Generating functions for certain types of permutations

Generating functions are obtained for certain types of permutations analogous to up-down and down-up permutations. In each case the generating function is a quotient of entire functions; the denominator in each case is φ₀²(x) ? φ₁(x)φ₃(x), where

$\text{φ}{}_{\mathrm{j}}\text{(x)=}\text{∑}\text{n=o}\text{∞}\text{x}{}^{\mathrm{4n+j}}\text{(4n+j)}\text{!.}$

     

Influence of interparticle interactions on the kinetics of self-assembly and mechanical strength of nanoparticulate aggregates

Computer simulations based on Discrete Element Method have been performed in order to investigate the influence of interparticle interactions on the kinetics of self-assembly and the mechanical strength of nanoparticle aggregates.Three different systems have been considered.In the first system the interaction between particles has been simulated using the JKR (Johnson,Kendall and Roberts) contact theory,while in the second and third systems the interaction between particles has been simulated using van der Waals and electrostatic forces respectively.In order to compare the mechanical behaviour of the three systems,the magnitude of the maximum attractive force between particles has been kept the same in all cases.However,the relationship between force and separation distance differs from case to case and thus,the range of the interparticle force.The results clearly indicate that as the range of the interparticle force increases,the self-assembly process is faster and the work required to produce the mechanical failure of the assemblies increases by more than one order of magnitude.     

Using Primary Source Projects to teach undergraduate mathematics content: analysis of instructor implementation and perceptions

ABSTRACT

This study investigates changes in instructor teaching tendencies, instructor’s perception of impact on student learning and dispositions, and methods of implementation of Primary Source Projects (PSPs). PSPs are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. In essence, they are a form of inquiry-based learning that incorporates the history of mathematics through original source texts. We provide an overview of results from two semesters of implementation reports and surveys administered at the beginning and end of the semester by instructors who implemented PSPs in their undergraduate mathematics classes.     

JT-60U装置温度分布剖面不变性的实验研究

在一系列H模放电条件下，建立了一个旨在研究等离子体温度分布剖面不变性的数据库。介绍了数据库建立过程中要解决的关键问题和所用软件，对等离子体温度分布剖面不变性及芯部约束与边缘参数的关系进行了研究。     

Enumeration of pairs of permutations

Let π = (a₁, a₂, …, a_n), ? = (b₁, b₂, …, b_n) be two permutations of $\text{Z}{}_{\mathrm{n}}\text{= {1, 2, …, n}}$. A rise of π is pair a_i, a_i+1 with a_i < a_i+1; a fall is a pair a_i, a_i+1 with a_i > a_i+1. Thus, for i = 1, 2, …, n ? 1, the two pairs a_i, a_i+1; b_i, b_i+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ω_n denotes the number of pairs with RR forbidden, it is proved that $\text{∑}{}^{\mathrm{\infty }}{}_0\text{ω}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}\text{=}\text{1}\text{?(z)}$, $\text{?(z) = ∑}{}^{\mathrm{\infty }}{}_0\text{(-1)}{}^{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}$. More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then $\text{1 + ∑}{}^{\mathrm{\infty }}{}_{\mathrm{n=1}}\text{z}{}^{\mathrm{n}}\text{n!n!}$$\text{∑}{}^{\mathrm{n?1}}{}_{\mathrm{k=0}}\text{ω(n, k)x}{}^{\mathrm{k}}\text{=}\text{(1 ? x)}\text{(?(z(1 ? x)) ? x)}$.     

Enumeration of pairs of sequences by rises,falls and levels

The paper is concerned with the enumeration of pairs of sequences with given specification according to rises, falls and levels. Thus there are nine possibilities RR, ..., LL. Generating functions in the general case are very complicated. However in a number of special cases simple explicit results are obtained.Supported in part by NSF grant GP-37924X.