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.     

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.

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

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.     

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.     

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 .     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

On conditions for the discreteness of the spectrum of a non-self-adjoint system of differential equations of the Dirac type

A first-order non-self-adjoint system of differential equations with 2n unknown functions and with certain conditions on the coefficients is considered on the entire axis. The system is similar to the well-known Dirac system. An operator , associated with the considered system, is introduced in the space L²(– , ) of 2n-component vector-valued functions. Conditions for the compactness of the resolvent of the operator are found, yielding sufficient conditions for the discreteness of its spectrum.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 3–24, 1992.     

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.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

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.     

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.     

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     

Varieties of lattice ordered groups that contain no non-abelian o-groups are solvable

It is shown that the variety _n of lattice ordered groups defined by the identity x ⁿ y ⁿ =y ⁿ x ⁿ , where n is the product of k (not necessarily distinct primes) is contained in the (k+1)st power A ^k+1 of the variety A of all Abelian lattice ordered groups. This implies, in particular, that _n is solvable class k + 1. It is further established that any variety V of lattice ordered groups which contains no non-Abelian totally ordered groups is necessarily contained in _n , for some positive integer n.This work was supported in part, by NSERC Grant A4044.     

Congruences of Clone Lattices, II

We continue the study of congruences of clone lattices _A , where A is finite, started in an earlier paper by the author and A. P. Semigrodskikh. We prove that each clone that either contains all unary operations or consists of essentially unary operations forms a one-element class of any non-trivial congruence of _A . As a consequence, we get that _A has the greatest non-trivial congruence provided the lattice is not simple, that _A is directly indecomposable, and that it has neither distributive nor dually distributive elements except for the trivial ones.For |A|>2, no example of a non-trivial congruence is known so far. We exhibit some reasons why such congruences are not easy to find.     

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.     

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.     

Zur Darstellung der Automorphismengruppe einer kinematischen Kettengeometrie im Quadrikenmodell

There is a unique projective representation of the group of automorphisms of a geometry () [1] over a kinematic [3] algebra which is compatible with the quadric model [2].     

Some properties of metrics in a study on convergence to normality

Summary This paper is concerned with the use of the ₁ and metrics in a study of certain properties and implications of convergence rates in the central limit theorem for sums of independent and identically distributed random variables which belong to the domain of attraction of the normal distribution. Also, some general convergence rate results on the metric obtained under the assumption of a finite second moment are used as a vital tool in a new proof of the classical iterated logarithm law and in extending the scope of classical methods for the proof of other similar results of a more general kind.     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.