On the one-dimensional scattering of plane waves on an arbitrary potential

Theoretical and Mathematical Physics -     

The perfectly Matchable Subgraph Polytope of an arbitrary graph

The Perfectly Matchable Subgraph Polytope of a graphG=(V, E), denoted byPMS(G), is the convex hull of the incidence vectors of thoseXV which induce a subgraph having a perfect matching. We describe a linear system whose solution set isPMS(G), for a general (nonbipartite) graphG. We show how it can be derived via a projection technique from Edmonds' characterization of the matching polytope ofG. We also show that this system can be deduced from the earlier bipartite case [2], by using the Edmonds-Gallai structure theorem. Finally, we characterize which inequalities are facet inducing forPMS(G), and hence essential.     

On the distance function of a connected graph

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.     

On the spectrum of an infinite graph

We exhibit an example of an infinite locally finite graph such that its adjacency operator is not self-adjoint. This gives a negative answer to a conjecture of Mohar [1].     

Making an H $H$-free graph k $k$-colorable

We study the following question: How few edges can we delete from any $H$-free graph on $n$ vertices to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For $H$ any fixed odd cycle, we determine the answer up to a constant factor when $n$ is sufficiently large. We also prove an upper bound when $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.     

On the Toeplitz embedding of an arbitrary matrix

It has been shown by Delosme and Morf that an arbitrary block matrix can be embedded into a block Toeplitz matrix; the dimension of this embedding depends on the complexity of the matrix structure compared to the block Toeplitz structure. Due to the special form of the embedding matrix, the algebra of matrix polynomials relative to block Toeplitz matrices can be interpreted directly in terms of the original matrix and therefore can be extended to arbitrary matrices. In fact, these polynomials turn out to provide an appropriate framework for the recently proposed generalized Levinson algorithm solving the general matrix inversion problem.     

On the Laplacian spectrum of an infinite graph

In this paper, we introduce the notion of Laplacian spectrum of an infinite countable graph in a different way than in the papers by B. Mohar. We prove some basic properties of this type of spectrum. The approach used is in line with our approach to the limiting spectrum of an infinite graph. The technique of the Laplacian spectrum of finite graphs is essential in this approach.     

On the spectral density function of the Laplacian of a graph

Let X$X$ be a finite graph. Let |V|$\left|V\right|$ be the number of its vertices and d$d$ be its degree. Denote by _F1(X)$F_1\left(X\right)$ its first spectral density function which counts the number of eigenvalues ≤λ²$\le {\lambda }^2$ of the associated Laplace operator. We provide an elementary proof for the estimate _F1(X)(λ)−_F1(X)(0)≤2⋅(|V|−1)⋅d⋅λ$F_1\left(X\right)\left(\lambda \right)-F_1\left(X\right)\left(0\right)\le 2\cdot (\left|V\right|-1)\cdot d\cdot \lambda$ for 0≤λ<1$0\le \lambda <1$ which has already been proved by Friedman (1996) [3] before. We explain how this gives evidence for conjectures about approximating Fuglede–Kadison determinants and L²$L^2$-torsion.     

On stable refinable function vectors with arbitrary support

Refinable function vectors with arbitrary support are considered. In particular, necessary conditions for stability are given and a characterization of the symbol associated with a stable refinable function vector in terms of the transfer operator is provided: this is a generalization of Gundy’s theorem to the vector case. The proof adapts the tools provided in [S. Saliani, On stability and orthogonality of refinable functions, Appl. Comput. Harmon. Anal. 21 (2006) 254–261]. Though complications arise from noncommuting matrix products, the fundamental ideas are the same.     

On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution

In this paper, we consider the Sparre Andersen risk model with an arbitrary interclaim time distribution and a fairly general class of distributions for the claim sizes. Via a two-step procedure which involves a combination of a probabilitic and an analytic argument, an explicit expression is derived for the Gerber-Shiu discounted penalty function, subject to some restrictions on its form. A special case of Sparre Andersen risk models is then further analyzed, whereby the claim sizes’ distribution is assumed to be a mixture of exponentials. Finally, a numerical example follows to determine the impact on various ruin related quantities of assuming a heavy-tail distribution for the interclaim times.     

Higher order optimality conditions with an arbitrary non-differentiable function

In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Fréchet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.     

On the Schwarz-Neumann method with an arbitrary number of domains

In this paper, a generalization of the Schwarz–Neumannmethod to more than two domains is given. We prove the convergenceand the numerical stability of the algorithm. The results applyto both bounded and unbounded domains, and are given for theweak solution of an elliptic problem with mixed boundary conditions.Numerical results are given for both bounded and unbounded domains.     

A slender dielectric body embedded in an arbitrary external potential

This paper determines the electrostatic potential and field taking place both inside and outside a slender dielectric body embedded in a given potential ₀. This task is actually reducedto the determination of the occurring polarization surface-chargedensity qwhich depends on ₀, on the body shape but also onthe ratio = ₂/₁of the dielectic constants (₂outside and ₁inside the body). The adopted procedure consists in asymptoticallyexpanding and inverting (with respect to the small slendernessratio of the body) the well-known Fredholm boundary integralequation of the second kind governing the function q. The technicaldifficulties such an approach encounters are bypassed by employinga systematic formula in getting the asymptotic estimate ofcertain integrals depending upon a small parameter. Contraryto other works in the field, this method authorizes us to handlethe case of non-axisymmetric slender bodies. As an illustrationthe theory is applied to a body of elliptical cross-sectionand comparisons are presented for a slender dielectric ellipsoidembedded in a special potential ₀for which the exact densityqis obtained in a closed form.