LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART I): A field-theoretic treatment of variational problems in n independent variables {x^j} and N dependent variables {ψ^A)} is presented that differs substantially from the standard field theories, such as those of Carathéodory [4] and Weyl [10], inasmuch as it is not stipulated ab initio that the Lagrangian be everywhere positive. This is accomplished by the systematic use of a canonical formalism. Since the latter must necessarily be prescribed by appropriate Legendre transformations, the construction of such transformations is the central theme of Part I.—The underlying manifold is M = M_n x M_N, where M_n, M_N are manifolds with local coordinates {x^j}, {ψ^A}, respectively. The basic ingredient of the theory consists of a pair of complementary distributions D_n, D_N on M that are defined respectively by the characteristic subspaces in the tangent spaces of M of two sets of smooth 1-forms {π^A:A = 1,…, N}, {π^j = 1,…, n} on M. For a given local coordinate system on M the planes of 4, have unique (adapted) basis elements B_j = (?/?_x ^j) + B^A _j (?/?ψ^A), whose coefficients B^A _j will assume the role of derivatives such as ?ψ^A/?x^j in the final analysis of Part II. The first step toward a Legendre transformation is a stipulation that prescribes B^A _j as a function of the components {π^j _h,π^j _A} of {π^j}—these components being ultimately the canonical Variables—in such a manner that B^A _j is unaffected by the action of any unimodular transformation applied to the exterior system {π^j}. A function H of the canonical variables is said to be an acceptable Hamiltonian if it satisfies a similar invariance requirement, together with a determinantal condition that involves its Hessian with respect to π^j _A. The second part of the Legendre transformation consists of the identification in terms of H and the canonical variables of a function L that depends solely on B^A _j and the coordinates on M. This identification imposes a condition on the Hessian of L with respect to B^A _j. Conversely, any function L that satisfies these requirements is an acceptable Lagrangian, whose Hamiltonian is uniquely determined by the general construction.     

INTRINSIC FUNCTIONAL ANALYSIS OF SEQUENTIAL CONVERGENCE SPACES

We study the functional analysis intrinsic to sequential convergence spaces with particular attention to reflexiveness of canonical function spaces and the categorical Banach-Steinhaus principle.     

NONLINEAR HOMOTOPY AND PRE-POINCARÉ LEMMAS

Abstract

Most homotopies considered in the literature are linear homotopies of the form h ⁱ (λ) = λx ⁱ + (1—λ)y ⁱ , 0 ≤ λ ≤ 1. Although these prove to be adequate in most instances, they lack direct geometric significance because {h ⁱ (λ) | 0 ≤ λ ≤ 1} are not orbits of a vector field. On the other hand, the nonlinear homotopy g ⁱ (s) = e ^s x ⁱ + (1—e ^s )y ⁱ ,—∞ ≤ s ≤ 0, are orbits of a vector field (i.e., dg ⁱ /ds = g ⁱ —y ⁱ , g ⁱ (0) = x ⁱ ), and thus have direct geometric significance. This suggests that useful results can be obtained by replacing linear homotopy by transport along flows of smooth vector fields. The purpose of this paper is to elaborate on this simple idea. We define prehomotopy operators induced by vector fields on a manifold. These allow us to obtain finite transport relations and pre-Poincaré lemmas that generalize the classical results. They are shown to reproduce the classical results as asymptotic limits and to obtain representations of all solutions of complete systems of exterior differential equations on a star shaped region of a manifold.     

NETS AND GRAPHS AS PINAL COMPLETIONS AND THE RELATIONS BETWEEN THEM

Abstract

Nets and graphs, both used in Computer Science, are studied from a categorical point of view. It is shown that they may be constructed via final completions of very simple small concrete categories and that their nice properties, namely to form topological categories which are quasitopoi with concrete powers such that products of final maps are final, depend on this fact. Furthermore, the relations between them can be described by means of adjoint functors.     

CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES

Abstract

The notion of Cauchy point (= regular Cauchy filter) and the corresponding Cauchy spectrum, for a nearness frame (= uniform without the star-refinement condition) are investigated in various directions, including basic motivation, several functorial aspects, and the recognition of the Cauchy spectrum as the ordinary spectrum of the completion, after the unique existence of the latter is obtained as a central new result in this context.     

HOMOGENEOUS FUNCTIONS DETERMINED BY CYCLIC SUBMODULES

Abstract

For a unital module V over a commutative ring R, let C denote the collection of cyclic submodules. The ring ?_R(V;C) = {f ε End_R V |f(C) ?C, ?C ε_R (V;C) has been the object of several recent studies in which the structure of ?_R(V;C) is related to the triple (V, R,C). Here we introduce a new ring H_R(V;C) containing ?(V;C) and investigate its structure in terms of the parameters (V, R, C).     

The Schur Lie-multiplier of Leibniz algebras

Abstract

For a free presentation 0 → τ → → → 0 of a Leibniz algebra , the Baer invariant is called the Schur multiplier of relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal of a Leibniz algebra , we construct a four-term exact sequence relating the Schur Lie-multipliers of and /, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.     

A new factor theorem for absolute matrix summability

Abstract

In this paper, we have extended a known theorem dealing with |, p_n|k summability factors to the |A, p_n|k summability under weaker conditions by using an almost increasing sequence     

ON ORBITS OF POLYNOMIAL MAPS IN BANACH SPACES

Abstract

We obtain some results concerning the notions of hypercyclicity and supercyclicity for continuous homogeneous polynomial maps in Banach spaces. We also present a generalization of a theorem due to Beauzamy concerning the existence of orbits tending to infinity for continuous linear operators with spectral radius > 1.