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.     

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.     

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.     

Dual-lattice theorems in the geometric approach

A one-to-one correspondence is shown to exist between the lattice of all self-bounded (A, )-controlled invariants contained in and the lattice of all self-hidden (A, )-conditioned invariants containing . This correspondence, stated herein as the main dual-lattice theorem, allows a straightforward derivation of the universal bounds of the lattices, particularly when additional constraints are imposed, such as to contain a given subspace for the elements of the former lattice and to be contained in a given subspace for the elements of the latter. Then, two further minor dual-lattice theorems, dual to each other, are presented, and some connections and applications of the new theory to standard control and observation problems are briefly discussed.     

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.     

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.     

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     

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.     

Pasting of R 2-planes

The notion pasting sum (P _i , _i ) of two R ²-planes (or Salzmann planes) (P _i , _i ) is developed. Necessary and sufficient conditions for it to be an R ²-plane again are given. The notion is applied to classify all flat projective planes whose collineation group contains a sub-group with (isomorphism type of , fixed element configuration)=(², x).     

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.     

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.     

Cusp structures of alternating links

Summary An alternating link is canonically associated with every finite, connected, planar graph . The natural ideal polyhedral decomposition of the complement of is investigated. Natural singular geometric structures exist onS ³– , with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection. For the class of balanced graphs, this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles. Examples of links containing immersed, closed ₁-injective surfaces in their complements are given. These surfaces persist after most surgeries on the link, the resulting closed 3-manifolds consequently being determined by their fundamental groups.Oblatum 20-V-1991     

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.     

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.     

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.     

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.     

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.     

Hilbert lattices with the extension property

Separable Hilbert lattices are constructed that enjoy the following property: each ortho-isomorphism between arbitrary interval sublattices [0, a] and [0, b] of height at least 3 (a, b ) extends to an ortho-automorphism of the lattice #x2112.     

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.     

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.