along the axis (as long as we are relatively far from the coil) is given by:
\[|\mathbf{B}| = \dfrac{\mu_0 NI {r_0}^2}{2 z^3}\]
Now, what if we’re instead interested in the magnetic field at point \(Q\), which resides in the plane of the coil? Again we turn to the right-hand-rule for currents. Notice that this time not all parts of the loop contribute to the \(\mathbf{B}\)-field in the same direction. The side of the loop closest to \(Q\) gives a contribution to the field in the
down direction, while the side farthest from \(Q\) contributes in the up direction. Since the field strength drops with distance, the closest side has the larger contribution, and the overall field is in the down direction. But one consequence is that, if \(Q\) and \(P\) were at similar distances away from the coil, the magnitude of \(\mathbf{B}\) at \(Q\) (in the coil plane) is significantly less than the magnitude of \(\mathbf{B}\) at \(P\) (along the axis).
The overall magnetic field lines are shown in the picture below, with points \(P\) and \(Q\) labeled.
Note the similarity between the field lines here and the field lines of a bar magnet. The similarity demonstrates that the magnetic field of a permanent magnet really
can be described as the result of many small current loops lined up with each other. |
Fred Kline
Contact: fred.kline.98104ATgmailDOTcom
I donate regularly to the The OEIS Foundation.
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1 Will there be an update fixing the rendering? Jun 26 '13 |
Assume that the underlying $S$ is some index, hence the risk-return $\mu=0$, where $S$ meets $$d S = \sigma S d W_t.$$
Let $V$ denote the price of the corresponding call option. To construct the related BS formula, I construct a portfolio $\Pi=V-\Delta S$, after setting a correct value of $\Delta$, I want the portfolio to be risk-free. That is $$d \Pi = r\Pi d t= r(V-\Delta S) d t,$$ where $r$ is risk-free rate.
hence by the Ito formula, I can get the BS-equation.
However, someone told me that the identity $d \Pi = r(V-\Delta S) d t$ should be $$d \Pi = (r*V-\mu*\Delta S) d t,$$ and then get another equation.
Since in my opinion, in the risk-neutral world, $\mu$ turns to be $r$ after applying the Girsonov transformation, and making the portfolio to be risk-free is under risk-neutral world. I agree with the first identity.
So my question is which one is correct? If is the latter one, what's meaning of risk-neutral pricing?
Thank you very much!
Added 2016/10/26 10:39AM(+8)
Thanks for @MJ73550. I am sure the first one is right now.
However, if we distinguish the funding and lending rate for unsecurities (denote as $r_F$) and stock collateral(denote as $r_R$). Then maybe the identity $d \Pi = r(V-\Delta S) d t$ should be $$d \Pi = (r_F*V-r_R*\Delta S) d t,$$
Is this equation right?
Thanks again. |
It certainly can be done and the commonest method is simply to shove the Ansatz you cite into whatever the relevant wave equation is. When we do this with Maxwell's equations, this leads, with the
slowly varying envelope approximation (I'll say exactly what this is below), to the Eikonal equation, which is equivalent to raytracing with Snell's law and Fermat's principle of least time in a variable refractive index medium, if the refractive index variation is spatially slow, i.e. $|\nabla n(\mathbf{r})| \ll k\, n(\mathbf{r})$ where $n(\mathbf{r})$ is the refractive index as a function of position $\mathbf{r}$ and $k$ the freespace wavenumber for the light in question (or the freespace wavenumber of the light with the longest important wavelength in a spectrum of light of different wavelengths).
This kind of procedure with the Schrödinger equation shows that the Hamilton Jacobi equation (see Wiki page with this name) can be thought of as an Eikonal approximation (see Wiki page with this name), or, equivalently, the Hamilton-Jacobi equation of classical mechanics can be thought of as the limit of the Schrödinger equation as $\hbar\to0$. Thus your ansatz with the Eikonal equation yields insight into how classical mechanics is an approximation embedded in the more general quantum theory and how we can continuously deform one into the other with $\hbar$ continuously running between nought and its experimentally witnessed value.
You can see the Wiki pages on how the ansatz plays out in quantum mechanics. Here's how it happens with the Maxwell equations. We write the electomagnetic fields in the form $\mathbf{E}\left(\mathbf{r}\right) = \mathbf{e}\left(\mathbf{r}\right) e^{i\,\varphi\left(\mathbf{r}\right)}$, $\mathbf{H}\left(\mathbf{r}\right) = \mathbf{h}\left(\mathbf{r}\right) e^{i\,\varphi\left(\mathbf{r}\right)}$ where $\mathbf{e}$ and $\mathbf{h}$ are slowly varying vectors and the phase $\varphi\left(\mathbf{r}\right)$ is real valued. Then Faraday's and Ampère's laws become:
$$\begin{array}{lcl}\nabla \wedge \mathbf{e} + i\, \nabla \varphi \wedge \mathbf{e} &=& -\mu\,\partial_t \mathbf{h}\\\nabla \wedge \mathbf{h} + i\, \nabla \varphi \wedge \mathbf{h} &=& \epsilon\,\partial_t \mathbf{e}\end{array}\quad\quad\quad\quad(1)$$
So far there is no approximation; one then makes the
slowly varying envelope approximation: that the envelopes $ \mathbf{e}$ and $ \mathbf{h}$ vary much more slowly with position than does the phase, {\it i.e.} $\left|\mathbf{e}\right|^{-1} \left|\nabla \wedge \mathbf{e}\right| \ll \left|\nabla \varphi_e\right| \approx \left|k\right|$ and $\left|\mathbf{h}\right|^{-1} \left|\nabla \wedge \mathbf{h}\right| \ll \left|\nabla \varphi_h\right| \approx \left|k\right|$ and we also assume a monochromatic (time-harmonic) field. The equations then become:
$$\begin{array}{lcl}\nabla \varphi \wedge \mathbf{e} &\approx& \omega\,\mu\,\mathbf{h}\\\nabla \varphi \wedge \mathbf{h} &\approx& -\omega\,\epsilon\, \mathbf{e}\end{array}\quad\quad\quad\quad(2)$$
Both of these equations individually say that $\mathbf{e}$ and $\mathbf{h}$ are orthogonal. On forming $-\omega^2\,\mu\,\epsilon\, \mathbf{e}\wedge\mathbf{h} = \left(\nabla \varphi \wedge \mathbf{h}\right)\wedge\left(\nabla \varphi \wedge \mathbf{e}\right)$ and simplifying, one gets:
$$\omega^2\,\mu\,\epsilon\, \mathbf{e}\wedge\mathbf{h} = \nabla \varphi \cdot \left(\mathbf{e}\wedge\mathbf{h}\right) \nabla \varphi\quad\quad\quad\quad(3)$$
so that $\nabla \varphi$ is orthogonal to both $\mathbf{e}$ and $\mathbf{h}$ and aligned with $\mathbf{e}\wedge\mathbf{h}$, whence $\omega^2\,\mu\,\epsilon\, \left|\mathbf{e}\wedge\mathbf{h}\right| = \left|\nabla \varphi\right|^2 \left|\mathbf{e}\wedge\mathbf{h}\right| $, therefore $\mathbf{e}$, $\mathbf{h}$ and $\nabla \varphi$ in that order are a mutually orthogonal, right-handed triple and:
$$ \left|\nabla \varphi\right|^2 = \omega^2\,\mu\,\epsilon = \left|\mathbf{k}\right|^2\quad\quad\quad\quad(4)$$
which is the
Eikonal Equation: the fundamental equation defining raytracing. More fully, we can summarise everything inferred from Eq.(2) as the "Eikonal Equations" as follows:
$$\begin{array}{lcl}\mathbf{k}\left(\mathbf{r}\right) &\stackrel{def}{=}& \nabla \varphi\left(\mathbf{r}\right)\\n\left(\mathbf{r}\right) &\stackrel{def}{=}& \sqrt{\frac{\mu\left(\mathbf{r}\right)\,\epsilon\left(\mathbf{r}\right)}{\mu_0\,\epsilon_0}} = \sqrt{\mu\left(\mathbf{r}\right) \epsilon\left(\mathbf{r}\right)} \;c\\\mathcal{Z}\left(\mathbf{r}\right) &\stackrel{def}{=}& \sqrt{\frac{\mu\left(\mathbf{r}\right)}{\epsilon\left(\mathbf{r}\right)}}\\\mathcal{Y}\left(\mathbf{r}\right) &\stackrel{def}{=}& \mathcal{Z}\left(\mathbf{r}\right)^{-1} = \sqrt{\frac{\epsilon\left(\mathbf{r}\right)}{\mu\left(\mathbf{r}\right)}}\\k\left(\mathbf{r}\right) &=& \left|\mathbf{k}\left(\mathbf{r}\right)\right| = \frac{\omega}{c} \, n\left(\mathbf{r}\right) \\\mathbf{\hat{k}}\left(\mathbf{r}\right) &\stackrel{def}{=}& \frac{\mathbf{k}\left(\mathbf{r}\right)}{k\left(\mathbf{r}\right)} = \frac{ \nabla \varphi\left(\mathbf{r}\right) }{\left|\nabla \varphi\left(\mathbf{r}\right)\right|}\\\mathbf{e}\wedge\mathbf{h} &=& \mathcal{Y} \left|\mathbf{e}\right|^2 \mathbf{\hat{k}}= \mathcal{Z} \left|\mathbf{h}\right|^2 \mathbf{\hat{k}} \\\mathbf{e} &=& -\mathcal{Z}\, \mathbf{\hat{k}} \wedge\mathbf{h} \\\mathbf{h} &=& \mathcal{Y}\, \mathbf{\hat{k}} \wedge\mathbf{e} \\\end{array}\quad\quad\quad\quad(5)$$
These equations are
exactly fulfilled when $\mathbf{e}$ and $\mathbf{h}$ are constant (independent of $\mathbf{r}$) vectors and $\varphi\left(\mathbf{r}\right) = \mathbf{k} \cdot \mathbf{r}$ for some constant wavevector $\mathbf{k}$, when they define a plane wave, and plane waves are the only exact solution of the Eikonal equations. The uniqueness of plane waves in fulfilling the Eikonal equations exactly is another way of stating the strict contradictory nature of a ray - a true, exact ray represents a wholly delocalised photon and only the axial component of $\mathbf{k} \cdot \mathbf{r}$ is important for setting its phase. The assertion that the Eikonal equations approximately describe more general waves is the intuitive one that $\mathbf{e}\left(\mathbf{r}\right)$ and $\mathbf{h}\left(\mathbf{r}\right)$ vary slowly enough with $\mathbf{r}$ that they locally differ only slightly from plane waves. |
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 |
Kepler's Laws of Planetary Motion/Third Law Contents Physical Law The square of the period of the orbit of a planet around the sun is proportional to the cube of its average distance from the sun. Proof $(1): \quad r = \dfrac {h^2 / k} {1 + e \cos \theta}$
where $k = G M$.
From Equation of Ellipse in Reduced Form: Cartesian Frame, the equation of the orbit can also be given as:
$(2): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
where the foci are placed at $\tuple {\pm c, 0}$.
$a^2 - b^2 = c^2$
and also:
$e = \dfrac c a$ Thus:
\(\displaystyle e^2\) \(=\) \(\displaystyle \dfrac {a^2 - b^2} {a^2}\) \((3):\quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle b^2\) \(=\) \(\displaystyle a^2 \paren {1 - e^2}\) Then mean distance $a$ of $p$ from the focus $F$ is half the sum of the least and greatest values of $r$.
So $(1)$ and $(3)$ give:
\(\displaystyle a\) \(=\) \(\displaystyle \dfrac 1 2 \paren {\dfrac {h^2 / k} {1 + e} + \dfrac {h^2 / k} {1 - e} }\) \(\displaystyle \) \(=\) \(\displaystyle \dfrac {h^2} {k \paren {1 - e^2} }\) \(\displaystyle \) \(=\) \(\displaystyle \dfrac {h^2 a^2} {k b^2}\) \((4):\quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle b^2\) \(=\) \(\displaystyle \dfrac {h^2 a} k\) Let $T$ be the orbital period of $p$. $\mathcal A = \pi a b$
From Kepler's Second Law of Planetary Motion it follows that:
$\dfrac {h T} 2 = \pi a b$
and so from $(4)$:
\(\displaystyle T^2\) \(=\) \(\displaystyle \dfrac {4 \pi^2 a^2 b^2} {h^2}\) \(\displaystyle \) \(=\) \(\displaystyle \paren {\dfrac {4 \pi^2} k} a^3\)
We have that:
$k = G M$
Hence the result.
$\blacksquare$
$T^2 = a^3$
These units are derived specifically from the nature of the revolution of the Earth in its orbit, for which $T = 1 \, \mathrm{yr}$ and $a = 1 \, \mathrm {au}$.
Let $P$ be:
$\text{(a)}: \quad$ Twice as far away from $S$ as the Earth $\text{(b)}: \quad$ $3$ times as far away from $S$ as the Earth $\text{(c)}: \quad$ $25$ times as far away from $S$ as the Earth. Then the orbital period of $P$ is: $\text{(a)}: \quad$ approximately $2.8$ years $\text{(b)}: \quad$ approximately $5.2$ years $\text{(c)}: \quad$ $125$ years. Also see Source of Name
This entry was named for Johannes Kepler.
Kepler derived his three laws of planetary motion in the early $1600$s from a concentrated study over the course of $20$ years of the colossal wealth of observational data which had been made previously by Tycho Brahe of the behavior of the planets of the solar system, and in particular Mars.
The first two of these results he published in his gigantic work
Astronomia Nova.
The third appears some ten years later in his
Harmonices Mundi of $1619$. Sources 1937: Eric Temple Bell: Men of Mathematics... (previous) ... (next): Chapter $\text{VI}$: On the Seashore 1972: George F. Simmons: Differential Equations... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: $(22)$ 1992: George F. Simmons: Calculus Gems... (previous) ... (next): Chapter $\text {A}.10$: Kepler ($1571$ – $1630$) 1992: George F. Simmons: Calculus Gems... (previous) ... (next): Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation 2008: Ian Stewart: Taming the Infinite... (previous) ... (next): Chapter $8$: The System of the World 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics(5th ed.) ... (previous) ... (next): Entry: Kepler's laws of planetary motion(iii) |
New option
Wavefront/Taper analysis is implemented. The new option allows you to estimate the influence of inhomogeneities of the deposition on the spectral characteristics and on the wavefront of the reflected/transmitted wave. The option is available at Analysis -->More-->Wavefront/Taper
Phase computations include total path of the beam including extra space in the incident medium due to changed total thickness of the coating at different positions (Fig. 1).
Example:
Fig. 1. Schematic of the coating with thickness non-uniformity. To calculate the effect of a lack of uniformity on the wavefront, an extra incident material (gray region) is added to the front surface of the coating, so that the reference surface for phase calculations is completely free from non-uniformity.
Fig. 2. Schematic of the multilayer at the central position (left) and the multilayer at the position x 1 (right).
Phase at different positions can be calculated as phase of the amplitude reflection coefficient in the following way:
\[ \varphi (x=0):\;\;\; r(d_1,...,d_m)\]
\[ \varphi (x=x_1):\;\;\; r(d_1(x_1),...,d_m(x_m))\]
Wavefront can be calculated and plotted vs. the wavelength or relative coordinate \(x\) :
\[ \delta R(\lambda)=\frac{\varphi(\lambda)}{2\pi} \]
\[ \delta R(x)=\frac{\varphi(x)}{2\pi} \]
Fig. 3. Dependence of the reflected wavefront vs. relative coordinate calculated for parabolic taper interpolation specified in Fig. 3. The wavelength can be varied using the slider on the bottom of the window.
In the window
Taper/Wavefron parameters available by pressing Parameters button, you can choose between Simple taper function \(f(x)\) and more complicated taper dependence specified in Environments tab.
Example. \(f(x)\) are parabolic functions defined through taper coefficients \(a\) and \(b\) for high- and low-index materials, respectively (Fig. 4).
Fig. 4. Schematic of the parabolic non-uniformity and correspondence window in OptiLayer.
Example. \(f(x)\) can be specified as a linear function through \(a\) and \(b\) for high- and low-index materials, respectively (Fig. 5).
Fig. 5. Schematic of the parabolic non-uniformity and correspondence window in OptiLayer.
Fig. 6. Dependence of the reflected wavefront vs. relative coordinate calculated for linear taper interpolation specified in Fig. 5. The wavelength can be varied using the slider on the bottom of the window.
Fig. 7. Specification of taper coefficients in the environment manager.
Reflected/transmitted wavefront can be calculated for more complicated taper interpolations specified through
Taper coefficients in the Environments manager (Data --> Environments manager) (Fig. 7) and Relative Positions (Fig. 8).
Fig. 8. Schematic of a complicated taper interpolation and the correspondence window in OptiLayer.
OptiLayer allows you to calculate
dependence of spectral characteristics (reflectance, transmittance, phase, GD, and GDD) on the relatvive coordinate and on the wavlength.
Fig. 9. Schematic of a coating with layer non-uniformity.
Fig. 10. Schematic of the coating with layer non-uniformity at central position (left) and at position \(x_1\) (right).
Computations of spectral characteristics (reflectance, transmittance, phase, GD, and GDD) are performed at each relative coordinate/wavelength using standard formulas.
Fig. 11. Wavelength dependence of reflectance at a relative coordinate of 0.45. You can vary the relative position using the slider. Also, you can vary the angle of incidence and polarization (s, p or average (both)).
Fig. 12. Dependence of the coating reflectance on the position at the wavelength of 552nm. You can vary the wavelength with the help of the slider. Also, you can vary the angle of incidence and polarization (s, p or average (both)). |
Although it is well known that the Ward identities prohibit anomalousdimensions for conserved currents in local field theories, a claim from certainholographic models involving bulk dilaton couplings is that the gauge fieldassociated with the boundary current can acquire an anomalous dimension. Weresolve this conundrum by showing that all the bulk actions that produceanomalous dimensions for the conserved current generate non-local actions atthe boundary. In particular, the Maxwell equations are fractional. To provethis, we generalize to p-forms the Caffarelli/Silvestre (CS) extension theorem.In the context of scalar fields, this theorem demonstrates that second-orderelliptic differential equations in the upper half-plane in ${\mathbbR}_+^{n+1}$ reduce to one with the fractional Laplacian, $\Delta^\gamma$, whenone of the dimensions is eliminated. From the p-form generalization of the CSextension theorem, we show that at the boundary of the relevant holographicmodels, a fractional gauge theory emerges with equations of motion of the form,$\Delta ^\gamma A^t= 0$ with $\gamma$ $\in R$ and $A^t$ the boundary componentsof the gauge field. The corresponding field strength $F = d_\gamma A^t= d\Delta^\frac{\gamma-1}{2} A^t$ is invariant under $A^t \rightarrow A^t+ d_\gamma\Lambda$ with the fractional differential given by $d_\gamma \equiv (\Delta)^\frac{\gamma-1}{2}d$, implying that $[A^t]=\gamma$which is in general not unity.
We show that it is possible to construct a Virasoro algebra as a centralextension of the fractional Witt algebra generated by non-local operators ofthe form, $L_n^a\equiv\left(\frac{\p f}{\p z}\right)^a$ where $a\in {\mathbbR}$. The Virasoro algebra is explicitly of the form, \beq[L^a_m,L_n^a]=A_{m,n}L^a_{m+n}+\delta_{m,n}h(n)cZ^a \eeq where $c$ is thecentral charge (not necessarily a constant), $Z^a$ is in the center of thealgebra and $h(n)$ obeys a recursion relation related to the coefficients$A_{m,n}$. In fact, we show that all central extensions which respect thespecial structure developed here which we term a multimodule Lie-Algebra, areof this form. This result provides a mathematical foundation for non-localconformal field theories, in particular recent proposals in condensed matter inwhich the current has an anomalous dimension.
Using holographic renormalization coupled with theCaffarelli/Silvestre\cite{caffarelli} extension theorem, we calculate theprecise form of the boundary operator dual to a bulk scalar field rather thanjust its average value. We show that even in the presence of interactions inthe bulk, the boundary operator dual to a bulk scalar field is an anti-localoperator, namely the fractional Laplacian. The propagator associated with suchoperators is of the general power-law (fixed by the dimension of the scalarfield) type indicative of the absence of particle-like excitations at theWilson-Fisher fixed point or the phenomenological unparticle construction.Holographic renormalization also allows us to show how radial quantization canbe extended to such non-local conformal operators.
We generalize the descriptions of vortex moduli spaces in \cite{Br} to morethan one section with adiabatic constant $s$. The moduli space is topologicallyindependent of $s$ but is not compact with respect to $C^\infty$ topology.Following \cite{PW}, we construct a Gromov limit for vortices of fixed energy,an attempt to compactify the moduli space.
We show explicitly that the full structure of IIB string theory is needed toremove the non-localities that arise in boundary conformal theories that borderhyperbolic spaces on AdS$_5$. Specifically, using theCaffarelli/Silvestri\cite{caffarelli}, Graham/Zworski\cite{graham}, andChang/Gonzalez\cite{chang:2010} extension theorems, we prove that the boundaryoperator conjugate to bulk p-forms with negative mass in geodesically completemetrics is inherently a non-local operator, specifically the fractionalconformal Laplacian. The non-locality, which arises even in compact spaces,applies to any degree p-form such as a gauge field. We show that the boundarytheory contains fractional derivatives of the longitudinal components of thegauge field if the gauge field in the bulk along the holographic directionacquires a mass via the Higgs mechanism. The non-locality is shown to vanishonce the metric becomes incomplete, for example, either 1) asymptotically byadding N transversely stacked Dd-branes or 2) exactly by giving the boundary abrane structure and including a single transverse Dd-brane in the bulk. Theoriginal Maldacena conjecture within IIB string theory corresponds to theformer. In either of these proposals, the location of the Dd-branes places anupper bound on the entanglement entropy because the minimal bulk surface in theAdS reduction is ill-defined at a brane interface. Since the branesingularities can be circumvented in the full 10-dimensional spacetime, weconjecture that the true entanglement entropy must be computed from the minimalsurface in 10-dimensions, which is of course not minimal in the AdS$_5$reduction.
We prove a regularity result for Monge-Amp\`ere equations degenerate alongsmooth divisor on Kaehler manifolds in Donaldson's spaces of $\beta$-weightedfunctions. We apply this result to study the curvature of Kaehler metrics withconical singularities along divisors and give a geometric sufficient conditionon the divisor for its boundedness.
In this paper we investigate the differential geometric and algebro-geometricproperties of the noncollapsing limit in the continuity method that wasintroduced by the first two named authors in \cite{LaTi14}.
We introduce a new continuity method which provides an alternative way ofcarrying out the Analytic Minimal Model Program introduced by G. Tian and J.Song and G. Tian. This equation -- unlike the Ricci flow -- has the advantageof having Ricci curvature bounded from below along the deformation, so that thecompactness theory of Cheeger-Colding-Tian for Kaehler manifolds and thepartial $C^0$-estimate, can be applied.
We prove a conjecture of Gromov's to the effect that manifolds with isotropiccurvature bounded below by 1 (after possibly rescaling) are macroscopically1-dimensional on the scales greater than 1. As a consequence we prove thatcompact manifolds with positive isotropic curvature have virtually freefundamental groups. Our main technique is modeled on Donaldson's version ofH\"ormander technique to produce (almost) holomorphic sections which we use toconstruct destabilizing sections.
We investigate how to obtain various flows of K\"ahler metrics on a fixedmanifold as variations of K\"ahler reductions of a metric satisfying a givenstatic equation on a higher dimensional manifold. We identify static equationsthat induce the geodesic equation for the Mabuchi's metric, the Calabi flow,the pseudo-Calabi flow of Chen-Zheng and the K\"ahler-Ricci flow. In the lattercase we re-derive the V-soliton equation of La Nave-Tian.
In this paper, we first show an interpretation of the K\"ahler-Ricci flow ona manifold $X$ as an exact elliptic equation of Einstein type on a manifold $M$of which $X$ is one of the (K\"ahler) symplectic reductions via a (non-trivial)torus action. There are plenty of such manifolds (e.g. any line bundle on $X$will do). Such an equation is called $V$-soliton equation, which can beregarded as a generalization of K\"ahler-Einstein equations or K\"ahler-Riccisolitons. As in the case of K\"ahler-Einstein metrics, we can also reduce the$V$-soliton equation to a scalar equation on K\"ahler potentials, which is ofMonge-Ampere type. We then prove some preliminary results towards establishingexistence of solutions for such a scalar equation on a compact K\"ahlermanifold $M$. One of our motivations is to apply the interpretation to studyingfinite time singularities of the K\"ahler-Ricci flow.
We discuss some classification results for Ricci solitons, that is, selfsimilar solutions of the Ricci Flow. Some simple proofs of known results willbe presented. In detail, we will take the equation point of view, trying toavoid the tools provided by considering the dynamic properties of the Ricciflow.
In this note we propose to show that the K\"ahler-Ricci flow fits naturallywithin the context of the Minimal Model Program for projective varieties. Inparticular we show that the flow detects, in finite time, the contractiontheorem of any extremal ray and we analyze the singularities of the metric inthe case of divisorial contractions for varieties of general type. In case onehas a smooth minimal model of general type (i.e., the canonical bundle is nefand big), we show infinite time existence and analyze the singularities.
In this note we show how to find the stable model of a one-parameter familyof elliptic surfaces with sections. More specifically, we perform the log Minimal Model Program in an explicitmanner by means of toric geometry, in each such one parameter family. This waywe obtain an explicit combinatorial description of the surfaces that may occurat the boundary of moduli (as well as a new proof of the completeness of themoduli space) |
In my most recent research, I’m working on finding
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 “Minimax Lower Bounds” .I pretend to make some review in the general method and introduce some bounds depending on the divergence between two probability measures. Also, I want to study the classic results of Le Cam, Fanoand Assouad. I hope that these publications are very educational for all of us.
$latex \displaystyle \sup_{f\in\mathcal{F}}\mathbb E\left[d^{2}(\hat{f}_{n},f)\right]\leq C\psi_{n} &fg=000000$
where $latex {\psi_{n}^{2}}&fg=000000$ is going to zero (for the density estimation case $latex {\psi_{n}=n^{-4/5}}&fg=000000$) and $latex {C<\infty}&fg=000000$. We would like to know if these rates are tight, in the sense that there is no other estimator that is better. To make this, we define a lower bound like
$latex \displaystyle \inf_{\hat{f}}\sup_{f\in\mathcal{F}}\mathbb E\left[d^{2}(\hat{f}_{n},f)\right]\geq c\psi_{n}. &fg=000000$
We will assume the following three element:
A class of functions $latex {\mathcal{F}}&fg=000000$ , containing the “true” function $latex {f}&fg=000000$. For example, $latex {\mathcal{F}}&fg=000000$ could be a Hölder class $latex {\Sigma(\beta,L)}&fg=000000$ or a Sobolev class $latex {W(\beta,L)}&fg=000000$. A probability measure family $latex {\{\mathbb{P}_{f},f\in\mathcal{F}\}}&fg=000000$, indexed by $latex {\mathcal{F}}&fg=000000$ on a measurable space $latex {(\mathcal{X},\mathcal{A})}&fg=000000$ associated with the data. For example, in the density model, $latex {\mathbb{P}_{f}}&fg=000000$ is the probability measure associated with a sample $latex {(X_{1},\ldots,X_{n})}&fg=000000$ when the $latex {X_{i}}&fg=000000$’s density is $latex {f}&fg=000000$. A distance $latex {d(\cdot,\cdot)\geq0}&fg=000000$ which compares the performance between the estimate $latex {\hat{f}_{n}}&fg=000000$ and the real value $latex {f}&fg=000000$. For example $latex {d(f,g)=\Vert f-g\Vert_{2}}&fg=000000$, $latex {d(f,g)=\vert f(x_{0})-g(x_{0})\vert}&fg=000000$ for $latex {x_{0}}&fg=000000$ fix. In fact, it is enough use a semi-distance which satisfies the triangle inequality.
Define, for a stochastic model $latex {\{\mathbb{P}_{f},f\in\mathcal{F}\}}&fg=000000$ and a distance $latex {d}&fg=000000$, the
minimax riskas
$latex \displaystyle \mathcal{R}^{*}\triangleq\inf_{\hat{f}}\sup_{f\in\mathcal{F}}\mathbb E\left[d^{2}(\hat{f}_{n},f)\right] &fg=000000$
where the $latex {\inf_{\hat{f}}}&fg=000000$ is taken over all estimators. The main goal is to get a result of the form
$latex \displaystyle \mathcal{R}^{*}\geq c\psi_{n} &fg=000000$
for $latex {c>0}&fg=000000$ and $latex {\psi_{n}\rightarrow0}&fg=000000$.
Suppose that we have proved that
and if for a particular $latex {\hat{f}_{n}}&fg=000000$
$latex \displaystyle \limsup_{n\rightarrow\infty}\sup_{f\in\mathcal{F}}\psi_{n}^{-1}\mathbb E\left[d^{2}(\hat{f}_{n},f)\right]\leq C. &fg=000000$
It implies directly that
$latex \displaystyle \limsup_{n\rightarrow\infty}\psi_{n}^{-1}\mathcal{R}^{*}\leq C. \ \ \ \ \ (2)&fg=000000$
If inequalities (1) and (2) are satisfied simultaneously, we say that $latex {\psi_{n}}&fg=000000$ is the optimal rate of convergence for this problem and that $latex {\hat{f}_{n}}&fg=000000$ attains that rate.
Remark 1 Two rates of convergence $latex {\psi_{n}}&fg=000000$ and $latex {\psi_{n}^{\prime}}&fg=000000$ are equivalent (we write $latex {\psi_{n}\asymp\psi_{n}^{\prime}}&fg=000000$) if
$latex \displaystyle 0<\liminf_{n\rightarrow\infty}\frac{\psi_{n}}{\psi_{n}^{\prime}}\leq\limsup_{n\rightarrow\infty}\frac{\psi_{n}}{\psi_{n}^{\prime}}<\infty. &fg=000000$
The big question here is:
How can we prove the equation (1) if it is necessary bound $latex {\mathcal{R}^{*}}&fg=000000$ for all the estimators $latex {\hat{f}_{n}}&fg=000000$ of $latex {f}&fg=000000$?
At a first glance, it will be an impossible task, just imagine the massive quantity of possible estimators for $latex {f}&fg=000000$. Fortunately, we can apply a “reduction procedure” in order to simplify the problem and find the bound required. We will study it the next post.
As always any thoughts, suggestions or improvements are welcome in the commentaries.
Related articles |
Suppose I am pushing a box on table with $10$ N force due to friction it is not moving and if i applied $20$ N the box started accelerating. Then, how does the box apply $20$ N force on me?Well actually,I really know fundamental laws are certainly true.But from where the opposing force on us is coming?
closed as unclear what you're asking by Dvij Mankad, David Z♦ Jun 17 at 8:50
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The difficulty you are having is quite common when first learning Newtons laws because you think Newton’s third law means the forces always cancel each other and nothing should accelerate. They don’t. You have to look at the net force acting on each body individually and apply Newton's second law to each individually.
When you were pushing on the box, say to the right, with a force of 10 N, an equal static friction force of 10 N was acting on the box to the left in opposition to your force. The net force on the box was zero and therefore it did not accelerate.
Per Newton's third law the box was exerting an equal and opposite force of 10 N on you to the left. You did not accelerate because of an equal and opposite static friction force between your feet and the ground that acted to the right. Therefore you do not accelerate.
Now when you were pushing with a 20 N force to the right the box accelerated because your force apparently exceeded the maximum static friction force between the box and ground acting to the left. The box is now sliding and it is the kinetic friction force, which is generally less than the static friction force, between the box and the ground that is now opposing your force. But since the box is accelerating your 20 N force to the right was greater than the kinetic friction force to the left, meaning there was a net force to the right, $F_{net}$, causing the box to accelerate per Newton's second law. Then, per Newton's second, the acceleration of the box to the right is $a=\frac{F_{net}}{m}$, where $m$ is the mass of the box.
Now the box also exerts an equal and opposite 20 N force on you per Newton's third law. What keeps you from accelerating? It is the static friction force between your feet and the ground acting to the right that is equal to the force the box exerts on you. The net force on you is zero as long as you don't push too hard that the maximum static friction force between your feet and the ground is exceeded, in which case you will start slipping.
Hope this helps.
Imagine that you have springs instead of your hands. If you push the box, the springs are compressed, shortened by the force from the box. (No box, no compress as you move.)
The side of box on which you push is slightly deformed - imagine that it is from very springy material, so you can see it. As you deform it more and more, it acts to you with greater and greater force - similarly as a drawn bow.
You may stop pushing the box when it starts accelerating - but the box immediately stops accelerating. To keep it in the accelerated motion, you have to keep pushing on it - and then the box will keep pushing on your hands - see 1. and 2.
Suppose the box is half your mass and you stand on a skateboard, and just think of the second part of your experiment. If you think that pushing the box will also make you move backward, then you agree (at least qualitatively) with Newton's third law. :)
I’d like to note here that there is an intuition that might lead one astray here: when you’re actually pushing on a static object that is stuck to the ground due to friction, it usually becomes easier to push it immediately after it slips. This is because the coefficient of static friction tends to be greater than that of kinetic friction.
So if you’re actually pushing a big heavy box along the floor and you find it becomes easier when it starts to slip, this isn’t because Newton’s third law is incorrect, but because you can
continue pushing the moving box with a force that is less than that which you had to exert to start moving the box. E.g., if the box is 50kg and the coefficient of status friction is .8 while the coefficient of kinetic friction is .4, then you may find that it takes
$$F = \mu_s m g \sim 400 N$$
to get the box moving, but then once you do you can relax and push with only a force of
$$F = \mu_k m g \sim 200 N$$
to keep the box moving at constant speed. So it gets easier, and there is no contradiction with Newton’s Third. |
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Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 N/dydp_T$ at midrapidity |y|<0.5 in ...
Transverse sphericity of primary charged particles in minimum bias proton-proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV
(Springer, 2012-09)
Measurements of the sphericity of primary charged particles in minimum bias proton--proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV with the ALICE detector at the LHC are presented. The observable is linearized to be ...
Pion, Kaon, and Proton Production in Central Pb--Pb Collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012-12)
In this Letter we report the first results on $\pi^\pm$, K$^\pm$, p and pbar production at mid-rapidity (|y|<0.5) in central Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, measured by the ALICE experiment at the LHC. The ...
Measurement of prompt J/psi and beauty hadron production cross sections at mid-rapidity in pp collisions at root s=7 TeV
(Springer-verlag, 2012-11)
The ALICE experiment at the LHC has studied J/ψ production at mid-rapidity in pp collisions at s√=7 TeV through its electron pair decay on a data sample corresponding to an integrated luminosity Lint = 5.6 nb−1. The fraction ...
Suppression of high transverse momentum D mesons in central Pb--Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV
(Springer, 2012-09)
The production of the prompt charm mesons $D^0$, $D^+$, $D^{*+}$, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at the LHC, at a centre-of-mass energy $\sqrt{s_{NN}}=2.76$ TeV per ...
J/$\psi$ suppression at forward rapidity in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012)
The ALICE experiment has measured the inclusive J/ψ production in Pb-Pb collisions at √sNN = 2.76 TeV down to pt = 0 in the rapidity range 2.5 < y < 4. A suppression of the inclusive J/ψ yield in Pb-Pb is observed with ...
Production of muons from heavy flavour decays at forward rapidity in pp and Pb-Pb collisions at $\sqrt {s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012)
The ALICE Collaboration has measured the inclusive production of muons from heavy flavour decays at forward rapidity, 2.5 < y < 4, in pp and Pb-Pb collisions at $\sqrt {s_{NN}}$ = 2.76 TeV. The pt-differential inclusive ...
Particle-yield modification in jet-like azimuthal dihadron correlations in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012-03)
The yield of charged particles associated with high-pT trigger particles (8 < pT < 15 GeV/c) is measured with the ALICE detector in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV relative to proton-proton collisions at the ...
Measurement of the Cross Section for Electromagnetic Dissociation with Neutron Emission in Pb-Pb Collisions at √sNN = 2.76 TeV
(American Physical Society, 2012-12)
The first measurement of neutron emission in electromagnetic dissociation of 208Pb nuclei at the LHC is presented. The measurement is performed using the neutron Zero Degree Calorimeters of the ALICE experiment, which ... |
Spring 2018, Math 171 Week 5 Reversibility/Detailed Balance Condition
Show that Ehrenfest’s chain is reversible. \[P(x,y)= \begin{cases}\frac{N-x}{N}, & \mathrm{if\ } y=x+1 \cr \frac{x}{N}, & \mathrm{if\ } y=x-1 \cr 0, & \mathrm{otherwise}\end{cases}\]
(Answer) Show KCC on simple cycles: All simple cycles are length 2, and cycles of length 2 always satisfy KCC.
Consider the Markov chain \(\{X_n:n \ge 0\}\) on \(\mathcal{S} = \{1, 2, \dots, N\}\) with a transition matrix of the form \[P(x,x-1)=q(x), \; P(x,x)=r(x), \; P(x,x+1)=p(x)\]
Find conditions on \(q, r, p\) which guarantee the chain will be irreducible.
(Answer) \(q(x) \neq 0\) and \(p(x) \neq 0\) \(\forall x\)
Show the chain is reversible
(Answer) Show KCC on simple cycles: All simple cycles are length 2, and cycles of length 2 always satisfy KCC.
Find the stationary distribution, assuming the chain is irreducible
(Discussed) Consider the Markov chain \(\{X_n:n \ge 0\}\) on \(\mathcal{S} = \{1, 2, \dots, N\}\) with a transition matrix of the form \[P(x,y)= \begin{cases}q(x) & \mathrm{if\ } x \neq y \cr 1-(N-1)q(x) & \mathrm{if\ } x = y \end{cases}\] Verify that such chains are reversible
Suppose \(P\) is the transition matrix for a reversible Markov chain. Fix \(i_0, j_0 \in \mathcal{S}\) Define a new transition matrix \[P'(x,y)= \begin{cases}aP(i_0, j_0) & \mathrm{if\ } x=i_0, y=j_0 \cr aP(j_0, i_0) & \mathrm{if\ } x=j_0, y=i_0 \cr P(i_0, i_0) + (1-a)P(i_0, j_0) & \mathrm{if\ } x=y=i_0 \cr P(j_0, j_0) + (1-a)P(j_0, i_0) & \mathrm{if\ } x=y=j_0 \cr P(x,y) & \mathrm{otherwise} \end{cases}\]
Verify that the Markov chain defined by the new transition matrix will be reversible
How will the stationary distribution of the new Markov chain relate to the stationary distribution of the original?
Limiting Behavior
(Discussed) Consider Ehrenfest’s chain \(\{X_n:n \ge 0\}\) subject to the transition probabilities. \[P(x,y)= \begin{cases}\frac{N-x}{N}, & \mathrm{if\ } y=x+1 \cr \frac{x}{N}, & \mathrm{if\ } y=x-1 \cr 0, & \mathrm{otherwise}\end{cases}\]
Compute the period of \(\{X_n:n \ge 0\}\)
(Answer) 2
Show that \(Y_n=X_{2n}\) is a Markov chain. Is it irreducible?
(Answer) No. The state space has been split into two parts which don’t communicate with each other.
Consider \(Y_n=X_{2n}\) under the restriction \(X_0 \in \{0,\ 2,\ \dots,\ 2\lfloor\frac{N}{2}\rfloor\}\). Compute its stationary distribution. Explain why \(Y_n\) converges in distribution to the stationary distribution as \(n \to \infty\).
(Discussed) Consider the Markov chain \(\{X_n:n \ge 0\}\) on \(\mathcal{S} = \{1, 2, \dots, N\}\) with a transition matrix of the form \[P(x,x-1)=q(x), \; P(x,x)=r(x), \; P(x,x+1)=p(x)\]
Find conditions on \(q, r, p\) which lead to a period of 2
(Answer) \(r(x) = 0\)
Suppose \(q(x)=0\). Find a formula for \(\mathbb{E}_x[N(y)]\) for \(x < y\)
Consider the Markov chain \(\{X_n:n \ge 0\}\) on \(\mathcal{S} = \{1, 2, \dots, N\}\) with a transition matrix of the form \[P(x,y)= \begin{cases}p, & \mathrm{if\ } y=x+1, \; x<N \cr 1-p, & \mathrm{if\ } x=0, \; y=0 \cr r & \mathrm{if\ } y = x, \; 0<y<N \cr q, & \mathrm{if\ } y=x-1, \; x>0 \cr 1-q, & \mathrm{if\ } x=N, \; y=N \cr 0, & \mathrm{otherwise}\end{cases}\]
Find a system of equations which could be solved to find \(\mathbb{E}_x[T_N]\) for any \(x\).
Compute \(\mathbb{E}_x[T_N]\) when \(r=0\) and \(q=1-p\) |
If it's not what You are looking for type in the equation solver your own equation and let us solve it.
Solution for 50t-4t^2=120 equation: 50t-4t^2=120
We move all terms to the left:
50t-4t^2-(120)=0
a = -4; b = 50; c = -120; Δ = b 2-4ac Δ = 50 2-4·(-4)·(-120) Δ = 580 The delta value is higher than zero, so the equation has two solutions We use following formulas to calculate our solutions:
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$
The end solution:
$\sqrt{\Delta}=\sqrt{580}=\sqrt{4*145}=\sqrt{4}*\sqrt{145}=2\sqrt{145}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(50)-2\sqrt{145}}{2*-4}=\frac{-50-2\sqrt{145}}{-8}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(50)+2\sqrt{145}}{2*-4}=\frac{-50+2\sqrt{145}}{-8}
$ |
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:-) |
Authors Index,
Methods Funct. Anal. Topology
22 (2016), no. 4, 295-310
We investigate general elliptic boundary-value problems in Hörmander inner product spaces that form the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces with respect to the Sobolev Hilbert scale. We prove that the operator corresponding to an arbitrary elliptic problem is Fredholm in appropriate couples of the Hörmander spaces and induces a collection of isomorphisms on the extended Sobolev scale. We obtain a local a priory estimate for generalized solutions to this problem and prove a theorem on their local regularity in the Hörmander spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of the solutions are continuous.
Methods Funct. Anal. Topology
22 (2016), no. 2, 95-116
The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinskiĭ in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions.
Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem
Methods Funct. Anal. Topology
22 (2016), no. 4, 311-329
It is proved that the Poisson measure is a spectral measure of some family of commuting selfadjoint operators acting on a space constructed from some generalization of the moment problem.
Methods Funct. Anal. Topology
22 (2016), no. 1, 1-47
The article is devoted to a study of Bogoliubov functionals by using methods of the operator spectral theory being applied to the classical power moment problem. Some results, similar to corresponding ones for the moment problem, where obtained for such functionals. In particular, the following question was studied: under what conditions a sequence of nonlinear functionals is a sequence of Bogoliubov functionals.
Methods Funct. Anal. Topology
22 (2016), no. 1, 74-80
We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of spectra of the Dirichlet-Neumann and Dirichlet-Dirichlet problems on $[0,a/2]$ and parts of spectra of the Dirichlet-Newman and Dirichlet-Dirichlet problems on $[a/2,a]$, we find the potential of the Sturm-Liouville equation.
Methods Funct. Anal. Topology
22 (2016), no. 2, 137-151
We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A=A+\alpha\left\langle\cdot,\omega_1\right\rangle\omega_2$, $\omega_1\not=\omega_2$, $\alpha\in{\mathbb C}$, in a general case $\omega_1,\omega_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.
Methods Funct. Anal. Topology
22 (2016), no. 4, 387-392
In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated.
Methods Funct. Anal. Topology
22 (2016), no. 3, 256-265
We give some new characterizations of almost weak Dunford-Pettis operators and we investigate their relationship with weak Dunford-Pettis operators.
Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$
Methods Funct. Anal. Topology
22 (2016), no. 4, 330-345
We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.
Methods Funct. Anal. Topology
22 (2016), no. 3, 210-219
Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\mathcal{S}_V(f) = \{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the set $\pi_0\partial \mathcal{E}_V$ of connected components of the boundary of $\mathcal{E}_V.$ Therefore we have a homomorphism $\phi:\mathcal{S}(f)\to \mathrm{Aut}(\pi_0\partial \mathcal{E}_V)$. Let also $G = \phi(\mathcal{S}(f))$ be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}(\pi_0\partial \mathcal{E}_V).$ Suppose that the inclusion $\partial \mathcal{E}_V\subset M\setminus V$ induces a bijection $\pi_0 \partial \mathcal{E}_V\to\pi_0(M\setminus V).$ Let $H$ be a subgroup of $G.$ We present a sufficient condition for existence of a section $s:H\to \mathcal{S}_V(f)$ of the homomorphism $\phi,$ so, the action of $H$ on $\partial \mathcal{E}_V$ lifts to the $H$-action on $M$ by $f$-preserving diffeomorphisms of $M$. This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$ having the following property: for each critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple linear factors.
Methods Funct. Anal. Topology
22 (2016), no. 3, 220-244
A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.
Methods Funct. Anal. Topology
22 (2016), no. 4, 346-374
Two coupled spatial birth-and-death Markov evolutions on $\mathbb{R}^d$ are obtained as unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive a system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology.
Methods Funct. Anal. Topology
22 (2016), no. 2, 117-136
We solve the inverse problem for non-Abelian Coxeter double Bruhat cells in terms of the matrix Weyl functions. This result can be used to establish complete integrability of the non-Abelian version of nonlinear Coxeter-Toda lattices in $GL_n$.
On approximation of solutions of operator-differential equations with their entire solutions of exponential type
Methods Funct. Anal. Topology
22 (2016), no. 3, 245-255
We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to $0$ of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator $A$ is generated by a second order elliptic differential expression in the space $L_{2}(\Omega)$ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a certain boundary condition.
Methods Funct. Anal. Topology
22 (2016), no. 2, 152-168
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics via studying the longtime behavior of the strongly continuous contraction semigroup solving the associated Kolmogorov (backward) equation as an abstract Cauchy problem. This hypocoercivity result is proven in previous works before by Dolbeault, Mouhot and Schmeiser in the corresponding dual Fokker-Planck framework, but without including domain issues of the appearing operators. In our elaboration, we include the domain issues and additionally compute the rate of convergence in dependence of the damping coefficient. Important statements for the complete elaboration are the m-dissipativity results for the Langevin operator established by Conrad and the first named author of this article as well as the essential selfadjointness results for generalized Schrödinger operators by Wielens or Bogachev, Krylov and Röckner. We emphasize that the chosen Kolmogorov approach is natural. Indeed, techniques from the theory of (generalized) Dirichlet forms imply a stochastic representation of the Langevin semigroup as the transition kernel of diffusion process which provides a martingale solution to the Langevin equation. Hence an interesting connection between the theory of hypocoercivity and the theory of (generalized) Dirichlet forms is established besides.
Methods Funct. Anal. Topology
22 (2016), no. 2, 184-196
We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}\alpha_i(x)$$ if the asymptotic frequency $\nu_j(x)$ of at least one digit does not exist, were $$ \nu_j(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$
Methods Funct. Anal. Topology
22 (2016), no. 3, 197-209
We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.
Methods Funct. Anal. Topology
22 (2016), no. 1, 81-93
We investigate the dynamical systems modeling conflict processes between a pair of opponents. We assume that opponents are given on a common space by distributions (probability measures) having the similar or self-similar structure. Our main result states the existence of the controlled conflict in which one of the opponents occupies almost whole conflicting space. Besides, we compare conflicting effects stipulated by the rough structural approximation under controlled redistributions of starting measures.
Methods Funct. Anal. Topology
22 (2016), no. 3, 266-282
Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.
On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators
Methods Funct. Anal. Topology
22 (2016), no. 2, 169-183
A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of the general criterion corresponding to a certain rapidly growing defining sequence is observed.
A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems
Methods Funct. Anal. Topology
22 (2016), no. 4, 375-386
We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}.$ The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.
Methods Funct. Anal. Topology
22 (2016), no. 1, 48-61
An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.
Methods Funct. Anal. Topology
22 (2016), no. 1, 62-73
In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.
Methods Funct. Anal. Topology
22 (2016), no. 3, 283-294
Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $\pi_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}(F)$ is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of $\pi_0 H^{+}(F)$ under an assumption that the corresponding space of leaves of $F$ has a structure similar to a rooted tree of finite diameter. |
Authors Index,
Methods Funct. Anal. Topology
22 (2016), no. 4, 295-310
We investigate general elliptic boundary-value problems in Hörmander inner product spaces that form the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces with respect to the Sobolev Hilbert scale. We prove that the operator corresponding to an arbitrary elliptic problem is Fredholm in appropriate couples of the Hörmander spaces and induces a collection of isomorphisms on the extended Sobolev scale. We obtain a local a priory estimate for generalized solutions to this problem and prove a theorem on their local regularity in the Hörmander spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of the solutions are continuous.
Methods Funct. Anal. Topology
22 (2016), no. 2, 95-116
The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinskiĭ in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions.
Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem
Methods Funct. Anal. Topology
22 (2016), no. 4, 311-329
It is proved that the Poisson measure is a spectral measure of some family of commuting selfadjoint operators acting on a space constructed from some generalization of the moment problem.
Methods Funct. Anal. Topology
22 (2016), no. 1, 1-47
The article is devoted to a study of Bogoliubov functionals by using methods of the operator spectral theory being applied to the classical power moment problem. Some results, similar to corresponding ones for the moment problem, where obtained for such functionals. In particular, the following question was studied: under what conditions a sequence of nonlinear functionals is a sequence of Bogoliubov functionals.
Methods Funct. Anal. Topology
22 (2016), no. 1, 74-80
We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of spectra of the Dirichlet-Neumann and Dirichlet-Dirichlet problems on $[0,a/2]$ and parts of spectra of the Dirichlet-Newman and Dirichlet-Dirichlet problems on $[a/2,a]$, we find the potential of the Sturm-Liouville equation.
Methods Funct. Anal. Topology
22 (2016), no. 2, 137-151
We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A=A+\alpha\left\langle\cdot,\omega_1\right\rangle\omega_2$, $\omega_1\not=\omega_2$, $\alpha\in{\mathbb C}$, in a general case $\omega_1,\omega_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.
Methods Funct. Anal. Topology
22 (2016), no. 4, 387-392
In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated.
Methods Funct. Anal. Topology
22 (2016), no. 3, 256-265
We give some new characterizations of almost weak Dunford-Pettis operators and we investigate their relationship with weak Dunford-Pettis operators.
Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$
Methods Funct. Anal. Topology
22 (2016), no. 4, 330-345
We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.
Methods Funct. Anal. Topology
22 (2016), no. 3, 210-219
Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\mathcal{S}_V(f) = \{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the set $\pi_0\partial \mathcal{E}_V$ of connected components of the boundary of $\mathcal{E}_V.$ Therefore we have a homomorphism $\phi:\mathcal{S}(f)\to \mathrm{Aut}(\pi_0\partial \mathcal{E}_V)$. Let also $G = \phi(\mathcal{S}(f))$ be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}(\pi_0\partial \mathcal{E}_V).$ Suppose that the inclusion $\partial \mathcal{E}_V\subset M\setminus V$ induces a bijection $\pi_0 \partial \mathcal{E}_V\to\pi_0(M\setminus V).$ Let $H$ be a subgroup of $G.$ We present a sufficient condition for existence of a section $s:H\to \mathcal{S}_V(f)$ of the homomorphism $\phi,$ so, the action of $H$ on $\partial \mathcal{E}_V$ lifts to the $H$-action on $M$ by $f$-preserving diffeomorphisms of $M$. This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$ having the following property: for each critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple linear factors.
Methods Funct. Anal. Topology
22 (2016), no. 3, 220-244
A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.
Methods Funct. Anal. Topology
22 (2016), no. 4, 346-374
Two coupled spatial birth-and-death Markov evolutions on $\mathbb{R}^d$ are obtained as unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive a system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology.
Methods Funct. Anal. Topology
22 (2016), no. 2, 117-136
We solve the inverse problem for non-Abelian Coxeter double Bruhat cells in terms of the matrix Weyl functions. This result can be used to establish complete integrability of the non-Abelian version of nonlinear Coxeter-Toda lattices in $GL_n$.
On approximation of solutions of operator-differential equations with their entire solutions of exponential type
Methods Funct. Anal. Topology
22 (2016), no. 3, 245-255
We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to $0$ of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator $A$ is generated by a second order elliptic differential expression in the space $L_{2}(\Omega)$ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a certain boundary condition.
Methods Funct. Anal. Topology
22 (2016), no. 2, 152-168
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics via studying the longtime behavior of the strongly continuous contraction semigroup solving the associated Kolmogorov (backward) equation as an abstract Cauchy problem. This hypocoercivity result is proven in previous works before by Dolbeault, Mouhot and Schmeiser in the corresponding dual Fokker-Planck framework, but without including domain issues of the appearing operators. In our elaboration, we include the domain issues and additionally compute the rate of convergence in dependence of the damping coefficient. Important statements for the complete elaboration are the m-dissipativity results for the Langevin operator established by Conrad and the first named author of this article as well as the essential selfadjointness results for generalized Schrödinger operators by Wielens or Bogachev, Krylov and Röckner. We emphasize that the chosen Kolmogorov approach is natural. Indeed, techniques from the theory of (generalized) Dirichlet forms imply a stochastic representation of the Langevin semigroup as the transition kernel of diffusion process which provides a martingale solution to the Langevin equation. Hence an interesting connection between the theory of hypocoercivity and the theory of (generalized) Dirichlet forms is established besides.
Methods Funct. Anal. Topology
22 (2016), no. 2, 184-196
We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}\alpha_i(x)$$ if the asymptotic frequency $\nu_j(x)$ of at least one digit does not exist, were $$ \nu_j(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$
Methods Funct. Anal. Topology
22 (2016), no. 3, 197-209
We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.
Methods Funct. Anal. Topology
22 (2016), no. 1, 81-93
We investigate the dynamical systems modeling conflict processes between a pair of opponents. We assume that opponents are given on a common space by distributions (probability measures) having the similar or self-similar structure. Our main result states the existence of the controlled conflict in which one of the opponents occupies almost whole conflicting space. Besides, we compare conflicting effects stipulated by the rough structural approximation under controlled redistributions of starting measures.
Methods Funct. Anal. Topology
22 (2016), no. 3, 266-282
Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.
On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators
Methods Funct. Anal. Topology
22 (2016), no. 2, 169-183
A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of the general criterion corresponding to a certain rapidly growing defining sequence is observed.
A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems
Methods Funct. Anal. Topology
22 (2016), no. 4, 375-386
We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}.$ The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.
Methods Funct. Anal. Topology
22 (2016), no. 1, 48-61
An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.
Methods Funct. Anal. Topology
22 (2016), no. 1, 62-73
In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.
Methods Funct. Anal. Topology
22 (2016), no. 3, 283-294
Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $\pi_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}(F)$ is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of $\pi_0 H^{+}(F)$ under an assumption that the corresponding space of leaves of $F$ has a structure similar to a rooted tree of finite diameter. |
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Now showing items 1-2 of 2
Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions, as well as give insight into the initial state of such collisions and ...
Investigations of anisotropic collectivity using multi-particle correlations in pp, p-Pb and Pb-Pb collisions
(Elsevier, 2017-11)
Two- and multi-particle azimuthal correlations have proven to be an excellent tool to probe the properties of the strongly interacting matter created in heavy-ion collisions. Recently, the results obtained for multi-particle ... |
For a given Hamiltonian with spin interaction, say Ising model $$H=-J\sum_{i,j} s_i s_j$$ in which there are no external magnetic field. The Hamiltonian is invariant under transformation $s_i \rightarrow -s_i$, so there are always two spin states with exactly same energy.
For the magnetization $M = \sum_i s_i$, we can take the ensemble average $$\left\langle M\right\rangle =\sum_{\{s_{i}\}}M\exp(-\beta E)$$ and the result should be $\left\langle M\right\rangle = 0$ because the two states with all spin flipped will exactly cancel each other. There is an argument for this in the wiki.
So the question: how is this situation handled for finite and infinite lattice? How can they obtain the non-zero magnetization for the 2D Ising model?
$$M=\left(1-\left[\sinh\left(\log(1+\sqrt{2})\frac{T_{c}}{T}\right)\right]^{-4}\right)^{1/8}$$
Some information: for a 1D Ising model with external magnetic field, one can solve the Hamiltonian$$H=-J\sum_{i,j} s_i s_j - \mu B \sum_i s_i$$and obtains the magnetization as$$\left\langle M\right\rangle =\frac{N\mu\sinh(\beta\mu B)}{\left[\exp(-4\beta J)+\sinh^{2}(\beta\mu B)\right]^{1/2}}$$It gives the result $\left\langle M\right\rangle \rightarrow 0$ when $B \rightarrow 0$ for any temperature and this match the definition of the magnetization above. However, it gives $\left\langle M\right\rangle \rightarrow N\mu$ when we take the limit of $T \rightarrow 0$. It suggests the ordering of limit is important, but we still get $\left\langle M\right\rangle = 0$ when there is no external magnetic field. Reminder: There are some precautions needed to care when you run computer simulation using the definition of magnetization above directly, otherwise, you will always get 0. These methods are similar to create a spontaneous symmetry breaking manually, for the Ising model, the following may be used: Use the $\left\langle |M|\right\rangle$ instead Fix the state of one spin so that the system will close to either $M=+1$ or $M=-1$ at low temperature.
In general, the finite size scaling should be used because we are likely interested in the thermodynamic limit of the system.
Update:Visualization should explain the problem better. Here is the figure of the canonical distribution as a function of magnetization $M$ and temperature $T$ for 3D Ising model ($L = 10$).
At a fixed temperature $T \lesssim T_c$, there are two symmetric peaks with opposite magnetization. If we blindly use the $\left\langle M\right\rangle$ defined before, we will get $\left\langle M\right\rangle = 0$. So how do we deal with this situation?
For a finite system, there is a finite probability that the transition between two peaks can occur. However, the "valley" between two peaks will become deeper and deeper when the size of the system $L$ increase. When $L \rightarrow \infty$, the transition probability tends to zero and those two configuration space should be separated. Note that the "flat mountain" in the figure at $L = 10$ will also become a very very sharp peak when $L \rightarrow \infty$.
One method discussed in the answers below is to consider the average of $M$ for separate configuration spaces. This seems reasonable for infinite system, but becomes a problem for finite system. Another problem raised here is that how to find each separated configuration spaces?
Thanks people try to give the answers to this question. In the following discussion, Kostya gives the typical treatment of spontaneous symmetry breaking. Marek discusses the ensemble average below and above the critical temperature. Greg Graviton gives an analogue for the real space spontaneous symmetry breaking.
If anyone can explain better to the problem of configuration space and how to take the average for Ising model, other spin models or general case, you are welcome to leave answer here. |
I know that this question has been submitted several times (especially see How are anyons possible?), even as a byproduct of other questions, since I did not find any completely satisfactory answers, here I submit another version of the question, stated into a very precise form using only
very elementary general assumptions of quantum physics. In particular I will not use any operator (indicated by $P$ in other versions) representing the swap of particles.
Assume to deal with a system of a couple of identical particles, each moving in $R^2$. Neglecting for the moment the fact that the particles are indistinguishable, we start form the Hilbert space $L^2(R^2)\otimes L^2(R^2)$, that is isomorphic to $L^2(R^2\times R^2)$. Now I divide the rest of my issue into several elementary steps.
(1) Every element $\psi \in L^2(R^2\times R^2)$ with $||\psi||=1$ defines a state of the system, where $|| \cdot||$ is the $L^2$ norm.
(2) Each element of the class $\{e^{i\alpha}\psi\:|\; \psi\}$ for $\psi \in L^2(R^2\times R^2)$ with $||\psi||=1$ defines the same state, and a state is such a set of vectors.
(3) Each $\psi$ as above can be seen as a complex valued function defined, up to zero (Lebesgue) measure sets, on $R^2\times R^2$.
(4) Now consider the "swapped state" defined (due to (1)) by $\psi' \in L^2(R^2\times R^2)$ by the function (up to a zero measure set):
$$\psi'(x,y) := \psi(y,x)\:,\quad (x,y) \in R^2\times R^2$$
(5) The physical meaning of the state represented by $\psi'$ is that of a state obtained form $\psi$ with the role of the two particles interchanged.
(6) As the particles are identical, the state represented by $\psi'$ must be the same as that represented by $\psi$.
(7) In view of (1) and (2) it must be: $$\psi' = e^{i a} \psi\quad \mbox{for some constant $a\in R$.}$$
Here physics stops. I will use only mathematics henceforth.
(8) In view of (3) one can equivalently re-write the identity above as
$$\psi(y,x) = e^{ia}\psi(x,y) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [1]\:.$$
(9) Since $(x,y)$ in [1] is every pair of points up to a zero-measure set, I am allowed to change their names obtaining
$$\psi(x,y) = e^{ia}\psi(y,x) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [2]$$
(Notice the zero measure set where the identity fails remains a zero measure set under the reflexion$(x,y) \mapsto (y,x)$, since it is an isometry of $R^4$ and Lebesgues' measure is invariant under isometries.)
(10) Since, again, [2] holds almost everywhere for every pair $(x,y)$, I am allowed to use again [1] in the right-hand side of [2] obtaining:
$$\psi(x,y) = e^{ia}e^{ia}\psi(x,y) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\:.$$
(This certainly holds true outside the union of the zero measure set $A$ where [1] fails and that obtained by reflexion $(x,y) \mapsto (y,x)$ of $A$ itself.)
(11) Conclusion:
$$[e^{2ia} -1] \psi(x,y)=0 \qquad\mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [3]$$
Since $||\psi|| \neq 0$, $\psi$ cannot vanish everywhere on $R^2\times R^2$.If $\psi(x_0,y_0) \neq 0$, $[e^{2ia} -1] \psi(x_0,y_0)=0$ implies $e^{2ia} =1 $ and so:
$$e^{ia} = \pm 1\:.$$
And thus, apparently, anyons are not permitted.
Where is the mistake?
ADDED REMARK. (10) is a completely mathematical result. Here is another way to obtain it. (8) can be written down as $\psi(a,b) = e^{ic} \psi(b,a)$ for some fixed $c \in R$ and all $(a,b) \in R^2 \times R^2$ (I disregard the issue of negligible sets). Choosing first $(a,b)=(x,y)$ and then $(a,b)=(y,x)$ we obtain resp. $\psi(x,y) = e^{ic} \psi(y,x)$ and $\psi(y,x) = e^{ic} \psi(x,y)$. They immediately produce [3] $\psi(x,y) = e^{i2c} \psi(x,y)$.
So the
physical argument (4)-(7) that we have permuted again the particles and thus a further new phase may appear does not apply here.
2nd ADDED REMARK. It is clear that as soon as one is allowed to write
$\psi(x,y) = \lambda \psi(y,x)$ for a
constant $\lambda\in U(1)$ and all $(x,y) \in R^2\times R^2$
the game is over:
$\lambda$ turns out to be $\pm 1$ and anyons are forbidden.This is just mathematics however. My guess for a way out is that the true configuration space is not $R^2\times R^2$ but some other space whose $R^2 \times R^2$ is the universal covering.
An idea (quite rough) could be the following. One should assume that particles are indistinguishable from scratch already defining the configuration space, that is something like $Q := R^2\times R^2/\sim$ where $(x',y')\sim (x,y)$ iff $x'=y$ and $y'=x$. Or perhaps subtracting the set $\{(z,z)\:|\: z \in R^2\}$ to $R^2\times R^2$ before taking the quotient to say that particles cannot stay at the same place. Assume the former case for the sake of simplicity. There is a (double?) covering map $\pi : R^2 \times R^2 \to Q$. My guess is the following. If one defines wavefunctions $\Psi$ on $R^2 \times R^2$, he automatically defines many-valued wavefunctions on $Q$. I mean $\psi:= \Psi \circ \pi^{-1}$. The problem of many values physically does not matter if the difference of the two values (assuming the covering is a double one) is just a phase and this could be written, in view of the identification $\sim$ used to construct $Q$ out of $R^2 \times R^2$: $$\psi(x,y)= e^{ia}\psi(y,x)\:.$$ Notice that the identity cannot be interpreted literally because $(x,y)$ and $(y,x)$ are the same point in $Q$, so my trick for proving $e^{ia}=\pm 1$ cannot be implemented. The situation is similar to that of $QM$ on $S^1$ inducing many-valued wavefunctions form its universal covering $R$. In that case one writes $\psi(\theta)= e^{ia}\psi(\theta + 2\pi)$.
3rd ADDED REMARK I think I solved the problem I posted focusing on the model of a couple of anyons discussed on p.225 of this paper matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp42116.pdf suggested by Trimok. The model is simply this one:$$\psi(x,y):= e^{i\alpha \theta(x,y)} \varphi(x,y)$$ where $\alpha \in R$ is a constant, $\varphi(x,y)= \varphi(y,x)$, $(x,y) \in R^2 \times R^2$ and $\theta(x,y)$ is the angle with respect to some fixed axis of the segment $xy$. One can pass to coordinates $(X,r)$, where $X$ describes the center of mass and $r:= y-x$. Swapping the particles means $r\to -r$. Without paying attention to mathematical details, one sees that, in fact: $$\psi(X,-r)= e^{i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{i \alpha \pi} \psi(y,x)\quad (A)$$for an anti clock wise rotation. (For clock wise rotations a sign $-$ appears in the phase, describing the other element of the braid group $Z_2$. Also notice that, for $\alpha \pi \neq 0, 2\pi$ the function vanishes for $r=0$, namely $x=y$, and this corresponds to the fact that we removed the set $C$ of coincidence points $x=y$ from the space of configurations.)
However a closer scrutiny shows that the situation is more complicated:The angle $\theta(r)$ is not well defined without fixing a reference axis where $\theta =0$. Afterwards one may assume, for instance, $\theta \in (0,2\pi)$,
otherwise $\psi$ must be considered multi-valued. With the choice $\theta(r) \in (0,2\pi)$, (A) does not hold everywhere. Consider an anti clockwise rotation of $r$. If $\theta(r) \in (0,\pi)$ then (A) holds in the form$$\psi(X,-r)= e^{+ i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{+ i \alpha \pi} \psi(y,x)\quad (A1)$$ but for $\theta(r) \in (\pi, 2\pi)$, and always for a anti clockwise rotation one finds$$\psi(X,-r)= e^{-i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{- i \alpha \pi} \psi(y,x)\quad (A2)\:.$$Different results arise with different conventions. In any cases it is evident that the phase due to the swap process is a function of $(x,y)$ (even if locally constant) and not a constant. This invalidate my "no-go proof", but also proves that the notion of anyon statistics is deeply different from the standard one based on the groups of permutations, where the phases due to the swap of particles is constant in $(x,y)$. As a consequence the swapped state is different from the initial one, differently form what happens for bosons or fermions and against the idea that anyons are indistinguishable particles. [Notice also that, in the considered model, swapping the initial pair of bosons means $\varphi(x,y) \to \varphi(y,x)= \varphi(x,y)$ that is $\psi(x,y)\to \psi(x,y)$. That is, swapping anyons does not mean swapping the associated bosons, and it is correct, as it is another physical operation on different physical subjects.]
Alternatively one may think of the anyon wavefunction $\psi(x,y)$ as a
multi-valued one, again differently from what I assumed in my "no-go proof" and differently from the standard assumptions in QM. This produces a truly constant phase in (A). However, it is not clear to me if, with this interpretation the swapped state of anyons is the same as the initial one, since I never seriously considered things like (if any) Hilbert spaces of multi-valued functions and I do not understand what happens to the ray-representation of states. This picture is physically convenient, however, since it leads to a tenable interpretation of (A) and the action of the braid group turns out to be explicit and natural.
Actually a last possibility appears. One could deal with (standard complex valued) wavefunctions defined on $(R^2 \times R^2 - C)/\sim$ as we know (see above, $C$ is the set of pairs $(x,y)$ with $x=y$) and we define the swap operation in terms of phases only (so that my "no-go proof" cannot be applied and the transformations do not change the states):
$$\psi([(x,y)]) \to e^{g i\alpha \pi}\psi([(x,y)])$$
where $g \in Z_2$. This can be extended to many particles passing to the braid group of many particles. Maybe it is convenient mathematically but is not very physically expressive.
In the model discussed in the paper I mentioned, it is however evident that, up to an unitary transformation, the Hilbert space of the theory is nothing but a standard bosonic Hilbert space, since the considered wavefunctions are obtained from those of that space by means of a unitary map associated with a singular gauge transformation,and just that singularity gives rise to all the interesting structure! However, in the initial bosonic system the singularity was pre-existent: the magnetic field was a sum of Dirac's delta. I do not know if it makes sense to think of anyons independently from their dynamics.And I do not know if this result is general. I guess that moving the singularity form the statistics to the interaction and This post imported from StackExchange Physics at 2014-04-11 15:20 (UCT), posted by SE-user V. Moretti
vice versa is just what happens in path integral formulation when moving the external phase to the internal action, see Tengen's answer. |
ProjectedDensityOfStates¶ class
ProjectedDensityOfStates(
configuration, kpoints=None, projections=None, energies=None, energy_zero_parameter=None, bands_above_fermi_level=None, spectrum_method=None)¶
Class for calculating the projected density of states for a configuration.
Parameters: configuration(
BulkConfiguration|
MoleculeConfiguration) – The configuration with an attached calculator for which to calculate the projected density of states.
kpoints– The k-points for which to calculate the projected density of states. Default:The k-point sampling used for the self-consistent calculation. projections(list of
Projection|
Projection|
ProjectionGenerator) – The projections used for the calculating the weights.
Default:
[Projection(spin=Spin.Up), Projection(spin=Spin.Down)].
energies(PhysicalQuantity of type energy) – The energies for which to calculate the projected density of states. The energies are relative to the zero of energy specified in
energy_zero_parameter.
Default:
numpy.arange(-2.0, 2.0, 0.01) * eV
energy_zero_parameter(
FermiLevel|
AbsoluteEnergy) – Specifies the choice for the energy zero.
Default:
FermiLevel
bands_above_fermi_level(int |
All) – The number of bands above the Fermi level per principal spin channel. Must be a non-negative integer.
Default:
All(All bands are included).
spectrum_method(
GaussianBroadening|
TetrahedronMethod|
Automatic) – The method to use for calculating the projected density of states. If
Automaticis set then the tetrahedron method is used if there are more than 10 k-points in any direction, else the Gaussian broadening method is used.
Default:
TetrahedronMethod
bandsAboveFermiLevel()¶
Returns: The number of bands above the Fermi level per principal spin channel. Return type: int
densityOfStates()¶
Returns: The full density of states as a vector of the length of the number of energies. Return type: PhysicalQuantity of type reciprocal energy
evaluate(
projection_index=None)¶
The projected density of states for a given projection.
Parameters: projection_index( int) – The index of the projection to query the projected density of states for. Negative indexing can be used such that e.g. projection_index=-1 will return the projected density of states for the last projection. Default:The projected density of states for each projection is returned. Returns: The projected density of states as a vector of the length of the number of energies, e. If
projection_indexis not specified, an array of shape (
p, e) is returned, where pis the number of projections. Return type: PhysicalQuantity of type reciprocal energy
fermiLevel(
spin=None)¶ Parameters: spin(
Spin.Up|
Spin.Down|
Spin.All) – The spin the Fermi level should be returned for. Must be either
Spin.Up,
Spin.Down, or
Spin.All. Only when the band structure is calculated with a fixed spin moment will the Fermi level depend on spin.
Default:
Spin.Up
Returns: The Fermi level in absolute energy. Return type: PhysicalQuantity of type energy
kpoints()¶
Returns: The k-point sampling used for calculating the projected density of states. Return type:
MonkhorstPackGrid|
RegularKpointGrid
metatext()¶
Returns: The metatext of the object or None if no metatext is present. Return type: str | unicode | None
nlprint(
stream=None)¶
Print a string containing an ASCII table useful for plotting the AnalysisSpin object.
Parameters: stream( python stream) – The stream the table should be written to. Default:
NLPrintLogger()
projections()¶
Returns: The projections used for the calculated projected density of states. Return type: list of class:~.Projection | class:~.Projection
setMetatext(
metatext)¶
Set a given metatext string on the object.
Parameters: metatext( str | unicode | None) – The metatext string that should be set. A value of “None” can be given to remove the current metatext. Usage Example¶
Calculate the Projected Density of States for bulk gallium arsenide, by projecting on each shell per each element
lattice = FaceCenteredCubic(5.6537*Angstrom)elements = [Gallium, Arsenic]fractional_coordinates = [[ 0. , 0. , 0. ], [ 0.25, 0.25, 0.25]]bulk_configuration = BulkConfiguration( bravais_lattice=lattice, elements=elements, fractional_coordinates=fractional_coordinates )# -------------------------------------------------------------# Calculator# -------------------------------------------------------------k_point_sampling = MonkhorstPackGrid(7, 7, 7)numerical_accuracy_parameters = NumericalAccuracyParameters( k_point_sampling=k_point_sampling, )calculator = LCAOCalculator( numerical_accuracy_parameters=numerical_accuracy_parameters, )bulk_configuration.setCalculator(calculator)bulk_configuration.update()# -------------------------------------------------------------# ProjectedDensityOfStates# -------------------------------------------------------------pdos = ProjectedDensityOfStates( configuration=bulk_configuration, projections=ProjectOnShellsByElement, kpoints=MonkhorstPackGrid(11, 11, 11), energies=numpy.arange(-4.0, 4.0, 0.01) * eV )
The same resuls can be obtained by defining explicitely the list of projections:
# -------------------------------------------------------------# ProjectedDensityOfStates# -------------------------------------------------------------projections = [Projection(l_quantum_numbers=[0], atoms=[Gallium]), Projection(l_quantum_numbers=[1], atoms=[Gallium]), Projection(l_quantum_numbers=[2], atoms=[Gallium]), Projection(l_quantum_numbers=[0], atoms=[Arsenic]), Projection(l_quantum_numbers=[1], atoms=[Arsenic]), Projection(l_quantum_numbers=[2], atoms=[Arsenic]),]pdos = ProjectedDensityOfStates( configuration=bulk_configuration, projections=projections, kpoints=MonkhorstPackGrid(11, 11, 11), energies=numpy.arange(-4.0, 4.0, 0.01) * eV)
More flexible projections can be achieved with algebraic operations, refer to the the documentation of Projection.
Notes¶
The ProjectedDensityOfStates is used to visualize the contribution of different orbitals to the density of states.
Recalling that the density of states can be written as
\[D(\epsilon) = \sum_{n} \delta \left(\epsilon - \epsilon _{n} \right)\]
where \(n\) includes all the quantum numbers of the system, we can define the Projected Density of States associated to a given projection M as
\[D _{M}(\epsilon) = \sum_{n} \delta \left(\epsilon - \epsilon _{n} \right) \langle \psi_{n} | \hat{\bf P}_M | \psi_{n } \rangle\]
\(\psi_{n }\) are the eigenstates and \(\hat{\bf P}_M\) is a projection operator defined according to the description contained in the documentation of Projection.
Note
The ProjectedDensityOfStates will always be positive (in units of \(\mathrm{eV}^{-1}\)) for projections on up/down spin, but might be both positive and negative (in units of \(\frac{\hbar}{2} \mathrm{eV}^{-1}\)) for projections involving the Pauli spin matrices. See the Usage Example in Projection for some more information. |
I'm having trouble in understanding Choquet-Bruhat's definition of a strongly causal spacetime ("GR and the Einstein Equations", OUP, sec. XII.10). Here she defines a strongly causal spacetime as a time-oriented Lorentz manifold $(M,g)$ such that
for any $x\in M$ and any neighbourhood $\Omega$ of $x$ there is a neighbourhood $U\subseteq \Omega$ such that $I_{x}^{+}\cap U$ is connected
Here $I_{x}^{+}$ is as usual the chronological future of $x$.
Now, aside from the fact that $U$ could be interpreted either as a neighbourhood of $x$ or as a generic neighbourhood, i.e. a generic open set, not necessarily containing $x$ (I'm leaning towards the first possibility, but different definitions of the strong causality condition make me wonder whether this is in fact so), it seems to me that even the Minkowski torus has this property, and as a Minkowski torus possesses closed causal curves, this cannot be possible. By Minkowski torus I mean the set $[-1,+1]\times [-1,+1]$ with metric $g=-dt^2+dx^2$ quotiented on its sides as in the usual construction for the one-torus, and orientation given by $\partial/\partial t$.
The proof of the validity of the property for the torus goes as follows. Consider the point $p=(0,0)$ and any other point $q=(t_{q},x_{q})$ with $|x_{q}|<1$. Construct a piecewise smooth curve by joining the timelike future-directed geodesic that goes from $p$ to $q'=(1,x_{q})$ and the timelike future-directed geodesic that goes from $q''=(-1,x_{q})$ to $q$. Modulo identifications that come from the quotient, $q'=q''$, and the curve is well defined and timelike. As for $q=(t_{q},±1)$, construct a piecewise-smooth curve by first going from $p$ to the side $t=1$ by means of a timelike future-directed geodesic, then going from the $t=-1$ side to the $x=1$ (or $x=-1$) side again through a timelike future-directed geodesic, then, if needed, going vertically up the $x=1$ (or $x=-1$) side. It is easy to see that by adjusting the inclination of the geodesics and modulo the identifications given by the quotient, any point on the $x=\pm 1$ side can be reached by means of a timelike future-directed piecewise smooth curve. This shows that $I_{p}^{+}$ is equal to the whole torus: in the Minkowski torus, every point is in the chronological future of $p$ (of course one can go from $p$ to $p$ too, by following a vertical future-directed closed geodesic). Hence, as $a\,\partial /\partial t + b\, \partial/\partial x$ with $a$ and $b$ arbitrary constants is Killing for the metric, the Minkowski torus is $\Bbb{R}^{2}$-homogeneous and the same must be true for any $p$ in the torus. It follows that $I_{p}^{+}\cap V=V$ for any $p$ and any subset $V$ in the torus, so that for any $p$ and any $\Omega$ the connected component of $\Omega$ containing $x$ is the connected neighbourhood we are looking for. I can't see why this proof should fail, but as I'm not an expert in the field there could well be something I'm missing.
I can't understand what is going wrong. I even though that there could be an error in the book (as I found many before). Elsewhere, I found definitions of strongly causal spacetimes in very different terms and I understood most of them (e.g. Penrose's definition in terms of causally convex neighbourhoods). Does anybody know how Choquet-Bruhat's definition is equivalent to the more common ones? |
Assume the limit $\lim_{x \to 0} f(1/x) = a$ exists. Let $\varepsilon > 0$. Now there exist $\delta > 0$ such that\begin{align}|x| < \delta & \implies |f(1/x) - a| < \varepsilon\,.\end{align}This is equivalent with$$|y| > 1/\delta \implies |f(y) - a | < \varepsilon\,.$$Thus for all $y \in (1/\delta, \infty)$ we have $f(y) \in (a - \varepsilon, a + \varepsilon)$.
Because $f(y-l) = f(y)$ this implies that for all $y \in (1/\delta -l, \infty)$ we have $f(y) \in (a- \varepsilon, a + \varepsilon)$. Continuing this shows that $f(y) \in (a - \varepsilon, a + \varepsilon)$ for all $y \in \mathbb{R}$.
Since $\varepsilon$ was arbitrary, we must have $f(y) = a$ for all $y$.
So if the limit exists, $f$ is constant. Thus if $f$ is not constant, the limit does not exist. |
It is well known that a parametric form of the parabola $y^2=4ax$ is $(at^2, 2at)$.
What are possible parametric forms of the general parabola $$(Ax+Cy)^2+Dx+Ey+F=0$$ ?
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It is well known that a parametric form of the parabola $y^2=4ax$ is $(at^2, 2at)$.
What are possible parametric forms of the general parabola $$(Ax+Cy)^2+Dx+Ey+F=0$$ ?
Bézier curves are a convenient way to produce parameterizations of parabolas: a quadratic Bézier is a (part of a) parabola. If $P_0$ and $P_2$ are points on the parabola and $P_1$ the intersection of the tangents at those points, the quadratic Bézier curve they define is given by $$\phi:t\mapsto(1-t)^2P_0+2t(1-t)P_1+t^2P_2.\tag{1}$$ (The parameter $t$ is usually taken to range from $0$ to $1$ for a Bézier patch.)
We can reproduce your parametrization by taking the vertex $P_0(0,0)$ and an end of the latus rectum $P_2(p,2p)$ as the points on the parabola. (Here I use the conventional name $p$ for this parameter instead of the $a$ in the question.) The tangent at the end of the latus rectum meets the parabola’s axis at a 45° angle, so our third control point will be $P_1(0,p)$. Plugging these into (1) we get $$(1-t)^2(0,0)+2t(1-t)(0,p)+t^2(p,2p)=(pt^2,2pt),$$ as required. As described here, parametrization of a parabola by a pair of quadratic polynomials has a nice symmetry about the vertex. Choosing the vertex as our first control point makes this symmetry quite simple.
To obtain the corresponding parameterization for a general parabola, you can either rotate and translate these three points to match the position and orientation of the given parabola, or compute them from other information that you have about the parabola. For example, if we have a parabola with vertex $P_0(x_0,y_0)$, focal length $p$ and axis direction $\theta$, we will have $P_1=P_0+(-p\sin\theta,p\cos\theta)$ and $P_2=P_0+(p\cos\theta-2p\sin\theta,2p\cos\theta+p\sin\theta)$, which gives the parameterization $$\begin{align}x&=x_0-2pt\sin\theta+pt^2\cos\theta \\ y&= y_0+2pt\cos\theta+pt^2\sin\theta.\end{align}$$
I’ll leave working out this parameterization for the general-form equation to you. As a hint, remember that for the parabola $y=ax^2+bx+c$, $p={1\over4a}$ and that a parabola’s vertex is halfway between its focus and directrix.
This solution to my other question on the axis of symmetry of a general parabola gives the following:
Axis of symmetry: $$Ax+Cy+t^*=0$$ Tanget at vertex: $$(D-2At^*)x+(E-2Ct^*)y+F-{t^*}^2=0$$ where $t^* \left(=\frac {AD+CE}{2(A^2+C^2)}\right)$ is chosen for both lines to be perpendicular.
Solving for the intersection of the two lines gives the coordinates of the vertex as $$\left(-\frac{C{t^*}^2-Et^*+CF}{CD-AE}, \frac{A{t^*}^2-Dt^*+AF}{CD-AE}\right)$$
Replacing $t^*$ with the general parameter $t$ gives a parametric form for the general parabola $(Ax+Cy)^2+Dx+Ey+F=0$ as
$$\color{red}{\left(-\frac{Ct^2-Et+CF}{CD-AE}, \frac{At^2-Dt+AF}{CD-AE}\right)}$$ which is the same as $$\color{red}{\left(\frac{Ct^2-Et+CF}{AE-CD}, -\frac{At^2-Dt+AF}{AE-CD}\right)}$$
For the special case where $A=C$, $$t^*=\frac {D+E}{4A}$$ Axis of Symmetry: $$Ax+Ay+\frac {D+E}{4A}=0$$ or $$x+y+\frac {D+E}{4A^2}=0$$ Vertex: $$\left(\frac{{t^*}^2-\frac EA t^*+F}{E-D}, -\frac{{t^* }^2-\frac DA t^*+F}{E-D}\right)$$ |
In mathematics, logarithmic functions is an inverse function to exponentiation. The logarithmic function is defined as
For x > 0 , a > 0, and a\(\neq\)1,
y= log
a x if and only if x = a y
Then the function is given by
f(x) = log
a x
The base of the logarithm is a. This can be read it as log base a of x. The most 2 common bases used in logarithmic functions are base 10 and base e.
Common Logarithmic Function: The logarithmic function with base 10 is called the common logarithmic function and it is denoted by log 10or simply log.
f(x) = log
10 x Natural Logarithmic Function :The logarithmic function to the base e is called the natural logarithmic function and it is denoted by log e .
f(x) = log
e x Logarithmic Functions Properties
Logarithmic Functions have some of the properties that allows you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below.
Product Rule: log bMN = log bM + log bN
Multiply two numbers with the same base, then add the exponents.
Example : log 30 + log 2 = log 60
Quotient Rule :log bM/N = log bM – log bN
Divide two numbers with the same base, subtract the exponents.
Example : log
8 56 – log 8 7 = log 8(56/7)=log 88 = 1 Power Rule :Raise an exponential expression to a power and multiply the exponents.
Log
b M p = P log b M
Example : log 100
3 = 3. Log 100 = 3 x 2 = 6 Zero Exponent Rule :log a1 = 0. Change of Base Rule :log b(x) = ln x / ln b or log b(x) = log 10x / log 10b Log bb = 1 Example : log 1010 = 1 Log bb x= x Example : log 1010 x= x \(b^{\log _{b}x}=x\) . Substitute y= log bx , it becomes b y= x
There are also some of the logarithmic function with fractions. It has a useful property to find the log of a fraction by applying the identities
ln(ab)= ln(a)+ln(b) ln(a x) = x ln (a)
We also can have logarithmic function with fractional base.
Consider an example,\(3\log _{\frac{4}{9}}\sqrt[4]{\frac{27}{8}}=\frac{3}{4}\log _{\frac{4}{9}}\frac{27}{8}\)
By the definition, log
a b = y becomes a y = b
(4/9)
y = 27/8
(2
2/3 2) y = 3 3 / 2 3
(⅔)
2y = (3/2) 3 Sample Example
Here you are provided with some logarithmic functions example.
Question 1 :
Use the properties of logarithms to write as a single logarithm for the given equation: 5 log
9 x + 7 log 9 y – 3 log 9 z Solution :
By using the power rule , Log
b M p = P log b M, we can write the given equation as
5 log
9 x + 7 log 9 y – 3 log 9 z = log 9 x 5 + log 9 y 7 – log 9 z 3
From product rule, log
b MN = log b M + log b N
5 log
9 x + 7 log 9 y – 3 log 9 z = log 9 x 5y 7 – log 9 z 3
From Quotient rule, log
b M/N = log b M – log b N
5 log
9 x + 7 log 9 y – 3 log 9 z = log 9 (x 5y 7 / z 3 )
Therefore, the single logarithm is 5 log
9 x + 7 log 9 y – 3 log 9 z = log 9 (x 5y 7 / z 3 ) Question 2 :
Use the properties of logarithms to write as a single logarithm for the given equation: 1/2 log
2 x – 8 log 2 y – 5 log 2 z Solution :
By using the power rule , Log
b M p = P log b M, we can write the given equation as
1/2 log
2 x – 8 log 2 y – 5 log 2 z = log 2 x 1/2 – log 2 y 8 – log 2 z 5
From product rule, log
b MN = log b M + log b N
Take minus ‘- ‘ as common
1/2 log
2 x – 8 log 2 y – 5 log 2 z = log 2 x 1/2 – log 2 y 8z 5
From Quotient rule, log
b M/N = log b M – log b N
1/2 log
2 x – 8 log 2 y – 5 log 2 z = log 2 (x 1/2 / y 8z 5 )
The solution is
1/2 log
2 x – 8 log 2 y – 5 log 2 z = \(\log _{2}\left ( \frac{\sqrt{x}}{y^{8}z^{5}} \right )\)
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eISSN:
2163-2480 Evolution Equations & Control Theory
December 2015 , Volume 4 , Issue 4
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Abstract:
In this paper, an abstract nonsimple thermoelastic problem involving higher order gradients of displacement is considered with Dirichlet boundary conditions. We prove that the linear operator of the proposed system generates a strongly continuous semigroup which decays exponentially to zero. The optimal decay rate is determined explicitly by the physical parameters of the problem. Then we show the approximate controllability of the considered problem.
Abstract:
We give relatively simple sufficient conditions on a Fourier multiplier so that it maps functions with the Hölder property with respect to a part of the variables to functions with the Hölder property with respect to all variables. By using these these sufficient conditions we prove solvability in Hölder classes of the initial-boundary value problems for the linearized Cahn-Hilliard equation with dynamic boundary conditions of two types. In addition, Schauders estimates are derived for the solutions corresponding to the problem under study.
Abstract:
In this paper, we undertake a comprehensive study for the Schrödinger-Hartree equation \begin{equation*} iu_t +\Delta u+ \lambda (I_\alpha \ast |u|^{p})|u|^{p-2}u=0, \end{equation*} where $I_\alpha$ is the Riesz potential. Firstly, we address questions related to local and global well-posedness, finite time blow-up. Secondly, we derive the best constant of a Gagliardo-Nirenberg type inequality. Thirdly, the mass concentration is established for all the blow-up solutions in the $L^2$-critical case. Finally, the dynamics of the blow-up solutions with critical mass is in detail investigated in terms of the ground state.
Abstract:
The aim of this paper is to highlight some recent developments and outcomes in the mathematical analysis of partial differential equations describing nonlinear sound propagation. Here the emphasis lies on well-posedness and decay results, first of all for the classical models of nonlinear acoustics, later on also for some higher order models. Besides quoting results, we also try to give an idea on their derivation by showning some of the crucial energy estimates. A section is devoted to optimization problems arising in the practical use of high intensity ultrasound.
While this review puts a certain focus on results obtained in the context of the mentioned FWF project, we also provide some important additional references (although certainly not all of them) for interesting further reading.
Abstract:
General nonlinear time-varying differential systems with delay are considered. Several new explicit criteria for exponential stability are given. A discussion of the obtained results and two illustrative examples are presented.
Abstract:
In this paper, we study the existence and controllability for fractional evolution inclusions in Banach spaces. We use a new approach to obtain the existence of mild solutions and controllability results, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Finally, two examples are given to illustrate our theoretical results.
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Polynomial equations are one of the major concepts of Mathematics, where the relation between numbers and variables are explained in a pattern. In Maths, we have studied a variety of equations formed with algebraic expressions. When we talk about polynomials, it is also a form of the algebraic equation.
What is a Polynomial Equation?
The equations formed with variables, exponents and coefficients are called as polynomial equations. It can have a number of different exponents, where the higher one is called the degree of the exponent. We can solve polynomials by factoring them in terms of degree and variables present in the equation.
A polynomial function is an equation which consists of a single independent variable, where the variable can occur in the equation more than one time with different degree of the exponent. Students will also learn here to solve these polynomial functions. The graph of a polynomial function can also be drawn using turning points, intercepts, end behaviour and the Intermediate Value Theorem.
Example of polynomial function: f(x) = 3x 2 + 5x + 19 Read More: Polynomial Functions Polynomial Equations Formula
Usually, the polynomial equation is expressed in the form of a
n(x n). Here a is the coefficient, x is the variable and n is the exponent. As we have already discussed in the introduction part, the value of exponent should always be a positive integer.
If we expand the polynomial equation we get;
F(x) = a nxn + an-1xn-1 + an-2xn-2 + …….. + a1x +a0
This is the general expression and is also a polynomial equation solver. It can also be expressed as;
F(x) = \(\sum_{k=0}^{n}a_{k}n^k\)
Example of a polynomial equation is: 2x
2 + 3x + 1 = 0, where 2x 2 + 3x + 1 is basically a polynomial expression which has been set equal to zero, to form a polynomial equation. Types of Polynomial Equation
Polynomial equation is basically of four types;
Monomial Equations Binomial Equations Trinomial or Cubic Equations Quadratic Polynomial Equations Monomial Equation:
An equation which has only one variable term is called a Monomial equation. This is also called a
linear equation. It can be expressed in the algebraic form of; ax + b = 0
For Example:
4x+1=0 5y=2 8z-3=0 Binomial Equations:
An equation which has only two variable terms and is followed by one variable term is called a Monomial equation. This is also in the form of the
quadratic equation. It can be expressed in the algebraic form of; ax 2 + bx + c = 0
For Example:
2x 2+ 5x + 20 = 0 3x 2– 4x + 12 = 0 Trinomial Equations:
An equation which has only three variable terms and is followed by two variable and one variable term is called a Monomial equation. This is also called a
cubic equation. In other words, a polynomial equation which has a degree of three is called a cubic polynomial equation or trinomial polynomial equation.
Since the power of the variable is maximum up to 3, therefore, we get three values for a variable, say x.
It is expressed as;
a 0 x3 + a1x2 + a2x + a3 = 0, a ≠ 0
or
ax 3 + bx2 + cx + d = 0 For Example: 3x 3+ 12x 2– 8x – 10 = 0 9x 3+ 5x 2– 4x – 2 = 0
To get the value of x, we generally use, trial and error method, in which we start putting the value of x randomly, to get the given expression as 0. If for both sides of the polynomial equation, we get a 0 ,then the value of x is considered as one of the roots. After then we can find the other two values of x.
Let us take an example: Problem: y 3 – y 2 + y – 1 = 0 is a cubic polynomial equation. Find the roots of it. Solution: y 3 – y 2 + y – 1 = 0 is the given equation.
By trial and error method, start putting the value of x.
If y = -1, then,
(-1)
3 – (-1) 2 -1 +1 = 0
-1 + 1 -1 + 1 = 0
-4 ≠ 0
If y = 1, then,
1
3 – 1 2 + 1 – 1 = 0
0 = 0
Therefore, one of the roots is 1.
y = 1
(y – 1) is one of the factors.
Now dividing the given equation by x+1 on both sides, we get,
y
3 – y 2 + y – 1 = 0
Dividing both sides by y-1, we get
(y-1) (y
2 + 1) = 0
Therefore, the roots are y = 1 which is a real number and y
2 + 1 gives complex numbers or imaginary numbers. Quadratic Polynomial Equation
A polynomial equation which has a degree as two is called a quadratic equation. The expression for the quadratic equation is:
ax 2 + bx + c = 0 ; a ≠ 0
Here, a,b, and c are real numbers. The roots of quadratic equations give two values for the variable x.
x = \(\frac{-b\pm \sqrt{b^2-4ac}}{2a}\)
Also Check: Polynomial Equation Solver
Related Topics Quadratic Formula & Quadratic Polynomial Multiplying Polynomials Polynomial Formula Factorization Of Polynomials Frequently Asked Questions What are Quartic Polynomials?
Polynomials of degree 4 are known as quartic polynomials. A quartic polynomial can have 0 to 4 roots.
How Polynomial Equations are Represented?
A polynomial equation is represented of the form:
F(x) = a nxn + an-1xn-1 + an-2xn-2 + …….. + a1x + a0 Can a Polynomial have no Real Zeroes?
Yes, a polynomial can have no real zeroes. An example of a polynomial having no zero is x2 – 2x + 5.
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I want to prove the following:
Let $(M,d)$ be a metric space. Let $A\subseteq V\subseteq M$.
1) $A$ is open in $V \Leftrightarrow A = C\cap V$ (for a certain open $C$ in $M$)
2) $A$ is closed in $V \Leftrightarrow A = C\cap V$ (for a certain closed $C$ in $M$)
Questions:
Could someone check the proof?
'
for a certain open $C$ in $\color{Blue}{M}$.'
Would this proof also work for a more specific choice of $C$? Like for a certain open $C$ in $\color{blue}{V}$. I don't really see the added value of choosing $M$ over $V$.
Could some give me some pointers on how to prove $2, \Rightarrow$? Proof 1)
$\Leftarrow$: Choose $a\in A$.
$$\begin{array}{rl} & a \in A = C\cap V\\ \Rightarrow & a \in C\\ \Rightarrow & (\exists r > 0)(B_M(a,r)\subseteq C)\\ \Rightarrow & (\exists r > 0)(B_M(a,r)\cap V \subseteq C\cap V)\\ \Rightarrow & (\exists r> 0) (B_V(a,r)\subseteq A \end{array}$$
$\Rightarrow$: Choose $a\in A$.
$$\begin{array}{rl} \Rightarrow & (\exists r_a >0)(B_V(a,r_a) \subseteq A) \end{array}$$
Consider all $a\in A$ then:
$$\begin{array}{rl} & A = \bigcup_{a\in A} B_V(a,r_a)\\ \Rightarrow & A = \bigcup_{a\in A} \left[ V\cap B_M(a,r_a)\right]\\ \Rightarrow & A = V\cap\left[ \bigcup_{a\in A} B_M(a,r_a)\right] \end{array}$$
Let $$\left[ \bigcup_{a\in A} B_M(a,r_a)\right] = C$$ which is open as a union of open sets.
Proof 2)
$\Leftarrow$:
$$\begin{array}{rrl} & V\setminus A &= V\setminus(C\cap V)\\ \Rightarrow & & = (V\setminus C)\cup (V\setminus V)\\ \Rightarrow && = V\setminus C \end{array}$$
Since $C$ is closed then $V\setminus C$ is open and so is $V\setminus A$. Then $A$ is closed in $V$.
$\Rightarrow$: How? |
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