Some Multiply Derived Translation Planes with SL(2,5) as an Inherited Collineation Group in the Translation Complement

Finite translation planes having a collineation group isomorphic to SL(2,5) occur in many investigations on minimal normal non-solvable subgroups of linear translation complements. In this paper, we are looking for multiply derived translation planes of the desarguesian plane which have an inherited linear collineation group isomorphic to SL(2,5). The Hall plane and some of the planes discovered by Prohaska [10], see also [1], are translation planes of this kind of order q ²;, provided that q is odd and either q ²; 1 mod 5 or q is a power of 5. In this paper the case q ² -1 mod 5 is considered and some examples are constructed under the further hypothesis that either q 2 mod 3, or q 1 mod 3 and q 1 mod 4, or q -1 mod 4, 3 q and q 3,5 or 6 mod 7. One might expect that examples exist for each odd prime power q. But this is not always true according to Theorem 2.     

Tables on fifth power residuacity

In this paper 2 ^p 1 (modq),q=10p+1,p 3 (mod 4),p andq prime, is expressed uniquely (except for changes in sign and interchange ofx, y) in the formq=w ²+25 (x ²+y ²)/2+125z ², 4wz=y ²–x ²–4xy, withw, x, y, z odd, forp<10⁵. For 10⁵<p<10⁶, allp such that 2 ^p 1 (mod 10p + 1),p 3 (mod 4),p and 10p + 1 prime, are listed.     

ExplicitN-polynomials of 2-power degree over finite fields,I

For an odd prime powerq the infinite field GF(q ² )= _n0 GF (q ²ⁿ ) is explicitly presented by a sequence (f _n)₁ ofN-polynomials. This means that, for a suitably chosen initial polynomialf ₁, the defining polynomialsf _nGF(q)[x] of degrees2 ⁿ are constructed by iteration of the transformation of variablexx+1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the caseq1 (mod 4) is considered and the caseq3 (mod 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].     

Congruences for Quadratic Units of Norm −1

Let D 1 (mod 4) be a positive integer. Let R be the ring {x + y(1 + )/2 : x, y }. Suppose that R contains a unit of norm –1 as well as an element of norm 2, and thus an element of norm –2. It is not hard to see that ±1(mod ²). In this paper we determine modulo ³ and modulo ³ using only elementary techniques. This determination extends a recent result of Mastropietro, which was proved using class field theory.     

Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

Quasi-symmetric designs and the Smith Normal Form

We obtain necessary conditions for the existence of a 2 – (, k, ) design, for which the block intersection sizes s ₁, s ₂, ..., s _n satisfy s ₁ s ₂ ... s _n s (mod p ^e ),where p is a prime and the exponent e is odd. These conditions are obtained from restriction on the Smith Normal Form of the incidence matrix of the design. We also obtain restrictions on the action of the automorphism group of a 2 – (, k, ) design on points and on blocks.     

C k -factorization of complete bipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of aC _k-factorization ofK _m,n is (i)m = n 0 (mod 2), (ii)k 0 (mod 2),k 4 and (iii) 2n 0 (modk) with precisely one exception, namely m =n = k = 6.     

Über die Quadratwurzel-Schranke für Quadratische-Rest-Codes

The minimal distanced of any QR-Code of lengthn 3mod4 over a prime fieldGF (p) with p3 mod4 satisfies the improved square root bound d(3d-2)4(n–1).

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Helmut Karzel zum 60. Geburtstag gewidmet     

Non-symmetric 2-designs modulo 2

Necessary conditions are obtained for the existence of a 2 – (v, k, ) design, for which the block intersection sizess ₁,s ₂, ...,s _n satisfys ₁ s ₂ ... s _n s (mod 2 ^e ), wheree is odd. These conditions are obtained by combining restrictions on the Smith Normal Form of the incidence matrix of the design with some well known properties of self-orthogonal binary codes with all weights divisible by 4.Research done at AT&T Bell Laboratories.     

Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq ² havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq ⁿ , whereq is an odd prime power andn 3 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq ⁿ , whereq is an odd prime power andn 2 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq ⁿ such thatn 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].The author is grateful to the referee for many helpful comments.     

Green's theorem from the viewpoint of applications

Making use of a line integral defined without use of the partition of unity, Green's theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces W^1,p () H^1,p () (1 p < ).     

Blocking Sets in (υ,{2, 4}, 1)-Designs

The existence of blocking sets in (, {2, 4}, 1)-designs is examined. We show that for 0, 3, 5, 6, 7, 8, 9, 11 (mod 12>), blocking sets cannot exist. We prove that for each 1, 2, 4 (mod 12) there is a (, {2, 4}, 1)-design with a blocking set with three possible exceptions. The case 10 (mod 12) is still open; we consider the first four values of in this situation.     

Existence of 1-Rotational <Emphasis Type="Italic">k</Emphasis>-Cycle Systems of the Complete Graph

We explicitly solve the existence problem for 1-rotational k-cycle systems of the complete graph K_v with v1 or k (mod 2k). For v1 (mod 2k) we have existence if and only if k is an odd composite number. For any odd k and vk (mod 2k), (except k3 and v15, 21 (mod 24)) a 1-rotational k-cycle system of K_v exists.Final version received: June 18, 2003     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .     

A Solution of a Problem of A. E. Brouwer

Let q=p^e 1(mod 4) be a prime power, and let ^(q) be the Paley graph over the finite field . Denote by ^(q) the subgraph of ^(q) induced on the set of non-zero squares of . In this paper the full automorphism group of ^(q) is determined affirming the conjecture of Brouwer [Des. Codes Cryptograph. 21, 69–76 (2000)]. The proof combines spectral and Schur ring techniques.     

Largek-arcs in inversive planes of odd order

An infinite family of largek-arcs in the inversive plane over a finite field GF(q), withq 1 (mod 3),q71 orq {17,23, 27,29,41,47,49,53,59} is constructed.Research supported by G.N.S.A.G.A. of C.N.R., project Applicazioni della matematica per la tecnologia e la società, subproject Calcolo simbolico.     

Rate of expansion of an inhomogeneous branching process of brownian particles

Summary Let X be the (B ⁰, {q _n (x)})-branching diffusion where B ⁰is the exp -subprocess of BM(R¹) and q _n (x) is the probability that a particle dying at x produces n offspring, q ₀ q ₁0. Put m(x) = nq _n (x). We assume q _n , n2, m and k are all continuous (but m is not necessarily bounded). If k(x)m(x)0 as ¦x¦, then we prove that R _t /t( ₂/2)^1/2, as t, a.s. and in mean (of any order) where R _t is the position of the rightmost particle at time t and ₀ is the largest eigenvalue of (1/2)d ²/dx ² + Q, Q(x) = k(x)(m(x)–1).This work was supported in part by a grant from the National Science Foundation # MCS-8201470.     

Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q ²-1-r lines, then r=s( +1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3, ). We also discuss maximal partial spreads in PG(3,p ³), p=p ₀ ^h , p ₀ prime, p ₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p ²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p ³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency p ²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1, ), which implies that 0 (mod +1) and, when (2t+1)( -1) <q-1, that 2( +1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2, ) and this implies 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.     

On a property of symmetric designs of order n≡2 (mod 4)

Summary U. Ott, during his visit in Rome (spring 1985), by using the theory of even unimodular lattices, proved that a (v,k,) symmetric design of order n2 (mod 4) satisfies the congruence v ±1 (mod 8). He asked me the question whether this is a consequence of the Bruck-Ryser-Chowla's theorem or not. In this paper we prove that the answer to this question is affirmative. As a consequence of this, we have that the conjecture according to which the Bruck-Ryser-Chowla's theorem and the identity k²–v=n imply the existence of a (v,k,) symmetric design is still open.     

Cyclic Steiner Quadruple Systems and Köhler's orbit graphs

In this article we are concerned with the problem of the existence of strictly cyclic Steiner Quadruple Systems sSQS(v), where v 2, 10 (24). E. Köhler (cf. (Köhler 1978)) used an orbit graph approach to handle such systems and obtained the result that in case p is a prime number with p 53, 77 (120) then sSQS(v) exists provided that the associated orbit graph OKG(p) is bridgeless. We continue these investigations by classifying the orbit graphs OKG(p) with p 5 (12), where the ones with p 53, 77 (120) constitute one out of four classes and thus show that sSQS(2p), p 5 (12) exists if OKG(p) or a reduced graph of it is bridgeless by discussing the four classes separately. Subsequent to this discussion we use the proof of Theorem 2 (Siemon 1991) to state that the bridgelessness of the graphs in all classes is equivalent to the number theoretic claim (3.1).Dedicated to Hanfried Lenz on the occasion of his 75th birthday.