Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.     

Equi-Attraction and Continuous Dependence of Strong Attractors of Set-Valued Dynamical Systems on Parameters

Equi-attraction properties of the global attractors of a family of set-valued dynamical systems {G_λ, λ∈Λ} with respect to a parameter λ∈Λ, where Λ is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equi-attraction of the family is equivalent to the continuity of in λ with respect to the Hausdorff distance. An example involving differential inclusions is given. Mathematics Subject Classifications (2000) Primary 47H17, 58F03, 34C20. Desheng Li: Supported by NNSF of China (10251002) and NNSF of Gansu province (3ZS041-A25-006).     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

To solving problems of algebra for two-parameter matrices. III

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem of solving an equation F(λ, μ)x = 0 with a polynomial two-parameter matrix F(λ, μ) to solving an equation of the form D(λ, μ)y = 0, where D(λ, μ) = A(μ)-λB(μ) is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix F(λ, μ) are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 166–207.     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

Structures and Representations of Generalized Path Algebras

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {e _ii } _I with each ē _ii in the center of if Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over a perfect field k (e.g., a field with characteristic zero or a finite field) if Λ is isomorphic to a generalized path algebra k (D, Ω, ρ) of finite directed graph with weak relations and dim < ∞; Λ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents if Λ is isomorphic to a path algebra with relations. Presented by Idun Reiten.     

Random walks,Kleinian groups,and bifurcation currents

Let (ρ _λ ) _λ∈Λ be a holomorphic family of representations of a finitely generated group G into PSL(2,ℂ), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λ describing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on ℙ¹. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μ on G.     

The normalizer of the automorphism group of a module arising under extension of the base ring

Let Λ te an arbitrary associative ring with unity and let R be its unital subring contained in the center of Λ. Further, let M=λ M be a left free Λ-module of finite rank. In this paper, the normalizer of the subgroupAut(λM) of automorphisms of the module λM in the groupAut(_RM) of automorphisms of the module_RM is computed. If the ring Λ is additively generated by its invertible elements, then the above normalizer coincides with the semidirect product of the normal subgroupAut(λM) and a subgroup isomorphic to the groupAut(Λ/R) of all ring automorphisms of the ring Λ that are identical on R. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 133‐135. Translated by A. I. Skopin.     

On Translation and Affine Systems Spanning L1(ℝ)

We show that the discrete translation parameter sets Λ ⊂ ℝ for which some φ ∈ L¹(ℝ) exists such that the translates φ(x − λ), λ ∈ Λ, span L¹(ℝ) are exactly the uniqueness sets for certain quasianalytic classes, and give explicit constructions of such generators φ. We also consider a similar situation for affine systems of the type φ(μx − λ), μ ∈ Γ, λ ∈ Λ.     

Inverse Scattering at Fixed Energy in de Sitter�CReissner�CNordstr?m Black Holes

In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter–Reissner–Nordstr?m black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ ≠ 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ ≠ 0 is the fixed energy and n ? \mathbbN^*{n \in \mathbb{N}^{*}} denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M, the square of the charge Q ² and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T(λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all n ? L{n \in {\mathcal{L}}} where L{\mathcal{L}} is a subset of \mathbbN^*{\mathbb{N}^{*}} that satisfies the Müntz condition ?_{n ? L}\frac1n = +￥{\sum_{n \in{\mathcal{L}}}\frac{1}{n} = +\infty} . Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the “unphysical” scattering matrix S(λ, z) in the complex variable z. We show, in particular, that the quantities \frac1T(l,z){\frac{1}{T(\lambda,z)}}, \fracR(l,z)T(l,z){\frac{R(\lambda,z)}{T(\lambda,z)}} and \fracL(l,z)T(l,z){\frac{L(\lambda,z)}{T(\lambda,z)}} belong to the Nevanlinna class in the region ${\{z \in \mathbb{C}, Re(z) > 0 \}}${\{z \in \mathbb{C}, Re(z) > 0 \}} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstruction formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.     

On the Category of Koszul Modules of Complexity One of an Exterior Algebra

In this paper, we prove that there is a natural equivalence between the category F1（x） of Koszul modules of complexity 1 with filtration of given cyclic modules as the factor modules of an exterior algebra A = ∧V of an m-dimensional vector space, and the category of the finite-dimensional locally nilpotent modules of the polynomial algebra of m - 1 variables.     

Majorants and uniqueness of series in the Franklin system

It is proved that if a series in the Franklin system converges almost everywhere to a function f(t) and the distribution function of the majorant of partial sums satisfies the condition mes{t∈[0,1]∶s(t)>λ}=o(1/λ) as λ→∞, then this series is a Fourier series for Lebesgue integrable functions f(t). In the general case the coefficients of the series are reconstructed by means of anA-integral. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 520–544, April, 1996.     

Infinite Töplitz–Lipschitz matrices and operators

We introduce a class of infinite matrices , which are asymptotically (as |s| + |s′| → ∞) close to Hankel–T?plitz matrices. We prove that this class forms an algebra, and that flow-maps of nonautonomous linear equations with coefficients from the class also belong to it.     

Hyperbolic branching Brownian motion

Summary. Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane ? ² , and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ: For λ≦ 1/8 the number of particles in any compact region of ? ² is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ≦ 1/8) the set Λ of all limit points in ∂? ² (the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ≦ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ is δ = (1−√1−8 λ)/2. Received: 2 November 1995 / In revised form: 22 October 1996     

Sparse exponential systems: Completeness with estimates

According to A. Beurling and H. Landau, if an exponential system {e ^iλt }_λ∈Λ is a frame in L ² on a set S of positive measure, then Λ must satisfy a strong density condition. We replace the frame concept by a weaker condition and prove that if S is a finite union of segments then the result holds. However, for “generic” S, very sparse sequences Λ are admitted. Supported in part by the Israel Science Foundation.     

Vanishing integrals for Hall–Littlewood polynomials

It is well known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent paper of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However, at q = 0 (the Hall–Littlewood level), these approaches do not work, although one can obtain the results by taking the appropriate limit. In this paper, we develop a direct approach for dealing with this special case. This technique allows us to prove some identities that were not amenable to the Hecke algebra approach. Moreover, we are able to generalize some of the identities by introducing extra parameters. This leads us to a finite-dimensional analog of a recent result of Warnaar, which uses the Rogers–Szegő polynomials to unify some existing summation type formulas for Hall–Littlewood functions.     

A Magnus- and Fer-Type Formula in Dendriform Algebras

We provide a refined approach to the classical Magnus (Commun. Pure Appl. Math. 7:649–673, [1954]) and Fer expansion (Bull. Classe Sci. Acad. R. Belg. 44:818–829, [1958]), unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X=1+λ a ≺ X and Y=1−λ Y ≻ a in A[[λ]] is provided, where (A,≺,≻) is a dendriform algebra. Then we present the solutions to these equations as an infinite product expansion of exponentials. Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.      

On a class of spaces of analytic functions

The spacesb (p, q, λ) (0<p<q⩽∞, 0<λ⩽∞) of functions, analytic in the circle |z|< 1, are introduced, and an unimprovable estimate is obtained for the Taylor coefficients of a functionf∃ b (p, q, λ). It is shown that B(p, q, λ) is the space of fractional derivatives f(α) of order α (−∞<α<1/p−1/q) of a function f of B(s, q, λ), where s=p/(1−αp). Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 141–150, February, 1977.