On a class of small 2-designs over gf(q)

A 2 ? (v,k,λ;q) design is a pair (V, B) of a v-dimensional vector space V over GF(q) and a collection B of k-dimensional subspaces of V such that each 2-dimensional subspace of V is contained in exactly λ members of B. Assuming transitivity of their automorphism groups on the nonzero vectors of V, we give a classification of nontrivial such designs for v = 7, q = 2,3 with small λ, together with the nonexistence proof of those designs for v ? 6. © 1995 John Wiley & Sons, Inc.     

The blue and minque in Gauss-Markoff model with linear transformation of the observable variables

For a singular linear model A = (y, Xβ, σ2 V) and its transformed model AF = (Fy, FXβ, σ2FVF'), where V is nonnegative definite and X can be rank-deficient,the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala (1981) are strengthened and consequences in Puntanen and Nurhonen (1992), and Puntanen (1996) are extended.     

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A_n-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1  ? k ? n − 1). With each hyperplane H of A_n-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧^n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A_n-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A_2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A_5,3(F).     

Designs on vector spaces constructed using quadratic forms

Let N be the set of nonnegative integers, let , t, v be in N and let K be a subset of N, let V be a v-dimensional vector space over the finite field GF(q), and let W _Kbe the set of subspaces of V whose dimensions belong to K. A t-[v, K, ; q]-design on V is a mapping : W _K N such that for every t-dimensional subspace, T, of V, we have (B)=. We construct t-[v, {t, t+1}, ; q-designs on the vector space GF(q ^v) over GF(q) for t2, v odd, and q ^t(q–1)² equal to the number of nondegenerate quadratic forms in t+1 variables over GF(q). Moreover, the vast majority of blocks of these designs have dimension t+1. We also construct nontrivial 2-[v, k, ; q]-designs for v odd and 3kv–3 and 3-[v, 4, q ⁶+q ⁵+q ⁴; q]-designs for v even. The distribution of subspaces in the designs is determined by the distribution of the pairs (Q, a) where Q is a nondegenerate quadratic form in k variables with coefficients in GF(q) and a is a vector with elements in GF(q ^v) such that Q(a)=0.This research was partly supported by NSA grant #MDA 904-88-H-2034.     

General Construction of Non-Dense Disjoint Iteration Groups on the Circle

Let be a disjoint iteration group on the unit circle , that is a family of homeomorphisms such that F ^v1 ○ F ^v2 = F ^v1+v2 for v ₁, v ₂ ∈ V and each F ^v either is the identity mapping or has no fixed point ((V, +) is a 2-divisible nontrivial Abelian group). Denote by the set of all cluster points of {F ^v (z), v ∈ V} for . In this paper we give a general construction of disjoint iteration groups for which .     

Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

Let (V_n, g) be a C^∞ compact Riemannian manifold. For a suitable function on V_n, let us consider the change of metric: g′ = g + Hess(), and the function, as a ratio of two determinants, M() = ¦g′¦ ¦g¦⁻¹. Using the method of continuity, we first solve in C^∞ the problem: Log M() = λ + ƒ, λ > 0, ƒ ε C^∞. Then, under weak hypothesis on F, we solve the general equation: Log M() = F(P, ), F in C^∞(V_n × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article.     

On Transformations and Determinants of Wishart Variables on Symmetric Cones

Let x and y be independent Wishart random variables on a simple Jordan algebra V. If c is a given idempotent of V, write for the decomposition of x in where V(c,) equals the set of v such that cv=v. In this paper we compute E(det(ax+by)) and some generalizations of it (Theorems 5 and 6). We give the joint distribution of (x ₁, x ₁₂, y ₀) where and P is the quadratic representation in V. In statistics, if x is a real positive definite matrix divided into the blocks x ₁₁, x ₁₂, x ₂₁, x ₂₂, then y ₀ is equal to . We also compute the joint distribution of the eigenvalues of x (Theorem 9). These results have been known only when V is the algebra of Hermitian matrices with entries in the real or the complex field. To obtain our results, we need to prove several new results on determinants in Jordan algebras. They include in particular extensions of some classical parts of linear algebra like Leibnitz's determinant formula (Proposition 2) or Schur's complement (Eqs. (3.3) and (3.6)).     

Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

Some new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD₁, …, D_nbe the normalized spacings associated with independent exponential random variablesX₁, …,X_n, whereX_ihas hazard rateλ_i,i=1, 2, …, n. LetD*₁, …, D*_nbe the normalized spacings of a random sampleY₁, …, Y_nof sizenfrom an exponential distribution with hazard rateλ=∑ⁿ_i=1 λ_i/n. It is shown that for anyn2, the random vector (D₁, …, D_n) is greater than the random vector (D*₁, …, D*_n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofX_j:n−X_1:nis Schur convex.     

Linear Codes and Character Sums

Let V be an rn-dimensional linear subspace of . Suppose the smallest Hamming weight of non-zero vectors in V is d. (In coding-theoretic terminology, V is a linear code of length n, rate r and distance d.) We settle two extremal problems on such spaces.First, we prove a (weak form) of a conjecture by Kalai and Linial and show that the fraction of vectors in V with weight d is exponentially small. Specifically, in the interesting case of a small r, this fraction does not exceed .We also answer a question of Ben-Or and show that if , then for every k, at most vectors of V have weight k.Our work draws on a simple connection between extremal properties of linear subspaces of and the distribution of values in short sums of -characters.* Supported in part by grants from the Israeli Academy of Sciences and the Binational Science Foundation Israel-USA. This work was done while the author was a student in the Hebrew University of Jerusalem, Israel.     

De Bruijn digraphs and affine transformations

Let be the additive group of 1×n row vectors over . For an n×n matrix T over  and , the affine transformation F_T,ω of sends x to xT+ω. Let α be the cyclic group generated by a vector . The affine transformation coset pseudo-digraph has the set of cosets of α in as vertices and there are c arcs from x+α to y+α if and only if the number of zx+α such that F_T,ω(z)y+α is c. We prove that the following statements are equivalent: (a)  is isomorphic to the d-nary (n−1)-dimensional De Bruijn digraph; (b) α is a cyclic vector for T; (c)  is primitive. This strengthens a result conjectured by C.M. Fiduccia and E.M. Jacobson [Universal multistage networks via linear permutations, in: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, ACM Press, New York, 1991, pp. 380–389]. Under the further assumption that T is invertible we show that each component of is a conjunction of a cycle and a De Bruijn digraph, namely a generalized wrapped butterfly. Finally, we discuss the affine TCP digraph representations for a class of digraphs introduced by D. Coudert, A. Ferreira and S. Perennes [Isomorphisms of the De Bruijn digraph and free-space optical networks, Networks 40 (2002) 155–164].     

The Structure of Disjoint Iteration Groups on the Circle

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle that is, families of homeomorphisms such that and each F ^veither is the identity mapping or has no fixed point ((V, +) is an arbitrary 2-divisible nontrivial (i.e., card V> 1) abelian group).     

The Groups H3(F(X)/FM) and CH2(X) for Generic Splitting Varieties of Quadratic Forms

Let F be a field of characteristic different from 2 and be a quadratic form over F. Let X be an arbitrary projective homogeneous generic splitting variety of . For example, we can take X to be equal to the variety X_,m of totally isotropic m-dimensional subspaces of V, where V is the quadratic space corresponding to and < dim V. In this paper, we study the groups CH²(X) and H³(F(X)/F) = ker(H ³(F) H ³(F(X))). One of the main results of this paper claims that the group Tors CH²(X) is always zero or isomorphic to . In many cases we prove that Tors CH²(X) = 0 and compute the group H ³(F(X)/F) completely. As an application of the main results, we give a criterion of motivic equivalence of eight-dimensional forms except for the case where the Schur indices of their Clifford algebras equal 4.     

Sphere and Dot Product Representations of Graphs

A graph G is a k-sphere graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the distance between v _i and v _j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the dot product of v _i and v _j is at least 1.     

On Universal Representation of Random Graphs

It is shown that every probability measure on the interval [0, 1] gives rise to a unique infinite random graph g on vertices {v₁, v₂, . . .} and a sequence of random graphs gn on vertices {v₁, . . . , v_n} such that . In particular, for Bernoulli graphs with stable property Q, can be strengthened to: probability space (, F, P), set of infinite graphs G(Q) , F with property Q such that .AMS Subject Classification: 05C80, 05C62.     

The incidence structure of subspaces with well-scaled frames

In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S - F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular.Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±2⁰, ±2¹, ±2², …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.     

Adjacent vertices on the b-matching polyhedron

Given a graph G = (V,E) and an integer vector $\text{b?}\text{N}{}^{\mathrm{v}}$, a b-matching is a set of edges F?E such that any vertex v?V is incident to at most b_v edges in F. The adjacency on the convex hull of the incidence vectors of the b-matchings is characterized by a very general adjacency criterion, the coloring criter on, which is at least sufficient for all 0–1-polyhedra and which can be checked in the b-matching case by a linear algorithm.     

Localisation des résidus de Baum-Bott, courbes généralisées et K-théorie (I: feuilletages dans \Bbb C2 {\Bbb C}^2 )

Let v be a holomorphic vector field in a neighborhood of a point m ₀ in , which is a non dicritical isolated singularity. Let f = 0 be a reduced equation of the maximal separatrix V through m ₀, v _f the vector field , and the union of separatrices and pseudo-separatrices (i.e. the set of points where v and v _f are colinear). Assuming the foliations defined by v and v _f to be distinct, we prove that the Baum-Bott residue BB(c ₁ ² , v) of v at m ₀, as well as the difference PH(v) - μ between the Poincaré-Hopf index and the Milnor number of V at m ₀, are "localised" near the separatrices and pseudo-separatrices. (The particular case of generalized curves has already been studied in details in [CLS] and [Br]). We also interpret in K-theory the difference PH - μ as well as the GSV index of Gomez Mont-Seade-Verjovski, and we give a caracterisation of generalized curves in this framework, which will enable us to extend this concept in higher dimension. Received: August 25, 2000     

A functional model for the tensor product of level 1 highest and level −1 lowest modules for the quantum affine algebra

Let V(Λ_i) (resp., V(−Λ_j)) be a fundamental integrable highest (resp., lowest) weight module of . The tensor product V(Λ_i)V(−Λ_j) is filtered by submodules , n≥0, n≡i−j mod 2, where v_iV(Λ_i) is the highest vector and is an extremal vector. We show that F_n/F_n+2 is isomorphic to the level 0 extremal weight module V(n(Λ₁−Λ₀)). Using this we give a functional realization of the completion of V(Λ_i)V(−Λ_j) by the filtration (F_n)_n≥0. The subspace of V(Λ_i)V(−Λ_j) of -weight m is mapped to a certain space of sequences (P_n,l)_{n≥0,n≡i−jmod2,n−2l=m}, whose members P_n,l=P_n,l(X₁,…,X_lz₁,…,z_n) are symmetric polynomials in X_a and symmetric Laurent polynomials in z_k, with additional constraints. When the parameter q is specialized to , this construction settles a conjecture which arose in the study of form factors in integrable field theory.     

Finite Interpolation with Minimum Uniform Norm in

Given a finite sequence a{a₁, …, a_N} in a domain Ω ⁿ, and complex scalars v{v₁, …, v_N}, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying f(a_j)=v_j for all j. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough so that its A(Ω)-hull contains a subset of the original a large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which contains the hull (in an appropriate, weaker, sense) of a measure supported on the maximum modulus set. An example is given to show that the inclusions can be strict.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).