Congruence-preserving extensions of finite lattices to sectionally complemented lattices

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice.

     

Monotonic dynamical systems under spatial discretization

We estimate the probability of replicating the asymptotic behaviour of a dynamical system generated by a monotonic mapping for randomly centered roundoff lattices.

     

A Brunn-Minkowski inequality for the integer lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.

     

A lower bound for rank 2 lattice rules

We give a lower bound for a quality measure of rank 2 lattice rules which shows that an existence result of Niederreiter is essentially best possible.

     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Compactness properties of operators dominated by AM-compact operators

We study several properties about the problem of domination in the class of positive AM-compact operators, and we obtain some interesting consequences on positive compact operators. Also, we give a sufficient condition under which a Banach lattice is discrete.

     

Semi-compactness of positive Dunford-Pettis operators on Banach lattices

We investigate Banach lattices on which each positive Dunford-Pettis operator is semi-compact and the converse. As an interesting consequence, we obtain Theorem 2.7 of Aliprantis-Burkinshaw and an element of Theorem 1 of Wickstead.

     

Sublattices of the banach envelope of weak

We prove that every separable Banach lattice is lattice isometric to a closed sublattice of the Banach envelope of Weak

     

Lipschitz functions with maximal Clarke subdifferentials are generic

We show that on a separable Banach space most Lipschitz functions have maximal Clarke subdifferential mappings. In particular, the generic nonexpansive function has the dual unit ball as its Clarke subdifferential at every point. Diverse corollaries are given.

     

Orbits and Lattices for Linear Random Number Generators with Composite Moduli

In order to analyze certain types of combinations of multiple recursive linear congruential generators (MRGs), we introduce a generalized spectral test. We show how to apply the test in large dimensions by a recursive procedure based on the fact that such combinations are subgenerators of other MRGs with composite moduli. We illustrate this with the well-known RANMAR generator. We also design an algorithm generalizing the procedure to arbitrary random number generators.

     

A solution to a problem on invertible disjointness preserving operators

We construct an invertible disjointness preserving operator on a normed lattice such that is not disjointness preserving.

     

Lattice computations for random numbers

We improve on a lattice algorithm of Tezuka for the computation of the -distribution of a class of random number generators based on finite fields. We show how this is applied to the problem of constructing, for such generators, an output mapping yielding optimal -distribution.

     

Efficient lattice assessment for LCG and GLP parameter searches

In the present paper we show how to speed up lattice parameter searches for Monte Carlo and quasi-Monte Carlo node sets. The classical measure for such parameter searches is the spectral test which is based on a calculation of the shortest nonzero vector in a lattice. Instead of the shortest vector we apply an approximation given by the LLL algorithm for lattice basis reduction. We empirically demonstrate the speed-up and the quality loss obtained by the LLL reduction, and we present important applications for parameter selections.

     

A natural lattice basis problem with applications

Integer lattices have numerous important applications, but some of them may have been overlooked because of the common assumption that a lattice basis is part of the problem instance. This paper gives an application that requires finding a basis for a lattice defined in terms of linear constraints. We show how to find such a basis efficiently.

     

Compact orthoalgebras

We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated is of finite height. We then focus on stably ordered TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated is determined by that of its space of atoms.

     

Cycles representing the Todd class of a toric variety

In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number to each rational polyhedral cone in the lattice, such that for any toric variety with fan in the lattice, we have

This constitutes an improved answer to an old question of Danilov.

In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli.

Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first author.

     

An intractability result for multiple integration

We prove that the problem of multiple integration in the Korobov class is intractable since the number of function evaluations required to achieve a worst case error less than is exponential in the dimension.

     

Distribution Properties of Multiply-with-Carry Random Number Generators

We study the multiply-with-carry family of generators proposed by Marsaglia as a generalization of previous add-with-carry families. We define for them an infinite state space and focus our attention on the (finite) subset of recurrent states. This subset will, in turn, split into possibly several subgenerators. We discuss the uniformity of the -dimensional distribution of the output of these subgenerators over their full period. In order to improve this uniformity for higher dimensions, we propose a method for finding good parameters in terms of the spectral test. Our results are stated in a general context and are applied to a related complementary multiply-with-carry family of generators.

     

Modularity prevents tails

We establish a direct correspondence between two congruence properties for finite algebras. The first property is that minimal sets of type have empty tails. The second property is that congruence lattices omit pentagons of type .