Parabolic equations and Feynman-Kac formula on general bounded domains

Parabolic equations on general bounded domains are studied. Using the refined maximum principle, existence and the semigroup property of solutions are obtained. It is also shown that the solution obtained by PDE’s method has the Feynmann-Kac representation for any bounded domains.     

Kolmogorov＇s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov＇s ε-entropy for the bounded sets with small ＂tail＂ in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov＇s ε-entropy of the global attractor for lattice systems.     

Fridman’s Invariant,Squeezing Functions,and Exhausting Domains

We show that if a bounded domain Ω is exhausted by a bounded strictly pseudoconvex domain D with C2 boundary, then Ω is holomorphically equivalent to D or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman’s invariant has certain growth condition near the boundary.     

Dissipativity of differential equations and the corresponding difference equations

We establish conditions under which the existence of a bounded solution of a difference equation yields the existence of a bounded solution of the corresponding differential equation. We investigate the relationship between the dissipativities of differential and difference equations in terms of Lyapunov functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1249–1256, September, 2006.     

Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term

The aim of this paper is to establish the existence of at least one solution for a general inequality of quasi-hemivariational type, whose solution is sought in a subset K of a real Banach space E. First, we prove the existence of solutions in the case of compact convex subsets and the case of bounded closed and convex subsets. Finally, the case when K is the whole space is analyzed and necessary and sufficient conditions for the existence of solutions are stated. Our proofs rely essentially on the Schauder’s fixed point theorem and a version of the KKM principle due to Ky Fan (Math Ann 266:519–537, 1984).     

The LIL for <Emphasis Type="Italic">U</Emphasis>-Statistics in Hilbert Spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest. R. Adamczak’s research partially supported by MEiN Grant 2 PO3A 019 30. R. Latała’s research partially supported by MEiN Grant 1 PO3A 012 29.     

Inequality constrained maximum likelihood estimation

Summary The existence of an estimator constrained to lie in a certain type of bounded set is established for a fairly wide class of probability density functions. The necessary and sufficient conditions thus obtained provide a convenient means of finding such an estimator by mathematical programming methods. This result is a generalization of Cramer’s demonstration of the existence of an unconstrained maximum likelihood estimator and of Aitchison and Silvey’s demonstration of the existence of a maximum likelihood estimator constrained to satisfy certain equations.     

On the convergence of circle packings to the Riemann map

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary. Oblatum 15-V-1995 & 13-XI-1995     

Regimes of more and more frequent switchings in the optimal control problem of oscillations of <Emphasis Type="Italic">n</Emphasis> oscillators

This paper considers the control system of n oscillators executing forced oscillations under the action of a scalar-valued control force common for all oscillators whose module is bounded. The author proves the existence of an optimal singular regime and the assertion that the optimal control has at least countably many switchings that accumulate to a conjunction point of the singular and nonsingular parts of the trajectory. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle,fixed point theorems and parametric optimization problems

In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder’s fixed point theorem are also given.     

Conformal invariants and higher-order schwarz lemmas

We derive a generalization of the Grunsky inequalities using the Dirichlet principle. As a corollary, sharp distortion theorems for bounded univalent functions are proven for invariant differential expressions which are higher-order versions of the Schwarzian derivative. These distortion theorems can be written entirely in terms of conformai invariants depending on the derivatives of the hyperbolic metric, and can be interpreted as ’Schwarz lemmas’. In particular, sharp estimates on distortion of the derivatives of geodesic curvature of a curve under bounded univalent maps are given.     

Resonance and nonlinearity: A survey

This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and piecewise-linear oscillators, as well as of some classes of planar Hamiltonian systems. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 190–205, February, 2007.     

Large Normally Hyperbolic Cylinders in a priori Stable Hamiltonian Systems

We prove the existence of normally hyperbolic cylinders in a priori stable Hamiltonian systems the size of which is bounded from below independently of the size of the perturbation. This result should have applications to the study of Arnold’s diffusion.     

Provably recursive functions of constructive and relatively constructive theories

In this paper we prove conservation theorems for theories of classical first-order arithmetic over their intuitionistic version. We also prove generalized conservation results for intuitionistic theories when certain weak forms of the principle of excluded middle are added to them. Members of two families of subsystems of Heyting arithmetic and Buss-Harnik’s theories of intuitionistic bounded arithmetic are the intuitionistic theories we consider. For the first group, we use a method described by Leivant based on the negative translation combined with a variant of Friedman’s translation. For the second group, we use Avigad’s forcing method.     

Spetral sets II

Sono studiati l’insiemi spettrali delle elementi di algebra di Banach negli termini della esistenza delle partizioni della unita di algebra. Circolarmente isolati l’insiemi spettrali sono caratterizati et una nuova decomposizione di Mbekhta per operatori lineari limitati e ottenuta. We study spectral sets of elements of a Banach algebra in terms of the existence of idempotent partitions of the unit of the algebra. Circularly isolated spectral sets are characterized, and a new Mbekhta decomposition for bounded linear operators is obtained.     

Variational inclusions problems with applications to Ekeland’s variational principle,fixed point and optimization problems

In this paper, we prove the existence theorems of two types of systems of variational inclusions problem. From these existence results, we establish Ekeland’s variational principle on topological vector space, existence theorems of common fixed point, existence theorems for the semi-infinite problems, mathematical programs with fixed points and equilibrium constraints, and vector mathematical programs with variational inclusions constraints.     

On the existence of almost-pure-strategy Nash equilibrium in <Emphasis Type="Italic">n</Emphasis>-person finite games

This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.     

On Ekeland��s variational principle

For proper lower semicontinuous functionals bounded from below which do not increase upon polarization, an improved version of Ekeland’s variational principle can be formulated in Banach spaces, which provides almost symmetric points.     

Hermite-Birkhoff interpolation of scattered data by radial basis functions

For Hermite-Birkhoff interpolation of scattered multidimensional data by radial basis function φ, existence and characterization theorems and a variational principle are proved. Examples include φ(r)=r^b, Duchon’s thin-plate splines, Hardy’s multiquadrics, and inverse multiquadrics.     

Vector variational principle

We prove an Ekeland’s type vector variational principle for monotonically semicontinuous mappings with perturbations given by a convex bounded subset of directions multiplied by the distance function. This generalizes the existing results where directions of perturbations are singletons.