Abstract: The supermanifold analogue of the Kodaira-Nirenberg-Spencer existence theorem for deformations of complex structures is given. It is shown that every complex supermanifold is a deformation of a vector bundle. 0. Supermanifolds, first used by physicists for modelling quantum gravity, have emerged as objects of independent interest. This paper will concentrate on supermanifolds with a complex structure, though the results also yield a simple and transparent proof of the fact that any supermanifold with only its C?? structure is the sheaf of sections of a vector bundle [1]. Let X be a complex manifold, with sheaf of holomorphic functions (9. Let 4' be a locally free sheaf of ?7-modules. Then A&, the sheaf of exterior algebras of & over ?, is an example of a complex supermanifold. Ag is, among other things, a sheaf of supercommutative algebras. This means that A&is Z2-graded and ab = (_1)lbI Iaba for a and b of definite parity. A& is also a sheaf of ?-modules, and a sheaf of Z-graded algebras, but for supersymmetry one is concerned only with the Z2-grading. This leads to the following more general definition: DEFINITION. A complex supermanifold of dimension (m, n) is a sheaf (M, -W) of supercommutative algebras over C such that (1) (M, i/AX) is an m-dimensional complex manifold. (X is the ideal of nilpotent elements of -.) (2) The sheaves (M, -V) and (M, AC0 X V/AX) are locally isomorphic as sheaves of Z2-graded commutative algebras over C. Set ?9 = -/Xand & = I/X42. Then &is an ?-module, and it follows from (2) that &is locally free. That is, &is the sheaf of sections of a holomorphic vector bundle. By writing _A'X2 as the direct sum of its even and odd parts, one obtains an exact sequence of sheaves of vector spaces (*~~~~~~~0_). X y2 __* _V __ (9 ED? _ If there is a splitting

Abstract: Our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring. Before stating the result we establish some notation and conventions. Rings are associative and with unit. If A is a ring then GL(A)= Uk?0GLk(A) is its infinite general linear group. An A-bimodule is an abelian group B which is both a left A-module and a right A-module and satisfies (alb)a2 = al(ba2) for ai eA, be B (for example B = A). If B is an A-bimodule, then M(B) = Uk?oMk(B) is the infinite additive group of matrices with entries in B. Conjugation defines an action (the adjoint action) of GL(A) on M(B). Note that an A ? Q-bimodule is just an A-bimodule which is also a rational vector space. If B is an A ? Q-bimodule, then HJ(A C) Q; B) denotes the Hochschild homology of A ? Q with coefficients in B. Our main result (it appears in slightly more detailed form as Theorem V.3) is:

Abstract: In this paper we introduce different notions of approximate extremal points of sets in ordered vector spaces. Following this we state duality relationships related to these types of approximate solutions of vector valued convex optimization problems, and we also mention some applications of these results. Finally we present a vector valued version of Ekeland's famous variational principle, a tool with so many applications in scalar optimization.

Abstract: Let τ denote a point of the Siegel upper-half space be a (column) vector in Cg and m a vector with m′ resp. m″ in Zg as its first resp. second entry vectors, then the series in which , represents an holomorphic function on the product and it is called the theta function of characteristic m.

Abstract: A theorem of Ky Fan is reformulated to yield a Farkas type result for infinite systems of sublinear functionals. By applying this result, necessary optimality conditions are established for minimization problems in normed vector spaces with objective and constraint functionals that are not necessarily convex or differentiable.

Abstract: The notion of approximate solutions or e-solutions emerged early in the development of modern convex analysis. An analogue of the well-known statement concerning the minimum of a convex function and its subgradient also holds in the approximate case: a convex function f has an e-approximate minimum at x if and only if 0 ∈ ∂ 0 f(x), where ∂ 0 f (x) is the e-subdifferential of f at x. Particular attention has been paid to e-subdifferentials (see Hiriart-Urruty, 1982; Demyanov, 1981). This has resulted in the construction of a new class of optimization procedures, the e-subgradient methods. The virtually complete set of calculation rules derived for the e-subdifferential has made possible the study and characterization of constrained convex optimization problems in both the real-valued and vector-valued cases, as in Strodiot et al. (1983), or for ordered vector spaces (Kutateladze, 1978).

Abstract: This paper addresses the issue of 'conservatism' in the time domain stability robustness bounds obtained by Lyapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of Lyapunov stability condition with respect to the basis of the vector space in which the Lyapunov function is constructed. Improved bounds are obtained, using transformation, on elemental and vector norm of perturbations (i.e., structured perturbations) as well as on matrix norm of perturbations (i.e. unstructured perturbations). For the case of diagonal transformation, an algorithm is proposed to find the 'optimal' transformation. Several examples are presented to illustrate the proposed analysis.

Abstract: The singular-value-decomposition (SVD) method has been utilized in an algebraic image-reconstruction problem. The effect of noise on a reconstructed image can be understood from the expansion coefficients of an object in singular vector space calculated by the SVD algorithm. We propose a method in which the coefficients corresponding to small singular values are estimated by using a priori information about the object. Results of computer simulations for iterative restoration of an image subjected to low-pass frequency filtering show the utility of the method.

Abstract: Several theorems on estimation and verification of linear hypotheses in some Zyskind-Martin (ZM) models are given. The assumptions are as follows. Let y = Xβ + e or (y, Xβ, σ2V) be a fixed model where y is a vector of n observations, X is a known matrix nXp with rank r(X) = r ≦ p < n, where p is a number of coordinates of the unknown parameter vector β, e is a random vector of errors with covariance matrix σ2V, where σ2 is unknown scalar parameter, V is a known non-negative definite matrix such that R(X) ⊂ R(V). Symbol R(A) denotes a vector space generated by columns of matrix A. The expected value of y is Xβ. In this paper four following Zyskind-Martin (ZM) models are considered: ZMd, ZMa, ZMc and ZMqd (definitions in sec. 1) when vector yy1y2 involves a vector y1 of m missing values and a vector y2 with (n — m) observed values. A special transformation of ZM model gives again ZM model (cf. theorem 2.1). Ten properties of actual (ZMa) and complete (ZMc) Zyskind-Martin models with missing values (cf. theorem 2.2) test functions F are given in (2.11)) are presented. The third propriety constitutes a generalization of R. A. Fisher's rule from standard model (y, Xβ, σ2I) to ZM model. Estimation of vector y1 (cf. 3.3) of vector β (cf. th. 3.2) and of scalar σ2 (cf. th. 3.4) in actual ZMa model and in diagonal quasi-ZM model (ZMqd) are presented. Relation between y 1 and β is given in theorem 3.1. The results of section 2 are illustrated by numerical example in section 4.

Abstract: I The Problem of Formulating an Axiomatics for Quantum Mechanics.- 1 Is There an Axiomatic Basis for Quantum Mechanics?.- 2 Concepts Unsuitable in a Basis for Quantum Mechanics.- 3 Experimental Situations Describable Solely by Pretheories.- 4 Mathematical Problems.- 5 Progress to More Comprehensive Theories.- II Pretheories for Quantum Mechanics.- 1 State Space and Trajectory Space.- 2 Preparation and Registration Procedures.- 2.1 Statistical Selection Procedures.- 2.2 Preparation Procedures.- 2.3 Registration Procedures.- 2.4 Dependence of Registration on Preparation.- 3 Trajectory Preparation and Registration Procedures.- 3.1 Trajectory Effects.- 3.2 Trajectory Ensembles.- 3.3 The Dynamic Laws and the Objectivating Manner of Description.- 3.4 Dynamically Continuous Systems.- 4 Transformations of Preparation and Registration Procedures.- 4.1 Time Translations of the Trajectory Registration Procedures.- 4.2 Time Translations of the Preparation Procedures.- 4.3 Further Transformations of Preparation and Registration Procedures.- 5 The Macrosystems as Physical Objects.- III Base Sets and Fundamental Structure Terms for a Theory of Microsystems.- 1 Composite Macrosystems.- 2 Preparation and Registration Procedures for Composite Macrosystems.- 3 Directed Interactions.- 4 Action Carriers.- 5 Ensembles and Effects.- 5.1 The Problem of Combining Preparation and Registration Procedures.- 5.2 Physical Systems.- 5.3 Mixing and De-mixing of Ensembles and Effects.- 5.4 Re-elimination of the Action Carrier.- 6 Objectivating Method of Describing Experiments.- 6.1 The Method of Describing Composite Macrosystems in the Trajectory Space.- 6.2 Trajectory Effects of the Composite Systems.- 6.3 Trajectory Ensembles of the Composite Systems.- 6.4 The Structure of the Trajectory Measures for Directed Action.- 6.5 Complete Description by Trajectories.- 6.6 Use of the Interaction for the Registration of Macrosystems.- 6.7 The Relation Between the Two Forms of an Axiomatic Basis.- 7 Transport of Systems Relative to Each Other.- IV Embedding of Ensembles and Effect Sets in Topological Vector Spaces.- 1 Embedding of K, L in a Dual Pair of Vector Spaces.- 2 Uniform Structures of the Physical Imprecision on K and L.- 3 Embedding of K and L in Topologically Complete Vector Spaces.- 4 ?, ?', D, D' Considered as Ordered Vector Spaces.- 5 The Faces of K and L.- 6 Some Convergence Theorems.- V Observables and Preparators.- 1 Coexistent Effects and Observables.- 1.1 Coexistent Registrations.- 1.2 Coexistent Effects.- 1.3 Observables.- 2 Mixture Morphisms.- 3 Structures in the Class of Observables.- 3.1 The Spaces ? (?) and ?' (?) Assigned to a Boolean Ring ?.- 3.2 Mixture Morphism Corresponding to an Observable.- 3.3 The Kernel of an Observable.- 3.4 De-mixing of Observables.- 3.5 Measurement Scales of Observables and Totally Ordered Subsets of L.- 4 Coexistent and Complementary Observables.- 5 Realization of Observables.- 6 Coexistent De-mixing of Ensembles.- 7 Complementary De-mixings of Ensembles.- 8 Realizations of De-mixings.- 9 Preparators and Faces of K.- 10 Physical Objects as Action Carriers.- 11 Operations and Transpreparators.- VI Main Laws of Preparation and Registration.- 1 Main Laws for the Increase in Sensitivity of Registrations.- 1.1 Increase in Sensitivity Relative to Two Effect Procedures.- 1.2 Some Experimental and Intuitive Indications for the Law of Increase in Sensitivity.- 1.3 Decision Effects.- 1.4 The Increase in Sensitivity of an Effect.- 2 Relations Between Preparation and Registration Procedures.- 2.1 Main Law for the De-mixing of Ensembles and Related Possibilities of Registering.- 2.2 Some Consequences of Axiom AV2.- 3 The Lattice G.- 4 Commensurable Decision Effects.- 5 The Orthomodularity of G.- 6 The Main Law for Not Coexistent Registrations.- 6.1 Experimental Hints for Formulating the Main Law for Not Coexistent Registrations.- 6.2 Some Important Equivalenees.- 6.3 Formulation of the Main Law and Some Consequences.- 7 The Main Law of Quantization.- 7.1 Intuitive Indications for Formulating the Main Law of Quantization.- 7.2 Simple Consequences of the Main Law of Quantization.- VII Decision Observables and the Center.- 1 The Commutator of a Set of Decision Effects.- 2 Decision Observables.- 3 Structures in That Class of Observables Whose Range also Contains Elements of G.- 4 Commensurable Decision Observables.- 5 Decomposition of ? and ?' Relative to the Center Z.- 5.1 Reduction of the Elements of ?' by the Elements of G.- 5.2 Reduction by Center Elements.- 5.3 Classical Systems.- 5.4 Decomposition into Irreducible Parts.- 6 System Types and Super Selection Rules.- VIII Representation of ?, ?' by Banach Spaces of Operators in a Hilbert Space.- 1 The Finite Elements of G.- 2 The General Representation Theorem for Irreducible G.- 3 Some Topological Properties of G.- 4 The Representation Theorem for K, L.- 4.1 The Representation Theorem for G.- 4.2 The Ensembles and Effects.- 4.3 Coexistence, Commensurability, Uncertainty Relations, and Commutability of Operators.- 5 Some Theorems for Finite-dimensional and Irreducible ?.- A II Banach Lattices.- A III The Axiom AVid and the Minimal Decomposition Property.- A IV The Bishop-Phelps Theorem and the Ellis Theorem.- List of Frequently Used Symbols.- List of Axioms.

Abstract: We study the characterization of the 13 cases obtained in the classification of the irreducible linear–antilinear representations of semigroups in a finite‐dimensional vector space X over an algebraically closed field K with a conjugation j (generalized Frobenius–Schur–Wigner, or FS×W, classification). It has already been shown that each case can be characterized by various equivalent properties, some of which can be endowed with a physical interpretation. We show here that, whenever K is the complex field C, each case can be characterized by the structure (in the sense specified by the Weyl theorem on the structure of the matrix algebras and their commutators) of a pair of suitable operator algebras over the real field R. This characterization coincides with the one given by Dyson for each case of his classification of symmetry groups. Thus, the latter classification is recovered under more general assumptions and in a generalized framework, and its one‐to‐one correspondence with the generalized FS×W cla...

Abstract: Let V be a finite dimensional vector space over the field Fand φ (x)∊F[x].Letx V → V be a linear operator. Let Sφ be the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S φ to be a subspace.

Abstract: The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grossencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grossencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grossencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.

Abstract: It is proved that the minimum number of generators of the lattice of all subspaces of a finite-dimensional vector space over a field is finite if and only if the field is finitely generated over the prime field. An upper bound is given for this number, which does not depend on the dimension of the space and is linearly dependent on the number of elements generating the field.

Abstract: LetG be a vector space over the field of rational numbers andf, g:G → ℝ ℚ-linear mappings. ℝ equipped with the usual norm topology. Denote byτf,τg the initial topologies onG induced byf respectivelyg.