Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls

10.37236/1733 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Edward A. Bender ◽  
William J. Helton ◽  
L. Bruce Richmond
Keyword(s):  
Functional Analysis ◽  
Integral Equation ◽  
Eigenvalue Problem ◽  
Asymptotic Formula ◽  
Fredholm Integral Equation ◽  
Random Variables ◽  
Periodic Patterns ◽  
Asymptotic Results ◽  
Alternating Permutations ◽  
Asymptotic Number

Ehrenborg obtained asymptotic results for nearly alternating permutations and conjectured an asymptotic formula for the number of permutations that have a nearly periodic run pattern. We prove a generalization of this conjecture, rederive the fact that the asymptotic number of permutations with a periodic run pattern has the form $Cr^{-n}\,n!$, and show how to compute the various constants. A reformulation in terms of iid random variables leads to an eigenvalue problem for a Fredholm integral equation. Tools from functional analysis establish the necessary properties.

scholarly journals The iterated projection solution for the Fredholm integral equation of second kind

1981 ◽  
Vol 22 (4) ◽  
pp. 439-451 ◽  
Author(s):  
Françoise Chatelin ◽  
Rachid Lebbar
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Fredholm Integral Equation ◽  
Mathematical Analysis ◽  
Orthogonal Projection ◽  
Inhomogeneous Equation ◽  
Function Type ◽  
Piecewise Polynomials ◽  
Gauss Points ◽  
Global Superconvergence

AbstractWe are concerned with the solution of the second kind Fredholm equation (and eigenvalue problem) by a projection method, where the projection is either an orthogonal projection on a set of piecewise polynomials or an interpolatory projection at the Gauss points of subintervals.We study these cases of superconvergence of the Sloan iterated solution: global superconvergence for a smooth kernel, and superconvergence at the partition points for a kernel of “Green's function” type. The mathematical analysis applies for the solution of the inhomogeneous equation as well as for an eigenvector.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

2011 ◽  
Vol 255-260 ◽  
pp. 1830-1835 ◽  
Author(s):  
Gang Cheng ◽  
Quan Cheng ◽  
Wei Dong Wang
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Free Vibration ◽  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
Integral Equation Method ◽  
Free Vibrations ◽  
Equation Method ◽  
Concentrated Mass

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

Sign in / Sign up

Export Citation Format

Share Document