Ergodicity and mixing of non-commuting epimorphisms

We study mixing properties of epimorphisms of a compact connectedfinite-dimensional abelian group X. In particular, we show thata set F, with |F| > dim X, of epimorphisms of X is mixingif and only if every subset of F of cardinality (dim X) + 1is mixing. We also construct examples of free non-abelian groupsof automorphisms of tori which are mixing, but not mixing oforder 3, and show that, under some irreducibility assumptions,ergodic groups of automorphisms contain mixing subgroups andfree non-abelian mixing subsemigroups.     

Quasi-Permutation Representations of p-Groups of Class 2

If G is a finite linear group of degree n, that is, a finitegroup of automorphisms of an n-dimensional complex vector space(or, equivalently, a finite group of non-singular matrices oforder n with complex coefficients), we shall say that G is aquasi-permutation group if the trace of every element of G isa non-negative rational integer. The reason for this terminologyis that, if G is a permutation group of degree n, its elements,considered as acting on the elements of a basis of an n-dimensionalcomplex vector space V, induce automorphisms of V forming agroup isomorphic to G. The trace of the automorphism correspondingto an element x of G is equal to the number of letters leftfixed by x, and so is a non-negative integer. Thus, a permutationgroup of degree n has a representation as a quasi-permutationgroup of degree n. See [8].     

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let _x denote the Dirac measureat a point x. A function in L²(R)) is called a difference oforder 1 if it is of the form g-_x * g for some x R and g L²(R)).Also, a difference of order 2 is a function of the form for some x R and g L²(R)). In fact,the concept of a ‘difference of order s’ may bedefined in a similar manner for each s 0. If f denotes the Fouriertransform of f, it is known that a function f in L²(R)) is afinite sum of differences of order s if and only if , and the vector space of all suchfunctions is denoted by D_s (L²(R)). Every function in D_s (L²(R))is a sum of int(2s) + 1 differences of order s, where int(t)denotes the integer part of t. Thus, every function in D₁ (L²(R))is a sum of three first order differences, but it was provedin 1994 that there is a function in D₁ (L(R)) which is neverthe sum of two first order differences. This complemented, forthe group R, the corresponding result for first order differencesobtained by Meisters and Schmidt in 1972 for the circle group.The results show that there is a function in L₂ R such that,for each s 1/2, this function is a sum of int (2s) + 1 differencesof order s but it is never the sum of int (2s) differences oforder s. The proof depends upon extending to higher dimensionsthe following result in two dimensions obtained by Schmidt in1972 in connection with Heilbronn's problem: if x₁, x__n arepoints in the unit square, Following on from the work of Meisters and Schmidt, this workfurther develops a connection between certain estimates in combinatorialgeometry and some questions of sharpness in harmonic analysis.2000 Mathematics Subject Classification 42A38 (primary), 52A40(secondary).     

The Accurate Numerical Inversion of Laplace Transforms

Numerical inversion of almost arbitrary Laplace transforms,for any value of t, to any prescribed accuracy up to atleastthree-quarters of the computer precision, is effected by trapezoidalintegration along a special contour. The required number ofpoints depends on t, the accuracy, and the transform singularitypositions, and for moderate t is typically 11 for errors oforder 10^–6, 18 for order 10^–10, 35 for order 10^–20(withdouble precision working).     

One-Factorizations of Complete Graphs with a Doubly Transitive Automorphism Group

It is shown that a 1-factorization of K_n with a doubly transitiveautomorphism group on vertices is either the affine line-parallelismof AG(d, 2), or one of three ‘sporadic’ exampleswith n = 6, 12 or 28. The full automorphism groups are respectivelyAGL (d, 2) (the holomorph of an elementary abelian group oforder 2^d), PGL(2,5), PSL(2,11) and PL(2,8).     

Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes

New lower bounds are given for the size of a point set in aDesarguesian projective plane over a finite field that containsat least a prescribed number s of points on every line. Thesebounds are best possible when q is square and s is small comparedwith q. In this case the smallest set is shown to be the unionof disjoint Baer subplanes. The results are based on new resultson the structure of certain lacunary polynomials, which canbe regarded as a generalization of Rédei's results inthe case when the derivative of the polynomial vanishes.     

Necessary and Sufficient Tauberian Conditions, Under Which Convergence Follows From Summability (C, 1)

Let (s_k: k = 0, 1, ...) be a sequence of real numbers whichis summable (C, 1) to a finite limit. We prove that (s_k) isconvergent if and only if the following two conditions are satisfied: where _n denotes the integer part of the productn. Both conditions are clearly satisfied if (s_k) is slowly decreasingin the sense of R. Schmidt and G. H. Hardy. The symmetric counterparts of the conditions above are thosewhen ‘limsup’ and ‘liminf’ are interchangedon the left-hand sides, while the inequality sign ‘ ’is changed for the opposite ‘ ’ in them. Next, let (s_k) be a sequence of complex numbers which is summable(C, 1) to a finite limit. We prove that (s_k) is convergent ifand only if one of the following conditions is satisfied: We also prove a general Tauberian theorem forsequences in ordered linear spaces.     

A Characterization of Finite Soluble Groups by Laws in Two Variables

Define a sequence (s_n) of two-variable words in variables x,y as follows: s₀(x, y) = x, s_n+1(x,y)=[s_n(x, y]^–y, s_n(x,y)for n 0. It is shown that a finite group G is soluble if andonly if s_n is a law of G for all but finitely many values ofn. 2000 Mathematics Subject Classification 20D10, 20D06.     

The norm of the Riemann-Liouville operator on Lp[0,1]: a probabilistic approach

We obtain explicit lower and upper bounds for the norm of theRiemann–Liouville operator V^s on L^p[0, 1] which are asymptoticallysharp, thus completing previous results by Eveson. Similar statementsare shown with respect to the norms ||V^s f||_p, whenever f satisfiescertain smoothness properties. It turns out that the correctrate of convergence of ||V^s f||_p as s depends both on theinfimum of the support of f and on the degree of smoothnessof f. We use a probabilistic approach which allows us to giveunified proofs.     

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k 3 be an integer. For 0<s<1, let D_s R² be the setthat is constructed iteratively as follows. Take a regular openk-gon with sides of unit length, attach regular open k-gonswith sides of length s to the middles of the edges, and so on.At each stage of the iteration the k-gons that are added area factor s smaller than the previous generation and are attachedto the outer edges of the family grown so far. The set D_s isdefined to be the interior of the closure of the union of allthe k-gons. It is easy to see that there must exist some s_k> 0 such that no k-gons overlap if and only if 0 < s s_k. We derive an explicit formula for s_k. The set D_s is open, bounded, connected and has a fractal polygonalboundary. Let denote the heat content of D_s at time t when D_s initially has temperature 0and D_s is kept at temperature 1. We derive the complete short-timeexpansion of up to terms that are exponentially small in 1/t. It turns out that there arethree regimes, corresponding to 0<s<1/(k–1), s=1/(k–1),and 1/(k–1)<s s_k. For s 1/(k–1) the expansionhas the form where p_s is a log (1/s²)-periodic function, d_s=log (k–1)/log(1/s) is a similarity dimension, A_s and B are constants relatedto the edges and vertices, respectively, of D_s, and r_s is anerror exponent. For s=1/(k–1), the t^1/2-term carries anadditional log t. 1991 Mathematics Subject Classification: 11D25,11G05, 14G05.     

One Cubic Diophantine Inequality

Suppose that G(x) is a form, or homogeneous polynomial, of odddegree d in s variables, with real coefficients. Schmidt [15]has shown that there exists a positive integer s₀(d), whichdepends only on the degree d, so that if s s₀(d), then thereis an x Z^s\{0} satisfying the inequality |G(x)|<1. (1) In other words, if there are enough variables, in terms of thedegree only, then there is a nontrivial solution to (1). Lets₀(d) be the minimum integer with the above property. In thecourse of proving this important result, Schmidt did not explicitlygive upper bounds for s₀(d). His methods do indicate how todo so, although not very efficiently. However, in fact muchearlier, Pitman [13] provided explicit bounds in the case whenG is a cubic. We consider a general cubic form F(x) with realcoefficients, in s variables, and look at the inequality |F(x)|<1. (2) Specifically, Pitman showed that if s(1314)²⁵⁶–1, (3) then inequality (2) is non-trivially soluble in integers. Wepresent the following improvement of this bound.     

A Bound for Solutions of a Fourth-order Dynamical System

We consider the homogeneous linearized equation of yawing motionof a spinning projectile ⁿ+A)s)' + B(s) = 0, where A and B are complex functions of the real independentvariable s. It is shown that where K is a positive constant and v(s) is a real function ofs.     

(k, l)-Algebraic Stability of Gauss Methods

It is well known that the s-stage Gauss Runge-Kutta methodsof order 2s are algebraically stable, or equivalently (1, 0)-algebraicallystable. In this paper, we show that there exists some l_s >0 such that the Gauss methods are (k, l) algebraically stablefor l [0, l_s) with k(l)=e^2l+O(l^p+1, where p=2s if s=1 or s=2,and p=2 if s>3.     

Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

Let Y be a reduced irreducible projective curve of arithmeticgenus g 2 with at most ordinary nodes as singularities. Fora subsheaf F of rank r', degree d' of a torsion-free sheaf Eof rank r, degree d, let s(E,F) = r'd-rd'. Define s_r'(E) = mins(E,F), where the minimum is taken over all subsheaves of Eof rank r'. For a fixed r', s_r' defines a stratification ofthe moduli space U(r,d) of stable torsion-free sheaves of rankr, degree d by locally closed subsets U_r',s. We study the nonemptinessand dimensions of the strata. We show that the general elementin each nonempty stratum is a vector bundle and it has onlyfinitely many (respectively unique) maximal subsheaves of rankr' for s r'(r-r')(g – 1) (respectively s < r'(r-r')(g– 1)). We prove that the tensor product of two generalstable vector bundles on an irreducible nodal curve Y is nonspecial.     

Supraconvergence of a finite difference scheme for solutions in Hs(0, L)

^** Email: silvia{at}mat.uc.pt^*** Email: ferreira{at}mat.uc.pt^**** Email: grigo{at}math.tu-berlin.de In this paper we study the convergence of a centred finite differencescheme on a non-uniform mesh for a 1D elliptic problem subjectto general boundary conditions. On a non-uniform mesh, the schemeis, in general, only first-order consistent. Nevertheless, weprove for s (1/2, 2] order O(h^s)-convergence of solution andgradient if the exact solution is in the Sobolev space H^1+s(0,L), i.e. the so-called supraconvergence of the method. It isshown that the scheme is equivalent to a fully discrete linearfinite-element method and the obtained convergence order isthen a superconvergence result for the gradient. Numerical examplesillustrate the performance of the method and support the convergenceresult.     

Profinite Groups with Multiplicative Probabilistic Zeta Function

To a finitely generated profinite group G, a formal Dirichletseries P_G(s)=_na_n/n^s is associated, where a_n = _|G:H|=n µ_G(H).It is proved that G is prosoluble if and only if the sequence{a_n}_nN is multiplicative, that is, a_rs = a_ra_s for any pairof coprime positive integers r and s. This extends the analogousresult on the probabilistic zeta function of finite groups.     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

On the division of polynomial matrices

The main purpose of this paper is to determine two new algorithmsfor the division of the polynomial matrix B(s) R[s]^pxq by A(s) R[s]^pxp (a) based on the Laurent matrix expansion at s = =of the inverse of A(s), i.e. A(s)^–1, and (b) in a waysimilar to the one presented by Gantmacher (1959).     

On Finite Elation Generalized Quadrangles with Symmetries

We study the structure of finite groups G which act as elationgroups on finite generalized quadrangles and contain a fullgroup of symmetries about some line through the base point.Such groups are related to the translation groups of translationtransversal designs with parameters depending on those of thequadrangles. Using results on the structure of p-groups which act as translationgroups on transversal designs and results on the index of theHughes subgroups of finite p-groups, we can show how restrictedthe structure of elation groups of finite generalized quadrangleswith symmetries is. One of our main results is that G is necessarily an elementaryabelian 2-group, provided that G has even cardinality. In particular,the elation generalized quadrangle coordinatized by G is a translationgeneralized quadrangle with G as translation group, that is,G contains full groups of symmetries about every line throughthe base point.     

A Note on the WKB Method for Difference Equations

We consider the asymptotic solution of the second-order differenceequation y_n + 1 –2y_n + y_n–1 + Q_ny_n = 0, where Q_n= N^–Q(n/N), 0 < < 2, Q(s) being a differentiablefunction of s, and N a large parameter such that Q(n/N) variesby order unity as n varies by order N. A discrete WKB methodis proposed, the form of the asymptotic expansion being similarto that used in the conventional WKB method. A particular Q(s)is studied, for which results of the discrete WKB method arein agreement with the results from the approach due to Bremmer(1951).