On controllable variants of Richardson’s model in political science

It is proved that the family of bi-Lipschitz classes of Delone sets in Euclidean space of dimension at least 2 has the cardinality of the continuum.     

On the existence of business cycles in Asada’s two-regional model

A two-regional five-dimensional model describing the development of income, capital stock and money stock, which was introduced by Asada (2004) [2] is analysed. Sufficient conditions for the existence of one pair of purely imaginary eigenvalues and three eigenvalues with negative real parts in the linear approximation matrix of the model are found. Formulae for the calculation of the bifurcation coefficients of the model are derived. The theorem on the existence of business cycles is presented. A numerical example illustrating the gained results is given.     

On “P” property and the column-W property

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D, F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D + FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m ? 1. A decision algorithm of complexity O(m²) to check whether the given pair of matrices (D, F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m ? 1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k + 1)-tuple. That is, given a matrix (k + 1)-tuple (D₁, … ,D_k, F), where D_js are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D₁ + FG₁, … ,D_k + FG_k) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F) = m ? 1. Using these conditions a decision algorithm of complexity O(km²) to check whether the given matrix (k + 1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m ? 1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.     

Maintenance of diversity in a hierarchical host–parasite model with balancing selection and reinfection

Inspired by DNA data of the human cytomegalovirus we propose a model of a two-type parasite population distributed over its hosts. The parasite is capable to persist in its host till the host dies, and to reinfect other hosts. To maintain type diversity within a host, balancing selection is assumed.For a suitable parameter regime we show that in the limit of large host and parasite populations the host state frequencies follow a dynamical system with a globally stable equilibrium, guaranteeing that both types are maintained in the parasite population for a long time on the host time scale.     

On deformations of linear systems of differential equations and the Painlevé property

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement with that approach.     

On the variation detracting property of operators of Balázs and Szabados

In this note we spotlight the linear and positive operators of discrete type \({{{(R_n)}_{n\geqq1}}}\) known as Balázs–Szabados operators. We prove that this sequence enjoys the variation detracting property. The convergence in variation of \({{{(R_{n}f)}_{n\geqq1}}}\) to f is also proved. A generalization in Kantorovich sense is constructed and boundedness with respect to BV-norm is revealed.     

On the sufficient descent property of the Shanno’s conjugate gradient method

Satisfying in the sufficient descent condition is a strength of a conjugate gradient method. Here, it is shown that under the Wolfe line search conditions the search directions generated by the memoryless BFGS conjugate gradient algorithm proposed by Shanno satisfy the sufficient descent condition for uniformly convex functions.     

On the oscillation property of Green’s function of a fourth-order discontinuous boundary-value problem

The paper deals with conditions under which the Green function of a multipoint boundary-value problem for fourth-order equations describing small strains of a rod fastened to a solid elastic basement and additionally fixed by “concentrated” elastic supports at separate points has the oscillation property. It is shown that the condition that the Green function is positive is necessary and sufficient for the Green function to have the oscillation property.     

On entropy weak solutions of Hughes’ model for pedestrian motion

We consider a generalized version of Hughes’ macroscopic model for crowd motion in the one-dimensional case. It consists in a scalar conservation law accounting for the conservation of the number of pedestrians, coupled with an eikonal equation giving the direction of the flux depending on pedestrian density. As a result of this non-trivial coupling, we have to deal with a conservation law with space–time discontinuous flux, whose discontinuity depends non-locally on the density itself. We propose a definition of entropy weak solution, which allows us to recover a maximum principle. Moreover, we study the structure of the solutions to Riemann-type problems, and we construct them explicitly for small times, depending on the choice of the running cost in the eikonal equation. In particular, aiming at the optimization of the evacuation time, we propose a strategy that is optimal in the case of high densities. All results are illustrated by numerical simulations.     

On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

We establish that, in ZF (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice AC), the statement RLT: Given a set I and a non-empty set \({\mathcal{F}}\) of non-empty elementary closed subsets of 2 ^I satisfying the fip, if \({\mathcal{F}}\) has a choice function, then \({\bigcap\mathcal{F} \ne \emptyset}\) , which was introduced in Morillon (Arch Math Logic 51(7–8):739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem (see Sect. 1 for terminology). The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon (2012) and, on the other hand, a negative answer—in the setting of ZFA (i.e., ZF with the axiom of extensionality weakened to permit the existence of atoms)—to the question in Morillon (2012) of whether RLT is equivalent to Rado’s selection lemma.     

On the strong Brillinger-mixing property of α-determinantal point processes and some applications

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C(x, y) defining an α-determinantal point process (DPP). Assuming absolute integrability of the function C₀(x) = C(o, x), we show that a stationary α-DPP with kernel function C₀(x) is “strongly” Brillinger-mixing, implying, among others, that its tail-σ-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of α-DPPs.     

On Hindman spaces and the Bolzano–Weierstrass property

We examine relationships between two classes of topological spaces defined with the aid of the Hindman ideal. We also do the same for another ideal—instead of sums, as in the Hindman ideal, we consider differences.     

The decidability of craig’s interpolation property in well-composed J-logics

Under study are the extensions of Johansson??s minimal logic J. We find sufficient conditions for the finite approximability of J-logics in dependence on the form of their axioms. Using these conditions, we prove the decidability of Craig??s interpolation property (CIP) in well-composed J-logics. Previously all J-logics with weak interpolation property (WIP) were described and the decidability of WIP over J was proved. Also we establish the decidability of the problem of amalgamability of well-composed varieties of J-algebras.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.