Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

Finiteness of integrals of functions of Levy processes

We prove necessary and sufficient conditions for the almostsure convergence of the integrals

and thus of ,where M_t = sup{|X_s|: s t} is the two-sided maximum processcorresponding to a Lévy process (X_t)_{t 0}, a(·)is a non-decreasing function on [0, ) with a(0) = 0, g(·)is a positive non-increasing function on (0, ), possibly withg(0 + ) = , and f(·) is a positive non-decreasing functionon [0, ) with f(0) = 0. The conditions are expressed in termsof the canonical measure, (·), of the process X_t. Thespecial case when a(x) = 0, f(x) = x and g(·) is equivalentto the tail of (at zero or infinity) leads to an interestingcomparison of M_t with the largest jump of X_t in (0, t]. Some results concerning the convergence at zero and infinityof integrals like t g(a(t) + |X_t|) dt, t g(S_t) dt,and t g(R_t) dt, where S_t is the supremum process and R_t= S_t – X_t is the process reflected in its supremum, arealso given. We also consider the convergence of integrals suchas , etc.     

Rates of Convergence of Means of Euclidean Functionals

Let L be the Euclidean functional with p-th power-weighted edges. Examples include the sum of the p-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (Ann. Appl. Probab. 4, 1057–1073, 1994, Stoch. Process. Appl. 61, 289–304, 1996) have shown that for n i.i.d. sample points {X ₁,…,X _n } from [0,1] ^d , L({X ₁,…,X _n })/n ^(d−p)/d converges a.s. to a finite constant. Here we bound the rate of convergence of EL({X ₁,…,X _n })/n ^(d−p)/d . Y. Koo supported by the BK21 project of the Department of Mathematics, Sungkyunkwan University. S. Lee supported by the BK21 project of the Department of Mathematics, Yonsei University.     

Filtration of components of processes of random evolution

The problem of estimation of a nonobservable component θ_t for a two-dimensional process (θ_t, ξ_t) of random evolution (θ _t,ξ_t);x_t, 0≤t≤T, is investigated on the basis of observations of ξ_s. s≤t, where x _t is a homogeneous Markov process with infinitesimal operator Q. Applications to stochastic models of a (B,S)-market of securities is described under conditions of incomplete market. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1701–1705, December, 1998.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

Systems of sets of lengths of Puiseux monoids

In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of ${\mathbb{Q}}_{\ge 0}$). We begin by presenting a BF-monoid M with full system of sets of lengths, which means that for each subset S of ${\mathbb{Z}}_{\ge 2}$ there exists an element $x\in M$ whose set of lengths $\mathsf{L}(x)$ is S. It is well known that systems of sets of lengths do not characterize numerical monoids. Here, we prove that systems of sets of lengths do not characterize non-finitely generated atomic Puiseux monoids. In a recent paper, Geroldinger and Schmid found the intersection of systems of sets of lengths of numerical monoids. Motivated by this, we extend their result to the setting of atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux monoid $P=〈1/p|p\text{is prime}〉$ with the Goldbach's conjecture; in particular, we show that $\mathsf{L}(2)$ is precisely the set of Goldbach's numbers.     

Rotational analysis of some of the bands of the orange system of FeO

The orange bands of FeO are excited in a low pressure arc in oxygen and photographed at dispersions of 0·8 and 0·5 Å/mm. respectively. Rotational analysis of five of the bands shows that they involve a¹ Σ-¹ Σ transition. The vibrational and rotational constants (in cm.^?1) of the upper and lowerΣ states are found to be

	Lower state	Upper state
B e	0·3491	0·3063
α e	0·0029	0·0032
w e	870·7	(ΔG½=770·0)
w e x e	4·39	..

     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.     

Minima of initial segments of infinite sequences of reals

Suppose that 〈x_k〉_k∈? is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA₀ suffices to prove the existence of a sequence of reals 〈u_k〉_k∈? such that for each k, u_k is the minimum of {x₀, x₁, …, x_k}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈x_k〉_k∈?, the stronger subsystem WKL₀ is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Methods of geometry of differential equations in analysis of integrable models of field theory

In this paper, we investigate algebraic and geometric properties of hyperbolic Toda equations u _xy = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogues of the potential modified Korteweg-de Vries equation u _t = u _xxx + u ₃ ^x is constructed and its relationship with the hierarchy for the Korteweg-de Vries equation T _t = T _xxx + TT _x is established. Group-theoretic structures for the dispersionless (2 + 1)-dimensional Toda equation u _xy = exp(?u _zz ) are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems Ψ_t = iΨ_xx + i f(|Ψ|) Ψ (multi-soliton complexes) are described.     

The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces

Letf be aC ^r diffeomorphism,r≥2, of a two dimensional manifoldM ², and let Λ be a horseshoe off (i.e. a transitive and isolated hyperbolic set with topological dimension zero). We prove that there exist aC ^r neighborhoodU off and a neighbourhoodU of Λ such that forg∈U, the Hausdorff dimension of ∩ _n g ⁿ (U) is aC ^r−1 function ofg.     

Limits of chromatic zeros of some families of maps

Two families of maps are considered, one consisting of maps with two pentagons separated by n 5-rings, the other of maps with two n-gons separated by two n-rings. For each family, a homogeneous linear recursion is derived for the corresponding family of chromatic polynomials. It is shown that B₅, B₇, and B₁₀ are limits of sequences of zeros from one or another of the families, where $\text{B}{}_{\mathrm{n}}\text{= 2(1+}\text{cos}\text{2π}\text{n}\text{)}$.     

Polarizations of Prym varieties of pairs of coverings

To any pair of coverings f_i:X → X_i, i=1, 2, of smooth projective curves one can associate an abelian subvariety of the Jacobian J_X, the Prym variety P(f₁, f₂) of the pair (f₁, f₂). In some cases we can compute the type of the restriction of the canonical principal polarization of J_X. We obtain 2 families of Prym-Tyurin varieties of exponent 6. Received: 2 September 2004