Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

General interpolation on the lattice h{\Bbb Z}^s: Compactly supported fundamental solutions

Summary. In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions of compactly supported solutions for cardinal interpolation to gain compactly supported fundamental solutions for the general interpolation problem. The general interpolation problem admits the interpolation of the functional and derivative values under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructions of compactly supported fundamental solutions for cardinal interpolation are discussed together with algorithms for their construction that make use of MAPLE. Another construction based on finite decomposition and reconstruction for spline spaces is also provided. Ideas used in the latter construction are lifted to provide a general construction of compactly supported fundamental solutions for cardinal interpolation in the multivariate case. Examples are provided, several in the context of some general interpolation problem to illustrate how easy is the transition from cardinal interpolation to general interpolation. Received May 11, 1993 / Revised version received August 16, 1994     

基于广义\mbox{\boldmath $Z_{4}$}线性Calderbank-McGuire码的对偶

本文讨论了广义Z4线性Calderbank-McGuire码的对偶的迹表示及2-adic表示, 同时给出了其最小Lee权估计.     

On a problem of cyclic permutations of integers

We consider a problem of cyclic permutations of integers arising in the field of memory technologies for computer systems, and we solve it in the more general case of reals.     

Some observations on the arithmetic of Eisenstein series for the modular group SL(2, \Bbb Z)

For even integers k\geqq4k\geqq4, let j_k(X)\varphi_k(X) be the separable rational polynomial that encodes the j-invariants of non-elliptic zeroes of the Eisenstein series E_k for the modular group SL(2,Bbb Z)(2,{Bbb Z}). We prove Kummer-type congruence properties for the j_k\varphi_k and, based on extensive calculations, make observations about the Galois group, the discriminant, and the distribution of zeroes of j_k(X)\varphi_k(X).     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

Positive Isometric Averaging Operators on {\ell^2(\mathbb{Z}, \mu)}

We show that positive isometric averaging operators on the sequence space \({\ell^2(\mathbb{Z}, \mu)}\) are determined by very subtle arithmetic conditions on \({\mu}\) (even for very simple examples), contrary to what happens in the continuous case \({L^2({\mathbb{R}}^+)}\), where any possible average value is realized by a suitable positive isometry.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

Minimal factorizations of permutations into star transpositions

We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form . This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner.     

CLIFFORD MARTINGALES Φ-EQUIVALENCE BETWEEN S（f） AND f^＊

ＣＬＩＦＦＯＲＤＭＡＲＴＩＮＧＡＬＥＳΦ－ＥＱＵＩＶＡＬＥＮＣＥＢＥＴＷＥＥＮＳ（ｆ）ＡＮＤｆ￥ＬＯＮＧＲＵＬＩＮ；ＱＩＡＮＴＡＯ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ１０００８０，Ｃｈｉｎａ．）...     

Affine lines on {\mathbb Q}-homology planes and group actions

This paper is a supplement to the papers [KiKo] and [GMMR]. We show the role of group actions in the classification of affine lines on -homology planes.     

Isomorphic classification of the spaces $H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R)$

We give a representation of the spaces H^k_loc(\Bbb R)?L²(\Bbb R)H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R) as Köthe sequence spaces and use some recent topological invariants to investigate their isomorphic classification.     

On the fundamental group of {{\rm Hom}({\mathbb Z}^k,G)}

Let G be a compact Lie group. Consider the variety ${{\rm Hom}({\mathbb Z}^{k},G)}$ of representations of ${{\mathbb Z}^k}$ into G. We can see this as a based space by taking as base point the trivial representation . The goal of this paper is to prove that ${\pi_1({\rm Hom}({\mathbb Z}^k,G))}$ is naturally isomorphic to π ₁(G) ^k .     

Subordinators and the actions of permutations with quasi-invariant measure

We introduce a class of probability measures in the space of virtual permutations associated with subordinators (i.e., processes with stationary positive independent increments). We prove that these measures are quasiinvariant under both left and right actions of the countable symmetric group , and a simple formula for the corresponding cocycle is obtained. In case of a stable subordinator, we find the value of the spherical function of a constant vector on the class of transpositions. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 181–218. Translated by N. Tsilevich.     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).