<Emphasis Type="Italic">C</Emphasis><Superscript>1,<Emphasis Type="Italic">α</Emphasis></Superscript> Theory for the Prescribed Mean Curvature Equation with Dirichlet Data

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. With such a solution we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a C ^1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C ^1,α .     

Maximum-cover source location problems with objective edge-connectivity three

Given a graph G = (V, E), a set of vertices covers a vertex if the edge-connectivity between S and v is at least a given number k. Vertices in S are called sources. The maximum-cover source location problem, which is a variation of the source location problem, is to find a source set S with a given size at most p, maximizing the sum of the weight of vertices covered by S. In this paper, we show a polynomial-time algorithm for this problem in the case of k = 3 for a given undirected graph with a vertex weight function and an edge capacity function. Moreover, we show that this problem is NP-hard even if vertex weights and edge capacities are both uniform for general k.     

On the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in \mathbbR^N{\mathbb{R}^N} of given volume. When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k ≥ 3, we prove that in many cases a minimiser cannot be independent of the value of the constant in the boundary condition, or equivalently of the domain’s volume. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions.     

A quasi‐boundary value regularization method for determining the heat source

In this paper, we consider the inverse problem of determining the heat source, which depends only on spatial variable in one‐dimensional heat equation in a bounded domain where data is given at some fixed time. A conditional stability result is given, and a quasi‐boundary value regularization method is also provided. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable. Copyright © 2013 John Wiley & Sons, Ltd.     

A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in _boxclose^d{\mathbb{R}^{d}} and 1 < q < ∞, the solvability of the L ^q Dirichlet problem for Δ ² u = 0 in Ω with boundary data in WA ^1,q (∂Ω) is equivalent to that of the L ^p regularity problem for Δ ² u = 0 in Ω with boundary data in WA ^2,p (∂Ω), where \frac1p + \frac1q=1{\frac{1}{p} + \frac{1}{q}=1}. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L ^p regularity problem for d ≥ 4 and p in certain ranges.     

Difference operators as semigroup generators

In this paper we examine difference operators with constant coefficients. We show that the type of the generated semigroup is determined by a matrix \mathbbB\mathbb{B}, originating from the domain of the operator. Moreover, we provide necessary and sufficient conditions for exponential and polynomial stability of the semigroup in terms of the matrix \mathbbB\mathbb{B}, using results of A. Borichev and Y. Tomilov. We close the paper with an application of our results to flows in networks.     

Flow and shape reconstructions from remote measurements

We develop a point source method (PSM) to obtain flow field reconstructions from remote measurements. The PSM belongs to the class of decomposition methods in inverse scattering because it solves the nonlinear and ill‐posed inverse shape reconstruction problem by a decomposition into a linear ill‐posed problem and a nonlinear well‐posed problem. As a model problem, we investigate the reconstruction of the flow field of two‐dimensional stationary Oseen equation, which is obtained by linearizing the Navier–Stokes equation with kinematic viscosity μ > 0 around the constant velocity u₀. In contrast to acoustics or electromagnetics, the use of the PSM in fluid dynamics leads to a number of challenges in terms of the analysis and the proper setup of the scheme, in particular, because the null‐spaces of the integral operators under consideration are no longer trivial and the fundamental solution is not symmetric in its spatial coordinate. We provide a suitable formulation of the method and prove convergence of flow reconstructions by the PSM. For the realization of the reconstruction when the inclusions are not known, we employ domain sampling. We will demonstrate the feasibility of the method for reconstructing one or several objects by numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

An Ensemble Kalman Filter Approach Based on Level Set Parameterization for Acoustic Source Identification Using Multiple Frequency Information

In this paper, a reconstruction problem of the spatial dependent acoustic source from multiple frequency data is discussed. Suppose that the source function is supported on a bounded domain and the piecewise constant intensities of the source are known on the support. We characterize unknown domain by the level set technique. And the level set function can be modeled by a Hamilton-Jacobi system. We use the ensemble Kalman filter approach to analyze the system state. This method can avoid to deal with the nonlinearity directly and reduce the computation complexity. In addition, the algorithm can achieve the stable state quickly with the Hamilton-Jacobi system. From some numerical examples, we show these advantages and verify the feasibility and effectiveness.     

Identification of Refined ARMA Echelon Form Models for Multivariate Time Series

In the present article, we are interested in the identification of canonical ARMA echelon form models represented in a “refined” form. An identification procedure for such models is given by Tsay (J. Time Ser. Anal.10(1989), 357-372). This procedure is based on the theory of canonical analysis. We propose an alternative procedure which does not rely on this theory. We show initially that an examination of the linear dependency structure of the rows of the Hankel matrix of correlations, with originkin (i.e., with correlation at lagkin position (1, 1)), allows us not only to identify the Kronecker indicesn₁, …, n_d, whenk=1, but also to determine the autoregressive ordersp₁, …, p_d, as well as the moving average ordersq₁, …, q_dof the ARMA echelon form model by settingk>1 andk<1, respectively. Successive test procedures for the identification of the structural parametersn_i,p_i, andq_iare then presented. We show, under the corresponding null hypotheses, that the test statistics employed asymptotically follow chi-square distributions. Furthermore, under the alternative hypothesis, these statistics are unbounded in probability and are of the formNδ{1+o_p(1)}, whereδis a positive constant andNdenotes the number of observations. Finally, the behaviour of the proposed identification procedure is illustrated with a simulated series from a given ARMA model.     

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω ² u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.     

Packing congruent hyperspheres into a hypersphere

The paper considers a problem of packing the maximal number of congruent nD hyperspheres of given radius into a larger nD hypersphere of given radius where n = 2, 3, . . . , 24. Solving the problem is reduced to solving a sequence of packing subproblems provided that radii of hyperspheres are variable. Mathematical models of the subproblems are constructed. Characteristics of the mathematical models are investigated. On the ground of the characteristics we offer a solution approach. For n ≤ 3 starting points are generated either in accordance with the lattice packing of circles and spheres or in a random way. For n > 3 starting points are generated in a random way. A procedure of perturbation of lattice packings is applied to improve convergence. We use the Zoutendijk feasible direction method to search for local maxima of the subproblems. To compute an approximation to a global maximum of the problem we realize a non-exhaustive search of local maxima. Our results are compared with the benchmark results for n = 2. A number of numerical results for 2 ≤ n ≤ 24 are given.     

Blind source separation with nonlinear autocorrelation and non-Gaussianity

Blind source separation (BSS) is a problem that is often encountered in many applications, such as biomedical signal processing and analysis, speech and image processing, wireless telecommunication systems, data mining, sonar, radar enhancement, etc. One often solves the BSS problem by using the statistical properties of original sources, e.g., non-Gaussianity or time-structure information. Nevertheless, real-life mixtures are likely to contain both non-Gaussianity and time-structure information sources, rendering the algorithms using only one statistical property fail. In this paper, we address the BSS problem when source signals have non-Gaussianity and temporal structure with nonlinear autocorrelation. Based on the two statistical characteristics of sources, we develop an objective function. Maximizing the objective function, we propose a gradient ascent source separation algorithm. Furthermore, We give some mathematical properties for the algorithm. Computer simulations for sources with square temporal autocorrelation and non-Gaussianity illustrate the efficiency of the proposed approach.     

Graded central polynomials for the matrix algebra of order two

Let K be an infinite integral domain, and let A = M ₂(K) be the matrix algebra of order two over K. The algebra A can be given a natural \mathbbZ₂{\mathbb{Z}_2} -grading by assuming that the diagonal matrices are the 0-component while the off-diagonal ones form the 1-component. In this paper we study the graded identities and the graded central polynomials of A. We exhibit finite bases for these graded identities and central polynomials. It turns out that the behavior of the graded identities and central polynomials in the case under consideration is much like that in the case when K is an infinite field of characteristic 0 or p > 2. Our proofs are characteristic-free so they work when K is an infinite field, char K = 2. Thus we describe finite bases of the graded identities and graded central polynomials for M ₂(K) in this case as well.     

Unique Maximum Matching Algorithms

We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exists, in O(m log⁴ n) time. This algorithm uses a recent dynamic connectivity algorithm and an old result of Kotzig characterizing unique perfect matchings in terms of bridges. For the special case of planar graphs, we improve the algorithm to run in O(n log n) time. Second, given one perfect matching, we can test for the existence of another in linear time. This algorithm is a modification of Edmonds' blossom-shrinking algorithm implemented using depth-first search. A generalization of Kotzig's theorem proved by Jackson and Whitty allows us to give a modification of the first algorithm that tests whether a given graph has a unique f-factor, and find it if it exists. We also show how to modify the second algorithm to check whether a given f-factor is unique. Both extensions have the same time bounds as their perfect matching counterparts. For the weighted case, we can test in linear time whether a maximum-weight matching is unique, given the output from Edmonds' algorithm for computing such a matching. The method is an extension of our algorithm for the unweighted case.     

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s (d + 1 ) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Padé approximation in matrix form is given.     

The S′-convolution with singular kernels in the Euclidean case and the product domain case

We characterize those tempered distributions which are S′-convolvable with a given class of singular convolution kernels. We study both, the Euclidean case and the product domain case. In the Euclidean case, we consider a class of kernels that includes Riesz kernels, Calderón–Zygmund singular convolution kernels, finite part distributions defined by hypersingular convolution kernels, and Hörmander multipliers. In the product domain case, we consider a class of singular kernels introduced by Fefferman and Stein as a generalization of the n-dimensional Hilbert kernel.     

On mechanical waves and Doppler shifts from moving boundaries

We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, X_s(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source co‐located with a moving boundary x=X_s(t), where X_s(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple‐scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed‐form leading‐order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) for non‐uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the non‐uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.     

Colorful Strips

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k − 1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k − 1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.     

A simplified Tikhonov regularization method for determining the heat source

This paper is to discuss the inverse problem of determining a spacewise dependent heat source in one-dimensional heat equation in a bounded domain where data is given at some fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The regularization solution is given by a simplified Tikhonov regularization. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable.     

A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x|  ≥  1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x|  ≤  1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.