Riffs on the infinite ping-pong ball conundrum

This article presents a novel re-conceptualisation to a well-known problem – The Ping-Pong Ball Conundrum. We introduce a variant of this super-task by considering it through the lens of ‘measuring infinity’ – a conceptualisation of infinity that extrapolates measuring properties of numbers, rather than cardinal properties. This approach is consistent with a nonstandard analysis approach to infinite numbers, and gives credence to the intuitive (but otherwise normatively incorrect) resolution. We explore the mathematical motivation and consequences of this variant, as well as offer further ‘riffs’ on the infinite ball problem for consideration.     

Hilbert's Grand Hotel with a series twist

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her own variations.     

Hilbert's objectivity

Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly “meaningless” signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions ( and ). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates Hilbert's account of mathematical objectivity, axiomatics, idealization, and consistency.     

Hilbert's “Grundlagen der Geometrie”

A careful analysis is made, in the light of subsequent developments, of the background, content, aim, and influence of the ten German editions of Hilbert's “Grundlagen der Geometrie”. Especial attention is given to the sources of Hilbert's ideas, the critical reactions of contemporary geometers to the first edition, the latter's connections with Hilbert's famous problems, and its role as a model for later axiomatic approaches tho the foundations of mathematics and mathematical physics.     

Mathematics education graduate students’ understanding of trigonometric ratios

This study describes mathematics education graduate students’ understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students’ understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students’ responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students’ concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students’ understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.     

Hilbert's metric and the analytic hierarchy process

This paper explores some of the properties of Hilbert's projective metric as a measure of closeness between two ratio scales in the context of the Analytic Hierarchy Process. Smallperturbation arguments are used to contrast the sensitivity and the distributional behavior of this metric with the more traditional Euclidean distance function, in situations where the paired comparison of alternatives is subject to random perturbations, and priorities are estimated either by Saaty's eigenvalue method or by the logarithmic least squares principle. A pivotal property of Hilbert's metric has surfaced which allows for the construction of confidence regions for an underlying priority vector. These regions are seen to enjoy good coverage properties.     

Anti-affine algebraic groups

We say that an algebraic group G over a field is anti-affine if every regular function on G is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem.     

Alice Bache Gould: mathematician in search of war work, 1918

Alliances between American mathematics and the military were not well defined when the United States entered World War I in April of 1917. Although academic mathematicians – like other American faculty members and students – were among the strongest supporters of the cause, it took time to identify how and where to utilize their technical training in the war effort. This paper investigates the case of Alice Bach Gould, one of E H Moore's mathematics graduate students at the University of Chicago, and her efforts to support US military endeavors with her scientific expertise. Although Gould's substantial research accomplishments in Spanish archives eventually surpassed her mathematical achievements, her quest for war work in 1917 nonetheless illustrates the difficulty of contributing mathematical training to patriotic service in the United States during World War I.     

A Special Topics Course in Perturbation Methods

A special topics course dealing with perturbation methods in applied mathematics which was recently taught at Stevens Institute of Technology is described. Since the course enrolment was comprised of undergraduate and graduate mathematics students as well as graduate students in engineering, an unusual course philosophy had to be developed and implemented. The approach taken to achieve this in both teaching techniques and selection of subject matter is discussed in detail. Finally, conclusions are drawn from the students’ and the instructor's experiences with the course which hopefully will be of value to those considering offering special topics courses in the future.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

A Finite Separating Set for Daigle and Freudenburg's Counterexample to Hilbert's Fourteenth Problem

This article gives the first explicit example of a finite separating set in an invariant ring which is not finitely generated, namely, for Daigle and Freudenburg's 5-dimensional counterexample to Hilbert's Fourteenth Problem.     

Factorization of Cesàro and Hilbert matrices based on generalized Cesàro matrix

ABSTRACT

In this paper, we introduce factorizations for the Hilbert and Cesàro matrices based on generalized Cesàro matrix and as the application, we generalize the Hardy's inequality versus Hilbert's.     

Sometimes binary is better

This note uses material involving perfect numbers and Zeno's paradoxes to show that although most students prefer to use base 10 when working with mathematical concepts there are times when the binary system is best.     

Mathematical paradoxes as pathways into beliefs and polymathy: an experimental inquiry

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.     

The Peeking Order of the Elite: America's Leading Professional Schools

Advanced professional education is assuming ever-increasing importance in our post-industrial society. Enrollment in American colleges and universities has doubled every 15 years since 1870. The expansion of graduate and professional training has been even more rapid than that of undergraduate education While the number of undergraduates today is 10 times as large as it was half a century ago, the number of graduate students is no less than 50 times as large Of the 9.2 million college students today, over 1 million are now engaged in postgraduate work, with the number of PhD degrees awarded rising at a faster percentage rate than master's degrees, which in turn are increasing faster than the first bachelor's degree.     

Equivalent property of a more accurate half-discrete Hilbert's inequality

By using the weight functions, the idea of introducing parameters, and Hermite-Hadamard''s inequality, a more accurate half-discrete Hilbert''s inequality with the nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the best possible constant factor related to parameters, the operator expressions and some particular cases are considered. The cases of the relating homogeneous kernel are also deduced.     

Extension of Hilbert's tenth problem to some algebraic number fields

We extend the solution of Hilbert's tenth problem to algebraic number fields having one pair of complex conjugated embeddings. The proof is based on the extended method of J. Denef used for totally real algebraic number fields.     

Factors contributing to academic achievement: a Bayesian structure equation modelling study

In Iran, high school graduates enter university after taking a very difficult entrance exam called the Konkoor. Therefore, only the top-performing students are admitted by universities to continue their bachelor's education in statistics. Surprisingly, statistically, most of such students fall into the following categories: (1) do not succeed in their education despite their excellent performance on the Konkoor and in high school; (2) graduate with a grade point average (GPA) that is considerably lower than their high school GPA; (3) continue their master's education in majors other than statistics and (4) try to find jobs unrelated to statistics. This article employs the well-known and powerful statistical technique, the Bayesian structural equation modelling (SEM), to study the academic success of recent graduates who have studied statistics at Shahid Beheshti University in Iran. This research: (i) considered academic success as a latent variable, which was measured by GPA and other academic success (see below) of students in the target population; (ii) employed the Bayesian SEM, which works properly for small sample sizes and ordinal variables; (iii), which is taken from the literature, developed five main factors that affected academic success and (iv) considered several standard psychological tests and measured characteristics such as ‘self-esteem’ and ‘anxiety’. We then study the impact of such factors on the academic success of the target population. Six factors that positively impact student academic success were identified in the following order of relative impact (from greatest to least): ‘Teaching–Evaluation’, ‘Learner’, ‘Environment’, ‘Family’, ‘Curriculum’ and ‘Teaching Knowledge’. Particularly, influential variables within each factor have also been noted.     

Analyzing students’ communication and representation of mathematical fluency during group tasks

Findings discussed in this paper are from a larger research project exploring mathematical fluency characteristics, and teacher noticing and interpreting of mathematical fluency. The current study involved students from seven primary classes (Kindergarten – Grade 6, N = 63 students) and investigated students’ written work samples and oral discussions as they collaborated in small groups to solve mathematical tasks. Students displayed mathematical fluency both orally and in written/drawn form. Certain aspects of mathematical fluency were easier to identify orally (adaptive reasoning) particularly for younger students and when students did not provide any written reasoning. Analyzing the oral responses was often needed to identify mathematical fluency beyond knowledge of a correct procedure (strategic competence). Findings suggested that the various representations students used were valuable for observing mathematical fluency. These results suggest that oral assessments as a means to understand and interpret students’ mathematical fluency are necessary.     

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.