Restricted signed permutations counted by the Schröder numbers

Gire, West, and Kremer have found ten classes of restricted permutations counted by the large Schröder numbers, no two of which are trivially Wilf-equivalent. In this paper we enumerate eleven classes of restricted signed permutations counted by the large Schröder numbers, no two of which are trivially Wilf-equivalent. We obtain five of these enumerations by elementary methods, five by displaying isomorphisms with the classical Schröder generating tree, and one by giving an isomorphism with a new Schröder generating tree. When combined with a result of Egge and a computer search, this completes the classification of restricted signed permutations counted by the large Schröder numbers in which the set of restrictions consists of two patterns of length 2 and two of length 3.     

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.     

关于乘法分拆数的一个猜想和上界

本文证明了乘法分拆数的一个上界,由此证明了Hughes-Shallit的第二猜想,同时证明了对任意的正数a,存在一个自然数N,当n≥N时,n的乘法分拆数f(n)0,使这个集合中的自然数的乘法分拆数≤n~a。     

On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of a finite poset

Let P be a finite poset and H(P) be the hypergraph whose vertices are the points of P and whose edges are the maximal intervals in P. We study the domatic number d(G(P)) and the total domatic number d_t(G(P)) of the 2-section graph G(P) of H(P). For the subset P_l,u of P induced by consecutive levels of P, we give exact values of d(G(P_l,u)) when P is the chain product C_n1×C_n2. According to the values of l,u,n₁,n₂, the maximal domatic partition is exhibited. Moreover, we give some exact values or lower bounds for d(G(P∗Q)) and d_t(G(P_l,u)), when ∗ is the direct sum, the linear sum or the Cartesian product. Finally we show that the domatic number and the total domatic number problems in this class of graphs are NP-complete.     

A note on the weakly convex and convex domination numbers of a torus

The distanced_G(u,v) between two vertices u and v in a connected graph G is the length of the shortest (u,v) path in G. A (u,v) path of length d_G(u,v) is called a (u,v)-geodesic. A set X⊆V is called weakly convex in G if for every two vertices a,b∈X, exists an (a,b)-geodesic, all of whose vertices belong to X. A set X is convex in G if for all a,b∈X all vertices from every (a,b)-geodesic belong to X. The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G, while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G. In this paper we consider weakly convex and convex domination numbers of tori.     

Permutation Polyhedra and Minimisation of the Variance of Completion Times on a Single Machine

We consider the problem of minimising variance of completion times when n-jobs are to be processed on a single machine. This problem is known as the CTV problem. The problem has been shown to be difficult. In this paper we consider the polytope P _n whose vertices are in one-to-one correspondence with the n! permutations of the processing times [p ₁, p ₂, ..., p _n]. Thus each vertex of P _n represents a sequence in which the n-jobs can be processed. We define a V-shaped local optimal solution to the CTV problem to be the V-shaped sequence of jobs corresponding to which the variance of completion times is smaller than for all the sequences adjacent to it on P _n. We show that this local solution dominates V-shaped feasible solutions of the order of 2 ^n–3 where 2 ^n–2 is the total number of V-shaped feasible solutions.Using adjacency structure an P _n, we develop a heuristic for the CTV problem which has a running time of low polynomial order, which is exact for n 10, and whose domination number is (2 ^n–3). In the end we mention two other classes of scheduling problems for which also ADJACENT provides solutions with the same domination number as for the CTV problem.     

A parametric total order on fuzzy numbers and a fuzzy shortest route problem

A parametric total order relation is introduced in the form of a certain modification of the "fuzzy max" order on the class of fuzzy numbers generated by a shape function. By the parametric relation. fuzzy numbers unordered with respect of the fuzzy max order can be ordered according either to their value of center or the size of ambiguity. A fuzzy shortest route problem in which are distances are given by fuzzy numbers is discussed under the criterion of the parametric total order, and solved by the dynamic programming approach. A method is proposed to find all of fuzzy routes mimmal in the sense of the fuzzy max order.     

A theorem on planar continua and an application to automorphisms of the field of complex numbers

Let K be a continuum in the plane which does not lie on a line. Then the set of differences, K - K, contains an open set. Let ψ be an automorphism of the field of complex numbers which is bounded on an F_σ set of positive inductive dimension. Then ψ is continuous.     

Action of the group of classes on representations of numbers by a ternary quadratic form: The formulas of Siegel and Kneser-Hsia-Peters

We specify in explicit form the action of the group of classes of binary quadratic forms of determinant dm on the set of primitive representations of the number m by a ternary quadratic form. As a consequence, we obtain an elementary proof of Siegel's formulas for the weighted number of representations by a genus and formulas for the weighted number of representations by a spinor genus of ternary forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 141–158, 1986.     

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

Variations of the complex structure of Riemann surfaces by vector fields on a contour and objects of the KP theory. The Krichever-Novikov problem of the action on the Baker-Akhieser functions

L. D. Landau Theoretical Physics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 1, pp. 72–73, January–March, 1990.     

The strong law of large numbers and the Shannon–McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree

In this paper, we study the strong law of large numbers and the Shannon–McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree. This article generalizes the relative results of level nonhomogeneous Markov chains indexed by a Cayley tree.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .      

从流体通过曲面的流量看高斯公式

采用类比和元素法的思想,通过流体流过曲面的流量推测高斯公式的形成过程,并通过一个具体的例子,验证高斯公式的正确性和有效性.     

A note on the empty balls left by a critical branching Wiener process

Summary In this note, we partially confirm some conjectures of P. Révész [10] on the critical branching Wiener process.     

外p-自同构对p-群生成元个数的一个限制

阶为某素数p的方幂的非内自同构,称为外p-自同构.研究了有限p-群G的任意外p-自同构φ满足|C_G(φ)|≤p^k时群G生成元的个数.     

A Note on a Paper by P. Amodio and F. Mazzia

In 1999 Amodio and Mazzia presented a new backward error analysis for LU factorization and introduced a new growth factor _n . Their very interesting approach allowed them to obtain sharp error bounds. In particular, they derive nice results assuming that partial pivoting is used. However, the forward error bound for the solution of a linear system whose coefficient matrix A is an M-atrix given in Theorem 4.1 of that paper is not correct. They first obtain a bound for the condition number (U) assuming that one has the LU factorization of an M-matrix and then they apply the bounds obtained when partial pivoting is used. But if P is the permutation associated with partial pivoting then PA = LU can fail to be an M-atrix and the bound for (U) can be false, as shown in our Example 1.1. We also prove that, for a pivoting strategy presented in the paper, the growth factor of an M-matrix A is _n(A) = 1 and (U) (A), where U is the upper triangular matrix obtained after applying such a pivoting strategy.This revised version was published online in October 2005 with corrections to the Cover Date.