Representable Lexicographic Products

A linear ordering is said to be representable if it can be order-embedded into the reals. Representable linear orderings have been characterized as those which are separable in the order topology and have at most countably many jumps. We use this characterization to study the representability of a lexicographic product of linear orderings. First we count the jumps in a lexicographic product in terms of the number of jumps in its factors. Then we relate the separability of a lexicographic product to properties of its factors, and derive a classification of representable lexicographic products. This revised version was published online in September 2006 with corrections to the Cover Date.     

Contractive Korovkin subsets in weighted spaces of continuous functions

We characterize those particular subsets of weighted spaces of continuous functions defined on locally compact Hausdorff spaces having the property that any net of positive linear contractions strongly converges to the identity operator provided it converges pointwise on them. Some variants and applications are indicated as well.     

Characterizations of ultrabarrelledness and barrelledness involving the singularities of families of convex mappings

Summary The paper reveals that ultrabarrelled spaces (respectively barrelled spaces) can be characterized by means of the density of the so-called weak singularities of families consisting of continuous convex mappings that are defined on an open absolutely convex set and take values in a locally full ordered topological linear space (respectively locally full ordered locally convex space). The idea to establish such characterizations arose from the observation that, in virtue of well-known results, the density of the singularities of families of continuous linear mappings allows to characterize both the ultrabarrelled spaces and the barrelled spaces.     

Lexicodes over finite principal ideal rings

Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module \(R^n\) is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides

Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. In this article, for any locally Lipschitz continuous mapping between finite-dimensional Euclidean spaces, Nesterov’s lexicographic derivatives are shown to be elements of the plenary hull of the (Clarke) generalized Jacobian whenever they exist. It is argued that in applications, and in several established results in nonsmooth analysis, elements of the plenary hull of the generalized Jacobian of a locally Lipschitz continuous function are no less useful than elements of the generalized Jacobian itself. Directional derivatives and lexicographic derivatives of solutions of parametric ordinary differential equation (ODE) systems are expressed as the unique solutions of corresponding ODE systems, under Carathéodory-style assumptions. Hence, the scope of numerical methods for nonsmooth equation-solving and local optimization is extended to systems with nonsmooth parametric ODEs embedded.     

Continuous maps on triangular matrices that preserve commutativity

We investigate continuous injective maps defined on the space of upper triangular matrices over a field F that preserve commutativity of matrices in both directions. We show that every such map is a composition of either an inner automorphism and a locally polynomial map or the two latter and one more automorphism of upper triangular matrices.     

Intersections of intrinsic submanifolds in the Heisenberg group

In the first Heisenberg group, we show that the intersection of two intrinsic submanifolds with linearly independent horizontal normals locally coincides with the image of an injective continuous curve. The key tool is a chain rule that relies on a recent result by Dafermos.     

Lexicographically ordered trees

We characterize trees whose lexicographic ordering produces an order isomorphic copy of some sets of real numbers, or an order isomorphic copy of some set of ordinal numbers. We characterize trees whose lexicographic ordering is order complete, and we investigate lexicographically ordered ω-splitting trees that, under the open-interval topology of their lexicographic orders, are of the first Baire category. Finally we collect together some folklore results about the relation between Aronszajn trees and Aronszajn lines, and use earlier results of the paper to deduce some topological properties of Aronszajn lines.     

Foundations of lexicographic utility

This paper presents an alternative mathematical characterization of lexicographic utility to the one given by Chipman (1960). A natural constructivistic procedure on imposing a lexicographic ordering on the product space of natural numbers is pursued. The consequences on the topological structure of such a space are examined.     

Continuous selections as parametrically defined integrals

An analog of the classical Michael theorem on continuous single-valued selections of lower semicontinuous maps whose values are closed and convex in a Fréchet space is proved for maps into metrizable (non-locally-convex) vector spaces. It turns out that, instead of the local convexity of the whole space containing these values, it is sufficient to require that the family of values of the map be uniformly locally convex. In contrast to the standard selection theorems, the proof bypasses the process of successively improving the approximations, and the desired selection is constructed as the result of pointwise integration with respect to a suitable probability distribution.     

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.     

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.      

On homomorphisms between generalized polygons

We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphism is either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Dedicated to Prof. Dr. H. R. Salzmann on the occasion of his 65th anniversary     

Generalized Derivatives of Lexicographic Linear Programs

Lexicographic linear programs are fixed-priority multiobjective linear programs that are a useful model of biological systems using flux balance analysis and for goal-programming problems. The objective function values of a lexicographic linear program as a function of its right-hand side are nonsmooth. This work derives generalized derivative information for lexicographic linear programs using lexicographic directional derivatives to obtain elements of the Bouligand subdifferential (limiting Jacobian). It is shown that elements of the limiting Jacobian can be obtained by solving related linear programs. A nonsmooth equation-solving problem is solved to illustrate the benefits of using elements of the limiting Jacobian of lexicographic linear programs.     

Theory of regularizability in topological vector spaces

Two known definitions of regularizability for topological vector spaces are found to be equivalent. Regularizability in the sense of Tikhonov is considered in reflexive linear metric spaces. In particular, an example is presented of a linear continuous injective operator on a reflexive Frécnet space whose inverse cannot be regularized. The latter indicates the sharp difference between regularizability in Fréchet spaces and in Banach spaces, respectively.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 777–781, June, 1990.     

A refinement of Stone duality to skew Boolean algebras

We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean -algebras whose morphisms are proper skew Boolean -algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.     

Semigroups of locally injective maps and transfer operators

We consider semigroups of continuous, surjective, locally injective maps of a compact metric space, and whether such semigroups admit a transfer operator.     

Block Orderings for Tensor-Product Grids in Two and Three Dimensions

We consider two-line and two-plane orderings for a convection–diffusion model problem in two and three dimensions, respectively. These strategies are aimed at introducing dense diagonal blocks, at the price of a slight increase of the bandwidth of the matrix, compared to natural lexicographic ordering. Comprehensive convergence analysis is performed for the block Jacobi scheme. We then move to consider a two-step preconditioning technique, and analyze the numerical properties of the linear systems that are solved in each step of the iterative process. For the 3-dimensional problem this approach is a viable alternative to the Incomplete LU approach, and may be easier to implement in parallel environments. The analysis is illustrated and validated by numerical examples.     

Homogeneous approximation property for continuous wavelet transforms

The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L² sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed.