Investigating secondary students beliefs about mathematical problem-solving

Many studies over the past 30 years have highlighted the important role of students’ beliefs for successful problem-solving in mathematics. Given the recent emphasis afforded to problem-solving on the reformed Irish secondary school mathematics curriculum, the main aim of this study was to identify Irish students’ (n = 975) beliefs about the field. A quantitative measure of these beliefs was attained through the use of the Indiana Mathematical Belief Scale, an existing 30-item (five-scale) self-report questionnaire. A statistical analysis of the data revealed that students who were further through their secondary education had a stronger belief that not all problems could be solved by applying routine procedures. In contrast, the same students held less positive beliefs than their younger counterparts that they could solve time-consuming problems and that conceptual understanding was important. The analysis also indicated that gender had a significant impact on three of the five belief scales.     

LABELLING THE CHARACTER TABLES OF SYMMETRIC AND ALTERNATING GROUPS

Let X be a character table of the symmetric group S_n. It isshown that unless n = 4 or n = 6, there is a unique way to assignpartitions of n to the rows and columns of X so that for all and , X is equal to (), the value of the irreducible characterof S_n labelled by on elements of cycle type . Analogous resultsare proved for alternating groups, and for the Brauer charactertables of symmetric and alternating groups.     

A solvable group whose character degree graph has diameter

We show that there is a solvable group so that the character degree graph of has diameter .

     

Comparison theorems and orbit counting in hyperbolic geometry

In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both ``thermodynamic' ergodic theory and the automaton associated to strongly Markov groups.

     

Linkage and sums of ideals

It is shown (under mild conditions) that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection. For a sum of geometrically linked Cohen-Macaulay ideals, we compute the depths of the conormal module and the first Koszul homology. As applications, we construct general examples of ideals which are strongly Cohen-Macaulay, strongly nonobstructed but not in the linkage class of a complete intersection, and Gorenstein ideals which are strongly nonobstructed but not syzygetic.

     

Short separating geodesics for multiply connected domains

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We prove that, although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible geodesics which separate the complement in this fashion.     

Hysteresis effects under CIR interest rates

Most decision making research in real options focuses on revenue uncertainty assuming discount rates remain constant. However, for many decisions revenue or cost streams are relatively static and investment is driven by interest rate uncertainty, for example the decision to invest in durable machinery and equipment. Using interest rate models from Cox et al. (1985b), we generalize the work of Ingersoll and Ross (1992) in two ways. Firstly, we include real options on perpetuities (in addition to zero coupon cash flows). Secondly, we incorporate abandonment or disinvestment as well as investment options, and thus model interest rate hysteresis (parallel to revenue uncertainty in Dixit (1989a)). Under stochastic interest rates, economic hysteresis is found to be significant, even for small sunk costs.     

A randomized algorithm for the decomposition of matrices

Given an $m\times n$ matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying $A^{\mathrm{T}}$ to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and $A^{\mathrm{T}}$ can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as $\sqrt{lm}$ times the $(k+1)$st greatest singular value ${\sigma }_{k+1}$ of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when $l?k$ is a fixed small nonnegative integer. For example, according to one of our estimates for $l?k=20$, the probability that the spectral norm $6A?Z6$ is greater than $10\sqrt{(k+20)m}{\sigma }_{k+1}$ is less than ${10}^{?17}$. The paper contains a number of estimates for $6A?Z6$, including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and $A^{\mathrm{T}}$ can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.     

A comparison of two dissimilarity functions for mixed-type predictor variables in the $$\delta $$ -machine

Advances in Data Analysis and Classification - The $$\delta $$ -machine is a statistical learning tool for classification based on dissimilarities or distances between profiles of the observations...