Parity considerations for drops in cycles on {1,2,…,n}

In 2019, A. Lazar and M. L. Wachs conjectured that the number of cycles on $[2n]$ with only even-odd drops equals the n-th Genocchi number. In this paper, we restrict our attention to a subset of cycles on $[n]$ such that in all drops in the cycle, the latter entry is odd. We deduce two bivariate generating functions for such a subset of cycles with an extra variable introduced to count the number of odd-odd and even-odd drops, respectively. One of the generating function identities confirms Lazar and Wachs' conjecture, while the other identity implies that the number of cycles on $[2n?1]$ with only odd-odd drops equals the $(n?2)$-th Genocchi median.     

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本文利用构造鞅的方法, 研究了Cayley树图上奇偶马氏链场的强极限定理, 给出了Cayley树图上奇偶马氏链场关于状态和状态序偶出现频率的强大数定律, 推广了一个已知结果.     

Cycles Avoiding a Color in Colorful Graphs

The Ramsey numbers of cycles imply that every 2‐edge‐colored complete graph on n vertices contains monochromatic cycles of all lengths between 4 and at least . We generalize this result to colors by showing that every k‐edge‐colored complete graph on vertices contains ‐edge‐colored cycles of all lengths between 3 and at least .     

Genocchi polynomials associated with the Umbral algebra

The aim of this paper is to study on the Genocchi polynomials of higher order on P, the algebra of polynomials in the single variable x over the field C of characteristic zero and P^∗, the vector spaces of all linear functional on P. By using the action of a linear functional L on a polynomial p(x) Sheffer sequences and Appell sequences, we obtain some fundamental properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order.     

Computations of form and stability of rotating drops with finite elements

^** Email: claus{at}mathematik.uni-freiburg.de We consider the numerical computation of equilibrium shapesof rotating drops and their bifurcations, depending on the angularvelocity. The drops are subject to centrifugal forces and surfacetension alone. We present a path-tracking algorithm which isbased on the discretisation of a continuously formulated Newton'siteration. Our algorithm uses parametric finite elements andavoids artificial symmetry assumptions. The results of our numericalexperiments extend those published by Brown & Scriven (1980,The shape and stability of rotating liquid drops. Proc. R. Soc.Lond. Ser. A, 371, 331– 357) and cover drops of spheroidalas well as annular shape.     

Orthogonal polynomials on the unit circle associated with the Laguerre polynomials

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.

     

Shortest Path Problem on a Network with Imprecise Edge Weight

A network with its arc lengths as imprecise number, instead of a real number, namely, interval number and triangular fuzzy number is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. To obtain a fuzzy shortest path from a source vertex to all other vertices, a common algorithm is developed which works well on both types of imprecise numbers under consideration. In the proposed algorithm, a decision-maker is to negotiate with the obtained fuzzy shortest paths according to his/her view only when the means are same but the widths are different of the obtained paths. Otherwise, a fuzzy optimal path is obtained to which the decision-maker always satisfies with different grades of satisfaction. All pairs fuzzy shortest paths can be found by repeated use of the proposed algorithm.     

Cycles with a chord in dense graphs

A cycle of order $k$ is called a $k$-cycle. A non-induced cycle is called a chorded cycle. Let $n$ be an integer with $n\ge 4$. Then a graph $G$ of order $n$ is chorded pancyclic if $G$ contains a chorded $k$-cycle for every integer $k$ with $4\le k\le n$. Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order $n$ satisfying ${deg}_{G}u+{deg}_{G}v\ge n$ for every pair of nonadjacent vertices $u$,  $v$ in $G$ is chorded pancyclic unless $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, the Cartesian product of $K_3$ and $K_2$. They also conjectured that if $G$ is Hamiltonian, we can replace the degree sum condition with the weaker density condition $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ and still guarantee the same conclusion. In this paper, we prove this conjecture by showing that if a graph $G$ of order $n$ with $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ contains a $k$-cycle, then $G$ contains a chorded $k$-cycle, unless $k=4$ and $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, Then observing that $K_{\frac{n}{2},\frac{n}{2}}$ and $K_3\square K_2$ are exceptions only for $k=4$, we further relax the density condition for sufficiently large $k$.     

带弦的圈是协调图

称具有ｅ条边的简单图Ｇ为协调图，若存在由Ｇ的顶点集到模ｅ的整数群Ｚｅ的一个单射ｈ，使得导出映射ｈ＾＊：ｈ＾＊（ｕｖ）≡ｈ（ｕ）＋ｈ（ｖ）（ｍｏｄ ｅ）是一个由Ｇ的边集到Ｚｅ的双射，带弦的圈Ｃ′ｎ是由含ｎ个顶点的圈Ｃｎ上添一条连结两个不相邻顶点的边而得到的图。本文中证明了，除了ｎ＝６且弦端点在Ｃｎ上的距离为２的情况外，所有带弦的圈都是协调图。     

Some Polynomial Systems Associated with a Certain Family of Differential Operators

The authors aim at presenting several (presumably new) classes of linear, bilinear, and mixed multilateral generating functions for some general systems of polynomials which are defined by means of a certain family of differential operators. Some of the generating functions considered here are associated with the Stirling numbers of the second kind. Many (known or new) consequences and applications of the results obtained in this paper are also indicated.     

Asymptotic Approximation with a Sequence of Positive Linear Operators

The rate of pointwise convergence for a sequence of positive linear operators L_n approximating continuous functions on a finite interval is considered. The complete asymptotic expansion for the operators L_n as n tends to infinity is presented. It turns out that the central factorial numbers of first and second kind play an important role in the asymptotic expansion. The present work is an extension to that reported by Ivan and Raa.     

带弦有向圈的2级合成

考察带一条弦的有向圈的2级合成图,得到了它们强连通的充要条件,这些结果被用于判定布尔矩阵组合合成的本原性。     

Cycles embedding on folded hypercubes with faulty nodes

Let F_Fv$F{F}_v$ be the set of faulty nodes in an n$n$-dimensional folded hypercube F_Qn$F{Q}_n$ with |F_Fv|≤n−2$\left|F{F}_v\right|\le n-2$. In this paper, we show that if n≥3$n\ge 3$, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every even length from 4$4$ to 2ⁿ−2|F_Fv|$2^n-2\left|F{F}_v\right|$, and if n≥2$n\ge 2$ and n$n$ is even, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every odd length from n+1$n+1$ to 2ⁿ−2|F_Fv|−1$2^n-2\left|F{F}_v\right|-1$.     

On the Distribution in Residue Classes of Integers with a Fixed Sum of Digits

We give an asymptotic formula for the distribution of those integers n in a residue class, such that n has a fixed sum of base-g digits, with some uniformity over the choice of the modulus and g. We then use this formula to solve the problem of I. Niven of giving an asymptotic formula for the distribution of those integers n divisible by the sum of their base-g digits. Our results also allow us to give a stronger form of a result of M. Olivier dealing with the distribution of integers with a given gcd with their sum of base-g digits.For our friend Jean-Louis Nicolas on his sixtieth birthdayResearch partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T029759, and “Balaton” French-Hungarian exchange program F-18/00.2000 Mathematics Subject Classification: Primary—11A63     

一类具有13个参数的7次系统的极限环分支

研究了一类具有幂零奇点的7次多项式微分系统的极限环分支与中心问题.借助于数学软件MATHEMATICA,推导出系统在原点的前14个拟Lyapunov常数,从而得到了系统的原点为中心的充要条件,证明了系统在3阶幂零奇点处可以分支出14个极限环,给出了7次李雅谱诺夫系统在3阶幂零奇点处的环性数的下界.     

Analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods

In this note we show the analytic solution of a class of fractional differential equations with variable coefficients by using operatorial methods. Taking inspiration from previous papers by Dattoli et al. [4], [5] and [6] about spectral properties of Laguerre derivative, we here generalize some of their results to fractional evolution equations. Besides that, we have two interesting generalized examples. One is about telegraph equation with time dependent coefficient. The other, that could be of some interest for realistic applications, is the fractional diffusion with a space-dependent diffusion coefficient.     

Existence of Limit Cycles for a Cubic Kolmogorov System with a Hyperbolic Solution

When we discuss some problems in bio-mathematics,we often meet cubicKolmogorov systems.For general type of Kolmogorov systems,the qualitative analysisis very difficult. Usually,one discusses some special type of cubic Kolmogorovsystems,for example,cubic Kolmogorov systems with an algebraic curve solution.Firstly,the existence oflimitcycles fora cubic Kolmogorov system with quadratic curvesolution was studied.In article[1 ] ,though the author proved the cubic Kolmogorovsystem with a solution…     

On the condition numbers associated with the polar factorization of a matrix

We are interested in the calculation of explicit formulae for the condition numbers of the two factors of the polar decomposition of a full rank real or complex m × n matrix A, where m ≥ n. We use a unified presentation that enables us to compute such condition numbers in the Frobenius norm, in cases where A is a square or a rectangular matrix subjected to real or complex perturbations. We denote by σ₁ (respectively σ _n) the largest (respectively smallest) singular value of A, and by K(A) = σ₁/σ _n the generalized condition number of A. Our main results are that the absolute condition number of the Hermitian polar factor is √2(1 + K(A)²)^1/2/(1 + K(A)) and that the absolute condition number of the unitary factor of a rectangular matrix is 1/σ _n. Copyright © 2000 John Wiley & Sons, Ltd.