Profile polytopes of some classes of families

The profile vector of a family F of subsets of an n-element set is (f ₀,f ₁,…,f _n ) where f _i denotes the number of the i-element members of F. The extreme points of the set of profile vectors for some class of families has long been studied. In this paper we introduce the notion of k-antichainpair families and determine the extreme points of the set of profile vectors of these families, extending results of Engel and P.L. Erd?s regarding extreme points of the set of profile vectors of intersecting, co-intersecting Sperner families. Using this result we determine the extreme points of the set of profile vectors for some other classes of families, including complement-free k-Sperner families and self-complementary k-Sperner families. We determine the maximum cardinality of intersecting k-Sperner families, generalizing a classical result of Milner from k = 1.     

有限酉群作用下子空间轨道按和生成的格

设F_q2⁽ⁿ⁾是 F_q2上的 n 维行向量空间, U_n( F_q2)是 F_q2上的 n 阶酉群. 设M(m, r; n)是U_n(F_q2}作用下的一个子空间轨道, L(m, r; n)是 M (m, r; n)中子空间的和生成的集合.该文讨论了各个轨道生成的集合L(m, r; n)之间的包含关系, 给出了一个子空间是属于给定的由M(m, r; n)生成的集合L(m, r, n)中的一个元素的条件, 以及L}(m, r; n)做成几何格的条件.     

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U₀=0, U₁=1, Un=PUn−1−QUn−2 (n?2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,…,7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,…,12, the only non-degenerate sequences where gcd(P,Q)=1 and Un(P,Q)=□, are given by U₈(1,−4)=²¹², U₈(4,−17)=⁶²⁰², and U₁₂(1,−1)=¹²².     

A Katona-type proof of an Erd?s-Ko-Rado-type theorem

Let p?1/2 and let μ_p be the product measure on {0,1}ⁿ, where μ_p(x)=p^∑x_i(1-p)^n-∑x_i. Let A⊂{0,1}ⁿ be an intersecting family, i.e. for every x,y∈A there exists 1?i?n such that x_i=y_i=1. Then μ_p(A)?p. Our proof uses a probabilistic trick first applied by Katona to prove the Erd?s-Ko-Rado theorem.     

Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r₁,…, r_m) and S = (s₁,…, s_n) be nonnegative integral vectors, and let $\text{U}$(R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of $\text{U}$(R, S) is a position whose entry is the same for all matrices in $\text{U}$(R, S). The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{U}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r_i ≤ n ? 1 (i = 1,…, m) and 1 ≤ s_j ≤ m ? 1 (j = 1,…, n), G(R, S) is prime if and only if $\text{U}$(R, S) has no invariant positions.     

U(n) Wigner coefficients,the path sum formula,and invariant G-functions

We prove the path sum formula for computing the U(n) invariant denominator functions associated to stretched U(n) Wigner operators. A family of U(n) invariant polynomials G_[λ]⁽ⁿ⁾ is then defined which generalize the _μG_q⁽ⁿ⁾ polynomials previously studied. The G_[λ]⁽ⁿ⁾ polynomials are shown to satisfy a number of difference equations and have symmetry properties similar to the _μG_q⁽ⁿ⁾ polynomials. We also give a direct proof of the important transposition symmetry for the G_[λ]⁽ⁿ⁾ polynomials. To enable the non-specialist to understand the foundations for these remarkable polynomials, we provide an exposition of the boson calculus and the construction of the multiplicity-free U(n) Wigner operators.     

On the combinatorics of Galois numbers

We define interval decompositions of the lattice of subspaces of a finite-dimensional vector space. We show that such a decomposition exists if and only if there exists a family of linear forms with certain properties. As applications we prove that all finite-dimensional real vector spaces admit an interval decomposition, while GF(2)ⁿ has an interval decomposition if and only if n≤4. On the other hand, we present an interval decomposition of GF(3)⁵. This partially answers a question of Faigle and Kruse (2004) [1] and [4].     

On Samelson products in p-localized unitary groups

The unitary group U(n) has elements εi∈π2i+1(U(n)) (0?i?n−1) of its homotopy groups in the stable range. In this paper we show that certain multi Samelson products of type 〈εi,〈εj,εk〉〉 are non-trivial. This leads us to the result that the nilpotency class of the group of the self homotopy set [SU(n),SU(n)] is no less than 3, if 4?n. Also by the power of generalized Samelson products, we can see the further result that, for a prime p and an integer n=pk, nil[SU(n),SU(n)]_(p)?3, if (1) p?7 or (2) p=5 and n≡0 or 1mod4.     

The universality of the quantum Fourier transform in forming the basis of quantum computing algorithms

The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(·), both of which are 2×2 unitary matrices as operators on the two-dimensional 1-qubit space. In this paper, we show that H and P(·) suffice to generate the unitary group U(2) and, consequently, through controlled-U operations and their concatenations, the entire unitary group U(2ⁿ) on n qubits can be generated. Since any quantum computing algorithm in an n-qubit quantum computer is based on operations by matrices in U(2ⁿ), in this sense we have the universality of the QFT.     

Some simple in-place merging algorithms

The theoretical presentation and analysis is given for two families of simple in-place merging algorithms and their limiting cases. The first family merges stably inO(k·n) time andO(n ^1/k ) additional space with a limiting case running inO(n logn) time and constant space. The second family merges unstably inO (k ·n) time andO(log ^k n) space with a limiting case running inO(nG(n)) time and constant space. HereG(n) is the leastk such thatF(k) n whereF(0)=1 andF(i)=2 ^F(i–1) fori1. Each algorithm gives rise to a corresponding merge sort.     

Padé-type approximants with orthogonal generating polynomials

The interpolation of the function x → 1/(1 ? xt) generating the series f(t) = ∑^∞_{i = 0}c_itⁱ at the zeros of an orthogonal polynomial with respect to a distribution d α satisfying some conditions will give us a process for accelerating the convergence of f_n(t) = ∑ⁿ_{i = 0}c_itⁱ. Then, we shall see that the polynomial of best approximation of x → 1/(1 ? xt) over some interval or its development in Chebyshev polynomials T_n or U_n are only particular cases of the main theorem.At last, we shall show that all these processes accelerate linear combinations with positive coefficients of totally monotonic and oscillating sequences.     

Generation of classical groups over finite fields

Let U_n(V) and Sp_n(V) denote the unitary group and the symplectic group of the n dimensional vector space V over a finite field of characteristic not 2, respectively. Assume that the hyperbolic rank of U_n(V) is at least one. Then U_n(V) is generated by 4 elements and Sp_n(V) by 3 elements. Further, U_2m+1(V) is generated by 3 elements and Sp_4m(V) by 2 elements.     

Pointwise bounded approximation and Dirichlet algebras

Those open sets U of S² for which A(U) is pointwise boundedly dense in H^∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in C_R(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H^∞(U).     

Approximation by light maps and parametric Lelek maps

The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈AP(n,0) if for every ε>0 and a map g:In→M there exists a 0-dimensional map g^′:In→M which is ε-homotopic to g. It is shown that this class has very nice properties. For example, if Mi∈AP(ni,0), i=1,2, then M₁×M₂∈AP(n₁+n₂,0). Moreover, M∈AP(n,0) if and only if each point of M has a local base of neighborhoods U with U∈AP(n,0). Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of n-dimensional Lelek maps.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

Modification of a Finsler space by a normalized semi-parallel vector field

LetF _n be a Finsler space with metric functionF(x, y). M. Matsumoto [6] has defined a modified Finsler spaceF _n ^* whose metric functionF ^*(x, y) is given byF ^*2 = = F² + (X_i(x)yⁱ)², whereX _i are the components of a covariant vector which is a function of coordintae only. Since a concurrent vector is a function of coordinate only, Matsumoto and Eguchi [9] have studied various properties of the modified Finsler spaceF _n ^* under the assumption thatX _i are the components of a concurrent vector field inF _n. In this paper we shall introduce the concept of semi-parallel vector field inF _n and study the properties of modified Finsler spaceF _n ^* .     

The bitwisted Cartesian model for the free loop fibration

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration ΩY→ΛY→Y is obtained. As an application we establish an algebra isomorphism H^∗(ΛY;Z)≈S(U)⊗Λ(s−1U) for the polynomial cohomology algebra H^∗(Y;Z)=S(U).     

ADMISSIBLE WAVELETS ASSOCIATED WITH THE TRANSFORM GROUP ON R

Let p be the transform group on R, then P has a natural unitary representation U onL2 (R^n). Decompose L2(R^n) into the direct sum of irreducible invariant closed subspace,s. The re-striction of U on these suhspaces is square-intagrable. In this paper the characterization of admissi-ble condition in tarrns of the Fourier transform is given. The wavelet transform is defined, and theorthogorml direct sum decomposition of function space L2 (P,du1) is obtained.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

Nombres de ramsey dans le cas orienté

Given two directed graphs G₁, G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any partition {U₁,U₂} of the arcs of the complete symmetric directed graph K¹_n, there exists an integer i such that the partial graph generated by U_i contains G_i as a subgraph. In this article, we determine R(P?_m,D?_n) and R(D?_m,D?_n) for some values of m and n, where P?_m denotes the directed path having m vertices and D?_m is obtained from P?_m by adding an arc from the initial vertex of P?_m to the terminal vertex.