Local and improper Daniell-Loomis integrals

In this paper we start from previous results obtained in [7] on the abstract space of Daniell-Loomis integrable functionsL, which is constructed like to the Daniell extension process, but without continuity assumptions on the elementary integral. The localized integral is used to prove thatL consists of those functions whose local upper and lower integrals are equal and finite, or thatL is closed with respect to improper integration. Our results are also holded in integration with respect to finitely additive measures.     

Measure theory of statistical convergence

The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)     

Simplicial join via tensor product

Let be a field, and M and N two finitely generated graded modules over standard graded -algebras A and B, respectively. We will study generalized, sequentially, almost, and approximately Cohen–Macaulay as well as clean, and pretty clean properties of the -module through the corresponding properties of M and N. The behavior of these properties with respect to the simplicial join of two simplicial (multi)complexes will be revealed as corollaries.     

A generalized Henstock-Stieltjes integral involving division functions

We can consider the Riemann-Stieltjes integral dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.      

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

A Cohen-type theorem for Artinian modules

We show that a finitely embedded module M over a commutative ring R is Artinian if the factor module M/(0 :_M P) is finitely embedded for every prime ideal P of R. Received: 10 June 2005     

The extent of non-conglomerability of finitely additive probabilities

Summary An arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition. If the probability is defined on a -field, the bound is sharp. Hence, non-conglomerability (or equivalently non-disintegrability) characterizes finitely as opposed to countably additive probability. Nonetheless, there exists a PFA probability which is simultaneously conglomerable over an arbitrary finite set of partitions.Neither conglomerability nor non-conglomerability in a given partition is closed under convex combinations. But the convex combination of PFA ultrafilter probabilities, each of which cannot be made conglomerable in a common margin, is singular with respect to any finitely additive probability that is conglomerable in that margin.     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

Completezza degli spaziL p per misure finitamente additive

Summary A criterion of completeness of metric (not necessarily normed) L^p spaces with respect to bounded finitely additive measures is established (L ^p spaces are defined differently that it is usually done in the setting of finitely additive measures). This result hinges on the Stone representation theorem for Boolean algebras.     

Fubini-integral metrics

In this paper, by using an integral extension of Lebesgue power with local integral metrics, we stablish abstract Fubini type theorems, which subsume most known situations of integration with respect to finitely additive measures.     

Weakly Integrally Closed Numerical Monoids and Forbidden Patterns

An integral domain D is weakly integrally closed if whenever x is in the quotient field of D, and J is a nonzero finitely generated ideal of D such that xJ ? J ², then x is in D. We define weakly integrally closed (WIC) numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. The characteristic function of a numerical monoid M can be thought of as an infinite binary string s(M). A pattern of finitely many 0's and 1's is called forbidden if whenever s(M) contains it, then M is not weakly integrally closed. The pattern 11011 is forbidden. We show that a numerical monoid M is WIC if and only if s(M) contains no forbidden patterns. We also show that for every finite set S of forbidden patterns, there exists a numerical monoid M that is not WIC and for which s(M) contains no stretch (in a natural sense) of a pattern in S.     

On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.     

Expansive Convergence Groups are Relatively Hyperbolic

Let a discrete group G act by homeomorphisms of a compactum in a way that the action is properly discontinuous on triples and cocompact on pairs. We prove that such an action is geometrically finite. The converse statement was proved by P. Tukia [T3]. So, we have another topological characterisation of geometrically finite convergence groups and, by the result of A. Yaman [Y2], of relatively hyperbolic groups. Further, if G is finitely generated then the parabolic subgroups are finitely generated and undistorted. This answer to a question of B. Bowditch and eliminates restrictions in some known theorems about relatively hyperbolic groups. Received: April 2007, Revision: May 2008, Accepted: August 2008     

Projections in free product C $^*$–algebras, II

Let M be a connected sum of complete Riemannian manifolds satisfying the volume doubling condition and the Poincaré inequality. We prove that the space of polynomial growth harmonic functions on M is finite dimensional whenever M has finitely many ends and satisfies the finite covering condition on each end. This result directly generalizes that of Tam, and it also partially generalizes that of Colding and Minicozzi II. Received July 17, 1998; in final form December 2, 1998     

Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

Let (X, ~) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ~) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l ^p (X) or c_0(X), where (X, ~) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case and in [42] in case X = G is a general finitely generated discrete group. Submitted: May 21, 2007. Revised: September 25, 2007. Accepted: November 5, 2007.     

Forcing Linearity Numbers for Multiplication Modules

In this article, we prove that for any multiplication module M, the forcing linearity number of M, fln(M), belongs to {0,1,2}, and if M is finitely generated whose annihilator is contained in only finitely many maximal ideals, then fln(M) = 0. Also, the forcing linearity numbers of multiplication modules over some special rings are given. We also show that every multiplication module is semi-endomorphal.     

An introduction to presentations of monoid acts: quotients and subacts

The purpose of this article is to introduce the theory of presentations of monoids acts. We aim to construct “nice” general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M-act A and a subact B of A, on the one hand, we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand, we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has a finite complement in an M-act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.     

Another Gorenstein analogue of flat modules

A right R-module M is called glat if any homomorphism from any finitely presented right R-module to M factors through a finitely presented Gorenstein projective right R-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat right R-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ring R. Then we obtain that a right R-module M is glat if and only if M is a direct limit of finitely presented Gorenstein projective right R-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules.     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

Integral formulas for closed spacelike hypersurfaces in anti-de Sitter space <Emphasis Type="Italic">H</Emphasis><Stack><Subscript>1</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript></Stack> (−1)

In this paper, we study closed k-maximal spacelike hypersurfaces M ⁿ in anti-de Sitter space H ₁ ⁿ⁺¹ (−1) with two distinct principal curvatures and give some integral formulas about these hypersurfaces. The first author was supported by Japan Society for Promotion of Science. The third author was supported by grant Proj. No. R17-2008-001-01000-0 from Korea Science & Engineering Foundation.