Gibbs measures for the HC Blume–Capel model with countably many states on a Cayley tree

Theoretical and Mathematical Physics - We study the Blume–Capel model with a countable set $$mathbb Z$$ of spin values and a force $$Jin mathbb R$$ of interaction between the nearest...     

Localization of translation-invariant Gibbs measures for the Potts and “solid-on-solid” models on a Cayley tree

We study Potts and “solid-on-solid” models with q ≥ 2 states on the Cayley tree of order k ≥ 1. For any values of the parameter q in the Potts model and q ≤ 6 in the “solid-on-solid” model, we find sets containing all translation-invariant Gibbs measures.     

Extremality of Gibbs Measures for the $$HC$$ -Blume–Capel Model on the Cayley Tree

Mathematical Notes - In this paper, we consider translation-invariant Gibbs measures (TIGMs) for the $$HC$$ -Blume–Capel model in case of “wands” with chemical potential with...     

Translation-invariant <Emphasis Type="Italic">p</Emphasis>-adic quasi-Gibbs measures for the Ising–Vannimenus model on a Cayley tree

We consider the p-adic Ising–Vannimenus model on the Cayley tree of order k = 2. This model contains nearest-neighbor and next-nearest-neighbor interactions. We investigate the model using a new approach based on measure theory (in the p-adic sense) and describe all translation-invariant p-adic quasi-Gibbs measures associated with the model. As a consequence, we can prove that a phase transition exists in the model. Here, “phase transition” means that there exist at least two nontrivial p-adic quasi-Gibbs measures such that one is bounded and the other is unbounded. The methods used are inapplicable in the real case.     

On the lattice structure of the set of stable matchings for a many-to-one model ∗

For the many-to-one matching model with firms having substitutable and q-separable preferences we propose two very natural binary operations that together with the unanimous partial ordering of the workers endow the set of stable matchings with a lattice structure. We also exhibit examples in which, under this restricted domain of firms' preferences, the classical binary operations may not even be matching     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

Given a finite simple graph G with n vertices, we can construct the Cayley graph on the symmetric group S _n generated by the edges of G, interpreted as transpositions. We show that, if G is complete multipartite, the eigenvalues of the Laplacian of Cay (G) have a simple expression in terms of the irreducible characters of transpositions and of the Littlewood–Richardson coefficients. As a consequence, we can prove that the Laplacians of G and of Cay (G) have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous’s conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.     

Stability on a cone in terms of two measures for impulsive differential equations with “supremum”

The stability of nonlinear impulsive differential equations with “supremum” is studied. A special type of stability, combining two different measures and a dot product, is defined. The definition is a generalization of several types of stability known in the literature. Razumikhin’s method as well as a comparison method for scalar impulsive ordinary differential equations have been employed.     

A modification of the method of linearization for solving a mathematical programming problem on a simple set of “parallelepiped” type

We propose a modification of B. N. Pshenichnyi's method of linearization for solving mathematical programming problems on a simple set of parallelepiped type. We study in detail the computational aspects and the efficiency of the approach in question. Two tables. Bibliography: 3 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 24–28.     

On a “Separating” set for higher-order equations of elliptic type with discontinuous nonlinearities

We study the existence of semiregular solutions of the Dirichlet problem for a higher-order equation of elliptic type with a spectral parameter and a discontinuous nonlinearity. By a variational method, we obtain a theorem on the existence of solutions and on the properties of a “separating” set for this problem.     

Finding the “Wasp waist” in a graph of partial ordering

For a graph of strict partial ordering the concept of wasp waist is introduced; it is a vertex of the graph which is the maximal independent set of its vertices. An algorithm that finds in a graph a wasp waist, if it exists, is given. The complexity of the algorithm is O(n²), where n is the number of the vertices of a graph.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 41–43, 1983.     

Supplement to the paper “the averaging operator with respect to a countable partition on a minimal symmetric ideal of the space L1(0, 1)”

Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let tL₁(0,1), let N_t be the smallest rearrangement-invariant order ideal vector lattice in L₁(0,1), containing t. In the paper one investigates the properties of the image E(N_t¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, gL₁(0,1), such that E(N_t¦A)N_g. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(tA)QE(t*A*), where is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1Q2⁵. One formulates the problem of the smallest value of Q.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 137–141, 1986.     

Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data

In this paper, we discuss the problem of testing the hypothesis that the underlying regression is a partial linear model. A test statistic, which is based on the quadratic form of a cusum process of residuals, is proposed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. The number simulation shows that the test is available.     

Optimality of (σS)-type policies for a stationary multi-product inventory model with a canonical diffusion process as demand process

The presented multi-product inventory problem is a special impulse-control problem of canonical diffusion processes and is considered within a general, time-discrete, Markovian decision model. A proof of the optimality of a (σS) type policy is given by using a policy iteration method     

Global existence of solutions for the one-dimensional motions of a compressible viscous gas with radiation: An “infrarelativistic model”

We consider an initial–boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled with radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients, we prove that the problem admits a unique weak solution.     

Discrete spectrum of a model Schr?dinger operator on the half-plane with Neumann conditions

We study the eigenpairs of a model Schr?dinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line that makes the angle ?? with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit ?? ?? 0, and we prove a full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.     

Dynamic complexity of a host–parasitoid ecological model with the Hassell growth function for the host

This paper investigates a discrete-time host–parasitoid ecological model with Hassell growth function for the host by qualitative analysis and numerical simulation. Local stability analysis of the system is carried out. Many forms of complex dynamics are observed, including chaotic bands with periodic windows, pitchfork and tangent bifurcations, attractor crises, intermittency, supertransients, and non-unique dynamics (meaning that several attractors coexist). The largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors.