On Fučik Spectra and Periodic Solutions

In this paper, we continue extending the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand side. To this end, we construct a new version of the method of shifts along trajectories. We compare the results obtained by the new approach and those obtained by the method of Fuik spectra.     

An Estimate for a Fundamental Solution of a Parabolic Equation with Drift on a Riemannian Manifold

We construct a fundamental solution for a parabolic equation with drift on a Riemannian manifold of nonpositive curvature. We obtain some estimates for this fundamental solution that depend on the conditions on the drift field.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

New Component of the Moduli Space M(2;0,3) of Stable Vector Bundles on the Double Space P 3 of Index Two

We study the moduli scheme M(2;0,n) of rank-2 stable vector bundles with Chern classes c ₁=0, c ₂=n, on the Fano threefold X – the double space P ³ of index two. New component of this scheme is produced via the Serre construction using certain families of curves on X. In particular, we show that the Abel–Jacobi map :HJ(X) of any irreducible component H of the Hilbert scheme of X containing smooth elliptic quintics on X into the intermediate Jacobian J(X) of X factors by Stein through the quasi-finite (probably birational) map g:M of (an open part of) a component M of the scheme M(2;0,3) to a translate of the theta-divisor of J(X).     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.     

The Problem of Aliasing in Identifying Finite Parameter Continuous Time Stochastic Models

This note exposits the problem of aliasing in identifying finite parameter continuous time stochastic models, including econometric models, on the basis of discrete data. The identification problem for continuous time vector autoregressive models is characterised as an inverse problem involving a certain block triangular matrix, facilitating the derivation of an improved sufficient condition for the restrictions the parameters must satisfy in order that they be identified on the basis of equispaced discrete data. Sufficient conditions already exist in the literature but these conditions are not sharp and rule out plausible time series behaviour.     

Stress and heat flux stabilization for viscous compressible medium equations with a nonmonotone state function

We study a system of quasilinear equations describing one-dimensional flow of a viscous compressible heat-conducting medium with a nonmonotone state function and mass force. The large-time behavior of solutions is considered for arbitrarily large initial data. In spite of possible nonuniqueness and discontinuity of the stationary solution, we prove L²-stabilization for the stress and heat flux as t → ∞ along with corresponding global energy estimates for them. The new method of proof utilizes a combination of energy type equalities for the stress and heat flux. Consequently, H¹-stabilization of the velocity and temperature along with global estimates for their derivatives are valid as well.     

Regularity of Pressure in the Neighbourhood of Regular Points of Weak Solutions of the Navier-Stokes Equations

In the context of the weak solutions of the Navier-Stokes equations we study the regularity of the pressure and its derivatives in the space-time neighbourhood of regular points. We present some global and local conditions under which the regularity is further improved.     

Monodromy Approach to the Scaling Limits in Isomonodromy Systems

The isomonodromy deformation method is applied to the scaling limits in the linear N×N matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves that describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated with the n-large asymptotics in the sequence of the biorthogonal polynomials with cubic potentials.