Rings of quotients of f-rings by gabriel filters of ideals

The article examines the role of Gabriel filters of ideals in the ontext of semiprime f-rings. It is shown that for every 2-convex semiprime f-ring Aand every multiplicative filter B of dense ideals the ring of quotients of A by B, namely the direct limit of the Hom _A (I, A) over all I∈ B, is an l-subring of QA, the maximum ring of quotients. Relative to the category of all commutative rings with identity, it is shown that for every 2-convex semiprime f-ring A qA, the classical ring of quotients, is the largest flat epimorphic extension of A. If Ais also a Prüfer ring then it follows that every extension of Ain qA is of the form S ^-1A for a suitable multiplicative subset S. The paper also examines when a Utumi ring of quotients of a semiprime f-ring is obtained from a Gabriel filter. For a ring of continuous functions C(X), with Xcompact, this is so for each C(U) and C ^*(U), when Uis dense open, but not for an arbitrary direct limit of C(U),taken over a filter base of dense open sets. In conclusion, it is shown that, for a complemented semiprime f-ring A, the ideals of Awhich are torsion radicals with respect to some hereditary torsion theory are precisely the intersections of minimal prime ideals of A.     

On the number of regular vertices of the union of jordan regions

Let \C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in \C cross exactly twice, then their intersection points are called regular vertices of the arrangement \A(\C) . Let R(\C) denote the set of regular vertices on the boundary of the union of \C . We present several bounds on |R(\C)| , depending on the type of the sets of \C . (i) If each set of \C is convex, then |R(\C)|=O(n ^1.5+\eps ) for any \eps>0 . (ii) If no further assumptions are made on the sets of \C , then we show that there is a positive integer r that depends only on s such that |R(\C)|=O(n ^2-1/r ) . (iii) If \C consists of two collections \C ₁ and \C ₂ where \C ₁ is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and \C ₂ is a collection of polygons with a total of n sides, then |R(\C)|=O(m ^2/3 n ^2/3 +m +n) , and this bound is tight in the worst case. Received December 4, 1998, and in revised form June 3, 2000. Online publication Feburary 1, 2001.     

Picard Groups of Rings of Coinvariants

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C). Presented by: Claus Ringel.     

On the number of subgraphs of prescribed type of graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For a graphH=〈V(H),E(H)〉 and forS ⊂V(H) defineN(S)={x ∈V(H):xy ∈E(H) for somey ∈S}. Define alsoδ(H)= max {|S| − |N(S)|:S ⊂V(H)},γ(H)=1/2(|V(H)|+δ(H)). For two graphsG, H letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also forl>0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We investigate the asymptotic behaviour ofN(l, H) for fixedH asl tends to infinity. The main results are:Theorem A. For every graph H there are positive constants c ₁, c₂ such that {fx116-1}. Theorem B. If δ(H)=0then {fx116-2},where |AutH|is the number of automorphisms of H. (It turns out thatδ(H)=0 iffH has a spanning subgraph which is a disjoint union of cycles and isolated edges.) This paper forms part of an M.Sc. Thesis written by the author under the supervision of Prof. M. A. Perles from the Hebrew University of Jerusalem.     

Plane Sections of Convex Bodies of Maximal Volume

Let K = {K ₀ ,... ,K _k } be a family of convex bodies in R ⁿ , 1≤ k≤ n-1 . We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k -dimensional plane A _k (subset, dbl equals) R ⁿ , called a common maximal k-transversal of K , such that, for each i∈ {0,... ,k} and each x∈ R ⁿ , where V _k is the k -dimensional Lebesgue measure in A _k and A _k +x . Given a family K = {K _i } _i=0 ^l of convex bodies in R ⁿ , l < k , the set C _k ( K ) of all common maximal k -transversals of K is not only nonempty but has to be ``large' both from the measure theoretic and the topological point of view. It is shown that C <sub>k</sub> ( K ) cannot be included in a ν -dimensional C <sup>1</sup> submanifold (or more generally in an ( H <sup>ν</sup> , ν) -rectifiable, H <sup>ν</sup> -measurable subset) of the affine Grassmannian AGr <sub>n,k</sub> of all affine k -dimensional planes of R <sup>n</sup> , of O(n+1) -invariant ν -dimensional (Hausdorff) measure less than some positive constant c <sub>n,k,l</sub> , where ν = (k-l)(n-k) . As usual, the ``affine' Grassmannian AGr _n,k is viewed as a subspace of the Grassmannian Gr _n+1,k+1 of all linear (k+1) -dimensional subspaces of R ⁿ⁺¹ . On the topological side we show that there exists a nonzero cohomology class θ∈ H ^n-k (G _n+1,k+1 ;Z ₂ ) such that the class θ ^l+1 is concentrated in an arbitrarily small neighborhood of C _k ( K ) . As an immediate consequence we deduce that the Lyusternik—Shnirel'man category of the space C _k ( K ) relative to Gr _n+1,k+1 is ≥ k-l . Finally, we show that there exists a link between these two results by showing that a cohomologically ``big' subspace of Gr _n+1,k+1 has to be large also in a measure theoretic sense. Received May 22, 1998, and in revised form March 27, 2000. Online publication September 22, 2000.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Mar enko and Pastur, is presented. Here, X(N × n), T(n × n), and A(N × N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N → c > 0 as N → ∞. Under additional assumptions on the eigenvalues of A and T, almost sure convergence of the empirical distribution function of the eigenvalues of A + XTX* is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.     

Quotient polytopes of cyclic polytopes part II: Stability of thef-vector and of thek-skeleton

This paper is the continuation of an earlier paper on quotient polytopesC(v, 2m)/F of cyclic polytopes and the associated quotient complexesC(V, 2m)/J. Here, we study mainly what changes in the faceJ do not affect thef-vector of the quotientC(V, 2m)/J. In the last section we examine the corresponding question fork-skeleta, i.e., what changes inJ do not affect the isomorphism type of skel _k C(V, 2m)/J.     

The structure of the 3-separations of 3-connected matroids

Tutte defined a k-separation of a matroid M to be a partition (A,B) of the ground set of M such that |A|,|B|k and r(A)+r(B)−r(M)<k. If, for all m<n, the matroid M has no m-separations, then M is n-connected. Earlier, Whitney showed that (A,B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M.     

Approximation of Projections of Random Vectors

Let X be a d-dimensional random vector and X _θ its projection onto the span of a set of orthonormal vectors {θ ₁,…,θ _k }. Conditions on the distribution of X are given such that if θ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from X _θ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of d, k, and the distribution of X, allowing consideration not just of fixed k but of k growing with d. The results are applied in the setting of projection pursuit, showing that most k-dimensional projections of n data points in ℝ ^d are close to Gaussian, when n and d are large and k=clog (d) for a small constant c.     

Regularity of weak solutions of semilinear parabolic systems of arbitrary order

Letu be a weak solution of the initial boundary value problem for the semilinear parabolic system of order 2m:u′(t)+Au(t)+f(t,.,u,..., ▽ ^m u)=0. Letf satisfy controllable growth conditions. Thenu is smooth. This result is proved by a kind of continuity method, where the timet is the parameter of continuity.     

On the chromatic spectrum of acyclic decompositions of graphs

If G is any graph, a G‐decomposition of a host graph H = (V, E) is a partition of the edge set of H into subgraphs of H which are isomorphic to G. The chromatic index of a G‐decomposition is the minimum number of colors required to color the parts of the decomposition so that two parts which share a node get different colors. The G‐spectrum of H is the set of all chromatic indices taken on by G‐decompositions of H. If both S and T are trees, then the S‐spectrum of T consists of a single value which can be computed in polynomial time. On the other hand, for any fixed tree S, not a single edge, there is a unicyclic host whose S‐spectrum has two values, and if the host is allowed to be arbitrary, the S‐spectrum can take on arbitrarily many values. Moreover, deciding if an integer k is in the S‐spectrum of a general bipartite graph is NP‐hard. We show that if G has c > 1 components, then there is a host H whose G‐spectrum contains both 3 and 2c + 1. If G is a forest, then there is a tree T whose G‐spectrum contains both 2 and 2c. Furthermore, we determine the complete spectra of both paths and cycles with respect to matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 83–104, 2007     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD(B _n ) of partial contracting transformations of a Boolean, the semigroup TD(B _n ) of full contracting transformations of a Boolean, and the inverse semigroup ISD(B _n ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD(B _n ) and TD(B _n ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD(B _n ), we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases.     

The group inverse of the combinations of two idempotent matrices

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈ ? and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ.     

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

Probabilities of high excursions of Gaussian fields

Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ(t) ≡ 0, D ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval T ì \mathbbR^m T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e^−Q  + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)).     

Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a tessellation if there is a discrete subgroup of G, such that acts properly discontinuously on G/H, and the double-coset space \G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples.It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, Kulkarni and Kobayashi constructed examples that are not obvious when G=SO(2, 2n)° or SU(2, 2n). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G=SO(2, 2n)°. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G=SU(2, 2n). This includes the construction of another family of examples.The main results are obtained from methods of Benoist and Kobayashi: we fix a Cartan decomposition G=K A ⁺ K, and study the intersection (K H K)A ⁺. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.     

The Combinatorics of a Class of Representation Functions

In the study of the irreducible representations of the unitary groupU(n), one encounters a class of polynomials defined onn²indeterminatesz_ij, 1i, jn, which may be arranged into ann×nmatrix arrayZ=(z_ij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrixD(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable “group multiplication property”D(XY)=D(X) D(Y) for arbitrary matricesXandY, even though these matrices may be singular. ForZ=UU(n), these matrices give irreducible unitary representations ofU(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted byL_{α, β}(Z), whereαandβare integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equivalent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n−jcoefficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties.