Let Vn(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of Vn(q) is a partition of Vn(q) if every nonzero vector in Vn(q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the
same size. A partition of Vn(q) containing ai subspaces of dimension ni for 1 ≤ i ≤ k induces a uniformly resolvable design on qn points with ai parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions
that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable
designs on qn points where corresponding partitions of Vn(q) do not exist.
A. D. Blinco—Part of this research was done while the author was visiting Illinois State University. 相似文献
Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 £ r £ q - 1 0 \leq r \leq q - 1 . For 1 £ i £ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 £ i £ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 £ j £ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 £ j £ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well. 相似文献
LetP be a finite classical polar space of rankr, r2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceWn(q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q–(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q2) arising from a non-singular Hermitian variety inPG(n, q2)n even andn > 2, have no ovoids.LetS be a generalized hexagon of ordern (1). IfV is a pointset of order n3 + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG2(q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=32h+1, has an ovoid, and thatH(q), q even, has no ovoid.A regular system of orderm onH(3,q2) is a subsetK of the lineset ofH(3,q2), such that through every point ofH(3,q2) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3). 相似文献
This paper studies the heavily trimmed sums (*)
[ns] + 1[nt]Xj(n)
, where {Xj(n)
}
j = 1n
are the order statistics from independent random variables {X1,...,Xn} having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like
j = 1[nt]J(j/n)Xj(n)
are also discussed. 相似文献
Let ηi, i=1,…,n, be iid Bernoulli random variables, taking values ±1 with probability . Given a multiset V of n integers v1,…,vn, we define the concentration probability as A classical result of Littlewood–Offord and Erd?s from the 1940s asserts that, if the vi are non-zero, then ρ(V) is O(n−1/2). Since then, many researchers have obtained improved bounds by assuming various extra restrictions on V.About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the inverse Littlewood–Offord problem. In the inverse problem, one would like to characterize the set V, given that ρ(V) is relatively large.In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen the previous result of Tao and Vu, obtaining an optimal characterization for V. This immediately implies several classical theorems, such as those of Sárközy and Szemerédi and Halász.The method also applies to the continuous setting and leads to a simple proof for the β-net theorem of Tao and Vu, which plays a key role in their recent studies of random matrices.All results extend to the general case when V is a subset of an abelian torsion-free group, and ηi are independent variables satisfying some weak conditions. 相似文献
We study the algebraic independence of values of the Ramanujan q-series $A_{2j+1}(q)=\sum_{n=1}^{\infty}n^{2j+1}q^{2n}/(1-q^{2n})$ or S2j+1(q) (j≥0). It is proved that, for any distinct positive integers i, j satisfying $(i,j)\not=(1,3)$ and for any $q\in \overline{ \mathbb{Q}}$ with 0<|q|<1, the numbers A1(q), A2i+1(q), A2j+1(q) are algebraically independent over $\overline{ \mathbb{Q}}$ . Furthermore, the q-series A2i+1(q) and A2j+1(q) are algebraically dependent over $\overline{ \mathbb{Q}}(q)$ if and only if (i,j)=(1,3). 相似文献
Let i be an i-tb population with a probability density function f(· | i) with one dimensional unknown parameter i = 1, 2, ... , k. Let ni sample be drawn from each i. The likelihood ratio criteria j|(j–1) for testing hypothesis that the first j parameters are equal against alternative hypothesis that the first (j – 1) parameters are equal and the j-th parameter is different with the previous ones are defined, j = 2, 3, ... , k. The paper shows the asymptotic independence of j|(j–1)'s up to the order 1/n under a hypothesis of equality of k parameters, where n is a number of total samples. 相似文献
Let {Vi,j;(i,j)∈ℕ2} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent
random products
In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x2 = y3 = 1. Since PSL(2,q) is a factor group of Gk,l,m if -1 is a quadratic residue in the finite field Fq, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40. 相似文献
Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdXj(t)=ndBj(t) –DjØn(X1,...,Xn)dt,j=1, 2,...,n. Here theBjare independent Brownian motions in d, and Øn(X1,...,Xn)=nijV(XiXj) + niU(X1). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x2/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444 相似文献
A real polynomial of one real variable is (strictly) hyperbolic if it has only real (and distinct) roots. There are 10 (resp. 116) possible non-degenerate configurations between the roots of a strictly hyperbolic polynomial of degree 4 (resp. 5) and of its derivatives (i.e., configurations without equalities between roots). The standard Rolle theorem allows 12 (resp. 286) such configurations. The result is based on the study of the hyperbolicity domain of the family P(x,a)=xn+a1xn-1+...+an for n=4,5 (i.e., of the set of values of an for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P(i),P(j))=0, 0 i < jn-1. 相似文献
We consider P(G is connected) when G is a graph with vertex set Z+ = {1,2, …}, and the edge between i and j is present with probability p(i, j) = min(λ h(i, j), 1) for certain functions h(i, j) homogeneous of degree -1. It is known that there is a critical value λc of λ such that . We show that the probability, at the critical point λc, that n1, and n2 are connected satisfies a power law, in the sense that for n2 ≧ nt ≧ 1 for any δ > 0 and certain constants c1 and c2. 相似文献
Summary LetX be a non-negative random variable with probability distribution functionF. SupposeXi,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(Xi+1,n−Xi,n) and (n−j)(Xj+1,n−Xj,n) for somei, j andn, (1≦i<j<n).
The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil. 相似文献