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.     

Construction of orthogonal starters and corollaries

A method is proposed for constructing a system of (v–1)/2 pairwise disjoint orthogonal starters of order v for v6k+17 (mod 12)pn²+n+1/t such that the number 3 is one of the primitive roots of the Galois field of prime order p (k is prime, k 2, and n and t are positive integers). The starters occurring in this system satisfy certain additional conditions. The construction of a series of combinatorial structures, including some not previously known, is a consequence of this result.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 654–662, May, 1992.     

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).     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

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.     

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.     

Pointwise decomposable sets

We show that, under the conditionala<0, every recursively enumerable (r.e.) A bia has a pointwise decomposable complement. If A T^B, A and ¯B are r.e. co-retraceable sets, and f(x)=f^B(x), then there exists a r.e. co-retraceable C, such thatA(c),BT ^C , (A n) (f(n) <c _n), where ¯C=C ₀<C ₁<C ₂<....Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 893–898, June, 1973.The author thanks A. N. Degtev for his interest in this work.     

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     

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.     

On dominant operators

This paper is devoted to the study of dominant operators with an emphasis on their spectral properties. In particular the equation (T–)f() x (T a dominant or hyponormal operator on the Hilbert space ,x andf a function from the open setU to ) is investigated in an effort to discover necessary and/or sufficient conditions for the analyticity off.Supported in part by the National Science Foundation.     

Necessary and sufficient conditions for the convergence of convolution products of non-identical distributions on finite abelian semigroups

Given a sequence of probability measures ( _n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products _{k,n k+1}*···* _n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( _n ).     

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.     

Odd Systems of Vectors and Related Lattices

We consider uniform odd systems, i.e. sets of vectors of constant odd norm with odd inner product, and the lattice L(V) linearly generated by a uniform odd system V of odd norm 2t+1. If uu p (mod 4) for all u V, one has v² p (mod 4) if v² is odd and v² 0 (mod 4) if v² is even, for any vector v L(V). The vectors of even norm form a double even sublattice L₀(V) of L(V), i.e. is an even lattice. The closure of V, i.e. all vectors of L(V) of norm 2t+1, are minimal vectors of L(V) for t=1, and they are almost always minimal for t=2. For such t, the convex hull of vectors of the closure of V is an L-polytope of L₀V and the contact polytope of L(V). As an example, we consider closed uniform odd systems of norm 5 spanning equiangular lines.     

Simple minimum coverings ofK n with copies ofK 4 −e

Summary AK ₄–e design of ordern is a pair (S, B), whereB is an edge-disjoint decomposition ofK _n (the complete undirected graph onn vertices) with vertex setS, into copies ofK ₄–e, the graph on four vertices with five edges. It is well-known [1] thatK ₄–e designs of ordern exist for alln 0 or 1 (mod 5),n 6, and that if (S, B) is aK ₄–e design of ordern then |B| =n(n – 1)/10.Asimple covering ofK _n with copies ofK ₄–e is a pair (S, C) whereS is the vertex set ofK _n andC is a collection of edge-disjoint copies ofK ₄–e which partitionE(K_n)P, for some . Asimple minimum covering ofK _n (SMCK _n) with copies ofK ₄–e is a simple covering whereP consists of as few edges as possible. The collection of edgesP is called thepadding. Thus aK ₄–e design of ordern isSMCK _n with empty padding.We show that forn 3 or 8 (mod 10),n 8, the padding ofSMCK _n consists of two edges and that forn 2, 4, 7 or 9 (mod 10),n 9, the padding consists of four edges. In each case, the padding may be any of the simple graphs with two or four edges respectively. The smaller cases need separate treatment:SMCK ₅ has four possible paddings of five edges each,SMCK ₄ has two possible paddings of four edges each andSMCK ₇ has eight possible paddings of four edges each.The recursive arguments depend on two essential ingredients. One is aK ₄–e design of ordern with ahole of sizek. This is a triple (S, H, B) whereB is an edge-disjoint collection of copies ofK ₄–e which partition the edge set ofK _n\K_k, whereS is the vertex set ofK _n, and is the vertex set ofK _k. The other essential is acommutative quasigroup with holes. Here we letX be a set of size 2n 6, and letX = {x ₁, x₂, ..., x_n} be a partition ofX into 2-element subsets, calledholes of size two. Then a commutative quasigroup with holesX is a commutative quasigroup (X, ) such that for each holex _i X, (x_i, ) is a subquasigroup. Such quasigroups exist for every even order 2n 6 [4].     

A multiple recursive non-linear congruential pseudo random number generator

On-linear multiple recursive congruential pseudo random number generator with prime modulus p is introduced. Let x, n0, be the sequence generated by a usual linear (r+1)-step recursive congruential generator with prime modulus p and denote by N(n), n0, the sequence of non-negative integers with x_N(n)0 (mod p). The non-linear generator is defined by z_nx_N(n)+1·x _N(n) ^–1 (mod p), n0, where x _N(n) ^–1 denotes the inverse element of x_N(n) in the Galois field GF(p). A condition is given which ensures that the generated sequence is purely periodic with period length p^r and all (p–1)^r r-tupels (y₁,...,y_r) with 1y₁,...,y_rp are generated once per period when r-tupels of consecutive numbers of the generated sequence are formed. For r=1 this generator coincides with the generator introduced by Eichenauer and Lehn [2].     

Subalgebras that are cyclic as submodules

Let R be an associative, commutative, unital ring. By a R-algebra we mean a unital R-module A together with a R-module homomorphism : _R ⁿ AA (n2). We raise the question whether such an algebra possesses either an idempotent or a nilpotent element. In section 1 an affirmative answer is obtained in case R=k is an algebraically closed field and dim_kA<, as well as in case R=, dimS<, and n0(2). Section 2 deals with the case of reduced rings R and R-algebras which are finitely generated and projective as R-modules. In section 3 we show that the generic algebra over an integral domain D fails to have nilpotent elements in any integral domain extending its base ring D_n,m, and thus acquires an idempotent element in some integral domain extending D_n,m.Partially supported by National Science Foundation Grant GP-38229.     

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     

Operatori lineariT-invarianti rispetto ad un gruppo di congruenze

Summary Let be an open subset of ⁿ, W_m() the linear space of m-vector valued functions defined on , G{} a group of orthogonal matrices mapping onto itself and T{T()} a linear representation of order m of G. A suitable groupC(G,T) of linear operators of W_m(), which leads to a general definition of T-invariant linear operator with respect to G, is here introduced. Characterization theorems concerning the linear differential and integral T-invariant operators are also given. When G is a finite group, projection operators are explicitly obtained; they define a «maximal» decomposition of W_m() into a direct sum of subspaces each of them invariant with respect to any T-invariant linear operator of W_m(). Some examples are givenc.

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Lavoro eseguito nell'ambito del progetto nazionale di ricerca «Analisi numerica e matematica computazionale» nell'anno 1985–86.     

Some complements and corrections to my papers on the theory of attractors for abstract semigroups

In Sec. 1 a correction is given of the estimate of the Hausdorff dimension and an estimate of the fractal dimension of a bounded subset of a Hilbert space, semiinvariant with respect to a flattening transformation. In Sec. 2 the results, proved by the author for semigroups with a continuous group parameter tR⁺[0, ), are carried over to the case when t runs through the semigroup ⁺{tt0} of some additive group R=(–, ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 102–112, 1990.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.