Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc.     

The Distribution of Reciprocal Pairs Modulo Polynomials over a Finite Field

Given a finite field F_q of order q, a fixed polynomial g in –F_q[X] of positive degree, and two elements u and v in the ring of polynomials in R = F_q [X]/gF_q[X], the question arises: How many pairs (a, 6) are there in R × R so that ab ? 1 mod g and so that a is close to u while b is close to v ? The answer is, about as many as one would expect. That is, there are no favored regions in R × R where inverse pairs cluster. The error term is quite sharp in most cases, being comparable to what would happen with random distribution of pairs. The proof uses Kloosterman sums and counting arguments. The exceptional cases involve fields of characteristic 2 and composite values of g. Even then the error term obtained is nontrivial. There is no computational evidence that inverses are in fact less evenly distributed in this case, however.     

Gaussian Property of the Rings R(X) and R〈X〉

The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. A commutative ring R is said to be Gaussian if c(fg) = c(f)c(g) for every polynomials f and g in R[X]. A number of authors have formulated necessary and sufficient conditions for R(X) (respectively, R?X?) to be semihereditary, have weak global dimension at most one, be arithmetical, or be Prüfer. An open question raised by Glaz is to formulate necessary and sufficient conditions that R(X) (respectively, R?X?) have the Gaussian property. We give a necessary and sufficient condition for the rings R(X) and R?X? in terms of the ring R in case the square of the nilradical of R is zero.     

Some results on the achromatic number

Let G be a simple graph. The achromatic number ψ(G) is the largest number of colors possible in a proper vertex coloring of G in which each pair of colors is adjacent somewhere in G. For any positive integer m, let q(m) be the largest integer k such that ≤ m. We show that the problem of determining the achromatic number of a tree is NP-hard. We further prove that almost all trees T satisfy ψ (T) = q(m), where m is the number of edges in T. Lastly, for fixed d and ϵ > 0, we show that there is an integer N₀ = N₀(d, ϵ) such that if G is a graph with maximum degree at most d, and m ≥ N₀ edges, then (1 - ϵ)q(m) ≤ ψ (G) ≤ q(m). © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 129–136, 1997     

Small cliques in random graphs

For d ≥ 1, a d-clique in a graph G is a complete d-vertex subgraph not contained in any larger complete subgraph of G. We investigate the limit distribution of the number of d-cliques in the binomial random graph G(n, p), p = p(n), n→∞.     

On two classes of finite supersoluble groups

Let ? be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ?-S-semipermutable if H permutes with every Sylow p-subgroup of G in ? for all p?π(H); H is said to be ?-S-seminormal if it is normalized by every Sylow p-subgroup of G in ? for all p?π(H). The main aim of this paper is to characterize the ?-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ? are ?-S-semipermutable in G and the ?-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ? are ?-S-seminormal in G.     

Associativity of the <Emphasis Type="Italic">∇</Emphasis>-operation on bands in rings

Given a multiplicative band of idempotents S in a ring R, for all e,f∈S the ∇-product e ∇ f=e+f+fe−efe−fef is an idempotent that lies roughly above e and f in R just as ef and fe lie roughly below e and f. In this paper we study ∇-bands in rings, that is, bands in rings that are closed under ∇, giving various criteria for ∇ to be associative, thus making the band a skew lattice. We also consider when a given band S in R generates a ∇-band.     

Algorithms for ham-sandwich cuts

Given disjoint setsP ₁,P ₂, ...,P _d inR ^d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR ^d−1 . With the current estimates, we get complexity close toO(n ^3/2 ) ford=3, roughlyO(n ^8/3 ) ford=4, andO(n ^d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR ³ when the three sets are suitably separated. A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Algebraic group actions on noncommutative spectra

Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the set Spec R of all prime ideals of R, viewed as a topological space with the Jacobson–Zariski topology, and on the subspace Rat R ⊆ Spec R consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid is equal to the base field. Our results generalize the work of Mœglin and Rentschler and of Vonessen to arbitrary associative algebras while also simplifying some of the earlier proofs. The map P ↦ ⋂ _{g ∈ G} g.P gives a surjection from Spec R onto the set G-Spec R of all G-prime ideals of R. The fibers of this map yield the so-called G-stratification of Spec R which has played a central role in the recent investigation of algebraic quantum groups, in particular, in the work of Goodearl and Letzter. We describe the G-strata of Spec R in terms of certain commutative spectra. Furthermore, we show that if a rational ideal P is locally closed in Spec R then the orbit G.P is locally closed in Rat R. This generalizes a standard result on G-varieties. Finally, we discuss the situation where G-Spec R is a finite set. Research supported in part by NSA Grant H98230-07-1-0008.     

On the Cyclic Homology of an Algebra Associated with a Cyclic Quiver and a Monic Polynomial

In this article, we compute the Hochschild homology group of A = KΓ/(f(X ^s )), where KΓ is the path algebra of the cyclic quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K, and X is the sum of all arrows in KΓ. Moreover, we compute the cyclic homology group of A in the case f(x) = (x ? a) ^m , where a ∈ K, so that we can determine the cyclic homology of A in general when K is an algebraically closed field.     

Approximations from Subspaces of C0(X)

We show among other things that if B is a linear space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, for which there is a continuous function h defined in a neighbourhood of 0 in the real line which is non-affine in every neighbourhood of 0 and satisfies |h(t)|k |t| for all t, such that hb is in B whenever b is in B and the composite function is defined, then every function in C₀(X) which can be approximated on every pair of points in X by functions in B can be approximated uniformly by functions in B.     

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.     

A Sufficient Condition for the Existence of Large Empty Convex Polygons

   Abstract. Let P be a set of points in general position in the plane. We say that P is k -convex if no triangle determined by P contains more than k points of P in the interior. We say that a subset A of P in convex position forms an empty polygon (in P ) if no point of P \ A lies in the convex hull of A . We show that for any k,n there is an N=N(k,n) such that any k -convex set of at least N points in general position in the plane contains an empty n -gon. We also prove an analogous statement in R ^d for each odd d≥ 3 .     

Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations

The main result of this paper (which is completely new, apart from our previous and less general result proved in reference [9]) states that the nonlinear system of equations (1.11) (or, equivalently, (1.10)) that describes the motion of an inviscid, compressible (barotropic) fluid in a bounded domain Ω, gives rise to a strongly well-posed problem (in the Hadamard classical sense) in spaces H^k(Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if (a_n, ?_n) → (a, ?) in H^k × H^k and if f_n → f in ??²(0, T;H^k), then (v_n, g_n) → (v, g) in ?? (0, T; H^k × H^k). The method followed here (see also [8]) also applies to the non-barotropic case p = p(ρ, s) (see [10]) and to other nonlinear problems. These results are based upon an improvement of the structural-stability theorem for linear hyperbolic equations. See Theorem 1.2 below. Added in proof: The reader is referred to [29], Part I, for a concise explanation of some fundamental points in the method followed here. © 1993 John Wiley & Sons, Inc.     

Convergence of Weighted Sums and Laws of Large Numbers in D([0,1]; E)

Convergence properties of weighted sums of functions in D([0, 1]; E) (E a Banach space) are investigated. We show that convergence in the Skorokhod J₁-topology of a sequence (x_n) in D([0, 1]; E) does not imply convergence of a sequence ( _n) of averages. Convergence in the J₁-topology of a sequence ( _n) of averages is shown, under the growth condition x_n ∞ = o(n), to be equivalent to the convergence of ( _n) in the uniform topology. Convergence of a sequence (x_n,) is shown to imply convergence of the sequence ( _n) of averages in the M₁ and M₂ topologies. The strong law of large numbers in D[0, 1] is considered and an example is constructed to show that different definitions of the strong law of large numbers are nonequivalent.     

On the desking completions,theta completions and theta pairs for maximal subgroups *

Given a maximal subgroup M of a finite group G,a θ completion of M in G is any subgroup C such that C is not contained in M while M_G , the core of M in G, is contained in C and C/M_G has no propor normal subgroup of G/M_G . By using this concept we can reveal the relationship between the concepts of completions and θ-pairs introduced respectively by Deskins, Mukherjee and Bhattacharya. The concept of maximal θ-completions offers a convenience for us to study the Deskins completions in inductions. We obtain in this paper several results which imply a group to be solvable, supersolvable and nilpotent.     

Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.X _Y =X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forX _Y under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenX _Y is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofX _Y . Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system. This work is supported by the National Natural Science Foundation of China.     

On the behavior of the product of independent random variables

For two independent non-negative random variables X and Y, we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y. We also show that if the distribution of X is in class L(γ) (γ>0) and continuous, then the product XY is in L if and only if Y is unbounded.