Ukrainian Mathematical Journal最新文献

A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rule with a decay term and contribute to changes in the states after a short delay. The lattice system is first reformulated as a coupled functional-ordinary differential equation system on an appropriate product space. Then it is shown that the solution of the system exists and is unique. Furthermore, it is demonstrated that the system of equations generates a continuous random dynamical system. Finally, the existence of random attractors for the random dynamical system generated by the Dong–Hopfield model is established.

We outline key points of the concept of ideal turbulence offering novel scenarios for distributed chaos based not on the geometric-dynamical complexity of the attractor but on the extremely complex spatial structure of elements of the attractor. Ideal turbulence is observed in idealized (without internal resistance) models of various processes related to electromagnetic or acoustic oscillations. This idealization significantly simplifies the analysis and, at the same time, in many cases, provides a quite adequate description of real processes.

We revisit, in a didactic manner and by using stochastic analysis, the parametrix method and its application to unbiased simulation. We consider, in particular, the case of one-dimensional diffusions without drift.

Let X be a diffusion in a cone with oblique reflection at the boundary. We study the question whether X reaches the vertex of the cone for a finite time with positive probability. We propose a new probabilistic method of investigation connected with the long-term behavior of the diffusion reflected in a cylinder.

We establish the classical solvability of a certain conjugation problem for one-dimensional (with respect to a spatial variable) Kolmogorov backward equation with discontinuous coefficients and some versions of the general nonlocal Feller–Wentzell boundary condition given on nonsmooth boundaries of the considered curvilinear domains. In addition, we prove, that the two-parameter Feller semigroup defined by the solution of this problem describes some inhomogeneous diffusion process with moving membranes on the given region of the real line. We also show the relationship between the constructed process and the generalized diffusion in a sense of M. I. Portenko.

We consider a large population of haploid sexually reproducing individuals. It is assumed that one individual initially carries a very strongly advantageous mutation at a single locus. We study the long-term contribution of this initial individual to the genome of the population.

We introduce a new type of submersions, which are called pointwise hemi-slant Riemannian submersions, as a generalization of slant Riemannian submersions, hemi-slant submersions, and pointwise slant submersions from Kaehler manifolds onto Riemannian manifolds. We obtain some geometric interpretations of this kind of submersions with respect to the total manifold, base manifold, and fibers. Moreover, we present nontrivial illustrative examples in order to demonstrate the existence of submersions of this kind. Finally, we obtain some curvature equalities and inequalities with respect to a certain basis.

For a function f from the Sobolev space W^1,p(C), where C ⊂ ℝ^d is an open convex cone, we establish a sharp inequality estimating ∥f∥ _L∞ via the L_p-norm of its gradient and a seminorm of the function. With the help of this inequality, we prove a sharp inequality estimating the L_∞-norm of the Radon–Nikodym derivative of a charge defined on Lebesgue measurable subsets of C via the L_p-norm of the gradient of this derivative and the seminorm of the charge. In the case where C = ℝ₊^m× ℝ^d−m, 0 ≤ m ≤ d, we obtain inequalities estimating the L_∞-norm of a mixed derivative of the function f : C → ℝ via its L_∞-norm and the L_p-norm of the gradient of mixed derivative of this function.

We investigate the sharp bound of certain coefficient functionals associated with a Hankel determinant of the second kind for the inverse function when f belongs to the class of starlike functions with respect to symmetric points.

We study the existence and regularity results for degenerate parabolic problems in the presence of strongly increasing regularizing lower-order terms and L^m-data/Dirac mass.