On a class of partially ordered sets and their linear invariants

Let (, <) be a finite partially ordered set with rank function. Then is the disjoint union of the classes _k of elements of rank k and the order relation between elements in _k and _k+1 can be represented by a matrix S _k. We study partially ordered sets which satisfy linear recurrence relations of the type S _k (S _k ^T ) – c _k (S _{k – 1})^T S _{k – 1} = d _k ⁺ – c _k d _k ^– ) Id for all k and certain coefficients d _k ⁺, d _k ^- and c _k.     

Convexity in cristallographical lattices

A subset of a (cristallographical) lattice ⁿ is called convex whenever it is the intersection of the lattice with a convex set of the affine space containing ⁿ. We give a characterization of the convex sets which is intrinsic to the lattice and do the same for other related notions, e.g. the boundary of a convex set of ⁿ. A statement analogous to Helly's theorem is also proved.     

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.     

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).     

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.

---

---

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

Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic

Given ringsR with prime power characteristicp ^k , quasivarieties (R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides ideald _R ofR. The main technical result is that (R) (S) makes all implications of the formd _s =S d_R=R true, for any such expressiond. The proof makes use of the known equivalence between (R) (S) and existence of an exact embedding functorR-Mod S -Mod. Fork 2, the ordered setW(p ^k ) of all lattice quasivarieties (R),R having characteristic p _K , is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(p ^k ) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerp ^k ,k 2, a ringR can be constructed so that (R) and (R ^op) for the opposite ringR ^op are distinct elements ofW(p ^k ).Presented by R. Freese.Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.     

On representing contexts in line arrangements

Acontext is defined to be a triple (G, M, J) of setsG, M and an incidence relationJ G×M.A finite set ofn oriented lines in general position in the euclidean plane induces a cell decomposition of the plane. For a givenk-element subset of cells of dimension 2, we define an incidence relationJ × as follows:t _i andl _j are incident if and only ift _i lies on the positive side with respect tol _j .We call a context (G, M, J)represented in a line arrangement if and only if there are relation preserving bijections betweenG and ,M and , respectively. We study conditions for a context to be representable in a line arrangement.Especially, we provide a non-trivial infinite class of contexts which can not be represented in a line arrangement.     

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.     

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     

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.     

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.     

On the convergence of the projected gradient method

We study the projected gradient algorithm for linearly constrained optimization. Wolfe (Ref. 1) has produced a counterexample to show that this algorithm can jam. However, his counterexample is only ¹( ⁿ ), and it is conjectured that the algorithm is convergent for ²-functions. We show that this conjecture is partly right. We also show that one needs more assumptions to prove convergence, since we present a family of counterexamples. We finally give a demonstration that no jamming can occur for quadratic objective functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada     

Smoothability of the Conformal Boundary of a Lorentz Surface Implies 'Global Smoothability'

In 1985, Kulkarni defined the conformal boundary of a simply connected and time-oriented Lorentzian surface . He also introduced a notion of 'smoothability' of this boundary, depending only on local properties of . In this paper we show that smoothability of is in fact a global property of . In doing so, we classify Lorentzian surfaces with smoothable boundaries up to conformal homeomorphism. To be specific, suppose that the Minkowski plane E ² ₁ is the x,y-plane with metric dxdy. Our main theorem states that if is smoothable then is conformally homeomorphic to the interior U of a Jordan curve in E ² ₁ that is locally the graph of a continuous function over either the x-axis or the y-axis at each point of U.     

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     

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.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

A Favard theorem for orthogonal rational functions on the unit circle

We consider forn=0, 1,... the nested spaces _n of rational functions of degreen at most with given poles . Given a finite measure supported on the unit circle, we associate with it a nested orthogonal basis of rational functions ₀,..., _n for _n ,n=0, 1,.... These _n satisfy a recurrence relation that generalizes the recurrence for Szeg polynomials.In this paper we shall prove a Favard type theorem which says that if one has a sequence of rational functions _{n n} which are generated by such a recurrence, then there will be a measure supported on the unit circle to which they are orthogonal. We shall give a sufficient condition for the uniqueness of this measure.     

Itô correction terms for the radial parts of semimartingales on manifolds

Summary We consider functions,F, of a semimartingale,X, on a complete manifold which fail to beC ² only on, and are sufficiently well-behaved near, a codimension 1 subset . We obtain an extension of the Itô formula which is valid for all time by adding a continuous predictable process given explicitly in terms of two geometric local times ofX on and the Gâteaux derivative ofF. We then examine the cut locus of a point of the manifold in sufficient detail to show that this result applies to give a corresponding expression for the radial part of the semimartingale.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Recurrence-Transience for Self-similar Additive Processes Associated with Stable Distributions

For any stable distribution on the line, recurrence-transience of the selfsimilar additive process {X _t ,t0} with (X ₁)= is determined. Comparison with the stable Lévy process {Y _t ,t0} with (Y ₁)= is made: if is not strictly stable, then {Y _t } is transient but {X _t } is recurrent except the obviously transient case of monotone sample functions.