Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L₁-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L₁-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.      

Cwikel and Quasi-Szegö Type Estimates for Random Operators

We consider Schrödinger operators with nonergodic random potentials. Specifically, we are interested in eigenvalue estimates and estimates of the entropy for the absolutely continuous part of the spectral measure. We prove that increasing oscillations in the potential at infinity have the same effect on the properties of the spectrum as the decay of the potential.     

Admissibility for Nonhomogeneous Linear Estimates on Multivariate Random Regression Coefficients and Parameters

61..IntroductionLetA,BaretwomatriceslthesignA>odenotesthatAisanon-negativedefinitesquarematrixFA>BdenotesA-B>OIA>odenotesthatAisaPoSitivedefinitesquarematrix;A>BdenotesA-B>OFp(A)denotesthelinearspaceextendedbycolumnvectorsofAFA denoteSMcore-PenrosegeneralinversionofA,i.e.A satisfies:AA A=A,A AA =A ,A AandAA botharesymmetriclA-denotesthesignu-"generalinversionofA,i.e.A-satisfies:AA-A=A3Adenotestherow-drawingstraightvectorofA;A@BdenotestheKrcrneckerpreductofAandB3tr(A)denot…     

Error Estimates for Rayleigh–Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation

Error estimates are derived for the computation of eigenvalues and eigenvectors of infinite tridiagonal matrices by the Rayleigh–Ritz method. The results are applied to the Mathieu and spheroidal wave equation.     

The Error Estimates of Semi-discrete Galerkin Methods for some Non-linear Parabolic Equations

1. Introduction Concerning the finite element analysis for linear and nonlinear parabolic equa tions there are a lot of papers, however, only a few of them devoted to the parabolic problems with mixed boundary conditions. In[1],[2] and [3], the finite element methods for some non-linear parabolic problems are systematically considered, but they are mainly restriced to the cases in which the boundary conditions are of Dirichlet     

On Kolmogorov SLLN Under Rearrangements for “Orthogonal” Random Variables in a B-Space

It is well-known that the Kolmogorov SLLN (non-i.i.d. case) fails for pair-wise independent random variables. However, as shown in the paper it can be saved even for orthogonal random variables if one allows permutations. We prove it in the setup of Banach space valued random variables.     

Schauder Estimates for Elliptic and Parabolic Equations

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.     

Energy Error Estimates for the Projection-Difference Method with the Crank–Nicolson Scheme for Parabolic Equations

We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.     

The Elastic–Plastic Torsion Problem: A Posteriori Error Estimates for Approximate Solutions

We consider the variational inequality that describes the torsion problem for a long elasto-plastic bar. Using duality methods of the variational calculus, we derive a posteriori estimates of functional type that provide computable and guaranteed upper bounds of the energy norm of the difference between the exact solution and any function from the corresponding energy space that satisfies the Dirichlet boundary condition.     

Numerical Solution of Riemann–Hilbert Problems: Random Matrix Theory and Orthogonal Polynomials

Recently, a general approach to solving Riemann–Hilbert problems numerically has been developed. We review this numerical framework and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with the approach of Bornemann to compute Fredholm determinants, we are able to calculate spectral densities and gap statistics for a broad class of finite-dimensional unitary invariant ensembles. We show that the accuracy of the numerical algorithm for approximating orthogonal polynomials is uniform as the degree grows, extending the existing theory to handle g-functions. As another example, we compute the Hastings–McLeod solution of the homogeneous Painlevé II equation.     

Estimates for Wallis’ ratio and related functions

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function.     

Trial and Error

Trial and error is a problem-solving strategy everyone uses at one time or another.In trial and error,you try ananswer.(The word trial comes from try.)If the answer is in error,you try something else.You keep trying untilyou get a correct answer.Trial and error is a particularly good strategy if a question has only a few possible answers.     

Error Estimates for the Velocity Tracking Problem for Navier–Stokes Flows Based on the Artificial Compressibility Formulation

ABSTRACT

The velocity tracking problem for Navier–Stokes flows is studied based on the artificial compressibility formulation. Several results concerning the analysis and convergence as ? → 0 are presented. Symmetric error estimates for semidiscrete finite element approximations for the artificially compressible optimality system are derived.     

Matching based very large-scale neighborhoods for?parallel machine scheduling

In this paper we study very large-scale neighborhoods for the minimum total weighted completion time problem on parallel machines, which is known to be strongly $\mathcal{NP}$ -hard. We develop two different ideas leading to very large-scale neighborhoods in which the best improving neighbor can be determined by calculating a weighted matching. The first neighborhood is introduced in a general fashion using combined operations of a basic neighborhood. Several examples for basic neighborhoods are given. The second approach is based on a partitioning of the job sets on the machines and a reassignment of them. In a computational study we evaluate the possibilities and the limitations of the presented very large-scale neighborhoods.     

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

A Reilly Formula and Eigenvalue Estimates for Differential Forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally, we also obtain, as a byproduct of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.     

On Weyl’s Asymptotics and Remainder Term for the Orthogonal and Unitary Groups

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups \(\mathbf{SO}(N)\), \(\mathbf{SU}(N)\), \(\mathbf{U}(N)\) and \(\mathbf{Spin}(N)\) it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for \(\mathbf{U}(N)\)]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.     

Estimates for Green's Functions ofElliptic Equations in Non-DivergenceForm with Continuous Coe?cientsSeick Kim?and Sungjin Lee

正(1.1)where the coe?cient A:=(aij) are symmetric and satisfy the uniform ellipticity condition. Namely,aij=aji,λ|ξ|2≤aij(x)ξiξj≤Λ|ξ|2,(1.2)     

Strong Consistency for the Kernal Estimates of the Random Window Width of the Density Function and its Derivatives Under Φ—Mixing Samples

In the paper,we study the strong uniform consistency for the kernal estimates of random window width of density function and its derivatives under the condition that the sequence｛Xa｝of the sample are the identically Φ-mixing random variables.