Intersection patterns of convex sets

LetK ₁,…K_n be convex sets inR ^d. For 0≦i denote byf _ithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{K _i∶i∈S}≠Ø. We prove:Theorem.If f _d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK ₁=…=K_r=R^d andK _r+1,…,K_n aren?r hyperplanes in general position inR ^d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Characterizing compact unions of two starshaped sets inR d

SetS inR ^d has propertyK ₂ if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c₁,c₂ (depending onF) such that each point ofF is clearly visible viaS from at least one c_i,i = 1,2. The following characterization theorem is established: Let , d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S _j } converging toS (relative to the Hausdorff metric) such that each setS _j satisfies propertyK ₂. For , the sufficiency of the condition above still holds, although the necessity fails.     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.     

The radius of convergence of Poincaré series of loop spaces

LetR _S (resp.R _A) be the radius of convergence of the Poincaré series of a loop space (S) (resp. of the Betti-Poincaré series of a noetherian connected graded commutative algebraA over a field of characteristic zero).IfS is a finite 1-connected CW-complex, the rational homotopy Lie algebra ofS is finite dimensional if and only ifR _S-1. OtherwiseR _S<1.There is an easily computable upper bound (usually less than 1) forR _S ifS is formal or coformal.On the other handR _A=+ if and only ifA is a polynomial algebra andR _A=1 if and only ifA is a complete intersection (Golod and Gulliksen conjecture). OtherwiseR _A<1 and the sequence dim Tor _p ^H grows exponentially withp.     

Identities of semigroup rings over semilattices of completely (0-) simple semigroups

LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.     

Smallest maximal snakes of translates of convex domains

LetK be a convex domain. A maximal snake of sizen is a set of non-overlapping translatesK ₁, ...,K _N ofK, whereK _i touchesK _j if and only if |i –j|=1 and no translate ofK can touchK ₁ orK _n without intersecting an additionalK _i ,i=1, ...,n. The size of the smallest maximal snake is proved to be 11 ifK is a parallelogram and to be 10 otherwise.     

Homology of free Lie powers and torsion in groups

LetG be a group that is given by a free presentationG=F/R, and let_γ4 R denote the fourth term of the lower central series of R. We show that ifG has no elements of order 2, then the torsion subgroup of the free central extensionF/[_γ4 R,F] can be identified with the homology groupR _γ6(G, ℤ/2ℤ). This is a consequence of our main result which refers to the homology ofG with coefficients in Lie powers of relation modules.     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

The ring of endomorphisms of a finite dimensional module

LetS denote the ring of endomorphisms of a finite dimensional moduleM _R. Necessary and sufficient conditions for a nil subring ofS to be nilpotent are given. We place conditions onM _R so that every nil subring ofS will be nilpotent.     

Asymptotic formulas for the hyperġeometric function2 F 1 of matrix argument,useful in multivariate analysis

Summary LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of for large n=n₁+n₂ is derived, when∑ ₁∑ ₂ ⁻¹ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ₂ ⁻¹ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].     

Connected components of arithmetic graphs

Let . LetG _m (R) be the graph whose vertices are the numbers 1, 2, ...,m and whose edges are all pairs {a, b} such thata+br (modm) for somerR. LetC _m (R) be the number of connected components ofG _m (R). Letd be the greatest common divisor ofm and the differencesr _j –r _j or allr _i ,r _j R. ThenC _m (R) is equal to (i) (d+1)/2 ifd is odd, (ii)d/2 ifd is even andr is odd for allrR, or (iii) (d/2)+1 ifd is even andr is even for allrR.This research was supported in part by the National Science Foundation under grant No. MCS78-07908.     

On some invariants in numerical semigroups and estimations of the order bound

Let S={s _i } _i∈??? be a numerical semigroup. For s _i ∈S, let ν(s _i ) denote the number of pairs (s _i ?s _j ,s _j )∈S ². When S is the Weierstrass semigroup of a family $\{\mathcal{C}_{i}\}_{i\in\mathbb{N}}Let S={s _i } _i∈ℕ⊆ℕ be a numerical semigroup. For s _i ∈S, let ν(s _i ) denote the number of pairs (s _i −s _j ,s _j )∈S ². When S is the Weierstrass semigroup of a family {C_i}_{i ? \mathbbN}\{\mathcal{C}_{i}\}_{i\in\mathbb{N}} of one-point algebraic-geometric codes, a good bound for the minimum distance of the code C_i\mathcal{C}_{i} is the Feng and Rao order bound d _ORD (C _i ). It is well-known that there exists an integer m such that d _ORD (C _i )=ν(s _i+1) for each i≥m. By way of some suitable parameters related to the semigroup S, we find upper bounds for m and we evaluate m exactly in many cases. Further we conjecture a lower bound for m and we prove it in several classes of semigroups.     

Fields for which the projective schur subgroup is the whole Brauer group

LetBr(K) denote the Brauer group of a fieldK andPS(K) the projective Schur subgroup. 1. LetK be a finitely generated infinite field. ThenPS(K)=Br(K) if and only ifK is a global field. 2.|LetK be a finitely generated infinite field, and letK((t)) denote the field of formal power series int overK. ThenPS(K((t)))=Br(K((t))) if and only ifK=ℚ.     

Projective and indecomposableS-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which contains_s S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP≅ ∐ _{jϕ I} Se _i ,e _i ϕE. In particular,P is a projective, indecomposable and unitary object if and only ifP ≅Se for somee ∈E. These generalize some results obtained by Knauer and Talwar. Research partially supported by a UGC (HK) (Grant No. 2160092).     

Projective and indecomposableS-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which contains_s S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP≅ ∐ _{jϕ I} Se _i ,e _i ϕE. In particular,P is a projective, indecomposable and unitary object if and only ifP ≅Se for somee ∈E. These generalize some results obtained by Knauer and Talwar.     

A theorem on arrangements of lines in the plane

LetA be an arrangement ofn lines in the plane. IfR ₁, …,R _r arer distinct regions ofA, andR _i is ap _i-gon (i=1, …,r) then we show that . Further we show that for allr this bound is the best possible ifn is sufficiently large. Financial support for this research was provided by the Carnegie Trust for the Universities of Scotland.     

On the evolution of islands

Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked cells. Let ξ _k denote the random number of islands afterk cells have been marked. We give explicit expressions for moments of products of ξ _k ’s and for moments of products of 1/ξ _k ’s. These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4.     

Coherent random allocations, and the Ewens-Pitman formula

Assume that there is a random number K of positive integer random variables S₁, …, S_K that are conditionally independent given K and all have identical distributions. A random integer partition N = S₁ + S₂ + … + S_K arises, and we denote by P_N the conditional distribution of this partition for a fixed value of N. We prove that the distributions {P_N} _N=1 ^∞ form a partition structure in the sense of Kingman if and only if they are governed by the Ewens-Pitman Formula. The latter generalizes the celebrated Ewens sampling formula, which has numerous applications in pure and applied mathematics. The distributions of the random variables K and S_j belong to a family of integer distributions with two real parameters, which we call quasi-binomial. Hence every Ewens-Pitman distribution arises as a result of a two-stage random procedure based on this simple class of integer distributions. Bibliography: 25 titles. This paper is an edited and actualized version of the unpublished PDMI preprint 21/1995. Further development of the ideas of this work can be found in [21, 25]. A number of detected misprints was fixed without notice, the bibliography was extended beyond the original 19 references, and a few comments were added as footnotes. (Comments by Alexander Gnedin.) __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 127–145.