The Existence of Partitioned Generalized Balanced Tournament Designs with Block Size 3

A generalized balanced tournament design, GBTD(n, k), defined on a kn-set V, is an arrangement of the blocks of a (kn, k, k – 1)-BIBD defined on V into an n × (kn – 1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. Suppose we can partition the columns of a GBTD(n, k) into k + 1 sets B1, B2,..., Bk + 1 where |Bi| = n for i = 1, 2,..., k – 2, |Bi| = n–1 for i = k – 1, k and |Bk+1| = 1 such that (1) every element of V occurs precisely once in each row and column of Bi for i = 1, 2,..., k – 2, and (2) every element of V occurs precisely once in each row and column of Bi Bk+1 for i = k – 1 and i = k. Then the GBTD(n, k) is called partitioned and we denote the design by PGBTD(n, k). The spectrum of GBTD(n, 3) has been completely determined. In this paper, we determine the spectrum of PGBTD(n,3) with, at present, a fairly small number of exceptions for n. This result is then used to establish the existence of a class of Kirkman squares in diagonal form.     

On the divisibility properties concerning sums of binomial coefficients

For any integer \(n> 1,\) we prove

$$\begin{aligned} 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(3k+1){2k\atopwithdelims ()k}^3(-8)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(6k+1){2k\atopwithdelims ()k}^3(-512)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(42k+5){2k\atopwithdelims ()k}^3 4096^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(20k^2+8k+1){2k\atopwithdelims ()k}^5(-4096)^{n-1-k}. \end{aligned}$$

The first three results confirm three divisibility properties on sums of binomial coefficients conjectured by Z.-W. Sun.

     

Spherical multipliers

It is proven in the paper that if functionf(x)L^p(Rⁿ), where 1/p> 1/2 + 1/(n + 1), then the restriction of the Fourier transform f() to the unit sphere S^n–1 lies in L²(S^n–1). As was shown by Fefferman [1], it follows from this that, when > (n –1)/(2(n + 1)), the Riesz-Bochner multiplier acts in LP(Rⁿ) if (n –1–2)/(2n) <1/p < (n + 1 + 2)/(2n).Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 105–112, January, 1978.The author wishes to thank B. S. Mityagin for his attention to this work.     

Singular Links of Almost Metastable Dimensions

The objects studied are singular links of spheres of dimensions p₁,..., p_r, and p in the n-sphere. A theory of such singular links for the case where max{p₁,..., p_r} < min{2n/3 − 1, n − (p + 5)/3} is constructed. The theory generalizes (as far as it is possible) the theory of singular links of spheres of dimensions k,..., k, and p in the (2k + 1)-sphere, where k > 1, developed in the author's recent papers. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 287–294.     

Gauss Sum of Index 4： （1） Cyclic Case

Let p be a prime, m ≥ 2, and （m,p（p - 1）） = 1. In this paper, we will calculate explicitly the Gauss sum G（X） = ∑x∈F＊qX（x）ζ^Tp^（x） in the case of [（Z/mZ）＊ ： （p）] = 4, and -1 （不属于） （p）, where q P^f, f =φ（m）/4, X is a multiplicative character of Fq with order m, and T is the trace map for Fq/Fp. Under the assumptions [（Z/mZ）＊ ： （p）] = 4 and 1（不属于） （p）, the decomposition field of p in the cyclotomic field Q（ζm） is an imaginary quartic （abelian） field. And G（X） is an integer in K. We deal with the case where K is cyclic in this oaDer and leave the non-cvclic case to the next paper.     

Estimates of the independence number of a hypergraph and the Ryser conjecture

For the Legendre-Sobolev orthonormal polynomials depending on the parametersM≥0,N≥0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials forM>0,N>0 decay at the rate ofn ^−3/2 at the points 1 and −1. The values of are calculated. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 871–880, December, 1997. Translated by N. K. Kulman     

Several identities in the Catalan triangle

In this paper, we first establish several identities for the alternating sums in the Catalan triangle whose (n, p) entry is defined by B _{n, p} = $ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) $ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) . Second, we show that the Catalan triangle matrix C can be factorized by C = FY = ZF, where F is the Fibonacci matrix. From these formulas, some interesting identities involving B _{n, p} and the Fibonacci numbers F _n are given. As special cases, some new relationships between the well-known Catalan numbers C _n and the Fibonacci numbers are obtained, for example:

Connectivity of cages

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer v such there exists a (k; g)-graph with v vertices. A (k; g)-cage is a (k; g)-graph with f(k; g) vertices. In this paper we prove that the cages are monotonic in that f(k; g₁) < f(k; g₂) for all k ≥ 3 and 3 ≥ g₁ < g₂. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k. © 1997 John Wiley & Sons, Inc.     

Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the d-dimensional lattice ℤ^d. We find a sufficient condition for the positivity of a family of operators h(k) appearing after the “separation of the center of mass” of a system of two particles depending on the values of the total quasimomentum k ∈ T^d (where T^d is a d-dimensional torus). We use the obtained result to show that the operator h(k) has positive eigenvalues for nonzero k ∈ T^d. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 381–387, December, 2007.     

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

Cubic elliptic functions

The function

具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

该文首先研究具有脉冲的线性Dirichlet边值问题 $\left\{ \begin{array}{ll} x'(t)+a(t)x(t)=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m) $ 给出该Dirichlet边值问题仅有零解的两个充分条件, 其中$a:[0,T]\rightarrow R$, $c_{k}, d_{k}, k=1,2,$ $\cdots,m$是常数, 该文首先研究具有脉冲的线性Dirichlet边值问题 $$\left\{ \begin{array}{ll} x'(t)+a(t)x(t)=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m) $$ 给出该Dirichlet边值问题仅有零解的两个充分条件, 其中$a:[0,T]\rightarrow R$, $c_{k}, d_{k}, k=1,2,$ $\cdots,m$是常数, $0<\tau_{1}<\tau_{2}\cdots<\tau_{m}<T$为脉冲时刻. 其次利用上面的线性边值问题仅有零解这个性质和Leray-Schauder度理论, 研究具有脉冲的非线性Dirichlet边值问题 $$\left\{ \begin{array}{ll} x'(t)+f(t,x(t))=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=I_{k}(x(\tau_{k})), \ \Delta x'(\tau_{k})=M_{k}(x(\tau_{k})), \ x(0)=x(T)=0 \end{array} \right. (k=1,2\cdots,m) $$ 解的存在性和唯一性, 其中 $f\in C([0,T]\times R,R)$, $I_{k},M_{k}\in C(R, R),k=1,2,\cdots,m$. 该文主要定理的一个推论将经典的Lyaponov不等式比较完美地推广到脉冲系统.     

On the Derived Series of Coxeter Groups

Let G be a group and let φ(G) be the least integer k such that G^(k) = G^(k+1). If no such k exists, then φ(G) = ∞ and we write G ∈ 𝒰. We are interested in the questions which Coxeter groups are in 𝒰 and how large can finite φ(G) be for Coxeter groups. The second author answered these questions for 3-generator and 4-generator Coxeter groups. This article begins the study for the 5-generator case.     

微分多项式的零点及其相关的正规定则

对于平面区域D上的亚纯函数族F,F中的每个函数的极点重数至少为k,零点重数至少为s.设a,b为两个有限复数a≠0.若对于F中的每对函数f(z),g(z)∈F,f~((k))-af~3和g~((k))-ag~3分担b,则F在区域D内正规,其中k是正整数,k≥2.当k=2,有s=3;当k≥3时,有s=k.     

An Lp-Lq-Version of Morgan’s Theorem for the Dunkl-Bessel Transform

In this paper, we give an L^p-L^q-version of Morgans theorem for the Dunkl-Bessel transform on More precisely, we prove that for all and then for all measurable function f on the conditions and imply f = 0, if and only if where are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004     

Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.     

Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate as . We also state the equations of the curves on which the zeros of , lie asymptotically as . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica. This revised version was published online in June 2006 with corrections to the Cover Date.     

P<Subscript>4k-1</Subscript>-factorization of complete bipartite graphs

Let K_m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P_v-factorization of K_m,n is a set of edge-disjoint P_v-factors of K_m,n which partition the set of edges of K_m,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of P_v-factorization of K_m,n. When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P_4k-1-factorization of K_m,n is (1) (2k - 1)m ⩽ 2kn, (2) (2k - 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k - 1), (4) (4k - 1)mn/[2(2k - 1)(m + n)] is an integer.     

Curved Hexagonal Packings of Equal Disks in a Circle

For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and one of the 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called ``billiards' simulation algorithm. A curved hexagonal packing pattern is invariant under a rotation. For , the density (covering fraction) of curved hexagonal packings tends to . The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present the new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness' of the curved hexagonal pattern for k ≤ 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality. Received May 15, 1995, and in revised form March 5, 1996.     

An inequality for steiner systems II

It is proved that if a Steiner systemS(t, k, v), witht < k < v, hast odd andv > (t + 1)(k – t + 1), thenv (t + 2)(k – t + 1) (Theorem 1). The caset is even is also considered and a characterization ofS(t, t + 1, 2t + 3), S(t, t + 1, 2t + 5), S(t, t + 2, 3t + 6) andS(t, t + 2, 3t + 7) is given (Theorem 2).