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.     

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.     

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 .     

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.     

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.     

Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

For a given convex body K in \Bbb R³{\Bbb R}^3 with C ² boundary, let P ^c _n be the circumscribed polytope of minimal volume with at most n edges, and let P ⁱ _n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P ^c _n and P ⁱ _n are asymptotically regular triangles and squares, respectively, in a suitable sense.     

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.     

图和线图的谱性质

Let G be a simple connected graph with n vertices and m edges,Lo be the line graph of G and λ1(LG)≥λ2 (LG)≥...≥λm(LG) be the eigenvalues of the graph LG,.. In this paper, the range of eigenvalues of a line graph is considered. Some sharp upper bounds and sharp lower bounds of the eigenvalues of Lc. are obtained. In oarticular,it is oroved that-2cos(π/n)≤λn-1(LG)≤n-4 and λn(LG)=-2 if and only if G is bipartite.     

On the structure of cluster point sets of iteratively generated sequences

LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x _i} be a sequence generated iteratively bya(·) fromx ₀ inX, i.e.,x _i+1 =a(x _i),i=0, 1, 2, ...; and letQ(x ₀) be the cluster point set of {x _i}. In this paper, we prove that, if there exists a pointz inQ(x ₀) such that (i)z is isolated with respect toQ(x ₀), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x ₀) containsp points, (v) the sequence {x _i} is contained in a sequentially compact set, and (vi) every point inQ(x ₀) possesses properties (i) and (ii). The application of the preceding results to the caseF=E ⁿ leads to the following: (vii) ifQ(x ₀) contains one and only one point, then {x _i} converges; (viii) ifQ(x ₀) contains a finite number of points, then {x _i} is bounded; and (ix) ifQ(x ₀) containsp points, then every point inQ(x ₀) is a periodic point ofa(·) of periodp.     

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.     

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.     

Study of families of monotone continuous functions on Tychonoff spaces

The main object of study is the space of all monotone continuous functions CM(X) on a connected Tychonoff space X endowed with the topology of pointwise (CM _p (X)) or uniform (CM(X)) convergence. Technical questions concerning restriction and extension of monotone functions are considered in Sec. 2. Conditions for CM(X) to separate the points of X and for CM(X) to contain only constant functions are found in Sec. 3. In Sec. 4, the linear structure of CM(X) is studied and all linear subspaces of CM(X) for a certain class of spaces X are described. In Sec. 5, conditions under which CM(X) is closed and nowhere dense in C _p (X) and C(X) are determined. The metrizability of CM _p (X) is considered in Sec. 6; necessary and sufficient metrizability conditions for various classes of spaces X are obtained. In Sec. 7, criteria for σ-compactness and the Hurewicz property in the class of spaces CM _p (X) are given. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

On the upper bounds of the minimum number of rows of disjunct matrices

A 0-1 matrix is d-disjunct if no column is covered by the union of any d other columns. A 0-1 matrix is (d; z)-disjunct if for any column C and any d other columns, there exist at least z rows such that each of them has value 1 at column C and value 0 at all the other d columns. Let t(d, n) and t(d, n; z) denote the minimum number of rows required by a d-disjunct matrix and a (d; z)-disjunct matrix with n columns, respectively. We give a very short proof for the currently best upper bound on t(d, n). We also generalize our method to obtain a new upper bound on t(d, n; z). The work of Y. Cheng and G. Lin is supported by Natural Science and Engineering Research Council (NSERC) of Canada, and the Alberta Ingenuity Center for Machine Learning (AICML) at the University of Alberta. The work of D.-Z. Du is partially supported by National Science Foundation under grant No.CCF0621829.     

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.     

Asymptotic Primes of Delta Closures of Ideals

Abstract

For an ideal H in a Noetherian ring R let H? = ∪{H ⁱ⁺¹ : _R H ⁱ | i ≥ 0} and for a multiplicatively closed set Δ of nonzero ideals of R let H _Δ = ∪{HK: _R K | K ? Δ}. It is shown that four standard results concerning the associated prime ideals of the integral closure (bR)_a of a regular principal ideal bR do not hold for certain Δ closures (bR)_Δ of bR. To do this it is first shown that if I is an ideal in R such that height (I) ≥ 1, then each radical ideal J of R containing I is of the form J = K? :_R cR for some ideal K closely related to I, and if I _a :_R J ? U = ∪{I?R _P ∩ R | P is a minimal prime divisor of J} (where I _a is the integral closure of I), then J = I _Δ :_R CR and I ? I _Δ ? I _a).     

Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C* and D* may overlap. When T: V W is linear and K V and D W are cones, these results will be applied to C = T(K) and D, giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D. The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T(X) = AX + X* A yields new and known results about the existence of block diagonal X's satisfying the Lyapunov condition: T(X) is an interior point of D. For the same V, W and D, T(X) = X – B* XB will be studied for certain cones K of entry-wise nonnegative X's.     

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)).