Two new bounds on the size of binary codes with a minimum distance of three

We prove two new upper bounds on the size of binary codes with a minimum distance of three, namelyA(10, 3)76 andA(11, 3)152.     

Presentations for Certain Chevalley Groups

Let G = P SL_n(K), n 3, K a division ring or D_n(K), n 4 or E_n(K), 6 n 8, K a field. Then two types of presentations for G are given. In the first, G is generated by SL₂(K)'s which satisfies relations according to the Dynkin diagram of G. In the second, G is generated by a set {A_i | i I } of Abelian groups, which satisfy relations similar to the root subgroups of G.     

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations E_m,n(f)p with undetermined n nodes in the metric of the space L_p(0, 1) (1p) is studied in a class of functionsf(x) for which f ^m+1 (x)L_q(0, 1)1(1qt8) or var {f(m) (x); 0, 1}1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n , and asymptotic formulas are obtained for p= and 1q in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 31–42, January, 1970.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

A generalization of the Bernoulli model occurring in order statistics. I.

Let X=(x₁, ..., x_n) and Y=(y₁, ..., y_m) be independent samples from populations G_x and G_y, x⁽¹⁾ ,... x⁽ⁿ⁾ be ordered statistics constructed from the sample X. A model of trials associated with the occurrence of dependent events A_k={y_k (x⁽ⁱ⁾}, x^(j), i < j, k=1, 2, ..., m, where x⁽ⁱ⁾, x^(j) are order statistics, is considered. This model is a generalization of the Bernoulli model. Distribution of frequencies of occurrences of events A_k and the limit theorems which describe asymptotic properties of these frequencies are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 518–528, April, 1990.     

On the entropic regularization method for solving min-max problems with applications

Consider a min-max problem in the form of min _xX max_1im {f _i (x)}. It is well-known that the non-differentiability of the max functionF(x) max_1im {f _i (x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximationF _p(x) that uniformly converges toF(x) overX with a difference bounded by ln(m)/p, forp > 0. In this way, withp being sufficiently large, minimizing the smooth functionF _p(x) overX provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems.This research work was supported in part by the 1995 NCSC-Cray Research Grant and the National Textile Center Research Grant S95-2.     

Sur un problème de Rényi

Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev _q (x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv _q (x); this improves on previous results.

     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

On Large Values of the Divisor Function

Let d(n) denote the divisor function, and let D(X) denote the maximal value of d(n) for n X. For 0 < z 1, both lower and upper bounds are given for the number of integersn with n X, zD(X) d(n).     

Some functionals over a compact Minkovskii space

In this paper we obtain estimates which are order-exact for the projection and Macphail constants of an arbitrary n-dimensional Banach space: 1(X)n, 1/n₁(X)1/n.Translated from Matematicheskii Zametki, Vol. 10, No. 4, pp. 453–457, 1971.     

Bounds for binary multiple covering codes

We study codes that are multiple coverings of the Hamming space and discuss lower and upper bounds onK(n, r, ), the minimum cardinality of a binary code of lengthn such that the Hamming spheres of radiusr centered at the codewords cover at least times. We also study the similar problem of multiple coverings containing repeated words. A table of bounds forn16,r4, 4 is given.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Arcs fixed byA 5 andA 6

The complete list of thek-arcsK inPG(n, q) fixed by a projective group isomorphic toA ₅ orA ₆, which acts primitively on the points ofK, is presented. This leads to new classes of 10-arcs inPG(n, q), 3 n 5. Our results also show that the non-classical 10-arc inPG(4, 9), discovered by D.G. Glynn [3], belongs to an infinite class of 10-arcs inPG(4, 3^h),h 2, fixed by a projective group isomorphic toA ₆.The first author wishes to thank the Belgian National Fund for Scientific Research for financial supportSenior research assistant of the Belgian National Fund for Scientific ResearchDedicated to Professor M. Scafati Tallini on the occasion of her sixtyfifth birthday     

Trees in tournaments

Letf(n) be the smallest integer such that every tournament of orderf(n) contains every oriented tree of ordern. Sumner has just conjectures thatf(n)=2n–2, and F. K. Chung has shown thatf(n)(1+o(1))nlog₂ n. Here we show thatf(n)12n andf(n)(4+o(1))n.     

Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

Improved Bounds for Ternary Linear Codes of Dimension 8 Using Tabu Search

In this paper, new codes of dimension 8 are presented which give improved bounds on the maximum possible minimum distance of ternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. Twenty three codes are given which improve or establish the bounds for ternary codes. In addition, a table of upper and lower bounds for d ₃(n, 8) is presented for n 200.