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1. CMB Online first

Haase, Christian; Hofmann, Jan
 Convex-normal (pairs of) polytopes In 2012 Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between $k$- and $(k+1)$-convex-normality (for $k\geq 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement of the normal fan of $Q$. If every edge $e_P$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap \mathbb{Z}^d = (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$. Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopesCategories:52B20, 14M25, 90C10

2. CMB Online first

Li, Boyu
 Normal Extensions of Representations of Abelian Semigroups A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation on an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extend a result due to Athavale to representations on abelian lattice ordered semigroups. Keywords:subnormal operator, normal extension, regular dilation, lattice ordered semigroupCategories:47B20, 47A20, 47D03

3. CMB 2015 (vol 59 pp. 123)

Jensen, Gerd; Pommerenke, Christian
 Discrete Space-time and Lorentz Transformations Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild's number-theoretic arguments, the subject is also interesting when isolated from its physical background. The paper of Schild is not easy to understand. Therefore we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers. Keywords:Lorentz transformation, integer lattice, Gaussian integersCategories:22E43, 20H99, 83A05

4. CMB 2014 (vol 58 pp. 182)

Tărnăuceanu, Marius
 On Finite Groups with Dismantlable Subgroup Lattices In this note we study the finite groups whose subgroup lattices are dismantlable. Keywords:finite groups, subgroup lattices, dismantlable lattices, planar lattices, crownsCategories:20D30, 20D60, 20E15

5. CMB 2014 (vol 57 pp. 277)

Elkholy, A. M.; El-Latif, M. H. Abd
 On Mutually $m$-permutable Product of Smooth Groups Let $G$ be a finite group and $H$, $K$ two subgroups of G. A group $G$ is said to be a mutually m-permutable product of $H$ and $K$ if $G=HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group which is a mutually m-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth. Keywords:permutable subgroups, $m$-permutable, smooth groups, subgroup latticesCategories:20D10, 20D20, 20E15, 20F16

6. CMB 2012 (vol 57 pp. 132)

Mubeena, T.; Sankaran, P.
 Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\operatorname{SL}(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}), \operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature. Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groupsCategory:20E45

7. CMB 2011 (vol 56 pp. 659)

Yu, Zhi-Xian; Mei, Ming
 Asymptotics and Uniqueness of Travelling Waves for Non-Monotone Delayed Systems on 2D Lattices We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed 2D lattice equations with non-monotone birth functions. First, with the help of Ikehara's Theorem, the a priori asymptotic behavior of travelling wave is exactly derived. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. These results complement earlier results in the literature. Keywords:2D lattice systems, traveling waves, asymptotic behavior, uniqueness, nonmonotone nonlinearityCategory:35K57

8. CMB 2011 (vol 54 pp. 645)

Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega
 An Extension of Craig's Family of Lattices Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p-1)$-dimensional and are geometrical representations of the integral $\mathbb Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p-1)(q-1)$-dimensional lattices from the integral $\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties. Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fieldsCategories:11H31, 11H55, 11H50, 11R18, 11R04

9. CMB 2011 (vol 54 pp. 277)

Farley, Jonathan David
 Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order Let $L$ be a finite distributive lattice. Let $\operatorname{Sub}_0(L)$ be the lattice $$\{S\mid S\text{ is a sublattice of }L\}\cup\{\emptyset\}$$ and let $\ell_*[\operatorname{Sub}_0(L)]$ be the length of the shortest maximal chain in $\operatorname{Sub}_0(L)$. It is proved that if $K$ and $L$ are non-trivial finite distributive lattices, then $$\ell_*[\operatorname{Sub}_0(K\times L)]=\ell_*[\operatorname{Sub}_0(K)]+\ell_*[\operatorname{Sub}_0(L)].$$ A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved. Keywords:(distributive) lattice, maximal sublattice, (partially) ordered setCategories:06D05, 06D50, 06A07

10. CMB 2008 (vol 51 pp. 15)

Aqzzouz, Belmesnaoui; Nouira, Redouane; Zraoula, Larbi
 The Duality Problem for the Class of AM-Compact Operators on Banach Lattices We prove the converse of a theorem of Zaanen about the duality problem of positive AM-compact operators. Keywords:AM-compact operator, order continuous norm, discrete vector latticeCategories:46A40, 46B40, 46B42

11. CMB 2004 (vol 47 pp. 191)

Grätzer, G.; Schmidt, E. T.
 Congruence Class Sizes in Finite Sectionally Complemented Lattices The congruences of a finite sectionally complemented lattice $L$ are not necessarily \emph{uniform} (any two congruence classes of a congruence are of the same size). To measure how far a congruence $\Theta$ of $L$ is from being uniform, we introduce $\Spec\Theta$, the \emph{spectrum} of $\Theta$, the family of cardinalities of the congruence classes of $\Theta$. A typical result of this paper characterizes the spectrum $S = (m_j \mid j < n)$ of a nontrivial congruence $\Theta$ with the following two properties: \begin{enumerate}[$(S_2)$] \item[$(S_1)$] $2 \leq n$ and $n \neq 3$. \item[$(S_2)$] $2 \leq m_j$ and $m_j \neq 3$, for all $j Keywords:congruence lattice, congruence-preserving extensionCategories:06B10, 06B15 12. CMB 2002 (vol 45 pp. 483) Baake, Michael  Diffraction of Weighted Lattice Subsets A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice$\varGamma$inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diffraction measure is periodic, with the dual lattice$\varGamma^*\$ as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base. Keywords:diffraction, Dirac combs, lattice subsets, homometric setsCategories:52C07, 43A25, 52C23, 43A05
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