Topologically Inseparable Functions I: Finitary Case

   Abstract. Given a finite set A and a distinguished function

A contractible Levi-flat hypersurface which is a determining set for pluriharmonic functions

We find a real analytic Levi-flat hypersurface in C² containing a bounded contractible domain which is a determining set for pluriharmonic functions.     

Semi inherited bivariate interpolation

The bivariate interpolation in two dimensional space R2 is more complicated than that in one dimensional space R, because there is no Haar space of continuous functions in R2. Therefore, the bivariate interpolation has not a unique solution for a set of arbitrary distinct pairwise points. In this work, we suggest a type of basis which depends on the points such that the bivariate interpolation has the unique solution for any set of distinct pairwise points. In this case, the matrix of bivariate interpolation has the semi inherited factorization.     

An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets

Abstract

This paper contains a generalized Gordan-type alternative theorem for set-valued maps which characterizes set relations without any convexity assumptions using certain evaluation functions. As a direct consequence and as a good example, we discuss robustness (or stability) of linear programming problems for modelling error. Moreover, this theorem can be utilized for that of general vector optimization problems in special cases due to reformation of the evaluation functions.     

Choice Functions and Extensive Operators

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider classes of heritage choice functions satisfying conditions M, N, W, and C, or combinations of these conditions. In terms of extensive operators these classes can be considered as generalizations of symmetric, anti-symmetric and transitive binary relations. Among these classes we meet the well-known classes of matroids and convex geometries. Using a ‘topological’ language we discuss these classes of monotone extensive operators (or heritage choice functions) in terms of neighborhood systems. A remarkable inversion on the set of choice functions is introduced. Restricted to the class of heritage choice functions the inversion turns out to be an involution, and under this involution the axiom N is auto-inverse, whereas the axioms W and M change places. This research was supported in part by NWO–RFBR grant 047.011.2004.017, by RFBR grant 05-01-02805 CNRSL_a, and the grant NSh-929.2008.6, School Support. We want to thank B. Monjardet, the editor and a referee for discussions and useful suggestions.     

Outer limits of subdifferentials for min–max type functions

Abstract

We generalize the outer subdifferential construction suggested by Cánovas, Henrion, López and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential posed by Li, Meng and Yang.     

An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for L ₁-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in L ₁. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.     

Topologically Inseparable Functions II: Infinitary Case

   Abstract. Given a set A and a function

Nonseparable Radial Frame Multiresolution Analysis in Multidimensions

Abstract

In this article we present a nonseparable multiresolution structure based on frames which is defined by radial frame scaling functions. The Fourier transform of these functions is the indicator (characteristic) function of a measurable set. We also construct the resulting frame multiwavelets, which can be isotropic as well. Our construction can be carried out in any number of dimensions and for a big variety of dilation matrices.     

Open-source machine learning: R meets Weka

Two of the prime open-source environments available for machine/statistical learning in data mining and knowledge discovery are the software packages Weka and R which have emerged from the machine learning and statistics communities, respectively. To make the different sets of tools from both environments available in a single unified system, an R package RWeka is suggested which interfaces Weka’s functionality to R. With only a thin layer of (mostly R) code, a set of general interface generators is provided which can set up interface functions with the usual “R look and feel”, re-using Weka’s standardized interface of learner classes (including classifiers, clusterers, associators, filters, loaders, savers, and stemmers) with associated methods.     

Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising

Abstract

An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l ₁ penalty term on the coefficients for the basis functions. Use of an l ₁ penalty instead of l ₂ has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l ₁ penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are the finite union of sets of orthonormal basis functions (e.g., wavelet packets). We show that the BCR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems.     

Comparing two methods of resolving homogeneous differential inclusions

Given a compact set we consider the differential inclusion We show how to use the main idea of the method of convex integration [ N], [G], [K] (to control convergence of the gradients of a sequence of approximate solutions by appropriate selection of the sequence) to obtain an optimal existence result. We compare this result with the ones available by the Baire category approach applied to the set of admissible functions with topology. A byproduct of our result is attainment in the minimization problems with integrands L having quasiaffine quasiconvexification that was, in fact, the reason of our interest to differential inclusions. This result can be considered as a first step towards characterization of those minimization problems which are solvable for all boundary data. This problem was solved in [S1] in the scalar case m=1. Received November 5, 1998 / Accepted July 17, 2000 / Published online December 8, 2000     

Induced modules over group algebras of metabelian groups of finite rank

ABSTRACT

Investigation of multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for functions is superposition (composition), but there are some other naturally defined operations, which are also worth of consideration. For example, the operation of set-theoretic intersection and the operation of projections. In this paper we find an abstract characterization of the set of multiplace functions which are closely related to these three operations.     

A set scalarization function based on the oriented distance and relations with other set scalarizations

ABSTRACT

The aim of this paper is, in the setting of normed spaces with a cone K non necessarily solid, to study new relations among set scalarization functions that are extensions of the oriented distance of Hiriart-Urruty. Moreover, we deal with a set scalarization function of sup-inf type, we investigate its relation to the cone-properness and cone-boundedness and it is related to other set scalarizations existing in the literature. In particular, with the norm induced by the Minkowski's functional, we obtain relations with a set scalarization which is an extension of the so called Gerstewitz's scalarization function.     

An optimal multivariate spline method of recovery of twice differentiable functions

A continuous quadratic polynomial spline of several variables is constructed. It solves the optimal recovery problem studied by V.F. Babenko, S.V. Borodachov, and D.S. Skorokhodov for the class of functions defined on a convex polytope in R ^d , whose second derivatives in any direction are uniformly bounded, and for a periodic analogue of this class. The information consists of the values and gradients of the function at some finite set of nodes in R ^d .     

Embedding relations connected with strong approximation of Fourier series

We consider the embedding relation between the class W ^q H _β ^ω , including only odd functions and a set of functions defined via the strong means of Fourier series of odd continuous functions. We establish an improvement of a recent theorem of Le and Zhou [Math. Inequal. Appl. 11(4) (2008) 749–756] which is a generalization of Tikhonov’s results [Anal. Math. 31 (2005) 183–194]. We also extend the Leindler theorem [Anal. Math. 31 (2005) 175–182] concerning sequences of Fourier coefficients.     

A note on meromorphic functions that share a set with their derivatives

In this paper, we study a problem of meromorphic functions that share an arbitrary set having three elements with their derivatives. A uniqueness result is derived which is an improvement of some related theorems given by Fang and Zalcman (J. Math. Anal. Appl. 280 (2003), 273–283) and Chang, Fang, and Zalcman (Arch. Math. 89 (2007), 561–569). As an application, we generalize the famous Brück conjecture with the idea of sharing a set.     

Convex-cone-based comparisons of and difference evaluations for fuzzy sets

Abstract

This paper focuses on how to compare two fuzzy sets and, from the viewpoint of set optimization, proposes eight types of fuzzy-set relations based on a convex cone as new comparison criteria of fuzzy sets. Then, difference evaluation functions for fuzzy sets are introduced. Under suitable assumptions of certain compactness and stability of fuzzy sets, we show that these functions correspond well to the fuzzy-set relations. In addition, through transforming these functions stepwise, we deal with numerical calculation methods of them in particular cases. Consequently, we can judge whether each fuzzy-set relation holds or not for given two fuzzy sets with the aid of computers.     

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category F≀FIn

We give a new proof for the Littlewood-Richardson rule for the wreath product F?S_n where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F?S_n to F?S_n+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F?FI_n where FI_n is the category of all injective functions between subsets of an n-element set.