The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

On the Weight of Nowhere Dense Subsets in Compact Spaces

We study a new cardinal-valued invariant ndw(X) (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points.     

LEFT T-IDEMPOTENCE AND LEFT T-STABILITY OF SEMIGROUP RINGS

A ring R is called left T-idempotent (left T-stable) if for any sequence r ₁,r ₂,… in R there exists a positive integer n such that r ₁ r ₂… r_n is an idempotent (r ₁ r ₂… r_n = r ₁ r ₂… r _n+1 = …). In this paper we characterize left T-idempotent rings and left T-stable rings and describe left T-idempotent semigroup rings and left T-stable semigroup rings.     

Estimating the order of mean-square derivatives with quadratic variations

We consider a real Gaussian process X with unknown smoothness r ₀ where r ₀ is a none negative integer and the mean-square derivative X^(r₀){X^{(r_{\rm 0})}} is supposed to be locally stationary. From n + 1 equidistant observations, we propose and study an estimator of r ₀ based on results for quadratic variations of the underlying process. We derive an exponential inequality for IP ([^(r)]_n 1 r₀){{\rm I}\kern-2.1pt{\rm P} (\widehat{r}_n \neq {r_{\rm 0}})} implying the strong consistency of [^(r)]_n{\widehat{r}_n} . An application to interpolation of processes of unknown regularity is also provided.     

Tearing a strip off the plane

In this article we examine some homomorphic properties of certain subgraphs of the unit-distance graph. We define G_r to be the subgraph of the unit-distance graph induced by the subset (−∞, ∞) × [0, r] of the plane. The bulk of the article is devoted to examining the graphs G_r, when r is the minimum width such that G_r contains an odd cycle of given length. We determine for each odd n the minimum width r_n such that contains an n-cycle C_n, and characterize the embeddings of C_n in $G_{r_{n}}$. We then show that is homomorphically equivalent to C_n when n ≡ 3 (mod 4), but is a core when n ≡ 1 (mod 4). We begin by showing that G_r is homomorphically compact for each r ≥ 0, as defined in [1]. We conclude with some other interesting results and open problems related to the graphs G_r. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 17–33, 1998     

Remote points in large products

A point p ∈ βX\X is a remote point of X if p? cl_βXD for any nowhere dense D ? X. Van Douwen, and independently Chae and Smith, have shown that each non-pseudocompact space of countable π-weight has a remote point. Van Mill showed that many spaces of π-weight ω₁, such as ω×2^ω₁ also have remote points.We show that arbitrarily large products of spaces with countable π-weight which are not pseudocompact have remote points. In particular, ω×2^? for any infinite cardinal ?.     

Disjoint cycles in star-free graphs

A graph is claw-free if it does not contain K_1.3 as an induced subgraph. It is K_1.r-free if it does not contain K_1.r as an induced subgraph. We show that if a graph is K_1.r-free (r ≥ 4), only p + 2r − 1 edges are needed to insure that G has two disjoint cycles. As an easy consequence we get a well-known result of Pósa's. © 1996 John Wiley & Sons, Inc.     

Wiener��s Lemma for Infinite Matrices II

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of B(l^q_w){\mathcal{B}}(\ell^{q}_{w}), the algebra of all bounded linear operators on the weighted sequence space l^q_w\ell^{q}_{w}, for any 1≤q<∞ and any discrete Muckenhoupt A _q -weight w.     

The Length of a Shortest Closed Geodesic on a Two-Dimensional Sphere and Coverings by Metric Balls

In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold M diffeomorphic to the standard two-dimensional sphere. The first result is that the length of a shortest closed geodesic l(M) is bounded from above by 4r , where r is the radius of M . (In particular that means that l(M) is bounded from above by 2d, when M can be covered by a ball of radius d/2, where d is the diameter of M.) The second result is that l(M) is bounded from above by 2( max{r₁,r₂}+r₁+r₂), when M can be covered by two closed metric balls of radii r₁,r₂ respectively. For example, if r₁ = r₂= d/2 , thenl(M) 3d. The third result is that l(M) 2(max{r₁,r₂r₃}+r₁+r₂+r₃), when M can be covered by three closed metric balls of radii r₁,r₂,r₃. Finally, we present an estimate for l(M) in terms of radii of k metric balls covering M, where k 3, when these balls have a special configuration.     

Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets D=(r_1D)∪(r_2D (1 r_1-r_2-r_3)/2)∪(r_3D 1 r_3) and E=(r_1E)∪(r_2E 1-r_2- r_3)∪(r_3E 1-r_3),and proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r_1~m=r_3~n.     

Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture

Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K _r,r (for odd r) and K _r+1. If true, this would be a strengthening of the Hajnal-Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r ≥ 6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains K _r,r and V(G) partitions into subsets V ₀, …, V _t such that G[V ₀] = K _r,r and for each 1 ≤ i ≤ t, G[V _i ] = K _r . We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen-Lih-Wu Conjecture by induction.     

On numerical experiments with symmetric semigroups generated by three elements and their generalization

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r₁²,r₁r₂+r₁²k,r₃-r₁²k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r ₁,r ₂,r ₃∈ℤ⁺, r ₁≥2 and gcd(r ₁,r ₂)=gcd(r ₁,r ₃)=1, and calculate their universal Frobenius number Φ(r ₁,r ₂,r ₃) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r₁²,r₃-r₁²k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation.     

New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants

Let G be a finite abelian group (written additively) of rank r with invariants n ₁, n ₂, . . . , n _r , where n _r is the exponent of G. In this paper, we prove an upper bound for the Davenport constant D(G) of G as follows; D(G) ≤ n _r + n _r-1 + (c(3) − 1)n _r-2 + (c(4) − 1) n _r-3 + · · · + (c(r) − 1)n ₁ + 1, where c(i) is the Alon–Dubiner constant, which depends only on the rank of the group \mathbb Z_nrⁱ{{\mathbb Z}_{n_r}^i}. Also, we shall give an application of Davenport’s constant to smooth numbers related to the Quadratic sieve.     

Regular factors in K1,n free graphs

A graph is said to be K_1,n-free, if it contains no K_1,n as an induced subgraph. We prove that for n ? 3 and r ? n ?1, if G is a K_1,n-free graph with minimum degree at least (n²/4(n ?1))r + (3n ?6)/2 + (n ?1)/4r, then G has an r-factor (in the case where r is even, the condition r ? n ?1 can be dropped).     

On the geometric realization of Albanese's inequality

If an algebraic curve C (irreducible and reduced) moving in a family in projective n-space specializes into a curve C ₀, having associated cycle Z=m ₁ B ₁+···+m _r B _r , then the geometric genera g, g ₁,....,g _r of C, B ₁,...,B _r respectively and the coefficients m ₁,...,m _r must satisfy a certain inequality (found by Albanese). The realization (or existence) problem asks whether an inequality of this type actually arises from an algebraic family of curves. In this paper some results are obtained concerning the strong version of the problem, where one specifies the cycle Z; with these an affirmative solution of the weak version (where the components B _i are not specified) is obtained.     

A unique arithmetic labeling of hexagonal lattices

Consider an r-layer hexagon consisting of 3r² + 3r + 1 hexagonal cells. Can one label the cells by the set N_r = {0,1,…, 3r² + 3r} such that each line of adjacent cells are labeled by numbers forming an arithmetic progression modulo (3r² + 3r + 1) (in proper ordering)? We show that for each r there exists such a labeling unique up to equivalence. We also study some other related issues. © 1995 John Wiley & Sons, Inc.     

Indecomposable r-graphs and some other counterexamples

An r-graph is any graph that can be obtained as a conic combination of its own 1-factors. An r-graph G(V, E) is said to be indecomposable when its edge set E cannot be partitioned as E = E₁ ∪ E₂ so that G_i(V, E_i) is an r_i-graph for i = 1, 2 and, for some r₁, r₂. We give an indecomposable r-graph for every integer r ≥ 4. This answers a question raised in [Seymour, Proc London Math Soc 38 (1979, 423–460], and has interesting consequences for the Schrijver System of the T-cut polyhedron to be given in [Rizzi, 1997, to appear]. A graph in which every two 1-factors intersect is said to be poorly matchable. Every poorly matchable r-graph is indecomposable. We show that for every r ≥ 4 that “being indecomposable” does not imply “being poorly matchable.” Next we give a poorly matchable r-graph for every r ≥ 4. The article provides counterexamples to some conjectures of Seymour. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 1–15, 1999     

Weights of the l-adic cohomology of a proper toric variety

We describe the weights of the l-adic cohomology groups of a toric variety over a finite field k in terms of the Ishida complexes of Z-modules. As a consequence, we conclude that, for an r-dimensional proper toric variety X, the m-th cohomology group H^m (X?_k[kbar],Q_l) is of pure weight if m = 0,1,2,3,2r - 3,2r - 2,2r - 1,2r.Furthermore, we show that, for any m such that 3 < m < 2r - 3,there exists an r-dimensional proper toric variety whose m-th cohomology group H^m (X?_k[kbar],Q _l ) is not pure.     

A criterion for regular sequences

LetR be a commutative noetherian ring and ƒ₁, …, ƒ_r ∃ R. In this article we give (cf. the Theorem in §2) a criterion for ƒ₁, …, ƒ_r to be regular sequence for a finitely generated module overR which strengthens and generalises a result in [2]. As an immediate consequence we deduce that if V(g ₁, …,g _r ) ⊆ V(ƒ₁, …, ƒ_r) in SpecR and if ƒ₁, …, ƒ_r is a regular sequence inR, theng ₁, …,g _r is also a regular sequence inR.     

Permutations with roots

We prove that the probability p₂(n) that a random permutation of length n has a square root is monotonically nonincreasing in n. More generally, we prove that the probability p_r(n) that a random permutation of length n has an rth root, r prime, is monotonically nonincreasing in n. We also show for all r≥2 that p_r(n)→0 as n→∞. While doing this, we combinatorially prove that p_r(n)=p_r(n+1) for r prime and for all n not congruent to −1 mod r, and we construct several bijections for sets of permutations defined by modular class restrictions on the cycle lengths. We also include a simple probabilistic proof that, for r≥2, p_r(n)→0 as n→∞. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 157–167, 2000