Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Limit theorems for branching Markov processes

We establish almost sure limit theorems for a branching symmetric Hunt process in terms of the principal eigenvalue and the ground state of an associated Schrödinger operator. Here the branching rate and the branching mechanism can be state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying Hunt process X. The almost sure limit theorems are established under the assumption that the associated Schrödinger operator of X has a spectral gap. Such an assumption is satisfied if the underlying process X is a Brownian motion, a symmetric α-stable-like process on or a relativistic symmetric stable process on .     

Symmetry and classification of certain regular group divisible designs

In this paper it is shown that a regular group divisible (GD) design, with parametersv, b, r, k, ₁, ₂ satisfyingrk – ₂ v + 1 and ₂ = ₁ + 1, must be symmetric (i.e.,v + b). Furthermore, the parameters of such symmetric regular GD designs can be expressed in terms of only two integral parameters.Supported in part by Grant 59540043 (C), Japan.     

On semi-translation blocks

In this paper we generalize the concept of (P,l)-transitivity from finite projective planes to arbitrary symmetric designs with > 1. We define (x y, x)-transitivity, and show that if a symmetric design D is (x y, x)-transitive then it is also (x z, x)-transitive, for all blocks z x, such that z x y, and x is necessarily a good block. In addition we show that D has the parameters of aP _n,q , for some integer n and prime power q, and that if q = 2, x must be a translation block. However, we do give examples of symmetric designs with a proper semi-translation block (i.e. one in which x is not a translation block). Finally, we give a classification of all symmetric designs with > 1 which contain more than one semi-translation block, and show that there are exactly six types to consider, of which only three can possibly be proper. The results in this paper form part of a Ph.D. thesis submitted by the author to the University of London in 1981.     

Galois theory for the family of partial differential equationsΔ α ψ=0

Summary The partial differential fields most suited for the purpose of construction of Galois theory for the family (1) are endowed with the symmetric bilinear form (2iv) and are called -differential fields. In Section 1 are defined certain algebraic notions related to the symmetric bilinear form (2iv) and which are necessary for the construction of any Galois theory. Necessary and sufficient condition for the extension of the domain of the operator (this operator is not a derivation although it commutes with the partial derivations of the -differential field) from an -differential fieldK to a finitely generated -differential extension field is given in Theorem 1.Section 2 defines the notion of -differential mapping as linear mappings which preserve the symmetric bilinear form and commute with the partial derivations. The group properties of the set of -differential mappings are discussed and the Galois correspondence theorems set up for -differential fields.Section 3 sets up the notion of -Liouvillian extensions of -differential fields and briefly discusses the Galois groups associated with these -Liouvillian extension fields.Section 4 points to the procedure for the algebraic characterization of -simple--differential field extensions by elementary solutions of the partial differential equation _m =0.     

How large delays build up in a GI/G/1 queue

LetW _k denote the waiting time of customerk, k 0, in an initially empty GI/G/1 queue. Fixa> 0. We prove weak limit theorems describing the behaviour ofW _k /n, 0kn, given W_n >na. LetX have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold forX, and queues for whichX has regularly varying positive tail.The results can also be interpreted as conditional limit theorems, conditional on large maxima in the partial sums of random walks with negative drift.Research supported by the NSF under Grant NCR 8710840 and under the PYI Award NCR 8857731.     

Existence theorems for abstract multidimensional control problems

In the present paper, the author discusses an abstract formulation of control problems involving general operators :S V, :S Y from a Banach spaceS into spaceV andY of vector functions in a fixed domain with components inL _p ,p1. For this general formulation, the author states closure theorems, lower closure theorems, and existence theorems for an optimal solution. It is then shown that the problems of control involving Dieudonné-Rashevski partial differential equations previously considered by the author are particular cases of the present formulation. Finally, it is shown by examples that problems of control involving usual partial differential equations, linear or not, as well as other functional relations, can be framed in the present formulation. The present work concerns problems withdistributed controls. Work concerning problems withdistributed as well as boundary controls is forthcoming.Parts of this paper were read at the International Conference of Optimal Control, Tbilisi, Georgia, USSR, 1969, and at the Conference on Optimal Control, Ann Arbor, Michigan, 1969 (Tenth Annual Meeting of the Society for Natural Philosophy). This research was partially supported by AFOSR Research Project No. 69-1662.     

Algorithms for Computing Characters for Symmetric Spaces

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces G _k /K _k , where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and G _k resp. K _k the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice of the symmetric space in terms of the characters of the weight lattice of the group. These algorithms can easily be implemented in a computer algebra package. The root system of the symmetric space can be described as the image of the root system of the group under a projection π derived from an involution θ on . This implies that . Using these formulas for the characters of each of these lattices we show that in fact . A.G. Helminck is partially supported by N.S.F. Grant DMS-0532140.     

Packing of partial designs

We say that two hypergraphsH ₁ andH ₂ withv vertices eachcan be packed if there are edge disjoint hypergraphsH ₁ andH ₂ on the same setV ofv vertices, whereH _i is isomorphic toH _i.It is shown that for every fixed integersk andt, wheretk2t–2 and for all sufficiently largev there are two (t, k, v) partial designs that cannot be packed. Moreover, there are twoisomorphic partial (t, k, v)-designs that cannot be packed. It is also shown that for every fixedk2t–1 and for all sufficiently largev there is a (₁,t,k,v) partial design and a (₁,t,k,v) partial design that cannot be packed, where _{1 2}O(v ^k–2t+1 logv). Both results are nearly optimal asymptotically and answer questions of Teirlinck. The proofs are probabilistic.     

Some monotonicity properties of symmetric pólya densities and their exponential families

Summary In this note we observe that for independent symmetric random variables X and Y, when the pdf of X is PF, the conditional distributions of ¦Y¦ given S = X + Y form a MLR family. We then show that for a function : R ⁿR that is symmetric in each coordinate and increasing on (0, )ⁿ, E((S₁,...,S_n)¦S_n = s) is even and increasing in ¦s¦. Here S₁,...,S_n are partial sums with independent symmetric PF summands. Application is made to sequential tests that minimize the maximum expected sample size when the model is a one-parameter exponential family generated by a symmetric PF density.Work supported by NSF grants MPS 72-05082 AO2 and MCS 75-23344     

Convergence of Associated Continued Fractions Revised

This is an expository article which contains alternative proofs of many theorems concerning convergence of a continued fraction to a holomorphic function. The continued fractions which are studied are continued fractions of the form

Linearization and connection coefficients of orthogonal polynomials

Let {P _n } _n ^=0/ be a system of orthogonal polynomials.Lasser [5] observed that if the linearization coefficients of {P _n } _n ^=0/ are nonnegative then each of theP _n (x) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomialsP _n can be expressed in terms ofQ _n with nonnegative coefficients, where {Q _n } _n ^=0/ is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.     

A family of relative difference sets associated with Hjelmslev structures over finite projective spaces

In this note, we construct a new family of relative difference sets, with parameters n=q^d, m=q^d+...+q+1, k=q^d-1(q^d-1), ₁ =q^d-1(q^d-q^d-1-1) and ₂ =q^d-2(q-1)(q^d-1-1) where q is a prime power and d 2 an integer. The associated symmetric divisible designs admit natural epimorphisms onto the symmetric designs formed by points and hyperplanes in the corresponding projective spaces PG(d,q). As in the theory of Hjelmslev planes, points with the same image can be recognized from having the larger of the two possible joining numbers, and dually. More formally, these symmetric divisible designs are balanced c-H-structures (in the sense of Drake and Jungnickel [2]) with parameters c=q^d-2 (q-1)² and t=q^d-1 (q-1) over PG_d-1(d,q). These are the first examples of balanced non-uniform c-H-structures of type 2; they can be used in known constructions to obtain new balanced c-H-structures (for suitable c) of arbitrary type. In fact, all these results are special cases of a more general construction involving arbitrary difference sets.The author gratefully acknowledges the hospitality of the University of Waterloo and the financial support of NSERC under grant IS-0367.     

Wachstumsverhalten zufÄlliger Potenzreihen

Summary Using the theorems collected in Chapter 2 the paper discusses growth and divergence properties of random power series (rps's) with — generally — independent coefficients a _n. In Chapter 3 assertions about rps's with a _n symmetric relative to 0 are transferred to rps's for which the medians a _n of a _n are 0. Chapter 4 treats analogies between rps's with symmetrized coefficients a _s ⁿ and rps's with centered coefficients a _n -a _n ; they concern growth of random entire functions defined by rps's with arbitrarily dependent coefficients, and local behaviour near the circle of convergence. For some kinds of singular behaviour, e.g. unboundedness of characteristic in every sector of the circle of convergence, Chapter 5 gives necessary and sufficient conditions in connection with the notion of essential divergence. The theorems hold also for -quantiles a _n ( fixed, 0<<1) instead of medians a _n .Herrn Professor Dr. F. Lösch zum 65. Geburtstag gewidmet.     

Existence of solutions for a class of nonlinear control problems

We consider problems of control and problems of optimal control, monitored by an abstract equation of the formEx=N _u x in a finite interval [0,T]; here,x is the state variable with values in a reflexive Banach space;u is the control variable with values in a metric space;E is linear and monotone; andN _u is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]×G,G ⁿ , with Cauchy data fort=0 and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of Kemochi for reflexive Banach spaces.     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

Uniqueness and radial symmetry of least energy solution for a semilinear Neumann problem

Consider the following Neumann problem
d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,
where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.     

Two results concerning symmetric bi-derivations on prime rings

Summary LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R R is called a symmetric bi-derivation if, for any fixedy R, the mappingx D(x, y) is a derivation. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, ifD ₁ andD ₂ are symmetric bi-derivations on a prime ring of characteristic different from two and three such thatD ₁(x, x)D ₂(x,x) = 0 holds for allx R, then eitherD ₁ = 0 orD ₂ = 0. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that [[D(x, x),x],x] Z(R) holds for allx R, whereZ(R) denotes the center ofR, forcesR to be commutative.     

Cocyclic Development of Designs

We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic and -cyclic development are both special cases of cocyclic development.Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic.We derive a characterization of cocyclic development which allows us to generate all matrices which are cocyclic over G. Any cocyclic matrix is equivalent to one obtained by entrywise action of an asymmetric matrix and a symmetric matrix on a G-developed matrix. The symmetric matrix is a Kronecker product of back -cyclic matrices, and the asymmetric matrix is determined by the second integral homology group of G. We believe this link between combinatorial design theory and low-dimensional group cohomology leads to (i) a new way to generate combinatorial designs; (ii) a better understanding of the structure of some known designs; and (iii) a better understanding of known construction techniques.