Analytic k-spaces

We study sequential convergence in spaces with analytic topologies avoiding thus a number of standard pathologies. For example, we identify bisequentiality of an analytic space as the Fréchet property of its square. We show that a countable Fréchet group is metrizable if and only if its topology is analytic. We also investigate the diagonal sequence properties and show their productiveness in the class of analytic spaces.     

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and K-analyticity, thereby adding to the ‘Baire space recognition literature’ (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak α-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by K-analytic (in particular analytic) sets that are ‘heavy’ (everywhere large in the sense of some σ-ideal). From this perspective fine-topology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy-Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multiple-target form of the Choquet Banach-Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in K-analytic form.     

On the Continuability of Multivalued Analytic Functions to an Analytic Subset

In the paper, it is shown that a germ of a many-valued analytic function can be continued analytically along the branching set at least until the topology of this set is changed. This result is needed to construct the many-dimensional topological version of Galois theory. The proof heavily uses Whitney's stratification.     

Analytic Operator Rings and Analytic J-Self Adjoint Operator Ring

Let A be a linear bounded operator in Hilbert space H with polar respresentation A =J(A^*A)^{1/2} where J^2=I, J^* = J. we use \pho_J(A) to denote the set of all complex \lambda, such that for any $\lambda \in \rho_J(A)$ there exist an bounded inverce R_J(A,\lambda) of (A—\lambda J)and \sigma_J(A)to complement of \rho_J(A). Let S be a closed Cauchy domain, S\supset \sigma_J(A) and f (z) an analytic function on S. We define $f(A)=\frac{1}{2\pi i}\oint\limits_{2s} {f(\zeta ){R_J}(A,\zeta )d\zeta }$, the set of all such f(A)is denoted M. If f(z) be analytic on S and symmetrical for real axis then f(A) is J-self adjoint. The set of all such f(A)is denoted M'. Let A\otimes B = AJB for A, B \in M(or M'). We have Theorem, the ring of functions analytic on S (or analytic symmetrical for real axis on S) is a algebra homomorphism of M (or M'). The constant function 1 or z corresponds to operator J or A^* respectively. Let $M_J={JB|B \in M}$ and $M'_J={JB|B \in M'}$ If the spectrum of (A^*A)^{1/2} is detached, we have Theorem. M_J has common non-trivial reducing subspace and it is true for M_J.     

The Analytic Caratheodory Conjecture

The aim of this article is to provide the reader with a real possibility of becoming confident that the index of an isolated umbilic point of an analytic surface is never greater than one. For a surface homeomorphic to a sphere, this means in particular that on the surface there necessarily exist at least two umbilic points as it was conjectured by Caratheodory.     

蕴含系分析法

针对决策科学中的策略集生成问题,提出蕴含系分析法.该方法基于蕴含系的概念与性质所建立,用于诊断问题形成的根本原因、发掘与问题相关的潜在与隐含因素、搜索求解问题的切入点,提供了一种分析问题和引导问题求解策略生成的形式化方法.     

Analytic continuation by summation-methods

The paper deals with the analytic continuation of the geometric series by a family of linear transformations into some special domains of the complex plane.     

The Analytic Polynomial-Time Hierarchy

Motivated by results on interactive proof systems we investigate an ?-?hierarchy over P using word quantifiers as well as two types of set quantifiers. This hierarchy, which extends the (arithmetic) polynomial-time hierarchy, is called the analytic polynomial-time hierarchy. It is shown that every class of this hierarchy coincides with one of the following Classes: ∑, Π (k?0), PSPACE, ∑ or Π (k?1). This improves previous results by Orponen [6] and allows interesting comparisons with the above mentioned results on inter-active proof systems.     

On Effective Analytic Continuation

Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. It would be interesting to apply a similar spirit of “exact computations” to the field of mathematical analysis. One important step for such a project is the ability to compute with computable complex numbers and computable analytic functions. Such computations include effective analytic continuation, the exploration of Riemann surfaces and the study of singularities. This paper aims at providing some first contributions in this direction, both from a theoretical point of view (such as precise definitions of computable Riemann surfaces and computable analytic functions) and a practical one (how to compute bounds and analytic continuations in a reasonably efficient way). We started to implement some of the algorithms in the MMXLIB library. However, during the implementation, it became apparent that further study was necessary, giving rise to the present paper. The paper was originally written using GNU (see www.texmacs.org). Unfortunately, Birkh?user insists on the use of with its own style files. Any insufficiencies in the typesetting quality should therefore be imputed to Birkh?user.     

Analytic families of semigroups

The Cauchy Problem for linear partial differential equations with coefficients depending analytically on a parameter, motivates the study of analytic families of infinitesimal generators of semigroups. The relations between the three natural ways to understand “analyticity” of the family are clarified. This is then applied to obtain some general results on the stability of the analyticity of the family under perturbations of the generators.     

Analytic Tridiagonal Reproducing Kernels

The paper characterizes the reproducing kernel Hilbert spaceswith orthonormal bases of the form {(a_n,0+a_n,1z+...+a_n,Jz^J)zⁿ,n 0}. The primary focus is on the tridiagonal case where J= 1, and on how it compares with the diagonal case where J =0. The question of when multiplication by z is a bounded operatoris investigated, and aspects of this operator are discussed.In the diagonal case, M_z is a weighted unilateral shift. Itis shown that in the tridiagonal case, this need not be so,and an example is given in which the commutant of M_z on a tridiagonalspace is strikingly different from that on any diagonal space.     

Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

Pseudoconvexity and Analytic Discs

The purpose of this paper is to investigate the geometry of pseudoconvex hypersurfaces and their interrelation with analytic discs. The method of analytic discs is a very powerful tool in the study of pseudoconvexity. One way to use this tool is via the so-called type function introduced by Dwilewicz and Hill for arbitrary Cauchy–Riemann manifolds. In the hypersurface case, and moreover, under the assumption of pseudoconvexity, it has additional properties which are given in this paper. As an application, boundary estimates of plurisubharmonic functions are given.     

Analytic normal basis theorem

Let p be a prime number, ℚ _p the field of p-adic numbers, and a fixed algebraic closure of ℚ _p . We provide an analytic version of the normal basis theorem which holds for normal extensions of intermediate fields ℚ _p ⊆ K ⊆ L ⊆ .      

Analytic methods in quantum computing

Here we consider a model of quantum computation, based on the monodromy representation of a Fuchsian system. The rôle of local and entangling operators in monodromic quantum computing is played by monodromy matrices of connections with logarithmic singularities acting on the fiber of a holomorphic vector bundle as on the space of qubits. The leading theme is the problem of constructing a set of universal gates as monodromy operators induced from a connection with logarithmic singularity. In the formal scheme developed by us, already known models — topological and holonomic — can be incorporated.     

Analytic iteration and differential equations

In this paper we study some mapping properties of analytic iterationsW(a, z). Our purpose is to establish a sufficient condition forW(a, z) to be conformal and univalent inz forz ∈D, whereD is a given domain and for sufficiently small |a |. To this end we consider the differential equationϱW(a, z)/ϱa=L[W(a, z)] with the conditionW(O, z)=z. A sufficient condition for the solutionW(a, z) of this system to be conformal and univalent inD for |a |<a ₀ (for somea ₀>0), and to satisfy the iteration equation, is established. This paper is based on a part of the author’s thesis towards the D.Sc. degree, under the guidance of Professor E. Jabotinsky at the Technion Israel Institute of Technology, Haifa.