{2, 3}‐perfect m‐cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only

An m‐cycle system (S,C) of order n is said to be {2,3}‐perfect provided each pair of vertices is connected by a path of length 2 in an m‐cycle of C and a path of length 3 in an m‐cycle of C. The class of {2,3}‐perfect m‐cycle systems is said to be equationally defined provided, there exists a variety of quasigroups V with the property that a finite quasigroup (Q, , \, /) belongs to V if and only if its multiplicative (Q, ) part can be constructed from a {2,3}‐perfect m‐cycle system using the 2‐construction (a a = a for all a ∈ Q and if a ≠ b, a b = c and b a = d if and only if the m‐cycle (…, d, x, a, b, y, c, …) ∈ C). The object of this paper is to show that the class of {2,3}‐perfect m‐cycle systems cannot be equationally defined for all m ≥ 10, m ≠ 11. This combined with previous results shows that {2, 3}‐perfect m‐cycle systems are equationally defined for m = 5, 7, 8, 9, and 11 only. © 2004 Wiley Periodicals, Inc.     

Pairwise intersections and forbidden configurations

Let f_m(a,b,c,d) denote the maximum size of a family of subsets of an m-element set for which there is no pair of subsets with

By symmetry we can assume a≥d and b≥c. We show that f_m(a,b,c,d) is Θ(m^a+b−1) if either b>c or a,b≥1. We also show that f_m(0,b,b,0) is Θ(m^b) and f_m(a,0,0,d) is Θ(m^a). The asymptotic results are as m→∞ for fixed non-negative integers a,b,c,d. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.     

Another Note on the Greatest Prime Factors of Fermat Numbers

For every positive integer k > 1, let P(k) be the largest prime divisor of k. In this note, we show that if F_m = 2^2m + 1 is the mth Fermat number, then P(F_m) 2^m+2(4m + 9) + 1 for all m 4. We also give a lower bound of a similar type for P(F_a,m), where F_a,m = a^2m + 1 whenever a is even and m a¹⁸.AMS Subject Classification (1991) 11A51 11J86     

Surfaces with simple geodesics through a point

We study compact connected surfaces inm-dimensional Euclidean spaceE ^m (3 m 5) with a point through which every geodesic is aW-curve regarded as a curve in E^m.     

P 3-factorization of complete multipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of aP ₃-factorization ofK _m ⁿ is (i)mn 0(mod 3) and (ii) (m – 1)n 0(mod 4).     

Optimal pairs of score vectors for positional scoring rules

Letw=(w ₁,,w _m ) andv=(v ₁,,v _m-1 ) be nonincreasing real vectors withw ₁>w _m andv ₁>v _m-1 . With respect to a lista ₁,,a _n of linear orders on a setA ofm3 elements, thew-score ofaA is the sum overi from 1 tom ofw _i times the number of orders in the list that ranka inith place; thev-score ofaA{b} is defined in a similar manner after a designated elementb is removed from everya _j .We are concerned with pairs (w, v) which maximize the probability that anaA with the greatestw-score also has the greatestv-score inA{b} whenb is randomly selected fromA{a}. Our model assumes that linear ordersa _j onA are independently selected according to the uniform distribution over them linear orders onA. It considers the limit probabilityP _m (w, v) forn that the element inA with the greatestw-score also has the greatestv-score inA{b}.It is shown thatP _m (m,v) takes on its maximum value if and only if bothw andv are linear, so thatw _i –w _i+1=w _i+1–w _i+2 forim–2, andv _i –v _i+1 =v _i+1 –v _i+2 forim–3. This general result for allm3 supplements related results for linear score vectors obtained previously form{3,4}.     

Two Conditions for Infinite Groups to Satisfy Certain Laws

Let G be an infinite group and m {2^k | k N^*}. In this paper, we prove that G satisfies the law [x_m, y_m] = 1 if and only if in any two infinite subsets X and Y of G, there exist a X and b Y such that [a^m,b^m] = 1. We also prove that G satisfies the law (x₁^mx₂^m x_n^m)² = 1 if and only if in any n infinite subsets X₁, X₂,..., X_n, there exist a_i X_i (i = 1,..., n) such that (a₁^ma₂^m a_n^m)² = 1.2000 Mathematics Subject Classification: 20F99     

On indecomposable definite hermitian forms

In this paper, for any given natural numbersn anda, we can construct explicitly positive definite indecomposable integral Hermitian forms of rankn over with discriminanta, with the following ten exceptions:n=2,a=1, 2, 4, 10;n=3,a=1, 2, 5;n=4,a=1;n=5,a=1; andn=7,a=1. In the exceptional cases there are no forms with the desired properties. The method given here can be applied to solving the problem of constructing indecomposable positive definite HermitianR _m -lattices of any given rankn and discriminanta, whereR _m is the ring of algebraic integers in an imaginary quadratic field with class number unity.     

On x 6 + x + a in Characteristic Three

Let F be a field of characteristic 3 and 0 a F. We show that the 10 ways to factor x ⁶ + x + a into two cubics over the algebraic closure F are in natural Galois bijection with the 10 roots of x ¹⁰ + ^ax + 1. We use this to (1) prove the two polynomials have the same splitting field; (2) prove that a difference set constructed by Arasu and Player using the polynomial x ⁶ + x + a is isomorphic to a difference set constructed by Dillon using the polynomial x ¹⁰ + x + a; (3) obtain a natural realization for the accidental isomorphism between the alternating group A ₆ and the special linear group PSL₂(9); and (4) characterize how x ⁶ + x + a factors when F = GF (3^m) with m odd. For example, x ⁶ + x + a is irreducible if and only if a can be written as ^{– 36} + ⁴ with F ^× and Tr(⁵) 0.     

Gabor dual spline windows

A method is presented for constructing dual Gabor window functions that are polynomial splines. The spline windows are supported in [−1,1], with a knot at x=0, and can be taken C^m smooth and symmetric. The translation and modulation parameters satisfy a=1 and 0<b1/2. The full range 0<ab<1 is handled by introducing an additional knot. Many explicit examples are found.     

Statistical topography. I. Fractal dimension of coastlines and number-area rule for Islands

Summary Statistical topography involves the study of the geometrical properties of the iso-sets (contour lines or surfaces) of a random potential (r). Previous work [1,2] has addressed coastlines on a random relief (x, y) that possess a single characteristic spatial scale with topography belonging to the universality class of the random percolation problem. In the present paper this previous analytical approach is extended to the case of a multiscale random function with a power spectrum of scales, ^H , in a wide range of wavelengths, ₀ < < _m . It is found that the pattern of the coastline differs significantly from that of a monoscale landscape provided that –3/4 <H < 1, with the case –3/4 <H < 0 corresponding to the long-range correlated percolation and 0 <H < 1 to the fractional Brownian relief. The expression for the fractal dimension of an individual coastline is derived,D _h = (10 – 3H)/7, the maximum value of whichD _h = 7/4, corresponds to the monoscale relief. The distribution functionF(a) of level lines over their sizea is calculated:F(a) a ^–4(1-H)/7, for ₀ a _m . A comparison of the theoretical results with geographical data is presented.     

Approximation of analytic sets by Nash tangents of higher order

The aim of this paper is to show that for every locally analytic subset X of C ^m and every there exist a neighborhood V of a in C ^m and a sequence of Nash subsets of V converging to such that X _ν and X satisfy a certain condition for tangency of order ν. Next it is shown that this condition implies that for sufficiently large ν the multiplicities of X _ν and X at a are equal.      

Almost universal graphs

We study the question as to when a random graph with n vertices and m edges contains a copy of almost all graphs with n vertices and cn/2 edges, c constant. We identify a “phase transition” at c = 1. For c < 1, m must grow slightly faster than n, and we prove that m = O(n log log n/log log log n) is sufficient. When c > 1, m must grow at a rate m = n^1+a, where a = a(c) > 0 for every c > 1, and a(c) is between and for large enough c. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Indescribable cardinals without diamonds

Summary We show that form, n1 the existence of a _n ^m indescribable cardinal is equiconsistent with the failure of the combinatorial principle at a _n ^m indescribable cardinal together with the Generalized Continuum Hypothesis.     

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number (G) and the chromatic number (G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each >0 and eachm2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+. This answers another question asked by Abbott and Zhou.     

Relative Hyperbolicity and Artin Groups

Let G=&langle; a₁,&ldots; , a _n | a _i a _j a _i &ldots; = a_ja_ia_j,&ldots; ,i>j&rangle; be an Artin group and let m _ij =m _ji be the length of each of the sides of the defining relation involving a _i and a _j . We show if all m_ij &ges; 7 then G is relatively hyperbolic in the sense of Farb with respect to the collection of its two-generator subgroups a _i , a _j for which m_ij >&infty;.     

Positive periodic solutions of higher-dimensional functional difference equations with a parameter

By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T-periodic solutions to the higher-dimensional functional difference equations of the form where A(n)=diag[a₁(n),a₂(n),…,a_m(n)], h(n)=diag[h₁(n),h₂(n),…,h_m(n)], a_j,h_j :Z→R⁺, τ :Z→Z are T -periodic, j=1,2,…,m, T1, λ>0, x :Z→R^m, f :R₊^m→R₊^m, where R₊^m={(x₁,…,x_m)^TR^m, x_j0, j=1,2,…,m}, R⁺={xR, x>0}.     

Strong solutions of the Navier-Stokes equation in Morrey spaces

It is shown that the nonstationary Navier-Stokes equation (NS) in ⁺× ^m is well posed in certain Morrey spacesM _p, (⁺× ^m ) (see the text for the definition: in particularM _p,0=L ^p ifp>1 andM _1,0 is the space of finite measures), in the following sense. Given a vectoraM _p,m-p with diva=0 and with certain supplementary conditions, there is a unique local (in time) solution (velocity field)u(t, ·)M _{p, m-p}, which is smooth fort>0 and takes the initial valuea at least in a weak sense.u is a global solution ifa is sufficiently small. Of particular interest is the spaceM _1,m–1, which admits certain measures; thusa may be a surface measure on a smooth (m–1)- dimensional surface in ⁺× ^m . The regularity of solutions and the decay of global solutions are also considered. The associated vorticity equation (for the vorticity =u) can similarly be solved in (tensor-valued)M _1,m–2, which is also a space of measures of another kind.Dedicated to Felix E. Browder     

j-partitions for visible shorelines

Summary LetC be a compact set inR ². A setS R ² C is said to have aj-partition relative toC if and only if there existj or fewer pointsc ₁,, c _j inC such that each point ofS sees somec _i via the complement ofC. Letm, j be fixed integers, 3 m, 2 j, and writem (uniquely) asm = qj + r, where 1 r j. Assume thatC is a convexm-gon in R², withS R ² C. Forq = 0 orq = 1, the setS has aj-partition relative toC. Forq 2,S has aj-partition relative toC if and only if every (qj + 1)-member subset ofS has aj-partition relative toC, and the Helly numberqj + 1 is best possible.IfC is a disk, no such Helly number exists.