Kp,q-factorization of complete bipartite graphs

Let K_(m,n) be a complete bipartite graph with two partite sets having m and nvertices, respectively. A K_(p,q)-factorization of K_(m,n) is a set of edge-disjoint K_(p,q)-factorsof K_(m,n) which partition the set of edges of K_(m,n). When p=i and q is a prime number,Wang, in his paper "On K_(1,k)-factorizations of a complete bipartite graph" (Discrete Math,1994, 126; 359-364), investigated the K_(1,q)-factorization of K_(m,n) and gave a sufficientcondition for such a factorization to exist. In the paper "K_(1,k)-factorizations of completebipartite graphs" (Discrete Math, 2002, 259: 301-306), Du and Wang extended Wang'sresult to the case that q is any positive integer In this paper, we give a sufficient conditionfor K_(m,n) to have a K_(p,q)-factorization. As a special case, it is shown that the Martin's BACconjecture is true when p: q=k: (k+1) for any positive integer k.     

{beta}-Informational Sufficient Partition

This paper deals with the construction of a ß-sufficientpartition of the sample space, in the sense that the relativeloss of information induced by the partition is equal to 1 –ß. In particular, the case ß = 1, i.e. asufficient partition, is considered. A necessary and sufficientcondition for a partition to be sufficient has been derived.Several illustrative examples are given, together with a comparativestudy of the relative loss of information when three differentinformation measures are used.     

Partitions of multisets

A multiset is a set with repeated elements. There are four distinct partition numbers to consider, unlike the classical set partition case which involves only Stirling numbers of the second kind. Using inclusion-exclusion, we obtain generating functions when each element appears exactly r = 1, 2 or 3 times. The case r = 1 is classical and r = 2 was studied by Comtet and Baróti using other methods. Our approach also leads to asymptotic formulae for the total number of partitions of multisets in which the repetition of elements is bounded. Another approach to multiset enumeration, using de Brujin's theorem for group reduced distributions, is described.     

α-可分解4-圈系统

An m-cycle system of order ν and index λ, denoted by rn-CS(ν, λ), is a collection of cycles of length m whose edges partition the edges of λκ_v. An m-CS(ν, λ) is α-resolvable if its cycles can be partitioned into classes such that each point of the design occurs in precisely α cycles in each class. The necessary conditions for the existence of such a design are m|λν(ν-1)/2,2|λ(ν-1), m|αν,α|λ(ν-1)/2 It is shown in this paper that these conditions are also sufficient when m = 4.     

CAUCHY微分中值定理的推广

设Δn:a=x0 相似文献   

The computational complexity of two‐state spin systems

The subject of this article is spin‐systems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hard‐core gas model. There are three degrees of freedom, corresponding to our parameters β, γ, and μ. Informally, β represents the weights of edges joining pairs of “spin blue” sites, γ represents the weight of edges joining pairs of “spin green” sites, and μ represents the weight of “spin green” sites. We study the complexity of (approximately) computing the partition function in terms of these parameters. We pay special attention to the symmetric case μ = 1. Exact computation of the partition function Z is NP‐hard except in the trivial case βγ = 1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time. We show that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the “ferromagnetic” region βγ ≥ 1, but (unless RP = NP) there is no FPRAS in the “antiferromagnetic” region corresponding to the square defined by 0 < β < 1 and 0 < γ < 1. Neither of these “natural” regions—neither the hyperbola nor the square—marks the boundary between tractable and intractable. In one direction, we provide an FPRAS for the partition function within a region which extends well away from the hyperbola. In the other direction, we exhibit two tiny, symmetric, intractable regions extending beyond the antiferromagnetic region. We also extend our results to the asymmetric case μ ≠ 1. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 133–154, 2003     

Partition graphs for finite symmetric groups

This paper outlines an investigation of a class of arc-transitive graphs admitting a finite symmetric group S_n acting primitively on vertices, with vertex-stabilizer isomorphic to the wreath product S_m wr S_r (preserving a partition of {1,2,…n} into r parts of equal size m). Several properties of these graphs are considered, including their correspondence with r × r matrices with constant row- and column-sums equal to m, their girth, and the local action of the vertex-stabilizer. Also, it is shown that the only instance where S_n acts transitively on 2-arcs occurs in the case m = r = 2. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 107–117, 1997     

The likelihood ratio criterion and the asymptotic expansion of its distribution

Summary Asymptotic expansion of the distribution of the likelihood ratio criterion (LRC) for testing a composite hypothesis is derived under null hypothesis and a correction factor ρ which makes the term of order 1/n in the asymptotic expansion of the distribution of it vanish is obtained. The problem is extended to the case of a general composite hypothesis and of Pitman's local alternatives. The asymptotic distribution of LRC for a simple hypothesis is studied under a fixed alternative. The Institute of Statistical Mathematics     

Analogues of Ramanujan’s partition identities

Ramanujan discovered that $$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$ where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for $p_{[c^{l}d^{m}]}(n)$ .     

Moment inequalities and the Riemann hypothesis

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫ ₀ ^∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments \(\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}\) satisfy the Turán inequalities (*) $$(\hat b_m (\lambda ))^2 > \left( {\frac{{2m - 1}}{{2m + 1}}} \right)\hat b_{m - 1} (\lambda )\hat b_{m + 1} (\lambda )(m \geqslant 1,\lambda \geqslant 0).$$ We give here a constructive proof that log \(\Phi (\sqrt t )\) is strictly concave for 0 <t < ∞, and with this we deduce in Theorem 2.4 a general class of moment inequalities which, as a special case, establishes that the inequalities (*) are in fact valid for all real λ. As the case λ=0 of (*) corresponds to the Pólya conjecture of 1927, this gives a new proof of the Pólya conjecture.     

P_(4k-1)-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,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 P4k-1-factorization of Km,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.     

On Eckhoff's conjecture for Radon numbers; or how far the proof is still away

Eckhoff's conjecture for the Τ-Radon numbers r(Τ) of a convexity space. (X,C) says r(Τ) ≦ (r?1)(Τ?1)+1, with r = r(2). The main result of this note is that Eckhoff's conjecture is true in case ¦X¦ ≦ 2r and Τ = 3, i.e. each (2r?1)-set in a space with 2r?1 or 2r elements has a 3-Radon partition.     

THE EXISTENCE AND STABILITY OF THE HETEROCLINIC CYCLE IN A KIND OF BIOLOGICAL SYSTEM

IIntroductlonConsider the n－spedes biological systemlit\1．＝IJ!厂．＋》*i,工上D、忍＝上,’··。n．ti)Ifffi＝1,it is S S－SpSCllS LOthaka－VoltOOYY SystSS．Iftti＝2,It Is S S－sPeCieSKolmongorov system．As to the n－spedes Gause－Lotb－Volterra system矿ti 乙工．＝T;Ii、y Qiil．I。Ti 7 U,on M U,t6)174 AnnofDiff Eqs．VO18M叫 and Leonard[1],Ho凡aner and Sigmund问 have studied this system forthe case n二 3 respectlvelyand noted that thereprobably exists aheterocllnlccyclefor…     

Cluster Partition Function and Invariants of 3-Manifolds

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G.The author focuses on the case of G =SL(N,C) and M being a knot complement:M =S3 \ k.The main result presented in this note is the cluster partition function,a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G =SL(N,C).He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.     

Loss of Smoothness and Inherent Instability of 2D Inviscid Fluid Flows

The paper is focused on the loss of smoothness hypothesis which claims that vorticity (or vorticity gradients in the 2D case) grows unboundedly for the substantial part of the inviscid incompressible flows. At least, every steady flow is supposed to belong to the closure of this set (relative to a reasonably strong topology). We approach the problem involving both direct Lyapunov method and some sort of the linearization. We present new (and rather wide) classes of 2D flows in a generic domains which admit the loss of smoothness and related phenomena.     

The new test criterion for the homogeneity of parameters of several populations

Summary The new test criterion for testing the homogeneity of parameters of several populations is proposed and the test properties of it is discussed. The asymptotic expansions of the distributions of test criterion are discussed under (i) null hypothesis, (ii) fixed alternative hypothesis and (iii) local alternative hypothesis converging to the null hypothesis with appropriate rate of convergence as the sample size increases. As a particular case the asymptotic theory of a statistic for a homogeneity of variances of normal populations is also discussed and the exact moments of it under a null hypothesis can be used to obtain a percentage point by a Pearsonian curve fitting. This Institute of statistical Mathematics     

Izergin-Korepin Determinant at a Third Root of Unity

We consider the partition function of the inhomogeneous six-vertex model defined on an n×n square lattice. This function depends on 2n spectral parameters x_i and y_i attached to the respective horizontal and vertical lines. In the case of the domain-wall boundary conditions, it is given by the Izergin-Korepin determinant. For q being an Nth root of unity, the partition function satisfies a special linear functional equation. This equation is particularly simple and useful when the crossing parameter is η = 2π/3, i.e., N = 3. It is well known, for example, that the partition function is symmetric in both the x and the y variables. Using the abovementioned equation, we find that in the case of η = 2π/3, it is symmetric in the union {x} ∪ {y}! In addition, this equation can be used to solve some of the problems related to enumerating alternating-sign matrices. In particular, we reproduce the refined alternating-sign matrix enumeration discovered by Mills, Robbins, and Rumsey and proved by Zeilberger, and we obtain formulas for the doubly refined enumeration of these matrices. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 65–76, January, 2006.     

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.     

q-形变Lévy-Meixner过程的重整化矩(英文)

本文研究了q-形变Lévy-Meixner分布的重整化矩问题.利用集合划分的方法,获得了这种重整化矩的显式表达式,并验证了它们与经典情形的一致性.     

P4κ-1-factorization of bipartite multigraphs

LetλKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pv-factorization of λKm,n is a set of edge-disjoint Pv-factors of λKm,n which partition the set of edges of λKm,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization of λKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v= 3. In this paper we will show that the conjecture is true when v= 4k- 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of λKm,n is (1) (2κ - 1)m ≤ 2kn, (2) (2k - 1)n ≤ 2km, (3) m + n ≡0 (mod 4κ - 1), (4) λ(4κ - 1)mn/[2(2κ - 1)(m + n)] is an integer.