Model-theory of vector-spaces over unspecified fields

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed fields is the model-completion of the theory of vector spaces.      

Semilocal rings with <Emphasis Type="Italic">n</Emphasis>-Engel multiplicative group

An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R* of a semilocal ring R generated by R* satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.Received: 21 January 2003     

A Note on the Multiplicative Semigroups Z n Whose Bi-ideals Are Quasi-ideals

We show that for any positive integer n, the multiplicative semigroup Z_n has the property that the set of bi-ideals and the set of quasi-ideals coincide if and only if either n = 4 or n is square-free.AMS Subject Classification (1991): 20M10     

Sensitivity and regionally proximal relation in minimal systems

A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n 1)-sensitive(n≥2)is determined.     

Representations of graphs modulo n

A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erd?s and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.     

On some subclasses of well-covered graphs

A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well-covered graphs are classified by the W_n property: For a positive integer n, a graph G belongs to class W_n if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of W_n graphs containing arbitrarily large independent sets. A characterization of W_n graphs in terms of well-covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in W_n graphs.     

Power-substitution,exchange rings and unit II-regularity

In this paper,we investigate power-substitution over exchange rings.We show that an exchange ring R satisfies power-substitution if and only if for any regular x ∈ R, there exists a positive integer n such that xI _n is unit πregular in M _n(R).     

Positive definite <Emphasis Type="Italic">n</Emphasis>-regular quadratic forms

A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases.     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Relative coherent modules and semihereditary modules

Given a positive integer n, a left R-module M is called n-coherent (resp. n-semihereditary) if every n-generated submodule of M is finitely presented (resp. projective). We investigate the properties of n-coherent modules and n-semihereditary modules. Various results are developed, many extending known results.     

Comparison of Some Purities,Flatnesses and Injectivities

In this article, we compare (n, m)-purities for different pairs of positive integers (n, m). When R is a commutative ring, these purities are not equivalent if R does not satisfy the following property: there exists a positive integer p such that, for each maximal ideal P, every finitely generated ideal of R _P is p-generated. When this property holds, then the (n, m)-purity and the (n, m′)-purity are equivalent if m and m′ are integers ≥np. These results are obtained by a generalization of Warfield's methods. There are also some interesting results when R is a semiperfect strongly π-regular ring. We also compare (n, m)-flatnesses and (n, m)-injectivities for different pairs of positive integers (n, m). In particular, if R is right perfect and right self (?₀, 1)-injective, then each (1, 1)-flat right R-module is projective. In several cases, for each positive integer p, all (n, p)-flatnesses are equivalent. But there are some examples where the (1, p)-flatness is not equivalent to the (1, p + 1)-flatness.     

Quasi-compactness and uniform ergodicity of positive operators

The following converse to the Yosida-Kakutani theorem is proved: IfT is a positive operator on a Banach lattice with ‖T ⁿ‖/n → 0, thenT is quasi-compact if (and only if) the averages of its iterates converge uniformly to a finite-dimensional projection.     

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x ₁…x _n+1 ∈ I for x ₁,…, x _n+1 ∈ R (resp., I ₁…I _n+1 ? I for ideals I ₁,…, I _n+1 of R), then there are n of the x _i 's (resp., n of the I _i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.     

Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n ≥ 1, let G_n denote the complete graph of order n, and for all even integers n ≥ 2 let G_n denote the complete graph of order n with the edges of a 1‐factor removed. It is shown that for all non‐negative integers h and t and all positive integers n, G_n can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G_n. © 2004 Wiley Periodicals, Inc.     

Scalar curvature,non-abelian group actions,and the degree of symmetry of exotic spheres

It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ∑ ⁿ is an exoticn-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ∑ ⁿ are tori of dimension ≤[(n+1)/2]. Research partially supported by the Sloan Foundation and NSF Grant GP-34785X.     

An LMI description for the cone of Lorentz-positive maps II

Let L _n be the n-dimensional second-order cone. A linear map from ? ^m to ? ⁿ is called positive if the image of L _m under this map is contained in L _n . For any pair (n,?m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n???1)(m???1) that describes this cone.     

Inertially arbitrary patterns

An n×n sign pattern matrix A is an inertially arbitrary pattern (IAP) if each non-negative triple (r s t) with r+s+t=n is the inertia of a matrix with sign pattern A. This paper considers the n×n(n2) skew-symmetric sign pattern S_n with each upper off-diagonal entry positive, the (1,1) entry negative, the (n n) entry positive, and every other diagonal entry zero. We prove that S_n is an IAP.     

On the n-minimal subgroups

Let G be a finite group and n be a positive integer. An n-minimal subgroup H of G is called to be exactly n-minimal if no proper subgroup of H is n-minimal. In this paper, we study the solvability of G under the assumption that all exactly n-minimal subgroups of G are S-permutable.     

Building a completely positive factorization

A symmetric matrix of order n is called completely positive if it has a symmetric factorization by means of a rectangular matrix with n columns and no negative entries (a so-called cp factorization), i.e., if it can be interpreted as a Gram matrix of n directions in the positive orthant of another Euclidean space of possibly different dimension. Finding this factor therefore amounts to angle packing and finding an appropriate embedding dimension. Neither the embedding dimension nor the directions may be unique, and so many cp factorizations of the same given matrix may coexist. Using a bordering approach, and building upon an already known cp factorization of a principal block, we establish sufficient conditions under which we can extend this cp factorization to the full matrix. Simulations show that the approach is promising also in higher dimensions.

     

Closed polylines inscribed in a closed space curve

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.