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ISSN: 1531-3492 eISSN: 1553-524X All Issues Discrete & Continuous Dynamical Systems - B July 2010 , Volume 14 , Issue 1 Select all articles Export/Reference: Abstract: The purpose of this paper is to study the relations between different concepts of dispersive solution for the Vlasov-Poisson system in the gravitational case. Moreover we give necessary conditions for the existence of partially and totally dispersive solutions and a sufficient condition for the occurence of statistical dispersion. These conditions take the form of inequalities involving the energy, the mass and the momentum of the solution. Examples of dispersive and non-dispersive solutions-steady states, periodic solutions and virialized solutions-are also considered. Abstract: This paper introduces a simplified dynamical systems framework for the study of the mechanisms behind the growth of cooperative learning in large communities. We begin from the simplifying assumption that individual-based learning focuses on increasing the individual's "fitness" while collaborative learning may result in the increase of the group's fitness. It is not the objective of this paper to decide which form of learning is more effective but rather to identify what types of social communities of learners can be constructed via collaborative learning. The potential value of our simplified framework is inspired by the tension observed between the theories of intellectual development (individual to collective or vice versa) identified with the views of Piaget and Vygotsky. Here they are mediated by concepts and ideas from the fields of epidemiology and evolutionary biology. The community is generated from sequences of successful "contacts'' between various types of individuals, which generate multiple nonlinearities in the corresponding differential equations that form the model. A bifurcation analysis of the model provides an explanation for the impact of individual learning on community intellectual development, and for the resilience of communities constructed via multilevel epidemiological contact processes, which can survive even under conditions that would not allow them to arise. This simple cooperative framework thus addresses the generalized belief that sharp community thresholds characterize separate learning cultures. Finally, we provide an example of an application of the model. The example is autobiographical as we are members of the population in this "experiment". Abstract: The purpose of this paper is to develop a numerical procedure for the determination of frequencies and amplitudes of a quasi--periodic function, starting from equally-spaced samples of it on a finite time interval. It is based on a collocation method in frequency domain. Strategies for the choice of the collocation harmonics are discussed, in order to ensure good conditioning of the resulting system of equations. The accuracy and robustness of the procedure is checked with several examples. The paper is ended with two applications of its use as a dynamical indicator. The theoretical support for the method presented here is given in a companion paper [21]. Abstract: In a previous paper [6], a numerical procedure for the Fourier analysis of quasi-periodic functions was developed, allowing for an accurate determination of frequencies and amplitudes from equally-spaced samples of the input function on a finite time interval. This paper is devoted to a complete error analysis of that procedure, from which computable bounds are deduced. These bounds are verified and further discussed in examples. Abstract: In this work we consider the existence of traveling plane wave solutions of systems of delayed lattice differential equations in competitive Lotka-Volterra type. Employing iterative method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a speed, c , and show the existence of traveling plane wave solutions connecting two different equilibria when the wave speeds are large than c . Abstract: Stability and dynamic bifurcation in the ac-driven complex Ginzburg-Landau (GL) equation with periodic boundary conditions and even constraint are investigated using central manifold reduction procedure and attractor bifurcation theory. The results show that the bifurcation into an attractor near a small-amplitude limit cycle takes place on a two dimensional central manifold, as bifurcation parameter crosses a critical value. Furthermore, the component of the bifurcated attractor is analytically described for the non-autonomous system. Abstract: The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In [18, 21] the method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets. Abstract: Recently a discrete-time prey-predator model with Holling type II was discussed for its bifurcations so as to show its complicated dynamical properties. Simulation illustrated the occurrence of invariant cycles. In this paper we first clarify the parametric conditions of non-hyperbolicity, correcting a known result. Then we apply the center manifold reduction and the method of normal forms to completely discuss bifurcations of codimension 1. We give bifurcation curves analytically for transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation separately, showing bifurcation phenomena not indicated in the previous work for the system. Abstract: In this article, we study the stability of weak solutions to the stochastic two dimensional (2D) primitive equations (PEs) with multiplicative noise. In particular, we prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. Abstract: In this work, we generalize the idea of Ginzburg-Landau approximation to study the existence and asymptotic behaviors of global weak solutions to the one dimensional periodical fractional Landau-Lifshitz equation modeling the soft micromagnetic materials. We apply the Galerkin method to get an approximate solution and, to get the convergence of the nonlinear terms we introduce the commutator structure and take advantage of special structures of the equation. Abstract: The paper presents an SEIQHRS model for evaluating the combined impact of quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms) on the spread of a communicable disease. Rigorous analysis of the model, which takes the form of a deterministic system of nonlinear differential equations with standard incidence, reveal that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. Using a Krasnoselskii sub-linearity trick, it is shown that the unique endemic equilibrium is locally-asymptotically stable for a special case. A nonlinear Lyapunov function of Volterra type is used, in conjunction with LaSalle Invariance Principle, to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations, using a reasonable set of parameter values (consistent with the SARS outbreaks of 2003), show that the level of transmission by individuals isolated in hospitals play an important role in determining the impact of the two control measures (the use of quarantine and isolation could offer a detrimental population-level impact if isolation-related transmission is high enough). Abstract: We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equilibria of the model lie on an invariant coordinate axis, both the saddle-node and the transcritical bifurcations are of codimension one. Their interaction can be associated with either a single or a double zero eigenvalue. We show that in the former case, the local bifurcation diagram is given by a nonversal unfolding of the cusp bifurcation whereas in the latter case it is a nonversal unfolding of a degenerate Bogdanov-Takens bifurcation. We present a simple model for each of the two cases to illustrate the possible unfoldings. We analyse the consequences of the generic phase portraits for the Lotka-Volterra system. Abstract: Backward stochastic Volterra integral equations (BSVIEs in short) are studied. We introduce the notion of adapted symmetrical solutions (S-solutions in short), which are different from the M-solutions introduced by Yong [16]. We also give some new results for them. At last a class of dynamic coherent risk measures were derived via certain BSVIEs. Abstract: In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, this paper perturbs the Lotka--Volterra system $\dot{x}(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))$ into the corresponding stochastic system $dx(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))[(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))dt+\beta dw(t)].$ This paper obtains one condition under which the above stochastic system has a global almost surely positive solution and gives the asymptotic pathwise estimation of this solution. This paper also shows that when the noise is sufficiently large, the solution of this stochastic system will converge to zero with probability one. This reveals that the sufficiently large noise may make the population extinct. Abstract: The dynamics of Leslie-Gower predator-prey models with constant harvesting rates are investigated. The ranges of the parameters involved in the systems are given under which the equilibria of the systems are positive. The phase portraits near these positive equilibria are studied. It is proved that the positive equilibria on the $x$-axis are saddle-nodes, saddles or unstable nodes depending on the choices of the parameters involved while the interior positive equilibria in the first quadrant are saddles, stable or unstable nodes, foci, centers, saddle-nodes or cusps. It is shown that there are two saddle-node bifurcations and by computing the Liapunov numbers and determining its signs, the supercritical or subcritical Hopf bifurcations and limit cycles for the weak centers are obtained. Readers Authors Editors Referees Librarians More Email Alert Add your name and e-mail address to receive news of forthcoming issues of this journal: [Back to Top]
Difference between revisions of "Middle attic" Line 7: Line 7: * [[correct]] cardinals, [[reflecting \| $V_\delta\prec V$]] and the [[reflecting#Feferman theory \| Feferman theory]] * [[correct]] cardinals, [[reflecting \| $V_\delta\prec V$]] and the [[reflecting#Feferman theory \| Feferman theory]] * [[reflecting#$\Sigma_2$ correct cardinals \| $\Sigma_2$ correct]] and [[reflecting \| $\Sigma_n$-correct]] cardinals * [[reflecting#$\Sigma_2$ correct cardinals \| $\Sigma_2$ correct]] and [[reflecting \| $\Sigma_n$-correct]] cardinals − * [[0-extendible]] cardinal + * [[0-extendible]] cardinal * [[extendible#$\Sigma_n$-extendible cardinals \| $\Sigma_n$-extendible ]] cardinal * [[extendible#$\Sigma_n$-extendible cardinals \| $\Sigma_n$-extendible ]] cardinal * [[beth#beth_fixed_point \| $\beth$-fixed point]] * [[beth#beth_fixed_point \| $\beth$-fixed point]] Revision as of 11:21, 23 July 2013 Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory. into the upper attic correct cardinals, $V_\delta\prec V$ and the Feferman theory [[reflecting#$\Sigma_2$ correct cardinals \| $\Sigma_2$ correct]] and $\Sigma_n$-correct cardinals 0-extendible cardinal [[extendible#$\Sigma_n$-extendible cardinals \| $\Sigma_n$-extendible ]] cardinal $\beth$-fixed point the beth numbers and the $\beth_\alpha$ hierarchy $\beth_\omega$ and the strong limit cardinals $\Theta$ the continuum cardinal characteristics of the continuum the descriptive set-theoretic cardinals $\aleph$-fixed point the aleph numbers and the $\aleph_\alpha$ hierarchy $\aleph_\omega$ and singular cardinals $\aleph_2$, the second uncountable cardinal uncountable, regular and successor cardinals $\aleph_1$, the first uncountable cardinal cardinals, infinite cardinals $\aleph_0$ and the rest of the lower attic
Search Now showing items 1-10 of 33 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 ...
SCM Repository View of /branches/vis12/test/unicode-cheatsheet.diderot Revision File size: 782 byte(s) 1927- ( download) ( annotate) Sat Jun 23 18:09:16 2012 UTC(7 years, 3 months ago) by jhr File size: 782 byte(s) converting to use "image" instead of "load" for image nrrd loading /* useful unicode characters for Diderot ⊛ convolution, as in field#2(3)[] F = bspln3 ⊛ image("img.nrrd"); LaTeX: \circledast is probably typical, but \varoast (with \usepackage{stmaryrd}) is slightly more legible × cross product, as in vec3 camU = normalize(camN × camUp); LaTeX: \times π Pi, as in real rad = degreesπ/360.0; LaTeX: \pi ∇ Del, as in vec3 grad = ∇F(pos); LaTeX: \nabla • dot product, as in real ld = norm • lightDir; LaTeX: \bullet, although \cdot more typical for dot products ⊗ tensor product, as in tensor[3,3] Proj = identity[3] - norm⊗norm LaTeX: \otimes ∞ Infinity, as in output real val = -∞; LaTeX: \infty / strand blah (int i) { output real out = 0.0; update { stabilize; } } initially [ blah(i) \| i in 0..0 ]; [email protected] ViewVC Help Powered by ViewVC 1.0.0
Advanced Monitoring Strategy The algorithm incorporated in OptiLayer takes various criteria into account: The strategy is available at: Results --> Monitor --> Strategy button --> Strategy 4 \[ A=V_\max-V_\min \] \[ S_{in}= \frac{V_\max-V_{in}}{A}\cdot 100\%, \]if the first extremum is maximum;\[ S_{in}= \frac{V_{in}-V_\min}{A}\cdot 100\%, \]otherwise. \[ S_{fin}= \frac{V_\max-V_{fin}}{A}\cdot 100\%, \]if the last extremum is maximum; \[ S_{fin}= \frac{V_{fin}-V_\min}{A}\cdot 100\%, \]otherwise, where $V_{in}$ is the signal level at the start of the layer deposition, $V_{fin}$ is the termination level. In the case of a thin layer, having no extrema of the monitoring signal inside, OptiLayer considers continuously increasing thickness of the monitored layer above its nominal thickness until the next two extrema appears on the monitoring curve. These two virtual extrema are considered as the first and the last one in the conditions above. Evidently, swing values close to 0% and 100% mean that signal is close to extremum values. Initial swing for the first layer is always equal to zero. Another important parameter is the difference between the termination level $V_{fin}$ and the next signal extremum $E_{next}$: \[\Delta=\|V_{fin}-E_{next}\|\] OptiLayer chooses monitoring wavelengths in order to satisfy the following five monitoring conditions: The values $a_1, a_2, b_1, b_2, \epsilon, \delta$ are the parameters of the algorithm. The conditions are connected with the requirement that termination levels should be located at signal slopes with enough steepness. For example, reasonable values of these parameters are $a_1=a_2=15\%,$ $b_1=b_2=85\%$, $\epsilon=4\%$, $\delta=4\%$. Generally, it may be not possible to find a sequence of wavelengths so that all conditions are satisfied simultaneously for all coating layers. Our algorithm chooses monitoring wavelength(s) so that the conditions are satisfied as close as possible. It may happen that some conditions are more important than the others. For such cases, condition weights are introduced. The weights allow you to adjust relative importance of different conditions. Bad Monitoring control allows you to visualize layers having monitoring signal with bad quality for reliable monitoring purposes. Illustrating example: a model coating consisting of two layers on a Suprasil substrate; layer refractive indices equal to 2.35 and 1.45, layer physical thicknesses equal to 150 nm and 100 nm. The corresponding signal for the case when monitoring wavelength is 500 nm is shown in Fig. 1 (upper panel). The corresponding parameters for Layer 1 are: \[ A=93.21\%-64.63\%=28.58\%>\epsilon=4\%,\] \[ S_{in}=0, \; \mbox{(excluded)}, \] \[ S_{fin}=\frac{93.21\%-66.21\%}{28.58\%}\cdot 100\%=94.47\%>a_2=85\%,\] \[\Delta=66.21\%-64.63\%=1.58\%<\delta=4\%.\] The parameters for Layer 2 are: \[ A=89.75\%-65.95\%=23.8\%>\epsilon=4\%,\] \[ S_{in}=\frac{66.23\%-65.95\%}{23.8\%}\cdot 100\%=1.2\%<a_1=15\%, \] \[ S_{fin}=\frac{89.75\%-89.25\%}{23.8\%}\cdot 100\%=2.1\%<b_1=15\%,\] \[\Delta=89.25\%-65.95\%=23.3\%>\delta=4\%.\] It is seen that some conditions are are satisfied, and others are violated; termination levels are very close to signal extrema. Application of our algorithm gives the monitoring wavelength of 588 nm. The corresponding monitoring signal is shown in Fig. 1 (lower pane). The parameters are now for Layer 1 as follows: \[ A=93.28\%-64.57\%=28.71\%>\epsilon=4\%,\] \[ S_{in}=0, \; \mbox{(excluded)}, \] \[ b_1\le S_{fin}=\frac{93.28\%-80.95\%}{28.71\%}\cdot 100\%=42.9\%\le b_2=85\%,\] \[\Delta=80.95\%-64.57\%=16.38\%>\delta=4\%;\] and for Layer 2: \[ A=95.56\%-78.27\%=17.29\%>\epsilon=4\%,\] \[ a_1=15\%\le S_{in}=\frac{80.95\%-78.27\%}{17.29\%}\cdot 100\%=15.5\%\le a_2=85\%, \] \[ b_1=15\%\le S_{fin}=\frac{91.52\%-78.27\%}{17.29\%}\cdot 100\%=76.6\%\le b_2=85\%,\] \[\Delta=95.56\%-91.52\%=4.04\%>\delta=4\%.\] All conditions for both layers are satisfied. A monitoring wavelength the wavelength of 700 nm is located in the middle of the antireflection spectral range.The corresponding signal is depicted in Fig. 2 (upper panel). The advanced algorithm proposes the wavelength of 621 nm (corresponding monitoring signal is plotted in the lower panel of Fig. 2. In order to compare the monitoring wavelengths of 700 nm and 621 nm from the practical point of view we perform a series of computational manufacturing experiments in order to estimate production yields. Production yields for the wavelengths of 700 nm and 621 nm are estimated as 0.2% and 98.6%, respectively. See the details in Monochromatic Monitoring Simulation.
NonEquilibriumHeatExchange¶ class NonEquilibriumHeatExchange( configuration, heating_power, heat_source, heat_sink, exchange_interval=1, update_profile_interval=0, profile_resolution=None)¶ A class that implements a heat flow by constant heating power exchange technique via a hook function. Parameters: configuration( BulkConfiguration) – The initial configuration on which the heat flow simulation will be performed. heating_power(PhysicalQuantity of type energy/time) – The thermal energy that is transferred from the cold to the hot region per time. heat_source( str \| list of ints) – The tag or list of indices defining the group of atoms in the hot region. heat_sink( str \| list of ints) – The tag or list of indices defining the group of atoms in the cold region. exchange_interval( int) – The interval to perform energy exchange by rescaling the velocities. Default:1 (every step). update_profile_interval( int) – The interval at which a measurement of the temperature profile is performed and added to the average profile. Set to zero to disable on-the-fly measurement. Default:0 profile_resolution(PhysicalQuantity of type length) – The spatial resolution for the on-the-fly calculation of the temperature profile. Default:2.0Angstrom callInterval()¶ Returns: The call interval of this hook function. Return type: int nlprint( stream=<_io.TextIOWrapper name='<stdout>' mode='w' encoding='UTF-8'>)¶ Print a formatted string with the average temperature profile, as well as the average heat current. Parameters: stream( A stream that supports strings being written to using 'write'.) – The stream the temperature profile is written to. Usage Example¶ Perform a non-equilibrium molecular dynamics simulation with a constant heating power on a silicon crystal with a single germanium layer, to obtain the thermal conductance through this impurity layer. # Add tagsbulk_configuration.addTags('heat_sink', [ 108, 109, 110, 111, 112, 113, 114, 115, 116, 225, 226, 227, 228, 229, 230, 231, 232, 233, 342, 343, 344, 345, 346, 347, 348, 349, 350, 459, 460, 461, 462, 463, 464, 465, 466, 467, 576, 577, 578, 579, 580, 581, 582, 583, 584, 693, 694, 695, 696, 697, 698, 699, 700, 701, 810, 811, 812, 813, 814, 815, 816, 817, 818, 927, 928, 929, 930, 931, 932, 933, 934, 935,])bulk_configuration.addTags('heat_source', [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 117, 118, 119, 120, 121, 122, 123, 124, 125, 234, 235, 236, 237, 238, 239, 240, 241, 242, 351, 352, 353, 354, 355, 356, 357, 358, 359, 468, 469, 470, 471, 472, 473, 474, 475, 476, 585, 586, 587, 588, 589, 590, 591, 592, 593, 702, 703, 704, 705, 706, 707, 708, 709, 710, 819, 820, 821, 822, 823, 824, 825, 826, 827,])# -------------------------------------------------------------# Calculator# -------------------------------------------------------------potentialSet = Tersoff_SiGe_1989()calculator = TremoloXCalculator(parameters=potentialSet)bulk_configuration.setCalculator(calculator)# -------------------------------------------------------------# Molecular Dynamics# -------------------------------------------------------------initial_velocity = MaxwellBoltzmannDistribution( temperature=300.0Kelvin, remove_center_of_mass_momentum=True)method = NVEVelocityVerlet( time_step=1femtoSecond, initial_velocity=initial_velocity,)# Use a heating power of 50 nW.heating_power = 50.0e-9Joule/Secondmomentum_exchange_hook = NonEquilibriumHeatExchange( configuration=bulk_configuration, heating_power=heating_power, heat_source='heat_source', heat_sink='heat_sink', update_profile_interval=10, profile_resolution=2.0*Ang)md_trajectory = MolecularDynamics( bulk_configuration, constraints=[], trajectory_filename='Si_w_Ge_layer_NEMD.hdf5', steps=100000, log_interval=500, post_step_hook=momentum_exchange_hook, method=method)# Print the information from the NEMD heat flow simulation.nlprint(momentum_exchange_hook) Here, only the molecular dynamics block of the script is shown. The full scriptcan be found found in the file nonequilibriumheatexchange.py. Notes¶ The NonEquilibriumHeatExchange method can be used to run MolecularDynamics simulations with a constant heat flux between the two selected reservoirs. The technique can therefore be used to simulate the thermal conductance of a given configuration (see e.g, [JJ99]). The reverse non-equilibrium heat exchange method is implemented as a class, which can be used as a post_step_hookin MolecularDynamics. The velocities in the two reservoir regions are rescaled in such a way that a well-defined amount kinetic energy is added to the atoms in the heat_sourceregion whereas the same amount is taken away from the kinetic energy in the heat_sinkregion. The magnitude of the kinetic energy that is transferred per time from the heat_sinkto the heat_sourcecan be specified by the heating_powerparameter. By default the rescaling is performed at every MD step, if desired, the interval can be given by the user via the parameter exchange_interval. If a non-zero update_profile_intervalis specified, a temperature profile is stored on-the-fly during the simulation. After the MD run, the profiles may be accessed using the method temperatureProfile. The thermal bulk conductivity can be obtained by plotting the temperature profile of the system using the MD-Analyzertool and fitting its gradient $\nabla T$ along the transport direction. The thermal conductivity is then obtained via Fourier’s law\[\kappa = - \frac{\dot{Q}}{A \nabla T}\] where $\dot{Q}$ is the specified heating_power and A is thecross-sectional area perpendicular to the transport direction. The thermal conductance across an interface G( Kapitza conductance) can be obtained as\[G = -\frac{\dot{Q}}{\Delta T}\] where $\Delta T$ is the temperature drop across the interface. [JJ99] Philippe Jund and Rémi Jullien. Molecular-dynamics calculation of the thermal conductivity of vitreous silica. Phys. Rev. B, 59:13707–13711, 1999. doi:10.1103/PhysRevB.59.13707.
Question: A {eq}5.00 g {/eq} bullet moving with an initial speed of {eq}410 \frac{m}{s} {/eq} is fired into and passes through a {eq}1.00 kg {/eq} block. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring of force constant {eq}910 \frac{N}{m} {/eq}. (a) If the block moves {eq}5.80 cm {/eq} to the right after impact, find the speed at which the bullet emerges from the block. {eq}(\frac{m}{s}) {/eq} (b) If the block moves {eq}5.80 cm {/eq} to the right after impact, find the energy lost in the collision. {eq}(J) {/eq} Conservation Of Mechanical Energy: As we know that energy neither be created nor be destroyed but can exchange its form for example when the bullet strikes the wooden block then all the kinetic energy stored in the bullet will be imparted to the block kinetic energy and the block kinetic energy will be transferred to the attached spring and spring will absorb all the energy in the form of spring potential energy. Answer and Explanation: When the bullet strikes the wooden block then all the kinetic energy of the block due to the bullet will be transferred to the spring and the spring will absorb the kinetic energy into spring potential energy: From the conservation of energy: {eq}K.E=P.E\\ \frac{1}{2}mv^2=\frac{1}{2}kx^2\\ \frac{1}{2}(1)(v)^2=\frac{1}{2}(910)\times (0.058)^2\\ 0.5v^2=1.530\\ v^2=3.06\\ v=1.74\ m/s {/eq} Thus, the velocity of the block after it is struck by the bullet is 1.74 m/s. This will be the velocity of the bullet after striking the wooden block. Now from the conservation of momentum: {eq}m_1u_1+m_2u_2=m_1v_1+m_2v_2\\ (0.005\times 410)+0=(0.005\times v_1)+(1\times 1.74 )\\ 2.05=(0.005\times v_1)+1.74\\ v_1=62\ m/s {/eq} Thus, the velocity of the bullet as it emerges from the block is 62 m/s. (B) The initial kinetic energy of the bullet: {eq}K.E_1=\frac{1}{2}mv^2\\ K.E_1=\frac{1}{2}(0.005)\times (410)^2\\ K.E_1=420.25\ J {/eq} Final kinetic energy for the bullet: {eq}K.E_2=\frac{1}{2}mv^2\\ K.E_2=\frac{1}{2}(0.005)\times (62)^2\\ K.E_2=9.61\ J {/eq} Kinetic energy lost in spring and block system: {eq}K.E_{sb}=\frac{1}{2}mv^2\\ K.E_{sb}=\frac{1}{2}(1)\times (1.74)^2\\ K.E_{sb}=3.0276\ J {/eq} For energy loss: {eq}\Delta K.E=420.25-(9.61+3.02)\\ \Delta K.E=407.62\ J {/eq} Thus, the energy loss is 407.62 J. Become a member and unlock all Study Answers Try it risk-free for 30 daysTry it risk-free Ask a question Our experts can answer your tough homework and study questions.Ask a question Ask a question Search Answers Learn more about this topic: from Geography 101: Human & Cultural GeographyChapter 13 / Lesson 9
OptiLayer Targets: Conventional Targets OptiLayer allows you to specify all possible target characteristics. Target Window can be used to enter and to edit target data. In the information fields you should specify the number of spectral points at which target data will be entered, the incidence angle(s) at which data will be entered, and the type of spectral characteristic(s) for which data will be supplied. For each angle of incidence (AOI) you can specify a separate spreadsheet (first column is a spectral unit, for example, wavelength). On the contrast, Additional features are target generator, interpolation of target function (increase the number of grid points), estimation of the required number of spectral points, User-Defined Target (UDT) and special editor for group delay/group delay dispersion (GD/GDD). OptiLayer provides wavelength grids of three different types. Each target spreadsheet contains: \[ MF^2=\frac 1K\sum\limits_{i=1}^K \frac 1L\sum\limits_{j=1}^L \left[\frac{S(X; \theta_i;\lambda_j)-\hat{S}(\theta_i;\lambda_j)}{\Delta_{i,j}}\right]^2,\] where $S$ is the theoretical spectral characteristic, $\hat{S}$ is the target spectral characteristic, $\{\lambda_j\}, j=1,...,L$ is the wavelength grid, $\{\theta_i\}, i=1,...,K$ is the angular grid, $\Delta_{i,j}$ are target function tolerance values, $X$ is the vector of design parameters. List of available target spectral characteristics includes: Example (right panel): Rs=Rp=0 in the range from 510 nm to 520 nm, Rs=Rp=100% in the range from 1000 nm to 1500 nm. Design tolerances equal to 0.1 in the vicinity of the wavelength 500 nm. It means that this spectral range is more important for the design problem. Wavelength grid is linear around the wavelength 500 nm and logarithmic in the spectral region from 1000 nm to 1500 nm. The number of spectral points is 11 in the first spectral range and it is 128 in the second one. In this example the merit function is calculated as follows: \[ MF^2=\frac 1{11}\sum\limits_{j=1}^{11}\left(\frac{R_s(X;\lambda_j)}{0.1}\right)^2+\frac 1{11}\sum\limits_{j=1}^{11}\left(\frac{R_p(X;\lambda_j)}{0.1}\right)^2+\frac 1{128}\sum\limits_{i=1}^{128}\left(R_s(X;\lambda_i)-100\%\right)^2+\frac 1{128}\sum\limits_{i=1}^{128}\left(R_p(X;\lambda_i)-100\%\right)^2 \] It is possible to estimate the number of spectral points required for the current spectral target in order to perform design synthesis efficiently. Using too small number of points will lead to bad approximation of the desired target requirements (undesirable "leaks", peaks in the spectral characteristics of interest). Using too many spectral points will slow down computations accordingly. OptiLayer provides linear, logarithmic and inversely proportional wavelength grids. The wavelength grid can be automatically specified with the help of a user-friendly grid generator: Logarithmic grid is more sparse in the longer wavelength range and more dense in the shorter wavelength region.
Woodin Cardinal Woodin cardinals are a generalization of the notion of strong cardinals and have been used to calibrate the exact proof-theoretic strength of the Axiom of Determinacy. Woodin cardinals are weaker than superstrong cardinals in consistency strength and fail to be weakly compact in general, since they are not $\Pi_1^1$ indescribable. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed. Contents Shelah Cardinals Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc.). In slightly more detail, Woodin had established that the Axiom of Determinacy (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ assuming the existence of a non-trivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point $<\lambda$. This axiom is known to be very strong and its use was first weakened to that of the existence of a supercompact cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the regularity properties of sets of reals mentioned above. A cardinal $\kappa$ is Shelah if, for every function $f:\kappa\to\kappa$ there is a non-trivial elementary embedding $j:V\prec M$ with $M$ a transitive class, $\kappa$ the critical point of $j$ and $M$ contains the initial segment $V_{j(f)(\kappa)}$. It turns out that Shelah cardinals have many large cardinals below them that suffice to establish the regularity properties, and as a result have mostly faded from view in the large cardinal research literature. Woodin Cardinals Woodin cardinals are a refinement of Shelah cardinals. The primary difference is the requirement of a closure condition on the functions $f:\kappa\rightarrow\kappa$ and associated embeddings. Woodin cardinals are not themselves the critical points of any of their associated embeddings and hence need not be measurable. They are, however, Mahlo cardinals (and hence also inaccessible) since the set of measurable cardinals below a Woodin cardinal must be stationary. Elementary Embedding Characterization A cardinal $\kappa$ is "Woodin" if, for every function $f:\kappa\to\kappa$ there is some $\gamma<\kappa$ such that $f$ is closed under $\gamma$ and there is an associated non-trivial elementary embedding $j:V\prec M$ with critical point $\gamma$ where $M$ contains the initial segment $V_{(j(f))(\gamma)}$. If $\kappa$ is Woodin then for any subset $A\subseteq V_\kappa$ some $\alpha <\kappa$ is $\gamma$-strong for every $\gamma <\kappa$. Intuitively this means that there are elementary embeddings $j_\gamma$ which preserve $A$ i.e., $A\cap V_{\alpha+\gamma}=j_\gamma(A)\cap V_{\alpha+\gamma}$, have critical point $\alpha$, and whose target transitive class contains the initial segment $V_{\alpha+\gamma}$. There is a hierarchy of Woodin-type cardinals Analogue of Vopenka's Principle Stationary Tower Forcing Role in $\Omega$-Logic and the Resurrection Theorem This article is a stub. Please help us to improve Cantor's Attic by adding information.
I know that bond angle decreases in the order $\ce{H2O}$, $\ce{H2S}$ and $\ce{H2Se}$. I wish to know the reason for this. I think this is because of the lone pair repulsion but how? Here are the $\ce{H-X-H}$ bond angles and the $\ce{H-X}$ bond lengths: \begin{array}{lcc} \text{molecule} & \text{bond angle}/^\circ & \text{bond length}/\pu{pm}\\ \hline \ce{H2O} & 104.5 & 96 \\ \ce{H2S} & 92.3 & 134 \\ \ce{H2Se}& 91.0 & 146 \\ \hline \end{array} The traditional textbook explanation would argue that the orbitals in the water molecule is close to being $\ce{sp^3}$ hybridized, but due to lone pair - lone pair electron repulsions, the lone pair-X-lone pair angle opens up slightly in order to reduce these repulsions, thereby forcing the $\ce{H-X-H}$ angle to contract slightly. So instead of the $\ce{H-O-H}$ angle being the perfect tetrahedral angle ($109.5^\circ$) it is slightly reduced to $104.5^\circ$. On the other hand, both $\ce{H2S}$ and $\ce{H2Se}$ have no orbital hybridization. That is, The $\ce{S-H}$ and $\ce{Se-H}$ bonds use pure $\ce{p}$-orbitals from sulfur and selenium respectively. Two $\ce{p}$-orbitals are used, one for each of the two $\ce{X-H}$ bonds; this leaves another $\ce{p}$-orbital and an $\ce{s}$-orbital to hold the two lone pairs of electrons. If the $\ce{S-H}$ and $\ce{Se-H}$ bonds used pure $\ce{p}$-orbitals we would expect an $\ce{H-X-H}$ interorbital angle of $90^\circ$. We see from the above table that we are very close to the measured values. We could fine tune our answer by saying that in order to reduce repulsion between the bonding electrons in the two $\ce{X-H}$ bonds the angle opens up a bit wider. This explanation would be consistent with the $\ce{H-S-H}$ angle being slightly larger than the corresponding $\ce{H-Se-H}$ angle. Since the $\ce{H-Se}$ bond is longer then the $\ce{H-S}$ bond, the interorbital electron repulsions will be less in the $\ce{H2Se}$ case alleviating the need for the bond angle to open up as much as it did in the $\ce{H2S}$ case. The only new twist on all of this that some universities are now teaching is that water is not really $\ce{sp^3}$ hybridized, the $\ce{sp^3}$ explanation does not fit with all of the experimentally observed data, most notably the photoelectron spectrum. The basic concept introduced is that "orbitals only hybridize in response to bonding." So in water, the orbitals in the two $\ce{O-H}$ bonds are roughly $\ce{sp^3}$ hybridized, but one lone pair resides in a nearly pure p-orbital and the other lone pair is in a roughly $\ce{sp}$ hybridized orbital. The question asks why water has a larger angle than other hydrides of the form $\ce{XH2}$ in particular $\ce{H2S}$ and $\ce{H2Se}$. There have been other similar questions, so an attempt at a general answer is given below. There are, of course, many other triatomic hydrides, $\ce{LiH2}$, $\ce{BeH2}$, $\ce{BeH2}$, $\ce{NH2}$, etc.. It turns out that some are linear and some are V shaped, but with different bond angles, and that the same general explanation can be used for each of these cases. It is clear that as the bond angle for water is neither $109.4^\circ$, $120^\circ$, nor $180^\circ$ that $\ce{sp^3}$, $\ce{sp^2}$ or $\ce{sp}$ hybridisation will not explain the bond angles. Furthermore, the UV photoelectron spectrum of water, which measures orbital energies, has to be explained as does the UV absorption spectra. The way out of this problem is to appeal to molecular orbital theory and to construct orbitals based upon $\ce{s}$ and $\ce{p}$ orbitals and their overlap as bond angle changes. The orbital diagram was worked out a long time ago is now called a Walsh diagram (A. D. Walsh J. Chem. Soc. 1953, 2262; DOI: 10.1039/JR9530002260). The figure below sketches such a diagram, and the next few paragraphs explain the figure. The shading indicates the sign (phase) of the orbital, 'like to like' being bonding otherwise not bonding. The energies are relative as are the shape of the curves. On the left are the orbitals arranged in order of increasing energy for a linear molecule; on the right those for a bent molecule. The orbitals labelled $\Pi_\mathrm{u}$ are degenerate in the linear molecule but not so in the bent ones. The labels $\sigma_\mathrm{u}$, $\sigma_\mathrm{g}$ refer to sigma bonds, the $\mathrm{g}$ and $\mathrm{u}$ subscripts refer to whether the combined MO has a centre of inversion $\mathrm{g}$ (gerade) or not $\mathrm{u}$ (ungerade) and derive from the irreducible representations in the $D_\mathrm{\infty h}$ point group. The labels on the right-hand side refer to representations in the $C_\mathrm{2v}$ point group. Of the three $\Pi_\mathrm{u}$ orbitals one forms the $\sigma_\mathrm{u}$, the other two are degenerate and non-bonding. One of the $\ce{p}$ orbitals lies in the plane of the diagram, the other out of the plane, towards the reader. When the molecule is bent this orbital remains non-bonding, the other becomes the $\ce{3a_1}$ orbital (red line) whose energy is significantly lowered as overlap with the H atom's s orbital increases. To work out whether a molecule is linear or bent all that is necessary is to put electrons into the orbitals. Thus, the next thing is to make a list of the number of possible electrons and see what diagram predicts. \begin{array}{rcll} \text{Nr.} & \text{Shape} & \text{molecule(s)} & \text{(angle, configuration)} \\ \hline 2 & \text{bent} & \ce{LiH2+} & (72,~\text{calculated})\\ 3 & \text{linear} & \ce{LiH2}, \ce{BeH2+} &\\ 4 & \text{linear} & \ce{BeH2}, \ce{BH2+} &\\ 5 & \text{bent} & \ce{BH2} & (131, \ce{[2a_1^2 1b_2^2 3a_1^1]})\\ 6 & \text{bent} & \ce{^1CH2} & (110, \ce{[1b_2^2 3a_1^2]})\\ & & \ce{^3CH2} & (136, \ce{[1b_2^2 3a_1 1b_1^1]})\\ & & \ce{BH2^-} & (102)\\ & & \ce{NH2+} & (115, \ce{[3a_1^2])}\\ 7 & \text{bent} & \ce{NH2} & (103.4, \ce{[3a_1^2 1b_1^1]})\\ 8 & \text{bent} & \ce{OH2} & (104.31, \ce{[3a_2^2 1b_1^2]})\\ & & \ce{NH2^-} & (104)\\ & & \ce{FH2^+} &\\ \hline \end{array} Other hydrides show similar effects depending on the number of electrons in $\ce{b2}$, $\ce{a1}$ and $\ce{b1}$ orbitals; for example: \begin{array}{ll} \ce{AlH2} & (119, \ce{[b_2^2 a1^1]}) \\ \ce{PH2} & (91.5, \ce{[b_2^2 a_1^2 b_1^1]}) \\ \ce{SH2} & (92)\\ \ce{SeH2} & (91)\\ \ce{TeH2} & (90.2)\\ \ce{SiH2} & (93)\\ \end{array} The agreement with experiment is qualitatively good, but, of course the bond angles cannot be accurately determined with such a basic model only general trends. The photoelectron spectrum (PES) of water shows signals from $\ce{2a1}$, $\ce{1b2}$, $\ce{3a1}$, $\ce{1b1}$ orbitals, ($21.2$, $18.7$, $14.23$, and $\pu{12.6 eV}$ respectively) the last being non-bonding as shown by the lack of structure. The signals from $\ce{3b2}$ and $\ce{3a1}$ orbitals show vibrational structure indicating that these are bonding orbitals. The range of UV and visible absorption by $\ce{BH2}$, $\ce{NH2}$, $\ce{OH2}$ are $600 - 900$, $450 - 740$, and $150 - \pu{200 nm}$ respectively. $\ce{BH2}$ has a small HOMO-LUMO energy gap between $\ce{3a1}$ and $\ce{1b1}$ as the ground state is slightly bent. The first excited state, is predicted to be linear as its configuration is $\ce{1b_2^2 1b_1^1}$ and this is observed experimentally. $\ce{NH2}$ has a HOMO-LUMO energy gap from $\ce{3a_1^2 1b_1^1}$ to $\ce{3a_1^1 1b_1^2 }$, so both ground and excited states should be bent, the excited state angle is approx $144^\circ$. Compared to $\ce{BH2}$, $\ce{NH2}$ is more bent so the HOMO-LUMO energy gap should be larger as observed. $\ce{OH2}$ has a HOMO-LUMO energy gap from $\ce{3a_1^2 1b_1^2}$ to $\ce{3a_1^2 1b_1^1 4a_1^1 }$, i.e. an electron promoted from the non-bonding orbital to the first anti-bonding orbital. The excited molecule remains bent largely due to the strong effect of two electrons in $\ce{3a1}$ counteracting the single electron in $\ce{4a1}$. The bond angle is almost unchanged at $107^\circ$, but the energy gap will be larger than in $\ce{BH2}$ or $\ce{NH2}$, again as observed. The bond angles of $\ce{NH2}$, $\ce{NH2-}$ and $\ce{NH2+}$ are all very similar, $103^\circ$, $104^\circ$, and $115^\circ$ respectively. $\ce{NH2}$ has the configuration $\ce{3a_1^2 1b_1^1}$ where the $\ce{b1}$ is a non bonding orbital, thus adding one electron makes little difference, removing one means that the $\ce{3a_1}$ orbital is not stabilised as much and so the bond angle is opened a little. The singlet and triplet state $\ce{CH2}$ molecules show that the singlet has two electrons in the $\ce{3a1}$ orbital and has a smaller angle than the triplet state with just one electron here and one in the non-bonding $\ce{b1}$, thus the triplet ground state bond angle is expected to be larger than the singlet. As the size of the central atom increases, its nucleus becomes more shielded by core electrons and it becomes less electronegative. Thus going down the periodic table the $\ce{X-H}$ bond becomes less ionic, more electron density is around the $\ce{H}$ atom thus the $\ce{H}$ nucleus is better shielded, and thus the $\ce{X-H}$ bond is longer and weaker. Thus, as usual with trends within the same family in the periodic table, the effect is, basically, one of atomic size. Molecules with heavier central atom, $\ce{SH2}$, $\ce{PH2}$, etc. all have bond angles around $90^\circ$. The decrease in electronegativity destabilises the $\Pi_\mathrm{u}$ orbital raising its energy. The $\ce{s}$ orbitals of the heavier central atoms are larger and lower in energy than those of oxygen, hence these orbitals overlap with the $\ce{H}$ atom's $\ce{s}$ orbital more weakly. Both these factors help to stabilise the linear $3\sigma_\mathrm{g}$ orbital and hence the $\ce{4a1}$ in the bent configuration. This orbital belongs to the same symmetry species as $\ce{3a1}$ and thus they can interact by a second order Jahn-Teller interaction. This is proportional to $1/\Delta E$ where $\Delta E$ is the energy gap between the two orbitals mentioned. The effect of this interaction is to raise the $\ce{4a1}$ and decrease the $\ce{3a1}$ in energy. Thus in going down the series $\ce{OH2}$, $\ce{SH2}$, $\ce{SeH2}$, etc. the bond angle should decrease which is what is observed. Example have been given for $\ce{XH2}$ molecules, but this method has also been used to understand triatomic and tetra-atomic molecules in general, such as $\ce{NO2}$, $\ce{SO2}$, $\ce{NH3}$, etc.. Adding a bit to the answers above, one factor that isn't shown in the Walsh diagram is that as the angle decreases, there is increased mixing between the central atom valence s and p orbitals, such that the 2a$_1$ orbital has increased p contribution and the 3a$_1$ has increased s. This is where one gets the result that Ron mentioned at the end of his answer that the lone pairs on water reside in a pure p (1b$_1$) and an sp (3a$_1$) orbital. That means the bonding orbitals shift from one pure s (2a$_1$) and one pure p (1b$_2$) to one sp (2a$_1$) and one p (1b$_2$) (ignoring the extreme case where 3a$_1$ actually gets lower in energy than 1b$_2$, which isn't really relevant). Mixing occurs to a greater extent in $\ce{SH2}$ relative to $\ce{OH2}$ because the 3s and 3p orbitals of S are closer in energy to each other than 2s and 2p on O. If we hybridize the two bonding orbitals so that they are equivalent and do the same for the two nonbonding orbitals, we find that they start as bonding = 50% s/50% p (ie $sp$ hybrid) and nonbonding = 100% p and shift towards an endpoint of bonding and nonbonding both being 25% s/75% p (ie $sp^3$ hybrid). Thus, the common introductory chemistry explanation that "bonding in $\ce{SH2}$ is pure p" is not supported by the MO analysis. Instead, $\ce{SH2}$ is closer to $sp^3$ than $\ce{H2O}$ is. The bonding orbitals in $\ce{H2O}$ are somewhere between $sp^2$ and $sp^3$. So it is correct to say that "the bonds in $\ce{SH2}$ have less s character than those in $\ce{OH2}$", but not to say that they are "pure p". The fact that the $\ce{SH2}$ bond angle is around 90 degrees is not because its bonds are made from p orbitals only. That coincidence is a red herring. Instead, the fact that the bond angle is smaller than the canonical $sp^3$ is because the bonding and nonbonding orbitals are not equivalent. That means that the particular p orbitals involved in each $sp^3$ group do not have to have the same symmetry as in, for example, a tetrahedral molecule like CH4. I will try to give u a most appropriate and short answer that u can understand easily See h20 has 104.5 degrees bond angle , h2s has 92degrees , h2se has 91degrees and h2te has 90degrees bond angles Draw diagrams of these u will find all of them have tetrahedral shape with 2 lone pairs , assume that no hybridization occurs and all these central atoms are using pure p orbitals for bonding then because of repulsions by lone pairs the bond angle should be 90degrees between 2 surrounding atoms , now according to dragos rules when central atom belongs to 3rd period or higher and electro negativity of surrounding atoms is 2.5 or less then central atom uses almost pure p orbitals . So the final answer is the extenend of hybridization decreases In this case which leads to decrease in bond angle . note that only in h2te no hybridization is observed. We know that as the electronegativity of central atom increases, the bond angles also increase. The relevant electronegative order is $$\ce{O > S > Se}\,,$$ hence the bond angle order of $$\ce{H2O>H2S>H2Se}\,.$$
Show, that every continious function $f:\mathbb{R}P^2\to S^1$ is a homotopy to a constant function. (Hint: Has $f$ a lift $\tilde{f}:\mathbb{R}P^2\to\mathbb{R}$) Hello, I want to solve this problem, but I am kinda stuck and the given hint just confuses me even more... I would appreciate, if someone could clarify this a bit. Thanks in advance. === Edit: I want to solve this by a way Eduard Longa gave: Let $p:\mathbb{R}→S^1$ be the standard covering map. Since $\mathbb{R}$ is contractible, the map $f$ is homotopic to a constant function via an homotopy $H$. Then, the composition $p\circ H$ provides an homotopy between $f$ and a constant function. I want to go through this, step by step. First of all I want to show, that $\mathbb{R}$ is contractible. Therefore I have to show, that $\mathbb{R}$ is homotopic to a set $\{x_0\}$. $f_1:\mathbb{R}\to\{x_0\}$, $f_1(x)=x_0$ $f_2:\{x_0\}\to\mathbb{R}$, $f_2(x)=x$ I have to show, that $f_1\circ f_2\sim id_{\{x_0\}}$ and $f_2\circ f_1\sim id_{\mathbb{R}}$ $(f_1\circ f_2)(x_0)=x_0$ $(f_2\circ f_1)(x)=x_0$ To show, that $f_1\circ f_2\sim id_{\{x_0\}}$ I give the homotopy $H_1:\{x_0\}\times[0,1]\to\{x_0\}$ simply by $H_1(x,t)=x_0$ and for $f_2\circ f_1\sim id_\mathbb{R}$ similar $H_2:\mathbb{R}\times [0,1]\to\mathbb{R}$, with $H_2(x,t)=(1-t)x+tx_0$. Hence $\mathbb{R}$ is contractible. (It is completly trivial) Am I right? The next step is to show, that $f$ is homotopic to a constant function via an homotopy $H$. Is here $\tilde{f}$ meant? Since $f:\mathbb{R}P^2\to S^1$ I do not know, why $\mathbb{R}$ contractible, has the consequence, that $f$ is homotopic to a constant function. Respectively how this observation about $\mathbb{R}$ helps.
I am working on a project to approximate numerically the solution $X_t$ of a stochastic diﬀerential equation (SDE) using the Euler method. I have do to this for the Brownian motion with drift. I am asked to stimulate $N$ paths under both the P and Q measure on the interval $[0,T]$. The pseudo code is as follows: for i to N-1 calculate the drift as function of previous stock price ($\mu$) calculate the volatility as function of previous stock price ($\sigma$) draw innovation from standard normal distribution ($\epsilon$) $S_{t+i} = S_t + \mu_t dt + \sigma_t \sqrt{dt } \epsilon_t$. next where $dt$ is defined as $(T-0)/N$. My current code is as follows: nr_runs = 1000; %number of simulation runsN = 1000; %compute N grid pointst0 = 0;T = 10;dt = (T - t0) / N;x0 = 0; %starting pointx = zeros(1000);mu = 0;sigma = zeros(1000); for i = 1:N sigma(i) = sqrt(idt); %under P measure, variance equal to time epsilon = normrnd(0,1); if i == 1 x(i) = x0 + mudt + sigma(i)* sqrt(dt)epsilon; else x(i+1) = x(i) + mudt + sigma(i)* sqrt(dt)*epsilon; end end M = mean(x); However, I know no idea how to calculate the drift ($\mu$) from the previous stock price. What is the formula? Thank you! Any help is appreciated.
Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= \begin{cases} 0 & \text{if $x=y$} \\ \frac{1}{k(x,y)} & \text{if $x \neq y$} \end{cases}$$ We know that $d$ is a metric on $X$. Now, I must prove that the metric space $(X,d)$ is complete. By definition, a metric space $(X,d)$ is complete if every Cauchy sequence in $X$ is convergent. On the real line, this is trivial, but I have trouble applying the concept to metric spaces. We know that a sequence is Cauchy if: $$(\forall \epsilon>0)(\exists N>0)(\forall m,n \geq N)(d(x_m,x_n)<\epsilon)$$ A Cauchy sequence also has the following properties: If a sequence $(x_n)$ converges, then it is Cauchy. Every Cauchy sequence is bounded: for all $a \in X$, there exists $C_a>0$ such that $d(a,x_n)<C_a$ for all $n$. If a Cauchy sequence $(x_n)$ has a converging subsequence $(x_{n_k})$ such that $\lim_{k \to \infty}x_{n_k}=x$, then $(x_n)$ converges to $x$. Here is my attempt thus far, although my reasoning feels wrong. Let $(x_n)_{n=1}^\infty$ be any Cauchy sequence in $X$. Take $\epsilon=\frac12$. Let $N$ be such that for $n,m \geq N$, we have $d(x_n,x_m)<\frac12$. So, $x_n=x_m$ for all $n,m \geq N$ and $x_n=x_N$ for $n \geq N$. Thus, $x_n$ is eventually constant and hence convergent, which proves that the metric space is complete. Any corrections or help on how to prove this would be appreciated. Thank you!
Consider $\sum_{n=1}^{\infty}nx^n\sin(nx)$. Find $R > 0$ such that the series is convergent for all $x\in(-R,R)$. Calculate the sum of the series. I could find the radius of convergence is $R=1$, hence for any $x\in (-1,1)$ the series is continuous and convergent, However, I have some problem in finding the exact sum of this series. To find $f(x)=\sum_{n=1}^{\infty}nx^n\sin(nx)$, I think it's reasonable to find $F(x)=\sum_{n=1}^{\infty}nx^ne^{inx}$ and the imaginary part of $F(x)$ is $f(x)$. So if $F(x)=\sum_{n=1}^{\infty}nx^ne^{inx}$, then $\frac{1}{2\pi}\int_{-\pi}^\pi F(x)e^{-inx}dx=nx^n$, but I don't know how to find $F(x).$
Short version: Is it possible to arrange the fluxes for the Kagomé lattice with triangle flux $\phi_\triangle=\frac{\pi}2$ and hexagon flux $\phi_{hex}=0$ using a single unit cell? Longer version: I am looking at fermionic mean field theories on the Kagomé lattice that describe a chiral spin liquid state for spin-1/2. Skipping the derivation (in 1 and 2), the mean field Hamiltonian is $H = -\sum\limits_{\langle i, j\rangle,\sigma}~\rho~e^{iA_{ij}}~f^\dagger_{i\sigma}~f_{j\sigma} + H.c.$ This basically says that $f$-fermions hop along nearest-neighbor links, picking up a phase $A_{ij}$ one way along the bond, and $-A_{ij}$ hopping in the reverse direction. Phases (aka gauge field) are assigned to each lattice link in a unit cell, often depicted with arrows. The flux $\phi$ is defined as the sum of phases on a closed lattice plaquette along, say, the counter-clockwise direction. In Marston's paper (Ref. 1), and Ran's paper (Ref. 2), they introduce a state with triangle flux $\frac{\pi}2$ and hexagon flux 0 (equivalent to Chua's (Ref. 5) (d)), and SL-[$\frac{\pi}2$,0] in Kim's diagram (Ref. 6). My problem is with that with Kim's (Ref. 6) phase assignment for the triangles, I don't see how it is possible to achieve zero (or 2$\pi$) hexagon flux, unless one uses a doubled magnetic unit cell, as is claimed in Chua's work (Ref. 5). Ran (Ref. 2) doesn't explicitly state whether the unit cell has to double or not. Am I missing something here?
VINAYAK B KAMBLE Articles written in Bulletin of Materials Science Volume 40 Issue 7 December 2017 pp 1291-1299 Monovalent ion doped lanthanum cobaltate La$_{1−x}$Na$_x$CoO$_3$ ($0 \leq x \leq 0.25$) compositions were synthesized by the nitrate–citrate gel combustion method. All the heat treatments were limited to below 1123 K, in order to retain the Na stoichiometry. Structural parameters for all the compounds were confirmed by the Rietveld refinement method usingpowder X-ray diffraction (XRD) data and exhibit the rhombhohedral crystal structure with space group R-3c (No. 167). Thescanning electron microscopy study reveals that the particles are spherical in shape and sizes, in the range of 0.2–0.5 $\mu$m.High temperature electrical resistivity, Seebeck coefficient and thermal conductivity measurements were performed on thehigh density hot pressed pellets in the temperature range of 300–800 K, which exhibit p-type conductivity of pristine anddoped compositions. The X-ray photoelectron spectroscopy (XPS) studies confirm the monotonous increase in Co$^{4+}$ withdoping concentration up to $x = 0.15$, which is correlated with the electrical resistivity and Seebeck coefficient values of thesamples. The highest power factor of 10 $\mu$WmK$^{−2}$ is achieved for 10 at% Na content at 600 K. Thermoelectric figure ofmerit is estimated to be $\sim$$1 \times 10^{−2}$ at 780 K for 15 at% Na-doped samples. Current Issue Volume 42 \| Issue 6 December 2019 Click here for Editorial Note on CAP Mode
Definition- An ellipse is a curve on a plane such that the sum of the distances to its two focal points is always a constant quantity from any chosen point on that curve. The ellipse belongs to the family of circles with both the focal points at the same location. In an ellipse if you make the minor and major axis of the same length with both foci F1 and F2 at the center then it results in a circle. Area of an Ellipse- Area= $\pi ab$ Where a and b denote the semi-major and semi-minor axes respectively. The above formula for area of the ellipse has been mathematically proven as shown below: We know that the standard form of an ellipse is: For Horizontal Major Axis- $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$, (where a>b) Or, $y = b.\sqrt{1-\left ( \frac{x}{a} \right )^{2}}$ For Vertical Major axis- $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1$, (where a>b) Or, $y = a.\sqrt{1-\left ( \frac{x}{b} \right )^{2}}$ Proof for area of an Ellipse- We know the general equation for an ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ $y = b.\sqrt{1-\left ( \frac{x}{a} \right )^{2}}$ $y = \frac{b}{a}\sqrt{a^{2} – x^{2}}$ As we can see the ellipse is divided into four quadrant. So calculating the area of 1 and multiplying by 4, we get the area of an ellipse. Area, $A = 4. \int_{0}^{a} y.dx$ $A = 4. \int_{0}^{a} \frac{b}{a} \sqrt{a^{2}-x^{2}} dx$ $= 4. \frac{b}{a}\int_{0}^{a} \sqrt{a^{2}-x^{2}} dx$ Substituting $x = a \sin t$ dx = $a \cos t . dt$ x = 0 changes to t = 0 and x = a changes to t = $\pi /2$ $A = 4. \frac{b}{a}\int_{0}^{\frac{\pi}{2}} \sqrt{a^{2}-a^{2} \sin^{2}t} .a \cos t .dt$ $A = 4. \frac{b}{a}\int_{0}^{\frac{\pi}{2}} a^{2} \cos^{2} t .dt$ A = $4ab \left ( \frac{t}{2} + \frac{\sin 2x}{4} \right )_{0}^{\pi /2}$ A = $4ab \times \frac{\pi}{4}$ A= $\pi ab$< Application of Ellipses: They have widespread applications in the field of engineering, physics, etc. For instance, all the planets revolve in their orbits which are elliptical in shape. Moreover, astronomy has a lot of use of this shape as many of the stars and planets are shaped as ellipsoids. This is all about the area of an ellipse. To know more about the various attributes of the ellipse and other geometrical figures, please do visit www.byjus.com or download BYJU’S-The Learning App.
I am studying a book about relativistic equations and special relativity, and I keep seeing $\sqrt{1-{v^2/c^2}}$ everywhere. It is not, as with most of the concepts in special relativity, simply a mathematical construct; it is a logical consequence of accepting the experimental fact that the speed of light is the same in every inertial reference frame. Why, then, is this expression so significant? 1) At least at low speeds, you expect $x'=x-vt$, just from elementary considerations. ($vt$ is, after all, the distance traveled in time $t$, so a person traveling at speed $v$ will have his origin displaced by the amount $vt$. 2) If you believe space and time should be treated symmetrically, then you are led to expect something like $t'=t-vx$. 3) So in matrix terms, our first guess is $$\pmatrix{x'\cr t'}=\pmatrix{1&-v\cr -v&1\cr}\pmatrix{x\cr t}$$ 4) But if the transformation matrix is to preserve geometric structure (or, pretty much equivalently, if you want the matrix associated to $-v$ to be the inverse of the matrix associated to $v$) you want its determinant to be $1$, whereas it currently has determinant $\Delta=1-v^2$. 5) So to fix the determinant (while making changes that are negligible when $v$ is small, you multiply the transformation matrix by the appropriate constant, which is $1/\sqrt{\Delta}$. That explains where the constant comes from. Let's confine ourselves to relativity in one spatial dimension: along the $x$-axis say. Velocities are then simply signed real numbers. Owing to the observer dependence of time and distance measurements that arise between relatively moving observers, "velocity" becomes an awkward and unnatural quantifier of relative motion between frames. A more natural quantification of relative motion is the rapidity $\eta = \mathrm{artanh} \frac{v}{c}$: this takes account of the observer dependence of time and distance measurements such that relative rapidities are linear additive at relativistic velocities in the same way that everyday velocities are at everyday speed. The co-ordinate transformation factors in the Lorentz transformation are all multiples of $\cosh\eta$ and $\sinh\eta$, which can be expanded: $$\cosh\eta = \frac{1}{\sqrt{1-\tanh^2 \eta}} = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ $$\sinh\eta = \frac{\tanh\eta}{\sqrt{1-\tanh^2 \eta}} = \frac{\frac{v}{c}}{\sqrt{1-\frac{v^2}{c^2}}}$$ and so the factor that mystifies you simply arises by dint of the identity $\cosh^2\eta - \sinh^2 \eta = 1$, given that the most natural measure of relative motion is related to velocity by $v = c \tanh \eta$. Here's a diagram to elaborate on what some of the other answers are saying. In relativity, where there is a lot of emphasis on "what observers measure",the computation of components of vectors is important. If a vector-of-interest makes an angle with a reference-vector, then the component of that vector-of-interest along the reference-direction is related to the cosine of that angle[in that geometry]. If, instead of the angle, you wish to use a slope, then expressions involving square-roots show up. In relativity, the geometry is Minkowski geometry. Let's draw a spacetime diagram. I'm using rotated graph paper so that it is easier to visualize the ticks along the legs of the triangle. Here the velocity (slope) $v=6/10$. The hypotenuse (of the vector-of-interest) is $T=8$. In terms of the slope, the components are as shown. $\frac{1}{\sqrt{1-v^2}}T$ is time component of that vector. It might be more geometrically-intuitive to express this in terms of the Minkowski-angle (called the rapidity) between the vector-of-interest and the observer's time-axis. If $v=\tanh\theta$, then it turns out that $(\cosh\theta)T$ is the time-component of that vector. That is, $$\cosh\theta=\frac{1}{\sqrt{1-v^2}}.$$ In addition, $\sinh\theta=(\tanh\theta)\cosh\theta= v\cosh\theta=\frac{v}{\sqrt{1-v^2}}$ Exercise: In ordinary Euclidean geometry, where $v=\tan\theta$, write $\cos\theta$ and $\sin\theta$ in terms of $v$. You know the pythagorean theorem? Take a right triangle of hypotenuse 1. Let $v/c$ be the length of one side. Then that formula is the length of the other side. Another way to look at it: suppose the hypotenuse is $c$, the speed of light, and $v$ is one side of the triangle, the speed something is moving. The closer $v$ comes to $c$, the smaller the third side is, and $v$ can never be greater than $c$ without breaking the right triangle.
(Sorry was asleep at that time but forgot to log out, hence the apparent lack of response) Yes you can (since $k=\frac{2\pi}{\lambda}$). To convert from path difference to phase difference, divide by k, see this PSE for details http://physics.stackexchange.com/questions/75882/what-is-the-difference-between-phase-difference-and-path-difference Yes you can (since $k=\frac{2\pi}{\lambda}$). To convert from path difference to phase difference, divide by k, see this PSE for details http://physics.stackexchange.com/questions/75882/what-is-the-difference-between-phase-difference-and-path-difference
Search Now showing items 1-10 of 108 Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV (Elsevier, 2013-04-10) The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (\|y\| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c with the ALICE experiment at the CERN LHC in ... Event-by-event mean pT fluctuations in pp and Pb–Pb collisions at the LHC (Springer, 2014-10) Event-by-event fluctuations of the mean transverse momentum of charged particles produced in pp collisions at s√ = 0.9, 2.76 and 7 TeV, and Pb–Pb collisions at √sNN = 2.76 TeV are studied as a function of the ... Anomalous evolution of the near-side jet peak shape in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV (American Physical Society, 2017-09-08) The measurement of two-particle angular correlations is a powerful tool to study jet quenching in a $p_{\mathrm{T}}$ region inaccessible by direct jet identification. In these measurements pseudorapidity ($\Delta\eta$) and ... Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions (Nature Publishing Group, 2017) At sufficiently high temperature and energy density, nuclear matter undergoes a transition to a phase in which quarks and gluons are not confined: the quark–gluon plasma (QGP)1. Such an exotic state of strongly interacting ... Multiplicity dependence of the average transverse momentum in pp, p-Pb, and Pb-Pb collisions at the LHC (Elsevier, 2013-12) The average transverse momentum <$p_T$> versus the charged-particle multiplicity $N_{ch}$ was measured in p-Pb collisions at a collision energy per nucleon-nucleon pair $\sqrt{s_{NN}}$ = 5.02 TeV and in pp collisions at ... Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider (American Physical Society, 2016-02) The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ... Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV (Elsevier, 2016-02) Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < \|\eta\| < 4.0$) and associated particles in the central range ($\|\eta\| < 1.0$) are measured with the ALICE detector in ... K$^{}(892)^{0}$ and $\phi(1020)$ meson production at high transverse momentum in pp and Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 2.76 TeV (American Physical Society, 2017-06) The production of K$^{}(892)^{0}$ and $\phi(1020)$ mesons in proton-proton (pp) and lead-lead (Pb-Pb) collisions at $\sqrt{s_\mathrm{NN}} =$ 2.76 TeV has been analyzed using a high luminosity data sample accumulated in ... Directed flow of charged particles at mid-rapidity relative to the spectator plane in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV (American Physical Society, 2013-12) The directed flow of charged particles at midrapidity is measured in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV relative to the collision plane defined by the spectator nucleons. Both, the rapidity odd ($v_1^{odd}$) and ... Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV (Springer, 2016-08) The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ...
Difference between revisions of "Wholeness axioms" Line 8: Line 8: language $\{\in,j\}$, augmenting the usual language of set language $\{\in,j\}$, augmenting the usual language of set theory $\{\in\}$ with an additional unary function symbol $j$ theory $\{\in\}$ with an additional unary function symbol $j$ − to represent the embedding. The base theory ZFC is + to represent the embedding. The base theory ZFC is expressed only in the smaller language $\{\in\}$. Corazza's expressed only in the smaller language $\{\in\}$. Corazza's original proposal, which we denote by $\text{WA}_0$, asserts original proposal, which we denote by $\text{WA}_0$, asserts Revision as of 00:51, 3 October 2017 The wholeness axioms, proposed by Paul Corazza [1, 2], occupy a high place in the upper stratosphere of the large cardinal hierarchy, intended as slight weakenings of the Kunen inconsistency, but similar in spirit. The wholeness axioms are formalized in thelanguage $\{\in,j\}$, augmenting the usual language of settheory $\{\in\}$ with an additional unary function symbol $j$to represent the elementary embedding. The base theory ZFC isexpressed only in the smaller language $\{\in\}$. Corazza'soriginal proposal, which we denote by $\text{WA}_0$, assertsthat $j$ is a nontrivial amenable elementary embeddingfrom the universe to itself. Elementarity is expressed bythe scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$runs through the formulas of the usual language of settheory; nontriviality is expressed by the sentence $\existsx j(x)\not=x$; and amenability is simply the assertionthat $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent tothe assertion that the separation axiom holds for$\Delta_0$ formulae in the language $\{\in,j\}$. The wholeness axiom WA, also denoted $\text{WA}_\infty$, asserts in addition that thefull separation axiom holds in the language $\{\in,j\}$. Those two axioms are the endpoints of the hierarchy of axioms $\text{WA}_n$, asserting increasing amounts of the separation axiom. Specifically, the wholeness axiom $\text{WA}_n$, where $n$ is amongst $0,1,\ldots,\infty$, consists of the following: (elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$. (separation) All instances of the Separation Axiom for $\Sigma_n$ formulae in the full language $\{\in,j\}$. (nontriviality) The axiom $\exists x\,j(x)\not=x$. Clearly, this resembles the Kunen inconsistency. What is missing from the wholeness axiom schemes, and what figures prominantly in Kunen's proof, are the instances of the replacement axiom in the full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with WA, and indeed, in every model of WA the critical sequence is unbounded in the ordinals. The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. [3] If $j:V_\lambda\to V_\lambda$ witnesses a rank into rank cardinal, then $\langle V_\lambda,\in,j\rangle$ is a model of the wholeness axiom. If the wholeness axiom is consistent with ZFC, then it is consistent with ZFC+V=HOD.[3] References Corazza, Paul. The Wholeness Axiom and Laver sequences.Annals of Pure and Applied Logic pp. 157--260, October, 2000. bibtex Corazza, Paul. The gap between ${\rm I}_3$ and the wholeness axiom.Fund Math 179(1):43--60, 2003. www DOI MR bibtex Hamkins, Joel David. The wholeness axioms and V=HOD.Arch Math Logic 40(1):1--8, 2001. www arχiv DOI MR bibtex
ISSN: 1930-8337 eISSN: 1930-8345 All Issues Inverse Problems & Imaging February 2010 , Volume 4 , Issue 1 Select all articles Export/Reference: Abstract: We extend the classical spectral estimation problem to the infinite-dimensional case and propose a new approach to this problem using the Boundary Control (BC) method. Several applications to inverse problems for partial differential equations are provided. Abstract: Let $z=Au+\gamma$ be an ill-posed, linear operator equation. Such a model arises, for example, in both astronomical and medical imaging, in which case $\gamma$ corresponds to background, $u$ the unknown true image, $A$ the forward operator, and $z$ the data. Regularized solutions of this equation can be obtained by solving $R_\alpha(A,z)= arg\min_{u\geq 0} \{T_0(Au;z)+\alpha J(u)\},$ where $T_0(Au;z)$ is the negative-log of the Poisson likelihood functional, and $\alpha>0$ and $J$ are the regularization parameter and functional, respectively. Our goal in this paper is to determine general conditions which guarantee that $R_\alpha$ defines a regularization scheme for $z=Au+\gamma$. Determining the appropriate definition for regularization scheme in this context is important: not only will it serve to unify previous theoretical arguments in this direction, it will provide a framework for future theoretical analyses. To illustrate the latter, we end the paper with an application of the general framework to a case in which an analysis has not been done. Abstract: In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact subset of a finite dimensional space. Abstract: We consider the interior transmission eigenvalue problem corresponding to the inverse scattering problem for an isotropic inhomogeneous medium. We first prove that transmission eigenvalues exist for media with index of refraction greater or less than one without assuming that the contrast is sufficiently large. Then we show that for an arbitrary Lipshitz domain with constant index of refraction there exists an infinite discrete set of transmission eigenvalues that accumulate at infinity. Finally, for the general case of non constant index of refraction we provide a lower and an upper bound for the first transmission eigenvalue in terms of the first transmission eigenvalue for appropriate balls with constant index of refraction. Abstract: It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable in the $L^p$ sense when the conductivities are uniformly bounded in any fractional Sobolev space $W^{\alpha,p}$ $\alpha>0, 1 < p < \infty$. Abstract: For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV operators with different "boundary condition'', and the problem of reconstructing a CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by unitary truncation. Abstract: We consider the tomography problem of recovering a covector field on a simple Riemannian manifold based on its weighted Doppler transformation over a family of curves $\Gamma$. This is a generalization of the attenuated Doppler transform. Uniqueness is proven for a generic set of weights and families of curves under a condition on the weight function. This condition is satisfied in particular if the weight function is never zero, and its derivatives along the curves in $\Gamma$ are never zero. Abstract: The inverse problem for time-harmonic acoustic wave scattering to recover a sound-soft obstacle from a given incident field and the far field pattern of the scattered field is considered. We split this problem into two subproblems; first to reconstruct the shape from the modulus of the data and this is followed by employing the full far field pattern in a few measurement points to find the location of the obstacle. We extend a nonlinear integral equation approach for shape reconstruction from the modulus of the far field data [6] to the three-dimensional case. It is known, see [13], that the location of the obstacle cannot be reconstructed from only the modulus of the far field pattern since it is invariant under translations. However, employing the underlying invariance relation and using only few far field measurements in the backscattering direction we propose a novel approach for the localization of the obstacle. The efficient implementation of the method is described and the feasibility of the approach is illustrated by numerical examples. Abstract: Three dimensional anisotropic attenuating and scattering media sharing the same albedo operator have been shown to be related via a gauge transformation. Such transformations define an equivalence relation. We show that the gauge equivalence is also valid in media with non-constant index of refraction, modeled by a Riemannian metric. The two dimensional model is also investigated. Abstract: We present a comparison of three methods for the solution of the magnetoencephalography inverse problem. The methods are: a linearly constrained minimum variance beamformer, an algorithm implementing multiple signal classification with recursively applied projection and a particle filter for Bayesian tracking. Synthetic data with neurophysiological significance are analyzed by the three methods to recover position, orientation and amplitude of the active sources. Finally, a real data set evoked by a simple auditory stimulus is considered. Abstract: Wavelet inpainting problem consists of filling in missed data in the wavelet domain. In [17], Chan, Shen, and Zhou proposed an efficient method to recover piecewise constant or smooth images by combining total variation regularization and wavelet representations. In this paper, we extend it to nonlocal total variation regularization in order to recover textures and local geometry structures simultaneously. Moreover, we apply an efficient algorithm framework for both local and nonlocal regularizers. Extensive experimental results on a variety of loss scenarios and natural images validate the performance of this approach. Readers Authors Editors Referees Librarians Email Alert Add your name and e-mail address to receive news of forthcoming issues of this journal: [Back to Top]
I would approach this by building a general equilibrium model of the economy. Here is the high-level idea: There is a single perishable consumption good and all prices are measured in terms of it. There is a fully equity financed firm with one unit of share outstanding. The firm pays a continuous dividend at a stochastic rate $\delta$. You could for example start with a geometric Brownian motion\begin{equation}\mathrm{d}\delta_t = \mu \delta_t \mathrm{d}t + \sigma \delta_t \mathrm{d}W_t\end{equation} There is a continuum of identical risk-averse individuals that maximize their expected lifetime utility\begin{equation}\mathbb{E}_{\mathbb{P}} \left[ \int_0^\infty e^{-\rho v} u \left( C_v \right) \mathrm{d}v \right]\end{equation}by, at each point in time, choosing their optimal consumption and portfolio compositions. Here, $u \left( C_t \right)$ is their utility of consumption and $\rho$ is their rate of time preference. You can treat these as one "representative agent". You can introduce additional derivatives into your economy that are in zero net supply and whose payoffs are measurable w.r.t. the filtration generated by $W$. A trivial such example are zero-coupon bonds. You find the competitive equilibrium that solves the agents' expected utility maximization problem. In this equilibrium, the representative agent holds the one unit of stock, no derivatives and his consumption is equal to the dividend $C_t = \delta_t$. In equilibrium you obtain the dynamics for the stock price as well as valuation functions for all derivatives (and thus a risk-free rate). In my above example, you will find that the stock price also follows a geometric Brownian motion. Note that there are many ways to setup such an economy and often your assumptions will be a bit reversed-engineered from the desired result. You could certainly add other things like production to make your setup more realistic. Here are some references to get you started: The above setup is the pure-diffusion version of the Naik and Lee (1990) exchange economy. They consider a Merton jump-diffusion model. Many variations of this model have been used to find economically motivated measure changes for Levy processes, see e.g. Milne and Madan (1991) for the variance gamma model or Kou (2002) for the double exponential jump-diffusion model. A good reference for the general topic of continuous time consumption and portfolio choice are the books by Back (2010) and Pennacchi (2008). One of the ground braking papers is Cox et al. (1985). They construct a very general multi-factor production economy. This paper is a tough read though. References Back, Kerry E. (2010) "Asset Pricing and Portfolio Choice Theory", Financial Management Association Survey and Synthesis Series: Oxford University Press Cox, John C., Jonathan E. Ingersoll and Stephen and Stephen A. Ross (1985) "An Intertemporal General Equilibrium Model of Asset Prices", Econometrica, Vol. 53, No. 2, pp. 363-384 Kou, Steven G. (2002) "A Jump-Diffusion Model for Option Pricing", Management Science, Vol. 48, No. 8, pp. 1086-1101 Milne, Frank and Dilip B. Madan (1991) "Option Pricing with V.G. Martingale Components", Mathematical Finance, Vol. 1, No. 4, pp. 39-55 Pennachi, George G. (2008) "Theory of Asset Pricing", Pearson
Basic Examples on Parabolas Set 3 Example – 12 A circle on any focal chord of a parabola as diameter cuts the curve again in P and Q. Show that PQ passes through a fixed point. Solution: In example -8, we wrote the equation of a circle described on any focal chord as diameter in terms of one of its end points ${t_0}$. \[(x - at_0^2)\left( {x - \frac{a}{{t_0^2}}} \right) + (y - 2a{t_0})\left( {y + \frac{{2a}}{{{t_0}}}} \right) = 0 \qquad\qquad\qquad \dots \dots \left( 1 \right)\] The intersection of this circle with the parabola can be evaluated by using the parametric form of the parabola : \[x = a{t^2},\,\,y = 2at \qquad\qquad\qquad\qquad\qquad \dots\dots \left( 2 \right)\] Using (2) in (1), we obtain \[\begin{align}&{a^2}({t^2} - t_0^2)\left( {{t^2} - \frac{1}{{t_0^2}}} \right) + 4{a^2}(t - {t_0})\left( {t + \frac{1}{{{t_0}}}} \right) = 0\\& \Rightarrow {t^4} + {t^2}\left( {4 - t_0^2 - \frac{1}{{t_0^2}}} \right) - 4t\left( {{t_0} - \frac{1}{{{t_0}}}} \right) - 3 = 0 \qquad \qquad\qquad\dots \left( 3 \right) \end{align}\] This has four roots in t, meaning that the circle intersects the parabola in four points, two of which are obviously ${t_0}$ and $ \begin{align}- \frac{1}{{{t_0}}}\end{align}$ (since these are the points using which the circle has been described in the first place !). Let the other two point P and Q be ${t_1}\;and{\text { }}{t_2}$. Thus ${t_0},\,\, - \begin{align}\frac{1}{{{t_0}}}\end{align},\,\,{t_1},\,\,{t_2}$ are the roots of (3): \[\begin{align}& \qquad \;\;{t_0} + \left( { - \frac{1}{{{t_0}}}} \right) + {t_1} + {t_2} = 0\,\,\,;\,\,\,{t_0} \cdot \left( { - \frac{1}{{{t_0}}}} \right) \cdot {t_1} \cdot {t_2} = - 3\\& \Rightarrow \qquad {t_1} + {t_2} = \frac{1}{{{t_0}}} - {t_0}{\rm{ }}\,and \,{\rm{ }}{t_1}{t_2} = 3 \qquad \qquad \qquad \qquad\qquad\qquad\dots\left( 4 \right)\end{align}\] The co-ordinates of P and Q are $(at_1^2,\,\,2a{t_1}){\rm{ \, and}} \, {\rm{ }}(at_2^2,\,\,2a{t_2}).$ Thus, the equation of PQ becomes : \[\begin{align}& \qquad \quad \;\;PQ:\frac{{y - 2a{t_1}}}{{x - at_1^2}} = \frac{{2a({t_1} - {t_2})}}{{a(t_1^2 - t_2^2)}}\\&\Rightarrow \qquad ({t_1} + {t_2})y = 2x + 2a{t_1}{t_2}\qquad \qquad\qquad\qquad \dots \left( 5 \right)\end{align}\] Using (4) in (5), we obtain \[\left( {\frac{1}{{{t_0}}} - {t_0}} \right)y = 2x + 6a\] This is always satisfied by (–3a, 0), no matter what the value of ${t_0}$ may be. Thus, PQ always passes through the fixed point (–3a, 0). Example - 13 Find the length of the intercept that the parabola ${y^2} = 4ax$ makes on the line $y = mx + c.$ Solution: Let the end-points of the intercept be A and B : The co-ordinates of A and B can be evaluated by simultaneously solving the equation of the line and the parabola : \[\begin{align}&\qquad\quad \;\;y = mx + c\\\\&\qquad \quad \;\;{y^2} = 4ax\\\\& \Rightarrow \qquad {(mx + c)^2} = 4ax\\\\ &\Rightarrow \qquad {m^2}{x^2} + (2mc - 4a)x + {c^2} = 0 \qquad \qquad \dots \left( 1 \right)\end{align}\] The first point to be noted from this quadratic is that the line will intersect the parabola in two distinct points only if the D of (1) is positive, because only then will two distinct values of x be obtained (the case when $m = \infty $ gives two distinct points of intersection for one value of x, but that can be considered separately; in that case, the quadratic (1) will not be formed). We thus have, \[\begin{align}& \qquad\qquad D > 0\\\\& \Rightarrow \qquad{{{(mc - 2a)}^2} > {m^2}{c^2}}\\\\&\ \Rightarrow \qquad {4amc < 4{a^2}} \\\\&\Rightarrow \qquad \boxed{mc < a} \qquad \qquad (a \text{ is +ve here)}\end{align}\] Thus, the line will intersect the parabola only if a > mc. If a = mc, the line will be a tangent to the parabola. If a < cm, the line will not intersect the circle at all. Assuming that D > 0, suppose that the roots of (1) are ${x_1}\;and\;{\text{ }}{x_2}:$ \[\begin{align}&x + {x_2}= \frac{{4a - 2mc}}{{{m^2}}},\,\,{x_1}{x_2} = \frac{{{c^2}}}{{{m^2}}}\\&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Rightarrow \!\!\quad {({x_1} - {x_2})^2} = {({x_1} + {x_2})^2} - 4{x_1}{x_2}\\\\&\qquad\;\;\;= \frac{{{{(4a - 2mc)}^2}}}{{{m^4}}} - \frac{{4{c^2}}}{{{m^2}}}\\\\& \qquad\;\;\;= \frac{{16{a^2} + 4{m^2}{c^2} - 16amc - 4{m^2}{c^2}}}{{{m^4}}}\\\\ &\qquad\;\;\;= \frac{{16a(a - mc)}}{{{m^4}}}\end{align}\] The corresponding y co-ordinates for ${x_1}\,and \,{\text{ }}{x_2}$ are \[{{y}_{1}}=m{{x}_{1}}+c\text{ }and~\text{ }{{y}_{2}}=m{{x}_{2}}+c\] so that ${y_2} - {y_1} = m({x_2} - {x_1})$ . The length AB (intercept) is now simply \[\begin{align}AB &= \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} \\\\ &= \sqrt {{{({x_1} - {x_2})}^2} + {m^2}{{({x_1} - {x_2})}^2}} \\\\ &= \frac{{4\sqrt {a(a - mc)} }}{{{m^2}}} \cdot \sqrt {1 + {m^2}} \end{align}\] This result need not be remembered. What is important is that you understand the underlying approach. From this example, we can deduce one more useful thing : for a non-zero variable $m \in \mathbb{R},$ the line $y = mx + \begin{align}\frac{a}{m}\end{align}$ will always be a tangent to the circle. Verify that the point of contact of this tangent with the parabola is $\begin{align}\left( {\frac{a}{{{m^2}}},\frac{{2a}}{m}} \right).\end{align}$ We’ll discuss tangents in more detail in the next section.
Difference between revisions of "Con ZFC" Line 1: Line 1: − {{DISPLAYTITLE: Con(ZFC)}} + {{DISPLAYTITLE: Con(ZFC)}} + − The + The Gödel theorem that $\ZFC$ is consistent, then it does not prove Con(ZFC), and so the addition of this axiom is strictly stronger than ZFCalone. − + − + == Consistency hierarchy == == Consistency hierarchy == Line 9: Line 8: The expression $\text{Con}^2(\text{ZFC})$ denotes the assertion $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$, and iterating this more generally, one may consider the assertion $\text{Con}^\alpha(\text{ZFC})$ whenever $\alpha$ itself is expressible. The expression $\text{Con}^2(\text{ZFC})$ denotes the assertion $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$, and iterating this more generally, one may consider the assertion $\text{Con}^\alpha(\text{ZFC})$ whenever $\alpha$ itself is expressible. − == Every model of ZFC contains a model of ZFC as an element == + == Every model of ZFCcontains a model of ZFCas an element == − Every model $M$ of ZFC has an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of ZFC, as viewed externally from $M$. This is clear in the case where $M$ is an [[omega model \| $\omega$-model]] of ZFC, since in this case $M$ agrees that ZFC is consistent and can therefore build a Henkin model of ZFC. In the remaining case, $M$ has nonstandard natural numbers. By the [[reflection theorem]] applied in $M$, we know that the $\Sigma_n$ fragment of ZFC is true in models of the form $V_\beta^M$, for every standard natural number $n$. Since $M$ cannot identify its standard cut, it follows that there must be some nonstandard $n$ for which $M$ thinks some $V_\beta^M$ satisfies the (nonstandard) $\Sigma_n$ fragment of ZFC. Since $n$ is nonstandard, this includes the full standard theory of ZFC, as desired. + Every model $M$ of ZFChas an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of ZFC, as viewed externally from $M$. This is clear in the case where $M$ is an [[omega model \| $\omega$-model]] of ZFC, since in this case $M$ agrees that ZFCis consistent and can therefore build a Henkin model of ZFC. In the remaining case, $M$ has nonstandard natural numbers. By the [[reflection theorem]] applied in $M$, we know that the $\Sigma_n$ fragment of ZFCis true in models of the form $V_\beta^M$, for every standard natural number $n$. Since $M$ cannot identify its standard cut, it follows that there must be some nonstandard $n$ for which $M$ thinks some $V_\beta^M$ satisfies the (nonstandard) $\Sigma_n$ fragment of ZFC. Since $n$ is nonstandard, this includes the full standard theory of ZFC, as desired. − The fact mentioned in the previous paragraph is occasionally found to be surprising by some beginning set-theorists, perhaps because naively the conclusion seems to contradict the fact that there can be models of $\text{ZFC}+\neg\text{Con}(\text{ZFC})$. The paradox is resolved, however, by realizing that although the model $N$ inside $M$ is actually a model of full ZFC, the model $M$ need not agree that it is a model of ZFC, in the case that $M$ has nonstandard natural numbers and hence nonstandard length axioms of ZFC. + The fact mentioned in the previous paragraph is occasionally found to be surprising by some beginning set-theorists, perhaps because naively the conclusion seems to contradict the fact that there can be models of $\text{ZFC}+\neg\text{Con}(\text{ZFC})$. The paradox is resolved, however, by realizing that although the model $N$ inside $M$ is actually a model of full ZFC, the model $M$ need not agree that it is a model of ZFC, in the case that $M$ has nonstandard natural numbers and hence nonstandard length axioms of ZFC. Revision as of 13:59, 11 November 2017 The assertion $\text{Con(ZFC)}$ is the assertion that the theory $\text{ZFC}$ is consistent. This is an assertion with complexity $\Pi^0_1$ in arithmetic, since it is the assertion that every natural number is not the Gödel code of the proof of a contradiction from $\text{ZFC}$. Because of the Gödel completeness theorem, the assertion is equivalent to the assertion that the theory $\text{ZFC}$ has a model $\langle M,\hat\in\rangle$. One such model is the Henkin model, built in the syntactic procedure from any complete consistent Henkin theory extending $\text{ZFC}$. In general, one may not assume that $\hat\in$ is the actual set membership relation, since this would make the model a transitive model of $\text{ZFC}$, whose existence is a strictly stronger assertion than $\text{Con(ZFC)}$. The Gödel incompleteness theorem implies that if $\text{ZFC}$ is consistent, then it does not prove $\text{Con(ZFC)}$, and so the addition of this axiom is strictly stronger than $\text{ZFC}$ alone. Consistency hierarchy The expression $\text{Con}^2(\text{ZFC})$ denotes the assertion $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$, and iterating this more generally, one may consider the assertion $\text{Con}^\alpha(\text{ZFC})$ whenever $\alpha$ itself is expressible. Every model of $\text{ZFC}$ contains a model of $\text{ZFC}$ as an element Every model $M$ of $\text{ZFC}$ has an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of $\text{ZFC}$, as viewed externally from $M$. This is clear in the case where $M$ is an $\omega$-model of $\text{ZFC}$, since in this case $M$ agrees that $\text{ZFC}$ is consistent and can therefore build a Henkin model of $\text{ZFC}$. In the remaining case, $M$ has nonstandard natural numbers. By the reflection theorem applied in $M$, we know that the $\Sigma_n$ fragment of $\text{ZFC}$ is true in models of the form $V_\beta^M$, for every standard natural number $n$. Since $M$ cannot identify its standard cut, it follows that there must be some nonstandard $n$ for which $M$ thinks some $V_\beta^M$ satisfies the (nonstandard) $\Sigma_n$ fragment of $\text{ZFC}$. Since $n$ is nonstandard, this includes the full standard theory of $\text{ZFC}$, as desired. The fact mentioned in the previous paragraph is occasionally found to be surprising by some beginning set-theorists, perhaps because naively the conclusion seems to contradict the fact that there can be models of $\text{ZFC}+\neg\text{Con}(\text{ZFC})$. The paradox is resolved, however, by realizing that although the model $N$ inside $M$ is actually a model of full $\text{ZFC}$, the model $M$ need not agree that it is a model of $\text{ZFC}$, in the case that $M$ has nonstandard natural numbers and hence nonstandard length axioms of $\text{ZFC}$.
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 $0$.
Definition:Set Theory Contents Definition There are several "versions" of set theory, all of which share the same basic ideas but whose foundations are completely different. A popular alternative (and inaccurate) definition describes this as a non-formalized definition of set theory which describes sets and the relations between them using natural language. Let: $\displaystyle V_1$ $=$ $\displaystyle \set {v_1, v_3, v_4}$ $\displaystyle V_2$ $=$ $\displaystyle \set {v_2, v_5}$ $\displaystyle V_3$ $=$ $\displaystyle \set {v_1, v_3}$ Then: $\displaystyle V_1 \cup V_2$ $=$ $\displaystyle \set {v_1, v_2, v_3, v_4, v_5}$ $\displaystyle V_1 \cup V_3$ $=$ $\displaystyle \set {v_1, v_3, v_4}$ $\displaystyle V_2 \cup V_3$ $=$ $\displaystyle \set {v_1, v_2, v_3, v_5}$ $\displaystyle V_1 \cap V_2$ $=$ $\displaystyle \O$ $\displaystyle V_1 \cap V_3$ $=$ $\displaystyle \set {v_1, v_3}$ $\displaystyle V_2 \cap V_3$ $=$ $\displaystyle \O$ Thus: Let: $\displaystyle A$ $=$ $\displaystyle \set {1, 2}$ $\displaystyle B$ $=$ $\displaystyle \set {1, \set 2}$ $\displaystyle C$ $=$ $\displaystyle \set {\set 1, \set 2}$ $\displaystyle D$ $=$ $\displaystyle \set {\set 1, \set 2, \set {1, 2} }$ Then: $\displaystyle A \cap B$ $=$ $\displaystyle \set 1$ $\displaystyle \paren {B \cap D} \cup A$ $=$ $\displaystyle \set {1, 2, \set 2}$ $\displaystyle \paren {A \cap B} \cup D$ $=$ $\displaystyle \set {1, \set 1, \set 2, \set {1, 2} }$ $\displaystyle \paren {A \cap B} \cup \paren {C \cap D}$ $=$ $\displaystyle \set {1, \set 1, \set 2}$ Let there exist $X \subseteq S$ such that: $A \cup \paren {X \cap B} = C$ $\paren {A \cup X} \cap B = D$ Then: $A \cap B \subseteq D \subseteq B$ and: $A \cup D = C$ $\blacksquare$ Also see Results about set theorycan be found here. Set theory arose from an attempt to comprehend the question: "What is a number?" The main initial development of the subject was in fact not directly generated as a result of trying to answer this question, but as a result of Georg Cantor's work around $1870$ to understand the nature of infinite series and related subjects. Cantor ....is usually considered the founder of set theory as a mathematical discipline ... Sources ... General set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it. 1975: T.S. Blyth: Set Theory and Abstract Algebra... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems It would be completely out of the question at this stage ... to attempt an axiomatisation of such topics ... 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set? In set theory, there is really only one fundamental notion:
Naidu, Thirupathi D and Kumar, AV and Chakradhar, RPS and Ratnakaram, YC (2006) Spectral studies of $Sm^{3+}$ and $Dy^{3+}$ doped lithium cesium mixed alkali borate glasses. In: International Symposium on Non Oxide and new Optical Glasses (ISNOG), 10 - 14 April, 2006, Indian Institue of Science, Bangalore. PDF Chakradhar2.pdf Download (80kB) Abstract The effect of host glass composition on the optical absorption and fluorescence spectra of $Sm^{3+}$ and $Dy^{3+}$ has been studied in mixed alkali borate glasses of the type $67B_2O_3{\cdot}xLi_2O{\cdot}(32-x)Cs_2O$ (x=8, 12, 16, 20 and 24). The Judd-Ofelt intensity parameters $(\Omega_2, \Omega_4\hspace{2mm}and\hspace{2mm}\Omega_6)$ are calculated. The radiative transition probabilities (A), radiative lifetimes $(\tau_R)$, branching ratios $(\beta)$ and integrated absorption cross sections $(\Sigma)$ are computed for certain excited states of $Sm^{3+}$ and $Dy^{3+}$ ions for different x values in the glass matrix. Stimulated emission cross sections (sp) are obtained for certain emission transitions of two ions in these mixed alkali borate glasses. These parameters are compared for different x values in the glass matrix. Variation of these parameters with x in the glass matrix has been studied. Item Type: Conference Poster Keywords: optical absorption;Emission;Sm3+ and Dy3+ ions;Lithium cesium mixed alkali borate glasses Department/Centre: Division of Physical & Mathematical Sciences > Physics Depositing User: Sreekanth Chakradhar Ph.D., Dr. R. P. Date Deposited: 10 Oct 2006 Last Modified: 19 Sep 2010 04:23 URI: http://eprints.iisc.ac.in/id/eprint/5408 Actions (login required) View Item
Calculate the longest wavelength of the electromagnetic radiation emitted by the hydrogen atom in undergoing a transition from the n = 7 level. Solution: Show me the final answer↓ The electron will go from n = 7 to the n = 6 level. We will use the following equation to figure out the frequency. (Where E is energy, h is Planck’s constant, and v is the frequency) Since we know the electron undergoes a transition from n = 7 to n = 6, we can write the following: (simplify by finding a common denominator)\dfrac{-R_H}{49}\,-\,\dfrac{-R_H}{36}\,=\,\dfrac{13R_H}{1764} We can now write the frequency emitted: v\,=\,\dfrac{13R_H}{1764h} v\,=\,\dfrac{13}{1764}\times \dfrac{2.179\times 10^{-18}\text{J}}{6.626\times 10^{-34}\text{J}\cdot \text{s}} v\,=\,2.42\times 10^{13} /s We can now figure out the wavelength by using the following equation: (Where \lambda is the wavelength, c is the speed of light, v is the frequency) \lambda\,=\,\dfrac{2.998\times 10^8\text{m/s}}{2.42\times 10^{13}/s} \lambda\,=\,1.238\times 10^{-5} m Final Answer:
This question already has an answer here: I was given the following problem: Let $f:[0,1] \to \Bbb R $ be a differentiable function on $[0,1]$ such that $f(0)=0$, and $\forall x \in[0,1]$ : $\lvert f'(x)\rvert \le \lvert f(x)\rvert$. Prove that $f(x)=0$ $\forall x \in [0,1]$. I tried to solve it using mean value theorem iteratively, and got to a point where I have a series that holds $f(x_1) \ge f(x_2) \ge f(x_3) \ge \dots $ pretty much stuck from here, is it a good start? any advice on how to move on / solve it differently? Thanks
Kodai Mathematical Journal Kodai Math. J. Volume 27, Number 3 (2004), 354-359. Mean growth of the derivative of a Blaschke product Abstract If $B$ is a Blaschke product with zeros $\{a_n\}$ and if $\sum_n(1-\|a_n\|)^{\alpha}$ is finite for some $\alpha \in (1/2,1]$, then limits are found on the rate of growth of $\int_0^{2\pi} \|B'(re^{it}\|^p\, dt$ in agreement with a known result for $\alpha \in (0,1/2)$. Also, a converse is established in the case of an interpolating Blaschke product, whenever $0<\alpha<1$. Article information Source Kodai Math. J., Volume 27, Number 3 (2004), 354-359. Dates First available in Project Euclid: 28 December 2004 Permanent link to this document https://projecteuclid.org/euclid.kmj/1104247356 Digital Object Identifier doi:10.2996/kmj/1104247356 Mathematical Reviews number (MathSciNet) MR2100928 Zentralblatt MATH identifier 1083.30033 Citation Protas, David. Mean growth of the derivative of a Blaschke product. Kodai Math. J. 27 (2004), no. 3, 354--359. doi:10.2996/kmj/1104247356. https://projecteuclid.org/euclid.kmj/1104247356
As whuber has commented: the Kolmogorov-Smirnov test is only valid as a comparison against a fully specified distribution. You cannot use it to compare an observed distribution against a distribution whose parameters have been estimated based on your observed sample. If you do so, your p-values will not be uniformly distributed under the null hypothesis, but show the exact pattern you have observed. This is unfortunately an extremely common error, which you can very often find in online tutorials. As a little illustration, let us simulate $x_1, \dots, x_{20}\sim N(0,1)$, then run a K-S test first against a fully specified $N(0,1)$ distribution, then against an estimated $N(\hat{\mu},\hat{\sigma}^2)$ distribution, where $\hat{\mu}$ and $\hat{\sigma}^2$ are estimated based on $x_1, \dots, x_{20}$. Record the $p$ value. Do this 10,000 times. Here are histograms of the $p$ values: As you see, the $p$ values of the tests against a fully specified distribution are uniformly distributed, as they should be, but the ones from a fitted distribution are anything but. n_sims <- 1e4 nn <- 20 pp_estimated <- pp_specified <- rep(NA,n_sims) pb <- winProgressBar(max=n_sims) for ( ii in 1:n_sims ) { setWinProgressBar(pb,ii,paste(ii,"of",n_sims)) set.seed(ii) sim <- rnorm(nn) pp_specified[ii] <- ks.test(sim,y="pnorm",mean=0,sd=1)$p pp_estimated[ii] <- ks.test(sim,y="pnorm",mean=mean(sim),sd=sd(sim))$p } close(pb) opar <- par(mfrow=c(1,2)) hist(pp_specified,main="Parameters specified",xlab="",col="lightgray") hist(pp_estimated,main="Parameters estimated",xlab="",col="lightgray") par(opar) If your hypothesized reference distribution is normal, but you need to estimate the mean and variance, then the Lilliefors test would be appropriate. Other approaches may work for other distribution types. You may want to ask a specific question for the distribution type you are interested in. I do not know of general framework for goodness-of-fit tests for fitted distributions. (As an extreme example, you could always use the empirical distribution of the data you observe. Of course, the fit would be perfect. But this would also likely not be very informative.) EDIT - I just asked the general question here: Goodness of fit to a fitted distribution.
Resolution Of Vectors RESOLUTION OF A VECTOR IN A GIVEN BASIS Consider two non-collinear vectors $\vec a\,\,{\text{and}}\,\,\vec b$; as discussed earlier, these will form a basis of the plane in which they lie. Any vector $\vec r$ in the plane of $\vec a\,\,{\text{and}}\,\,\vec b$ can be expressed as a linear combination of $\vec a\,\,{\text{and}}\,\,\vec b$: The vectors $\overrightarrow {OA} \,\,\,{\text{and}}\,\,\overrightarrow {OB} $ are called the components of the vector $\vec r$ along the basis formed by $\vec a\,\,{\text{and}}\,\,\vec b$ . This is also stated by saying that the vector $\vec r$ when resolved along the basis formed by $\vec a\,\,{\text{and}}\,\,\vec b$ , gives the components $\overrightarrow {OA} \,\,\,{\text{and}}\,\,\overrightarrow {OB} $ . Also, as discussed earlier, the resolution of any vector along a given basis will be unique. We can extend this to the three dimensional case: an arbitrary vector can be resolved along the basis formed by any three non-coplanar vectors, giving us three corresponding components. Refer to Fig - 20 for a visual picture. RECTANGULAR RESOLUTION Let us select as the basis for a plane, a pair of unit vector $\hat i\,\,{\text{and}}\,\,\hat j$ perpendicular to each other. Any vector $\vec r$ in this basis can be written as \[\begin{align}&\vec r = \overrightarrow {OA} + \overrightarrow {OB} \hfill \\&\;= \left( {\left\| {\vec r} \right\|\cos \theta } \right)\hat i + \left( {\left\| {\vec r} \right\|\sin \theta } \right)\hat j \hfill \\&\;= x\hat i + y\hat j \hfill \\ \end{align} \] where x and y are referred to as the x and y components of $\vec r$. For 3-D space, we select three unit vectors $\hat i,\hat j\,\,{\text{and}}\,\,\hat k$ each perpendicular to the other two. In this case, any vector $\vec r$ will have three corresponding components, generally denoted by x, y and z. We thus have \[\vec r = x\hat i + y\hat j + z\hat k\] The basis ($\hat i,\hat j$) for the two dimensional case and ($\hat i,\hat j,\,\,\hat k$) for the three-dimensional case are referred to as rectangular basis and are extremely convenient to work with. Unless otherwise stated, we’ll always be using a rectangular basis from now on. Also, we’ll always be implicitly assuming that we’re working in three dimensions since that automatically covers the two dimensional case.
It is already known, that for $latex { Y\in {\mathbb R} }&fg=000000$ and $latex { X \in {\mathbb R}^{p} }&fg=000000$, the regression problem $latex \displaystyle Y = f(\mathbf{X}) + \varepsilon, &fg=000000$ when $latex { p }&fg=000000$ is larger than the data available, it is well-known that the curse of dimensionality problem arises. Richard E. Bellman (see [1]) used this terminology when he was considering problems in dynamic optimization. He found that as the $latex { \mathbf{X} }&fg=000000$’s dimension increases, so the data sparsity as well. The Sliced Inverse Regression (SIR) method, which we have discussed before, is a technique to reduce the dimensionality for this kind of problems. This method focuses in estimate the eigenvector associated to largest eigenvalues of $latex \displaystyle \Lambda = \mathop{\mathrm{Cov}}(\mathbb E[\mathbf{X}\vert Y]). &fg=000000$ This post pretends to be a little review of the method presented by [6]. In this paper, they use a kernel method to estimate each matrix’s coefficient. Firstly, we use an Nadaraya-Watson kernel estimator to compute $latex { \mathbb E[X\vert Y] }&fg=000000$ for $latex { Y }&fg=000000$ fixed. Then with this quantity, we estimate empirically $latex { {\mathbb C}(\mathbb E [X\vert Y]) }&fg=000000$ (I’ll put some details below). Theorems about the asymptotic normality for this estimator are shown. Also, using perturbation theory for linear operators (see [2]), they show that the eigenvalues and eigenvector also converge to a normal distribution. To explain briefly the method, I shall introduce some notation. If someone gets lost in the middle, please let me know it in the comments. Write $latex {(\mathbf{X},Y)}&fg=000000$ and its independent copies $latex {(\mathbf{X}_{i},Y_i)}&fg=000000$ as $latex \displaystyle \begin{array}{rl} (\mathbf{X},Y) & =(X_{1},\ldots,X_{p},Y)^{\top}, \\ (\mathbf{X}_{j},Y_j) & =(X_{1j},\ldots,X_{pj},Y_j)^{\top},\quad j=1,\ldots,n. \end{array} &fg=000000$ Notice that if $latex { \mathbb E[\mathbf{X}] =\mathbf{0} }&fg=000000$, then $latex \displaystyle \Lambda = \mathop{\mathrm{Cov}}(\mathbf{X}\vert Y) = \mathbb E[\mathbb E[\mathbf{X}\vert Y ] \mathbb E[\mathbf{X}\vert Y ]^\top ] &fg=000000$ We denote $latex \displaystyle \mathbb E[\mathbf{X}\vert Y ] = \mathbf{R}(Y) = (R_1(Y), \ldots,R_p(Y)) &fg=000000$ Now, the trick is to use the Nadaraya-Watson estimator into every element of $latex { \mathbf{R}(Y) }&fg=000000$. Let us, $latex \displaystyle \hat{R}_i(Y)=\frac{\hat{g}_i(Y)}{\hat{f}(Y)} &fg=000000$ where $latex \displaystyle \begin{array}{rl} \hat{g}_{i}(Y) & = \displaystyle \frac{1}{nh}\sum_{k=1}^{n}X_{jk}K_{h} \left(\frac{Y-Y_{k}}{h}\right),\nonumber \\ \hat{f}(Y) & = \displaystyle \frac{1}{nh}\sum_{k=1}^{n}K_{h}\left(\frac{Y-Y_{k}}{h}\right), \end{array} &fg=000000$ and the function $latex {K(u)}&fg=000000$ satisfies the classical assumption for a kernel (See [4]). Defining $latex \displaystyle \begin{array}{rl} \mathbf{\hat{R}}(Y) & =(\hat{R}_{1}(Y),\ldots,\hat{R}_{p}(Y))^{\top} \end{array} &fg=000000$ it is possible to estimate $latex { \Lambda }&fg=000000$ by $latex \displaystyle \hat{\Lambda}_n =\frac{1}{n} \sum_{j=1}^{n} \left( \mathbf{\hat{R}}(Y_j) \right) \left( \mathbf{\hat{R}}(Y_j) \right)^\top &fg=000000$ Remark Actually, the estimator $latex { \hat{\Lambda}_n }&fg=000000$ takes account that $latex { f(Y) }&fg=000000$ and $latex { \hat{f}(Y) }&fg=000000$ do not get small values. We do not present all the details to keep the ideas simple. The principal theorem is that $latex { \hat{\Lambda}_n }&fg=000000$ is asymptotically normal. In other words, they show that $latex \displaystyle \sqrt{n} (\hat{\Lambda}_n – \Lambda) \Rightarrow \mathbf{H} &fg=000000$ where $latex \displaystyle \lambda^\top \mathop{\mathrm{vech}}(\mathbf{H}) \sim N(0,\sigma_\lambda^2 )\quad \forall \lambda\neq 0&fg=000000$ and $latex \displaystyle \begin{array}{rl} V(X,Y) &= \displaystyle\frac{1}{2}\left(X_i R_l(Y) + X_l R_i(Y) \right) \\ \sigma_\lambda^2 & = \lambda^\top \mathop{\mathrm{Cov}} (\mathop{\mathrm{vech}}(V(X,Y))) \lambda,\quad \lambda \in {\mathbb R} ^ {d(d+1)/2} \end{array} &fg=000000$ I am not going to present the results about the asymptotic normality for the eigenvalues and eigenvectors. I encourage to the interested reader to take a look to the paper. The proof for the $latex { \hat{\Lambda}_n }&fg=000000$’s asymptotic convergence basically consist in decompose it into several addends and show that most of them are significantly small with respect to the leading term $latex { V(X,Y) }&fg=000000$. To control those terms, the paper displays two interesting secondary results, $latex \displaystyle \begin{array}{rl} \displaystyle \sup_y \left\vert \hat{f}(y) – f(y) \right\vert & = \displaystyle O\left(h^4 +\frac{\log n}{n^{1/2} h}\right),\\ \displaystyle \sup_y \left\vert \hat{g}_i(y) – g_i(y) \right\vert & = \displaystyle O_p\left(h^4 +\frac{\log n}{n^{1/2} h}\right). \end{array} &fg=000000$ These results allow us to control $latex { \hat{f} }&fg=000000$ and $latex { \hat{g}_i }&fg=000000$ convergence at rate $latex { h^4 +n^{-1/2} h^{-1}\log n }&fg=000000$, the former punctually and the latter in probability. You can find all the proofs and the complete discussion in [5] and [3]. References Bellman, R. (1957). Dynamic programming. Princeton University Press, Princeton, N. J. Kato, T. (1995). Perturbation theory for linear operators, volume 132. Springer Verlag. Prakasa Rao, B. (1983). Nonparametric functional estimation. Probability and mathematical statistics. Academic Press, New York. Tsybakov, A. B. (2008). Introduction to nonparametric estimation. Springer series in statistics. Springer. Zhu, L.-X. (1993). Convergence rates of empirical processes indexed by classes of functions and their applications. J. Syst. Sci. Math. Sci, 13:3341. Zhu, L.-X. and Fang, K.-T. (1996). Asymptotics for kernel estimate of sliced inverse regression. The Annals of Statistics, 24(3):10531068. Related articles Paper’s review: Zhu & Fang, 1996. Asymptotics for kernel estimate of sliced inverse regression. (maikolsolis.wordpress.com)
December 14th, 2018, 08:02 AM # 1 Newbie Joined: Dec 2018 From: Germany Posts: 2 Thanks: 0 Ackermann function modulo calculations There's a programming puzzle at Programming Praxis in which one is to calculate values of the Ackermann function A(m, n). One comment was particularly helpful, you can see it . here The function values of A(m, n) are known to get gigantic for m >= 4. Therefore I'd like to approach it with modulo operations. With an additional function "tetration" the code in Python looks like this: Code: def tetration(a, b, mod): '' t0 = 1 while b > 0: t0 = pow(a, t0, mod) b -= 1 return t0 def hyper(n, a, b, mod): '' if n == 0: return (b + 1) % mod if n == 1: return (a + b) % mod if n == 2: return (a * b) % mod if n == 3: return pow(a, b, mod) if n == 4: return tetration(a, b, mod) if b == 0: return 1 x = a for _ in range(b - 1): x = hyper(n-1, a, x, mod) return x % mod def ackermann(m, n, mod): '' return hyper(m, 2, n + 3, mod) - (3 % mod) if __name__ == "__main__": '' print (ackermann(4, 2, 10*9 + 7)) Now, here's my question: Is there some (mathematical) way to considerably speed things up for values of m >= 5? The above code works well in terms of runtime for small moduli, but as they get bigger runtime increases unbearably. Already thanks for having a look and helping. Last edited by skipjack; December 14th, 2018 at 10:31 AM. December 19th, 2018, 06:37 AM # 3 Newbie Joined: Dec 2018 From: Germany Posts: 2 Thanks: 0 @billymac00.... I had already tried that link, but didn't find anything which might point me in the right direction. Perhaps you have an idea where to look closer? January 1st, 2019, 07:19 PM # 4 Newbie Joined: Dec 2018 From: Euclidean Plane Posts: 7 Thanks: 3 As indicated in your code, the Ackermann function is basically a power tower of twos when m >= 4: $\displaystyle \begin{align} A(4,0) + 3 &= 2^{2^2}\\ A(4, 1) + 3 &= 2^{2^{2^2}}\\ A(4, 2) + 3 &= 2^{2^{2^{2^2}}}\\ \end{align} $ and so forth. Now, I don't know how much number theory you know, but for a given modulus M, power towers will "stabilize" pretty quickly mod M. Here's an example of what I mean for M=19: $\displaystyle \begin{align} 2 \equiv 2 \text{ (mod }19\text{)}&\\ 2^2 \equiv 4 \text{ (mod }19\text{)}&\\ A(4,0) + 3 = 2^{2^2} \equiv 16 \text{ (mod }19\text{)}&\\ A(4, 1) + 3 = 2^{2^{2^2}} \equiv 5 \text{ (mod }19\text{)}&\\ A(4, 2) + 3 = 2^{2^{2^{2^2}}} \equiv 5 \text{ (mod }19\text{)}&\\ A(4, 3) + 3 = 2^{2^{2^{2^{2^2}}}} \equiv 5 \text{ (mod }19\text{)}& \end{align*} $ and in fact any power tower of twos with more than 3 twos will be congruent to 5 modulo 19. Proving this fact uses Euler's Theorem repeatedly, one version of which can be stated as follows: If $a$ is relatively prime to $M$, then $$a^x \equiv a^y \;\text{ mod }M \qquad \text{ if } \qquad x \equiv y \;\text{ mod }\phi(M)$$ where $\phi(M)$ is the Euler totient function. (I won't go into the details now since I'm not sure how much you already know, but I can provide further details if you like.) For your modulus, the power tower stabilizes after only 7 twos: For any n >= 4, $$A(4, n) \equiv A(4, 4) = 2^{2^{2^{2^{2^{2^2}}}}}-3 \equiv 661944223 \;\;\text{(mod } 10^9 + 7 \text{)}.$$ From here you can calculate all of the rest of the values of the Ackermann function modulo your prime: $$A(5, 0) \equiv A(4, 1) = 2^{2^{2^2}}-3 \equiv 65540\;\;\text{(mod } 10^9 + 7 \text{)}$$ and everything stabilizes after that point: For all n>0, $$A(5, n) = A(4, A(5,n-1)) \equiv A(4,4) \equiv 661944223 \;\;\text{(mod } 10^9 + 7 \text{)},$$ and similarly for all m > 5 $$A(m, n) \equiv A(4,4) \equiv 661944223 \;\;\text{(mod } 10^9 + 7 \text{)}.$$ Last edited by skipjack; January 2nd, 2019 at 04:46 AM. Tags ackermann, calculations, function, modulo Thread Tools Display Modes Similar Threads Thread Thread Starter Forum Replies Last Post Ackermann function and primitive recursive functions safyras Linear Algebra 4 May 11th, 2014 11:59 AM is modulo operation a bijective function? mathisdrama Number Theory 0 February 13th, 2012 11:13 AM Ackermann's function A (2,2) mathproblems Applied Math 0 December 4th, 2011 10:18 AM solution to complex hyper-operators/Ackermann function jamesuminator Number Theory 5 June 25th, 2011 12:16 PM Ackermann's Function aikiart Computer Science 6 August 26th, 2010 11:50 AM
Several online sources (e.g. Wikipedia, the nLab) assert that the Gelfand representation defines a contravariant equivalence from the category of (non-unital) commutative $C^{\ast}$-algebras to the category of locally compact Hausdorff (LCH) spaces. This seems wrong to me. The naive choice is to take all continuous maps between LCH spaces. This doesn't work. For example, the constant map $\mathbb{R} \to \bullet$ does not come from a morphism $\mathbb{C} \to C_0(\mathbb{R})$, the problem being that composing with the map $\bullet \to \mathbb{C}$ sending $\bullet$ to $1$ gives a function on $\mathbb{R}$ which doesn't vanish at infinity. It is necessary for us to restrict our attention to proper maps. But this still doesn't work. If $A, B$ are any commutative $C^{\ast}$-algebras we can consider the morphism $$A \ni a \mapsto (a, 0) \in A \times B.$$ This morphism does not define a map on Gelfand spectra; if $\lambda : A \times B \to \mathbb{C}$ is a character factoring through the projection $A \times B \to B$, then composing with the above morphism gives the zero map $A \to \mathbb{C}$. This contradicts the nLab's claim that taking Gelfand spectra gives a functor into locally compact Hausdorff spaces (if one requires that the morphisms are defined everywhere on the latter category). The correct statement appears to be that commutative $C^{\ast}$-algebras are contravariantly equivalent to the category $\text{CHaus}_{\bullet}$ of pointed compact Hausdorff spaces; the functor takes an algebra to the Gelfand spectrum of its unitization (we adjoin a unit whether or not the algebra already had one). There is an inclusion of the category of LCH spaces and proper maps into this category but it is not an equivalence because maps $(C, \bullet) \to (D, \bullet)$ in $\text{CHaus}_{\bullet}$ may send points other than the distinguished point of $C$ to the distinguished point of $D$. So do sources mean something else when they claim the equivalence with locally compact Hausdorff spaces?
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September 13th, 2018, 06:33 PM # 1 Newbie Joined: Aug 2018 From: United States Posts: 2 Thanks: 0 geometric transformations Matt wants to reflect a shape over the x-axis and then reflect it over the y-axis. Cathy says that this would be the same as rotating the shape 180 degrees. Do you agree or disagree with Cathy? Justify your answer and use matrix transformations as part of your justification. thanks for any help!! September 13th, 2018, 06:54 PM # 2 Senior Member Joined: Sep 2015 From: USA Posts: 2,553 Thanks: 1403 suppose we have a point $(x,y)$ Reflecting it would be accomplished by pre-multiplying by the matrix $ref_x = \begin{pmatrix}-1 &0 \\ 0 &1\end{pmatrix}$ similarly $ref_y = \begin{pmatrix}1 &0 \\ 0 &-1\end{pmatrix}$ multiplying these two reflections matrices $ref_y ref_x = \begin{pmatrix}-1 &0 \\ 0 &-1 \end{pmatrix}$ a rotation matrix of $\pi$ radians is given by $rot_\pi = \begin{pmatrix}\cos(\pi) &\sin(\pi) \\ -\sin(\pi) &\cos(\pi) \end{pmatrix} =\begin{pmatrix}-1 &0 \\ 0 &-1 \end{pmatrix}$ and we see that the two reflections are indeed equivalent to the single rotation by $\pi$ September 13th, 2018, 08:17 PM # 3 Global Moderator Joined: Dec 2006 Posts: 20,978 Thanks: 2229 Your $ref_x$, $ref_y$ and $rot$ are usually denoted by $\operatorname{Ref_y}$, $\operatorname{Ref_x}$ and $\operatorname{Rot}$ respectively. If the point $(x,~ y)$ is written as a row vector, use post-multiplication. If the point $(x,~ y)$ is written as a column vector, use pre-multiplication. Tags geometric, transformations Thread Tools Display Modes Similar Threads Thread Thread Starter Forum Replies Last Post Transformations u17159BR Trigonometry 2 May 31st, 2018 04:19 PM transformations laxus95 Linear Algebra 2 December 11th, 2013 05:40 PM exp - log transformations reto11 Applied Math 1 October 18th, 2010 10:08 PM Transformations Adrian Algebra 10 July 7th, 2010 04:40 PM Transformations math8553 Linear Algebra 1 December 2nd, 2008 05:34 PM
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?
Category:Zermelo-Fraenkel Class Theory In $\textrm{ZF}$, classes are written using class builder notation: $\left\{{x : P \left({x}\right)}\right\}$ $\displaystyle y \in \left\{ {x: P \left({x}\right)}\right\}$ $\quad \text{for} \quad$ $\displaystyle P \left({y}\right)$ $\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in y$ $\quad \text{for} \quad$ $\displaystyle \exists z \in y: \forall x: \left({x \in z \iff P \left({x}\right)}\right)$ $\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in \left\{ {y: Q \left({y}\right)}\right\}$ $\quad \text{for} \quad$ $\displaystyle \exists z: \left({Q \left({z}\right) \land \forall x: \left({x \in z \iff P \left({x}\right)}\right)}\right)$ where: $x, y ,z$ are variables of $\textrm{ZF}$ $P, Q$ are propositional functions. Through these "rules", every statement involving $\left\{{x : P \left({x}\right) }\right\}$ can be reduced to a simpler statement involving only the basic language of set theory. Subcategories This category has only the following subcategory. A ► Axiom of Foundation (7 P) Pages in category "Zermelo-Fraenkel Class Theory" The following 11 pages are in this category, out of 11 total.
Using the Flow Coefficient to Characterize the Performance of a Piping System Fluid flow through a piping system that consists of components such as valves, fittings, heat exchangers, nozzles, filters, and pipelines will result in a loss of energy due to the friction between the fluid and internal surfaces, changes in the direction of the flow path, obstructions in the flow path, and changes in the cross-section and shape of the flow path. The energy loss, or head loss, will be seen as a pressure drop across the piping system and each component in the system. There are different ways to characterize the impact of these factors with regard to the flow rate through the component and the resulting pressure drop. One way is to characterize the amount of that is offered by the system or components. The resistance can be given a numerical value as a resistance coefficient (K) or by the pipe length (or equivalent length), which can be used in the 2 resistance ndorder Darcy equation to calculate the amount of head loss: H_{L} = f \frac{L}{D}\frac{v^2}{2g} \text{ or, } H_{L} = K\frac{v^2}{2g} Where: H L = head loss (ft) f = Darcy friction factor (unitless) L eqv = pipe length (ft) D = pipe internal diameter (ft) v = fluid velocity (ft/sec) g = gravitational constant (ft/sec 2) K= resistance coefficient (unitless) The pressure drop resulting from the head loss is given by: dP = \frac{\rho H_L}{144} Where: dP = pressure drop (psi) rho = fluid density (lb/ft 3) The resistance of a piping system is constant when all the valves (including control valves) remain in one position. When the position of a valve is changed, the resistance of the valve and therefore the entire system is changed. The change in resistance results in a change of flow rate and pressure drop in the system. Conversely, if the resistance remains constant and the differential pressure changes, the effect on the flow rate can be calculated. Another way to characterize the performance of the system or a component is by its . The capacity can be expressed with a numerical value as a flow coefficient (C capacity V) that equates the amount of pressure drop to the flow rate by the following equation: Q = C_v\sqrt{\frac{dP}{SG}} \text{ therefore, } C_v = \frac{Q}{\sqrt{\frac{dP}{SG}}} Where: C v = flow coefficient (unitless) Q = flow rate (gpm) SG = fluid specific gravity The flow coefficient is commonly used by control valve manufacturers to characterize the performance of their valves with a change in valve position. Nozzle and sprinkler manufacturers also use the flow coefficient, except that they generally refer to it as a discharge coefficient. For a nozzle or sprinkler, when the inlet pressure changes, the flow rate through the nozzle will change proportionally, as given by the C V equation above and holding C V constant. Just as with the system resistance, the system capacity remains constant as long as the positions of the valves in the system are constant. A constant flow coefficient allows for calculating the flow rate when the differential pressure changes. This concept can be used to simplify piping systems for analysis using Engineered Software's PIPE-FLO or Flow of Fluids programs. If the pressure (or differential pressure) and flow rate at a point in the system is known and the system resistance downstream from this point is constant, the downstream system can be simplified and represented with a flow coefficient, C V. Consider the following simple system of water at 60 °F being fed from a 10 psig pressure source. The upper branch goes to a 5 psig boundary pressure and the lower branch goes to a boundary flow rate of 50 gpm. The inlet and both outlet boundary conditions are set by the user. PIPE-FLO calculates the flow rate in the upper branch (246.5 gpm) and the pressure at the outlet of the lower branch (7.847 psig). To determine what happens to the flow rate and pressure in the upper branch when the flow rate in the lower branch changes, assuming the configuration of the upper branch remains the same (i.e. valve positions in the upper branch are not changed), the system should be modified to obtain an equivalent system, as shown below. A short length of pipe (0.0001 ft) is added to the upper branch at the outlet with a boundary pressure of 0 psig and a fixed C V fitting installed in the pipe. The C V is calculated using the C V formula above, using SG=1.0 (for water at 60 °F), the calculated flow rate, and a differential pressure of 5 psid. C_v = \frac{Q}{\sqrt{\frac{dP}{SG}}} = \frac{246.5}{\sqrt{5-0}} = 110.238 The calculated model confirms that the two systems are equivalent since the calculated flow rate in the upper branch is 246.5 gpm and the calculated pressure at the node is 5 psig. Now the model can be evaluated for a change to the flow rate in the lower branch. When the flow rate is increased to 200 gpm, the pressure at the common junction decreased from 7.923 psig to 6.049 psig. This results in a lower differential pressure in the upper branch and therefore a lower flow rate in the upper branch. The flow rate in the upper branch goes from 246.5 to 214.8 gpm, and the pressure at the outlet goes from 5 psig to 3.797 psig. This method of simplifying a system can be used when the flow rate and pressure are known at a given point in the system and it is known that the system downstream of that point remains constant with regard to valve positions. It can also be used when the differential pressure is known for a given flow rate and the components within the boundaries that define the differential pressure remain constant. This method can also be used for evaluating the performance of a closed loop system. Consider the following system with three heat exchangers, each with a flow rate of about 200 gpm. The flow control valves are manually operated valves and have Cv data entered for the range of valve positions (each valve has different Cv values so they are not identical). The three heat exchangers have identical head loss curves. This is a common configuration for a plant with heat exchangers located throughout the facility. The top and middle branches from the inlet pipe of the heat exchanger to the outlet pipe of the control valve can be simplified using fixed C V fittings by calculating the values based on the above formula: \text{Top branch } C_v\text{:}\] \[C_v = \frac{Q}{\sqrt{dP}} = \frac{199.7 \text{ gpm}}{\sqrt{47.78-9.984 \text{ psid}}}=32.4829 \text{Middle branch } C_v\text{:}\] \[C_v = \frac{Q}{\sqrt{dP}} = \frac{199.2 \text{ gpm}}{\sqrt{50.09-7.673 \text{ psid}}}=30.5858 These C V values are entered as fixed C V fittings in a very short length pipe (0.0001 ft) to ensure all the pressure drop is across the fitting and not the pipe itself. The system below shows the simplified system with the C V values installed and the model calculated. Note that the flow rates in the top and middle branches are the same as the original system, as well as the inlet and outlet pressures of the branches. This confirms that the calculated C V values are equivalent to the original branches consisting of the pipes, heat exchanger, and control valve. Now a flow rate change in the bottom branch can be evaluated to see the impact on the flow rates and pressure drops in the top and middle branches. This is similar to an operator in one part of the plant making an adjustment to the flow rate through his heat exchanger, and operators of the heat exchangers in other parts of the plant seeing the impact on their flow rates. First, evaluate the flow rate change in the original system so it can be used as a comparison to the simplified model with the fixed C V fittings. The system below shows the bottom flow rate adjusted to 50 gpm (from the original 200 gpm). A lower flow rate in the bottom branch causes the inlet pressure to rise (also, this pressure increases due to increased pump total head at a lower pump flow rate) and the outlet pressure to drop. The inlet pressures to the middle and top branches increase in response and the outlet pressures decrease slightly. The resulting increase in differential pressure causes the flow rates to increase in the top and middle branches. The flow rate in the top branch has increased from 199.7 gpm to 208.9 gpm, with an inlet pressure of 51.23 psig and an outlet pressure of 9.786 psig. The flow rate in the middle branch has increased from 199.2 gpm to 208.3 gpm due to the inlet pressure increase to 53.74 psig and the outlet pressure decrease to 7.269 psig. The same impact can be seen in the model using fixed C V fittings to simplify the top and middle branches. The calculated results are shown below. The top branch flow rate is 209 gpm in the simplified model, and the middle branch flow rate is 208.5 gpm. Both flow rates are comparable to the original model with the reduced flow rate in the bottom branch. The inlet and outlet pressures are also very close to the original model. Conclusion The flow coefficient C V represents an important relationship between the flow rate and resulting differential pressure caused by the head loss across a pipeline, component, valve, fitting, or a portion of a system consisting of these devices. By calculating a C V from a given flow rate and pressure drop, a large system can be simplified to a single pipeline with a fixed C V fitting installed. This method can only be used if the resistance of the system being simplified remains constant with no changes in valve position.
I recently came across this in a textbook (NCERT class 12 , chapter: wave optics , pg:367 , example 10.4(d)) of mine while studying the Young's double slit experiment. It says a condition for the formation of interference pattern is$$\frac{s}{S} < \frac{\lambda}{d}$$Where $s$ is the size of ... The accepted answer is clearly wrong. The OP's textbook referes to 's' as "size of source" and then gives a relation involving it. But the accepted answer conveniently assumes 's' to be "fringe-width" and proves the relation. One of the unaccepted answers is the correct one. I have flagged the answer for mod attention. This answer wastes time, because I naturally looked at it first ( it being an accepted answer) only to realise it proved something entirely different and trivial. This question was considered a duplicate because of a previous question titled "Height of Water 'Splashing'". However, the previous question only considers the height of the splash, whereas answers to the later question may consider a lot of different effects on the body of water, such as height ... I was trying to figure out the cross section $\frac{d\sigma}{d\Omega}$ for spinless $e^{-}\gamma\rightarrow e^{-}$ scattering. First I wrote the terms associated with each component.Vertex:$$ie(P_A+P_B)^{\mu}$$External Boson: $1$Photon: $\epsilon_{\mu}$Multiplying these will give the inv... As I am now studying on the history of discovery of electricity so I am searching on each scientists on Google but I am not getting a good answers on some scientists.So I want to ask you to provide a good app for studying on the history of scientists? I am working on correlation in quantum systems.Consider for an arbitrary finite dimensional bipartite system $A$ with elements $A_{1}$ and $A_{2}$ and a bipartite system $B$ with elements $B_{1}$ and $B_{2}$ under the assumption which fulfilled continuity.My question is that would it be possib... @EmilioPisanty Sup. I finished Part I of Q is for Quantum. I'm a little confused why a black ball turns into a misty of white and minus black, and not into white and black? Is it like a little trick so the second PETE box can cancel out the contrary states? Also I really like that the book avoids words like quantum, superposition, etc. Is this correct? "The closer you get hovering (as opposed to falling) to a black hole, the further away you see the black hole from you. You would need an impossible rope of an infinite length to reach the event horizon from a hovering ship". From physics.stackexchange.com/questions/480767/… You can't make a system go to a lower state than its zero point, so you can't do work with ZPE. Similarly, to run a hydroelectric generator you not only need water, you need a height difference so you can make the water run downhill. — PM 2Ring3 hours ago So in Q is for Quantum there's a box called PETE that has 50% chance of changing the color of a black or white ball. When two PETE boxes are connected, an input white ball will always come out white and the same with a black ball. @ACuriousMind There is also a NOT box that changes the color of the ball. In the book it's described that each ball has a misty (possible outcomes I suppose). For example a white ball coming into a PETE box will have output misty of WB (it can come out as white or black). But the misty of a black ball is W-B or -WB. (the black ball comes out with a minus). I understand that with the minus the math works out, but what is that minus and why? @AbhasKumarSinha intriguing/ impressive! would like to hear more! :) am very interested in using physics simulation systems for fluid dynamics vs particle dynamics experiments, alas very few in the world are thinking along the same lines right now, even as the technology improves substantially... @vzn for physics/simulation, you may use Blender, that is very accurate. If you want to experiment lens and optics, the you may use Mistibushi Renderer, those are made for accurate scientific purposes. @RyanUnger physics.stackexchange.com/q/27700/50583 is about QFT for mathematicians, which overlaps in the sense that you can't really do string theory without first doing QFT. I think the canonical recommendation is indeed Deligne et al's Quantum Fields and Strings: A Course For Mathematicians , but I haven't read it myself @AbhasKumarSinha when you say you were there, did you work at some kind of Godot facilities/ headquarters? where? dont see something relevant on google yet on "mitsubishi renderer" do you have a link for that? @ACuriousMind thats exactly how DZA presents it. understand the idea of "not tying it to any particular physical implementation" but that kind of gets stretched thin because the point is that there are "devices from our reality" that match the description and theyre all part of the mystery/ complexity/ inscrutability of QM. actually its QM experts that dont fully grasp the idea because (on deep research) it seems possible classical components exist that fulfill the descriptions... When I say "the basics of string theory haven't changed", I basically mean the story of string theory up to (but excluding) compactifications, branes and what not. It is the latter that has rapidly evolved, not the former. @RyanUnger Yes, it's where the actual model building happens. But there's a lot of things to work out independently of that And that is what I mean by "the basics". Yes, with mirror symmetry and all that jazz, there's been a lot of things happening in string theory, but I think that's still comparatively "fresh" research where the best you'll find are some survey papers @RyanUnger trying to think of an adjective for it... nihilistic? :P ps have you seen this? think youll like it, thought of you when found it... Kurzgesagt optimistic nihilismyoutube.com/watch?v=MBRqu0YOH14 The knuckle mnemonic is a mnemonic device for remembering the number of days in the months of the Julian and Gregorian calendars.== Method ===== One handed ===One form of the mnemonic is done by counting on the knuckles of one's hand to remember the numbers of days of the months.Count knuckles as 31 days, depressions between knuckles as 30 (or 28/29) days. Start with the little finger knuckle as January, and count one finger or depression at a time towards the index finger knuckle (July), saying the months while doing so. Then return to the little finger knuckle (now August) and continue for... @vzn I dont want to go to uni nor college. I prefer to dive into the depths of life early. I'm 16 (2 more years and I graduate). I'm interested in business, physics, neuroscience, philosophy, biology, engineering and other stuff and technologies. I just have constant hunger to widen my view on the world. @Slereah It's like the brain has a limited capacity on math skills it can store. @NovaliumCompany btw think either way is acceptable, relate to the feeling of low enthusiasm to submitting to "the higher establishment," but for many, universities are indeed "diving into the depths of life" I think you should go if you want to learn, but I'd also argue that waiting a couple years could be a sensible option. I know a number of people who went to college because they were told that it was what they should do and ended up wasting a bunch of time/money It does give you more of a sense of who actually knows what they're talking about and who doesn't though. While there's a lot of information available these days, it isn't all good information and it can be a very difficult thing to judge without some background knowledge Hello people, does anyone have a suggestion for some good lecture notes on what surface codes are and how are they used for quantum error correction? I just want to have an overview as I might have the possibility of doing a master thesis on the subject. I looked around a bit and it sounds cool but "it sounds cool" doesn't sound like a good enough motivation for devoting 6 months of my life to it
Can someone give me suggestions how can I construct a 2-tape Turing machine which simulates PDA ? closed as unclear what you're asking by Evil, David Richerby, Rick Decker, Juho, hengxin Jul 9 '17 at 13:00 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. The first tape for input and the second for storing symbols (like on a stack). The first tape should be read-only so that the TM reads symbols one by one from left to right. Every time your PDA writes a symbol on the stack, your TM moves the second head right and writes a symbol on the rightmost empty cell. When the PDA removes a symbol from the stack the TM replaces the rightmost symbol with the blank symbol (erases) and moves the second head left. On each state transition the first head should always move right. In addition, your TM shouldn't be allowed to arbitrarily move heads right and left violating PDA rules. Consider the following formal definition of moves for PDA (assume deterministic for simplicity): UPDATE Definition of PDA moves $\delta(q, a, Z) = (p, \alpha)$: the PDA in state $q$ with input $a$ and $Z$ on the top of the stack. The PDA enters the state $p$ and replaces the top symbol $Z$ with the symbols of the string $\alpha$. So, $\delta(q, a, Z) = (p, \epsilon)$ means remove the top stack symbol (pop). Advance the input head one symbol. $\delta(q,\epsilon, Z) = (p, \alpha)$: the PDA in state $q$ with $Z$ on the top of the stack. Independent of the input symbol, PDA enters the state $p$ and replaces the top symbol $Z$ with the symbols of the strings $\alpha$. Input head does not advance in this case. Examples of translation of PDA moves into TM moves Example 1.PDA move $\delta(q_1, 0, A) = (q_2, B)$ is translated as: TM in state $q_1$, head 1 (input head on the tape 1) reads $0$, and head 2 (stack head on the tape 2) reads $A$. Replace $A$ with $B$ (head 2 writes the symbol $B$ while it is on $A$). Advance head 1 one symbol. Enter state $q_2$. Example 2.$\delta(q_1, 0, A) = (q_2, BC)$ is translated as: TM in state $q_1$, head 1 reads $0$, and head 2 reads $A$. Head 2 writes $B$ (replaces $A$), advances head 2 one symbol, TM enters state $q_{21}$. Then, while TM in state $q_{21}$ and head 1 reads $0$, and head 2 reads the blank symbol: head 2 writes symbol $C$, enters state $q_2$. Advance head 1 one symbol. The basic idea: in order to write $BC$ we introduced a new state $q_{21}$. Analogously, $\delta(q,\epsilon, Z) = (p, \alpha)$ means head 2 just writes the blank symbol and moves left. $\delta(q,\epsilon, Z) = (p, \alpha)$ means that you define the same transition for every input symbol. This is how a multitape Turing machine is defined.
Notes from 9 Sep REPL tip The repl.it environment requires there to be a main function, evenif you only want to experiment with other functions. This isn’t toomuch of a problem, you can just define main = putStrLn "Okay" but the minimal main function is actually main = return () so use that as a placeholder if you don’t need it to do anything at all. Operator sections This refers to specifying an infix operator with only one argument,which then becomes a function of the other argument. For example: (/3)is a section of the division operator, where the numerator is the next argument. (3/)is also a section of the division operator, but now the denominator is the next argument. Usage: λ> boo = (/3) λ> moo = (3/) λ> boo 10 3.3333333333333335 λ> moo 10 0.3 λ> boo 12 4.0 λ> moo 12 0.25 We can define functions this way, without ever referring to the arguments: half = (/2) square = (*2) twice = (2) Function composition In algebra, we sometimes use a small circle as an operator that can compose two functions. The rule is: \[ (f\circ g)(x) = f(g(x)) \] So it’s like a pipeline of the two functions – first apply $g$ to the argument $x$, and then apply $f$ to that result. So if: \[ f(x) = 2x+1 \] and \[ g(x) = \sqrt{\frac{x}{2}} \] Then: \[ (f\circ g)(x) = 2\sqrt{\frac{x}{2}} + 1\] and: \[ (g\circ f)(x) = \sqrt{\frac{2x+1}{2}} \] Note that the function to the right of the operator is applied first! In Haskell, the function composition operator is a single dot (period) character. We usually put spaces around the dot, although that isn’t strictly necessary. Here are some examples: λ> (half . square) 10 50.0 λ> (square . half) 10 25.0 λ> (square . half . square) 10 2500.0 λ> (half.square.half) 10 12.5 In some situations, it may be more intuitive for a sequence of functions to be applied left-to-right instead of right-to-left. We can define a different composition operator for that purpose, simply like this: (\|>) = flip (.) The pre-defined flip function takes any function with two parametersand switches the order of the parameters: λ> (-) 9 2 7 λ> flip (-) 9 2 -7 Once we flip the composition operator, the function on the left will be applied first: λ> (half \|> square) 10 25.0 λ> (square \|> half) 10 50.0 λ> (square \|> half \|> square) 10 2500.0 λ> (half \|> square \|> half) 10 12.5 For symmetry, we can provide an alias for composition in the standard right-to-left order: (<\|) = (.) Functions can be defined with multiple clauses – different expressionsthat apply to different cases. One way to do this is with patternguards, which are introduced with a pipe/bar character ( \|) before theequals sign: collatz n \| even n = n `div` 2 \| otherwise = 3n + 1 Following each bar is a Boolean expression, called a guard. The firstguard that results in True indicates the expression that is will beexecuted. The keyword otherwise will match if no previous guardmatched. Here are some derivations using the sample function above: collatz 18 ↪ -- plug in 18 for n in the guards -- evaluate: even 18 ↪ True 18 `div` 2 ↪ 9 collatz 5 ↪ -- plug in 5 for n in the guards -- evaluate: even 5 ↪ False -- move on to otherwise case 35 + 1 ↪ 15 + 1 ↪ 16 Let’s define a few more numeric functions this way: fact n \| n < 0 = error "Negative argument" \| n == 0 = 1 \| otherwise = n * fact (n-1) pow x n \| n < 0 = error "Negative argument" \| n == 0 = 1 \| otherwise = x * pow x (n-1) The error function signals a run-time error. It will look somethinglike this: *** Exception: Negative argument CallStack (from HasCallStack): error, called at <interactive>:60:1 in interactive:Ghci44 In Haskell we usually try pretty hard to avoid generating run-time errors. Later we can explore some techniques for ensuring they don’t happen. Let’s attempt to derive the values of expressions using these guarded recursive functions. fact 5 ↝ -- substitute 5 for n in guards -- evaluate: 5 < 0 ↝ False -- evaluate: 5 == 0 ↝ False -- move on to otherwise case 5 * fact (5-1) ↝ -- just arithmetic 5 * fact 4 ↝ -- substitute 4 for n in guards -- evaluate: 4 < 0 ↝ False -- evaluate: 4 == 0 ↝ False -- move on to otherwise case 5 * 4 * fact (4-1) ↝ -- arithmetic 5 * 4 * fact 3 ↝ -- otherwise case 5 * 4 * 3 * fact 2 ↝ -- arithmetic, otherwise 5 * 4 * 3 * 2 * fact 1 ↝ -- arithmetic, otherwise 5 * 4 * 3 * 2 * 1 * fact 0 ↝ -- arithmetic -- evaluate: 0 < 0 ↝ False -- evaluate: 0 == 0 ↝ True, use that case 5 * 4 * 3 * 2 * 1 * 1 ↝ 20 * 3 * 2 * 1 * 1 ↝ 60 * 2 * 1 * 1 ↝ 120 * 1 * 1 ↝ 120 * 1 ↝ 120 pow 2 6 ↝ 2 * pow 2 5 ↝ 2 * 2 * pow 2 4 ↝ 2 * 2 * 2 * pow 2 3 ↝ 2 * 2 * 2 * 2 * pow 2 2 ↝ 2 * 2 * 2 * 2 * 2 * pow 2 1 ↝ 2 * 2 * 2 * 2 * 2 * 2 * pow 2 0 ↝ 2 * 2 * 2 * 2 * 2 * 2 * 1 ↝ 4 * 2 * 2 * 2 * 2 * 1 ↝ 8 * 2 * 2 * 2 * 1 ↝ 16 * 2 * 2 * 1 ↝ 32 * 2 * 1 ↝ 64 * 1 ↝ 64 Exercise: why is it helpful to have a case for n < 0 in thesefunctions? What happens if we omit that case and we try to evaluate fact (-1)? Other tips for Assignment 1 For the palindrome function, you basically need reverse and theequality operator (=)= (double equals, like in many programminglanguages). The reverse function just reverses the order of a list(or string): λ> reverse [1..4] [4,3,2,1] λ> reverse "liart" "trail" Here are some of the operators that generate Booleans: &&is the logical AND (same as in C++) \|\|is logical OR (same as C++) notis logical negation ( !in C++) ==is equality of two values (same as C++) /=is not-equals ( !=in C++) <=, <, >, >=are inequality operators, as in C++ evenand oddtake integers and return Trueif they are even/odd respectively. To manipulate characters for the rot13 problem, we need to add orsubtract 13 from a character. You can’t directly add a character andan integer (like in C), but you could apply succ or pred 13 times: λ> ( succ . succ . succ . succ . succ . succ . succ . succ . succ . succ . succ . succ . succ) 'F' 'S' Of course, that’s pretty ugly. A more elegant way is to use ord and chr from Data.Char module, to convert to integer and back: Then you can add in between: λ> ord 'F' 70 λ> chr 70 'F' λ> chr (ord 'F' + 13) 'S' The other thing we’ll need for rot13 is to determine if the letteris in the front half or back half of the alphabet. The elem functionis a good tool for that, it simply determines whether an element iscontained in a list: λ> elem 5 [1..10] True λ> elem 10 [1..5] False λ> elem 'k' "apple" False λ> elem 'p' "apple" True λ> elem 'S' ['N'..'Z'] True
Trapping of Magnetic Flux in Bi-2223 Ceramic Superconductors Doped with α-Al 2 O 3 Nanoparticles 38 Downloads Abstract By combining experimental results and a simple model, we offer here an explanation for the role played by the low-angle grain boundaries and the doping with α-Al 2 O 3 nanoparticles in the trapping of the magnetic flux in Bi 1.65Pb 0.35Sr 2Ca 2Cu 3 O 10+ δ (Bi-2223) ceramic samples. Our model correlates the size of the nanoparticles, properties of the superconducting matrix, and the magnetic flux trapped at the intragranular planar defects, the so-called Abrikosov-Josephson vortices. The results indicate that the role played by the doping with α-Al 2 O 3nanoparticles is to pin the AJ vortices at the grain boundaries with misorientation angles $\theta \sim 10^{\circ }$. We also argue that a similar procedure for identifying conditions for the trapping of the magnetic flux may be applied to other superconducting materials doped with non-magnetic nanoparticles. KeywordsBi-based superconductors Trapped flux Doping with nanoparticles Low-angle grain boundaries Notes Acknowledgments The authors are indebted to Prof. E. Martínez-Guerra and César Cutberto Leyva Porras, from Centro Investigación en Materiales Avanzados S. C., Unidad Monterrey-PIIT and Chihuahua, respectively, for the HRTEM images. We also thank Prof. F. Solís-Pomar, from Centro de Investigación en Ciencias Físico-Matemáticas, FCFM, UANL, Monterrey, Nuevo León, México, for the AFM images. Finally, the authors would like to express their deepest gratitude to Prof. A. Gurevich for several important suggestions. Funding Information This work was supported by Brazil’s agencies FAPESP (Grant nos. 13/07296-2 and 14/19245-6), CNPq, and CAPES under Grant CAPES/MES no. 104/10. References 1. 2. 3. 4. 5. 6.Hernández-Wolpez, M., Cruz-García, A., Jardim, R. F., Muné, P.: Intragranular defects and Abrikosov-Josepson vortices in Bi-2223 bulk superconductors. J. Mater Sci:Mater Electron. 28, 15246 (2017)Google Scholar 7. 8. 9. 10.Hernández-Wolpez, M., Fernández-Gamboa, J. R., García-Fornaris, I., Govea-Alcaide, E., Pérez-Tijerina, E., Jardim, R. F., Muné, P.: Voltage relaxation and Abrikosov-Josepson vortices in Bi-2223 superconductors doped with α,-Al 2 O 3nanoparticles. J. Mater Sci:Mater Electron. 29, 5926 (2018)Google Scholar 11. 12. 13. 14. 15. 16. 17.This value of γ 2= 0.423 differs from the incorrect value γ 2= 1.118 of (Ref. [1])Google Scholar 18. 19. 20.Yavuz, S., Bilgili, O., Kocabas, K.: Effects of superconducting parameters of SnO 2nanoparticles addition on (Bi, Pb)-2223 phase. J. Mater. Sci: Mater. Electron. 27, 4526 (2016)Google Scholar
Seed If $j:V \to M$ is an elementary embedding and $a \in j(D)$ for some set $D$, then $a$ is a seed for the measure $\mu$ on $D$ defined by $X \in \mu \iff X \subseteq D$ and $a \in j(X)$. In this case, we say that $a$ generates $\mu$ via $j$. If $b=j(f)(a)$ for some function $f \in V$, then we say that $a$ generates $b$ via the embedding. If every element of $M$ is generated by $a$, then we will say that $a$ generates all of $M$ or all of the embedding $j$. This definition comes from Joel Hamkin's book "Forcing and Large Cardinals"
Does anyone here understand why he set the Velocity of Center Mass = 0 here? He keeps setting the Velocity of center mass , and acceleration of center mass(on other questions) to zero which i dont comprehend why? @amanuel2 Yes, this is a conservation of momentum question. The initial momentum is zero, and since there are no external forces, after she throws the 1st wrench the sum of her momentum plus the momentum of the thrown wrench is zero, and the centre of mass is still at the origin. I was just reading a sci-fi novel where physics "breaks down". While of course fiction is fiction and I don't expect this to happen in real life, when I tired to contemplate the concept I find that I cannot even imagine what it would mean for physics to break down. Is my imagination too limited o... The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe... not exactly identical however Also typo: Wavefunction does not really have an energy, it is the quantum state that has a spectrum of energy eigenvalues Since Hamilton's equation of motion in classical physics is $$\frac{d}{dt} \begin{pmatrix} x \\ p \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \nabla H(x,p) \, ,$$ why does everyone make a big deal about Schrodinger's equation, which is $$\frac{d}{dt} \begin{pmatrix} \text{Re}\Psi \\ \text{Im}\Psi \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \hat H \begin{pmatrix} \text{Re}\Psi \\ \text{Im}\Psi \end{pmatrix} \, ?$$ Oh by the way, the Hamiltonian is a stupid quantity. We should always work with $H / \hbar$, which has dimensions of frequency. @DanielSank I think you should post that question. I don't recall many looked at the two Hamilton equations together in this matrix form before, which really highlight the similarities between them (even though technically speaking the schroedinger equation is based on quantising Hamiltonian mechanics) and yes you are correct about the $\nabla^2$ thing. I got too used to the position basis @DanielSank The big deal is not the equation itself, but the meaning of the variables. The form of the equation itself just says "the Hamiltonian is the generator of time translation", but surely you'll agree that classical position and momentum evolving in time are a rather different notion than the wavefunction of QM evolving in time. If you want to make the similarity really obvious, just write the evolution equations for the observables. The classical equation is literally Heisenberg's evolution equation with the Poisson bracket instead of the commutator, no pesky additional $\nabla$ or what not The big deal many introductory quantum texts make about the Schrödinger equation is due to the fact that their target audience are usually people who are not expected to be trained in classical Hamiltonian mechanics. No time remotely soon, as far as things seem. Just the amount of material required for an undertaking like that would be exceptional. It doesn't even seem like we're remotely near the advancement required to take advantage of such a project, let alone organize one. I'd be honestly skeptical of humans ever reaching that point. It's cool to think about, but so much would have to change that trying to estimate it would be pointless currently (lol) talk about raping the planet(s)... re dyson sphere, solar energy is a simplified version right? which is advancing. what about orbiting solar energy harvesting? maybe not as far away. kurzgesagt also has a video on a space elevator, its very hard but expect that to be built decades earlier, and if it doesnt show up, maybe no hope for a dyson sphere... o_O BTW @DanielSank Do you know where I can go to wash off my karma? I just wrote a rather negative (though well-deserved, and as thorough and impartial as I could make it) referee report. And I'd rather it not come back to bite me on my next go-round as an author o.o
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than be $v(t) = 3t^2-12t+9$ how could I find the intervals Fix $c\in\{0,1,\dots\}$, let $K\geq c$ be an integer, and define $z_K=K^{-\alpha}$ for some $\alpha\in(0,2)$.I believe I have numerically discovered that$$\sum_{n=0}^{K-c}\binom{K}{n}\binom{K}{n+c}z_K^{n+c/2} \sim \sum_{n=0}^K \binom{K}{n}^2 z_K^n \quad \text{ as } K\to\infty$$but cannot ... So, the whole discussion is about some polynomial $p(A)$, for $A$ an $n\times n$ matrix with entries in $\mathbf{C}$, and eigenvalues $\lambda_1,\ldots, \lambda_k$. Anyways, part (a) is talking about proving that $p(\lambda_1),\ldots, p(\lambda_k)$ are eigenvalues of $p(A)$. That's basically routine computation. No problem there. The next bit is to compute the dimension of the eigenspaces $E(p(A), p(\lambda_i))$. Seems like this bit follows from the same argument. An eigenvector for $A$ is an eigenvector for $p(A)$, so the rest seems to follow. Finally, the last part is to find the characteristic polynomial of $p(A)$. I guess this means in terms of the characteristic polynomial of $A$. Well, we do know what the eigenvalues are... The so-called Spectral Mapping Theorem tells us that the eigenvalues of $p(A)$ are exactly the $p(\lambda_i)$. Usually, by the time you start talking about complex numbers you consider the real numbers as a subset of them, since a and b are real in a + bi. But you could define it that way and call it a "standard form" like ax + by = c for linear equations :-) @Riker "a + bi where a and b are integers" Complex numbers a + bi where a and b are integers are called Gaussian integers. I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd... Does anyone know if $T: V \to R^n$ is an inner product space isomorphism if $T(v) = (v)_S$, where $S$ is a basis for $V$? My book isn't saying so explicitly, but there was a theorem saying that an inner product isomorphism exists, and another theorem kind of suggesting that it should work. @TobiasKildetoft Sorry, I meant that they should be equal (accidently sent this before writing my answer. Writing it now) Isn't there this theorem saying that if $v,w \in V$ ($V$ being an inner product space), then $\|\|v\|\| = \|\|(v)_S\|\|$? (where the left norm is defined as the norm in $V$ and the right norm is the euclidean norm) I thought that this would somehow result from isomorphism @AlessandroCodenotti Actually, such a $f$ in fact needs to be surjective. Take any $y \in Y$; the maximal ideal of $k[Y]$ corresponding to that is $(Y_1 - y_1, \cdots, Y_n - y_n)$. The ideal corresponding to the subvariety $f^{-1}(y) \subset X$ in $k[X]$ is then nothing but $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$. If this is empty, weak Nullstellensatz kicks in to say that there are $g_1, \cdots, g_n \in k[X]$ such that $\sum_i (f^* Y_i - y_i)g_i = 1$. Well, better to say that $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$ is the trivial ideal I guess. Hmm, I'm stuck again O(n) acts transitively on S^(n-1) with stabilizer at a point O(n-1) For any transitive G action on a set X with stabilizer H, G/H $\cong$ X set theoretically. In this case, as the action is a smooth action by a Lie group, you can prove this set-theoretic bijection gives a diffeomorphism
Difference between revisions of "Linear representation theory of symmetric group:S5" (→Character table) (→Family contexts) (9 intermediate revisions by the same user not shown) Line 8: Line 8: ==Summary== ==Summary== + {\| class="sortable" border="1" {\| class="sortable" border="1" ! Item !! Value ! Item !! Value \|- \|- − \| [[Degrees of irreducible representations]] over a [[splitting field]] \|\| 1,1,4,4,5,5,6<br>[[maximum degree of irreducible representation\|maximum]]: 6, [[lcm of degrees of irreducible representations\|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes\|number]]: 7, [[sum of squares of degrees of irreducible representations equals order of group\|sum of squares]]: 120 + \| [[Degrees of irreducible representations]] over a [[splitting field]] \|\| 1,1,4,4,5,5,6<br>[[maximum degree of irreducible representation\|maximum]]: 6, [[lcm of degrees of irreducible representations\|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes\|number]]: 7, [[sum of squares of degrees of irreducible representations equals order of group\|sum of squares]]: 120 \|- \|- \| [[Schur index]] values of irreducible representations \|\| 1,1,1,1,1,1,1<br>[[maximum Schur index of irreducible representation\|maximum]]: 1, [[lcm of Schur indices of irreducible representations\|lcm]]: 1 \| [[Schur index]] values of irreducible representations \|\| 1,1,1,1,1,1,1<br>[[maximum Schur index of irreducible representation\|maximum]]: 1, [[lcm of Schur indices of irreducible representations\|lcm]]: 1 Line 23: Line 24: \| Smallest size [[splitting field]] \|\| [[field:F7]], i.e., the field of 7 elements. \| Smallest size [[splitting field]] \|\| [[field:F7]], i.e., the field of 7 elements. \|} \|} − + ==Family contexts== ==Family contexts== Line 29: Line 30: ! Family name !! Parameter values !! General discussion of linear representation theory of family ! Family name !! Parameter values !! General discussion of linear representation theory of family \|- \|- − \| [[symmetric group]] \|\| 5 \|\| [[linear representation theory of symmetric groups]] + \| [[symmetric group]] \|\| 5\|\| [[linear representation theory of symmetric groups]] \|- \|- − \| [[projective general linear group of degree two]] \|\| [[field:F5]] \|\| [[linear representation theory of projective general linear group of degree two]] + \| [[projective general linear group of degree two]] \|\| [[field:F5]]\|\| [[linear representation theory of projective general linear group of degree two ]] \|} \|} Line 81: Line 82: \| Unclear \|\| a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the property that <math>\varphi(x)^{q+1} = 1</math> for all <math>x</math>, and <math>\varphi</math> takes values other than <math>\pm 1</math>. Identify <math>\varphi</math> and <math>\varphi^q</math>. \|\| unclear \|\| <math>q - 1</math> \|\| 4 \|\| <math>(q-1)/2</math> \|\| 2 \|\| <math>(q-1)^3/2</math> \|\| 32 \|\| standard representation, product of standard and sign \| Unclear \|\| a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the property that <math>\varphi(x)^{q+1} = 1</math> for all <math>x</math>, and <math>\varphi</math> takes values other than <math>\pm 1</math>. Identify <math>\varphi</math> and <math>\varphi^q</math>. \|\| unclear \|\| <math>q - 1</math> \|\| 4 \|\| <math>(q-1)/2</math> \|\| 2 \|\| <math>(q-1)^3/2</math> \|\| 32 \|\| standard representation, product of standard and sign \|- \|- − + Total \|\| NA \|\| NA \|\| NA \|\| NA \|\| <math>q + 2</math> \|\| 7 \|\| <math>q^3 - q</math> \|\| 120 \|\| NA \|} \|} Line 87: Line 88: {{character table facts to check against}} {{character table facts to check against}} + + + + + + + + + + + + + + + + + + + + {\| class="sortable" border="1" {\| class="sortable" border="1" − ! Representation/conjugacy class representative and size !! <math>()</math> (size 1) !! <math>(1,2)</math> (size 10) !! <math>(1,2)(3,4)</math> (size 15) !! <math>(1,2,3)</math> (size 20) !! <math>(1,2,3, + ! Representation/conjugacy class representative and size !! <math>()</math> (size 1) !! <math>(1,2)</math> (size 10) !! <math>(1,2)(3,4)</math> (size 15) !! <math>(1,2,3)</math> (size 20) !! <math>(1,2,3,)</math> (size ) !! <math>(1,2,34,5)</math> (size ) !! <math>(1,2,3,4)</math> (size ) \|- \|- − \| trivial representation \|\| 1 \|\| + \| trivial representation \|\| 1 \|\| \|\| \|\| \|\| \|\| \|\| \|- \|- − \| sign representation \|\| 1 \|\| - + \| sign representation \|\| 1 \|\| -\|\| \|\| \|\| -\|\| \|\| \|- \|- − \| standard representation \|\| + \| standard representation \|\| \|\| \|\| 0 \|\| \|\| \|\| -\|\| \|- \|- − \| product of standard and sign representation \|\| + \| product of standard and sign representation \|\| \|\| -\|\| 0 \|\| \|\| \|\| \|\| \|- \|- − \| irreducible five-dimensional representation \|\| + \| irreducible five-dimensional representation \|\| \|\| \|\| \|\| \|\| \|\| \|\| \|- \|- − \| irreducible five-dimensional representation \|\| + \| irreducible five-dimensional representation \|\| \|\| \|\| \|\| \|\| \|\| \|\| \|- \|- − \| exterior square of standard representation \|\| + \| exterior square of standard representation \|\| \|\| 0 \|\| -\|\| 0 \|\| 0 \|\| \|\| \|} \|} Latest revision as of 05:41, 16 January 2013 This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S5. View linear representation theory of particular groups \| View other specific information about symmetric group:S5 This article describes the linear representation theory of symmetric group:S5, a group of order . We take this to be the group of permutations on the set . Summary Item Value Degrees of irreducible representations over a splitting field (such as or ) 1,1,4,4,5,5,6 maximum: 6, lcm: 60, number: 7, sum of squares: 120 Schur index values of irreducible representations 1,1,1,1,1,1,1 maximum: 1, lcm: 1 Smallest ring of realization for all irreducible representations (characteristic zero) -- ring of integers Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) -- hence it is a rational representation group Criterion for a field to be a splitting field Any field of characteristic not equal to 2,3, or 5. Smallest size splitting field field:F7, i.e., the field of 7 elements. Family contexts Family name Parameter values General discussion of linear representation theory of family symmetric group of degree linear representation theory of symmetric groups projective general linear group of degree two over a finite field of size , i.e., field:F5, so the group is linear representation theory of projective general linear group of degree two over a finite field Degrees of irreducible representations FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD: Divisibility facts: degree of irreducible representation divides group order \| degree of irreducible representation divides index of abelian normal subgroup Size bounds: order of inner automorphism group bounds square of degree of irreducible representation\| degree of irreducible representation is bounded by index of abelian subgroup\| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup Cumulative facts: sum of squares of degrees of irreducible representations equals order of group \| number of irreducible representations equals number of conjugacy classes \| number of one-dimensional representations equals order of abelianization Note that the linear representation theory of the symmetric group of degree four works over any field of characteristic not equal to two or three, and the list of degrees is . Interpretation as symmetric group Common name of representation Degree Corresponding partition Young diagram Hook-length formula for degree Conjugate partition Representation for conjugate partition trivial representation 1 5 1 + 1 + 1 + 1 + 1 sign representation sign representation 1 1 + 1 + 1 + 1 + 1 5 trivial representation standard representation 4 4 + 1 2 + 1 + 1 + 1 product of standard and sign representation product of standard and sign representation 4 2 + 1 + 1 + 1 4 + 1 standard representation irreducible five-dimensional representation 5 3 + 2 2 + 2 + 1 other irreducible five-dimensional representation irreducible five-dimensional representation 5 2 + 2 + 1 3 + 2 other irreducible five-dimensional representation exterior square of standard representation 6 3 + 1 + 1 3 + 1 + 1 the same representation, because the partition is self-conjugate. Interpretation as projective general linear group of degree two Compare and contrast with linear representation theory of projective general linear group of degree two over a finite field Description of collection of representations Parameter for describing each representation How the representation is described Degree of each representation (general odd ) Degree of each representation () Number of representations (general odd ) Number of representations () Sum of squares of degrees (general odd ) Sum of squares of degrees () Symmetric group name Trivial -- 1 1 1 1 1 1 trivial Sign representation -- Kernel is projective special linear group of degree two (in this case, alternating group:A5), image is 1 1 1 1 1 1 sign Nontrivial component of permutation representation of on the projective line over -- -- 5 1 1 25 irreducible 5D Tensor product of sign representation and nontrivial component of permutation representation on projective line -- -- 5 1 1 25 other irreducible 5D Induced from one-dimensional representation of Borel subgroup ? ? 6 1 36 exterior square of standard representation Unclear a nontrivial homomorphism , with the property that for all , and takes values other than . Identify and . unclear 4 2 32 standard representation, product of standard and sign Total NA NA NA NA 7 120 NA Character table FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field): Orthogonality relations: Character orthogonality theorem \| Column orthogonality theorem Separation results(basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions\|Character determines representation in characteristic zero Numerical facts: Characters are cyclotomic integers \| Size-degree-weighted characters are algebraic integers Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class\| Conjugacy class of more than average size has character value zero for some irreducible character \| Zero-or-scalar lemma Representation/conjugacy class representative and size (size 1) (size 10) (size 15) (size 20) (size 20) (size 24) (size 30) trivial representation 1 1 1 1 1 1 1 sign representation 1 -1 1 1 -1 1 -1 standard representation 4 2 0 1 -1 -1 0 product of standard and sign representation 4 -2 0 1 1 -1 0 irreducible five-dimensional representation 5 1 1 -1 1 0 -1 irreducible five-dimensional representation 5 -1 1 -1 -1 0 1 exterior square of standard representation 6 0 -2 0 0 1 0 Below are the size-degree-weighted characters, i.e., these are obtained by multiplying the character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers. Representation/conjugacy class representative and size (size 1) (size 10) (size 15) (size 20) (size 20) (size 24) (size 30) trivial representation 1 10 15 20 20 24 30 sign representation 1 -10 15 20 -20 24 -30 standard representation 1 5 0 5 -5 -6 0 product of standard and sign representation 1 -5 0 5 5 -6 0 irreducible five-dimensional representation 1 2 3 -4 4 0 -6 irreducible five-dimensional representation 1 -2 3 -4 -4 0 6 exterior square of standard representation 1 0 -5 0 0 4 0 GAP implementation The degrees of irreducible representations can be computed using GAP's CharacterDegrees function: gap> CharacterDegrees(SymmetricGroup(5)); [ [ 1, 2 ], [ 4, 2 ], [ 5, 2 ], [ 6, 1 ] ] This means that there are 2 degree 1 irreducible representations, 2 degree 4 irreducible representations, 2 degree 5 irreducible representations, and 1 degree 6 irreducible representation. The characters of all irreducible representations can be computed in full using GAP's CharacterTable function: gap> Irr(CharacterTable(SymmetricGroup(5))); [ Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, -1, 1, 1, -1, -1, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, -2, 0, 1, 1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, -1, 1, -1, -1, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 6, 0, -2, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, 1, 1, -1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, 2, 0, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ) ]
I am a Postdoc at the Faculty of Mathematics of the Bielefeld University. I am a member of the research group Groups and Geometry, and my mentor is Prof. Kai-Uwe Bux. I am supported by my own grant within the Schwerpunktprogram "Geometry at Infinity" of the DFG. Next year I will start to work on my project "Fibring" funded by the ERC Starting Grant. My research interests lie within the area of (broadly understood) Geometric Group Theory. I am a member of the Bielefeld Graduate School in Theoretical Sciences. I am also a member of the DMV and the EMS. We prove the $K$- and $L$-theoretic Farrell-Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $A \rtimes \mathbb{Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell-Jones Conjecture for free-by-cyclic groups. We construct examples of fibered three-manifolds with fibered faces all of whose monodromies extend to a handlebody. We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \geqslant 6$) and of $\mathrm{SL}_n(\mathbb Z)$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n> 6$. We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope. We use the agrarian invariants to solve the torsion-free case of a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, for such groups, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann. Finally, we prove that for agrarian groups of deficiency $1$, the agrarian polytope admits a marking of its vertices which controls the Bieri-Neumann-Strebel invariant of the group, improving a result of the second author and partially answering a question of Friedl-Tillmann. We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds. We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav. We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincaré duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl. We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the $L^2$-torsion polytope of Friedl-Lück is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lück-Tillmann. We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least $3$ without punctures is $\mathrm{Sp}_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on $\mathbb C$-linear representations of mapping class groups to projective representations over any field. We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: "A group G is amenable if and only if every cellular automaton with carrier G that has gardens of Eden also has mutually erasable patterns." This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Machì and Scarabotti. An appendix by Dawid Kielak proves that group rings without zero divisors are Ore domains precisely when the group is amenable, answering a conjecture attributed to Guba. We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is $\mathrm{PSL}_n(\mathbb Z/2 \mathbb Z)=\mathrm L_n(2)$, thus confirming a conjecture of Mecchia-Zimmermann. In the course of the proof we give an exponential (in $n$) lower bound for the cardinality of a set on which $\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can act non-trivially. We also offer new results on the representation theory of $\mathrm{SAut}(F_n)$ in small dimensions over small, positive characteristics, and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of Lie type and algebraic groups in characteristic $2$. We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szabó, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension. In the appendix, we justify optimality of the aforementioned result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces. We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $\Gamma$, and any finite group $G \leqslant \mathrm{U}^0(A_\Gamma)\leqslant \mathrm{Out}^0(A_\Gamma)$, we find a non-positively curved cube complex with fundamental group $A_\Gamma$ on which $G$ acts by isometries, realising the action on $A_\Gamma$. We investigate Friedl-Lück's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb Z$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$ has finitely many connected components. When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in $\mathrm{GL}(n,\mathbb Z)$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide. We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel-Mosher and on the outer space of a free product of Guirardel-Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained. We determine the precise conditions under which $\mathrm{SOut}(F_n)$, the unique index two subgroup of $\mathrm{Out}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac 1 2 \binom n 2$ vertices. We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph. Along the way we determine the minimal dimensions of non-trivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which $\mathrm{SOut}(F_n)$ can act non-trivially. We show that braid groups with at most $6$ strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type $A$. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond. We show that any finitely generated group $F$ with infinitely many ends is not a group of fractions of any finitely generated proper subsemigroup $P$, that is $F$ cannot be expressed as a product $PP^{−1}$. In particular this solves a conjecture of Navas in the positive. As a corollary we obtain a new proof of the fact that finitely generated free groups do not admit isolated left-invariant orderings. We study homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$, and $\mathrm{GL}(m,K)$ for $m<7$, where $K$ is a field of characteristic other than $2$ or $3$. We conclude that all $K$-linear representations of dimension at most $6$ of $\mathrm{Out}(F_3)$ factor through $\mathrm{GL}(3,\mathbb Z)$, and that all homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$ have finite image. We study the existence of homomorphisms between $\mathrm{Out}(F_n)$ and $\mathrm{Out}(F_m)$ for $n > 5$ and $m < n(n-1)/2$, and conclude that if $m$ is not equal to $n$ then each such homomorphism factors through the finite group of order $2$. In particular this provides an answer to a question of Bogopol'skii and Puga. In the course of the argument linear representations of $\mathrm{Out}(F_n)$ in dimension less than $n(n+1)/2$ over fields of characteristic zero are completely classified. It is shown that each such representation has to factor through the natural projection from $\mathrm{Out}(F_n)$ to $\mathrm{GL}(n,\mathbb Z)$ coming from the action of $\mathrm{Out}(F_n)$ on the abelianisation of $F_n$. We obtain similar results about linear representation theory of $\mathrm{Out}(F_4)$ and $\mathrm{Out}(F_5)$. Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type $\mathsf{A}_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments. For various values of $n$ and $m$ we investigate homomorphisms $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ and $\mathrm{Out}(F_n)\to \mathrm{GL}_m(K)$, i.e. the free and linear representations of $\mathrm{Out}(F_n)$ respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of $\mathrm{Out}(F_n)$ we prove that each homomorphism $\mathrm{Out}(F_n) \to \mathrm{GL}_m(K)$ factors through the natural map $\pi_n \colon \mathrm{Out}(F_n)\to \mathrm{GL}(H_1(F_n,\mathbb Z))\cong \mathrm{GL}_n(\mathbb Z)$ whenever $n= 3,m <7$ and $\mathrm{char}(K)\not\in \{2,3\}$, and whenever $n >5,m <\binom {n+1} 2$ and $\mathrm{char}(K) \not\in \{2,3,...,n+ 1\}$. We also construct a new infinite family of linear representations of $\mathrm{Out}(F_n)$ (where $n >2$), which do not factor through $\pi_n$. When $n$ is odd these have the smallest dimension among all known representations of $\mathrm{Out}(F_n)$ with this property. Using the above results we establish that the image of every homomorphism $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ is finite whenever $n= 3$ and $n < m <6$, and of cardinality at most $2$ whenever $n >5$ and $n < m < \binom n 2$. We further show that the image is finite when $\binom n 2 \leqslant m < \binom{n+1} 2$. We also consider the structure of normal finite index subgroups of $\mathrm{Out}(F_n)$. If $N$ is such then we prove that if the derived subgroup of the intersection of $N$ with the Torelli subgroup $\overline{\mathrm{IA}}_n < \mathrm{Out(F_n)}$ contains some term of the lower central series of $\overline{\mathrm{IA}}_n$ then the abelianisation of $N$ is finite. We will follow the notes of Holger Kammeyer. The main reference is the book `$L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory' by Wolfgang Lück.
A new version of Unicode Technical Note #28, UnicodeMath, a Nearly Plain-Text Encoding of Mathematics is now available. It updates several topics and references and uses the name UnicodeMath instead of Unicode linear format. Since there are several math linear formats, such as Nemeth braille, [La]TeX, and AsciiMath, having the name UnicodeMath clarifies the discussion nicely. The text has been polished in other ways too and some errors have been corrected. No notational constructs have been added, so the version number is only incremented to 3.1. Here’s a UnicodeMath example in case you don’t want to read the whole spec ☺ The formula sin θ=(e^iθ-e^-iθ)/2i displays as Operators and operator precedence are used to delimit arguments. A binary minus has lower precedence than the superscript operator ^ and the fraction operator /, but a unary minus has higher precedence than ^. This approach contrasts with LaTeX and AsciiMath which require that arguments consisting of more than one element be enclosed in {} or (), respectively. In LaTeX, the formula above is given by \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i} In AsciiMath, the formula is given by sin theta=(e^(i theta)-e^(-i theta))/(2i) In Microsoft Office apps, you can enter Unicode symbols in UnicodeMath using the corresponding [La]TeX controls words such as \theta, using names that you choose, or using symbol galleries.
The OP first stated the problem as: Knowing $K_\mathrm{s} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ in neutral solutions, calculate the minimum pH of an acidic solution in order to completely dissolve $\pu{10 mg}$ of $\ce{Fe(OH)3}$. Data: $V = \pu{0.1 L}$. I took that to mean that acid was added to a solution containing 10 mg of iron (iii) hydroxide and the pH of the solution when the iron (iii) hydroxide was totally dissolved was desired.* Knowing that $K_\mathrm{sp} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ then\begin{align} K_\mathrm{sp} &= \ce{[Fe^{3+}][OH^-]^3} = 6\cdot10^{-38},&\text{ so }&&\ce{[OH^-]} &=\sqrt[3]{\frac{6\cdot10^{-38}}{\ce{[Fe^{3+}]}}}\end{align} You need to solve for $\ce{[Fe^{3+}]}$ which you can calculating knowing that there is $\pu{10 mg}$ of $\ce{Fe(OH)_3}$ in 0.1 liters of solution. It has a molecular weight of $\pu{107 g/mol}$, so there is $$0.010/107 = \pu{9.35* 10^-5 mol//L}\text{ of }\ce{Fe(OH)_3}.$$ Going back to the equation for the $K_\mathrm{sp}$:$$\ce{[OH^-]} =\sqrt[3]{\frac{6\cdot10^{-38}}{\ce{[Fe^{3+}]}}} = \sqrt[3]{\frac{6\cdot10^{-38}}{9.35\cdot10^{-5}}} = 8.63\cdot10^{-12}$$ Knowing $\ce{[OH^-]}$ you can calculate $\ce{[H^+]}$ via $$\ce{[H^+]} = \frac{1\cdot10^{-14}}{\ce{[OH^-]}} = \frac{1\cdot10^{-14}}{8.63\cdot10^{-12}} = 1.16\cdot10^{-3}$$ and then pH: $$\mathrm{pH} = -\log\left(1.16\cdot10^{-3}\right) = 2.94.$$ Now the OP has modified the problem statement to be: Knowing $K_\mathrm{s} = 6\cdot10^{-38}$ for $\ce{Fe(OH)3}$ in neutral solutions, calculate the minimum pH of an acidic solution in order to completely dissolve $10\ \mathrm{mg}$ of $\ce{Fe(OH)3}$. Data: $V = \pu{0.1 L}$. We again end up with the final pH being just acidic enough to dissolve $\pu{10 mg}$ of iron (iii) hydroxide in $\pu{100 ml}$ of water, but obviously the solution must start out more acidic. The final solution will again have a pH of $2.94$ or a $\ce{[H^+]} = 1.16\cdot10^{-3}$. In $\pu{100 mL}$ of a solution of pH $2.94$ there are $\pu{0.116 mmol}$ of acid. We calculated before that there were $\pu{9.3510^{-5} mol}$ of $\ce{Fe^{3+}}$, but there are three $\ce{OH^-}$ anions for every cation of $\ce{Fe^{3+}}$, so there are $3 \times 9.35\cdot10^{-5} = \pu{2.8110^{-4} mol}$ of $\ce{OH^-}$. The total amount of substance (in millimoles) of acid needed is $$0.116 + 0.281 = 0.397.$$ For $\pu{100 ml}$ to contain $\pu{0.397 mmol}$ the solution would need to be 3.97 millimolar in $\ce{[H^+]}$, or the pH = $2.40$. If you start with $\pu{100 ml}$ of solution with pH $2.40$ and dump in $\pu{10 mg}$ of $\ce{Fe(OH)3}$, then the final pH will be $2.94$ and all of the $\ce{Fe(OH)3}$ will be just dissolved.
Among many fascinating sides of mathematics, there is one that I praise, especially in teaching at the boundary between undergraduate/graduate level : the parallels that can be drawn between a "Continuous world" and a "Discrete world". A concept, an explanation, etc. in one these worlds that can be "paralleled" into the other. I am looking for instances that bring a help to a global understanding... Disclaimer : my objective here is analogy, for didactic purposes, without the barrier of full rigor. Besides, I do not deny at all the interest of having a rigorous approach showing in particular in which sense the continuous "object" is the limit of the discrete one. I should appreciate if some colleagues can give examples of their own, in the style "my favorite one is...", or references to works about this theme. Let me provide five typical examples: 1st example: How to obtain the equations of certain epicycloids, here a nephroid : Consider a $N$-sided regular polygon $A_1,A_2,\cdots A_N$ with any integer $N$ large enough, say around $50$. Let us connect every point $A_k$ to point $A_{3k}$ by a line segment (we assume a cyclic numbering). As can be seen on Fig. 1, a certain "envelope curve" is generated (I should say "suggested"). Question : which (smooth) curve is behind this construction ? Let us consider two consecutive line segments like those represented on Fig. 1 with a larger width : the evolution speed of $A_{3k} \to A_{3k'}$ where $k'=k+1$ is three times the evolution speed of $A_{k} \to A_{k'}$, the pivoting of the line segment takes place at the point (of the line segment) which is 3 times closer to $A_k$ than to $A_{3k}$ (the ratio 3:1 of weights comes from the size ratio of ''homothetic'' triangles $P_kA_kA_k'$ and $P_kA_{3k}A_{3k'}$.) Said in an algebraic way : $$P_k=\tfrac{3}{4}e^{ika}+\tfrac{1}{4}e^{3ika}$$ ($A_k$ is identified with $e^{ika}$ with $a:=\tfrac{2 \pi}{N}$). Replacing now discrete values $ka$ by a continuous parameter $t$, we get $$z=\tfrac{3}{4}e^{it}+\tfrac{1}{4}e^{3it}$$ i.e., a parametric representation of the nephroid, or the equivalent real equations : $$\begin{cases}x=\tfrac{3}{4}\cos(t)+\tfrac{1}{4}\cos(3t)\\ y=\tfrac{3}{4}\sin(t)+\tfrac{1}{4}\sin(3t)\end{cases}$$ Fig. 1 : The nephroid as an envelope. It can be viewed as the trajectory of a point of a small circle with radius $\dfrac14$ rolling inside a circle with radius $1$. Remark: if, instead of connecting $A_k$ to $A_{3k}$, we had connected it to $A_{2k}$, we would have obtained a cardioid, with $A_{4k}$ an astroid, etc. 2nd example: Coupling ''second derivative $ \ \leftrightarrow \ \min \ $ kernel'' : All functions considered here are at least $C^2$, but function $K$. Let $f:[0,1] \rightarrow \mathbb{R}$ and $K:[0,1]\times[0,1]\rightarrow \mathbb{R}$ (a so-called "kernel") defined by $K(x,y):=\min(x,y)$. Let us associate $f$ with function $\varphi(f)=g$ defined by $$\tag{1}g(y)=\int_{t=0}^{t=1} K(t,y)f(t)dt=\int_{t=0}^{t=1} \min(t,y)f(t)dt$$ We can get rid of "$\min$" function by decomposing the integral into : $$\tag{2}g(y)=\int_{t=0}^{t=y} t f(t)dt+\int_{t=y}^{t=1} y f(t)dt$$ $$\tag{3}g(y)=\int_{t=0}^{t=y} t f(t)dt - y F(y)$$ where we have set $$\tag{4}F(y):=\int_{t=1}^{t=y}f(t)dt \ \ \ \ \ \ \ \ \text{Remark:} \ \ \ F'(y)=f(y)$$ Let us differentiate (4) twice : $$\tag{5}g'(y)=y f(y) - 1 F(y) - y f(y) = -F(y)$$ $$\tag{6}g''(y)= -f(y) \ \ \Longleftrightarrow \ \ f(y)=-g''(y)$$ Said otherwise, the inverse of transform $f \rightarrow \varphi(f)=g$ is: $$\tag{7}\varphi^{-1} = \text{opposite of the second derivative.}$$ This connexion with the second derivative is rather unexpected... Had we taken a discrete approach, what would have been found ? The discrete equivalents of $\varphi$ and $\varphi^{-1}$ are matrices : $$\bf{M}=\begin{pmatrix}1&1&1&\cdots&\cdots&1\\1&2&2&\cdot&\cdots&2\\1&2&3&\cdots&\cdots&3\\\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\1&2&3&\cdots&\cdots&n \end{pmatrix} \ \ \textbf{and}$$ $$\bf{D}=\begin{pmatrix}2&-1&&&&\\-1&2&-1&&&\\&-1&2&-1&&\\&&\ddots&\ddots&\ddots&\\&&&-1&2&-1\\&&&&-1&1 \end{pmatrix}$$ that verify matrix identity: $\bf{M}^{-1}=\bf{D}$ in analogy with (7). Indeed, Nothing to say about the connection of matrix $\bf{M}$ with coefficients $\bf{M}_{i,j}=min(i,j)$ with operator $K$. tridiagonal matrix $\bf{D}$ is well known (in particular by all people doing discretization) to be "the" discrete analog of the second derivative due to the classical approximation: $$f''(x)\approx\dfrac{1}{2h^2}(f(x-h)-2f(x)+f(x+h))$$ that can easily be obtained using Taylor expansions. The exceptional value $1$ at bottom right of $\bf{D}$ is explained by discrete boundary conditions. Remark: this correspondence between "min" operator and second derivative is not mine ; I have known it for such a long time that I am unable to trace back where I saw it at first (hopefully in a signal processing book). If somebody has a reference ? Connected : the eigenvalues of $D$ are remarkable (http://www.math.nthu.edu.tw/~amen/2005/040903-7.pdf) 3rd example : the Itô integral. One could think that the Lebesgue integral (1902) is the ultimate theory of integration, correcting the imperfections of the theory elaborated by Riemann some 50 years before. This is not the case. In particular, Itô has defined (1942) a new kind of integral which is now essential in probability and finance. Its principle, roughly said, is that infinitesimal "deterministic" increments "dt" are replaced by random increments of brownian motion type as formalized by Einstein (1905), then by Wiener (1923). Let us give an image of it. Let us first recall definitions of brownian motion $W(t)$ or $W_t$, ($W$ for Wiener), an informal one, and a formal one: Informal : A "particle" starting at $x=0$ at time $t$, jumps "at the next instant" $t+dt$, to a nearby position; either on the left or on the right, the amplitude and sign of the jump being governed by a normal distribution $N(x,\sigma^2)$ with an infinitesimal fixed standard deviation $\sigma.$ $\text{Formal}: \ \ W_t:=G_0 t+\sqrt{2}\sum_{n=1}^{\infty}G_n\dfrac{\sin(\pi n t)}{\pi n}$, with $G_n$ iid $N(0,1)$ random variables. (Other definitions exist. This one, under the form of a "random Fourier series" is handy for many computations). Let us now consider one of the fundamental formulas of Itô's integral, for a continuously differentiable function $f$: $$\tag{8}\begin{equation} \displaystyle\int_0^t f(W(s))dW(s) = \displaystyle\int_0^{W(t)} f(\lambda)d \lambda - \frac{1}{2}\displaystyle\int_0^t f'(W(s))ds. \end{equation}$$ Remark: The integral sign on the LHS of (8) defines Itô's integral, whereas the integrals on the RHS have to be understood in the sense of Riemann/Lebesgue. The presence of the second term on the RHS is rather puzzling, isnt'it ? Question: how can be understood/justified this second integral ? Szabados has proposed (1990) (see (https://mathoverflow.net/questions/16163/discrete-version-of-itos-lemma)) a discrete analog of formula (8). Here is how it runs: Theorem: Let $f:\mathbb{Z} \longrightarrow \mathbb{R}$. let us define : $$ \tag{9}\begin{equation} F(k)=\left\{ \begin{matrix} \dfrac{1}{2}f(0)+\displaystyle\sum_{j=1}^{k-1} f(j)+\dfrac{1}{2}f(k) & if & k \geq 1 & \ \ (a)\\ 0 & if & k = 0 & \ \ (b)\\ -\dfrac{1}{2}f(k)-\displaystyle\sum_{j=k+1}^{-1} f(j)-\dfrac{1}{2}f(0) & if & k \leq -1 & \ \ (c) \end{matrix} \right. \end{equation} $$ Remarks: 1) We will work only on (a) and its particular case (b). 2) (a) is nothing else than the "trapezoid formula" explaining in particular factors $\dfrac{1}{2}$ in front of $f(0)$ et $f(k)$. Let us now define a family of Random Variables $X_k$, $k=1, 2, \cdots $, iid on $\{-1,1\}$ with $P(X_k=-1)=P(X_k=1)=\frac{1}{2}$, and let $$ \begin{equation} S_n= \displaystyle\sum_{k=1}^n X_k. \end{equation} $$ Then $$ \tag{10}\begin{equation} \forall n, \ \ \displaystyle\sum_{i=0}^{n}f(S_i)X_{i+1} = F(S_{n+1})-\dfrac{1}{2}\displaystyle\sum_{i=0}^{n}\dfrac{f(S_{i+1})-f(S_{i})}{X_{i+1}} \end{equation} $$ Remark : Please note analogies : between $\frac{f(S_{i+1})-f(S_{i})}{X_{i+1}}$ and $f'(S_i)$. between $F(k)$ and $\displaystyle\int_{\lambda=0}^{\lambda=k}f(\lambda)d\lambda$. For example, a) If $f$ is identity function ($\forall k \ f(k)=k$), definition (9)(a) gives : $$ \begin{equation} F(k)=\frac{1}{2}(k-1)k+\frac{1}{2}k=\dfrac{1}{2}k^2. \tag{11} \end{equation} $$ which doesn't come as a surprise : the 'discrete antiderivative' of $k$ is $\frac{1}{2}k^2$... (the formula in (11) remains in fact the same for $k<0$). b) If $f$ is the "squaring function" ($\forall k, \ f(k)=k^2$), (9)(a) becomes : $$ \begin{equation} \text{If} \ k>0, \ \ \ F(k)=\frac{1}{6}(k-1)k(2k-1)+\frac{1}{2}k^2=\dfrac{1}{3}k^3+\dfrac{1}{6}k. \tag{12} \end{equation} $$ This time, a new term $\dfrac{1}{6}k$ has entered into the play. Proof of the Theorem: The definition allows to write : \begin{equation}F(S_{i+1})-F(S_i)=f(S_i)X_{i+1}+\frac{1}{2}\dfrac{f(S_{i+1})-f(S_i)}{X_{i+1}} \tag{13} \end{equation} In fact, proving (11) can be split into two cases: either $X_{i+1}=1$, or $X_{i+1}=-1$. Let us consider the first case (the second case is similar): the RHS of (13) becomes $f(S_i)+\frac{1}{2}(f(S_{i+1})-f(S_i))=\frac{1}{2}(f(S_{i+1})+f(S_i))$ which is the area variation in the trapezoid formula ; Summing all equations in (10) gives the desired identity. An example of application : Let $f(t)=t$ ; we get $$\displaystyle\sum_{i=0}^{n}S_iX_{i+1} = F(S_{n+1})-\frac{n}{2}=\dfrac{1}{2}S_{n+1}^2-\frac{n}{2}.$$ which appears as the discrete equivalent of the celebrated formula: $$ \begin{equation} \displaystyle\int_0^t W(s)dW(s) = \frac{1}{2}W(t)^2-\frac{1}{2}t. \end{equation} $$ One can establish the autocorrelation of $W_t$ process is $$cov(W_s,W_t)=E(W_sW_t)-E(W_s)E(W_t)=\min(s,t),$$ (see (Autocorrelation of a Wiener Process proof)) providing an unexpected connection with the second example... Last remark: Another kind of integral based on a discrete definition : the gauge integral (https://math.vanderbilt.edu/schectex/ccc/gauge/). 4th example (Darboux sums) : Here is a discrete formula : $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$ Has this formula a continuous "counterpart" ? Taking the logarithm on both sides, and dividing by $n$, we get : $$\tfrac1n \sum_{k=1}^n \ln \sin \tfrac{k \pi}{n}=\tfrac{\ln(n)}{n}-\ln(2)\tfrac{n-1}{n}$$ Letting now $n \to \infty$, we obtain the rather classical integral : $$\int_0^1 \ln(\sin(\pi x))dx=-\ln(2)$$ 5th example : bivariate cdfs (cumulative probability density functions). Let $(X,Y)$ a pair of Random Variables with pdf $f_{X,Y}$ and cdf : $$F_{X,Y}(x,y):=P(X \leq x \ \& \ Y \leq y).$$ Take a look at this formula : $$P(x_1<X \leq x_2, \ \ y_1<Y \leq y_2)=F_{XY}(x_2,y_2)-F_{XY}(x_1,y_2)-F_{XY}(x_2,y_1)+F_{XY}(x_1,y_1)\tag{14}$$ It is the discrete equivalent of the continuous definition of $F_{XY}$ as the mixed second order partial derivative of $F_{X,Y}$, under the assumption that $F$ is a $C^2$ function : $$f_{XY}(x,y)=\dfrac{\partial^2 F_{X,Y}}{\partial x \partial y}(x,y).\tag{15}$$ Do you see why ? Hint : make $x_2 \to x_1$ and make $y_2 \to y_1$ and assimilate the LHS of (14) with $f(x_1,y_1)dxdy$. Final remarks : 1) A remarkable text about this analogy in Physics : https://www.lptmc.jussieu.fr/user/lesne/MSCS-Lesne.pdf 2) There are many other tracks, e.g., connections with graphs (http://jimhuang.org/CDNDSP.pdf), discrete vs. continuous versions of the logistic equation (https://math.stackexchange.com/q/3328867), etc.
I can't seem to get anywhere with this problem. Any hints would be much appreciated: Suppose that $p$ and $q$ are distinct primes satisfying $p, q \equiv 1 \bmod{4}$. Show that the congruence $x^2 \equiv -1 \bmod {pq}$ has a solution. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community I can't seem to get anywhere with this problem. Any hints would be much appreciated: Suppose that $p$ and $q$ are distinct primes satisfying $p, q \equiv 1 \bmod{4}$. Show that the congruence $x^2 \equiv -1 \bmod {pq}$ has a solution. By supplemental laws of quadratic reciprocity, we know that $-1$ is a square mod $p$ exactly when $p \equiv 1 \pmod 4$. So we know there exist $x_p$ such that $x_p^2 \equiv -1 \pmod p$ and $x_q$ such that $x_q^2 \equiv -1 \pmod q$. By the Chinese Remainder Theorem, there is a solution $x$ to the system of congruences $$\begin{align} x &\equiv x_p \pmod p \\ x &\equiv x_q \pmod q \end{align}$$ and that this $x$ is defined $\bmod {pq}$. Since $x^2 \equiv -1$ both $\bmod p$ and $\bmod q$, this $x$ is the solution you're looking for. $\diamondsuit$
Examples On Vector Dot Product Set-3 Example – 21 If a, b, c are the lengths of the sides of $\Delta ABC$ opposite to the angles A, B and C respectively, prove using vector methods that \[a(1 + \cos A) + b(1 + \cos B) + c(1 + \cos C) = (a + b + c)(\cos A + \cos B + \cos C)\] Solutions: We have, by the triangle law, \[\begin{align}&\qquad\;\;\vec a + \vec b + \vec c = \vec 0 \hfill \\\\ &\Rightarrow \quad \vec a = - (\vec b + \vec c) \hfill \\\\ &\Rightarrow \quad \vec a \cdot \vec a = - \vec a \cdot (\vec b + \vec c) \hfill \\\\& \Rightarrow \quad {a^2} = ab\cos C + ac\cos B\left( \begin{gathered} \because \vec a \cdot \vec b = - ab\cos C \hfill \\\\\,\,\,\,\,\vec a \cdot \vec c = - ac\cos B \hfill \\\\ \end{gathered} \right) \hfill \\\\& \Rightarrow \quad a = b\cos C + c\cos B\qquad\qquad\qquad...\left( 1 \right) \hfill \\ \end{align} \] Similarly, \[\begin{align}&b = c\cos A + a\cos C\qquad\qquad\qquad...\left( 2 \right) \hfill \\\\& c = a\cos B + b\cos A\qquad\qquad\qquad...\left( 3 \right) \hfill \\ \end{align} \] Adding (1), (2) and (3), we have \[a + b + c = a(\cos B + \cos C) + b(\cos C + \cos A) + c(\cos A + \cos B)\] Adding $a\cos A + b\cos B + c\cos C$ on both sides, we have \[a(1 + \cos A) + b(1 + \cos B) + c(1 + \cos C) = (a + b + c)(\cos A + \cos B + \cos C)\] Example – 22 Find three-dimensional vectors ${\vec v_1},\;\;{\vec v_2}\;\;{\text{and}}\;\;{\vec v_3}$ satisfying the relations \[\begin{align}& {{\vec v}_1} \cdot {{\vec v}_1} = 4 && {{\vec v}_1} \cdot {{\vec v}_2} = - 2 && {{\vec v}_1} \cdot {{\vec v}_3} = 6 \hfill \\\\&{{\vec v}_2} \cdot {{\vec v}_2} = 2 && {{\vec v}_2} \cdot {{\vec v}_3} = - 5 & &{{\vec v}_3} \cdot {{\vec v}_3} = 29 \hfill \\ \end{align} \] Solutions: A reference frame for the vectors has not been specified; therefore, it is up to us to choose a reference frame and then use it consistently and evaluate the required vectors in that reference frame. Assume ${\vec v_1}$ to be along the x-direction, i.e. \[\begin{align}&\qquad\; {{\vec v}_1} = 2\hat i \hfill \\\\& Let\quad{{\vec v}_2} = a\hat i + b\hat j + c\hat k \hfill \\\\&\qquad\;\; {{\vec v}_3} = p\hat i + q\hat j + r\hat k \hfill \\ \end{align} \] Now we step by step use all the given relations to determine the unknown constraints: \[\begin{align}& {{\vec v}_1} \cdot {{\vec v}_2} = - 2 \quad \Rightarrow \quad 2a = - 2 \hfill \\\\&\qquad\qquad\qquad \Rightarrow\quad a = - 1\quad\qquad\qquad\;...\left( 1 \right) \hfill \\\\& {{\vec v}_2} \cdot {{\vec v}_2} = 2 \quad\;\; \Rightarrow\quad {a^2} + {b^2} + {c^2} = 2 \hfill \\\\& \qquad\qquad\qquad \Rightarrow \quad {b^2} + {c^2} = 1\quad\qquad\quad...\left( 2 \right){\text{ }}\left( {from{\text{ }}\left( 1 \right)} \right) \hfill \\\\&{{\vec v}_1} \cdot {{\vec v}_3} = 6 \quad \;\; \Rightarrow \quad 2p = 6 \hfill \\\\ &\qquad\qquad\qquad \Rightarrow \quad p = 3\qquad\qquad\qquad\;\;...\left( 3 \right) \hfill \\\\& {{\vec v}_2} \cdot {{\vec v}_3} = - 5 \quad \Rightarrow \quad ap + bq + cr = - 5 \hfill \\\\&\qquad\qquad\qquad\;\Rightarrow \quad bq + cr = - 2\qquad\;\;\;\;\;...\left( 4 \right){\text{ }}\left( {using{\text{ }}\left( 1 \right){\text{ }}and{\text{ }}\left( 3 \right)} \right) \hfill \\\\&\;\; {{\vec v}_3} \cdot {{\vec v}_3} = 29 \quad \Rightarrow \quad {p^2} + {q^2} + {r^2} = 29 \hfill \\\\&\qquad\qquad\qquad\;\Rightarrow \quad {q^2} + {r^2} = 20\qquad\;\;\;\;\;\;...\left( 5 \right){\text{ }}\left( {using{\text{ }}\left( 3 \right)} \right) \hfill \\ \end{align} \] Notice that (2), (4) and (5) are three equations in four unknowns. To get over this problem (it is not a problem actually! There will be an infinite set of vectors satisfying the given constraints. We have to find any one of them), when we chose ${{\vec v}_1}$ to be along the x-axis, we could also have adjusted the co-ordinate frame, so that ${\vec v_1}\;\;{\text{and}}\;\;{\vec v_2}$ lie in the x – z plane. This can always be done; since it is upto us to choose the frame of reference, we chose it so that the x – z plane co-insides with the plane of ${\vec v_1}\;\;{\text{and}}\;\;{\vec v_2}$. How does this help? Now we’ll have one unknown less, since the y-component of ${{\vec v}_2}$ is zero, i.e., b = 0. Thus, (2), (4) and (5) reduce to \[{c^2} = 1, \,\,\,\,\,\,cr = 0 - 2, \,\,\,\,\,\,{q^2} + {r^2} = 20\] \[ \Rightarrow \quad c = \pm \;1,\,\,\,\,\,\,\,\,r = \mp \;2, \,\,\,\,\,\,\,\,q = \pm 4\] Thus, the three dimensional vectors that satisfy the given constraints can be \[{\vec v_1} = 2\hat i \,\,\,\,\,\,\,\,\,{\vec v_2} = - \hat i + \hat k \,\,\,\,\,\,\,\,\, {\vec v_3} = 3\hat i \pm 4\hat j - 2\hat k\] OR \[{\vec v_1} = 2\hat i \,\,\,\,\,\,\,\,\,{\vec v_2} = - \hat i - \hat k \,\,\,\,\,\,\,\,\,{\vec v_3} = 3\hat i \pm 4\hat j + 2\hat k\] To emphasize once again, we were required to find vectors satisfying the given constraints. This meant that absolute positions of the vectors were not important; what mattered was their relative sizes and orientation; and thus the coordinate axes was our choice. We selected it in a way which made the calculations most convenient. TRY YOURSELF - II Q. 1 Determine the values of c possible so that for all real x, the vectors $cx\hat i - 6\hat j + 3\hat k\;and\;x\hat i + 2\hat j + 2cx\hat k$ and make an obtuse angle with each other. Q. 2 Constant forces ${F_1} \equiv (2\hat i - 5\hat j + 6\hat k)N\;and\;{\vec F_2} \equiv ( - \hat i + 2\hat j - \hat k)N$ act on a particle and the particle is displaced from $A \equiv (4\hat i - 3\hat j - 2\hat k)m\;to\;B \equiv (6\hat i + \hat j - 3\hat k)m$ . Find the total work done by the forces. Q. 3 Show that the diagonals of a rhombus bisect each other at right angles. Q. 4 Using vectors, prove the trigonometric relation \[\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B\] Q. 5 Prove that the perpendicular bisectors of the sides of a triangle are concurrent. Q. 6 In $\Delta ABC$ with sides a, b, c opposite to angles A, B, C respectively, prove that (i) ${a^2} = {b^2} + {c^2} - 2bc\;\;\cos A$ (ii) $\begin{align}a\cos B - b\cos A = \frac{{{a^2} - {b^2}}}{c}\end{align}$ Q. 7 Find the unit vector which makes equal angles with the vectors $(\hat i - 2\hat j + 2\hat k),\,\,( - 4\hat i - 3\hat k)\;\;{\text{and}}\;\;\hat j$ Q. 8 The lengths of the sides a, b, c of $\Delta ABC$ ( a, b, c opposite to A, B, C respectively) satisfy the relation ${a^2} + {b^2} = 5{c^2}$ . Prove that the medians drawn to the sides with lengths a and b, are perpendicular. Q. 9 Find the possible values of a for which the vector $\vec r = ({a^2} - 4)\,\hat i + 2\hat j - ({a^2} - 9)\hat k$ makes acute angles with the coordinate axes. Q. 10 Prove using vector methods that the angle in a semi-circle is a right angle.
Learning Objectives Explain the concepts of stress and strain in describing elastic deformations of materials Describe the types of elastic deformation of objects and materials A model of a rigid body is an idealized example of an object that does not deform under the actions of external forces. It is very useful when analyzing mechanical systems—and many physical objects are indeed rigid to a great extent. The extent to which an object can be perceived as rigid depends on the physical properties of the material from which it is made. For example, a ping-pong ball made of plastic is brittle, and a tennis ball made of rubber is elastic when acted upon by squashing forces. However, under other circumstances, both a ping-pong ball and a tennis ball may bounce well as rigid bodies. Similarly, someone who designs prosthetic limbs may be able to approximate the mechanics of human limbs by modeling them as rigid bodies; however, the actual combination of bones and tissues is an elastic medium. For the remainder of this section, we move from consideration of forces that affect the motion of an object to those that affect an object’s shape. A change in shape due to the application of a force is known as a deformation. Even very small forces are known to cause some deformation. Deformation is experienced by objects or physical media under the action of external forces—for example, this may be squashing, squeezing, ripping, twisting, shearing, or pulling the objects apart. In the language of physics, two terms describe the forces on objects undergoing deformation: stress and strain. Stress is a quantity that describes the magnitude of forces that cause deformation. Stress is generally defined as force per unit area. When forces pull on an object and cause its elongation, like the stretching of an elastic band, we call such stress a tensile stress. When forces cause a compression of an object, we call it a compressive stress. When an object is being squeezed from all sides, like a submarine in the depths of an ocean, we call this kind of stress a bulk stress (or volume stress). In other situations, the acting forces may be neither tensile nor compressive, and still produce a noticeable deformation. For example, suppose you hold a book tightly between the palms of your hands, then with one hand you press-and-pull on the front cover away from you, while with the other hand you press-and-pull on the back cover toward you. In such a case, when deforming forces act tangentially to the object’s surface, we call them ‘shear’ forces and the stress they cause is called shear stress. The SI unit of stress is the pascal (Pa). When one newton of force presses on a unit surface area of one meter squared, the resulting stress is one pascal: $$one\; pascal = 1.0\; Pa = \frac{1.0\; N}{1.0\; m^{2}} \ldotp$$ In the British system of units, the unit of stress is ‘psi,’ which stands for ‘pound per square inch’ (lb/in 2). Another unit that is often used for bulk stress is the atm (atmosphere). Conversion factors are $$1\; psi = 6895\; Pa\; and\; 1\; Pa = 1.450 \times 10^{-4}\; psi$$ $$1\; atm = 1.013 \times 10^{5}\; Pa = 14.7\; psi \ldotp$$ An object or medium under stress becomes deformed. The quantity that describes this deformation is called strain. Strain is given as a fractional change in either length (under tensile stress) or volume (under bulk stress) or geometry (under shear stress). Therefore, strain is a dimensionless number. Strain under a tensile stress is called tensile strain, strain under bulk stress is called bulk strain (or volume strain), and that caused by shear stress is called shear strain. The greater the stress, the greater the strain; however, the relation between strain and stress does not need to be linear. Only when stress is sufficiently low is the deformation it causes in direct proportion to the stress value. The proportionality constant in this relation is called the elastic modulus. In the linear limit of low stress values, the general relation between stress and strain is $$stress = (elastic\; modulus) \times strain \ldotp \label{12.33}$$ As we can see from dimensional analysis of this relation, the elastic modulus has the same physical unit as stress because strain is dimensionless. We can also see from Equation \ref{12.33} that when an object is characterized by a large value of elastic modulus, the effect of stress is small. On the other hand, a small elastic modulus means that stress produces large strain and noticeable deformation. For example, a stress on a rubber band produces larger strain (deformation) than the same stress on a steel band of the same dimensions because the elastic modulus for rubber is two orders of magnitude smaller than the elastic modulus for steel. The elastic modulus for tensile stress is called Young’s modulus; that for the bulk stress is called the bulk modulus; and that for shear stress is called the shear modulus. Note that the relation between stress and strain is an observed relation, measured in the laboratory. Elastic moduli for various materials are measured under various physical conditions, such as varying temperature, and collected in engineering data tables for reference (Table $\PageIndex{1}$). These tables are valuable references for industry and for anyone involved in engineering or construction. In the next section, we discuss strain-stress relations beyond the linear limit represented by Equation \ref{12.33}, in the full range of stress values up to a fracture point. In the remainder of this section, we study the linear limit expressed by Equation \ref{12.33}. Material Young’s modulus × 10 Bulk modulus × 10 Shear modulus × 10 Aluminum 7.0 7.5 2.5 Bone (tension) 1.6 0.8 8.0 Bone (compression) 0.9 Brass 9.0 6.0 3.5 Brick 1.5 Concrete 2.0 Copper 11.0 14.0 4.4 Crown glass 6.0 5.0 2.5 Granite 4.5 4.5 2.0 Hair (human) 1.0 Hardwood 1.5 1.0 Iron 21.0 16.0 7.7 Lead 1.6 4.1 0.6 Marble 6.0 7.0 2.0 Nickel 21.0 17.0 7.8 Polystyrene 3.0 Silk 6.0 Spider thread 3.0 Steel 20.0 16.0 7.5 Acetone 0.07 Ethanol 0.09 Glycerin 0.45 Mercury 2.5 Water 0.22 Tensile or Compressive Stress, Strain, and Young’s Modulus Tension or compression occurs when two antiparallel forces of equal magnitude act on an object along only one of its dimensions, in such a way that the object does not move. One way to envision such a situation is illustrated in Figure $\PageIndex{1}$. A rod segment is either stretched or squeezed by a pair of forces acting along its length and perpendicular to its cross-section. The net effect of such forces is that the rod changes its length from the original length L 0 that it had before the forces appeared, to a new length L that it has under the action of the forces. This change in length $\Delta$L = L − L 0 may be either elongation (when $L$ is larger than the original length $L_o$) or contraction (when L is smaller than the original length L 0). Tensile stress and strain occur when the forces are stretching an object, causing its elongation, and the length change $\Delta L$ is positive. Compressive stress and strain occur when the forces are contracting an object, causing its shortening, and the length change $\Delta L$ is negative. In either of these situations, we define stress as the ratio of the deforming force $F_{\perp}$ to the cross-sectional area A of the object being deformed. The symbol F $\perp$ that we reserve for the deforming force means that this force acts perpendicularly to the cross-section of the object. Forces that act parallel to the cross-section do not change the length of an object. The definition of the tensile stress is $$tensile\; stress = \frac{F_{\perp}}{A} \ldotp \label{12.34}$$ Tensile strain is the measure of the deformation of an object under tensile stress and is defined as the fractional change of the object’s length when the object experiences tensile stress $$tensile\; strain = \frac{\Delta L}{L_{0}} \ldotp \label{12.35}$$ Compressive stress and strain are defined by the same formulas, Equations \ref{12.34} and \ref{12.35}, respectively. The only difference from the tensile situation is that for compressive stress and strain, we take absolute values of the right-hand sides in Equation \ref{12.34} and \ref{12.35}. Young’s modulus $Y$ is the elastic modulus when deformation is caused by either tensile or compressive stress, and is defined by Equation \ref{12.33}. Dividing this equation by tensile strain, we obtain the expression for Young’s modulus: $$Y = \frac{tensile\; stress}{tensile\; strain} = \frac{\frac{F_{\perp}}{A}}{\frac{\Delta L}{L_{0}}} = \frac{F_{\perp}}{A} = \frac{L_{0}}{\Delta L} \ldotp \label{12.36}$$ Example $\PageIndex{1}$: Compressive Stress in a Pillar A sculpture weighing 10,000 N rests on a horizontal surface at the top of a 6.0-m-tall vertical pillar Figure $\PageIndex{1}$. The pillar’s cross-sectional area is 0.20 m 2 and it is made of granite with a mass density of 2700 kg/m 3. Find the compressive stress at the cross-section located 3.0 m below the top of the pillar and the value of the compressive strain of the top 3.0-m segment of the pillar. Strategy First we find the weight of the 3.0-m-long top section of the pillar. The normal force that acts on the cross-section located 3.0 m down from the top is the sum of the pillar’s weight and the sculpture’s weight. Once we have the normal force, we use Equation 12.34 to find the stress. To find the compressive strain, we find the value of Young’s modulus for granite in Table $\PageIndex{1}$ and invert Equation \ref{12.36}. Solution The volume of the pillar segment with height h = 3.0 m and cross-sectional area A = 0.20 m 2 is $$V = Ah = (0.20\; m^{2})(3.0\; m) = 0.60\; m^{3} \ldotp$$ With the density of granite $\rho$ = 2.7 x 10 3 kg/m 3, the mass of the pillar segment is $$m = \rho V = (2.7 \times 10^{3}\; kg/m^{3})(0.60\; m^{3}) = 1.60 \times 10^{3}\; kg \ldotp$$ The weight of the pillar segment is $$w_{p} = mg = (1.60 \times 10^{3}\; kg)(9.80\; m/s^{2}) = 1.568 \times 10^{4}\; N \ldotp$$ The weight of the sculpture is w s = 1.0 x 10 4 N, so the normal force on the cross-sectional surface located 3.0 m below the sculpture is $$F_{\perp} = w_{p} + w_{s} = (1.568 + 1.0) \times 10^{4}\; N = 2.568 \times 10^{4}\; N \ldotp$$ Therefore, the stress is $$stress = \frac{F_{\perp}}{A} = \frac{2.568 \times 10^{4}\; N}{0.20 m^{2}} = 1.284 \times 10^{5}\; Pa = 128.4\; kPa \ldotp$$ Young’s modulus for granite is Y = 4.5 x 10 10 Pa = 4.5 x 10 7 kPa. Therefore, the compressive strain at this position is $$strain = \frac{stress}{Y} = \frac{128.4\; kPa}{4.5 \times 10^{7}\; kPa} = 2.85 \times 10^{-6} \ldotp$$ Significance Notice that the normal force acting on the cross-sectional area of the pillar is not constant along its length, but varies from its smallest value at the top to its largest value at the bottom of the pillar. Thus, if the pillar has a uniform cross-sectional area along its length, the stress is largest at its base. Exercise $\PageIndex{2}$ Find the compressive stress and strain at the base of Nelson’s column. Example $\PageIndex{2}$: Stretching a Rod A 2.0-m-long steel rod has a cross-sectional area of 0.30 cm 2. The rod is a part of a vertical support that holds a heavy 550-kg platform that hangs attached to the rod’s lower end. Ignoring the weight of the rod, what is the tensile stress in the rod and the elongation of the rod under the stress? Strategy First we compute the tensile stress in the rod under the weight of the platform in accordance with Equation 12.34. Then we invert Equation 12.36 to find the rod’s elongation, using L 0 = 2.0 m. From Table 12.1, Young’s modulus for steel is Y = 2.0 x 10 11 Pa. Solution Substituting numerical values into the equations gives us $$\begin{split} \frac{F_{\perp}}{A} & = \frac{(550\; kg)(9.8\; m/s^{2})}{3.0 \times 10^{-5}\; m^{2}} = 1.8 \times 10^{8}\; Pa \\ \Delta L & = \frac{F_{\perp}}{A} \frac{L_{0}}{Y} = (1.8 \times 10^{8}\; Pa) \left(\dfrac{2.0\; m}{2.0 \times 10^{11}\; Pa}\right) = 1.8 \times 10^{-3}\; m = 1.8\; mm \ldotp \end{split}$$ Significance Similarly as in the example with the column, the tensile stress in this example is not uniform along the length of the rod. Unlike in the previous example, however, if the weight of the rod is taken into consideration, the stress in the rod is largest at the top and smallest at the bottom of the rod where the equipment is attached. Exercise $\PageIndex{2}$ A 2.0-m-long wire stretches 1.0 mm when subjected to a load. What is the tensile strain in the wire? Objects can often experience both compressive stress and tensile stress simultaneously Figure $\PageIndex{3}$. One example is a long shelf loaded with heavy books that sags between the end supports under the weight of the books. The top surface of the shelf is in compressive stress and the bottom surface of the shelf is in tensile stress. Similarly, long and heavy beams sag under their own weight. In modern building construction, such bending strains can be almost eliminated with the use of I-beams Figure $\PageIndex{4}$. Simulation A heavy box rests on a table supported by three columns. View this demonstration to move the box to see how the compression (or tension) in the columns is affected when the box changes its position. Contributors Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).
Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$.Let's denote the width of a sample by $h$ where $$h\rightarrow0$$Now, for finding the area under the curve between the bounds $a ~\& ~b $ we can a... @Ultradark You can try doing a finite difference to get rid of the sum and then compare term by term. Otherwise I am terrible at anything to do with primes that I don't know the identities of $\pi (n)$ well @Silent No, take for example the prime 3. 2 is not a residue mod 3, so there is no $x\in\mathbb{Z}$ such that $x^2-2\equiv 0$ mod $3$. However, you have two cases to consider. The first where $\binom{2}{p}=-1$ and $\binom{3}{p}=-1$ (In which case what does $\binom{6}{p}$ equal?) and the case where one or the other of $\binom{2}{p}$ and $\binom{3}{p}$ equals 1. Also, probably something useful for congruence, if you didn't already know: If $a_1\equiv b_1\text{mod}(p)$ and $a_2\equiv b_2\text{mod}(p)$, then $a_1a_2\equiv b_1b_2\text{mod}(p)$ Is there any book or article that explains the motivations of the definitions of group, ring , field, ideal etc. of abstract algebra and/or gives a geometric or visual representation to Galois theory ? Jacques Charles François Sturm ForMemRS (29 September 1803 – 15 December 1855) was a French mathematician.== Life and work ==Sturm was born in Geneva (then part of France) in 1803. The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg around 1760 - about 50 years before Charles-François's birth. His mother's name was Jeanne-Louise-Henriette Gremay. In 1818, he started to follow the lectures of the academy of Geneva. In 1819, the death of his father forced Sturm to give lessons to children of the rich in order to support his own family. In 1823, he became tutor to the son... I spent my career working with tensors. You have to be careful about defining multilinearity, domain, range, etc. Typically, tensors of type $(k,\ell)$ involve a fixed vector space, not so many letters varying. UGA definitely grants a number of masters to people wanting only that (and sometimes admitted only for that). You people at fancy places think that every university is like Chicago, MIT, and Princeton. hi there, I need to linearize nonlinear system about a fixed point. I've computed the jacobain matrix but one of the elements of this matrix is undefined at the fixed point. What is a better approach to solve this issue? The element is (24x_2 + 5cos(x_1)x_2)/abs(x_2). The fixed point is x_1=0, x_2=0 Consider the following integral: $\int 1/4(1/(1+(u/2)^2)))dx$ Why does it matter if we put the constant 1/4 behind the integral versus keeping it inside? The solution is $1/2\arctan{(u/2)}$. Or am I overseeing something? it should be du instead of dx in the integral *and the solution is missing a constant C of course Is there a standard way to divide radicals by polynomials? Stuff like $\frac{\sqrt a}{1 + b^2}$? My expression happens to be in a form I can normalize to that, just the radicand happens to be a lot more complicated. In my case, I'm trying to figure out how to best simplify $\frac{x}{\sqrt{1 + x^2}}$, and so far, I've gotten to $\frac{x \sqrt{1+x^2}}{1+x^2}$, and it's pretty obvious you can move the $x$ inside the radical. My hope is that I can somehow remove the polynomial from the bottom entirely, so I can then multiply the whole thing by a square root of another algebraic fraction. Complicated, I know, but this is me trying to see if I can skip calculating Euclidean distance twice going from atan2 to something in terms of asin for a thing I'm working on. "... and it's pretty obvious you can move the $x$ inside the radical" To clarify this in advance, I didn't mean literally move it verbatim, but via $x \sqrt{y} = \text{sgn}(x) \sqrt{x^2 y}$. (Hopefully, this was obvious, but I don't want to confuse people on what I meant.) Ignore my question. I'm coming of the realization it's just not working how I would've hoped, so I'll just go with what I had before.
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?
Examples on Point and Angle of Intersection of Two Straight Lines Example – 5 Find the equation to the straight line which passes through (3, –2) and is inclined at ${{60}^{\circ }}$ to the line $\sqrt 3 \,x + y = 1.$ Solution: Observe carefully that there will be two such lines. Denote the two lines by ${L_1}\,\,{\text{and}}\,\,{L_2}$ Let the slope of the line(s) we require be m. The slope of $\sqrt 3 \,x + y = 1$ is ${m_1} = - \sqrt 3 $ Since we want the acute angle between the two lines to be 60º, we must have by Art - 5, \[\begin{align} & \qquad\quad\tan 60{}^\text{o}=\left\| \frac{{{m}_{1}}-m}{1+m{{m}_{1}}} \right\| \\ & \Rightarrow \quad \sqrt{3}=\left\| \frac{-\sqrt{3}-m}{1-\sqrt{3}\,m} \right\| \\ & \Rightarrow \quad \frac{m+\sqrt{3}}{1-\sqrt{3}\,m}=\pm \,\sqrt{3} \\ & \Rightarrow\quad m+\sqrt{3}=\sqrt{3}-3m\,\,\text{or}\,\,m+\sqrt{3}=3m-\sqrt{3} \\ & \Rightarrow\quad m=0\,\,\text{or}\,\,\,m=\sqrt{3} \\ \end{align}\] Since we get two values of m, this confirms our earlier assertion that two such lines will exist. We now have the slope. We also know that the lines pass through (3, –2). We can therefore use the point-slope form to write down the required equations: \[\begin{align}& \qquad \qquad {{L}_{1}}\equiv y-\left( -2 \right)=0\left( x-3 \right);\,\,\,{{L}_{2}}\equiv y-\left( -2 \right)=\sqrt{3}\left( x-3 \right) \\ & \Rightarrow \qquad \,\,\,{{L}_{1}}\equiv y+2=0\,\,\,\,\text{and}\,\,\,{{L}_{2}}\equiv y-\sqrt{3}\,x\,\,+2+3\sqrt{3}=0 \end{align}\] Example – 6 Find the equation of the straight line which passes through the point $\left( a{{\cos }^{3}}\theta ,\,\,a{{\sin }^{3}}\theta \right)$ and is perpendicular to the straight line $x\sec \theta +y\,\text{cosec}\theta =a$ . Solution: The slope of the given line is $\begin{align}{{m}_{1}}=-\frac{\sec \theta }{\text{cosec}\theta }=-\tan \theta \end{align}$ Therefore, the slope of the line we require will be given by m 2 where \[\begin{align} & \qquad \,\,\,{{m}_{2}}=-\frac{1}{{{m}_{1}}} \\ & \Rightarrow \qquad {{m}_{2}}=\cot \theta \end{align}\] We now know the slope of the line and we are also given a fixed point through which the line passes. We can therefore use the point-slope form to determine its equation: \[\begin{align} & \qquad\qquad\quad\;\; y-a{{\sin }^{3}}\theta =\cot \theta \left( x-a{{\cos }^{3}}\theta \right) \\ & \Rightarrow \qquad x\cos \theta -y\sin \theta =a\left( -{{\sin }^{4}}\theta +{{\cos }^{4}}\theta \right) \\ & \qquad \qquad \qquad \qquad \qquad=a\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)\left( {{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right) \\ & \qquad \qquad \qquad \qquad \qquad=a\cos 2\theta \end{align}\] Thus, the required equation is \[x\cos \theta -y\sin \theta =a\cos 2\theta \]
Search Now showing items 1-1 of 1 Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (Elsevier, 2014-09) Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, \|y\| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
Search Now showing items 1-1 of 1 Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (Elsevier, 2014-09) Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, \|y\| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
I've wondered why the tape/tapes are not part of the formal definition of a Turing Machine. Consider, for example, the formal definition of a Turing machine on Wikipedia page. The definition, following Hopcroft and Ullman, includes: the finite set of states $Q$, the tape alphabet $\Gamma$, the blank symbol $b \in \Gamma$, the initial state $q_0\in Q$, the set of final states $F\subseteq Q$, and the the transition function $\delta:(Q\backslash F)\times \Gamma\rightarrow Q\times\Gamma\times\{L,R\}$. None of which is the tape itself. A Turing Machine is always considered to work on a tape, and the transition function is interpreted as moving its head, substitution of symbol and changing state. So, why is the tape left out of the mathematical definition of a Turing machine? From what I can see, the formal definition in itself doesn't seem to imply that the Turing Machine operates like it's often described informally (with a head moving around on a tape). Or does it?
Vector Equations Of Lines VECTOR EQUATIONS OF A LINE Consider a straight line passing through the point $A(\vec{a})$ and parallel to the vector $\vec{b}.$ Any point $\vec{r}$ on this line can be written in terms of real parameter $\lambda $. \[\begin{align}& \vec{r}=\overrightarrow{OA}+\overrightarrow{AR} \\\\ &\;\ =\vec{a}+\lambda \vec{b}\,\,\qquad\qquad\qquad\;where\,\,\lambda \in \mathbb{R} \\ \end{align}\] The equation \[\boxed{\vec r = \vec a + \lambda \vec b}\] can be viewed as the (vector) equation of this line. As we vary $\lambda $, we get varying position vectors $\vec{r}$ and hence varying points on this line. This form of the equation of a line is called the parametric form since it involves the use of a parameter $\lambda $. We could also have specified the equation in non-parametric form. Observe that since $\overrightarrow {AR} $ is parallel to $\vec b,$ we have \[\begin{align}&\qquad\;\left( {\vec r - \vec a} \right) \times \vec b = \vec 0 \hfill \\&\Rightarrow \quad \boxed{\vec r \times \vec b = \vec a \times \vec b} \hfill \\ \end{align} \] This is the required equation of the line. You must convince yourself that this equation is valid; in particular, understand that only points lying on the line and none other will satisfy this equation. We can use the equations obtained above to obtained the equation of a line passing through the points $A(\vec a)\,and\,B(\vec b).$ \[\boxed{\vec r = \vec a + \lambda (\vec b - \vec a)}\qquad \qquad Parametric{\text{ }}form\] OR \[\begin{align}& \qquad\;(\vec r - \vec a) \times (\vec b - \vec a) = \vec 0 \hfill \\\\&\Rightarrow \quad \boxed{\vec r \times (\vec b - \vec a) = \vec a \times \vec b}\qquad \qquad Non{\text{ }} - {\text{ }}parametric{\text{ }}form \hfill \\ \end{align} \]
The standard Newtonian centripetal acceleration is: $$g = \frac{V^2}{R}$$ where $V$ is the rectilinear velocity being bent into a circular motion and $R$ is the radius of the circular trajectory that it is being bent into. When the velocity is closer to lightspeed, the Lorentzian gamma-factor gets involved: $$\gamma^2 = \frac{1}{1-\beta^2}$$ where $\beta$ is the ratio of the velocity to the speed of light, $\frac{V}{c}$. The centripetal acceleration – as felt by the observer in circular motion – becomes: $$g = \frac{\gamma^2V^2}{R} = \frac{c^2(\gamma^2-1)}{R}$$ To get a feel for the numbers, at what velocity is the force per unit mass equivalent to the Newtonian case of the speed of light? In otherwords… $$g = \frac{c^2(\gamma^2-1)}{R} = \frac{c^2}{R}$$ i.e. $\gamma^2-1 = 1$ or $\gamma^2 = 2$, thus $\gamma = 1.4142…$ To convert from $\gamma$ to $\beta$ requires some hyperbolic functions: $$\beta = tanh(acosh(\gamma))$$ …which, in this case of $\gamma = 1.4142$ means $\beta = 0.70710$. In Stephen Baxter’s novel “Ring” the starship “The Great Northern” sets out in a huge loop to take it 5 million years into the future. It accelerates at 1 gee for the whole journey, for the sake of its human passengers, and its total trip time is about 1,000 years, meaning it must have a gamma-factor of about 5,000. Such a gamma-factor would require a loop of roughly 25 million light-years radius, as the above equation implies, thus the journey would be much longer than 5 million years. Reference Yongwan Gim, Hwajin Um, Wontae Kim, “Unruh temperatures in circular and drifted Rindler motions”, (Submitted on 28 Jun 2018) https://arxiv.org/abs/1806.11439 [accessed 02 August 2018]
Does anyone here understand why he set the Velocity of Center Mass = 0 here? He keeps setting the Velocity of center mass , and acceleration of center mass(on other questions) to zero which i dont comprehend why? @amanuel2 Yes, this is a conservation of momentum question. The initial momentum is zero, and since there are no external forces, after she throws the 1st wrench the sum of her momentum plus the momentum of the thrown wrench is zero, and the centre of mass is still at the origin. I was just reading a sci-fi novel where physics "breaks down". While of course fiction is fiction and I don't expect this to happen in real life, when I tired to contemplate the concept I find that I cannot even imagine what it would mean for physics to break down. Is my imagination too limited o... The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe... not exactly identical however Also typo: Wavefunction does not really have an energy, it is the quantum state that has a spectrum of energy eigenvalues Since Hamilton's equation of motion in classical physics is $$\frac{d}{dt} \begin{pmatrix} x \\ p \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \nabla H(x,p) \, ,$$ why does everyone make a big deal about Schrodinger's equation, which is $$\frac{d}{dt} \begin{pmatrix} \text{Re}\Psi \\ \text{Im}\Psi \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \hat H \begin{pmatrix} \text{Re}\Psi \\ \text{Im}\Psi \end{pmatrix} \, ?$$ Oh by the way, the Hamiltonian is a stupid quantity. We should always work with $H / \hbar$, which has dimensions of frequency. @DanielSank I think you should post that question. I don't recall many looked at the two Hamilton equations together in this matrix form before, which really highlight the similarities between them (even though technically speaking the schroedinger equation is based on quantising Hamiltonian mechanics) and yes you are correct about the $\nabla^2$ thing. I got too used to the position basis @DanielSank The big deal is not the equation itself, but the meaning of the variables. The form of the equation itself just says "the Hamiltonian is the generator of time translation", but surely you'll agree that classical position and momentum evolving in time are a rather different notion than the wavefunction of QM evolving in time. If you want to make the similarity really obvious, just write the evolution equations for the observables. The classical equation is literally Heisenberg's evolution equation with the Poisson bracket instead of the commutator, no pesky additional $\nabla$ or what not The big deal many introductory quantum texts make about the Schrödinger equation is due to the fact that their target audience are usually people who are not expected to be trained in classical Hamiltonian mechanics. No time remotely soon, as far as things seem. Just the amount of material required for an undertaking like that would be exceptional. It doesn't even seem like we're remotely near the advancement required to take advantage of such a project, let alone organize one. I'd be honestly skeptical of humans ever reaching that point. It's cool to think about, but so much would have to change that trying to estimate it would be pointless currently (lol) talk about raping the planet(s)... re dyson sphere, solar energy is a simplified version right? which is advancing. what about orbiting solar energy harvesting? maybe not as far away. kurzgesagt also has a video on a space elevator, its very hard but expect that to be built decades earlier, and if it doesnt show up, maybe no hope for a dyson sphere... o_O BTW @DanielSank Do you know where I can go to wash off my karma? I just wrote a rather negative (though well-deserved, and as thorough and impartial as I could make it) referee report. And I'd rather it not come back to bite me on my next go-round as an author o.o
Back in January this year I was commuting to work and routinely opened the daily coding problem email: “Good morning! Here’s your coding interview problem for today. […] Assume you have access to a function toss_biased() which returns 0 or 1 with a probability that’s not 50-50 (but also not 0-100 or 100-0). You do not know the bias of the coin. Write a function to simulate an unbiased coin toss.” As it turned out, this problem posed a much greater challenge than I expected it to. It took me about two more commutes and quite a few Jupyter Notebook cells to come up with a first solution. An admittedly very slow and inefficient one, relying on some deep tree structures and so on. Meanwhile, I have spent some more time on the problem, did some research, and can confidently say that in this year’s birthday post I present a clear and efficient solution to a more generalized version of the coding problem. Problem Statement Let us start by understanding exactly what we want to do here. Suppose somebody gave you a coin with some bias. Instead of landing on heads 50% and on tails 50% of the time it has some different probabilities for heads and tails, which are unknown to you. It might for instance land heads-up in 99% of all tosses. Your task is, to simulate another unbiased coin using only the first, biased coin that is provided to you. Simulating in a sense that somebody asks you to say “heads” or “tails” and you have to reply 50-50 with either, randomly. Now that’s what the coding problem asked us to do, we will, however, take this one step further and simulate a coin of any arbitrary bias using only the first coin. The remainder of this article is combining the mathematical side of the problem with a Python implementation and verbal explanations. The verbose, verbal formulation from above can be expressed more concisely, where $\operatorname{Ber}\left(\cdot\right)$ is the Bernoulli distribution: Derive $Y\sim\operatorname{Ber}\left(p_2\right)$ for any $p_2\in\left[0,1\right]$, given $X\sim\operatorname{Ber}\left(p_1\right)$ with unknown $p_1\in(0,1)$. In Python one could define a random variable following a Bernoulli distribution as a function mapping from nothing to either 0 or 1. The type is: RandomVariable = Callable[[], int] Next we define a higher-order function which creates random variables with any given bias. It is our biased coin generator. def any_x_rv(p_1: float = 0.5) -> RandomVariable: """Returns a function that follows Ber(p_1)""" return lambda: 1 if random.uniform(0, 1) <= p_1 else 0 Given some $p_1$, any_x_rv returns a function which, when called, returns 1 with probability $p_1$ and 0 with probability $(1-p_1)$. We are seeking for a function def any_y_rv(x_rv: RandomVariable, p_2: float) -> RandomVariable: pass which takes a random variable (e.g. some function returned by any_x_rv) and a second probability $p_2$. Note that any_y_rv does not know about the internals of x_rv. It can only evaluate the provided function. The return type of any_y_rv is again a random variable, i.e. a function that can be evaluated. It is supposed to return 1 with a probability of $p_2$. If you want to solve the problem by yourself, now is a good time to pause and ponder. Try to implement any_y_rv! We will tackle the problem in two steps: First, creating a fair coin from the coin that is given to us. Second, using the simulated fair coin to simulate any biased coin. Simulating a Fair Coin with a Biased Coin In the first step we construct a random variable $Z$ which follows $\operatorname{Ber}\left(0.5\right)$ from $X$. Recall that $\Pr\left[X=1\right]=p_1$ is unknown to us. Let $(X_1,X_2)$ be a random sample of $X$. The probabilities for all possible outcomes of the tuple are:\begin{align} \Pr\left[X_1=0,X_2=0\right]&=\left(1-p_1\right)^2\\ \Pr\left[X_1=0,X_2=1\right]&=\left(1-p_1\right)\times p_1\\ \Pr\left[X_1=1,X_2=0\right]&=p_1\times\left(1-p_1\right)\\ \Pr\left[X_1=1,X_2=1\right]&=p_1^2\,. \end{align} It is apparent that the probabilities for two of the four outcomes are equal, namely the ones where $X_1\ne X_2$, i.e. $\Pr\left[X_1=0,X_2=1\right]=\Pr\left[X_1=1,X_2=0\right]$. That fact comes in useful when constructing a fair coin. We define $Z$ to be 1 if $X_1=1,X_2=0$ and 0 if $X_1=0,X_2=1$. In all other cases, we draw from the tuple $(X_1,X_2)$ again. Since $X$ is non-deterministic, it is not always 0 or always 1, we are guaranteed to encounter the first case eventually. This idea was presented by John von Neumann in “Various techniques used in connection with random digits” (1951). Verbally, the instructions are: Toss the biased coin two times. If the two outcomes are identical, ignore them and go back to step (1). If the two outcomes differ, use the outcome of the first coin as the result of the fair coin toss. The Python implementation of the algorithm looks as follows: def z_rv(x_rv: RandomVariable) -> int: """Returns Z~Ber(0.5)""" while True: x_1, x_2 = x_rv(), x_rv() if x_1 != x_2: return x_1 The function z_rv represents $Z$. It takes another random variable $X$, which represents the biased coin. The function samples from $X$ until a sample tuple differs. Once that is the case, it returns the first element of the tuple, namely x_1. At this point we have access to a simulated fair coin which is based on any coin with bias. In order to toss it we simply call z_rv. Time for the second step. Simulating a Coin with Any Bias Given a Fair Coin Given the fair coin $Z\sim\operatorname{Ber}\left(0.5\right)$, we seek to simulate $Y\sim\operatorname{Ber}\left(p_2\right)$ for any $p_2$ using $Z$. The method propounded in this section is taken from the blog post “Arbitrarily biasing a coin in 2 expected tosses” by Александър Макелов. We start by defining the binary expansion $a_i\in\left\{0,1\right\}$, with $i\in\mathbb{N}$, of the provided probability $p_2$ as $$p_2=\sum_{i=1}^\infty\frac{a_i}{2^i}\,.$$ You can think of the values $a_1,a_2,\dots$ as the fractional digits of the binary representation of $p_2$: $p_2=0.a_1a_2\dots$. Since $p_2$ is given, the sequence is entirely known to us. Here is an example: Suppose $p_2=0.8$. It can be expressed as$$p_2=0.8=\frac{1}{2}+\frac{1}{4}+\frac{0}{8}+\frac{0}{16}+\frac{1}{32}+\dots\,,$$where $a=(1,1,0,0,1,\dots)$. $a$ can be infinite and periodic, but neither matters to us. $a$ is implemented as a Python function returning a generator so that the infinite stream of values can be read sequentially, on demand: def a(p: float) -> Iterable[int]: while True: yield 1 if p >= 0.5 else 0 p = 2 * p - int(2 * p) Let $Z_1,Z_2,\dots$ be a random sample of $Z$. We define $$\hat{Y}(i):=\begin{cases} 0&{\text{if }}Z_i>a_i\\ 1&{\text{if }}Z_i<a_i\\ \hat{Y}(i+1)&{\text{otherwise}\,,} \end{cases}$$ which iterates over the fractional digits of $p_2$ and uses the $i$-th sample of $Z$ for each of them. At every step it compares $Z_i$ to the digit $a_i$ and terminates if $Z_i\ne a_i$. This blog post by Alex Irpan provides a nice visual intuition for why this works. Now $Y$ can be defined as $Y:=\hat{Y}(1)$. In our implementation, $Y$ is a function that uses the two previously defined functions z_rv (the unbiased coin simulator) and a (the binary expansion of $p_2$): def y_rv() -> int: for a_i in a(): if z_rv() != a_i: return a_i Final Solution Bringing the pieces together we get the following Python snippet: import random from typing import Callable, Iterable RandomVariable = Callable[[], int] def any_x_rv(p_1: float = 0.5) -> RandomVariable: """Returns a function that follows Ber(p_1)""" assert 0 < p_1 < 1 return lambda: 1 if random.uniform(0, 1) <= p_1 else 0 def any_y_rv(x_rv: RandomVariable, p_2: float) -> RandomVariable: """Returns a function that follows Ber(p_2), based on x_rv""" assert 0 <= p_2 <= 1 if p_2 == 0 or p_2 == 1: return lambda: p_2 def a() -> Iterable[int]: p = p_2 while True: yield 1 if p >= 0.5 else 0 p = 2 * p - int(2 * p) def z_rv() -> int: """Returns Z~B(1,0.5)""" while True: x_1, x_2 = x_rv(), x_rv() if x_1 != x_2: return x_1 def y_rv() -> int: for a_i in a(): if z_rv() != a_i: return a_i return y_rv We can create a new, biased coin $X$, for instance with $p_1=0.1$ (this could be anything $\in(0,1)$) >>> x_rv = any_x_rv(0.1) and retrieve a new, biased coin $Y$ with some different bias, say $p_2=0.73$ >>> y_rv = any_y_rv(x_rv, 0.73) just to toss it 20 times: >>> [y_rv() for _ in range(20)] [0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1] And there we go: We have constructed a coin with an arbitrary bias based on another coin with unknown bias! Now it can be tossed: A big thank you goes to my colleague Lennart van der Goten for showing a ton of interest in the problem and solving it mathematically in an entirely different, yet elegant way. My fellow student Tanja Bien took the photos for this post which look gorgeous. Thank you very much as well! To evaluate whether the correctness of the solution, I simulated a coin for any $p_2\in\left[0,1\right]$ and tossed it many times: import numpy as np import matplotlib.pyplot as plt def determine_bias(biased_fn: Callable[[], int], n: int = 10000) -> float: """Determines p_1 of a biased function""" assert n > 0 return sum([biased_fn() for _ in range(n)]) / n p_1 = 0.1 x_rv = any_x_rv(p_1) p_2s = np.linspace(0,1,100) ys = [determine_bias(any_y_rv(x_rv, p_2)) for p_2 in p_2s] plt.plot(p_2s, ys, 'ro')
Potential energy From Academic Kids Potential energy ( U, or E p), a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. For example a mass released above the Earth has energy resulting from the gravitational attraction of the Earth which is transferred in to kinetic energy. Contents Types Gravitational potential energy This energy is stored as a result of the elevated position of an object such as a rock on top of a hill or water behind a dam. It is written as <math>U_g = m g h \,<math> where <math>m<math> is the mass of the object, <math>g<math> the acceleration due to gravity and <math>h<math> the height above a chosen reference level (typical units would be kilograms for <math>m<math>, metres/second 2 for <math>g<math>, and metres for <math>h<math>). In relation to spacecraft and astronomy g is not constant and the formula becomes an integral. In the case of a sphere of uniform mass (such as a planet), with h measured above the surface, the integral takes the form: <math>U_g = \int_{h_0}^{h + h_0} {GmM \over r^2} dr<math> Where <math>h_0<math> is the radius of the sphere, M is the mass of the sphere, and G is the gravitational constant. If h is instead taken to be the distance from the center of the sphere, then outside the sphere the potential energy relative to that at the center has two terms: <math>U_g = \int_{h_0}^h {GmM \over r^2} dr + \int_0^{h_0} {GmM \over h_0^2} {r \over h_0} dr<math> which evaluates to: <math>U_g = GmM \left[{1 \over h_0} - {1 \over h}\right] + {1 \over 2} {GmM \over h_0} = GmM \left[{3 \over 2h_0} - {1 \over h}\right]<math> [We may also want to link to an explanation of that second term (the gravitational forces created by hollow spherical shells)] A frequently adopted convention is that an object infinitely far away from an attracting source has zero potential energy. Relative to this, an object at a finite distance r from a source of gravitation has negative potential energy. If the source is approximated as a point mass, the potential energy simplifies to: <math>U_g = - {GmM \over r}<math> See also Gravitational binding energy. Elastic potential energy This energy is stored as the result of a deformed solid such as a stretched spring. As a result of Hooke's law, it is given by: <math>U_e = {1\over2}kx^2<math> where <math>k<math> is the spring constant (a measure of the stiffness of the spring), expressed in N/m, and <math>x<math> is the displacement from the equilibrium position, expressed in metres (see Main Article: Elastic potential energy). Chemical energy Chemical energy is a form of potential energy related to the breaking and forming of chemical bonds. Rest mass energy <math>E_0 = m c^2 \,<math> where E 0 is the rest mass energy, mis mass of the body, and cis the speed of light in a vacuum. (The subscript zero is used here to distinguish this form of energy from the others that follow. In most other contexts, the equation is written with no subscript.) The rest mass energy is the amount of energy inherent in the mass when it is at rest. This equation quantifies the equivalence of mass and energy: A small amount of mass is equivalent to a very large amount of energy. (i.e., 90 petajoule/kg ≈ 21 megatons/kg) Electrical potential energy The electrical potential energy per unit charge is called electrical potential. It is expressed in volts. The fact that a potential is always relative to a reference point is often made explicit by using the term potential difference. The term voltage is also common. The electrical potential energy between two charges <math>q_1<math> and <math>q_2<math> is: <math> U = - \frac{q_1 q_2} {4 \pi \epsilon_o r} <math> The electric potential generated by charges <math>q_1<math> (denoted <math>V_1<math>) and <math>q_2<math> (denoted <math>V_2<math>) is: <math> V_1 = - \frac{q_1} {4 \pi \epsilon_o r} <math> <math> V_2 = - \frac{q_2} {4 \pi \epsilon_o r} <math> Relation between potential energy and force Potential energy is closely linked with forces. If the work done going around a loop is zero, then the force is said to be conservative and it is possible to define a numerical value of potential associated with every point in space. A force field can be re-obtained by taking the vector gradient of the potential field. For example, gravity is a conservative force. The work done by a unit mass going from point A with <math>U = a<math> to point B with <math>U = b<math> by gravity is <math>(b - a)<math> and the work done going back the other way is <math>(a - b)<math> so that the total work done from <math>U_{A \to B \to A} = (b - a) + (a - b) = 0 \,<math> The nice thing about potential energy is that you can add any number to all points in space and it doesn't affect the physics. If we redefine the potential at A to be <math>a + c<math> and the potential at B to be <math>b + c<math> [where <math>c<math> can be any number, positive or negative, but it must be the same number for all points] then the work done going from <math>U_{A \to B} = (b + c) - (a + c) = b - a \,<math> as before. In practical terms, this means that you can set the zero of <math>U<math> anywhere you like. You might set it to be zero at the surface of the Earth or you might find it more convenient to set it zero at infinity. A thing to note about conservative forces is that the work done going from A to B does not depend on the route taken. If it did then it would be pointless to define a potential at each point in space. An example of a non-conservative force is friction. With friction, the route you take does affect the amount of work done, and it makes no sense at all to define a potential associated with friction. All the examples above are actually force field stored energy (sometimes in disguise). For example in elastic potential energy, stretching an elastic material forces the atoms very slightly further apart. Powerful electromagnetic forces try to keep the atoms at their optimal distance and so elastic potential is actually electromagnetic potential. Having said that, scientists rarely talk about forces on an atomic scale. Everything is phrased in terms of energy rather than force. You can think of potential energy as being derived from force or you can think of force as being derived from potential energy. A conservative force can be expressed in the language of differential geometry as an exact form. Because Euclidean space is contractible, its de Rham cohomology vanishes, so every exact form is closed, i.e., is the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field. Graphical representation A graph of a 1D or 2D potential function with the function value scale increasing upward is useful to visualize the potential field: a ball rolling to the lowest part corresponds to a mass or charge, etc. being attracted. See also da:Potentiel energi de:Potenzielle Energie fr:�nergie potentielle m�canique it:Energia potenziale ms:Tenaga Keupayaan nl:Potenti�le energie ja:位置エネルギー pl:Energia potencjalna sl:Potencialna energija fi:Potentiaalienergia zh:势能
I'm attempting to prove that for a closed, path-connected, volume $V$ of $\mathbb{R}^3$, $$\iiint_{V} (\operatorname{div} F) \, \text{d}V = \iint_{\partial V} F \cdot \mathbf{\hat n} \, \text{d}S$$ Where $F : \mathbb{R}^3 \to \mathbb{R}^3$ is any differentiable function. Is there anything wrong with my proof? Can I make it simpler? Consider a box $B$ containing $V$. Partition $B$ into nonoverlapping boxes $B_1, B_2, \ldots B_m$, such that diameter of the partition is $< 1/n$. Now consider "chunks" $C_k = B_k \cap V$. Clearly, $$\sum_{k} \left (\iint_{\partial C_k} F \cdot \mathbf{\hat n} \, \text{d}S \right) = \iint_{\partial V} F \cdot \mathbf{\hat n} \, \text{d}S$$ Moreover, by my generalized mean value theorem, there exists some $p_k \in C_k$ such that $$\iint_{\partial C_k} F \cdot \mathbf{\hat n} \, \text{d}S = \operatorname{div} F\|_{p_k}\cdot {\rm m}(C_k)\ .$$ Pick all of these $p_k$ and use them to form a Riemman sum $$S_n = \sum_k \operatorname{div} (F\|_{p_k}) \cdot \operatorname{m}(C_k) = \sum_{k} \left (\iint_{\partial C_k} F \cdot \mathbf{\hat n} \, \text{d}S \right) = \iint_{\partial V} F \cdot \mathbf{\hat n} \, \text{d}S$$ As we assumed $\operatorname{div} F$ was integrable, $$\lim_{n \to \infty} S_n = \iiint_{V} (\operatorname{div} F) \, \text{d}V$$ But $S_n$ is a constant sequence! Namely, $$S_n = \iint_{\partial V} F \cdot \mathbf{\hat n} \, \text{d}S$$ Thus, it is the only possibility that $$\iiint_{V} (\operatorname{div} F) \, \text{d}V = \iint_{\partial V} F \cdot \mathbf{\hat n} \, \text{d}S$$
Sum of Reciprocals in Base 10 with Zeroes Removed Jump to navigation Jump to search Theorem The infinite series $\displaystyle \sum_{P \left({n}\right)} \dfrac 1 n$ where $P \left({n}\right)$ is the propositional function: $\forall n \in \Z_{>0}: P \left({n}\right) \iff$ the decimal representation of $n$ contains no instances of the digit $0$ Proof Sources Oct. 1971: R.P. Boas, Jr. and J.W. Wrench, Jr.: Partial Sums of the Harmonic Series( Amer. Math. Monthly Vol. 78, no. 8: 864 – 870) www.jstor.org/stable/2316476
ECE 641 Fall 2008 Professor Bouman Hi everybody! Here is an example of a page_ECE641Fall2008bouman. Getting started Rhea works very much like Kiwi. So if you want to learn how to edit this Rhea wiki, you can watch the Kiwi instructional video. Configuration settings list MediaWiki FAQ MediaWiki release mailing list Errors in the notes Dear ECE 641 Student, It would GREAT if you could post errors you find in my notes here. Thanks. Prof. Bouman Errors version dated September 3 2008 Please swap the exponents in (Bernoulli)PMF equation on page 12. Please correct the posterior variance on page 19 (middle of the page): Assignment for "a" after the line "More specifically ...". Dalton Errors version dated September 5 2008 I believe the definition of the conditional expectation on page 8 is not true, possibly what was meant was: $ E[X\|Y=y]=\int_\mathbb{R} x dF(x\|Y=y) $. (Also, less significantly, the pdf/cdf definitions are swapped.) On page 15, it may be worthwhile to note that an estimator with minimum MSE is not necessarily efficient (since we are omitting the fact that efficiency is actually defined in terms of achieving the CRLB). I believe the expected cost (page 17) should be $ \hat{C}(x)=E[C(x,T(Y))\|X=x]=C\left(x,\int_{\mathbb{R}^n} T(y)p_{Y\|X}(y\|x)dy\right) $ Josh Errors version dated September 7 2008 Since the exponents in the PMF equation on Page 12 has been swapped, I believe the equation on the next page (Example 2.1.2) needs to have its exponents swapped also since it's the same PMF. Alfa Errors Lab 3 Note: the following discussion uses prefix notation instead of infix (so everything 'reads' like functions). It may be overkill, but I want to be notationally explicit. In case anyone was listening to my comments regarding the lab, I want to correct my stated results. To recap, the conditional from which $ W $ was drawn should indeed depend on the neighborhood using vales from $ X^{(k+1)} $ and $ X^{(k)} $ (the last pixel of the ordering will depend solely on a neighborhood of $ X^{(k+1)} $). This conclusion still holds (and it darn well better!). However, I noticed that I made a mistake based on the bad use of dirac functions. To clarify, if boolean operators are a function of two reals, ie $ \ne:\mathbb{R}^2\rightarrow \{0,1\} $ (with the typical interpretation) and $ \delta(x)=\begin{cases}0, & x\ne 0\\ 1, & x=0\end{cases} $ then $ \delta(\ne\!(x,y)) \text{ iff } =\!(x,y) $. Anyway, my point is that as it is written, and under my interpretation, it is the complement of what it should be. If you replace all the $ \delta $ of the lab with indicator function $ I $ then things are less ambiguous. One could equivalently use: $ 1-\delta(x-y) $. Bottom-line: the particular choice of parameters listed in the questions results in all zeros or all ones with high probability. Apologies for any confusion! Josh
ISSN: 1937-5093 eISSN: 1937-5077 Kinetic & Related Models September 2015 , Volume 8 , Issue 3 Issue on the workshop “Electromagnetics-Modelling, Simulation, Control and Industrial Applications" Select all articles Export/Reference: Abstract: This paper is devoted to the study of the inviscid Boussinesq equations. We establish the local well-posedness and blow-up criteria in Besov-Morrey spaces $N_{p,q,r}^s(\mathbb{R}^n)$ for super critical case $s > 1 + \frac{n}{p}, 1 < q \leq p < \infty, 1 \leq r\leq \infty$, and critical case $s=1+\frac{n}{p}, 1 < q \leq p < \infty, r=1$. Main analysis tools are Littlewood-Paley decomposition and the paradifferential calculus. Abstract: The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient $\epsilon$ is varied from $\epsilon=1$ (for 1D transport models) to $\epsilon=0$ (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit $\epsilon\to 0$, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams. Abstract: The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum $N$ -particle dynamics. We assume the integral of the microscopic interaction is zero and we assume $W^{4,1}$ per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below $W^{4,1}$. Abstract: The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some difficulties caused by the random noise. Abstract: We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity. Abstract: In plasma physics domain, the electrons transport can be described from kinetic and hydrodynamical models. Both methods present disadvantages and thus cannot be considered in practical computations for Inertial Confinement Fusion (ICF). That is why we propose in this paper a new model which is intermediate between these two descriptions. More precisely, the derivation of such models is based on an angular closure in the phase space and retains only the energy of particles as a kinetic variable. The closure of the moment system is obtained from a minimum entropy principle. The resulting continuous model is proved to satisfy fundamental properties. Moreover the model is discretized w.r.t the energy variable and the semi-discretized scheme is shown to satisfy conservation properties and entropy decay. Abstract: In this paper, we consider the time-asymptotic stability of a superposition of shock waves with contact discontinuities for the one dimensional Jin-Xin relaxation system with small initial perturbations, provided that the strengths of waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a delicate decay estimate on the heat kernel in Huang-Li-Matsumura [5]. Abstract: We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The finite-difference scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs). Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes. In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh. Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure higher order in time as well. Abstract: N/A Readers Authors Editors Referees Librarians Email Alert Add your name and e-mail address to receive news of forthcoming issues of this journal: [Back to Top]
ISSN: 1937-5093 eISSN: 1937-5077 Kinetic & Related Models December 2015 , Volume 8 , Issue 4 Issue on rate-independent evolutions and hysteresis modelling Select all articles Export/Reference: Abstract: This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in $L^2$ space hold true for moderately soft potential cases $ \gamma \in[-2, 0) $ without any smallness assumption on the initial data. For very soft potential cases $ \gamma \in[-3, -2) $, which cover in particular the Coulomb case $\gamma=-3$, we get local in time estimates of weak solutions in $L^{2}$. In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $ \gamma \in[-3, -2) $, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality. Abstract: We investigate a one-dimensional linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint, it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process. Abstract: In this paper, we study a free boundary problem for a class of parabolic type chemotaxis model in high dimensional symmetry domain $\Omega$. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain $\Omega$ with free boundary condition. Abstract: We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions. Abstract: In this paper, we are concerned with the simplified Ericksen-Leslie system （1）--（3）, modeling the flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0, d_0)\in {\bf H}\times H^1(\Omega, \mathbb{S}^2)$, with $d_0(\Omega)\subset\mathbb{S}^2_+$. We define a dissipation term $D(u,d)$ that stems from an eventual lack of smoothness in the solutions, and then obtain a local equation of energy for weak solutions of liquid crystals in dimensions three. As a consequence, we consider the 2D case and obtain $D(u,d)=0$. Abstract: We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision [10] cannot capture the correct energy-transport limit. This problem was addressed in [13], where a thresholded BGK penalization was introduced. Here we propose an alternative based on a splitting approach. It has the advantage of treating the collisions at different scales separately, hence is free of choosing threshold and easier to implement. Formal asymptotic analysis and numerical results validate the efficiency and accuracy of the proposed scheme. Abstract: We consider the relativistic transfer equations for photons interacting via emission absorption and scattering with a moving fluid. We prove a comparison principle and we study the non-equilibrium regime: the relativistic correction terms in the scattering operator lead to a frequency drift term modeling the Doppler effects. We prove that the solution of the relativistic transfer equations converges toward the solution of this drift diffusion equation. Abstract: We consider strong solutions to compressible barotropic viscoelastic flow in a domain $\Omega\subset\mathbb{R}^{3}$ and prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Inspired by the work of Kato and Lax, we use the contraction mapping principle to get the result. Abstract: We study a one--dimensional quasilinear system proposed by J. Tello and M. Winkler [27] which models the population dynamics of two competing species attracted by the same chemical. The kinetic terms of the interacting species are chosen to be of the Lotka--Volterra type and the boundary conditions are of homogeneous Neumann type which represent an enclosed domain. We prove the global existence and boundedness of classical solutions to the fully parabolic system. Then we establish the existence of nonconstant positive steady states through bifurcation theory. The stability or instability of the bifurcating solutions is investigated rigorously. Our results indicate that small intervals support stable monotone positive steady states and large intervals support nonmonotone steady states. Finally, we perform extensive numerical studies to demonstrate and verify our theoretical results. Our numerical simulations also illustrate the formation of stable steady states and time--periodic solutions with various interesting spatial structures. Readers Authors Editors Referees Librarians Email Alert Add your name and e-mail address to receive news of forthcoming issues of this journal: [Back to Top]
Search Now showing items 1-1 of 1 Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (Elsevier, 2014-09) Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, \|y\| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
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) $ace \neq 0$\[\begin{bmatrix}1/a & 0 & 0 \\ -b/ac & 1/c & 0\\bd/ace & -d/ce & 1/e\end{bmatrix}\]
I'm considering a Cox-Ingersoll-Ross (CIR) process $$ dx_{t} = \alpha\left(\theta - x_{t}\right)dt + \sigma \sqrt{x_{t}}\,dW_{t}\,,\qquad \alpha,\beta,\sigma > 0 $$ which by assumption has $2\alpha \theta < \sigma^{2}$ (violates the Feller condition) and can therefore reach $x_{t}=0$ for some $t$ . The conditional distribution is $$ f(x_{t+T} \vert x_{t}) = c e^{-u-v}\left(\frac{v}{u}\right)^{q/2}I_{q}\left(2\sqrt{uv}\right) $$ where $q = \tfrac{2\alpha\theta}{\sigma^{2}} - 1$ (note $q <0$ by assumption of violation of the Feller condition), $c = \frac{2\alpha}{\sigma^{2}\left(1-e^{-\alpha T}\right)}\,$, $u=cx_{t}e^{-\alpha T}$, $v = c x_{t+T}$ and $I_{q}$ is a modified Bessel function of the first kind of order $q\,$. I want to calibrate $\alpha,\theta,\sigma$ from certain historical observations $x_{i=1\ldots N}$. As explained in e.g. arXiv:0812.4210, in principle this can be done by minimizing minus the logarithm of the likelihood, i.e. $$ -\log (\text{Likelihood}) = -\log \prod_{i=1}^{N-1}f(x_{i+1}\vert x_{i})\,. $$ The tricky thing is that some of my historic observations $x_{i}$ are zero. Now, when $x_{t + T} \to 0$ ($v \to 0$ in the notation above), using the series expansion of the Bessel function one finds $$ f(x_{t +T} \to 0 \vert x_{t}) \to c e^{-u-v}\frac{v^{q}}{\Gamma(q+1)} $$ Since $q<0$ by assumption, the density blows up as $v\to 0\,$. In particular, if (say) the 9th observation is zero in my dataset of historic observations (i.e. $v_{9} = cx_{9}=0$), the likelihood would be $$ -\log (\text{Likelihood}) = -q\log v_{9} -\log\left(\frac{c e^{-u_{9}-v_{9}}}{\Gamma(q+1)}\right)-\log \prod_{i\neq 8}f(x_{i+1}\vert x_{i})\,,\quad \text{with } v_{9} \to 0 $$ and the term $-q\log v_{9} \to -\infty$ as $v_{9} \to 0$ and will spoil the minimization (a numeric solver ceases to converge, for example). Any ideas on how to calibrate the CIR process in such situations, namely when the historic data contains points (one or many) where the process hits zero?. Is Maximum Likelihood just not suited to this situation or is there a work-around?.
I want to solve the following ODE using NDSolveProcessEquations but in an iterative way. $\ddot{x}(t)+5\dot{x}(t)+3x(t)=2\cos(2\pi t)$ $\dot{x}(0)=0,\,x(0)=1$ I wrote the below code ClearAll["Global`"] (----------------------------------------------------------) dxdt0 = 0; x0 = 1; ti = 0; tf = 10; \[Delta]t = 0.01; ItrNo = Round[(tf - ti)/\[Delta]t]; acc = 12; eq1 = Derivative[2][x][t] + 5Derivative[1][x][t] + 3x[t] == 2Cos[2Pit]; ics = {Derivative[1][x][0] == dxdt0, x[0] == x0}; (----------------------------------------------------------) StateVar = First[NDSolve`ProcessEquations[{eq1, ics},{Derivative[1][x], x}, t, MaxSteps -> Infinity,PrecisionGoal -> acc, AccuracyGoal -> acc]]; solC = ConstantArray[0, ItrNo - 1];(----------------------------------------------------------) Do[ics = {Derivative[1][x][0] == dxdt0, x[0] == x0}; NewStateVar = First[NDSolve`Reinitialize[StateVar,ics]]; NDSolve`Iterate[NewStateVar, \[Delta]t]; solUC = NDSolve`ProcessSolutions[NewStateVar]; x0 = x[t] /. {t -> \[Delta]t} /. solUC; dxdt0 = Derivative[1][x][t] /. {t -> \[Delta]t} /. solUC; solC[[i]] = Flatten[{dxdt0, x0}]; If[Mod[Rationalize[\[Delta]t]i, Rationalize[1]] == 0, Print[" t = ", \[Delta]ti]], {i, ItrNo - 1}] The plots were generated using tData = Table[n, {n, \[Delta]t, tf - \[Delta]t, \[Delta]t}]; xData = solC[[All, 2]]; dxdtData = solC[[All, 1]]; xList = Partition[Riffle[tData, xData], 2]; dxdtList = Partition[Riffle[tData, dxdtData], 2]; ListLinePlot[{xList, dxdtList}, PlotRange -> All, Frame -> True, FrameStyle -> Directive[Black, Thick]] And I got some results To validate the result I checked with DSolve and NDSolve and instantly understood that they had produced the correct result, which obviously different from what NDSolveProcessEquations had produced. ClearAll["Global`"] dxdt0 = 0; x0 = 1; eq = Derivative[2][x][t] + 5Derivative[1][x][t] + 3x[t] == 2Cos[2Pit]; Ics = {Derivative[1][x][0] == dxdt0, x[0] == x0}; sol = First[DSolve[{eq, Ics}, x[t], t]]; y[t] = D[x[t] /. sol, t]; Plot[{y[t], x[t] /. sol}, {t, 0, 10}, PlotRange -> All, Frame -> True, FrameStyle -> Directive[Black, Thick]] sol1 = First[ NDSolve[{eq, Ics}, {Derivative[1][x][t], x[t]}, {t, 0, 10}]]; Plot[{Derivative[1][x][t] /. sol1, x[t] /. sol1}, {t, 0, 10}, PlotRange -> All, Frame -> True, FrameStyle -> Directive[Black, Thick]] What I have understood is that the term $2\cos(2\pi t)$ (where time $t$ appears explicitly ) is the main reason for two different results. In my code I have failed to incorporate the fact that with every iteration, time $t$ grows and this changes the value of function $2\cos(2\pi t)$ at every iteration steps. I did not also want to include NDSolveProcessEquations within the loop. Looking forward for any valuable help.
I'm using MCMC to simulation the distribution of some parameters in a Bayesian hierarchical model, which has the following form: $$\gamma_{ik} \sim Ber(\omega_{ik}).$$ Then I make a logit-transiformation, more specifically, $\theta_{ik}$ = $log(\frac{\omega_{ik}}{1-\omega_{ik}}).$ In order to incorporate covariates $x_{i}$, I use the following model, \begin{align} \left( \begin{array}{ccc} \theta_{i1} \\ \vdots \\ \theta_{iK} \end{array} \right)&=\mu + \beta x_{i} + \left( \begin{array}{c} \epsilon_{i1} \\ \vdots \\ \epsilon_{iK} \end{array} \right) \qquad i = 1,\ldots, I \\ \theta_{i} &\sim N(\mu + \beta x_{i},\Sigma) \\ \mu &\sim N(0,G) \\ \beta_{K \times p} &\sim MN(0,\Sigma,I) \\ \end{align} where $MN$ denotes the multivariate normal distribution. I try to update these parameters separately, but it results in high autocorrelation. See the figure below, Then I find out since $\mu$ and $\beta$ are not conditional independent given the other parameters, I should do block Gibbs sampling to avoid high autocorrelation between them. In order to do so, I need to calculate the covariance of $\beta$ and $\mu$ given the other parameters. I tried to do this in the following way: convert $\beta$ into a vector vec($\beta$) with length $K \times p$, but I failed to calculate cov($\mu$ , $\beta$ $\mid \cdots$). My question is 1: Is there any methods that I can use? 2: Is there any way to avoid high autocorrelation in this model? Any idea or suggestion would be appreciated.
The Exponential Family: Getting Weird Expectations! I spent quite some time delving into the beauty of variational inference in the recent month. I did not realize how simple and convenient it is to derive the expectations of various forms (e.g. logarithm) of random variables under variational distributions until I finally got to understand (partially, ) how to make use of properties of the exponential family. In the following, I would like to share knowledge of exponential family, which can help us obtain weird expectations that we may encounter in probabilistic graphical models. For complete coverage, I list some online materials that’s very helpful when I started to learn about these things. just enough Definition We say a random variable $x$ follows an exponential family of distribution if the probabilistic density function can be written in the following form: where: $\eta$ is a vector of parameters $T(x)$ is the sufficient statistics $A(\eta)$ is the cumulant function The presence of $T(x)$ and $A(\eta)$ actually confuses me a lot when I first start to read materials on exponential family of distributions. It turns out that they can be understood by getting a little bit deeper into this special formulation of density functions. Fanscinating properties We start by focusing on the property of $A(\eta)$. According to the definition of probabilities, the integral of density function in Equation \eqref{eq:pdf} over $x$ equals to 1: From the equation above, we can tell that $A(\eta)$ can be viewed as the logarithm of the normalizer. Its value depends on $T(x)$ and $h(x)$. An interesting thing happens now - if we take the derivative of Equation \eqref{eq:a_eta} with respect to $\eta$, we have: How neat it is! It turns out the first derivative of the cumulant function $A(\eta)$ is actually the expectation of the sufficient statistic $T(x)$! If we go further - take the second derivative - we will get the variance of $T(x)$: In fact, we can go deeper and deeper to generate higher order moment of $T(x)$. Examples Let’s end by taking a look at some familar probability distributions that belong to the exponential family. Dirichlet distribution Suppose a random variable $\theta$ is drawn from a Dirichlet distribution parameterized by $\alpha$, we have the following density function: This simple transformation turns Dirichlet density function into the form of exponential family, where: $\eta = \alpha - 1$ $A(\eta) = \sum_k log~\Gamma(\alpha_k) - log~\Gamma(\sum_k \alpha_k)$ $T(\theta) = log~\theta$ Such information is helpful when we want to compute the expectation of the log of a random variable that follows a Dirichlet distribution (e.g., this happens in the derivation of latent Dirichlet allocation with variational inference.) where $\Psi(\cdot)$ is the first derivative of log gamma. It is called the digamma function. Gamma distribution Gamma distribution has two parameters: $\alpha$ that controls the shape and $\beta$ that controls the scale. Therefore, we can find two sets of $A(\eta)$ and $T(x)$. Suppose $x \sim Gamma(\alpha, \beta)$, we have: Although it is obvious that $A(\eta) = log~\Gamma(\alpha) - \alpha log~\beta$ for both situations, the natural parameter $\eta$ is different. The first transformation helps us get the expectation of $x$ itself: Similarly, the second one helps get the expectation of $log~x$ Conclusion In fact, there are many more to explore and know about the exponential family! Important concepts such as convexity and sufficency are not discussed here. Finally, I would recommend the following excellent materials for getting to know this cool concept of unifying a set of probabilistic distributions: Chapter 4.2.4 and 10.4 of Pattern Recognition and Machine Learning http://www.cs.columbia.edu/~jebara/4771/tutorials/lecture12.pdf https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/other-readings/chapter8.pdf
ISSN: 1078-0947 eISSN: 1553-5231 All Issues Discrete & Continuous Dynamical Systems - A January 1996 , Volume 2 , Issue 1 Select all articles Export/Reference: Abstract: Here we established the partial regularity of suitable weak solutions to the dynamical systems modelling the flow of liquid crystals. It is a natural generalization of an earlier work of Caffarelli-Kohn-Nirenberg on the Navier-Stokes system with some simplifications due to better estimates on the pressure term. Abstract: Under quite general assumptions, we prove existence, uniqueness and regularity of a solution $U$ to the evolution equation $-U'(t)\in\partial(g\circ F)(U(t))$, $U(0)=u_0$, where $g:\mathbb{R}^q\rightarrow\mathbb{R}\cup\{+\infty\}$ is a closed proper convex function, $F:\mathbb{R}^p\rightarrow \mathbb{R}^q$ is a continuously differentiable mapping whose gradient is Lipschitz continuous on bounded subsets and $u_0\in\dom (g\circ F)$. We also study the asymptotic behavior of $U$ and give an application to nonlinear mathematical programming. Abstract: A nonlinear Volterra inclusion associated to a family of time-dependent $m$-accretive operators, perturbed by a multifunction, is considered in a Banach space. Existence results are established for both nonconvex and convex valued perturbations. The class of extremal solutions is also investigated. Abstract: In this paper, we study the normal forms and analytic conjugacy for a class of analytic quasiperiodic evolutionary equations including parabolic equations and Schrödinger equations. We first obtain a normal form theory. Then as a special case of the normal form theory, we show that if the frequency and the eigenvalues satisfy certain small divisor conditions then the nonlinear equation is locally analytically conjugated to a linear equation. In other words, the normal form is a linear equation. Abstract: Our aim in this article is to derive an upper bound on the dimension of the attractor for Navier-Stokes equations with nonhomogeneous boundary conditions. In space dimension two, for flows in general domains with prescribed tangential velocity at the boundary, we obtain a bound on the dimension of the attractor of the form $c\mathcal{R} e^{3/2}$, where $\mathcal{R} e$ is the Reynolds number. This improves significantly on previous bounds which were exponential in $\mathcal{R} e$. Abstract: A number of recent papers examine for a dynamical system $f: X \rightarrow X$ the concept of equicontinuity at a point. A point $x \in X$ is an equicontinuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that the orbit of initial points $\delta$ close to $x$ remains at all times $\epsilon$ close to the corresponding points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ implies $d(f^i(x),f^i(x_0)) \leq \epsilon$ for $i = 1,2,\ldots$. If we suppose that the errors occur not only at the initial point but at each iterate we obtain not the orbit of $x_0$ but a $\delta$-chain, a sequence $\{x_0,x_1,x_2,\ldots\}$ such that $d(f(x_i),x_{i+1}) \leq \delta$ for $i = 0,1,\ldots$. The point $x$ is called a chain continuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that all $\delta$ chains beginning $\delta$ close to $x$ remain $\epsilon$ close to the points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ and $d(f(x_i),x_{i+1}) \leq \delta$ imply $d(f^i(x),x_i) \leq \epsilon$ for $i = 1,2,\ldots$. In this note we characterize this property of chain continuity. Despite the strength of this property, there is a class of systems $(X,f)$ for which the chain continuity points form a residual subset of the space $X$. For a manifold $X$ this class includes a residual subset of the space of homeomorphisms on $X$. Abstract: The Lorenz equations are a system of ordinary differential equations $x' =s(y-x), \quad y'= Rx -y-xz, \quad z'= xy -qz,$ where $s$, $R$, and $q$ are positive parameters. We show by a purely analytic proof that for each non-negative integer $N$, there are positive parameters $s, q, $ and $R$ such that the Lorenz system has homoclinic orbits associated with the origin (i.e., orbits that tend to the origin as $t\to \pm \infty$) which can rotate around the $z$-axis $N/2$ times; namely, the $x$-component changes sign exactly $N$ times, the $y$-component changes sign exactly $N+1$ times, and the zeros of $x$ and $y$ are simple and interlace. Abstract: We present in this paper some results on continuous dependence for parameters in a groundwater flow model. These results are crucial for theoretical and computational aspects of least squares estimation of parameters. As is typically the case in field studies, the form of the data is pointwise observation of hydraulic head and hydraulic conductivity at a discrete collection of observation well sites. We prove continuous dependence results for the solution of the groundwater flow equation, with respect to conductivity and boundary values, under certain types of numerical approximation. Readers Authors Editors Referees Librarians More Email Alert Add your name and e-mail address to receive news of forthcoming issues of this journal: [Back to Top]
At room temperature, the electrical conductivity of PbS is 25 (Ω.m) -1 , whereas the electron and hole mobilities are 0.06 and 0.02 m 2/V.s, respectively. Calculate the intrinsic carrier concentration for PbS at room temperature. (AMIE, Material Science, Summer 2019) Solution \[\begin{array}{l}\sigma = 25{(\Omega m)^{ - 1}}\\{\mu _e} = 0.06{m^2}/Vs\\{\mu _h} = 0.02{m^2}/Vs\end{array}\] In the intrinsic region \[n = p = {n_i}\] So, \[\sigma = {n_i}\left\| e \right\|({\mu _e} + {\mu _h})\] \[\therefore {n_i} = \frac{\sigma }{{\left\| e \right\|({\mu _e} + {\mu _h})}} = \frac{{25}}{{1.6x{{10}^{ - 19}}(0.06 + 0.02)}} = 1.95x{10^{21}}{m^{ - 3}}\] Study material for AMIE Exams
Merit function \[MF=\left[\frac 1L \sum\limits_{j=1}^{L}\left(\frac{R(X,\lambda_j)-\hat{R}(\lambda_j)}{\Delta R_j}\right)^2\right]^{1/2} \] where $\Delta R_j$ are tolerances, $X$ is a vector of layer thicknesses. Optimization methods Sequential QP method is based on a sequential approximations of the optimization problem by a set of Quadratic Programming (QP) problems. It has a good convergence and can be recommended for complicated problems. Degree of bulk inhomogeneity $\delta$ $ \delta=\displaystyle\frac{n_i-n_0}{n}\cdot 100\%, \;\; n=\frac{n_i+n_0}{2} $where $n_i, n_0$ are refractive indices at the outer boundary and substrate boundary, respectively. Admittance $A$ $\displaystyle\frac{dA}{dz}=ik\left[n^2(z)-A^2\right], $ $r(k)=\displaystyle\frac{1-A(z_a,k)}{1+A(z_a,k)},\;\; A(0,k)=n_s $ where $r(k)$ is the amplitude reflectance, $n(z)$ is refractive index profile, $k$ is the wavenumber. $A(0,k)=n_s$ Admittance is a complex value. Geometrically it can be represented as a point on a complex plane (admittance phase plane). For more details, see, for example, this book , pages 34-46. Group delay (GD) and group delay dispersion (GDD) The first and the second derivatives of the total phase shift with respect to angular frequency: $GD=-\displaystyle\frac{d\varphi}{d\omega},\;\;\; GDD=-\frac{d^2\varphi}{d\omega^2} $
Generally speaking, atomic and molecular orbitals are not physical quantities, and generally they cannot be connected directly to any physical observable. (Indirect connections, however, do exist, and they do permit a window that helps validate much of the geometry we use.) There are several reasons for this. Some of them are relatively fuzzy: they present strong impediments to experimental observation of the orbitals, but there are some ways around them. For example, in general it is only the square of the wavefunction, $\|\psi\|^2$, that is directly accessible to experiments (but one can think of electron interference experiments that are sensitive to the phase difference of $\psi$ between different locations). Another example is the fact that in many-electron atoms the total wavefunction tends to be a strongly correlated object that's a superposition of many different configurations (but there do exist atoms whose ground state can be modelled pretty well by a single configuration). The strongest reason, however, is that even within a single configuration $-$ that is to say, an electronic configuration that's described by a single Slater determinant, the simplest possible many-electron wavefunction that's compatible with electron indistinguishability $-$ the orbitals are not recoverable from the many-body wavefunction, and there are many different sets of orbitals that lead to the same many-body wavefunction. This means that the orbitals, while remaining crucial tools for our understanding of electronic structure, are generally on the side of mathematical tools and not on the side of physical objects. OK, so let's turn away from fuzzy handwaving and into the hard math that's the actual precise statement that matters. Suppose that I'm given $n$ single-electron orbitals $\psi_j(\mathbf r)$, and their corresponding $n$-electron wavefunction built via a Slater determinant,\begin{align}\Psi(\mathbf r_1,\ldots,\mathbf r_n)& = \det\begin{pmatrix}\psi_1(\mathbf r_1) & \ldots & \psi_1(\mathbf r_n)\\\vdots & \ddots & \vdots \\\psi_n(\mathbf r_1) & \ldots & \psi_n(\mathbf r_n)\end{pmatrix}.\end{align} Claim If I change the $\psi_j$ for linear combinations of them, $$\psi_i'(\mathbf r)=\sum_{j=1}^{n} a_{ij}\psi_j(\mathbf r),$$ then the $n$-electron Slater determinant $$ \Psi'(\mathbf r_1,\ldots,\mathbf r_n) = \det \begin{pmatrix} \psi_1'(\mathbf r_1) & \ldots & \psi_1'(\mathbf r_n)\\ \vdots & \ddots & \vdots \\ \psi_n'(\mathbf r_1) & \ldots & \psi_n'(\mathbf r_n) \end{pmatrix}, $$ is proportional to the initial determinant, $$\Psi'(\mathbf r_1,\ldots,\mathbf r_n)=\det(a)\Psi(\mathbf r_1,\ldots,\mathbf r_n).$$ This implies that both many-body wavefunctions are equal under the (very lax!) requirement that $\det(a)=1$. The proof of this claim is a straightforward calculation. Putting in the rotated orbitals yields\begin{align}\Psi'(\mathbf r_1,\ldots,\mathbf r_n)&=\det\begin{pmatrix}\psi_1'(\mathbf r_1) & \cdots & \psi_1'(\mathbf r_n)\\\vdots & \ddots & \vdots \\\psi_n'(\mathbf r_1) & \cdots & \psi_n'(\mathbf r_n)\end{pmatrix}\\&=\det\begin{pmatrix}\sum_{i}a_{1i}\psi_{i}(\mathbf r_1) & \cdots & \sum_{i}a_{1i}\psi_{i}(\mathbf r_n)\\\vdots & \ddots & \vdots \\\sum_{i}a_{ni}\psi_{i}(\mathbf r_1) & \cdots & \sum_{i}a_{ni}\psi_{i}(\mathbf r_n)\end{pmatrix},\end{align}which can be recognized as the following matrix product:\begin{align}\Psi'(\mathbf r_1,\ldots,\mathbf r_n)&=\det\left(\begin{pmatrix}a_{11} & \cdots & a_{1n} \\\vdots & \ddots & \vdots \\a_{n1} & \cdots & a_{nn} \\\end{pmatrix}\begin{pmatrix}\psi_1(\mathbf r_1) & \cdots & \psi_1(\mathbf r_n)\\\vdots & \ddots & \vdots \\\psi_n(\mathbf r_1) & \cdots & \psi_n(\mathbf r_n)\end{pmatrix}\right).\end{align}The determinant then factorizes as usual, giving\begin{align}\Psi'(\mathbf r_1,\ldots,\mathbf r_n)&=\det\begin{pmatrix}a_{11} & \cdots & a_{1n} \\\vdots & \ddots & \vdots \\a_{n1} & \cdots & a_{nn} \\\end{pmatrix}\det\begin{pmatrix}\psi_1(\mathbf r_1) & \cdots & \psi_1(\mathbf r_n)\\\vdots & \ddots & \vdots \\\psi_n(\mathbf r_1) & \cdots & \psi_n(\mathbf r_n)\end{pmatrix}\\\\&=\det(a)\Psi(\mathbf r_1,\ldots,\mathbf r_n),\end{align}thereby proving the claim. Disclaimers The calculation above makes a very precise point about the measurability of orbitals in a multi-electron context. Specifically, saying things like the lithium atom has two electrons in $\psi_{1s}$ orbitals and one electron in a $\psi_{2s}$ orbital is exactly as meaningful as saying the lithium atom has one electron in a $\psi_{1s}$ orbital, one in the $\psi_{1s}+\psi_{2s}$ orbital, and one in the $\psi_{1s}-\psi_{2s}$ orbital, since both will produce the same global many-electron wavefunction. This does not detract in any way from the usefulness of the usual $\psi_{n\ell}$ orbitals as a way of understanding the electronic structure of atoms, and they are indeed the best tools for the job, but it does mean that they are at heart tools and that there are always alternatives which are equally valid from an ontology and measurability standpoint. However, there are indeed situations where quantities that are very close to orbitals become accessible to experiments and indeed get measured and reported, so it's worth going over some of those to see what they mean. The most obvious is the work of Stodolna et al. [ Phys. Rev. Lett. 110, 213001 (2013)], which measures the nodal structure of hydrogenic orbitals (good APS Physics summary here; discussed previously in this question and this one). These are measurements in hydrogen, which has a single electron, so the multi-electron effect discussed here does not apply. These experiments show that, once you have a valid, accessible one-electron wavefunction in your system, it is indeed susceptible to measurement. Somewhat more surprisingly, recent work has claimed to measure molecular orbitals in a many-electron setting, such as Nature 432, 867 (2004) or Nature Phys. 7, 822 (2011). These experiments are surprising at first glance, but if you look carefully it turns out that they measure the Dyson orbitals of the relevant molecules: this is essentially the overlap$$\psi^\mathrm{D}=⟨\Phi^{(n-1)}\|\Psi^{(n)}⟩$$between the $n$-electron ground state $\Psi^{(n)}$ of the neutral molecule and the relevant $(n-1)$-electron eigenstate $\Phi^{(n-1)}$ of the cation that gets populated. (For more details see J. Chem. Phys. 126, 114306 (2007) or Phys. Rev. Lett. 97, 123003 (2006).) This is a legitimate, experimentally accessible one-electron wavefunction, and it is perfectly measurable.
Let $ \Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $ \mathbb{R}^3$ . Consider an operator $ H$ on complex valued functions on $ \mathbb{R}^3$ $ $ H\psi=\Delta\psi(x) +i\sum_{p=1}^3A_p(x)\frac{\partial \psi(x)}{\partial x_p} +B(x)\psi(x),$ $ where $ A_i,B$ are smooth real valued functions. I am looking for a precise result of the following approximate form: (1) if $ A_i$ and $ B$ are ‘small’ then the discrete spectrum of $ H$ is non-positive. (2) If $ A_i,B$ are ‘large’ then the discrete spectrum of $ H$ contains necessarily a positive element. Solve the following using singular perturbation method: εy′′ – y = 0, y(0) = 1, y(1) = 0 ε is a small parameter. I’d like to perturbatively handle an eigenvalue problem similar to this: $ $ \lambda f = (\hat{H} + 1/\epsilon^2 \hat{V} + \epsilon {W}) f, $ $ where $ f$ is a function, $ \lambda$ is an eigenvalue, $ \epsilon$ is a small parameter, and the rest are linear (differential) operators. The problem is, that if one writes a series for the eigenvalue and the eigenfunction, $ $ f = f_0 + \epsilon f_1 + \epsilon^2 f_2 + …\ \lambda = \lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + …, $ $ one will get e.g. $ $ \lambda_0 f_0 = \hat{H} f_0 + \hat{V} f_2\ \lambda_1 f_0 + \lambda_0 f_1 = \hat{H} f_1 + \hat{W} f_0 + \hat{V} f_3\ … $ $ i.e. the different orders of the series start to mix. Is there a way to develop a systematic perturbation theory for this case? In game theory, a equilibrium is: Stable, if a perturbation form any of the players returns the equilibrium back to it’s original state Unstable, if a perturbation moves the equilibrium away from the original state Semi-stable if some of the players perturbations are acceptable. What do you call an equilibrium if a perturbation merely moves it slightly. Consider the following game: Two players must state a number. Each of them gets a payoff of $ 1 if they state the same number, otherwise there is no payoff. (1,1) is an equilibrium. If one of the players chooses a different strategy, say 1.01. Then the Equilibrium is (1.01, 1.01). This does not seem unstable – neither does this seem stable. What is this type of equilibrium called? How to solve using perturbation method for small e? $ \frac{d}{dx}((1-\frac{ey}{y+1})\frac{dy}{dx})=0$ $ x=0,y=0$ $ x=1,y=1$ Let $ (M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure $ $ Y \rightarrow M \overset{\pi}{\rightarrow} B $ $ where $ (Y,g_Y)$ and $ (B,g_B)$ are closed Riemannian manifolds such that $ \pi$ is a Riemannian submersion. Now we define $ g_{\epsilon}=\epsilon^{-2}\pi^*g_B+g_Y$ and $ \tilde g_{\epsilon}=g_{\epsilon}+\alpha_{\epsilon}$ for some error term $ \alpha_{\epsilon}=O(\epsilon^{\tau-2})$ for some $ \tau>0$ . If we denote the signature operators on $ M$ with respect to $ g_{\epsilon}$ and $ \tilde g_{\epsilon}$ by $ A_{\epsilon}$ and $ \tilde A_{\epsilon}$ respectively. Can we conclude that $ $ \lim_{\epsilon \to 0} \eta(A_{\epsilon})= \lim_{\epsilon \to 0} \eta(\tilde A_{\epsilon}), $ $ if the former exists? Let $ K$ be a compact self-adjoint operator on a Hilbert space $ H$ such that for some normalized $ x \in H$ and $ \lambda \in \mathbb C:$ $ \Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known that this implies that $ d(\sigma(K),\lambda) \le \varepsilon.$ However, I am wondering whether this implies also something about $ x.$ For example, it seems plausible that $ x$ cannot be orthogonal to the direct sum of eigenspaces of $ K$ with eigenvalues that are close to $ \lambda.$ In other words, are there any non-trivial restrictions on $ x$ coming from the spectral decomposition of $ K$ ? Let $ T$ be a selfadjoint operator on Hilbert space $ H$ . Then we know that there is a resolvent estimation $ $ \left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \rho(T).$ $ But what if we want to consider the compact perturbation, that is, operator $ A = T+ D,$ where $ D$ is a compact operator on $ H$ . Is there any known results on the resolvent estimate? Does anyone know some related references ? Thank you very much. Let $ T$ be a bounded linear operator acting on a complex Banach space. Suppose that $ T$ has spectral radius strictly less than $ 1$ . If we introduce an analytic perturbation to $ T$ , $ s\mapsto T_s$ for $ \|s\|<\epsilon$ (with $ T_0 = T$ ) then by upper-semicontinuity of the spectrum, assuming that $ \epsilon$ is sufficiently small, the spectral radius of each $ T_s$ is less than $ \rho$ for some $ \rho <1$ . My question is the following: Is it possible to find $ K>0$ and $ \rho <\rho ‘<1$ such that $ $ \\|T_s^n\\|\le K (\rho ‘)^n,$ $ for all $ \|s\|<\epsilon$ and $ n$ ? This certainly seems like it should be true but I can’t find a proof – I think I’m missing something obvious. Any help would be greatly appreciated – cheers! Here we assume that $ u=u^{\varepsilon}(x,t)$ satisfies the following hyperbolic PDE problem, $ $ \partial^2_t u^{\varepsilon}-\partial_x \big(a(x)\partial_x u^{\varepsilon} \big)+\varepsilon \partial_x u^{\varepsilon} =0,~(x\in\mathbb{R},t>0),$ $ $ $ u^{\varepsilon}(x,0)=0,\quad \partial_t u^{\varepsilon}(x,0)=F_0(x).$ $ Suppose that $ a(x)>0$ is a smooth even periodic function, i.e., $ a(x+1)=a(x),~a(-x)=a(x)$ . The initial condition $ F_0(x)\in C^{\infty}( \mathbb{R})\cap L^2(\mathbb{R})$ . If $ \varepsilon=0$ , we denote the above solution as $ u^0(x,t)$ . Our question is when $ \varepsilon>0$ , for any $ t\in[0,\frac1{\varepsilon}]$ , could we get the following estimate under the same initial condition $ $ \\|u^{\varepsilon}(x,t)\\|_{L^2(\mathbb{R})}\le C \sup_{\tau\in[0,\frac{1}{\varepsilon}]}\\|u^0(x,\tau)\\|_{L^2(\mathbb{R})} ~?$ $ Where $ C$ is independent of $ \varepsilon$ . When $ a(x)$ is a constant, namely $ \mathcal{L}=-\partial_x \big(a(x)\partial_x\big)$ is Laplacian, we can prove the above result by using Fourier tranform, Plancherel theorem and Gronwall inequality. How about the case of a general elliptic operator, is the result also the same? If so, does Riesz transform work here?
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 $latex \displaystyle \mathrm{MISE}\approx\frac{\left(\mu_{2}(K)\right)^{4}}{4}\left[\sum_{j=1}^{d}h_{j}^{4}\int f_{jj}^{2}(x)dx+\sum_{j\neq k}h_{j}^{2}h_{k}^{2}\int f_{jj}f_{kk}dx\right]+\frac{\left(\int K^{2}(x)dx\right)^{d}}{nh_{1}\cdots h_{d}} &fg=000000$ 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 […] Photos of Sergey Nikolskii from The Russian Academy of Sciences The MSE gives an error of the estimator $latex {\hat{p}_{n}}&fg=000000$ at an arbitrary point $latex {x_{0}}&fg=000000$, but it is worth to study a global risk for $latex {\hat{p} _{n}}&fg=000000$. The mean integrated squared error (MISE) is an important global measure, $latex \displaystyle \mathrm{MISE}\triangleq\mathop{\mathbb E}_{p}\int\left(\hat{p} _{n}(x)-p(x)\right)^{2}dx &fg=000000$ […] I will make a summary of ideas about nonparametric estimation, including some basics results to develop more advanced theory later. In the first post we talk something about the density estimation and the nonparametric regression. Later, in posts about histogram (I,II,III,IV) , we saw how the histogram is a nonparametric estimator and we studied its […] We are going to introduce the histogram as a simple nonparametric density estimator. I will divide this presentation in several posts for simplicity reasons. Let us $latex {X_1,\ldots,X_n}&fg=000000$ with pdf $latex {f}&fg=000000$. The histogram is the simplest nonparametric estimator of $latex {f}&fg=000000$. I would like to start this blog with some basic ideas about density estimation and nonparametric regression. The study of the probability density function (pdf) is called nonparametric estimation. This kind of estimation can serve as a block building in nonparametric regression. The typical regression problem is setting as follows. Assume that we have a […]
The golden ratio is well known and is associated with the Fibonacci sequence of numbers. $\varphi$ is the positive root of the quadratic equation $x^2-x-1 = 0$. The exact value is given by $\displaystyle \frac{1 + \sqrt5}{2} \approx 1.61803$ Its continued fraction is $[1; 1, 1, 1, …]$. I have written on the silver ratio and its association with the Pell sequence of numbers. $\delta_s$ is the positive root of the quadratic equation $x^2-2x-1 = 0$. The exact value is given by $1 + \sqrt2 \approx 2.41421$ Its continued fraction is $[2; 2, 2, 2, …]$. The plastic number (or ratio), denoted by $\rho$ (rho), is associated with the Padovan sequence of numbers. This sequence of numbers begins $1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, …$ The Padovan sequence $P(n)$ is defined by the initial values $P(0) = P(1) = P(2) = 1$ and the recurrence relation $P(n) = P(n-2) + P(n-3)$ For example, $P(12) = P(10) + P(9)$ which is $16 = 9 + 7$. In the above image is a visual proof that the sequence also satisfies the recurrence relation $P(n) = P(n-1) + P(n-5)$ Some values of the convergents are $\begin{array}{l\|l} Ratio & Decimal\\ \hline 1/1&1\\ 1/1 &1\\ 2/1 &2\\ 2/2&1\\ 3/2&1.5\\ 4/3&1.\overline 3\\ 5/4&1.25\\ 7/5&1.4\\ 9/7&1.\overline{285714}\\ 12/9&1.\overline3\\ 16/12&1.\overline3\\ 21/16&1.3125\\ 28/21&1.\overline3\\ 37/28&1.32\overline{142857}\\ 49/37&1.\overline{324} \end{array} $ The limit of the convergents of consecutive Padovan numbers is $\displaystyle \lim \limits_{n \to \infty} \frac{P(n)}{P(n-1)} = \rho $ $\rho$ is the real root of the cubic equation $x^3-x-1 = 0$. A cubic equation either has three real roots ( which may be repeated) or one real root and two complex roots. $\rho$ has the exact value $\displaystyle \rho = \frac{\sqrt[3]{108 + 12\sqrt69} + \sqrt[3]{108 -12\sqrt69}}{6}$ with an approximate value of $1.324 \, 717 \, 957 \,…$ Its continued fraction is $[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, …]$ The powers of $\rho$ $\begin{array}{l\|l} \rho^n & Approx. Value\\ \hline \rho^{-2} & 0.569840\\ \rho^{-1} & 0.754878\\ \rho^0 & 1\\ \rho^1 & 1.324717957\\ \rho^2 & 1.754878\\ \rho^3 & 2.324718\\ \rho^4 & 3.079596\\ \rho^5 & 4.079596 \end{array}$ The following diagram shows both of the above given recurrence relations and the plastic ratio in one dimension (the division of the vertical line on the RHS – $\rho^2 : \rho^3 = 1 : \rho$) and the plastic ratio in two dimensions (the rectangle on the LHS – $\rho^4 : \rho^5 = 1 : \rho$). © OldTrout $2018$ No Audio file – it does not translate well. . .
Search Now showing items 1-5 of 5 Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV (Springer, 2015-05-20) The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement is performed in the central pseudorapidity ... Rapidity and transverse-momentum dependence of the inclusive J/$\mathbf{\psi}$ nuclear modification factor in p-Pb collisions at $\mathbf{\sqrt{\textit{s}_{NN}}}=5.02$ TeV (Springer, 2015-06) We have studied the transverse-momentum ($p_{\rm T}$) dependence of the inclusive J/$\psi$ production in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV, in three center-of-mass rapidity ($y_{\rm cms}$) regions, down to ... Measurement of charm and beauty production at central rapidity versus charged-particle multiplicity in proton-proton collisions at $\sqrt{s}$ = 7 TeV (Springer, 2015-09) Prompt D meson and non-prompt J/$\psi$ yields are studied as a function of the multiplicity of charged particles produced in inelastic proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV. The results ... Coherent $\rho^0$ photoproduction in ultra-peripheral Pb-Pb collisions at $\mathbf{\sqrt{\textit{s}_{\rm NN}}} = 2.76$ TeV (Springer, 2015-09) We report the first measurement at the LHC of coherent photoproduction of $\rho^0$ mesons in ultra-peripheral Pb-Pb collisions. The invariant mass and transverse momentum distributions for $\rho^0$ production are studied ... Inclusive, prompt and non-prompt J/ψ production at mid-rapidity in Pb-Pb collisions at √sNN = 2.76 TeV (Springer, 2015-07-10) The transverse momentum (p T) dependence of the nuclear modification factor R AA and the centrality dependence of the average transverse momentum 〈p T〉 for inclusive J/ψ have been measured with ALICE for Pb-Pb collisions ...
First My college book contains the following passage: the angle of incidence (relative to the fibre axis ) can't be too large else the ray would be refracting on the core\cladding boundary and transmitted outside the fiber and a very small percentage passes. Here's how I reasoned and I'd like to be corrected. The geometry I imagined was as follows: light falls with a very large angle $\alpha$ from air, light is refracted with angle $\theta$, $\theta$ < $\alpha$, light falls on cladding with large angle $\beta$ where: $ \alpha > \beta $ and $\beta > \theta c$ of the material of the cladding, light is simply totally internally reflected. Second I looked it up and found numerical aperture and acceptance angle articles but I'd like to be cleared out on the first matter first.
General case. In relativistic thermodynamics, inverse temperature $\beta^\mu$ is a vector field, namely the multipliers of the 4-momentum density in the exponent of the density operator specifying the system in terms of statistical mechanics, using the maximum entropy method, where $\beta^\mu p_\mu$ (in units where $c=1$) replaces the term $\beta H$ of the nonrelativistic canonical ensemble. This is done in C.G. van Weert, Maximum entropy principle and relativistic hydrodynamics, Annals of Physics 140 (1982), 133-162. for classical statistical mechanics and for quantum statistical mechanics in T. Hayata et al., Relativistic hydrodynamics from quantum field theory on the basis of the generalized Gibbs ensemble method, Phys. Rev. D 92 (2015), 065008. https://arxiv.org/abs/1503.04535 For an extension to general relativity with spin see also F. Becattini, Covariant statistical mechanics and the stress-energy tensor, Phys. Rev. Lett 108 (2012), 244502. https://arxiv.org/abs/1511.05439 Conservative case. One can define a scalar temperature $T:=1/k_B\sqrt{\beta^\mu\beta_\mu}$ and a velocity field $u^\mu:=k_BT\beta^\mu$ for the fluid; then $\beta^\mu=u^\mu/k_BT$, and the distribution function for an ideal fluid takes the form of a Jüttner distribution $e^{-u\cdot p/k_BT}$. For an ideal fluid (i.e., assuming no dissipation, so that all conservation laws hold exacly), one obtains the format commonly used in relativistic hydrodynamics (see Chapter 22 in the book Misner, Thorne, Wheeler, Gravitation). It amounts to treating the thermodynamics nonrelativistically in the rest frame of the fluid. Note that the definition of temperature consistent with the canonical ensemble needs a distribution of the form $e^{-\beta H - terms~ linear~ in~ p}$, conforming with the identification of the noncovariant $\beta^0$ as the inverse canonical temperature. Essentially, this is due to the frame dependence of the volume that enters the thermodynamics. This is in agreement with the noncovariant definition of temperature used by Planck and Einstein and was the generally agreed upon convention until at least 1968; cf. the discussion in R. Balescu, Relativistic statistical thermodynamics, Physica 40 (1968), 309-338. In contrast, the covariant Jüttner distribution has the form $e^{-u_0 H/k_BT - terms~ linear~ in~ p}$. Therefore the covariant scalar temperature differs from the canonical one by a velocity-dependent factor $u_0$. This explains the different transformation law. The covariant scalar temperature is simply the canonical temperature in the rest frame, turned covariant by redefinition. Quantum general relativity. In quantum general relativity, accelerated observers interpret temperature differently. This is demonstrated for the vacuum state in Minkowski space by the Unruh effect, which is part of the thermodynamics of black holes. This seems inconsistent with the assumption of a covariant temperature. Dissipative case. The situation is more complicated in the more realistic dissipative case. Once one allows for dissipation, amounting to going from Euler to Navier-Stokes in the nonrelativistic case, trying to generalize this simple formulation runs into problems. Thus it cannot be completely correct. In a gradient expansion at low order, the velocity field defined above from $\beta^\mu$ can be identified in the Landau-Lifschitz frame with the velocity field proportional to the energy current; see (86) in Hayata et al.. However, in general, this identification involves an approximation as there is no reason for these velocity fields to be exactly parallel; see, e.g., P. Van and T.S. Biró, First order and stable relativistic dissipative hydrodynamics, Physics Letters B 709 (2012), 106-110. https://arxiv.org/abs/1109.0985 There are various ways to patch the situation, starting from a kinetic description (valid for dilute gases only): The first reasonable formulation by Israel and Stewart based on a first order gradient expansion turned out to exhibit acausal behavior and not to be thermodynamically consistent. Extensions to second order (by Romatschke, e.g., https://arxiv.org/abs/0902.3663) or third order (by El et al., https://arxiv.org/abs/0907.4500) remedy the problems at low density, but shift the difficulties only to higher order terms (see Section 3.2 of Kovtun, https://arxiv.org/abs/1205.5040). A causal and thermodynamically consistent formulation involving additional fields was given by Mueller and Ruggeri in their book Extended Thermodynamics 1993 and its 2nd edition, called Rational extended Thermodynamics 1998. Paradoxes. Concerning the paradoxes mentioned in the original post: Note that the formula $\langle E\rangle = \frac32 k_B T$ is valid only under very special circumstances (nonrelativistic ideal monatomic gas in its rest frame), and does not generalize. In general there is no simple relationship between temperature and velocity. One can say that your paradox arises because in the three scenarios, three different concepts of temperature are used. What temperature is and how it transforms is a matter of convention, and the dominant convention changed some time after 1968; after Balescu's paper mentioned above, which shows that until 1963 it was universally defined as being frame-dependent. Today both conventions are alive, the frame-independent one being dominant. This post imported from StackExchange Physics at 2016-06-24 15:03 (UTC), posted by SE-user Arnold Neumaier
I am looking for an explanation of the following fact, which seems to be rather simple yet I am missing something. Say that $S_t$ is a stock following GBM $$ dS_t = r S_td_t + \sigma S_t dW_t,$$ and I want to price a derivative with payoff $max(S_T^2-K,0)$. I know how to do this using the risk-neutral expectation, but it's rather long and apparently unnecessary. I read in a book (Mark Joshi's Concepts...) another solution which uses the forward price. If $F_T(t)$ is the forward price of $S$ at time $t$, then one can write $$ F_T(T)^2 = (F_T(0)^2 e^{\sigma^2 T})e^{-\frac{\sigma'^2}{2} T + \sigma'\sqrt{T}N(0,1)},$$ with $\sigma'=2\sigma$, so then one can use Black's formula with forward equal to $F_T(0)^2 e^{\sigma^2T}$ and volatility $\sigma'$. But how is Black's formula justified here exactly? I assume that the formula can be applied for a call option on the forward price of an asset, but how do we know there exists a tradable asset with volatility $\sigma'$, for instance? Or is this even necessary?
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:-)
Neurons (Activation Functions)¶ They could be attached to any layers. The neuron of each layer will affect the output in the forward pass and the gradient in the backward pass automatically unless it is an identity neuron. A layer have an identity neuron by default [1]. class Neurons. Identity¶ An activation function that does nothing. class Neurons. ReLU¶ Rectified Linear Unit. During the forward pass, it inhibit all the negative activations. In other words, it compute point-wisely $y=\max(0, x)$. The point-wise derivative for ReLU is\[\begin{split}\frac{dy}{dx} = \begin{cases}1 & x > 0 \\ 0 & x \leq 0\end{cases}\end{split}\] Note ReLU is actually not differentialble at 0. But it has subdifferential$[0,1]$. Any value in that interval could be taken as a subderivative, and could be used in SGD if we generalize from gradient descent to subgradientdescent. In the implementation, we choose 0. class Neurons. Sigmoid¶ Sigmoid is a smoothed step function that produces approximate 0 for negative input with large absolute values and approximate 1 for large positive inputs. The point-wise formula is $y = 1/(1+e^{-x})$. The point-wise derivative is\[\frac{dy}{dx} = \frac{-e^{-x}}{\left(1+e^{-x}\right)^2} = (1-y)y\] [1] This is actually not true: not all layers in Mocha support neurons. For example, data layers currently does not have neurons, but this feature could be added by simply adding a neuron property to the data layer type. However, for some layer types like loss layers or accuracy layers, it does not make much sense to have neurons.
amp-mathml Displays a MathML formula. Required Script <script async custom-element="amp-mathml" src="https://cdn.ampproject.org/v0/amp-mathml-0.1.js"></script> Supported Layouts container Examples amp-mathml.amp.html Behavior This extension creates an iframe and renders a MathML formula. Example: The Quadratic Formula <amp-mathml layout="container" data-formula="\[x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\]"> </amp-mathml> Example: Cauchy's Integral Formula <amp-mathml layout="container" data-formula="\[f(a) = \frac{1}{2\pi i} \oint\frac{f(z)}{z-a}dz\]"> </amp-mathml> Example: Double angle formula for Cosines <amp-mathml layout="container" data-formula="$$ \cos(θ+φ)=\cos(θ)\cos(φ)−\sin(θ)\sin(φ) $$"> </amp-mathml> Example: Inline formula This is an example of a formula of <amp-mathml layout="container" inline data-formula="`x`"></amp-mathml>, <amp-mathml layout="container" inline data-formula="$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$"></amp-mathml> placed inline in the middle of a block of text. <amp-mathml layout="container" inline data-formula="$ \cos(θ+φ) $"></amp-mathml> This shows how the formula will fit inside a block of text and can be styled with CSS. Attributes data-formula (required) Specifies the formula to render. inline (optional) If specified, the component renders inline ( inline-block in CSS). Validation See amp-mathml rules in the AMP validator specification. You've read this document a dozen times but it doesn't really cover all of your questions? Maybe other people felt the same: reach out to them on Stack Overflow.Go to Stack Overflow Found a bug or missing a feature? The AMP project strongly encourages your participation and contributions! We hope you'll become an ongoing participant in our open source community but we also welcome one-off contributions for the issues you're particularly passionate about.Go to GitHub
Photos of Sergey Nikolskii from The Russian Academy of Sciences The MSE gives an error of the estimator $latex {\hat{p}_{n}}&fg=000000$ at an arbitrary point $latex {x_{0}}&fg=000000$, but it is worth to study a global risk for $latex {\hat{p} _{n}}&fg=000000$. The mean integrated squared error (MISE) is an important global measure, $latex \displaystyle \mathrm{MISE}\triangleq\mathop{\mathbb E}_{p}\int\left(\hat{p} _{n}(x)-p(x)\right)^{2}dx &fg=000000$ […] I will make a summary of ideas about nonparametric estimation, including some basics results to develop more advanced theory later. In the first post we talk something about the density estimation and the nonparametric regression. Later, in posts about histogram (I,II,III,IV) , we saw how the histogram is a nonparametric estimator and we studied its […] Today we will apply the ideas of the others post by a simple example. Before, we are going to answer the question of the last week. What is exactly the $latex {h_{opt}}&fg=000000$ if we assume that $latex \displaystyle \displaystyle f(x) = \frac{1}{\sqrt{2\pi}} \text{exp}\left(\frac{-x^2}{2}\right)? &fg=000000$ How $latex {f(x)}&fg=000000$ is the density of standard normal distribution. It is […] Before to continue with today’s post we will answer the question of last week, Is it $latex {\hat{f}_{h}(x)}&fg=000000$ a consistent estimator? The answer is yes. Because convergence in mean squared implies convergence in probability. We continue our presentation about the estimation of histograms and its statistical properties. Today we will start the theory for reducing the mean squared error. In order to study the statistical properties of $latex {\hat{f}_{h}(x)}&fg=000000$We will start introducing the concept of mean squared error (MSE) or quadratic risk. We define We are going to introduce the histogram as a simple nonparametric density estimator. I will divide this presentation in several posts for simplicity reasons. Let us $latex {X_1,\ldots,X_n}&fg=000000$ with pdf $latex {f}&fg=000000$. The histogram is the simplest nonparametric estimator of $latex {f}&fg=000000$. I would like to start this blog with some basic ideas about density estimation and nonparametric regression. The study of the probability density function (pdf) is called nonparametric estimation. This kind of estimation can serve as a block building in nonparametric regression. The typical regression problem is setting as follows. Assume that we have a […]
Bulletin of the American Physical Society APS March Meeting 2013 Volume 58, Number 1 Monday–Friday, March 18–22, 2013; Baltimore, Maryland Session G46: SPS Undergraduate V Hide Abstracts Sponsoring Units: SPS Chair: Kendra Redmond, American Institute of Physics Room: Hilton Baltimore Holiday Ballroom 5 Tuesday, March 19, 2013 11:15AM - 11:27AM G46.00001: Reverse Micelle Synthesis of Gadolinium Nanoparticles R.H. Fukuda, M.M. Castro, P.-C. Ho, S. Attar, M. Golden, D. Margosan Nanotechnology is an area of great interest due to its variety of applications such as nano-medicine. The reverse micelle method has been used to synthesize Gd nanoparticles by our research group. Through this method, a surfactant protectively cages particles of Gd in the presence of polar methanol and nonpolar hexane. This method can control particle size by growth temperature and the molar ratio of polar solvent to surfactant. The Gd was reduced from its chloride compound by using sodium borohydride. The final products have been derived either through a method of liquid liquid extraction or filtration. Scanning electron microscopy (SEM) paired with energy dispersive x-ray spectroscopy (EDX) was used to examine the size, shape, and composition of the products. The size and shape were also examined using a Leica light microscope between SEM analyses. We found that liquid liquid extraction does not work in the solvent combination of methanol-hexane due to the instability of the reverse micelles. Additionally, the process of carbon coating SEM samples may have destroyed the reverse micelle structures. [Preview Abstract] Tuesday, March 19, 2013 11:27AM - 11:39AM G46.00002: Ferromagnetic Nanoparticles for Biomedical Applications Frank Holder, Cristina Iftode, Tabbetha Dobbins This work examines the cytotoxicity of barium hexaferrite to fibroblast (HEK-293) cells and also the response of barium hexaferrite to magnetic fields. Cytotoxicity is a great way for pharmacies to measure for toxic compounds. Cytotoxicity assays are widely used by the pharmaceutical industry to screen new compounds which may be introduced to the cells. Results show the cytotoxicity of nanoparticles of barium hexaferrite. We chose barium hexaferrite because it is a magnetic material---so it can be driven using an applied magnetic field. This would be useful in biomedical applications where these particles may be added to direct treatment to various parts of the body and across the cell wall membrane by an applied magnetic field. [Preview Abstract] Tuesday, March 19, 2013 11:39AM - 11:51AM G46.00003: Morphological, Thermal, and Magnetic Analysis of Ball-Milled $\gamma $-Fe$_2$O$_3$ and Fe$_3$O$_4$ Nanoparticles for Biomedical Application Philip Burnham, Georgia C. Papaefthymiou, Arthur Viescas, Calvin Li, Norman Dollahon Superparamagnetic iron oxide nanoparticles are promising agents for hyperthermia cancer treatment, because, when exposed to an alternating magnetic field, they impart heat to surrounding tissue. A comparison of $\gamma $-Fe$_2$O$_3$ and Fe$_3$O$_4$ nanoparticles for such application is presented. The particles were obtained via surfactant-assisted high energy ball-milling in a hexane/oleic acid carrier-fluid environment. Particles with diameters of 5 to 16 nm were prepared with mass ratios (oleic acid):($\gamma $-Fe$_2$O$_3)$ of 0:1, 1:5, 1:10 and 1:20, with milling times of 3, 6, 9, and 12 hours. TEM micrographs revealed spherical morphology and the effect of oleic acid shells. Optimal size distributions were obtained for high oleic acid contents. At room temperature, a reduced internal magnetic field $\sim$480 kOe) was recorded via M\"{o}ssbauer spectroscopy compared to bulk $\gamma $-Fe$_2$O$_3$ $\sim$500 kOe), due to magnetic relaxation; Fe$_3$O$_4$ particles produced similar results. For the $\gamma $-Fe$_2$O$_3$ and Fe$_3$O$_4$ nanoparticles with 20{\%} oleic acid by mass, comparative ZFC/FC magnetization (H$_{\mathrm{app}}=$ 200 Oe in temperature range from 2 to 400 K) and hysteresis loops (T $=$ 2 K and 300 K up to H$_{\mathrm{app}}=$6 kOe) were obtained. Thermal transport characteristics were verified by Specific Absorption Rate (SAR) measurements using an AC magnetic field ($f=$282 kHz). Differences and similarities in behavior will be discussed. [Preview Abstract] Tuesday, March 19, 2013 11:51AM - 12:03PM G46.00004: Multi-scale Size Distributions of Colloidal Gold Clusters Measured by Ultrasmall Angle X-ray Scattering (USAXS) and Dynamic Light Scattering (DLS) Ashli Nieves, Jan Ilavsky, Tabbetha Dobbins Gold colloids are of interest as: (1) catalysts for energy conversion and (2) absorption agents for laser photothermal therapy. This research examines the agglomerate sizes (using DLS) and primary particle sizes (using USAXS) for gold nanoparticles synthesized by trisodium citrate reduction of gold chloroauric acid (HAuCl4). USAXS data was collected at the Advanced Photon Source, beamline 15ID-D. Model fitting of the data show primary particle sizes of 7nm to 14nm formed. DLS results show these particles to aggregate into a bimodal set of clusters centered on approximately 20nm and approximately 200nm. Preliminary results aimed at effectively breaking apart these aggregates are presented. [Preview Abstract] Tuesday, March 19, 2013 12:03PM - 12:15PM G46.00005: Dynamical properties of colloids immersed in a uniform electric field at high densities Matthew Wozniak, Manuel Valera, Athula Herat In light of the recent interest in the control of colloidal systems, we have explored specific properties of electrically interacting colloidal particles. We explored the structural and dynamical characteristics of mono-disperse systems of colloidal particles that are affected by dipole-dipole interactions while immersed in a uniform electric field and compared with the outcomes that could occur if different sizes of particles are mixed. We used molecular dynamics simulations to study the systems. We present results for the diffusion coefficient and other dynamical properties in the high density regime. [Preview Abstract] Tuesday, March 19, 2013 12:15PM - 12:27PM G46.00006: Synthesis and Characterization of Mg-doped ZnO Nanorods for Biomedical Applications H. Gemar, N.C. Das, A. Wanekaya, R. Delong, K. Ghosh Nanomaterials research has become a major attraction in the field of advanced materials research in the area of Physics, Chemistry, and Materials Science. Bio-compatible and chemically stable metal nanoparticles have biomedical applications that includes drug delivery, cell and DNA separation, gene cloning, magnetic resonance imaging (MRI). This research is aimed at the fabrication and characterization of Mg-doped ZnO nanorods. Hydrothermal synthesis of undoped ZnO and Mg-doped ZnO nanorods is carried out using aqueous solutions of Zn(NO$_{3})_{2}$ .6H$_{2}$O, MgSO$_{4}$, and using NH$_{4}$OH as hydrolytic catalyst. Nanomaterials of different sizes and shapes were synthesized by varying the process parameters such as molarity (0.15M, 0.3M, 0.5M) and pH (8-11) of the precursors, growth temperature (130$^{\circ}$C), and annealing time during the hydrothermal Process. Structural, morphological, and optical properties are studied using various techniques such as XRD, SEM, UV-vis and PL spectroscopy. Detailed structural, and optical properties will be discussed in this presentation. This work is partially supported by National Cancer Institute (1 R15 CA139390-01). [Preview Abstract] Tuesday, March 19, 2013 12:27PM - 12:39PM G46.00007: Study of Thermal Conductivity of Si Nanowires with micro-Raman Spectroscopy Bingqing Li, Kathryn F. Murphy, Daniel S. Gianola, X.M. Cheng Nanowires have played an increasingly important role in thermoelectric technology due to their high figure of merit ZT resulting from the reduced thermal conductivity, K, and good electrical conductivity. In this work, we report the measurement of K of individual silicon nanowires (SiNWs) by mapping Raman temperature profiles along the testing nanowires using a microelectromechanical system (MEMS) device and a micro-Raman system with a 530 nm laser beam. Thermal conductivity was measured as a function of uniaxial tensile stress applied to the SiNWs, which was varied from 0 to 1.2 GPa. The measured K results for the unstrained nanowires agree well with the predictions based on diffuse phonon boundary scattering. The dependence of SiNWs' thermal conductivity on engineering stress can provide significant information for nanowires fabrication. [Preview Abstract] Tuesday, March 19, 2013 12:39PM - 12:51PM G46.00008: Characterization of Carbon Nanotubes Synthesized Using Chemical Vapor Deposition Andrew Zeidell, Shawn Huston, Nathanael Cox, Brian Landi, Tonya Coffey, Phillip Russell, Brad Conrad Carbon Nanotubes were synthesized using a Chemical Vapor Deposition system with precursor Cyclopentadienyliron Dicarbonyl Dimer and were systematically characterized over a variety of growth conditions using several methods. Scanning Electron Microscopy (SEM) was used to investigate catalyst contamination, tube diameters, growth morphologies, and material alignment. Transmission Electron Microscopy (TEM) was employed to quantify nanotube wall crystallinity and sidewall defects. Raman Spectroscopy was used in conjunction with Thermo-Gravimetric analysis to ascertain the purity levels of each sample. Results are discussed in terms of related precursors and are used to evaluate the efficacy of the precursor and material quality. [Preview Abstract] Tuesday, March 19, 2013 12:51PM - 1:03PM G46.00009: Bi2Te3 Nanostructure Synthesis on Multiple Substrates Nicha Apichitsopa, Jerome T. Mlack, Nina Markovic The chalcogenide Bi2Te3 is a known and widely used thermoelectric material that has received renewed experimental interest due to the recent discovery of its topologically protected surface states. Nanodevices of this material are particularly interesting because of their high surface-to-volume ratio, which enhances surface-related transport properties by minimizing bulk contributions. Many synthesis processes for Bi2Te3 have been reported, such as as Au-catalyzed vapor-liquid-solid mechanism (VLS) and lithographically patterned galvanic displacement (LPGD). The VLS mechanism is much simpler than the highly-controlled LPGD; however, remnant of Au catalyst on the nanostructures can alter their electronic structure, resulting in modification of TI surface. We report the synthesis of Bi2Te3 nanostructures by VLS mechanism without using Au catalyst, which improves the quality of the nanostructures. [Preview Abstract] Tuesday, March 19, 2013 1:03PM - 1:15PM G46.00010: SAM surface domains of 1-mercaptoundecanoic acid and 1-dodecanethiol mixtures on Au(111) investigated via polarized probes Rose Pasquale, Reshani Senevirathne, Indrajith Senevirathne SAM (Self Assembled Monolayer) surfaces with --COOH terminus is bio active and therefore has many bioengineering applications. However complex devices patterned on surfaces require a deeper understanding of the surface domain architecture of SAMs with multi component mixtures of thiols. Varying concentration mixed solutions of 1-mercaptoundecanoic acid (hydrophilic -COOH end) and 1-dodecanethiol (hydrophobic --R), dissolved in 200 proof Ethanol with total 5mM concentration were prepared. These solutions were used in developing SAMs on clean flat Au(111) on mica. Resulting SAMs surfaces were investigated with regular and custom built positively and negatively polarized AFM (Atomic Force Microcopy) probes via contact, non contact and lateral force mode AFM with topography and phase imaging. Domains of distinct thiols were identified as selective self assembly on step edges and terraces. Surface roughness, corrugation and morphology at each domain were estimated. Total RMS surface roughness is estimated at $\sim$ 2.44nm for SAMs with 75{\%} 1-mercaptoundecanoic acid while for SAMs with 25{\%} 1-mercaptoundecanoic acid it is estimated at $\sim$ 2.68nm. [Preview Abstract] Tuesday, March 19, 2013 1:15PM - 1:27PM G46.00011: Characterization of Nanophosphors for Solid State Lighting Devices Grown by Microwave Plasma Assisted Deposition Process Jedidiah McCoy, Marek Merlak, Sarath Witanachchi Increasingly, greenhouse farming and urban agriculture are being looked at as a more efficient and more cost effective way to grow produce. Currently the lights used in greenhouses rely on light sources that emit a broad spectrum of light. However, only light at wavelengths around 460 nm (blue) and 670 nm (red) are absorbed by most plants for photosynthesis. Solid state lighting devices can be engineered to produce light to match the needs of the plant while reducing the energy cost. An investigation into the photoluminescence properties of the nanophosphor La$_2$O$_3$ doped with Bi was done in an effort to produce a phosphor emitting in blue wavelengths. The La$_2$O$_3$:Bi coatings were grown using a microwave plasma growth process. Microwave power and chamber pressure were varied to find the optimum synthesis conditions. Power was varied from 100Watts to 900Watts and chamber pressure was varied from 30Torr to 60Torr. The process utilized O$_2$ and CO$_2$ plasma. The nanophosphors were investigated by X-ray diffraction, electron microscopy, and photoluminescent spectroscopy. Photoluminescence was shown to be higher from samples synthesized in a CO$_2$ plasma. [Preview Abstract] Tuesday, March 19, 2013 1:27PM - 1:39PM G46.00012: Slip, Slide, or Roll? Mike Testa Using an atomic force microscope the research project, ``Slip, Slide, or Roll?'' investigates rolling and sliding friction on the nanoscale. The findings of this study may be used to develop improved mechanical lubricants and surfaces. Friction may seem like a simple idea that is familiar to everyone, yet scientific literature explaining what dictates the translational modes of nanoscale objects is surprisingly lacking. In the macroscopic world spherical objects energetically prefer rolling over sliding, for nanoscale objects this is not necessarily the case. We are testing the hypothesis that size, surface chemistry, and elastic modulus dictate whether spherical nanoscale objects will slide or roll when a lateral force is applied. In order to understand the conditions that cause nanoscale particles to transition between the two translational modes we precisely manipulate these variables and measure their effects. [Preview Abstract] Engage My APS Information for The American Physical Society (APS) is a non-profit membership organization working to advance the knowledge of physics. Headquarters1 Physics Ellipse, College Park, MD 20740-3844(301) 209-3200 Editorial Office1 Research Road, Ridge, NY 11961-2701(631) 591-4000 Office of Public Affairs529 14th St NW, Suite 1050, Washington, D.C. 20045-2001(202) 662-8700
Systems containing thousand atoms and more (e.g. polymers, reactions in solution, molecules on surfaces) are hard to describe fully ab initio. In many applications, however, the main attention focusses on a small subset A of the large system (active sites, the dissolved/adsorbed molecules). While the interaction of part A with the remaining part B cannot be neglected, one can often resort to treating B by less computationally expensive molecular mechanics (MM) models that interact with a quantum mechanical (QM) description of A. The usefulness of this hybrid QM/MM approach was recognized by the 2013 Nobel Prize in Chemistry, awarded to A. Warshel, M. Levitt and A. Karplus for “the development of multiscale models for complex chemical systems”. In this exercise, we are studying the adsorption of cyclohexaphenylene on Cu(111), following the work by Pignedoli et al.. We will start by examining the molecular mechanics model for the copper substrate, calculating the surface energy and the melting point. We then go on to add the cyclohexaphenylene molecule, which is described quantum mechanically, and study its interaction with the Cu(111) surface. The Cu(111) substrate will be described by the embedded atom model (EAM), a computationally simple, yet effective model for metals. Read through the first two pages of the paper by Foiles et al. describing the embedded atom model and give a summary of the EAM in your own words. (2P) Cutting a solid into pieces breaks bonds and thus costs energy. The surface energy is the energy required to create a unit of surface area. One way to simulate a surface within periodic boundary conditions is the so-called slab geometry.A slab has an upper and a lower surface and the total energy $E(N)$ of the slab contains the energetic contributionsfrom both surfaces. If the slab is too thin, upper and lower surface may interact, causing their energetic contributions to deviate from the surface energy of a semi-infinite slab. One therefore increases the number of layers $N$ until the energy $\triangle E(N) = E(N) - E(N - 1)$ for adding one layer to the slab converges. At this point, the energy for adding one layer has reached its bulk value $\triangle E(\infty)$ and the surface energy $\sigma$ can be extracted via the formula$$\sigma =\lim_{N\rightarrow\infty} \frac{1}{2} \frac{1}{A} (E(N)-N \triangle E(N))$$where $A$ is the surface area. We have prepared a set of Cu(111) slabs with thicknesses ranging from 1 to 9 layers. Use the provided script ./run-slabs.sh to optimize their geometry with CP2K. run-slabs.sh to see what it does.You may need to adapt the name of the CP2K executable. analyze-slabs.sh. Another important parameter of a potential for a metal is its melting temperature $T_m$. The naive way of determining $T_m$ would be to set up a bulk crystal, perform MD simulations at different temperatures and note at which temperature the system melts and the crystal structure is destroyed. The problem with this approach is that the lack of any surfaces makes it very hard for the liquid phase to nucleate in a small cell and within a reasonable simulation time. In such a simulation, melting will typically be observed only at temperatures significantly above $T_m$. Here, we will use the so-called phase coexistence technique developed by Ercolessi et al..We work with a bulk cell with length $l_z$ significantly larger than $l_x$ and $l_y$ and periodic boundary conditions (shown above with z-axis along horizontal direction). In a first step, we have molten the upper half of the crystal, keeping the atoms in the lower half artificially fixed. Starting from the structure of the half-molten cell, perform an MD simulation at constant energy with an initial temperature $T_0$ somewhere near where you expect the melting temperature. equilibrate.in and define the initial temperature. What type of ensemble are we using? Why aren't we using the NVE ensemble? (2P) equilibrate.in is 1 ns, which may be longer than required. While the simulation is running, monitor the temperature and the atomic structure (see hint below) to find out when the system reaches equilibrium. .xyz file containing a single frame (and tell VMD the cell for this frame): vmd -f half-molten.xyz run-CU-EQUIL-pos-1.dcd pbc set {10.2247 17.7098 100.1818} -first 0 -last 0 pbc wrap -all Chemically inert surfaces can influence the reaction pathways of adsorbed molecules by van der Waals interactions, making the molecule more planar. This is one of the motivations for studying the adsorption of cyclohexaphenylene (CHP), a molecular precursor for graphene, on Cu(111). We have prepared an input file geo.in for performing a geometry optimization of cyclohexaphenylene on Cu(111) in the QM/MM approach.Since this file is a bit more complicated than usual, let's have a look at its overall structure: ################################### ## CHP (PM6) on Cu(111) (EAM) ### ################################### @SET LIBDIR ../cp2klib &GLOBAL PROJECT run-GEO RUN_TYPE GEO_OPT # we want to perform a geometry optimization PRINT_LEVEL LOW &END GLOBAL &MOTION # Describes what geometry optimization algorithm to use ... # and which quantities to plot while doing so. OPTIMIZER LBFGS # The algorithm is completely independent of how ... # the forces are actually computed. &END MOTION &MULTIPLE_FORCE_EVALS # notify CP2K that this input file contains multiple MULTIPLE_SUBSYS .TRUE. # FORCE_EVAL sections FORCE_EVAL_ORDER 2 3 &END ######################## ## How to mix QM & MM ## ######################## &FORCE_EVAL # The first FORCE_EVAL section only has an organizational role. ... # It defines that the total energy of the combined QM/MM system MIXING_FUNCTION E1+E2 # should be obtained as a sum of the QM energy (E1, from FORCE_EVAL 2) ... # and the MM energy (E2, from FORCE_EVAL 3). END FORCE_EVAL ####################### ## QM - CHP with PM6 ## ####################### &FORCE_EVAL # The second FORCE_EVAL is responsible for the quantum-mechanical ... # description of the CHP molecule END FORCE_EVAL ############################################### ## MM - Cu(111) (EAM) + interaction with CHP ## ############################################### &FORCE_EVAL # The third FORCE_EVAL section describes the MM model ... # for the Cu(111) slab and the MM model for the interaction of the END FORCE_EVAL # Cu(111) slab with the adsorbed molecule. Use geo.in to perform a geometry optimization of cyclohexaphenylene adsorbed on the Cu(111) surface. geo.in, we are using a less computationally demanding method to treat the QM part. (2P) geo.in and describe its overall structure. How is the coupling of QM and MM represented in the input file? geo.in and remove the sections that you don't need. Make sure that the .xyz files you use in the input only contain the atoms you need for the respective calculation. RUNTYPE ENERGY will calculate the total energy without optimizing the geometry. set mol [atomselect top "element C or element H"] # select molecule measure center $mol # calculate geometric mean of coordinates in selection set lay [atomselect top "element Cu and z > 12"] # select surface layer of slab
Harmonic Series is Divergent Contents Theorem The harmonic series: $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n = \underbrace 1_{s_0} + \underbrace {\frac 1 2 + \frac 1 3}_{s_1} + \underbrace {\frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 7}_{s_2} + \cdots$ where $\displaystyle s_k = \sum_{i \mathop = 2^k}^{2^{k + 1} \mathop - 1} \frac 1 i$ From Ordering of Reciprocals: $\forall m, n \in \N_{>0}: m < n: \dfrac 1 m > \dfrac 1 n$ so each of the summands in a given $s_k$ is greater than $\dfrac 1 {2^{k + 1} }$. The number of summands in a given $s_k$ is $2^{k + 1} - 2^k = 2 \times 2^k - 2^k = 2^k$, and so: $s_k > \dfrac{2^k} {2^{k + 1} } = \dfrac 1 2$ Hence the harmonic sum $H_{2^m}$ satisfies the following inequality: $\displaystyle H_{2^m}$ $=$ $\displaystyle \sum_{n \mathop = 1}^{2^m} \frac 1 n$ $\displaystyle $ $>$ $\displaystyle \sum_{n \mathop = 1}^{2^m - 1} \frac 1 n$ $\displaystyle $ $=$ $\displaystyle \sum_{k \mathop = 0}^m \left({s_k}\right)$ $\displaystyle $ $>$ $\displaystyle 1 + \sum_{a \mathop = 0}^m \frac 1 2$ $\displaystyle $ $=$ $\displaystyle 1 + \frac m 2$ The result follows from the the Comparison Test for Divergence. $\blacksquare$ Observe that all the terms of the harmonic series are strictly positive. Hence the Cauchy Condensation Test can be applied, and we examine the convergence of: $\displaystyle \sum_{n \mathop = 1}^\infty 2^n \frac 1 {2^n}$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty 1$ This diverges, from the $n$th term test. Hence $\displaystyle \sum \frac 1 n$ also diverges. $\blacksquare$ We have that the Integral of Reciprocal is Divergent. $\blacksquare$ For all $N \in \N$: $\dfrac 1 N + \dfrac 1 {N + 1} + \cdots + \dfrac 1 {2 N} > N \cdot \dfrac 1 {2 N} = \dfrac 1 2$ Hence, by Cauchy's Convergence Criterion for Series, the Harmonic series is divergent. $\blacksquare$ Assume that for $G \ge 4$ that $\displaystyle \sum_{n = G}^\infty \frac{1}{n} = L < \infty$. Namely, that for some number $G$, there is a tail of the harmonic series which converges. Then from Definition:Series/Sequence of Partial Sums: $s_N := \sum_{n = G}^N \frac{1}{n}$ is the partial sum of the above series. Which yields the sequence $\{ s_N \}$ of partial sums. And, from Definition:Convergent Series we have that $\displaystyle \sum_{n = G}^\infty \frac{1}{n}$ converges iff $\{ s_N\}$ converges. From Combination Theorem for Sequences/Normed Division Ring/Constant Rule : The constant sequence $\{ G \}$ has limit $G$. Note: $\R$ is a normed division ring as it is a field. By Combination Theorem for Sequences/Real/Product Rule: The product of the sequences $\{ G \}$ and $\{ s_N\}$ has limit $GL$. Namely, the sequence $\{Gs_N\}$ has limit $GL$, by the opening assumption. $GL = \displaystyle \sum_{n = G}^\infty \frac{G}{n} = \underbrace {1}_{s_0} + \underbrace{ \frac{G}{G+1} + \ldots + \frac{G}{G+4}}_{s_1} + \underbrace { \frac{G}{G+5} + \ldots + \frac{G}{G+12} }_{s_2} + \underbrace { \frac{G}{G+13} + \ldots + \frac{G}{G+28} }_{s_3} + \ldots $ Where $s_0 = 1, s_1 = \frac{G}{G+1} + \ldots + \frac{G}{G+4}$ and for $k \ge 2, s_k = \displaystyle \sum_{i = 2^k + 2^{k-1} + \ldots 2^2 + 1}^{2^{k+1} + 2^{k} + \ldots 2^2} \frac{G}{G+i} $. From the above, $s_0 = 1, s_1 \ge \frac{4G}{G+4} \ge 1$ by inspection. And for $k \ge 2$ then $s_k \ge \frac{2^{k+1}G}{G+2^{k+2}}$ since $\frac{G}{G+2^{k+2}}$ is smaller than the smallest summand of $s_k$. If summed $2^{k+1}$ many times, $2^{k+1}$ being the number of summands in $s_k$, it yields a result less than $s_k$. Note: The smallest summand of $s_k$ is $\frac{G}{G+2^{k+1}+ \ldots + 2^2}$. Claim: For $k \ge 1, G \ge 4$ We have $\frac{2^{k+1}G}{G+2^{k+2}} \ge 1$ Proof: $\frac{2^{k+1}G}{G+2^{k+2}} \ge 1 \iff k + 1 \ge \log_2\frac{G}{G-2}$. If $G = 4$ then the righthand side of the second inequality is $\log_2(2) = 1$. If $G > 4$, then $ 1 < \frac{G}{G-2} < 2$. Namely as $G \uparrow$ we have $\frac{G}{G-2} \to 1$ meaning $\log_2\frac{G}{G-2} \to 0$. $\Box$ Now, $\displaystyle GL = s_0 + s_1 + s_2 + s_3 + \ldots \ge 1 + \frac{4G}{G+4} + \frac{8G}{G+16 } + \frac{16G}{G+32} + \ldots \ge 1 + 1 + 1 + 1 + \ldots \to \infty > GL$ Deriving a contradiction. Hence , the series does not converge which implies the sequence $\{s_N\}$ does not converge. Therefore by Tail of Convergent Sequence : A sequence $a_n$ converges iff the sequence $a_{n+N}, N \in \N$ converges. We have the tail of the harmonic series diverges for any $G$ thus the harmonic series will diverge. $\blacksquare$ Also see However, it was lost for centuries, before being rediscovered by Pietro Mengoli in $1647$. Some sources attribute its rediscovery to Jacob Bernoulli. Sources 1986: David Wells: Curious and Interesting Numbers... (previous) ... (next): $23 \cdotp 103 \, 45 \ldots$ 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests 1992: George F. Simmons: Calculus Gems... (previous) ... (next): Chapter $\text {B}.19$: The Series $\sum 1/ p_n$ of the Reciprocals of the Primes 1997: David Wells: Curious and Interesting Numbers(2nd ed.) ... (previous) ... (next): $23 \cdotp 10345 \ldots$
Search Now showing items 1-1 of 1 Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (Elsevier, 2014-09) Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, \|y\| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
Background For a system consisting of two molecules (monomers or fragments are also used) X and Y, the binding energy is $$\Delta E_{\text{bind}} = E^{\ce{XY}}(\ce{XY}) - [E^{\ce{X}}(\ce{X}) + E^{\ce{Y}}(\ce{Y})]\label{eq:sherrill-1} \tag{Sherrill 1}$$ where the letters in the parentheses refer to the atoms present in the calculation and the letters in the superscript refer to the (atomic orbital, AO) basis present in the calculation. The first term is the energy calculated for the combined X + Y complex (the dimer) with basis functions, and the next two terms are energy calculations for each isolated monomer with only their respective basis functions. The remainder of this discussion will make more sense if the complex geometry is used for each monomer, rather than the isolated fragment geometry. The counterpoise-corrected (CP-corrected) binding energy [1] to correct for basis set superposition error (BSSE) [2] is defined as $$\Delta E_{\text{bind}}^{\text{CP}} = E^{\ce{XY}}(\ce{XY}) - [E^{\ce{XY}}(\ce{X}) + E^{\ce{XY}}(\ce{Y})]\label{eq:sherrill-3} \tag{Sherrill 3}$$ where the monomer calculations are now performed in the dimer/complex basis. Let's explicitly state how this works for the $E_{\ce{XY}}(\ce{X})$ term. The first molecule X contributes nuclei with charges, basis functions (AOs) centered on those nuclei, and electrons that will count to the final occupied molecular orbital (MO) index into the MO coefficient array. There is no reason why additional AOs that are not centered on atoms can't be added to a calculation. Depending on their spatial location, if they're close enough to have significant overlap, they may combine with atom-centered MOs, increasing the variational flexibility of the calculation and lowering the overall energy. Put another way, place the AOs that would correspond to molecule Y at their correct positions, but don't put the nuclei there, and don't consider the number of electrons they would contribute to the total number of occupied orbitals. This means that for the full electronic Hamiltonian $$\hat{H}_{\text{elec}} = \hat{T}_{e} + \hat{V}_{eN} + \hat{V}_{ee}$$ calculating the electron-nuclear attraction $\hat{V}_{eN}$ term is now different. Considered explicitly in matrix form in the AO basis, $$\begin{align}V_{\mu\nu} &= \int \mathop{d\mathbf{r}_{i}} \chi_{\mu}(\mathbf{r}_{i}) \left( \sum_{A}^{N_{\text{atoms}}} \frac{Z_{A}}{\|\mathbf{r}_{i} - \mathbf{R}_{A}\|} \right) \chi_{\nu}(\mathbf{r}_{i}) \\&=\sum_{A}^{N_{\text{atoms}}} Z_{A} \left< \chi_{\mu} \middle\| \frac{1}{r_{A}} \middle\| \chi_{\nu} \right>\end{align}$$ there are now fewer terms in the summation, since the nuclear charges from molecule Y are zero (the atoms just aren't there), but the number of $\mu\nu$ are the same as for the XY complex. This and the $\hat{T}_{e}, \hat{V}_{ee}$ terms aren't really mathematically or functionally different then; this is more to show where the additional basis functions enter, or to show where nuclei appear in the equations [3]. These atoms that don't have nuclei or electrons, only basis functions, are called ghost atoms. Sometimes you also see the term ghost functions, ghost basis, or ghost {something} calculation. Adding the basis of monomer Y to make the full "dimer basis" means taking the monomer X and including basis functions at the nuclear positions for Y. Geometry optimization Now to calculate the molecular gradient, that is, the derivative of the energy with respect to the $3N$ nuclear coordinates. This is the central quantity in any geometry optimization. For the sake of simplicity, consider a steepest descent-type update of the nuclear coordinates$$R_{A,x}^{(n+1)} = R_{A,x}^{(n)} - \alpha \frac{\partial E_{\text{total}}^{(n)}}{\partial R_{A,x}}\label{eq:steepest-descent} \tag{Steepest Descent}$$where $n$ is the optimization iteration number, $\alpha$ is some small step size with units [length 2][energy], and the last term is the derivative of the total (not just electronic) energy with respect to a change in atom $A$'s $x$-coordinate. Even Newton-Raphson-type updates with approximate Hessians (2nd derivative of the energy with respect to nuclear coordinates, rather than the 1st) need the gradient, so we must formulate it. Formulation of the energy We're in a bit of trouble, because we want to replace $E_{\text{total}}$ in the gradient with $E_{\text{total}}^{\text{CP}}$, but all we have is $\Delta E_{\text{bind}}^{\text{CP}}$. The concept of CP correction can still be applied to a total energy, but the BSSE must be removed from each monomer. The BSSE correction itself for each monomer is$$\begin{split}E_{\text{BSSE}}(\ce{X}) &= E^{\ce{XY}}(\ce{X}) - E^{\ce{X}}(\ce{X}), \\E_{\text{BSSE}}(\ce{Y}) &= E^{\ce{XY}}(\ce{Y}) - E^{\ce{Y}}(\ce{Y}),\end{split}\label{eq:2}$$which, when subtracted from $\eqref{eq:sherrill-1}$, gives $\eqref{eq:sherrill-3}$. More correctly, considering that the geometry for each step is at the final cluster geometry and not the isolated geometry, the above is [4]$$\begin{split}E_{\text{BSSE}}(\ce{X}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{X}) - E_{\ce{XY}}^{\ce{X}}(\ce{X}), \\E_{\text{BSSE}}(\ce{Y}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{Y}) - E_{\ce{XY}}^{\ce{Y}}(\ce{Y}).\end{split}\label{eq:sherrill-10} \tag{Sherrill 10}$$ The CP-corrected total energy is the full dimer energy with BSSE removed from each monomer is then$$\begin{split}E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} &= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - E_{\text{BSSE}}(\ce{X}) - E_{\text{BSSE}}(\ce{Y}), \\&= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X}) - E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X}) \right] - \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y}) - E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y}) \right].\end{split}\label{eq:sherrill-15} \tag{Sherrill 15}$$Note that I have modified which geometry is used for each monomer in $\eqref{eq:sherrill-15}$. All monomers are calculated at the supermolecule geometry. This is convenient for two reasons: 1. We are only interested in removing the BSSE, not the effect of monomer deformation, and 2. a isolated monomer geometry without deformation doesn't make sense in the context of a geometry optimization. I also added the tilde to signify that the supermolecular/dimer geometry used may not be the final or minimum-energy geometry, as would be the case during a geometry optimization. We simply extract all structures consistently from a given geometry iteration. Perhaps $\ce{XY}(n)$ would be better notation. Formulation of the gradient As Pedro correctly states, the differentiation operator is a linear operator. Because there are no products in $\eqref{eq:sherrill-15}$, the total gradient needed for $\eqref{eq:steepest-descent}$ will be a sum of gradients [5]:$$\frac{\partial E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}}}{\partial R_{A,x}} = \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY})}{\partial R_{A,x}} - \left[ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X})}{\partial R_{A,x}} \right] - \left[ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y})}{\partial R_{A,x}} \right],$$so each step of a CP-corrected geometry optimization will require 5 gradient calculations rather than 1. Note that the nuclear gradient should be included for each term as well, which is a trivial calculation. Extension to other molecular properties Although not commonly done, counterpoise correction can be applied to any molecular property, not just energies or gradients. Simply replace $E$ or $\partial E/\partial R$ with the property of interest. For example, the CP-corrected polarizability $\alpha$ of two fragments is$$\alpha_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} = \alpha_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - \left[ \alpha_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X}) - \alpha_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X}) \right] - \left[ \alpha_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y}) - \alpha_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y}) \right]$$where I believe it now makes even less sense to have each individual fragment calculation not be at the cluster geometry. In papers that calculate CP-corrected properties, no mention is usually made of which geometry the individual calculations are performed at for this reason. References Boys, S. Francis; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553-566. Sherrill, C. David. Counterpoise Correction and Basis Set Superposition Error. 2010, 1-6. One implementation note: Most common quantum chemistry packages should allow for the usage of ghost atoms in energy and gradient calculations. However, as Sherrill states, they do not properly allow for composing the full gradient expression to perform CP-corrected geometry optimizations. Gaussian can, and Psi4 may. For programs that can calculate gradients with ghost atoms, Cuby can be used to drive the calculation of CP-corrected geometries and frequencies. There is a typo in the Sherrill paper; the subscripts for all 4 energy terms should be $AB$, which here are $\ce{XY}$. Simon, S.; Bertran, J.; Sodupe, M. Effect of Counterpoise Correction on the Geometries and Vibrational Frequencies of Hydrogen Bonded Systems. J. Chem. Phys. A 2001, 105,, 4359-4364.
In Feynman's Statistical Mechanics - A Set of Lectures, upon the introduction of the path integral, a series of approximations are made in order to calculate integrals. I am unsure how exactly to get to the following important approximation. Section 3.1 Path Integral Formulation of the Density Matrix: For low $\epsilon$, because $\rho_\text{free}$ is a very localized Gaussian, most of the contribution in the integral over $x''$ occurs near $x''=x_0$ with $$x_0=\frac{ux+(\epsilon-u)x'}{\epsilon}$$ So we can, for small $\epsilon$ , write $$\rho(x,x';\epsilon)\approx -\int\limits_{0}^{\epsilon}V(x_0)\underbrace{\rho_\text{free}(x,x';\epsilon)}_{…?}\ \text{d}u$$ with the density matrix of the free particle $$\rho_0(x,x',\epsilon)=\rho_{\text{free}}(x,x',\epsilon)=\sqrt{\frac{m}{2m\hbar \epsilon}}\exp\bigg(\frac{-m}{2\hbar \epsilon}(x-x')^2\bigg)$$ and the integral in question $$\rho(x,x';\epsilon)=-\int\limits_{0}^{\epsilon}\int\limits_{-\infty}^{+\infty}\rho_0(x,x'';\epsilon-u)V(x'')\rho_0(x'',x';u) \ \text{d}u\text{d}x''$$ I thought, that with $$ \int\limits_{-\infty}^{+\infty}\rho_0(x,x'';\epsilon-u)\rho_0(x'',x';u) \ \text{d}x''=\int\limits_{-\infty}^{+\infty}<x\|\rho_0(\epsilon-u)\underbrace{\|x''><x''\|}_{\mathbb{1}}\rho_0(u)\|x'> \ \text{d}x''$$ I might get further, that still leaves me with $$ \rho_0(\epsilon-u)\cdot\rho_0(u)$$
This shows you the differences between two versions of the page. Both sides previous revision Previous revision gibson:teaching:spring-2018:math445:lab11 [2018/05/01 06:20] gibson gibson:teaching:spring-2018:math445:lab11 [2018/05/01 06:23] (current) gibson Line 38: Line 38: - Problem 2: Determine the minimum initial ball speed and optimal angle that result in a home run, at sea level, and in Denver. You'll have to start with a guesses for $v_0$ and $\theta$ and tweak them by stages. For a starting point, recall that a good fastball clocks at 90 mph or roughly 40 m/s, and that 45 degrees is $\theta =\pi/4 \approx 0. 78$. + Problem 2: Determine the minimum initial ball speed and optimal angle that result in a home run, at sea level, and in Denver. You'll have to start with a guesses for $v_0$ and $\theta$ and tweak them by stages. For a starting point, recall that a good fastball clocks at 90 mph or roughly 40 m/s, and that the optimal angle is $\theta = \pi/4 \approx 0. 79$ when there is no air resistance. Note that Matlab's ode45 function will return the x,y positions of the trajectory points at discrete time intervals, and it's unlikely that any of these will line up exactly with the outfield fence. However you can use interpolation to get the ball height y at exactly at the fence, as follows. If you set up your Matlab code so that $x$ is //x(:,1)// and $y$ is //x(:,2)//, the following code will determine the height $y$ of the ball at the position of the fence, $x=120$. Note that Matlab's ode45 function will return the x,y positions of the trajectory points at discrete time intervals, and it's unlikely that any of these will line up exactly with the outfield fence. However you can use interpolation to get the ball height y at exactly at the fence, as follows. If you set up your Matlab code so that $x$ is //x(:,1)// and $y$ is //x(:,2)//, the following code will determine the height $y$ of the ball at the position of the fence, $x=120$.

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