Embedded Thick Finite Generalized Hexagons in Projective Space

We show that every embedded finite thick generalized hexagon of order (s, t) in PG(n,q) which satisfies the conditions

1. s = q
2. the set of all points of generates PG(n, q)
3. for anypoint x of , the set of all points collinear in withx is containedin a plane of PG(n, q)
4. for any point x of , the set of allpoints of not oppositex in is contained in a hyperplane ofPG,(n, q)

is necessarily the standard representation of H(q) in PG(6,q) (on the quadric Q(6, q)), the standard representation ofH(q) for q even in PG(5, q) (inside a symplectic space), orthe standard representation of H(q, ) in PG(7, q) (where the lines of are the lines fixed by a trialityon the quadric Q⁺(7, q)). This generalizes a result by Cameronand Kantor [3], which is used in our proof.     

On Hill's Equation with a Singular Complex-Valued Potential

In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x₀, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C². We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.     

Sur L'equation Diophantienne (xn-1)/(x-1)=yq, III

In this paper we study the diophantine equation of the title,which was first introduced by Nagell and Ljunggren during thefirst half of the twentieth century. We describe a method whichallows us, on the one hand when n is fixed, to obtain an upperbound for q, and on the other hand when n and q are fixed, toobtain upper bounds for x and y which are far sharper than thosederived from the theory of linear forms in logarithms. We alsoshow how these bounds can be used even when they seem too largefor a straightforward enumeration of the remaining possiblevalues of x. By combining all these techniques, we are ableto solve the equation in many cases, including the case whenn has a prime divisor less than 13, or the case when n has aprime divisor which is less than or equal to 23 and distinctfrom q. 2000 Mathematical Subject Classification: primary 11D41;secondary 11J86, 11Y50.     

On rational approximations with denominators from thin sets, ii

In the general case a real irrational number cannot be approximated by infinitely many rationalsp/q involving error terms less than q^-2 when the denominatorsq are taken from a given thin set of positive integers. The distribution of irrationals which are situated in close neighborhoods of infinitely many fractionsp/q, whereq is restricted to the elements of a thin set, depends on the asymptotic behaviour of theq’s and on their arithmetic properties.     

Distribution of a Sparse Set of Fractions Modulo q

The distribution on the torus R/Z of a set of fractions of theform is investigated, where q is a large integer, is the inverse of m modulo q, R(x) is a rationalfunction defined modulo q, and U, M, N are subsets of {1,...,q}.Under some natural assumptions, it is shown that the set R isuniformly distributed on R/Z.     

Multiple Blocking Sets and Arcs in Finite Planes

This paper contains two main results relating to the size ofa multiple blocking set in PG(2, q). The first gives a verygeneral lower bound, the second a much better lower bound forprime planes. The latter is used to consider maximum sizes of(k, n)-arcs in PG(2, 11) and PG(2, 13), some of which are determined.In addition, a summary is given of the value of m_n(2, q) forq 13.     

Hemisystems on the Hermitian Surface

The natural geometric setting of quadrics commuting with a Hermitiansurface of PG(3,q²), q odd, is adopted and a hemisystem on theHermitian surface H(3,q²) admitting the group P^–(4,q)is constructed, yielding a partial quadrangle PQ((q–1)/2,q²,(q–1)²/2) and a strongly regular graph srg((q³+1)(q+1)/2,(q²+1)(q–1)/2,(q–3)/2,(q–1)²/2).For q>3, no partial quadrangle or strongly regular graphwith these parameters was previously known, whereas when q=3,this is the Gewirtz graph. Thas conjectured that there are nohemisystems on H(3,q²) for q>3, so these are counterexamplesto his conjecture. Furthermore, a hemisystem on H(3,25) admitting3.A₇.2 is constructed. Finally, special sets (after Shult) andovoids on H(3,q²) are investigated.     

Improving the Stability of Predictor-Corrector Methods by Residue Smoothing

Residue smoothing is usually applied in order to acceleratethe convergence of iteration processes. Here, we show that residuesmoothing can also be used in order to increase the stabilityregion of predictor-corrector methods. We shall concentrateon increasing the real stability boundary. The iteration parametersand the smoothing operators are chosen such that the stabilityboundary becomes as large as c(m, q)m²4^g where m is the numberof right-hand side evaluations per step, q the number of smoothingoperations applied to each right-hand side evaluation, and c(m,q) a slowly varying function of m and q, of magnitude 1.3 ina typical case. Numerical results show that, for a variety oflinear and nonlinear parabolic equations in one and two spatialdimensions, these smoothed predictor-corrector methods are atleast competitive with conventional implicit methods.     

How many s-subspaces must miss a point set in PG(d, q)

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG(n, q) with the property that every s-subspace has a point in the set is (q ^n+1-s - 1)/(q - 1). We solve in finite projective spaces PG(n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ (q ^n+1-s - 1)/(q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ (q ^d+1 - 1)/(q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG(n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner.     

Specht Filtrations for Hecke Algebras of Type A

Let H_q(d) be the Iwahori–Hecke algebra of the symmetricgroup, where q is a primitive 1th root of unity. Using resultsfrom the cohomology of quantum groups and recent results aboutthe Schur functor and adjoint Schur functor, it is proved that,contrary to expectations, for l 4 the multiplicities in a Spechtor dual Specht module filtration of an H_q(d)-module are welldefined. A cohomological criterion is given for when an H_q(d)-modulehas such a filtration. Finally, these results are used to givea new construction of Young modules that is analogous to theDonkin–Ringel construction of tilting modules. As a corollary,certain decomposition numbers can be equated with extensionsbetween Specht modules. Setting q = 1, results are obtainedfor the symmetric group in characteristic p 5. These resultsare false in general for p = 2 or 3.     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

On the mean values of Dirichlet L-functions

We study the 2kth power moment of Dirichlet L-functions L(s,) at the centre of the critical strip , where the average is over all primitive characters (mod q). We extend to this case the hybrid Euler–Hadamardproduct results of Gonek, Hughes and Keating for the Riemannzeta function. This allows us to recover conjectures for themoments based on random matrix models, incorporating the arithmeticalterms in a natural way.     

Well matched systems

The product (3.10) on page 33 is incorrectly called a cartesianproduct on pages 33 and 35. This misnomer in effect amountsto a wrong definition. The product (3.10) should be definedso that the right-hand member of (3.10) is the set of all sums_f=1 f_j (not the set of all ordered q-tuples) such that f₁ F(m₁,d₁), ..., f_q F(m_q, d_q).     

Hurwitz Groups with Arbitrarily Large Centres

In this paper a new family of quotients of the triangle group is obtained. Each group in this family is constructed as centralproduct of the groups SL(2, q) for various prime-powers q, andin this way it is shown that for every positive integer s thereare infinitely many Hurwitz groups with a centre of size 2⁸.     

Two New Families of Commutative Semifields

Two new families of commutative semifields of order q⁴ withmiddle nucleus GF(q²), where q is an odd prime power, are defined.One family occurs when q 3(mod 4) and the other when q 1(mod4). 1991 Mathematics Subject Classification 17A35, 12K10.     

On the second weight of generalized Reed-Muller codes

Not much is known about the weight distribution of the generalized Reed-Muller code RM _q (s,m) when q > 2, s > 2 and m ≥ 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m – 1)(q – 1) we then find the first s + 1 – (m – 1)(q–1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.     

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

It is proved here that if is an elliptic function and q is the maximal multiplicity ofall poles of f, then the Hausdorff dimension of the Julia setof f is greater than 2 q/(q + 1), and the Hausdorff dimensionof the set of points that escape to infinity is less than orequal to 2q/(q + 1). In particular, the area of this latterset is equal to 0. 2000 Mathematics Subject Classification 37F35(primary); 37F10, 30D30 (secondary).     

Rates of Recurrence for Zq and Rq Extensions of Subshifts of Finite Type

In this note we give new asymptotic formulae for certain countingfunctions associated to the periodic behaviour of Z^q and R^qextensions of subshifts of finite type. In the case of the Z^qextensions, these strengthen previous estimates of Marcus andTuncel [9]. For both types of extension, our results complementthe central limit type results of Lalley [6]. Our proof requiresthe application of ideas from thermodynamic formalism. Whilstdeveloping this approach, in Section 2, we take the opportunityto present a counter-example to a related conjecture of Coelho-Filho[2].     

Congruences De Sommes De Chiffres De Valeurs Polynomiales

Let m, g, q N with q 2 and (m, q – 1) = 1. For n N,denote by s_n(n) the sum of digits of n in the q-ary digitalexpansion. Given a polynomial f with integer coefficients, degreed 1, and such that f(N) N, it is shown that there exists C= C(f, m, q) > 0 such that for any g Z, and all large N, In the special case m = q = 2 and f(n)= n², the value C = 1/20 is admissible. 2000 Mathematics SubjectClassification 11B85 (primary), 11N37, 11N69 (secondary).