A combinatorial approach to binary positional number systems

Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in the binary case – digits 0 and 1. A binary positional number system (binary radix system) with base equal to the golden ratio \((1+\sqrt{5}\,)/2\) is fairly well known. The main result of this paper is a construction of infinitely many binary radix systems, each one constructed combinatorially from a single pair of binary strings. Every binary radix system that satisfies even a minimal set of conditions that would be expected of a positional number system, can be constructed in this way.     

Affine Surfaces which are Both Affine Harmonic and Affine Maximal

We study affine surfaces which are both affine maximal and affine harmonic. We prove that an indefinite surface satisfying both conditions is affine equivalent to an open part $(u,{1\over 2}u^2,P_1(u)+\upsilon,P_2(u)+{1\over 2}\upsilon^2)$ , where P₁ and P₂ are arbitrary functions of one variable.     

Systems of iterated projective ordinal notations and combinatorial statements about binary labeled trees

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding impredicative Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate simple term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ(), for any primitive recursive limit wellordering. In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ() the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT() is a proper subsystem ofPRJ(), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID_< , for arbitrarily large limit ordinal. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]).     

Vector Hulls of Affine Spaces and Affine Bundles

Every affine space A can be canonically immersed as a hyperplane into a vector space  , which is called the vector hull of A. This immersion satisfies a universal property for affine functions defined on A. In the same way, every affine map between affine spaces has a linear prolongation to their vector hulls. Though not much known, this construction is greatly clarifying, both for affine geometry and for its applications. The goal of this paper is to perform a thorough study of the vector hull functor and to describe its counterpart in the framework of affine bundles. With this respect, it is shown that the vector hull of some interesting affine bundles, and more specifically some jet bundles, can be identified with certain vector bundles.      

Affine Quasi-Heredity of Affine Schur Algebras

In this paper we prove that the affine Schur algebra \(\widehat {S}(n,r)\) is affine quasi-hereditary. This result is then used to show that the centralizer subquotient algebras of \(\widehat {S}(n,r)\) are Laurent polynomial algebras. Moreover, we give a parameter set of simple \(\widehat {S}(n,r)\)-modules and identify this parameter set with that given in [7]. In the Appendix, the affine quasi-heredity of affine quantum Schur algebras is studied.     

Representations of Affine Multifunctions by Affine Selections

The paper deals with affine selections of affine (both convex and concave) multifunctions acting between finite-dimensional real normed spaces. It is proved that each affine multifunction with compact values possesses an exhaustive family of affine selections and, consequently, can be represented by its affine selections. Moreover, a convex multifunction with compact values possesses an exhaustive family of affine selections if and only if it is affine. Thus the existence of an exhaustive family of affine selections is the characteristic feature of affine multifunctions which differs them from other convex multifunctions with compact values. Besides a necessary and sufficient condition for a concave multifunction to be affine on a given convex subset is also proved. Finally it is shown that each affine multifunction with compact values can be represented as the closed convex hull of its exposed affine selections and as the convex hull of its extreme affine selections. These statements extend the Straszewicz theorem and the Krein–Milman theorem to affine multifunctions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Affine Frames of Multivariate Box Splines and their Affine Duals

We give a simple and explicit construction of primal and dual wavelet filters based on refinable multivariate splines (with respect to dilation matrices M) such that the corresponding wavelet functions generate dual affine frames of arbitrarily high regularity. Furthermore, the number of wavelets does not depend on the regularity. We apply the method also to generalized B-splines.     

An enumeration algorithm for combinatorial problems of the reliability analysis of binary coherent systems

In this paper the problem of the generation of all elements of a system of sets is investigated and a backtrack algorithm (Al) is given solving this problem.The algorithm is applied to combinatorial problems of reliability:

A t-graph is interpreted as a binary coherent system S and the system function of S is represented as the sum of Boolean orthogonal products. This approach is a simple and efficient alternative to a method suggested by Satyanarayana and N. Hagstbom [9].

A directed p-graph is also interpreted as a binary coherent system S and both the minimal cuts of S and the cocycles of G are determined.     

Affine and eclipsing multifunctions

In this paper, we want to compare two classes of multifunctions which can be used as approximating multifunctions in differentiability theory: affine and eclipsing multifunctions. We show how the notion of eclipsing multifunctions is an extension of affine multifunctions, and what kinds of difficulties arise in this extension.     

Euclidean Complete Affine Surfaces with Constant Affine Mean Curvature

The purpose of this paper is to prove that alocally strongly convex, Euclidean complete surface with constantaffine mean curvature is also affine complete. Consequently weobtain a classification of locally strongly convex, Euclideancomplete surfaces with constant affine mean curvature.     

胞腔代数和仿射胞腔代数简介

表示论中一个最基本的问题是确定不可约表示的参数集,这个问题至今没有完全解决.对于Graham和Lehrer引入的有限维胞腔代数,这个问题得到了完满解答,并被成功地应用于数学和物理中出现的许多代数.近来,人们引入仿射胞腔代数,将Graham和Lehrer有限维胞腔代数的表示理论框架推广到一类无限维代数上.仿射胞腔代数不仅包括有限维胞腔代数,也包括无限维的仿射Temperley-Lieb代数和Lusztig的A-型仿射Hecke代数.本文将对胞腔代数的发展历史和主要研究成果做一些综述,同时,对新引入的仿射胞腔代数及其最新成果做一点简介.