Arbitrary Vector Space Research Articles

Multi-affine and multi-Jensen functions and their connection with generalized polynomials

In 1987 Lyle Ramshaw ([R]) has introduced the concept of multi-affine functions, called blossoms. This idea unified some important approaches and methods in the theory of polynomial splines. The main observation is that any polynomial $F:\mathbb{R}^p \to \mathbb{R}^q $ of degree ≤ n is the diagonalization of a (unique) symmetric n-affine function $f:(\mathbb{R}^p )^n \to \mathbb{R}^q :\;F(x) = f(x, x, \ldots ,x)$ for all $x \in \mathbb{R}^p .$ Here we investigate these ideas with the aim to get similar characterizations for generalized polynomials F : V → W, where V, W are arbitrary vector spaces over $\mathbb{Q}.$

A generalization of the Chandler Davis convexity theorem

In 1957 Chandler Davis proved a theorem that a rotationally invariant function on symmetric matrices is convex if and only if it is convex on the diagonal matrices. We generalize this result to groups acting nonlinearly on convex subsets of arbitrary vector spaces thereby understanding the abstract mechanism behind the classical theorem. We apply the new theorem to a problem from the mathematical theory of composite materials and derive its corollaries in the Lie algebra setting. Using the latter, we show that the Pfaffian is log-concave.

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components. As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A 0 of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A 0-linear. We also show that in this case A is simple if and only if A 0 is a field.

Partitioning large vector spaces

The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝdto vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago.Sierpiński's Theorem.The Continuum Hypothesis is equivalent to: There is a partition{X, Y, Z}ofℝ3such that if ℓ is a line parallel to the x-axis[respectively:y-axis, z-axis]then X∩ℓ[respectively:Y∩ℓ, Z∩ℓ]is finite.The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝn+2inton+ 2 sets obtaining an equivalence with. The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.Kuratowski's Theorem.Let n < ω and A be any set. Then∣A∣ ≤ ℵnif and only if there is a partition P:An+2→n+ 2such that if i≤n+ 1and ℓ is a line parallel to the i-th coordinate axis, then{x∈ℓ:P(x) =i}is finite.

On C-sequences of operators

Invariant results are established for a considerably general multiplier convergence of operator series where the operators are defined on arbitrary topological vector spaces and valued in arbitrary locally convex spaces.

Continuous representability of homothetic preorders by means of sublinear order-preserving functions

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.

Pták Homomorphism Theorem Revisited

Rodrigues' extension (1989) of the classical Pták's homomoiphism theorem to a non-necessarily locally convex setting stated that a nearly semi-open mapping between a semi-B-complete space and an arbitrary topoiogical vector space is semi-open. In this paper we study this extension and, as a consequence of the results obtained, provide an improvement of Pták's homomorphism theorem.

Products of two idempotent transformations over arbitrary sets and vector spaces

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

Some fixed point theorems for nonconvex spaces

We give a theorem for nonconvex topological vector spaces which yields the classical fixed point theorems of Ky Fan, Kim, Kaczynski, Kelly and Namioka as immediate consequences, and prove a new fixed point theorem for set‐valued maps on arbitrary topological vector spaces.

Remarks on regular (LF)-spaces

Developing ideas found in a recent paper of Gilsdorf to arbitrary topological vector spaces (tvs) one shows that a Hausdorff (LB)tv-spaceE is regular provided every null-sequence inE has a series convergent subsequence inE.

Vector logics: The matrix-vector representation of logical calculus

We describe a kind of logical calculus based on matrix operators. This representation is inspired on a neural network model for a context-dependent associative memory. The binary operations of classical propositional calculus (e.g. conjunction, disjunction, implication, Sheffer's connective) are represented by rectangular matrices that operate over the Kronecker product of two vectors that belong to an arbitrary vector space. The basic matrix operators verify some of the classical laws of logical connectives, like De Morgan's Laws and the Law of Contraposition. We show that the matrices that execute the logical operations negation, conjunction and disjunction are also able to recognize the set operations complementation, intersection and union, respectively. In the presence of fuzzy data the basic binary matrix operators generate a many-valued logic.

Séparation et polarité

To every subset of an arbitrary vector space we introduce a new cone in the dual, We show that there is a connection to the polar cone and to the well-known prepolar cone. Then we prove the relationship to the polarity at the separation:the characterization of all separating planes is possible.

Products of idempotent linear transformations

SynopsisIn 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L ∞( M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's ( J. Econ. Theory 4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.

Further results on the asymptotic cone

We give new properties of the asymptotic cone \(\mathbb{A}\)(A) of a (non-necessarily convex) set A in an arbitrary real vector space E, as an answer to some questions raised by Jongmans [IV]. Especially, we characterize the asymptotic cone of a weakly star-shaped set and, in corollary, we show that the asymptotic cone of a convex set A coincides with the infinitude or the recession cone of the closure of A for the natural topology on E; last but not least, we find in Euclidean space a fairly general law expressing the polar of the barrier cone of an arbitrary set A as the closure of the convex hull of \(\mathbb{A}\)(A).

Doubly stochastic matrices over arbitrary vector spaces and the Birkhoff theorem

Doubly stochastic matrices are defined which have entries from an arbitrary vector space V. The extreme points of this convex set of matrices are studied, and convex subsets of V are identified for which these extreme matrices are of a permutation matrix type, i.e. for which a Birkhoff theorem holds.

Linear operators whose domain is locally convex

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on S separates the points of S. If X is a Banach space and T : X → F is a continuous linear operator, then T is quasi-convex if is quasi-convex, where U is the unit ball of X.

Criteres de separation pour polyedres convexes

We give a few results about the separation of two convex polyhedra in an arbitrary real vector space; so we generalize theorems of Cernikov, Klee, Rockafellar and Gallivan-Zaks.

Exhaustive measures in arbitrary topological vector spaces

Exhaustive measures in arbitrary topological vector spaces

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint