Even Unimodular Gaussian Lattices of Rank 12

We classify even unimodular Gaussian lattices of rank 12, that is, even unimodular integral lattices of rank 12 over the ring of Gaussian integers. This is equivalent to the classification of the automorphisms τ with τ²=−1 in the automorphism groups of all the Niemeier lattices, which are even unimodular (real) integral lattices of rank 24. There are 28 even unimodular Gaussian lattices of rank 12 up to equivalence.     

Extremal Even Unimodular Lattices of Rank 32 and Related Codes

In the paper the asymptotic behaviour of the solutions of a class of neutral differential equations with distributed delay is studied.     

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

In this paper, we investigate self-dual codes over finite rings, specifically the ring of integers modulo 2m. Type II codes over are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over are also studied.     

全实代数数域上正定幺模格的分类

本文推广了邻格方法,并利用局部格,正旋量种及种的分类理论,给出了维数4判别式1的二次空间上幺模种的个数公式,正定幺模格种的质量与邻格数的关系公式,以及幺模格种的邻格图,正定幺模格种质量的计算方法.还完成了判别式148的全实三次循环代数数域K_(148)上二次空间V≌〈1〉⊥〈1〉⊥〈1〉⊥〈1〉内的所有5个正定幺模格种的分类.     

Conference Matrices and Unimodular Lattices

Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by nonprincipal ideals yields simple constructions of further lattices including the Leech lattice.     

Double Circulant Codes over \mathbb{Z}_4 and Even Unimodular Lattices

With the help of some new results about weight enumerators of self-dual codes over we investigate a class of double circulant codes over , one of which leads to an extremal even unimodular 40–dimensional lattice. It is conjectured that there should be Nine more constructions of the Leech lattice     

Unimodular Lattices with Isometries of Large Prime Order II

In this paper first we establish the existence of extremal solutions for evolution inclusions driven by time dependent convex subdifferentials, and then show that they are dense for the topology of uniform convergence on T=[0, b] to the solutions of the original system. Then these results are applied to establish “bang-bang” type results for some classes of nonlinear infinite dimensional control systems. A few examples are also presented.     

Even Cycles in Graphs with Many Odd Cycles

It will be shown that if G is a graph of order n which contains a triangle, a cycle of length n or n−1 and at least cn odd cycles of different lengths for some positive constant c, then there exists some positive constant k=k(c) such that G contains at least kn ^1/6 even cycles of different lengths. Other results on the number of even cycle lengths which appear in graphs with many different odd length cycles will be given. Received: October 15, 1997     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

Algebraic Isomorphisms and Finite Distributive Subspace Lattices

Let L₁ and L₂ be finite distributive subspace lattices on realor complex Banach spaces. It is shown that every rank-preservingalgebraic isomorphism of AlgL₁ onto AlgL₂ is quasi-spatiallyinduced. If the algebraic isomorphism in question is known onlyto preserve the rank of rank one operators, then it inducesa lattice isomorphism between L₁ and L₂.     

Maeda-type Decomposition of CJ-generated Algebraic Lattices

We prove necessary and sufficient conditions for the decomposition of an arbitrary CJ-generated algebraic lattice into a direct product of subdirectly irreducible lattices. We generalize earlier results due to F. Maeda, T. Katriák and the present author. New structure theorems for two classes of CJ-generated algebraic lattices are also obtained.AMS Subject Classification (1991) 06B05 06B10Research partially supported by Hungarian National foundation for Scientific Research, Grant No. T029525 and T030243 and by János Bolyai Grant of Hungarian Academy of Science.     

Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

Let B(n) be the subset lattice of \({\{1,2,\dots, n\}.}\) Sperner’s theorem states that the width of B(n) is equal to the size of its biggest level. There have been several elegant proofs of this result, including an approach that shows that B(n) has a symmetric chain partition. Another famous result concerning B(n) is that its cover graph is hamiltonian. Motivated by these ideas and by the Middle Two Levels conjecture, we consider posets that have the Hamiltonian Cycle–Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. We show that the subset lattices have the HC-SCP property, and we obtain this result as a special case of a more general treatment.     

Algebraic Representation of Mappings Between Submodule Lattices

We discuss algebraic representations of mappings preserving arbitrary joins between submodule lattices. For a given join-preserving mapping $ \bar{g}:\mathfrak{L}\left( {_RM} \right)\to \mathfrak{L}\left( {_SN} \right) $ between submodule lattices, a representation is an R-balanced mapping h : B × M → N, where _S B _R is a bimodule such that $ \left\langle {h\left( {B\times U} \right)} \right\rangle =\bar{g}(U) $ for all $ U\in \mathfrak{L}\left( {_RM} \right) $ . We begin by posing the question in a general abstract context and by defining the canonical subrepresentation, which is a representation if and only if there exists a representation. The problem is to give easy and natural conditions for the existence of a representation. We consider a very general situation for the mappings and give sufficient criteria for the existence of a representation. We also consider lattice isomorphisms.     

A Note on Even Cycles and Quasirandom Tournaments

A cycle in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from to . In this short article, we show that for every fixed even integer , if close to half of the k‐cycles in a tournament T are even, then T must be quasirandom.This resolves an open question raised in 1991 by Chung and Graham 1991.     

Some Structure Theorems for Atomistic Algebraic Lattices

We prove that any atomistic algebraic lattice is a direct product of subdirectly irreducible lattices iff its congruence lattice is an atomic Stone lattice. We define on the set A(L) of all atoms of an atomistic algebraic lattice L a relation R as follows: for a, b A(L), (a, b) R ? θ(0, a) ∧ θ(0, b) ≠ ?_{Con L} . We prove that Con L is a Stone lattice iff R is transitive and we give a characterization of Cen (L) using R. We also give a characterization of weakly modular atomistic algebraic lattices.