Trace series on

Let K be a number field which contains the absolute Galois group of K. Let be the completion of relative to the spectral norm . A trace series and a trace function are associated to an element . A measure preserving function H:G{inK}→[0,1] is constructed and a Fourier type theory is considered for elements of .     

Variational method for precise asymptotic formulas for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem motivated by the perturbed elliptic sine-Gordon equation where p >1 is a constant and λ ∈ R is an eigenvalue parameter. Our aim is to clarify the asymptotic relationship between L ^p+1-norm, L ^∞-norm of the solutions and λ when these norms are large. To this end, we consider an associated variational problem with this equation on a manifold new parameter), and obtain a solution pair . We establish the precise asymptotic formulas for .     

Two Parameter Asymptotic Spectra

The two parameter eigenvalue problem in Hilbert space is discussed for selfadjoint operators T. with discrete spectrum and bounded symmetric V_jk satisfying “uniform right definiteness”. Then there are countably many eigenvalues in ?². Results are given relating the set of limit points of ) to a set defined entirely by the V_jk.     

Variations on ω-boundedness

Let be a property (or, equivalently, a class) of topological spaces. A space X is called -bounded if every subspace of X with (or in) has compact closure. Thus, countable-bounded has been known as ω-bounded and (σ-compact)-bounded as strongly ω-bounded. In this paper we present a systematic study of the interrelations of these two known “boundedness” concepts with -boundedness where is one of the further countability properties weakly Lindelöf, Lindelöf, hereditarily Lindelöf, and ccc.     

Orthogonality and Additivity Modulo Z

Let E be a real inner product space of dimension at least 2. Suppose ? : E → ? satisfies If there exist a neighbourhood U of the origin and γ ∈ (0, 1/4) such that ?(U) ? (?γ,γ) + ?, then there exist a real constant c and a continuous linear functional g : E → ? such that Suppose Φ : E → ? satisfies If there exist a neighbourhood U of the origin and β ∈ (0, +∞) such that ¦Φ(x)¦ ≤ β?(Φ(x)) for every x ∈ U, then either Φ vanishes on E? {0} or there exist additive functions a : ? → ? and A : E → ?, a real constant c and a continuous linear functional g : E → ? such that      

Isometric approximation property of unbounded sets

We give a necessary and sufficient quantitative geometric condition for an unbounded set A ? Rⁿ to have the following property with a given c > 0: For every ε ≥ 0 and for every map f: A → Rⁿ such that , there is an isometry T: A → Rⁿ such that ¦Tx?fx¦ ≤ cε for all x ∈ A.     

Base subsets in symplectic Grassmannians of small indices

Let V and V′ be n-dimensional vector spaces. Let also Ω and Ω′ be non-degenerate symplectic forms defined on V and V′, respectively. Denote by Π and Π′ the projective spaces associated with V and V′. We put for the sets of k-dimensional totally isotropic subspaces of Π and Π′. We study mappings of BF03322901 which transfer base subsets to base subsets and show that such mappings are induced by strong embeddings of Π to Π′ if 3k + 3 ≤ n.     

Expansions in Series of Suitable Systems of Functions

The paper is devoted to the study of expansions of functions ?, which are holomorphic in a circle Ω around 0, in a series of the form where the given system of functions (f_k) _k=0 ^? has the property that each function f_k has a representation as a power series of the form with a ₀ ^(k) = 1. It is the question if and where an expansion of the form (*) holds. We present sufficient conditions for the sequence (a _n ^(k) ) _k,n=0 ^∞ , under which there exists a subset Ω₀ of Ω such that for arbitrary ? the expansion (*) holds on Ω₀ and the series converges absolutely uniformly on compact subsets of Ω₀. The obtained results can be applied to prove expansions of analytic functions in a series of a suitable system of m-fold products of Bessel functions. Expansions of this type have been treated by many authors in special cases, but we are able to state a general expansion theorem.     

Further remarks on the non-degeneracy condition

The structure of the set of positive solutions of the semilinear elliptic boundary value problem depends on a certain non-degeneracy condition, which was proved by K.J. Brown [2] and T. Ouyang and J. Shi [12], with a shorter proof given later by P. Korman [8]. In this note we present a more general result, communicated to us by L. Nirenberg [13]. We also discuss the extensions in cases when the domain D is in R ², and it is either symmetric or convex.     

On the best convergence of trigonometric integrals

ОсНОВНОИ РЕжУльтАт Ё тОИ стАтьИ жАклУЧАЕт сь В слЕДУУЩЕМ. ЕслИ с ФИксИРОВАННыМr=2, 3,..., тО Дль тОгО, ЧтОБы ИМЕлО М ЕстО (*) гДЕ ДОстАтОЧНО ВыпОлНЕН ИЕ слЕДУУЩИх УслОВИИ: 1) ЕслИ $$1< \frac{{n_\nu + 1}}{{n_\nu }} = g_\nu \in N, \nu = 0,1,$$ тО Дль кАжДОгОΝ=0,1,... ИМЕ Ет МЕстО (**) 2) ЕслИ тО (***) $$n_{\nu + 1} \geqslant (2r - 1)n_\nu , \nu = 0,1,...$$ . ЕслИ И тО УслОВИь (**), (***) ьВльУтсь тАкжЕ НЕОБхОДИМыМИ Д ль (*).     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

The Approximation of The Continuous Functions by Means of Some Linear Positive Operators

In this paper we give positive answers to some problems posed by H.Gonska, respectively A.Lupa?. We show here that there exist positive linear operators H _n : C[0, 1] → π_n satisfying      

On inductive limits of measure spaces

We introduce a category of (topological) measure spaces in which inductive limitis exist and where the Banach spaces and (1≤p≤+∞) are isometric for arbitrary inductive systems of (topological) measure spaces.     

Martindale rings and H-module algebras with invariant characteristic polynomials

Under study is the category of the possibly noncommutative H-module algebras that are mapped homomorphically onto commutative algebras. The H-equivariant Martindale ring of quotients Q _H (A) is shown to be a finite-dimensional Frobenius algebra over the subfield of invariant elements Q _H (A) ^H and also the classical ring of quotients for A. We introduce a full subcategory of such that the algebras in are integral over its subalgebras of invariants and construct a functor ?? , which is left adjoined to the inclusion ?? .     

Polylogarithms in the Theory of Univalent Functions

Let a, b > ?1, p and q be non-zero real numbers such that p + q ≥ 1. In this paper, we study the geometric mapping properties of the (normalized) generalized polylogarithm defined in the unit disc      

Lelong numbers of positive plurisubharmonic currents

A (k,k)-current T on an open subset of C^N is plurisubharmonic if is positive. Positive plurisubharmonic currents admit Lelong numbers; we prove here that they are independent on the coordinates system. Moreover, if Y is an analytic subset of pure dimension, then the Lelong numbers of T on Y are given by a non negative weakly plurisubharmonic function.     

Zur Verteilung der Quadrate ganzer Zahlen in rationalen Quaternionenalgebren

For γ ∈ ?letQ 〈γ〉 = ?[i]+?[i]j. where j. is a hypercomplex number withj2 = γ, and define addition and multiplication formally with respect to $zj = j\overline z $ for all z ∈ ?[i], so thatQ〈γ〉 becomes a quaternion algebra over the rationals. Further fix γ s.t.Q 〈γ 〉 is a division algebra and define for real X ≥ 1 where |Re(α)|, |Im(α)|, |Re(β)|, |Im(ö)|≤ X and Generalizing former results concerning Hamilton’s quaternions (i.e. the case γ =- 1) we show that, as X → ∞, when γ < 0, when γ > 0, when γ < 0, wheny γ 0. Thereby δ(t) is any upper bound of the error term in Dirichlet’s divisor problem, e.g. δ(t) =t0.315, C_γ, D_γ > 0 are numerical constants, and c, d are given by c := π(1 + log 2 - 2η) and d := π2(1 + 4 log 4 - 4π)/8, where π = 0.577 … is Euler’s constant.     

Euler characteristics and Gysin sequences for group actions on boundaries

Let G be a locally compact group, let X be a universal proper G-space, and let be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup . Let . Assuming the Baum-Connes conjecture for G with coefficients and C(?X), we construct an exact sequence that computes the map on K-theory induced by the embedding . This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincaré duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic is non-zero, then the unit element of is a torsion element of order . Furthermore, we get a new proof of a theorem of Lück and Rosenberg concerning the class of the de Rham operator in equivariant K-homology.     

Segal-Bargmann transforms associated with finite Coxeter groups

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on ? ^N . We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space of holomorphic functions on with reproducing kernel equal to the Dunkl-kernel. The definition and properties of extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further, we prove a branching decomposition of as a unitary -module and a general version of Hecke's formula for the Dunkl transform.