Showing papers in "Mathematical Notes in 2011"

Well-posed problems for the Laplace operator in a punctured disk

Abstract: We give a complete description of well-posed solvable boundary-value problems for the Laplace operator in the disk and in the punctured disk. We present formulas for resolvents of wellposed problems for the Laplace operator in the disk.

Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain

Abstract: For an equation of mixed type, namely, $$ \left( {1 - \operatorname{sgn} t} \right)u_{tt} + \left( {1 - \operatorname{sgn} t} \right)u_t - 2u_{xx} = 0 $$ in the domain {(x, t) | 0 < x < 1, −α < t < β}, where α, β are given positive real numbers, we study the problem with boundary conditions $$ u\left( {0,t} \right) = u\left( {1,t} \right) = 0, - \alpha \leqslant t \leqslant \beta , u\left( {x, - \alpha } \right) - u\left( {x,\beta } \right) = \phi \left( x \right), 0 \leqslant x \leqslant 1. $$ . We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition φ(x).

Commutation of projections and trace characterization on von Neumann algebras. II

Abstract: We obtain new necessary and sufficient commutation conditions for projections in terms of operator inequalities. These inequalities are applied for trace characterization on von Neumann algebras for the class of all positive normal functionals.

Deformations of the Lie algebra o(5) in characteristics 3 and 2

Abstract: All finite-dimensional simple modular Lie algebras with Cartan matrix fail to have deformations, even infinitesimal ones, if the characteristic p of the ground field is equal to 0 or exceeds 3. If p = 3, then the orthogonal Lie algebra o(5) is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras br(2; α) appear in this family of deformations of the 10-dimensional Lie algebras, and therefore are not listed separately); moreover, the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra which has a Cartan matrix and admits a deformation. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of an algebraic group O(5) of automorphisms of the Lie algebra o(5) on the space H2(o(5); o(5)) of infinitesimal deformations and presented representatives of the isomorphism classes. We give here an explicit description of the global deformations of the Lie algebra o(5) and describe the deformations of a simple analog of this orthogonal algebra in characteristic 2. In characteristic 3, we have found the representatives of the isomorphism classes of the deformed algebras that linearly depend on the parameter.

Properties of sets admitting stable ɛ-selections

Abstract: Sets in which some convex subsets admit local (global) continuous ɛ-selections are studied. In particular, it is shown that if, for any number ɛ > 0, some neighborhood O(x) of a point x in a Banach space X contains a dense (in O(x)) convex set K admitting an upper semicontinuous acyclic (in particular, continuous single-valued) ɛ-selection to an approximatively compact set M ⊂ X, then x is a δ-sun point; if, in addition, X ∈ (R), then the set of all points nearest to x in M is a singleton.

Stochastic monotonicity and duality for one-dimensional Markov processes

Abstract: The theory of monotonicity and duality is developed for general one-dimensional Feller processes, extending the approach from [1]. Moreover it is shown that local monotonicity conditions (conditions on the Levy kernel) are sufficient to prove the well-posedness of the corresponding Markov semigroup and process, including unbounded coefficients and processes on the half-line.

On degeneration of the surface in the fitting compactification of moduli of stable vector bundles

Abstract: A new compactification of the moduli scheme of Gieseker-stable vector bundles with given Hilbert polynomial on a smooth projective polarized surface (S, H) over a field \(k = \bar k\) of zero characteristic was constructed in previous papers by the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the schemes which are certain modifications of S. We describe the class of modified surfaces that appear in the construction.

Toral Rank Conjecture for Moment-Angle Complexes

Abstract: We consider an operation K ↦ L(K) on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as K-powers). The main property of the doubling operation is that the moment-angle complex Open image in new window can be identified with the real moment-angle complex Open image in new window for the double L(K). By way of application, we prove the toral rank conjecture for the spaces Open image in new window by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes Open image in new window. This paper can be viewed as a continuation of the author’s previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.

Slowly synchronizing automata with zero and noncomplete sets

Abstract: Using the combinatorial properties of noncomplete sets in a free monoid, we construct a series of finite deterministic synchronizing automata with zero for which the shortest synchronizing word has length n2/4 + n/2 − 1, where n is the number of states.

Mathematical Solution of the Gibbs Paradox

Abstract: Since all the papers of the author dealing with this problem 1 were published in English, here we present a short survey of these papers in both Russian and English. The Gibbs paradox has been a stumbling block for many physicists, including Einstein, Gibbs, Planck, Fermi, and many others (15 Nobel laureates in physics studied this problem) as well as for two great mathematicians Von Neumann and Poincar ´

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

Abstract: On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m − 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.

Discrete wavelets and the Vilenkin-Chrestenson transform

Abstract: In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to construct new orthogonal wavelet bases defined by finite collections of parameters. Earlier similar bases were defined for the Cantor and Vilenkin groups by means of generalized Walsh functions. It is noted that similar constructions can be realized for biorthogonal wavelets as well as for the space l2(ℤ+).

On the properties of the Q -integral

Abstract: We study the Titchmarsh Q-integral, its generalization, and its elementary properties are studied; integrability criteria on sets of finite measure are obtained.

Homogenization of monotone operators under conditions of coercitivity and growth of variable order

Abstract: We obtain a homogenization procedure for the Dirichlet boundary-value problem for an elliptic equation of monotone type in the domain Ω ⊂ ℝ d . The operator of the problem satisfies the conditions of coercitivity and of growth with variable order p ɛ (x) = p(x/ɛ); furthermore, p(y) is periodic, measurable, and satisfies the estimate 1 0 tends to zero. Here α and β are arbitrary constants. Taking Lavrent’ev’s phenomenon into account, we consider solutions of two types: H- and W-solutions. Each of the solution types calls for a distinct homogenization procedure. Its justification is carried out by using the corresponding version of the lemma on compensated compactness, which is proved in the paper.

Squeezed States and Their Applications to Quantum Evolution

Abstract: In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrodinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.

On homogeneous mixtures of gases

Abstract: In the paper, relations for the critical values of a mixture of new ideal gases are presented and differential equations for a mixture of real gases are written out.

A new approach to probability theory and thermodynamics

Abstract: A new point of equilibrium thermodynamics is obtained as a consequence of the constancy of the gas density inside a closed vessel.

On the Spectral Properties of Differential Operators with Unbounded Operator Coefficients Determined by a Linear Relation

Abstract: Necessary and sufficient conditions for the invertibility and the Fredholm property of operators generated by a family of evolution operators and by the boundary conditions determined by a linear relation are obtained

On large values of the function S ( t ) on short intervals

Abstract: We prove a theorem on upper and lower bounds for the argument S(t) of the Riemann zeta function on short intervals of the critical line.

Some problems of approximation theory in the spacesLp on the line with power weight

Abstract: In the spaces Lp on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.

Bifurcation Problems for Equations of Elliptic Type with Discontinuous Nonlinearities

Abstract: We consider the problem of the existence of semiregular solutions to the main boundary-value problems for second-order equations of elliptic type with a spectral parameter and discontinuous nonlinearities. A variational method is used to obtain the theorem on the existence of solutions and properties of the “separating” set for the problems under consideration. The results obtained are applied to the Goldshtik problem.

Hyperbolic operator semigroups and Lyapunov’s equation

Abstract: We obtain necessary and sufficient conditions for the hyperbolicity of a semigroup of operators. In so doing, we use Lyapunov’s equation in operator form constructed from its generator.

Tunnel quantization of thermodynamics and critical exponents

Abstract: In the paper, it is shown that the critical indices of practical simple liquids, as well as many other thermodynamical effects, readily and naturally follow from the conception of tunnel quantization of contemporary thermodynamics.

Continuity of derivations on properly infinite *-algebras of τ-measurable operators

Abstract: Let A be an algebra over a field K. A linear mapping δ : A → A is referred to as a derivation if δ(ab) = δ(a)b + aδ(b) for any a, b ∈ A . For every element a ∈ A , the mapping δa(x) = [a, x] = ax−xa is a derivation on A . Derivations of this kind are said to be inner. An important problem in the theory of derivations on algebras is to find classes of algebras for which every derivation is inner. It is clear that every inner derivation in a topological algebra is continuous. In this connection, a natural problem arises: Distinguish the classes of topological algebras in which every derivation is continuous. As is well known, the answer to this problem is affirmative, for example, for arbitrary C∗-algebras [1]. However, in general, a derivation on an arbitrary C∗-algebra need not be inner [1]. At the same time, all derivations on W ∗-algebras are inner [1]. A similar problem was also considered for diverse algebras of measurable operators affiliated to von Neumann algebras. As was proved in [2], every derivation on the algebra S(M ) of measurable operators affiliated with a commutative von Neumann algebra M is zero if and only if the algebra M is atomic. All questions posed above for the algebra S(M ) of all measurable operators, the algebra S(M , τ) of all τ-measurable operators, and the algebra LS(M ) of all locally measurable operators were solved in the case of a von Neumann algebra M of type I [3] and for the case in which the von Neumann algebra M acts on a separable Hilbert space [4]. In particular, it was proved that, for a von Neumann algebra M of type I∞, every derivation on the algebras S(M ), S(M , τ), and LS(M ) is inner. Since the ∗-algebra S(M , τ) is a topological ∗-algebra with respect to the topology tτ of convergence in measure generated by a faithful normal semifinite trace τ , it follows that every derivation on S(M , τ) is continuous with respect to the topology tτ for any von Neumann algebra M of type I∞. In the present note, it is proved that a similar result remains valid in the case of arbitrary properly infinite von Neumann algebras M . We use the terminology and notation of von Neumann algebra theory ([5], [1]) and of the theory of measurable operators [6]. Let M be a von Neumann algebra with a faithful normal semifinite trace τ , and let P (M ) be the lattice of all projections in M . A densely defined linear operator a affiliated with M is said to be τ-measurable if τ(e|a|(λ,+∞)) < ∞ for some λ > 0, where e|a|(λ,+∞) stands for the spectral projection of |a| = (a∗a)1/2 corresponding to the interval (λ,+∞). The set S(M , τ) of all τ-measurable operators is a ∗-algebra with respect to strongly defined operations of addition and multiplication. The algebra S(M , τ) is equipped with a separated metrizable vector topology tτ of convergence in measure for which a basis of neighborhoods of zero is given by the sets U(ε, δ) = {a ∈ S(M , τ) : ∃p ∈ P (M ), τ(1 − p) ≤ δ, ap ∈ M , ‖ap‖ ≤ ε}, where ‖ · ‖ stands for the C∗-norm on M . As is known (see, e.g., [6]), (S(M , τ), tτ ) is a topological ∗-algebra and an F-space.

Best polynomial approximations in L2 of classes of 2π-periodic functions and exact values of their widths

Abstract: We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity ofmth order in the space L 2 as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L 2.

Gibbs paradox, liquid phase as an alternative to the bose condensate, and homogeneous mixtures of new ideal gases

Abstract: In this paper, relations for the liquid phase of a new ideal gas are given. The points of the spinodal of the liquid phase (i.e., the endpoints of metastable states) in the domains of positive and negative pressures are defined. Relations for the critical values of homogeneous mixtures are presented. We study the phase transition and critical indices from the point of view of the geometric quantization of thermodynamics.

The greatest possible lower type of entire functions of order ρ ∈ (0; 1) with zeros of fixed ρ-densities

Abstract: For ρ ∈ (0, 1), we obtain the supremum of lower ρ-types of entire functions whose sequence of roots has given lower and upper densities for the order ρ.

Estimates in Beurling-Helson type theorems: Multidimensional case

Abstract: We consider the spaces Ap(\(\mathbb{T}^m \)) of functions f on the m-dimensional torus \(\mathbb{T}^m \) such that the sequence of Fourier coefficients \(\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} \) belongs to lp(ℤm), 1 ≤ p < 2. The norm on Ap(\(\mathbb{T}^m \)) is defined by \(\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} \). We study the rate of growth of the norms \(\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} \) as |λ| → ∞, λ ∈ ℝ, for C1-smooth real functions φ on \(\mathbb{T}^m \) (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces Ap(ℝm).