The extent to which subsets are additively closed

Given a finite abelian group G (written additively), and a subset S of G, the size r(S) of the set may range between 0 and ²|S|, with the extremal values of r(S) corresponding to sum-free subsets and subgroups of G. In this paper, we consider the intermediate values which r(S) may take, particularly in the setting where G is Z/pZ under addition (p prime). We obtain various bounds and results. In the Z/pZ setting, this work may be viewed as a subset generalization of the Cauchy-Davenport Theorem.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

The extent to which triangular sub-patterns explain minimum rank

For any zero-nonzero pattern of a matrix, the minimum possible rank is at least the size of a sub-pattern that is permutation equivalent to a triangular pattern with nonzero diagonal. For certain numbers of rows and columns, the minimum rank of a pattern is k only when there is a k-by-k such triangle. Here, we complete the determination of such sizes by showing that an m-by-n pattern of minimum rank k must contain a k-triangle for m=5, k=4; m=6, k=5; and m=6, k=4. A table is given showing whether or not this happens for all m, n, k. In the process, a Schur complement approach to minimum rank is described and used, and simple ways to recognize the presence of triangles of sizes less than 7 are given.     

Stimulating student aesthetic response to mathematical problems by means of manipulating the extent of surprise

We investigated aesthetic responses of 60 middle school students as they engaged in a pair of similar looking geometry problems in one-on-one semi-structured interviews. The investigation was driven by three predictions. The first two predictions were about the association between the evaluative aesthetic response and surprise stemming from the solution to each problem. The third, main, prediction was that the problem with more surprising solution would be evaluated as more beautiful. The extent of surprise was manipulated by the order in which two problems were given. The third prediction came to be true in 90% of the cases, in which the first two predictions were fulfilled. The findings suggest that school students’ evaluative aesthetic response to mathematical problems can be stimulated in instructional settings. Implications for research and practice are drawn.     

Nonlinear scattering: The states which are close to a soliton

We assume that the nonlinear Schroedinger equation with sufficiently general nonlinearity admits solutions of the soliton type. The Cauchy problem with initial data close to a soliton is considered. We also assume that the linearization of the equation in the vicinity of the soliton possesses only a real spectrum. The main result claims that the asymptotic behavior of the solution as t→+∞ is given by the sum of a soliton with deformed parameters and a dispersive tail, i.e., a solution of the linear Schroedinger equation. The case of the minimal spectrum has been considered in the previous paper. Bibliography: 1 title. Dedicated to O. A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 38–50. Translated by G. S. Perelman     

Subrings which are closed with respect to taking the inverse

Let S be a subring of the ring R. We investigate the question of whether S∩U(R)=U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation.     

Proper maps which are Lipschitz α up to the boundary

In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cⁿ into the unit ball in C^N which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is C^k on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than C^N ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bⁿ → B^N that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C^∞ -boundary in Cⁿ, then we can find a map F: D → B^N, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f₁, …, F_N): Bⁿ → B^N that is Lipschitz α and f_N(z) = c(1 - Z₁)^1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bⁿ.     

Attractors which are homeomorphic to compact abelian groups

The roader should be warned that this term is used in the literature for a variety of different concepts     

Modules which are extending relative to module classes

Given a ring Rand a class χ of right R-modules, a right R-module Mcan be an extending module relative to χ in two different ways. Various general properties of such extending modules are given and, in case of specific classes of R-modules, we characterize them.     

Renewing textbooks to align with reformed curriculum in former colonies: Ugandan school mathematics textbooks

Several nations have reformed both their mathematics pedagogy and curriculum. The remaining challenge is to review teaching and learning resources to support the renewed pedagogy and curriculum. This paper responds to the following question: what pedagogy and curriculum are depicted in textbooks used in Uganda? Ugandan textbooks were analyzed in terms of mathematics content structure and genre, and presentation of written and non-written voice and looks (appearance). Whereas certain Ugandan mathematics textbooks used the narrative form and others chose to eliminate the use of extensive text, these textbooks include common characteristics such as spiral coverage of mathematics content. A few strides toward reform pedagogy, such as use of contexts familiar to learners in development of rules and concepts, were evinced among selected Uganda textbooks. More strides are needed in revising a majority of textbooks to align with the renewed curricula on certain aspects including integrating learning tools—digital and non-digital—within the textbook resources. A critical reflection on curriculum renewals adopted from other countries is needed when designing textbooks to match these renewals.     

Modules which are Weak Extending Relative to Module Classes

A module is extending relative to a module class in two different ways. The definition and general properties of such extending modules have been given in [3] and also in the case of specific classes of modules has been characterized. In this paper, we define weak extending modules in two different ways with respect to general classes of modules. We give the relations between extending and weak extending modules and characterize them. This revised version was published online in June 2006 with corrections to the Cover Date.     

On operations in which graphs are appended to trees

We consider the effects on the algebraic connectivity of various graphs when vertices and graphs are appended to the original graph. We begin by considering weighted trees and appending a single isolated vertex to it by adding an edge from the isolated vertex to some vertex in the tree. We then determine the possible set vertices in the tree that can yield the maximum change in algebraic connectivity under such an operation. We then discuss the changes in algebraic connectivity of a star when various graphs such as trees and complete graphs are appended to its pendant vertices.     

Optimal trajectories which are tangent to an affine distribution

In this work we study the trajectories which are tangent to an affine sub-bundle in the tangent bundle of a manifold and which minimize the “total energy”.We give some characterizations of such “regular” trajectories in terms of control theory and geometrical theory. We also build some sufficient conditions of existence for such curves. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)