Congruences for broken 3-diamond and 7 dots bracelet partitions

Andrews and Paule introduced broken k-diamond partitions by using MacMahon’s partition analysis. Later, Fu found a generalization which he called k dots bracelet partitions. In this paper, with the aid of Farkas and Kra’s partition theorem and a p-dissection identity of f(?q), we derive many congruences for broken 3-diamond partitions and 7 dots bracelet partitions.     

Coordinating Theories of Learning to Account for Second-Grade Children's Arithmetical Understandings

This article coordinates social constructivism and socioculturalism orientations to explain 2nd-grade children's reasoning with 2-digit quantities. From a social constructivist position, we illustrate how the classroom teacher and the students constituted what counted as an acceptable mathematical explanation. As children offered informal and conventional ways of interpreting problem situations, they were expected to reason with quantities in sensible ways. From a sociocultural position, we explain how the teacher's and students' contributions were situated within the mathematical ways of knowing constituted by the community at large. Particular children's contributions were clarified in terms of the ways in which they participated in socially organized activities. By coordinating these lenses, we argue the local classroom mathematical practices constrained and enabled the mathematical practices of the wider society.     

On filtering and estimation of a threshold stochastic volatility model

We derive a nonlinear filter and the corresponding filter-based estimates for a threshold autoregressive stochastic volatility (TARSV) model. Using the technique of a reference probability measure, we derive a nonlinear filter for the hidden volatility and related quantities. The filter-based estimates for the unknown parameters are then obtained from the EM algorithm.     

Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n₁,…,n_d, and let L denote the lattice of points (a₁,…,a_d)∈Z^d that satisfy 0≤a_i≤n_i for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting n_i=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.     

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C₀(X) into C₀(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ_n δ ?h_n for a (possibly finite) sequence {x_n }_n of distinct points in X and a norm null sequence {h_n }_n of mutually disjoint functions in C₀(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaos control of new Ikeda–Lorenz systems by GYC partial region stability theory

A new strategy to achieve chaos control by GYC partial region stability theory is proposed. By using the GYC partial region stability theory, the Lyapunov function is a simple linear homogeneous function of error states, the controllers are more simple and have less simulation error because they are in lower degree than that of traditional controllers. Simulation results for a new Ikeda–Lorenz system show the effectiveness of this strategy. Copyright © 2008 John Wiley & Sons, Ltd.     

The Relationship Between Stock Markets Of Major Developed Countries And Asian Emerging Markets

With the emergence of new capital markets and liberalization of stock markets in recent years, there has been an increase in investors' interest in international diversification. This is so because international diversification allows investors to have a larger basket of foreign securities to choose from as part of their portfolio assets, so as to enhance the reward-to-volatility ratio. This benefit would be limited if national equity markets tend to move together in the long run. This paper thus studies the issue of co-movement between stock markets in major developed countries and those in Asian emerging markets using the concept of cointegration. We find that there is co-movement between some of the developed and emerging markets, but some emerging markets do differ from the developed markets with which they share a long-run equilibrium relationship. Furthermore, it has been observed that there has been increasing interdependence between most of the developed and emerging markets since the 1987 Stock Market Crash. This interdependence intensified after the 1997 Asian Financial Crisis. With this phenomenon of increasing co-movement between developed and emerging stock markets, the benefits of international diversification become limited.     

An experiment to discover mathematical talent in a primary school in Kampong Air

It is often said that many pupils have hidden talent in mathematics. This hidden ability is rarely seen in a normal classroom teaching and learning situation if the focus of the teacher is on drilling with routine exercises. To allow pupils to display their mathematical talent and to break from mental set and fixation in mathematics, they must be given opportunity to think by themselves with mininum cue or guidance. The pupils could be left entirely on their own to show their mathematical creativity even on mathematical topics which have not been exposed to them. With this approach, five non-routine questions were administered one at a time to a standard 5 class. One out of the 25 pupils in the class consistently exhibited mathematical creativity and talent is answering the questions. Her responses were shown and discussed in this paper.