Open Convex Subset Research Articles

Periodic bounce solutions of dynamical conservative systems and their minimal period

SEVERAL important studies have been made on the problem of finding periodic solutions of a conservative dynamical system. An important part of such a study is concerned with the case in which the minimal period of the trajectory is given (see e.g. [l, 3, 6, 9, 14)). In this paper we consider the problem of existence of periodic trajectories with prescribed minimal period, when the material point moves in a convex billiard in I?” under the action of a conservative field and it bounces in a completely elastic way against the boundary, hitting it orthogonally. If Q is a bounded open convex subset of I?” (not necessarily regular), T > 0 and V is a real Cz-function in a neighbourhood of 0, we consider the nonconstant periodic orbits q : R” -+ Q (which are called “principal bounce trajectories” @.b.t.)) having period 2T such that:

On a Bornological Structure in Infinite-Dimensional Holomorphy

The bornology (b) of bounded subsets with respect to continuous convergence is used on spaces of holomorphic functions. It is shown that HomcoHb(U) ≅ U for a circled convex open subset U of a complete nuclear space. Exponential laws for spaces of holomorphic functions with bornological structures are proved and the connection with Colombeau's Silva holomorphic functions is established.

A chain rule involving vector functions of bounded variation

By f ϵ lbv( I, X), we mean that f is a function of a real interval I to a Banach space X, with bounded variation on every compact subinterval of I; to such f, an X-valued measure df, called its differential measure, classically corresponds. Let Ω be an open convex subset of X and γ: Ω → R . Two situations are investigated where the function γ ∘ f: t → γ ( f( t)) belongs to lbv( I R ) and some properties of the real measure d( γ ∘ f) are established. In the first case, γ is supposed convex and continuous in Ω. The subdifferential δγ is invoked in the sense of Convex Analysis; under the ordering of real measures, d( γ ∘ f) is shown to satisfy some inequalities. This generalizes previous results of one of the authors, aimed at deriving energy-like inequalities in nonsmooth mechanical evolution problems. In the second case, γ is supposed Lipschitz on every bounded subset of Ω and Clarke's generalized gradient of γ is used. In both situations, if γ happens to be Gâteaux-differentiable, and f ϵ lbv( I, X) continuous, a chain rule of the familiar form is found to hold. Finally, for γ Fréchet-differentiable, an expression of d( γ ∘ f) is obtained.

Hyperplane complements of large type

Let f2 be an open convex subset of a finite dimensional real vectorspace V and M a collection of linear hyperplanes. We shall suppose that M is locally finite on f2. This means that each point of f2 has a neighborhood meeting only finitely many hyperplanes in M. As in [2; Chap. 5] f2 is partitioned by M in facets. Recalling that a plinth is a codimension 2 facet we shall say that M is of large type on f2 if for each plinth P its multiplicity # {M~M; P c M } > 3. The main result of the paper is the following.

Einbettung von topologischen Raumgeometrien auf R3 in den reellen affinen Raum

Betten [1] had defined topological spatial geometries on R 3: In R 3 a system L of closed subsets homeomorphic to R (the lines) and a system ℰ of closed subsets homeomorphic to R 2 (the planes) are given such that through any two different points passes exactly one line and through any three non-collinear points passes exactly one plane. Furthermore, ℒ and ℰ carry topologies such that the operations of joining and intersection are continuous. It is proved that any topological spatial geometry on R 3 can be imbedded into R 3 as an open convex subset K such that the lines in ℒ (planes in ℰ) are mapped onto intersections of lines (planes) of R 3 with K. The collineation group of the geometry is isomorphic to the subgroup of the colineation group of real projective space consisting of the automorphisms that map K into itself. In particular, it is a Lie group of dimension ⩽12.

On the characterisation of Asplund spaces

AbstractA Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

Geometrical implications of upper semi-continuity of the duality mapping on a Banach space

For the duality mapping on a Banach space the relation between lower semi-continuity and upper semi-continuity properties is explored, upper semi-continuity is characterized in terms of slices of the ball and upper semi-continuity properties are related to geometrical properties which imply that the space is an Asplund space. The duality mapping is a natural set-valued mapping from the unit sphere of a normed linear space into subsets of its dual sphere, and which for an inner product space is the mapping associating an element of the unit sphere with the corresponding continuous linear functional given by the inner product. It is an example of a subdifferential mapping of a continuous convex function (in this case, the norm), which is in turn a special kind of maximal monotone mapping. Cudia [4, p. 298] showed that the duality mapping is always upper semi-continuous when the space has the norm and the dual space has the weak* topology, and Kenderov [10, p. 67] extended this to maximal monotone mappings. Bonsall, Cain, and Schneider [3] used the property to prove the connectedness of the numerical range of an operator on a normed linear space. Along with the activity which culminated in StegalΓs theorem [15] characterizing an Asplund space as one whose dual has the Radon-Nikodym property, there has been some interest in finding geometrical conditions sufficient for a space to be Asplund. A Banach space X is an Asplund space if every continuous convex function defined on an open convex subset of X is strongly differentiate on a dense Gδ subset of its domain. Ekeland and Lebourg [6, p. 204] have shown that a Banach space is Asplund if there exists a strongly differentiate real function on the space with bounded nonempty support, in particular, if the space can be equivalently renormed to have norm strongly differentiable on the unit sphere. Using StegalΓs theorem, a result of Diestel and Faires [5, p. 625] gives that a Banach space is Asplund if the space can be equivalently renormed to be very smooth, that is, to be smooth and to have the single valued duality mapping continuous when the space has the norm and the dual space has the weak topology. Recently Smith and Sullivan [13, Theorem 15] have exhibited a more general condition which is sufficient for a Banach space to be Asplund. We show that

Geometric convexity. III. Embedding

The straight line spaces of dimension three or higher which were considered by the first author in previous papers are shown to be isomorphic with a strongly open convex subset of a real vector space. To achieve this result we consider the classical descriptive geometry studied in various papers and textbooks by Pasch, Hilbert, Veblen, Whitehead, Coxeter, Robinson, and others, with the significant difference that the geometry considered here is not restricted to 3 dimensions. Our main theorem (which is well known in dimension 3) is that any such geometry is isomorphic to a strongly open convex subset of a real vector space whose “chords” play the role of lines.

Minimum-time isochronal surfaces for certain third-order systems

Explicit algebraic expressions for the minimum-time isochronal surfaces for certain third-order systems with real eigenvalues and a single saturable control input are presented in this paper. For a system with positive real eigenvalues in simple ratio the region of controllability (an open convex subset of states from which the state origin can be reached in finite time) is precisely defined

Amenability and invariant subspaces

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

Some properties of support functionals

The author considers convex functionalsf t defined on the open convex subset G of real B-space X and continuous at x0∈ G. All linear functionals on X which are support functionals of form $$\int_T {f_t (x)} \mu (dt) and \begin{array}{*{20}c} {max} \\ {t \in T} \\ \end{array} f_t (x)$$ . at x0 are described.

Rings of Continuous Functions on Open Convex Subsets of R n

Rings of Continuous Functions on Open Convex Subsets of R n

Rings of continuous functions on open convex subsets of 𝑅ⁿ

0. Introduction. If h is a homeomorphism from a topological space X onto another space Y, then one can easily show that the correspondence f-f o h-1 is an isomorphism from C(X), the ring of all real continuous functions on X, onto C( Y), the ring of all real continuous functions on Y. The converse problem-given an isomorphism 4 from C(X) onto C(Y) to show that there exists a homeomorphism h from X onto Y such that qf =f o h-l is more difficult and is one of the problems which has motivated much of the research on rings of continuous functions. The isomorphism 4 is usually constructed by using fixed, maximal ideals. An ideal M in C(X) is fixed if there exists a point xo of X such that f(xo) = 0 for all f in M. If X is completely regular, then the set M(x) of all functions in C(X) which are zero at x is a fixed, maximal ideal and all fixed, maximal ideals are of this form. If one can show that the property of a maximal ideal being fixed is invariant under ring isomorphisms and if X and Y are completely regular, then one can show that the correspondence

On the Graded Ring of Theta-Constants

Introduction. Let g be a positive integer and let T be a complex symmetric matrix of degree g with a positive-definite imaginary part. The set of such matrices forms a convex open subset of the 1 * g (g + 1) -dimensional complex vector space and (as a complex manifold) it is called the Siegel upper-half plane of degree (or genus) g. A standard notation for this is Sg. Observe that 25, is the upper-half of the ordinary complex plane. We know that the group Sp (g, R) operates transitively on (Big as