Grasping Infinity by Finite Sets

We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of (finite) wide sets.     

Infinite Rings with Planar Zero-Divisor Graphs

In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ?₂-graded algebras. Essentially, the results are the same as in the classical case, except that the notion of a ?₂-graded division algebra needs to be modified. We classify all finite dimensional ?₂-graded division algebras over ? and ?.     

On the Normal Structure of NonCommutative Association Schemes of Rank 6

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory. One of these generalizations is the observation that, similar to groups, association schemes of finite order are commutative if they have at most five elements and not necessarily commutative if they have six elements. While there is (up to isomorphism) only one noncommutative group of order 6, there are infinitely many pairwise non-isomorphic noncommutative association schemes of finite order with six elements. (Each finite projective plane provides such a scheme, and non-isomorphic projective planes yield non-isomorphic schemes.) In this note, we investigate noncommutative schemes of finite order with six elements which have a symmetric normal closed subset with three elements. We take advantage of the classification of the finite simple groups.     

Maximal Condorcet Domains

A Condorcet domain is a subset of the set of linear orders on a finite set of candidates (alternatives to vote), such that if voters preferences are linear orders belonging to this subset, then the simple majority rule does not yield cycles. It is well-known that the set of linear orders is the Bruhat lattice. We prove that a maximal Condorcet domain is a distributive sublattice in the Bruhat lattice. An explicit lattice formula for the simple majority rule is given. We introduce the notion of a symmetric Condorcet domain and characterize symmetric Condorcet domains of maximal size.     

Partially finite convex programming,Part I: Quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.     

Rectangular tileability and complementary tileability are undecidable

Does a given set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this problem. However, we present an algorithm for testing whether the complement of a finite region is tileable by a set of rectangles.     

The mean value property for harmonic functions on graphs and trees

Following the Euclidean example, we introduce the strong and weak mean value property for finite variation measures on graphs. We completely characterize finite variation measures with bounded support on radial trees which have the strong mean value property. We show that for counting measures on bounded subsets of a tree with root o, the strong mean value property is equivalent to the invariance of the subset under the action of the stabilizer of o in the automorphism group. We finally characterize, using the discrete Laplacian, the finite variation measures on a generic graph which have the weak mean value property and we give a non-trivial example. Received: July 21, 2000; in final form: March 13, 2001?Published online: March 19, 2002     

Decomposition of group rings and the converse of Schur’s lemma

In this work, we give necessary and su?cient conditions for a group ring (of finite group) to satisfy the converse of Schur’s lemma for group rings of finite groups. Rings considered here are commutative or noncommutative perfect. Some cases of twisted group rings are studied. Also we introduce the notion of semi-CSL ring.     

Mild solutions for a class of fractional SPDEs and their sample paths

We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset ${D\subset\mathbb{R}^{d}}We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset and driven by an infinite-dimensional fractional noise. We prove the existence of such a solution, establish its relation with the variational solution introduced by Nualart and Vuillermot (J Funct Anal 232:390–454, 2006) and the H?lder continuity of its sample paths when we consider it as an L ²(D)-valued stochastic process. When h is an affine function, we also prove uniqueness. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness. M. Sanz-Solé was supported by the grant MTM 2006-01351 from the Dirección General de Investigación, Ministerio de Educación y Ciencia, Spain.     

Multi-graded nilpotent tuples

We discuss multi-graded nilpotent tuples of multi-graded vector spaces which are a generalization of graded nilpotent pairs. The multi-grading yields a natural notion of a shape of such tuple and our main interest is to answer the question “Is the number of multi-graded nilpotent tuples of a fixed shape, up to base change in the homogeneous components, finite?” Our methods make use of a translation to the class of so-called “Multi-staircase algebras” and we classify their representation types.     

The finite cutset property

A cutset of H is a subset of ∪ H which meets every element of H. H has the finite cutset property if every cutset of H contains a finite one. We study this notion, and in particular how it is related to the compactness of H for the natural topology. MSC: 04A20, 54D30.     

Finite groups with sn-embedded or s-embedded subgroups

A number of authors have studied the structure of a finite group G under the assumption that some subgroups of?G are well located in?G. We will generalize the notion of s-permutable and s-permutably embedded subgroups and we will obtain new criterions of p-nilpotency and supersolvability of groups. We also generalize some known results.     

Decidability of absorption in relational structures of bounded width

The absorption theory of Barto and Kozik has proven to be a very useful tool in the algebraic approach to the Constraint Satisfaction Problem and the structure of finite algebras in general. We address the following problem: Given a finite relational structure \({\mathbb{A}}\) and a subset \({B \subseteq A}\) , is it decidable whether B is an absorbing subuniverse? We provide an affirmative answer in the case when \({\mathbb{A}}\) has bounded width (i.e., the algebra of polymorphisms of \({\mathbb{A}}\) generates a congruence meet semidistributive variety). As a by-product, we confirm that in this case the notion of Jónsson absorption coincides with the usual absorption. We also show that several open questions about absorption in relational structures can be reduced to digraphs.     

Zip模(英文)

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.     

Analytic subvarieties with many rational points

We give a generalization of the classical Bombieri–Schneider–Lang criterion in transcendence theory. We give a local notion of LG-germ, which is similar to the notion of E-function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let ${K \subset \mathbb{C}}We give a generalization of the classical Bombieri–Schneider–Lang criterion in transcendence theory. We give a local notion of LG-germ, which is similar to the notion of E-function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let K ì \mathbbC{K \subset \mathbb{C}} be a number field and X a quasi-projective variety defined over K. Let γ : M → X be an holomorphic map of finite order from a parabolic Riemann surface to X such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every p ? X(K)?g(M){p\in X(K)\cap\gamma(M)} the formal germ of M near P is an LG-germ, then we prove that X(K)?g(M){X(K)\cap\gamma(M)} is a finite set. Then we define the notion of conformally parabolic K?hler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact K?hler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that A is conformally parabolic variety of dimension m over \mathbbC{\mathbb{C}} with K?hler form ω and γ : A → X is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then m. Suppose that for every p ? X(K)?g(A){p\in X(K)\cap \gamma (A)} , the image of A is an LG-germ. then we prove that there exists a current T on A of bidegree (1, 1) such that ò_ATùw^m-1{\int_AT\wedge\omega^{m-1}} explicitly bounded and with Lelong number bigger or equal then one on each point in γ ⁻¹(X(K)). In particular if A is affine γ ⁻¹(X(K)) is not Zariski dense.     

Euclidean quotients of finite metric spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α?1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ?_p, and the particular case of the hypercube.     

Description of random fields by means of one-point finite-conditional distributions

The aim of this note is to investigate the relationship between strictly positive random fields on a lattice ? ^ν and the conditional probability measures at one point given the values on a finite subset of the lattice ? ^ν . We exhibit necessary and sufficient conditions for a one-point finite-conditional system to correspond to a unique strictly positive probability measure. It is noteworthy that the construction of the aforementioned probability measure is done explicitly by some simple procedure. Finally, we introduce a condition on the one-point finite conditional system that is sufficient for ensuring the mixing of the underlying random field.     

On quasi-amorphous sets

A set is said to be amorphous if it is infinite, but cannot be written as the disjoint union of two infinite sets. The possible structures which an amorphous set can carry were discussed in [5]. Here we study an analogous notion at the next level up, that is to say replacing finite/infinite by countable/uncountable, saying that a set is quasi-amorphous if it is uncountable, but is not the disjoint union of two uncountable sets, and every infinite subset has a countably infinite subset. We use the Fraenkel–Mostowski method to give many examples showing the diverse structures which can arise as quasi-amorphous sets, for instance carrying a projective geometry, or a linear ordering, or both; reconstruction results in the style of [1] are harder to come by in this case. Received: 8 April 1999 / Published online: 3 October 2001     

The Finite Dimensional Approximation Property and the AR-Property in Needle Point Spaces

We introduce the notion of the finite dimensional approximationproperty (the FDAP) and prove that if a subset X of a linearmetric space has the FDAP, then every non-empty convex subsetof X is an AR. As an application we show that every needle point space X containsa dense linear subspace E with the following properties: (i) E contains a non-empty compact convex set with no extremepoints; (ii) all non-empty convex subsets of E are AR.     

Relative local equivalence of Engel structures

The conditions to determine germs of Engel structures relative to arbitrary subsets are studied. We show that germs of Engel structures at a point relative to an arbitrary subset are determined by the algebraic restrictions of the Engel structures themselves to the subset, and the projected algebraic restrictions of the derived even-contact structures to the subset. When the subset is a smooth submanifold, algebraic restriction is equivalent to geometric restriction. Even when the subset is a smooth submanifold, we need a new stricter notion, projected algebraic restriction.