Subset Of Rn Research Articles

On a Class of Convex Sets and Functions

Given a convex subset C of ℝn, the set-valued mapping α↦α⋅C (where 0⋅C is, by convention, the recession cone of C) is increasing on ℝ+ if and only if C contains the origin, and decreasing on ℝ+ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ℝm with a convex subset of ℝn to produce a convex subset of ℝn+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.

Unconditional basis recursively generated in Lp on compact sets

We construct an unconditional basis in the Banach space L p(Ω) for p 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in ℝn. We also give an algorithm of how to construct an unconditional basis in L p(Ω p). At the end of this paper, we give the characterization of the functions in L p(Ω p) by using the wavelet coefficients.

More on convexity numbers of closed sets in ℝⁿ

The convexity number of a set S ⊆ R n S\subseteq \mathbb R^n is the least size of a family F \mathcal F of convex sets with ⋃ F = S \bigcup \mathcal F=S . S S is countably convex if its convexity number is countable. Otherwise S S is uncountably convex. Uncountably convex closed sets in R n \mathbb R^n have been studied recently by Geschke, Kubiś, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all n ≥ 2 n\geq 2 , it is consistent that there is an uncountably convex closed set S ⊆ R n + 1 S\subseteq \mathbb R^{n+1} whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of R n \mathbb R^n . Moreover, we construct a closed set S ⊆ R 3 S\subseteq \mathbb R^3 whose convexity number is 2 ℵ 0 2^{\aleph _0} and that has no uncountable k k -clique for any k > 1 k>1 . Here C ⊆ S C\subseteq S is a k k -clique if the convex hull of no k k -element subset of C C is included in S S . Our example shows that the main result of the above-named authors, a closed set S ⊆ R 2 S\subseteq \mathbb R^2 either has a perfect 3 3 -clique or the convexity number of S S is > 2 ℵ 0 >2^{\aleph _0} in some forcing extension of the universe, cannot be extended to higher dimensions.

Continuum of Zero Points of a Mapping on a Compact, Convex Set

Let X be a nonempty, compact and convex set in Rn and $\phi$ be an outer semicontinuous mapping from X to the collection of nonempty, compact convex subsets of Rn. We show that for any nonzero vector c in Rn there exists a set of stationary points of $\phi$ on X with respect to c connecting a point in the boundary of X at which $c^{\top} x$ is minimized on X to another point in the boundary of X at which $c^{\top}x$ is maximized on X. We provide several conditions on $\phi$ under which there exists a continuum of zero points of $\phi$ connecting two such points in the boundary of X, and an intersection result on a convex, compact set. An application to constrained equilibria is also given.

Intrinsic characterizations of Besov spaces on Lipschitz domains

AbstractThe aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ⊂ ℝn is a bounded Lipschitz open subset in ℝn. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ℝn. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ℝn described in Theorem 2.4 to the case of Lipschitz domains.

Regularity of Solutions to the Dirichlet Problem for Degenerate Elliptic Equation

In this paper, the author studies the regularity of solutions to the Dirichlet problem for equation Lu = f, where L is a second order degenerate elliptic operator in divergence form in Ω, a bounded open subset of Rn(n ≥ 3).

Relaxation in an L∞-optimization problem

Let Ω be an open bounded subset of Rn and f a continuous function on Ω̄ satisfying f(x) > 0 for all x ∈ Ω̄. We consider the maximization problem for the integral ∫Ωf(x)u(x)dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on ∂Ω and to the gradient constraint of the form H(Du(x)) ≤ 1, and prove that the supremum is ‘achieved’ by the viscosity solution of Ĥ(Du(x)) = 1 in Ω and u = 0 on ∂Ω, where Ĥ denotes the convex envelope of H. This result is applied to an asymptotic problem, as p → ∞, for quasi-minimizers of the integral An asymptotic problem as k → ∞ for inf is also considered, where the infimum is taken all over and the set K is given by {ξ | H(ξ) ≤ 1}.

The projection of a rectifiable Legendrian set is C2-rectifiable: a simplified proof

We provide a simplified proof of the following well-known theorem. The projection to Rn + 1of an n-dimensional rectifiable Legendrian subset of Rn+1 × Snis C2-rectifiable. Such a result generalizes the following simple fact. Let M be a hypersurface whose Gauss map has graph of class C1. Then M is of class C2.

Convexity numbers of closed sets in ℝⁿ

For n > 2 n>2 let I n \mathcal I_n be the σ \sigma -ideal in P ( n ω ) \mathcal P(n^\omega ) generated by all sets which do not contain n n equidistant points in the usual metric on n ω n^\omega . For each n > 2 n>2 a set S n S_n is constructed in R n \mathbb {R}^n so that the σ \sigma -ideal which is generated by the convex subsets of S n S_n restricted to the convexity radical K ( S n ) K(S_n) is isomorphic to I n \mathcal I_n . Thus cov ⁡ ( I n ) \operatorname {cov}(\mathcal I_n) is equal to the least number of convex subsets required to cover S n S_n — the convexity number of S n S_n . For every non-increasing function f : ω ∖ 2 → { κ ∈ card : cf ⁡ ( κ ) > ℵ 0 } f:\omega \setminus 2\to \{\kappa \in \operatorname {card}:\operatorname {cf}(\kappa )>\aleph _0\} we construct a model of set theory in which cov ⁡ ( I n ) = f ( n ) \operatorname {cov}(\mathcal I_n)=f(n) for each n ∈ ω ∖ 2 n\in \omega \setminus 2 . When f f is strictly decreasing up to n n , n − 1 n-1 uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of R n \mathbb {R}^n . It is conjectured that n n , but never more than n n , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of R n \mathbb {R}^n . This conjecture is true for n = 1 n=1 and n = 2 n=2 .

Alternative theorems for nonlinear projection equations and applications to generalized complementarity problems

Let K be a closed convex subset in Rn, and let f; g and h be three functions from Rn into itself. We consider the following nonlinear projection equation (NPE): h(x) = :K (g(x)− f(x)); x ∈ R; (1) where :K (·) is the orthogonal projection operator onto the set K . This equation provides a uni<ed formulation of several interesting and important special cases. In the case when h(x)= g(x) for any x∈Rn; then (1) reduces to the equation studied by Pang and Yao [17] under the name of generalized normal equation which is equivalent to the following generalized variational inequality (GVI): g(x) ∈ K; (y − g(x))f(x) ≥ 0 for all y ∈ K: (2)

On combinatorial design spaces for the configuration design of product families

For typical optimization problems, the design space of interest is well defined: It is a subset of Rn, where n is the number of (continuous) variables. Constraints are often introduced to eliminate infeasible regions of this space from consideration. Many engineering design problems can be formulated as search in such a design space. For configuration design problems, however, the design space is much more difficult to define precisely, particularly when constraints are present. Configuration design spaces are discrete and combinatorial in nature, but not necessarily purely combinatorial, as certain combinations represent infeasible designs. One of our primary design objectives is to drastically reduce the effort to explore large combinatorial design spaces. We believe it is imperative to develop methods for mathematically defining design spaces for configuration design. The purpose of this paper is to outline our approach to defining configuration design spaces for engineering design, with an emphasis on the mathematics of the spaces and their combinations into larger spaces that more completely capture design requirements. Specifically, we introduce design spaces that model physical connectivity, functionality, and assemblability considerations for a representative product family, a class of coffeemakers. Then, we show how these spaces can be combined into a “common” product variety design space. We demonstrate how constraints can be defined and applied to these spaces so that feasible design regions can be directly modeled. Additionally, we explore the topological and combinatorial properties of these spaces. The application of this design space modeling methodology is illustrated using the coffeemaker product family.

Existence of generalized equilibria

We consider the problem of the existence of equilibria, generalized equilibria, and 8xed-points of a correspondence F , de8ned on a compact subset M of Rn, with values in Rn, when the set M is neither assumed to be convex, nor smooth. 1 The general framework of the article is the following. Let F be a correspondence from M to Rn, that is, a map from M to the set of all the subsets of Rn; the correspondence F is said to be upper semicontinuous (u.s.c.), if the set {x∈M |F(x)⊂V} is open in M for every open set V ⊂Rn. Let N be a “normal cone” correspondence, i.e., at this stage, N is a correspondence from M to Rn such that, for every x∈M; N (x)

Box and packing dimensions of projections and dimension profiles

For E a subset of ℝn and s ∈ [0, n] we define upper and lower box dimension profiles, B-dimsE and B-dimsE respectively, that are closely related to the box dimensions of the orthogonal projections of E onto subspaces of ℝn. In particular, the projection of E onto almost all m-dimensional subspaces has upper box dimension B-dimmE and lower box dimension B-dimmE. By defining a packing type measure with respect to s-dimensional kernels we are able to establish the connection to an analogous packing dimension theory.

Topical and sub-topical functions, downward sets and abstract convexity

We study topical and sub-topical functions (i.e., functions which are increasing in the natural partial ordering of ℝn and additively homogeneous, respectively additively sub-homogeneous), and downward sets (i.e., subsets of ℝn which contain, along with each element, all smaller elements), in the framework of abstract convex analysis, with the aid of the additive min-type coupling function . We study primal and dual link between topical functions and closed downward sets, via plus-Minkowski gauges and ϕ-support functions of sets and level sets and ϕ-support sets of functions. Also, we give characterizations of topical functions in terms of their Fenchel-Moreau conjugates and biconjugates with respect to the above coupling function ϕ

L∞-Approximation via Subdominants

Given a vector u and a certain subset K of a real vector space E, the problem of l∞-approximation involves determining an element u in K nearest to u in the sense of the l∞-error norm. The subdominant u∗ of u is the upper bound (if it exists) of the set {x∈K:x≺u} (we let x≺y if all coordinates of x are smaller than or equal to the corresponding coordinates of y). We present general conditions on K under which a simple relationship between the subdominant of u and a best l∞-approximation holds. We specify this result by taking as K the cone of isotonic functions defined on a poset (X, ≺), the cone of convex functions defined on a subset of RN, the cone of ultrametrics on a set X, and the cone of tree metrics on a set X with fixed distances to a given vertex. This leads to simple optimal algorithms for the problem of best l∞-fitting of distances by ultrametrics and by tree metrics preserving the distances to a fixed vertex (the latter provides a 3-approximation algorithm for the problem of fitting a distance by a tree metric). This simplifies the recent results of Farach, Kannan, and Warnow (1995) and of Agarwala et al. (1996).

Local optimality subsets and global optimization a prospective approach

We set here the problem of introducing memory into global optimization processes involving the minimization of a convex function on some discrete subset of ℝn. In order to address this problem, we propose an analysis of the structure of the local optimality subsets induced by such a problem. We discuss potential interpretations of these results in the context of global optimization problems.

Th�or�me de division avec des conditions au bord

Let Ω be an open subset of ℝn and\(\mathcal{O}(\Omega )\) be a subalgebra of the algebra of analytic functions on Ω. We suppose that\(\mathcal{O}(\Omega )\) satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of\(\mathcal{O}(\Omega )\). We also suppose that\(\mathcal{O}(\Omega )\) satifies global £ojasiewicz's inequalities. We prove the following: Let\(f_{1, \ldots ,} f_p \in \mathcal{O}(\Omega )\) andf C∞ on\(\overline \Omega \) flat on ∂Ω; if for eacha ∈ Ω the Taylor's serie off ata, Taf, is in the ideal generated byTaf1,...,Tafp in the ring of formal power series, then there existα1,...,αp,C∞ on\(\overline \Omega \) flat on ∂Ω such that\(f = \sum\nolimits_{j = 1}^p {\alpha _j f_j } \). This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.

A Picard‐Maclaurin theorem for initial value PDEs

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) on ℜn with a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset of ℜn, we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.

New Existence Results For Singular Solutions Of Some Semilinear Elliptic Equations

Let ω be a smooth open bounded subset of RN with n≥4. Given σ⊂ω any closed, smooth k-dimensional submanifold, 1≤k<n−2, with smooth boundary, we prove the existence of infinitly many positive weak solutions of whose singular set is σ, provided the exponent p is larger than and close to

An isoperimetrical problem in n dimensions – an elementary proof

Isoperimetrical problems in ℝ2 and ℝ3 belong to the classical problems of geometry. As their name indicates one has to find geometrical bodies or areas with maximal volume, within a particular class of bodies or areas with equal surface measure. In the 19th century the isoperimetrical problem was solved for several polyhedron classes in ℝ3 [1, pp. 133-137]. Since then, a lot of research work has been devoted to more general problems where the admissable bodies were contained in the class of all compact convex subsets of ℝn or in a specific subclass, respectively. Their solutions usually can be characterized as zero sets of algebraic inequalities in surface measure and volume. The derivation of these inequalities and their solutions usually require sophisticated techniques. Of course, assertion (*) below is the expected generalisation of the well-known two-dimensional case.