Enumeration of three-dimensional convex polygons

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ ^d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ ^d are defined up to a translation, and the relevant statistic is their perimeter. A SAP on ℤ ^d is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension. We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤ ^d is always a quotient ofdifferentiably finite power series.     

Three enumeration formulas of standard Young tableaux of truncated shapes

In this paper we consider the enumeration of three kinds of standard Young tableaux (SYT) of truncated shapes by use of the method of multiple integrals. A product formula for the number of truncated shapes of the form (n^m, n ? r)\δ_k–1 is given, which implies that the number of SYT of truncated shape (n², 1)\(1) is the number of level steps in all 2-Motzkin paths. The number of SYT with three rows truncated by some boxes ((n + k)³)\(k) is discussed. Furthermore, the integral representation of the number of SYT of truncated shape (n^m)\(3, 2) is derived, which implies a simple formula of the number of SYT of truncated shape (nⁿ)\(3, 2).     

On a combinatorial problem related to permanents

Some numerical results will be presented concerning the largest and smallest values of square permanents containing the integers 1, 2, 3, ...,n ² n=1(1)8.Dedicated to Peter Naur on the occasion of his 60th birthday     

Norms and Generating Functions in Clifford Algebras

Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)⁻¹ as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t ² u ² + ...) = 1, provided the sum (1 + tu + t ² u ² + ...) exists, in which case u ^m is the Cliffordalgebraic coefficient of t ^m in the series expansion of (1 − tu)⁻¹. In this paper, conditions on for the existence of (1 − tu)⁻¹ are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)⁻¹ are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of A ^k . Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)⁻¹. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory.      

Maximum Number of Constant Weight Vertices of the Unit <Emphasis Type="Italic">n</Emphasis>-Cube Contained in a <Emphasis Type="Italic">k</Emphasis>-Dimensional Subspace

We introduce and solve a natural geometrical extremal problem. For the set E (n,w) = {x ⁿ {0,1} ⁿ : x ⁿ has w ones } of vertices of weight w in the unit cube of ⁿ we determine M (n,k,w) max{|U _k ⁿ E(n,w)|:U _k ⁿ is a k-dimensional subspace of ⁿ . We also present an extension to multi-sets and explain a connection to a higher dimensional Erds–Moser type problem.     

Maximal exponents of polyhedral cones (II)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rⁿ. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that the maximum value of γ(K) as K runs through all n-dimensional minimal cones (i.e., cones having n+1 extreme rays) is n²-n+1 if n is odd, and is n²-n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K-primitive matrices A such that γ(A) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.     

On the number of homogeneous subgraphs of a graph

Let us defineG(n) to be the maximum numberm such that every graph onn vertices contains at leastm homogeneous (i.e. complete or independent) subgraphs. Our main result is exp (0.7214 log² n) ≧G(n) ≧ exp (0.2275 log² n), the main tool is a Ramsey—Turán type theorem. We formulate a conjecture what supports Thomason’s conjecture R(k, k)^1/k = 2.     

Asymptotic enumeration ofN-free partial orders

LetN(n) andN ^* (n) denote, respectively, the number of unlabeled and labeledN-free posets withn elements. It is proved thatN(n)=2 ^{n logn+o(n logn)} andN ^*(n)=2^{2n logn+o(n logn)}. This is obtained by considering the class ofN-free interval posets which can be easily counted.     

Density results for uniform families

A familyF of subsets is calledk-dense if there exists ak-element setA such that all 2 ^k of its subsets can be obtained in the formAF for someFF. Best possible bounds are obtained for the maximum number of sets in notk-densek-partite families. This is a consequence of a new rank formula for inclusion matrices.     

AN EIGENSPACE CHARACTERIZATION OF LORENTZ-IMPROVING MEASURES ON COMPACT GROUPS

Abstract

A measure μ on a compact group is called Lorentz-improving if for some 1 > p > ∞ and 1 → q ₁ > q ₂ ∞ μ *L (p, q ₂) ? L(p, q ₁). Let T _μ denote the operator on L ² defined by T _μ(f) = μ * f. Lorentz-improving measures are characterized in terms of the eigenspaces of T _μ, if T _μ is a normal operator, and in terms of the eigenspaces of |T _μ| otherwise. This result generalizes our recent characterization of Lorentz-improving measures on compact abelian groups and is modelled after Hare's characterization of L ^p -improving measures on compact groups.     

含有快速增长非线性恢复力的梁方程的行波解

Abstract. On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:     

Some asymptotic results useful in enumeration problems

Suppose one is given two related generating functionsa(x) = a _n x ⁿ andb(x) = b _n x ⁿ , often it is of interest to determine the limiting behaviour of the quantitiesa _n /b _n We survey some earlier results of this nature and give some new ones     

Some asymptotic results useful in enumeration problems

Suppose one is given two related generating functionsa(x) = a _n x ⁿ andb(x) = b _n x ⁿ , often it is of interest to determine the limiting behaviour of the quantitiesa _n /b _n We survey some earlier results of this nature and give some new ones     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

On the homogeneous algebraic graphs of large girth and their applications

Families of finite graphs of large girth were introduced in classical extremal graph theory. One important theoretical result here is the upper bound on the maximal size of the graph with girth ?2d established in Even Circuit Theorem by P. Erdös. We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth ?2d is established. Getting the lower bound, we use the family of bipartite graphs D(n,K) with n?2 over a field K, whose partition sets are two copies of the vector space Kⁿ. We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory. Finally, we present a very short survey on applications of finite homogeneous algebraic graphs to coding theory and cryptography.     

The bias of three pseudo-random shuffles

Three schemes for shuffling a deck ofn cards are studied, each involving a random choice from [n] ⁿ . The shuffles favor some permutations over others sincen! does not dividen ⁿ . The probabilities that the shuffles lead to some simple permutations, for instance cycles left and right and the identity, are calculated. Some inequalities are obtained which lead to information about the least and most likely permutations. Numbers of combinatorial interest occur, notably the Catalan numbers and the Bell numbers.     

On products of additive functions (a third approach)

Summary Leta ₁, , a_s : G K be additive functions from an abelian groupG into a fieldK such thata ₁(g)··a_s(g) = 0 for allg G. If char(K) =0, then it is well known that one of the functions a₁ has to vanish. We give a new proof of this result and show that, if char(K) > 0, it is only valid under additional assumptions.     

SMALL RAMSEY NUMBERS II: CRITICAL COLOURINGS FOR r(C5 + e,K5)

Abstract

It is shown that if G is a graph of order 16 such that C₅+e ? G and K₅ ? [Gbar] then either 4K₄ ? G or G is one of 5 other graphs.     

Fcfs-scheduling in a hard real-time environment under rush-hour conditions

We investigate some real-time behaviour of a (discrete time) single server system with FCFS (first come first serve) task scheduling under rush-hour conditions. The main result deals with the probability distribution of a random variable SRD(T), which describes the time the system operates without violating a fixed task service time deadlineT.Relying on a simple general probability model, asymptotic formulas concerning the mean and the variance of SRD(T) are determined; for instance, if the average arrival rate is larger than the departure rate, the expectation of SRD(T) is proved to fulfilE[SRD(T)]=c ₁+O(T ^–3) forT, wherec ₁ denotes some constant. If the arrival rate equals the departure rate, we findE[SRD(T)]c ₂ T ⁱ for somei2.