Lattices of Optimal Finite Lattice Packings

We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.The second author was partially supported by a DAAD Postdoc fellowship and the hospitality of Peking University during his work.     

Large Finite Lattice Packings

A finite lattice packing of a centrally symmetric convex body K in ^d is a family C+K for a finite subset C of a packing lattice of K. For >0 the density (C;K,) is defined by (C;K,) = card C·V(K)/V(conv C+K). Assume that C ⁿ is the optimal packing with given n=card C, n large. It was known that conv C ⁿ is a segment if is less than the sausage radius _s (>0), and the inradius r(conv C ⁿ ) tends to infinity with n if is greater than the critical radius _c ( _s ). We prove that if > _c in ^d , then the shape of conv C ⁿ is not too far from being a ball. In addition, if r(conv C ⁿ ) is bounded but the radius of the largest (d–2)-ball in C ⁿ tends to infinity, then eventually C ⁿ is contained in some k–plane and its shape is not too far from being a k-ball where either k=d–1 or k=d–2. This yields in ³ that if _s << _c , then conv C ⁿ is eventually planar and its shape is not too far from being a disc. As an example, we show that _s = _c if K is a 3-ball, verifying the Strong Sausage Conjecture in this case. On the other hand, if K is the octahedron then _s < _c holds even for general (not only lattice) packings.     

Embedding Finite Lattices into Finite Biatomic Lattices

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively,– the class of all finite lattices;– the class of all finite lower bounded lattices (solved by the first author's earlier work);– the class of all finite join-semidistributive lattices (this problem was, until now, open).We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive lattices.     

Finite Packings of Spheres

We show that the sausage conjecture of László Fejes Tóth on finite sphere packings is true in dimension 42 and above. Received October 19, 1995, and in revised form May 28, 1996.     

On Large Lattice Packings of Spheres

The packing density of large lattice packings of spheres in Euclidean E ^d measured by the parametric density depends on the parameter and on the shape of the convex hull P of the sphere centers; in particular on the isoperimetric coefficient of P and on the second term in the Ehrhart polynomial of the lattice polytope P. We show in E ^d , d 2, that flat or spherelike polytopes generate less dense packings, whereas polytopes with suitably chosen large facets generate dense packings. This indicates that large lattice packings in E ³ of high parametric density may be good models for real crystals.     

On Extremal Finite Packings

Abstract. We show that in contrast to the classical infinite packing problem, even in the Euclidian plane, the solutions to several finite packing problems are non-lattice packings if the number of translates is large enough. This answers, in particular, a question by Paul Erdos [E].     

On Free Planes in Lattice Ball Packings

This note, by studying the relations between the length of theshortest lattice vectors and the covering minima of a lattice,proves that for every d-dimensional packing lattice of ballsone can find a four-dimensional plane, parallel to a latticeplane, such that the plane meets none of the balls of the packing,provided that the dimension d is large enough. Nevertheless,for certain ball packing lattices, the highest dimension ofsuch ‘free planes’ is far from d.     

Counting Finite Lattices

The correct values for the number of all unlabeled lattices on n elements are known for . We present a fast orderly algorithm generating all unlabeled lattices up to a given size n. Using this algorithm, we have computed the number of all unlabeled lattices as well as that of all labeled lattices on an n-element set for each . Received April 4, 2000; accepted in final form November 2, 2001. RID="h1" ID="h1" Presented by R. Freese.     

Lattices with Complemented Tolerance Lattice

We characterize lattices with a complemented tolerance lattice. As an application of our results we give a characterization of bounded weakly atomic modular lattices with a Boolean tolerance lattice.     

Finite Lattices as Lattices of Relative Congruences of Finite Unars and Abelian Groups

It is proved that every finite lattice is isomorphic to an R-congruence lattice of a finite unar (finite Abelian group), as well as to a lattice of R-varieties for some locally finite finitely axiomatizable quasivariety of unars (Abelian groups) R.     

On the Densities of Certain Lattice Packings by Parallelepipeds

Given any (non-degenerate) n-dimensional lattice L, let (L) denote the supremum of the numbers such that there exists a lattice packing Q + L of density where Q is some o-symmetric parallelepiped with faces parallel to the coordinate axes. Many efforts have been made to determine or estimate the minimal such density _n taken over all n-dimensional lattices. It is known that 0$$ " align="middle" border="0"> . Here we investigate a sequence of lattices L _n which are known to minimize the function (L) in dimensions n 3 and are likely to provide the minima _n = (L _n ) in certain higher dimensions. We establish the inequality (L _n ) n ^–n/2 which supports the conjecture that lim sup _n ( _n )^{1/(n log n)} is positive.     

Finite and Uniform Stability of Sphere Packings

The main purpose of this paper is to discuss how firm or steady certain known ball packing are, thinking of them as structures. This is closely related to the property of being locally maximally dense. Among other things we show that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small ε > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than ε , is the identity rearrangement. For example, the lattice packings D _d and A _d for d ≥ 3 in E ^d are all uniformly stable. The methods developed here can work for many other packings as well. We also give a construction to show that the densest cubic lattice ball packing in E ^d for d ≥ 2 is not uniformly stable. A packing of balls is called finitely stable if any finite subfamily of the packing is fixed by its neighbors. If a packing is uniformly stable, then it is finitely stable. On the other hand, the cubic lattice packings mentioned above, which are not uniformly stable, are nevertheless finitely stable. Received April 22, 1996, and in revised form October 11, 1996.     

On the Density of 2-Saturated Lattice Packings of Discs

 In a recent paper of G. Fejes Tóth, G. Kuperberg and W. Kuperberg [1] a conjecture has been published concerning the greatest lower bound of the density of a 2-saturated packing of unit discs in the plane. (A packing of unit discs is said to be 2-saturated if none of the discs could be replaced by two other ones of the same size to generate a new packing. A packing of the unit disc is a lattice packing if the centers form a point lattice.) In the present note we study this problem for lattice packings, however, in a more general form in which the removed unit disc is replaced by two discs of radius r. A corollary of our results supports the above conjecture proving that a lattice packing cannot be 2-saturated except if its density is larger than the conjectured bound. (Received 6 December 2000; in revised form March 29, 2001)     

Supermodular Functions on Finite Lattices

The supermodular order on multivariate distributions has many applications in financial and actuarial mathematics. In the particular case of finite, discrete distributions, we generalize the order to distributions on finite lattices. In this setting, we focus on the generating cone of supermodular functions because the extreme rays of that cone (modulo the modular functions) can be used as test functions to determine whether two random variables are ordered under the supermodular order. We completely determine the extreme supermodular functions in some special cases.     

关于剩余格中滤子的格结构

This paper is devoted to the discussion of filters in residuated lattices. The lattice structure of filters in residuated lattice was established. It is proved that the set of all filters forms a distributive lattice. Also, the concept of prime filter in residuated lattice was proposed and some equivalent conditions about prime filter were given.     

Whaley's Theorem for Finite Lattices

Whaley's Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|^1/3. It is also shown that that every finite modular lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|^1/2, and every finite distributive lattice L with |L|≥4 contains a proper sublattice S with |S|≥3/4|L|. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Family of Optimal Packings in Grassmannian Manifolds

A remarkable coincidence has led to the discovery of a family of packings of -dimensional subspaces of m-dimensional space, whenever m is a power of 2. These packings meet the orthoplex bound and are therefore optimal.