An inverse cavity problem for Maxwellʼs equations

Consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity embedded in a perfect electrically conducting infinite ground plane, where the electromagnetic wave propagation is governed by the Maxwell equations. The upper half-space is filled with a lossless homogeneous medium above the flat ground surface; while the interior of the cavity is assumed to be filled with a lossy homogeneous medium accounting for the energy absorption. The inverse problem is to determine the cavity structure or the shape of the cavity from the tangential trace of the electric field measured on the aperture of the cavity. In this paper, results on a global uniqueness and a local stability are established for the inverse problem. A crucial step in the proof of the stability is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the cavity.     

An improved algorithm for the $$L_2$$ – $$L_p$$ minimization problem

Mathematical Programming - In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the $$L_2$$ – $$L_p$$ minimization problem. We propose an...     

On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

Nakajimaʼs remark on Hennʼs proof

We fill up a gap in Hennʼs proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

An approximation algorithm for quadratic dynamic systems based on N. Chomsky’s grammar for Taylor’s formula

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.     

An exact algorithm for the type-constrained and variable sized bin packing problem

In this paper, we introduce an additional constraint to the one-dimensional variable sized bin packing problem. Practically, some of items have to be packed separately in different bins due to their specific requirement. Therefore, these items are labelled as different types. The bins can be used to pack either any type of items if they are empty originally or the same type of items as what they already have. We model the problem as a type-constrained and variable sized bin packing problem (TVSBPP), and solve it via a branch and bound method. An efficient backtracking procedure is proposed to improve the efficiency of the algorithm.     

An infinite-dimensional generalization of Zengerʼs lemma

We prove an infinite-dimensional generalization of Zenger?s lemma that was used in the proof of the fact that the convex hull of the point spectrum of a linear operator is contained in its numerical range. Two relevant examples are given, and possible application in the Arrow–Debreu model is also discussed.     

An improvement on Waring?CGoldbach problem for unlike powers

Let p _i be prime numbers. In this paper, it is proved that for any integer k?R5, with at most $O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)$ exceptions, all positive even integers up to N can be expressed in the form $p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}$ . This improves the result $O\big(\frac{N}{\log^{c}N}\big)$ for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1?C4] for the problem in the form $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ . This method can also be used for some other similar forms.     

On a problem for Wardʼs equation with a Mittag–Leffler potential

We solve a problem for a type of non-linear partial differential equation (“Ward?s equation”). This is an equation arising naturally in the study of Coulomb gases and random normal matrix ensembles [4]. In this paper, we consider a problem for Ward?s equation whose solutions are precisely the well-known Mittag–Leffler functions. Our solution to this problem generalizes certain results obtained in [4].     

Approximation formulas for Landauʼs constants

In this paper, we establish approximation formulas for evaluating Landau?s constants.