On Rubel’s Problem in the Class of Linear Operators on the Space of Analytic Functions

In this paper, we describe all derivation pairs of linear operators that act in spaces of functions analytic in domains.     

Coordinating Numeric and Linear Units: Elementary Students’ Strategies for Locating Whole Numbers on the Number Line

Two investigations of fifth graders’ strategies for locating whole numbers on number lines revealed patterns in students’ coordination of numeric and linear units. In Study 1, we investigated the effects of context on students’ placements of three numbers on an open number line. For one group (n?=?24), the line was presented in a thematic context as a “race course,” and, for a second group (n?=?24), the line was presented as a conventional number line. Most students in both groups placed consecutive whole numbers at appropriate linear distances, but the thematic context group was more likely to place nonconsecutive whole numbers at appropriate linear distances. In Study 2 (n?=?24), students placed numbers on lines marked with two numbers. Most students placed a third number appropriately when the marked numbers were consecutive whole numbers, but not when the labeled numbers were nonconsecutive whole numbers. The findings reveal fifth graders’ conceptual difficulties in coordinating numeric and linear units on the number line and a thematic context that can support this coordination.     

A Necessary and Sufficient Condition on Convergence of Bounded Linear Operators in the Space of Continuous Functions

Korovkin[1]gave a necessary and sufficient condition on lineax positiveapproximation in C[a,b].In this note,we give such a condition on general boundedapproximation in C[a,b]     

Influence of the multiplicity of the roots on the basins of attraction of Newton’s method

In this work, we develop and implement two algorithms for plotting and computing the measure of the basins of attraction of rational maps defined on the Riemann sphere. These algorithms are based on the subdivisions of a cubical decomposition of a sphere and they have been made by using different computational environments. As an application, we study the basins of attraction of the fixed points of the rational functions obtained when Newton’s method is applied to a polynomial with two roots of multiplicities m and n. We focus our attention on the analysis of the influence of the multiplicities m and n on the measure of the two basins of attraction. As a consequence of the numerical results given in this work, we conclude that, if m > n, the probability that a point in the Riemann Sphere belongs to the basin of the root with multiplicity m is bigger than the other case. In addition, if n is fixed and m tends to infinity, the probability of reaching the root with multiplicity n tends to zero.     

An analytical comparison of students’ reasoning in the context of Inquiry-Oriented Instruction: The case of span and linear independence

In this report we analyze differences in reasoning about span and linear independence by comparing written work of 126 linear algebra students whose instructors received support to implement a particular inquiry-oriented (IO) instructional approach compared to 129 students whose instructors did not receive that support. Our analysis of students’ responses to open-ended questions indicated that IO students’ concept images of span and linear independence were more aligned with the formal concept definition than the concept images of Non-IO students. Additionally, IO students exhibited more coordinated conceptual understandings and used deductive reasoning at higher rates than Non-IO students. We provide illustrative examples of systematic differences in how students from the two groups reasoned about span and linear independence.     

What Do Middle and High School Students Know About the Particulate Nature of Matter After Instruction? Implications for Practice

This study explores middle and high school students' understanding of the particulate nature of matter after they were taught the concept. A total of 87 students (41 high school and 46 middle school) participated in the study. Findings suggest that students held misconceptions about the law of conservation of matter, chemical composition of matter in different phases of matter, the process of condensation, and behaviors of molecules at a microscopic level. The discussion focuses on the implications of these findings for enhancing science teachers' pedagogical content knowledge.     

An extension of Motohashi’s observation on the zero-free region of the Riemann zeta-function

We give an extension of Yoichi Motohashi’s theorem saying that if the Riemann zeta-function on the line Re s = 1 attains very small values, then Vinogradov’s zero-free region can be improved.     

Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities

Contributing to a growing body of research addressing secondary students’ quantitative and covariational reasoning, the multiple case study reported in this article investigated secondary students’ quantification of ratio and rate. This article reports results from a study investigating students’ quantification of rate and ratio as relationships between quantities and presents the Change in Covarying Quantities Framework, which builds from Carlson, Jacobs, Coe, Larsen, and Hsu’s (2002) Covariation Framework. Each of the students in this study was consistent in terms of the quantitative operation he or she used (comparison or coordination) when quantifying both ratio and rate. Illustrating how students can engage in different quantitative operations when quantifying rate, the Change in Covarying Quantities Framework helps to explain why students classified as operating at a particular level of covariational reasoning appear to be using different mental actions. Implications of this research include recommendations for designing instructional tasks to foster students’ quantitative and covariational reasoning.     

A Generalization of Schep’s Theorem on the Positive Definiteness of a Piecewise Linear Function

Mathematical Notes - Schep proved that, for a piecewise linear function with nodes at integer points, positive definiteness on ? is equivalent to positive definiteness on ?. In this...     

An Answer to S. Simons’ Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification—that is, whether or not “the sum theorem” is true—is the most famous open problem in Monotone Operator Theory. In his 2008 monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons’ question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.      

Students’ conceptions of mathematics as sensible: Towards the SCOMAS framework

In this article I describe the development of a framework for considering students’ conceptions about the sensible nature of mathematics. I begin by using extant literature on conceptions of mathematics to develop a framework of action-oriented indicators that students’ conceive of mathematics as sensible. I then use classroom data to modify and illustrate the framework. The result is a coding framework, grounded in the literature, which can be used to assess the enacted conceptions of mathematics as sensible of a group of students. This work also provides a conceptual framework, grounded in classroom data, of the dimensions of these conceptions.     

Linear transformations on matrices: the invariance of generalized permutation matrices—III

Let F be a field, F1 be its multiplicative group, and $\text{H}$ = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let D_n be the dihedral group of degree n, H be a nontrivial group in $\text{H}$, and τ_n(H) = {α = (α₁, α₂,…, α_n):α_i?H}. For σ?D_n and α?τ_n(H), let P(σ, α) be the matrix whose (i,j) entry is α_iδ_iσ(j) (i.e., a generalized permutation matrix), and

$\text{P(D}{}_{\mathrm{n}}\text{, H) = {P(σ, α):σ?D}{}_{\mathrm{n}}\text{, α?τ}{}_{\mathrm{n}}\text{(H)}}$

. Let M_n(F) be the vector space of all n×n matrices over F and $\text{T}$P(D_n, H) = {T:T is a linear transformation on M_n (F) to itself and T(P(D_n, H)) = P(D_n, H)}. In this paper we classify all T in $\text{T}$P(D_n, H) and determine the structure of the group $\text{T}$P(D_n, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper.     

An Exponential Growth Learning Trajectory: Students’ Emerging Understanding of Exponential Growth Through Covariation

This article presents an Exponential Growth Learning Trajectory (EGLT), a trajectory identifying and characterizing middle grade students’ initial and developing understanding of exponential growth as a result of an instructional emphasis on covariation. The EGLT explicates students’ thinking and learning over time in relation to a set of tasks and activities developed to engender a view of exponential growth as a relation between two continuously covarying quantities. Developed out of two teaching experiments with early adolescents, the EGLT identifies three major stages of students’ conceptual development: prefunctional reasoning, the covariation view, and the correspondence view. The learning trajectory is presented along with three individual students’ progressions through the trajectory as a way to illustrate the variation present in how the participants made sense of ideas about exponential growth.     

Modeling the Influence of Public’s Memory on the Corruption–Popularity Dilemma in Politics

We consider the problem of a rational politician who gains benefit from both being popular and corrupt. In 1994, Feichtinger and Wirl studied this trade-off by means of an infinite-horizon optimal control approach. We reconsider the problem over a finite time horizon, to model the dilemma of a politician who stays in office for a limited period of time and wishes to be reelected. We also include explicitly the possibility that awareness of politician’s conduct to the general population is delayed and the effect of such delay on the politician’s optimal behavior is analyzed. We show the outcomes under several different scenarios, with particular reference to the role of the politician’s communication skills. One general conclusion is that the synergy of a good public’s memory and immediate knowledge of corruption may often help the people to protect themselves from political corruption. Moreover, this synergy may greatly enhance the popularity of a politician with poor communication skills, provided that he/she aims at maximizing benefit from popularity during his/her mandate.     

Calculus of Generalized Quasi-differentiable Functions I: Some Results on the Space of Pairs of Convex-Set Collections

A Riesz space K1 whose elements are pairs of convex-set collections is presented for the study on the calculus of generalized quasi-differentiable functions. The space K1 is constructed by introducing a well-defined equivalence relation among pairs of collections of convex sets. Some important properties on the norm and operations in K1 are given.     

Integer comparisons across the grades: Students’ justifications and ways of reasoning

This study is an investigation of students’ reasoning about integer comparisons—a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g., −5 > − 6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students’ justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students’ ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.     

An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions

We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.     

An advanced study on the solution of nanofluid flow problems via Adomian’s method

Nanofluid flow is one of the most important areas of research at the present time due to its wide and significant applications in industry and several scientific fields. The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with boundary conditions at infinity. These boundary conditions at infinity cause difficulties for any of the series method, such as Adomian’s method, the variational iteration method and others.The objective of the present work is to introduce a reliable method to overcome such difficulties that arise due to an infinite domain. The proposed scheme, that we will introduce, is based on Adomian’s decomposition method, where we will solve a system of nonlinear differential equations describing the boundary layer flow of a nanofluid past a stretching sheet.