In this paper we discuss the shape of the quadrature domain of a signed measure for harmonic functions. It is known that the quadrature domain of a positive measure with small support is like a ball if the total measure is large enough. We show that, on the contrary, if the measure is not positive then the quadrature domain can be close to an arbitrary domain. This follows from a lemma concerning linear combinations of harmonic measures.

This paper is based on the interpretation of the ring of additive polynomials in one variable over a finite field Fq, as a maximal R-order inside a certain skew-field D, R being a principal ideal domain isomorphic to Fp[T]. The classical (1930's) structure theory of maximal orders in global fields is used to solve enumeration questions involving the iteration of members of Pages from .

Using the Eichler-Shimura isomorphism and the action of the Hecke operator T2 on period polynomials, we shall give a simple and new proof of the following result (implicitly contained in the literature): let f be a normalized Hecke eigenform of weight k with respect to the full modular group with eigenvalues λp under the usual Hecke operators Tp (p a prime). Let Kf be the field generated over Q by the λp for all p. Let p be a prime of Kf lying above 5. Then

In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann – Schatten p-classes of compact operators. The ideas developed there can be used in ℓp-spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results.

The usual definition of hyperbolicity of a group G demands that all geodesic triangles in the Cayley graph of G should be thin. Using the theorem that a susbquadratic isoperimetric inequality implies a linear one, we show that it is in fact only necessary for all triangles from a given combing to be thin, thus giving a new criterion for hyperbolicity of finitely presented groups.

In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar localization properties.

We relax the growth condition in time for uniqueness of solutions of the Cauchy problem for the heat equation as follows: Let u(x, t) be a continuous function on ℝn × [0, T] satisfying the heat equation in ℝn × (0, t) and the following:

(i) There exist constants a > 0, 0 < α < 1, and C > 0 such that

(ii) u(x, 0) = 0 for x ∈ ℝn.

Then u(x, t)≡ 0 on ℝn × [0, T]

We also prove that the condition 0 < α < 1 is optimal.

We give an algorithm which for a given set of generators can decide whether a stem product of infinite cyclic groups is a one-relator group. We also generalize this to the case of one-relator products of groups.

For the real interpolation method, we identify the interpolated spaces of couples of classical Lorentz spaces through interpolation of the corresponding weighted Lp-spaces restricted to decreasing functions.

We study the relationship between the dual of the K-space defined by means of a polygon and the J-space generated by the dual N-tuple. The results complete the research started in [4]. Special attention is paid to the case when the N-tuple is formed by Banach lattices

Let (K, v) be a complete, rank-1 valued field with valuation ring Rv, and residue field kv. Let vx be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by The classical Hensel's lemma asserts that if polynomials F(x), G0(x), H0(x) in Rv[x] are such that (i) vx(F(x) – G0(x)H0(x)) > 0, (ii) the leading coefficient of G0(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of vx satisfying vx(A(x)G0(x) + B(x)H0(x) – 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)H(x), (b) deg G(x) = deg G0(x), (c) vx(G(x)–G0(x)) > 0, vx(H(x) – H0(x)) > 0. In this paper, our goal is to prove an analogous result when vx is replaced by any prolongation w of v to K(x), with the residue field of wa transcendental extension of kv.

In this paper we shall develop a new method for the computation of eigenvalues of singular Sturm-Liouville problems of the Bessel type. This new method is based on the interpolation of a boundary function in Paley-Wiener spaces. Numerical results are provided to illustrate the method.

In this note we show that if we have an exact sequence of AH algebras (AH stands for “approximately homogeneous”) 0 → I → A → B → 0, then A has the ideal property (i.e., any ideal is generated by its projections) if and only if I and B have the ideal property. Also, we prove that an extension of two AT algebras (AT stands for “approximately circle”) with the ideal property is an AT algebra with the ideal property if and only if the extension is quasidiagonal.

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

We give necessary and sufficient conditions on a general cone of positive functions to satisfy the Decomposition Property (DP) introduced in [5] and connect the results with the theory of interpolation of cones introduced by Sagher [9]. One of our main result states that if Q satisfies DP or equivalently is divisible, then for the quasi-normed spaces E0 and E1,

According to this formula, it yields that the interpolation theory for divisible cones can be easily obtained from the classical theory.

This is the first of three papers in which we generalise the classical McAlister structure theory for E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.

We prove two types of vanishing results for the Euler characteristic.

In this paper, finite travelling waves for the semilinear parabolic systems

are studied, where di > 0, ei > 0, mij ≥ 0 for all 1 ≤ i,j ≤ n, and for all 1 ≤ i ≤ n. Let M = (mij)n × n and A = I – M. It will be proved that (*) has finite travelling waves if and only if all principal minors of A are positive. Moreover, some asymptotic behaviours of finite travelling waves will be obtained.

This paper is concerned with the relationship between the properties of the subalgebra lattice ℒ(L) of a Lie algebra L and the structure of L. If the lattice ℒ(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice ℒ(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A ≤ L and B is maximal in A (Lie algebras satisfying this condition are called sχ-algebras). Our aim is to characterize lower semimodular Lie algebras and sχ-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modularmaximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.