Semimodular lattices with isomorphic graphs

Two discrete modular lattice and have isomorphic graphs if and only if is of the form A × and is of the form A × for some lattices A and and . We prove that for discrete semimodular lattices and this latter condition holds if and only if and have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.     

A construction of semimodular lattices

In this paper we prove that if is a finite lattice, and r is an integral valued function on satisfying some very natural conditions, then there exists a finite geometric (that is, semimodular and atomistic) lattice I containing as a sublattice such that r is the height function of restricted to . Moreover, we show that if, for all intervals [e, f] of , semimodular lattices I, of length at most r(f)-r(e) are given, then I can be chosen to contain I in its interval [e, f] as a cover preserving {0}-sublattice. As applications, we obtain results of R. P. Dilworth and D. T. Finkbeiner.     

Support functions of convex compacta

The properties of the space (X_x) of all sublinear functionals, defined on a space X' (topologically adjoint to a Hausdorff locally convex barrelled space X) and continuous in the Arens topology × (X, X), equipped with topology of uniform convergence on bounded subsets of X are studied. It is shown that completeness and separability of a space X are hereditary for (X_x). Criteria for the compactness of subsets of (X_x) and conditions for the metrizability of compacta in (X_x) are given. The topological isomorphism between (X_x) and the space of all nonempty convex compacta in X with the Vietoris topology is established. The results obtained here are applied for the study of the properties of multiple-valued integrals.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 203–213, August, 1977.The author thanks S. S. Kutateladze for useful discussions regarding this article.     

Adjungierte Funktoren in der Kategorie der p-Bornologischen Räume

In the category p b of p-convex vector spaces and linear maps preserving bounded sets a p-bornological topology will be introduced on the tensor product of two spaces, likewise on the spaces of morphisms Hom(E,F). Thus one gets a pair of adjoint functors from p to p , p being the category of p-bornological spaces and continuous linear maps, and the topologies being introduced will be characterized by extreme properties with respect to the adjoint transformations.

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Dieser Arbeit liegt ein Teil der Dissertation des Autors, Kiel 1967, zugrunde.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Normal bundles of rational curves inP 3

LetP ^r=P _k ^r be the projective space over an algebraically closed ground field k. Let X be a rational space cur ve of degree n with only ordinary singularities. Since X is rational, the normal bundleN of X inP ³ splits inN = _{1 2} where ₁, and ₂ are line bundles, and we have deg ₁ + deg ₂ = 4n – 2. We consider the non-negative integer defined by 2 = |deg ₁ – deg ₂|. The aim of this paper is to determine all possible values of and to describe the variety parametrizing all twisted rational curves inP ³ with only ordinary singularities for a fixed degree n and fixed .The paper was supported by C.N.R., while both authors were members of GNSAGA     

Tidy Noether lattices

A Noether lattice satisfying the union condition on primes which is not a domain and in which every nonzero principal element is integrally closed is characterized in terms of its direct summands. It is shown that either: (1) if has no proper nonzero direct summands, then every nonzero principal element of is integrally closed if and only if is a local Noether lattice whose maximal element is principal and has square zero; or (2) if has a proper nonzero direct summand, then every nonzero principal element of is integrally closed if and only if for each minimal direct summandA of, the quotient lattice [0,A] is an integrally closed domain.Presented by R. P. Dilworth.     

Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients

Summary Let be a weighted Schwartz's space of rapidly decreasing functions, the dual space and (t) a perturbed diffusion operator with polynomial coefficients from into itself. It is proven that (t) generates the Kolmogorov evolution operator from into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on : whereW(t) is a Brownian motion and ^*(t) is the adjoint of (t) and show a central limit theorem for interacting multiplicative diffusions.     

Flocks of a quadratic cone in PG(3, q), q ≤ 8

If (O) is a quadratic cone in PG(3,q), with vertex x, then a flock of (O) is a partition of (O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q ² and a generalized quadrangle of order (q ², q). Here we determine all flocks of (O) for q 8.     

Removability of singularities in potential theory

IfQ is a compact metric space, a system of its closed subsets andg: R a prescribed nonnegative function, the conditions ong, and a closedF Q are specified guaranteeing the existence of a nontrivial Borel measure with support inF such that (L)g(L), L.For some kernels in potential theory these conditions permit to characterize geometrically those sets which contain support of a nontrivial measure whose potential belongs to a given class of functions. Several applications concerning removability of singularities of partial differential equations are presented.     

Complete lattices of subgroups in compact groups

Summary LetG be a compact group and a sublattice of the lattice of all closed subgroups ofG. In Proposition 1 it is shown that is a complete lattice if it is a closed subset of the spaceG ^c of all closed non empty subsets ofG. In general the converse of this fact is not true (Example 3), but the following result can be obtained (Theorem 5): If is complete and if each element of is normalized by the connected component of the identity ofG, then is a closed, totally disconnected subset ofG ^c. We mention the following corollary: IfG is totally disconnected or abelian, then is complete if and only if it is a closed subset ofG ^c.While writing this paper the author was a fellow of the National Research Council (A 7171).     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

On a Laplace Transform Based Metric for Probabilities on a Hilbert Space

Let D be a closed subset of a real separable Hilbert space H. Let (D) denote the set of all Borel probability measures on D and (D) the set of all probabilities with integrable Laplace transform. A metric d, based on the Laplace transform, is defined on (D). Topological properties, viz., separability, connectedness, completeness, compactness and local compactness, of (D, d are investigated, and the d-topology is compared with the topology of weak convergence.     

Zur Krümmungskennzeichnung der kinematischen Laguerre-Ketten

In the case of certain geometries () [1] over kinematic algebras [5],[6] the -trace of the system of chains can be characterized by constant curvature. The curvature allows an affine interpretation of non-affine points.     

Weak relative pseudo-complements of closure operators

We define the notion of weak relative pseudo-complement on meet semi-lattices, and we show that it is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main result is that if a complete lattice is meet-continuous, then every closure operator on admits weak relative pseudo-complements with respect to continuous closure operators on .Presented by E. T. Schmidt.     

Chow rings of toric varieties defined by atomic lattices

We study a graded algebra over defined by a finite lattice and a subset in , a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice , as the Chow ring of a smooth toric variety that we construct from and . We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Gröbner basis of the relation ideal of D and a monomial basis of D.     

Commutativity of operators on the lattice of existence varieties

In a previous paper the authors introduced seven complete congruences on the lattice _ev(I of e-varieties of regular semigroups of the form _P :U _P VPU=PV, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form _P :U _P VPU=PV, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence on _ev(I) and eachU_ev(I), the -class ofU is an interval [U ,U ] so that there is associated with each such congruence an idempotent operatorUU on _ev(I). This paper establishes numerous results concerning the commutativity of operators of this form.This work was supported in part by NSERC Grant 4044.     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

Extending the Balas-Yu bounds on the number of maximal independent sets in graphs to hypergraphs and lattices

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most ^p+1, where is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors in the product of n lattices =₁××_n, where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into . We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors x=₁××_n for a system of polymatroid inequalities does not exceed max{Q,^logt/c(2Q,)}, where is the number of minimal feasible vectors for the system, , , and c(,) is the unique positive root of the equation 2^c(^c/log–1)=1. This bound is nearly sharp for the Boolean case ={0,1}ⁿ, and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides . This research was supported by the National Science Foundation (Grant IIS-0118635), and by the Office of Naval Research (Grant N00014-92-J-1375). The second and third authors are also grateful for the partial support by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science.Mathematics Subject Classification (2000):20E28, 20G40, 20C20     

Saturation and omitting types

Models which omit a fixed set of types are considered, and notions analogous to those of saturation and compactness are formulated in this context. Conditions are found under which such saturated models exist and are unique; and the preservation of compactness under direct products and homomorphisms is proven. Atomic and positive compactness are shown to be equivalent in this context.In [5] Morley and Vaught give conditions on a class of structuresK which ensure thatK has homogeneous universal models of certain powers. The work is based on the nature of the structures concerned and not on that of the language for which they are assumed to be models.Keisler [2] and [3] considered the notion of a saturated model and showed that for certain cardinals, a complete theoryT in elementary logic has a saturated model of power, which is homogeneous-universal in the sense of Morley and Vaught ifT is neat. Keisler's definition of saturated model is intimately connected with the language involved, and the results are dependent on the semantics of that language.In this paper we consider such problems as the existence of saturated models, and under what conditions they are homogeneous-universal, assuming that a specified set of types is omitted by all the models under consideration.In § 1 we formulate the definition of a-(, J, K)-saturated model for a cardinal, a set of types,J andK sets of formulae, and prove some basic results. In § 2 the existence of ⁺-(, J, K.)-saturated models of power 2 is proven and in § 3 a condition is given for the uniqueness of such a model. We examine (, J, K)-special models in § 4. This work is based on that of Bell and Slomson [1] and Wilmers [10]. In § 5-(, J)-K compact models are defined and we examine closure under direct products and homomorphisms. Lastly, in § 6-(, J)-atomic compactness and-(, J)-positive compactness are shown to be equivalent.