A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in . This orthonormal system is a generalization of the classical exponential system defining Fourier series.

     

Decay rates of Volterra equations on ℝ N

This note is concerned with the linear Volterra equation of hyperbolic type

Degenerate Nonlinear Elliptic Equations Lacking in Compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|~(p-2)u) in R~N, where ▽_pu =|▽u|~(p-2)▽u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.     

INVERSE PROBLEMS FOR A GENERAL MULTI－CONNECTED BOUNDED DRUM WITH APPLICATIONS IN PHYSICS

this paper studies the influence of a finite container on an ideal gas,The trace of the heat kernel Θ(t)=∑(μ=1)^∞ exp(-tλμ），where{λμ}(μ=1)^∞ are the eigenvalues of the negative Laplacian-△n=-∑(p=1)^n (a/axp)^2 in R^n(n=2 or 3) ,is studied for a general mult-connected bounded drum Ω which is surrounded by simply connected bounded domains Ωi with smooth boundaries aΩi(i=1,……,m) where the Dirichlet ,Neumann and Robin boundary Conditions on aΩi(i=1,……,m) are considered.Some geometrical properties of Ω are determined ,The theremodynamic quantities for an ideal gas encolosed in Ω are examined by using the asymptotic expansions of Θ(t) for short-sime t.It is shown that the ideal gas can not feel the shape of its container Ω，althought it can feel some geometrical properties of it.     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05     

Error analysis of collocation method based on reproducing kernel approximation

Solving partial differential equations (PDE) with strong form collocation and nonlocal approximation functions such as orthogonal polynomials, trigonometric functions, and radial basis functions exhibits exponential convergence rates; however, it yields a full matrix and suffers from ill conditioning. In this work, we discuss a reproducing kernel collocation method, where the reproducing kernel (RK) shape functions with compact support are used as approximation functions. This approach offers algebraic convergence rate, but the method is stable like the finite element method. We provide mathematical results consisting of the optimal error estimation, upper bound of condition number, and the desirable relationship between the number of nodal points and the number of collocation points. We show that using RK shape function for collocation of strong form, the degree of polynomial basis functions has to be larger than one for convergence, which is different from the condition for weak formulation. Numerical results are also presented to validate the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 554–580, 2011     

Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems

In this paper, the complex variable reproducing kernel particle (CVRKP) method and the finite element (FE) method are combined as the CVRKP-FE method to solve transient heat conduction problems. The CVRKP-FE method not only conveniently imposes the essential boundary conditions, but also exploits the advantages of the individual methods while avoiding their disadvantages, then the computational efficiency is higher. A hybrid approximation function is applied to combine the CVRKP method with the FE method, and the traditional difference method for two-point boundary value problems is selected as the time discretization scheme. The corresponding formulations of the CVRKP-FE method are presented in detail. Several selected numerical examples of the transient heat conduction problems are presented to illustrate the performance of the CVRKP-FE method.     

Analysis of variable coefficient advection-diffusion problems via complex variable reproducing kernel particle method

The complex variable reproducing kernel particle method （CVRKPM） of solving two-dimensional variable coefficient advection-diffusion problems is presented in this paper. The advantage of the CVRKPM is that the shape function of a two-dimensional problem is formed with a one-dimensional basis function. The Galerkin weak form is employed to obtain the discretized system equation, and the penalty method is used to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional variable coefficient advection-diffusion problems are obtained. Two numerical examples are given to show that the method in this paper has greater accuracy and computational efficiency than the conventional meshless method such as reproducing the kernel particle method （RKPM） and the element- free Galerkin （EFG） method.     

Information Flows of Diverse Autoencoders

Deep learning methods have had outstanding performances in various fields. A fundamental query is why they are so effective. Information theory provides a potential answer by interpreting the learning process as the information transmission and compression of data. The information flows can be visualized on the information plane of the mutual information among the input, hidden, and output layers. In this study, we examine how the information flows are shaped by the network parameters, such as depth, sparsity, weight constraints, and hidden representations. Here, we adopt autoencoders as models of deep learning, because (i) they have clear guidelines for their information flows, and (ii) they have various species, such as vanilla, sparse, tied, variational, and label autoencoders. We measured their information flows using Rényi’s matrix-based α-order entropy functional. As learning progresses, they show a typical fitting phase where the amounts of input-to-hidden and hidden-to-output mutual information both increase. In the last stage of learning, however, some autoencoders show a simplifying phase, previously called the “compression phase”, where input-to-hidden mutual information diminishes. In particular, the sparsity regularization of hidden activities amplifies the simplifying phase. However, tied, variational, and label autoencoders do not have a simplifying phase. Nevertheless, all autoencoders have similar reconstruction errors for training and test data. Thus, the simplifying phase does not seem to be necessary for the generalization of learning.     

Monotonicity properties of the Neumann heat kernel in the ball

A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U⊂Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball.