Energy transfer in the Kuramoto-Sivashinskii equation on a multidimensional torus with Riemannian metric

In this paper, we consider the Kuramoto-Sivashinskii equation on the multidimensional torus with a Riemannian metric: where . For this equation the theorem on energy transfer holds. It is formulated in the following way. Let be the Fourier expansion of the function u. Denote by P _N and P _N ^⊥ the operators of rejection of the “leading” and, respectively, “lowest” terms of the Fourier expansion (harmonics), i.e., . For any ρ > 0,N ∈ ℕ, s ≥ n/2+5, and λ ∈ (0, 1), there exists R such that for any function. ϕ ∈ lying outside the ball in the ball , there exists an instant of time t ∈ (0, 1) such that g _KS ^t ϕ=ψ and . Here, R is a constant depending on the metric (g), n _s is the sth Sobolev norm, and is the C ¹-norm. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Classical r-matrices with a parabolic carrier

Using a graphic presentation of the dual Lie algebra ^# (r) for a simple algebra , it is possible to show that there always exist solutions r_ech of the classical Yang—Baxter equation with a parabolic carrier. To get a closed-form expression for r_ech, we find dual coordinates in which the adjoint action of the carrier _c is reducible. This allows us to find the structure of Jordanian r-matrices r_J which are candidates for enlarging the initial full chain r_fch and realize the desired solution r_ech in the factorized form r_ech ≈ r_fch + r_J. We obtain a unique transformation: the canonical chain has to be replaced by a special kind of peripheric r-matrices, r_fch → r_rfch. To illustrate the method, the case of =sl(11) is considered in the full detail. Bibliography: 11 titles. To my friend Petr Petrovich Kulish __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 122–141.     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

Invariant and quasi-invariant transformations of measures that correspond to stable processes with independent increments

Let ξ(t), t ∈ [0, 1], be a strictly stable process with an index of stability α ∈ (0, 2). By we denote the law of ξ in the Skorokhod space [0,1]. For arbitrary strictly stable process ξ, we construct a semigroup of transformations of [0, 1] for which the measure is quasi-invariant. For strictly stable processes with positive and negative jumps, we construct a group of transformations of [0, 1] for which the measure is quasi-invariant. In the symmetric case, this group is a group of invariant transformations. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 135–149.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Compactness of the congruence group of measurable functions in several variables

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let be a measurable function defined on the product of finitely many standard probability spaces (X_i, , μ_i), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g₁, ..., g_n) of measure preserving automorphisms of the respective spaces (X_i, , μ_i) such that f(g₁ x ₁, ..., g_nx_n) = f(x₁, ..., x_n) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence, we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 57–67.     

Temperature-Dependence of Equilibrium States of a Two-Phase Elastic Medium with the Zero Surface Tension Coefficient

We consider the energy functional of a two-phase elastic medium with quadratic energy densities defined for such that ,where is a measurable characteristic function. Under some natural conditions on the data of the problem, we prove the existence of an interval (t ^-,t ⁺) of the change of temperature such that the energy functional has only a minimizer such that for or such that t^ + $$ " align="middle" border="0"> . The energy functional has no minimizers such that or if . We derive two-sided estimates for the numbers in terms of the characteristics of the two-phase elastic medium and the boundary condition. Bibliography: 3 titles.     

A Higher-Order Analog of the Linking Number for a Pair of Divergence-Free Vector Fields

Pairs B, of divergence-free vector fields with compact support in are considered higher-order analog M(B, c (of order 3) of the Gauss helicity number H(B, )= , curl(A)=B; (of order 1) is constructed, which is invariant under volume-preserving diffeomorphisms. An integral expression for M is given. A degree-four polynomial m(B(x₁), B(x₂), ( ₁), ( ₂)), x₁, x₂, _{1 2} , is defined, which is symmetric in the first and second pairs of variables separately. M is the average value of m over arbitrary configurations of points. Several conjectures clarifying the geometric meaning of the invariant and relating it to invariants of knots and links are stated. Bibliography: 11 titles.     

Transformation of measures generated by jump Lévy processes

Let ξ(t), t ∈ [0, 1], be a jump Lévy process. We denote by the law of ξ in the Skorokhod space [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of -preserving transformations of the space [0, 1]. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 175–188.     

On estimation and detection of a function of infinitely many variables

We observe an unknown function of infinitely many variables f = f(t), t = (t₁, ..., t_n, ... ) ∈, [0, 1]^∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension of the function f(t) to IR ^∞. Taking a quantity σ > 0 and a positive sequence a = {a_k}, we consider the set that consists of functions f such that . We consider the cases a_k = k^α and a_k = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈ or to test the null hypothesis H₀: f = 0 against the alternatives f ∈ , where the set consists of functions f ∈ such that ∥f∥₂ ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates f _ɛ ^* that provide distiguishability in the problems. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113.     

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data

The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system

Extremal Decomposition Problems in the Space of Riemann Surfaces

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface with a prescribed set of distinguished points to the case of the Teichmüller space corresponding to under quasiconformal homeomorphisms f. For the functional of our problem on extremal decomposition of a surface , we consider a function expressing the dependence of the extremal value of on a point . Differentiation formulas for the function are derived. These formulas are different and depend on the genus g of the surface . The case where the function is pluriharmonic is considered. Bibliography: 8 titles.     

Carleson measures and the heat equation

Let { }, where { } is the open unit disk on the complex plane { }. In G, we consider analytic solutions u(t, z) ({ }, { }) of the heat equation 2u_t=u_zz with initial data f(z)=u(0, z) belonging to the Fock space F, i.e., to the space of entire functions square summable with the weight e−|z|2.Conditions on a nonnegative measure μ on G are described under which for all f ∈ F we have { } Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 146–155. Translated by S. V. Kislyakov.     

Finite p-groups with automorphism of a special form

Research on finite solvable groups with C-closed invariant subgroups has given rise to groups structured as follows. Let p, q₁, q₂, ..., q_m be distinct primes, n_i be the exponent of p modulo q_i, and n be the exponent of p modulo . Then G = Pλ〈x〉, where P is a group and ; Z_i; here, Z_i and P/Z(P) are elementary Abelian groups of respective orders and pⁿ, |x| = r, the element x acts irreducibly on P/Z(P) and on each of the subgroups Z_i, and . We state necessary and sufficient conditions for such groups to exist. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 379–389, July–August, 2006.     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Construction of conjugate quadrature filters with specified zeros

Let ℂ denote the complex numbers and denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials , which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials satisfying the equation Q ₁ P ₁ + Q ₂ P ₂ = 1. We provide some information about the minimal length Laurent polynomials Q ₁ and Q ₂ with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every implies and . This revised version was published online in June 2006 with corrections to the Cover Date.     

Lower semicontinuity of some functionals under PDE constraints: An \mathcal{A}-quasiconvex pair

The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special class of functionals:

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.