Simplices with congruent k-faces

Let S be a non-degenerate simplex in $\mathbb{R}^{2}$. We prove that S is regular if, for some k $\in$ {1,...,n-2}, all its k-dimensional faces are congruent. On the other hand, there are non-regular simplices with the property that all their (n1)-dimensional faces are congruent.     

On the number of antipodal or strictly antipodal pairs of points in finite subsets ofR d,II

The paper is a continuation of [MM], namely containing several statements related to the concept of antipodal and strictly antipodal pairs of points in a subsetX ofR ^d , which has cardinalityn. The pointsx _i, x_jX are called antipodal if each of them is contained in one of two different parallel supporting hyperplanes of the convex hull ofX. If such hyperplanes contain no further point ofX, thenx _i, x_j are even strictly antipodal. We shall prove some lower bounds on the number of strictly antipodal pairs for givend andn. Furthermore, this concept leads us to a statement on the quotient of the lengths of longest and shortest edges of speciald-simplices, and finally a generalization (concerning strictly antipodal segments) is proved.Research (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1817     

Minimal zonotopes containing the crosspolytope

Motivated by the problem to improve Minkowski’s lower bound on the successive minima for the class of zonotopes we determine the minimal volume of a zonotope containing the standard crosspolytope. It turns out that this volume can be expressed via the maximal determinant of a ±1-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to ±1- matrices, we characterize all zonotopes attaining the minimal volume in dimension 3 and present related results in higher dimensions.     

Determinants and the volumes of parallelotopes and zonotopes

The Minkowski sum of edges corresponding to the column vectors of a matrix A with real entries is the same as the image of a unit cube under the linear transformation defined by A with respect to the standard bases. The geometric object obtained in this way is a zonotope, Z(A). If the columns of the matrix are linearly independent, the object is a parallelotope, P(A). In the first section, we derive formulas for the volume of P(A) in various ways as , as the square root of the sum of the squares of the maximal minors of A, and as the product of the lengths of the edges of P(A) times the square root of the determinant of the matrix of cosines of angles between pairs of edges. In the second section, we use the volume formulas to derive real-case versions of several well-known determinantal inequalities—those of Hadamard, Fischer, Koteljanskii, Fan, and Szasz—involving principal minors of a positive-definite Hermitian matrix. In the last section, we consider zonotopes, obtain a new proof of the decomposition of a zonotope into its generating parallelotopes, and obtain a volume formula for Z(A).     

On Continuous and Smooth Solutions of the Schröder Equation in Normed Spaces

The problem of the existence of continuous and smooth solutions of the Schr?der equation defined on normed spaces is examined. This research was supported by the Silesian University Mathematics Department (Linear functional equations program).     

Adhesivity of polymatroids

Two polymatroids are adhesive if a polymatroid extends both in such a way that two ground sets become a modular pair. Motivated by entropy functions, the class of polymatroids with adhesive restrictions and a class of selfadhesive polymatroids are introduced and studied. Adhesivity is described by polyhedral cones of rank functions and defining inequalities of the cones are identified, among them known and new non-Shannon type information inequalities for entropy functions. The selfadhesive polymatroids on a four-element set are characterized by Zhang-Yeung inequalities.     

Irreducibility of polynomials modulo p via Newton polytopes

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.     

Polynomials over ordered fields

In this paper we determine which polynomials over ordered fields have multiples with nonnegative coefficients and also which polynomials can be written as quotients of two polynomials with nonnegative coefficients. This problem is related to a result given by Pólya in [G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, England, 1952] (as a companion of Artin’s theorem) that asserts that if F(X₁,…,X_n)∈R[X₁,…,X_n] is a form (i.e., a homogeneous polynomial) s.t.  with ∑x_j>0, then F=G/H, where G,H are forms with all coefficients positive (i.e., every monomial of degree degG or degH appears in G or H, resp., with a coefficient that is strictly positive). In Pólya’s proof H is chosen to be H=(X₁+?+X_n)^m for some m.At the end we give some applications, including a generalization of Pólya’s result to arbitrary ordered fields.     

A note on number of orderings of higher level

The main aim of the paper is to find an explicit formula for the number of orderings of higher level. Among others, we discuss the relationship between the number of orderings of higher level and the number of orderings of level 1. We also construct a field with a given possible number of orderings of higher level. Received: 10 December 2004     

Cyclic polytopes, hyperplanes, and Gray codes

We consider cyclic d-polytopes P that are realizable with vertices on the moment curve $M_d:t\longrightarrow (t,t^2,\ldots,t^d)$ of order $d\geq 3$. A hyperplane H bisects a j-face of P if H meets its relative interior. For $\ell\geq 1$, we investigate the maximum number of vertices that P can have so that for some $\ell$ hyperplanes, each j-face of P is bisected by one of the hyperplanes. For $\ell > 1$, the problem translates to the existence of certain codes, or equivalently, certain paths on the cube $\{0,1\}^\ell$.     

On a vector space analogue of Kneser’s theorem

In a previous paper [X. Hou, K.H. Leung, Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002) 1-9], the following result was established: let E⊂K be fields such that the algebraic closure of E in K is separable over E. Let A,B be E-subspaces of K such that 0<dim_EA<∞ and 0<dim_EB<∞. Then dim_EAB?dim_EA+dim_EB-dim_EH(AB), where AB is the E-space generated by {ab:a∈A,b∈B} and H(AB)={x∈K:xAB⊂AB}. The separability assumption was essential in the proof of this result. However, even without the separability assumption, no counterexample is known. The present paper shows that no counterexample can be found if dim_EA?5.     

All 0–1 polytopes are traveling salesman polytopes

We study the facial structure of two important permutation polytopes in , theBirkhoff orassignment polytopeB _n , defined as the convex hull of alln×n permutation matrices, and theasymmetric traveling salesman polytopeT _n , defined as the convex hull of thosen×n permutation matrices corresponding ton-cycles. Using an isomorphism between the face lattice ofB _n and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices ofB _n is contained in a cubical face, showing faces ofB _n to be fairly special 0–1 polytopes. On the other hand, we show thatevery 0–1d-polytope is affinely equivalent to a face ofT _n , fordlogn, by showing that every 0–1d-polytope is affinely equivalent to the asymmetric traveling salesman polytope of some directed graph withn nodes. The latter class of polytopes is shown to have maximum diameter [n/3].Partially supported by NSF grant DMS-9207700.     

Some remarks concerning kissing numbers,blocking numbers and covering numbers

This article shows an inequality concerning blocking numbers and Hadwiger's covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwiger's covering numbers ford-dimensional centrally symmetric convex bodies to 3 ^d –1.     

Nice decompositions of R n entirely into nice sets are mostly impossible

While there are several interesting examples of partitions of R ³ into elements which individually are geometrically nice — circles or segments — the partitions themselves fail to be nice, in the sense of forming continuous or upper semicontinuous decompositions. We show that this is no accident: R ³ has no continuous decomposition into circles, and no open subset of R ⁿ has an upper semicontinuous decomposition into convex compact nonsingleton sets.     

Generalized sine equations,III

Summary We solve the equationf(x + y)f(x – y) = P(f(x), f(y)) under various conditions on the unknown functionsf, P.     

Cubic and quartic congruences modulo a prime

Let p>3 be a prime, and denote the number of solutions of the congruence . In this paper, using the third-order recurring sequences we determine the values of N_p(x³+a₁x²+a₂x+a₃) and N_p(x⁴+ax²+bx+c), and construct the solutions of the corresponding congruences, where a₁,a₂,a₃,a,b,c are integers.     

Projektive Skalen,Tschebyscheff-Polynome und Fibonacci-Zahlen

Summary. Let $\widehat{\widehat T}_n$ and $\overline U_n$ denote the modified Chebyshev polynomials defined by $\widehat{\widehat T}_n (x) = {T_{2n + 1} \left(\sqrt{x + 3 \over 4} \right) \over \sqrt{x + 3 \over 4}}, \quad \overline U_{n}(x) = U_{n} \left({x + 1 \over 2}\right) \qquad (n \in \mathbb{N}_{0},\ x \in \mathbb{R}).$ For all $n \in \mathbb{N}_{0}$ define $\widehat{\widehat T}_{-(n + 1)} = \widehat{\widehat T}_n$ and $\overline U_{-(n + 2)} = - \overline U_n$, furthermore $\overline U_{-1} = 0$. In this paper, summation formulae for sums of type $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k}(\nu; x)$ are given, where $\bigl(\mathbf a_{\mathbf k}(\nu; x)\bigr)^{-1} = (-1)^k \cdot \Bigl( x \cdot \widehat{\widehat T}_{\left[k + 1 \over 2\right] - 1} (\nu) +\widehat{\widehat T}_{\left[k + 1 \over 2\right]}(\nu)\Bigr) \cdot \Bigl(x \cdot \overline U_{\left[k \over 2\right] - 1} (\nu) + \overline U_{\left[k \over 2\right]} (\nu)\Bigr)$ with real constants $ x, \nu $. The above sums will turn out to be telescope sums. They appear in connection with projective geometry. The directed euclidean measures of the line segments of a projective scale form a sequence of type $(\mathbf a_{\mathbf k} (\nu;x))_{k \in \mathbb{Z}}$ where $ \nu $ is the cross-ratio of the scale, and x is the ratio of two consecutive line segments once chosen. In case of hyperbolic $(\nu \in \mathbb{R} \setminus] - 3,1[)$ and parabolic $\nu = -3$ scales, the formula $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k} (\nu; x) = {\frac{1}{x - q_{{+}\atop(-)}}} - {\frac{1}{x - q_{{-}\atop(+)}}} \eqno (1)$ holds for $\nu > 1$ (resp. $\nu \leq - 3$), unless the scale is geometric, that is unless $x = q_+$ or $x = q_-$. By $q_{\pm} = {-(\nu + 1) \pm \sqrt{(\nu - 1)(\nu + 3)} \over 2}$ we denote the quotient of the associated geometric sequence.

     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Characterizations of cyclic polytopes

Let P denote a simplicial convex 2m-polytope with n vertices. Then the following are equivalent: (i) P is cyclic; (ii) P satisfies Gale’s Evenness Condition; (iii) Every subpolytope of P is cyclic; (iv) P has at least 2m+2 cyclic subpolytopes with n−1 vertices if n ≥ 2m+5; (v) P is neighbourly and has n universal edges. We present an additional characterization based upon an easily described point arrangement property.