Mathematical Issues Research Articles

The Sociopolitical Turn in Mathematics Education

Over the past decade, the mathematics education research community has incorporated more sociocultural perspectives into its ways of understanding and examining teaching and learning. However, researchers who have a long history of addressing anti-racism and social justice issues in mathematics have moved beyond this sociocultural view to espouse sociopolitical concepts and theories, highlighting identity and power at play. This article highlights some promising conceptual tools from critical theory (including critical race theory/Latcrit theory) and post-structuralism and makes an argument for why taking the sociopolitical turn is important for both researchers and practitioners. Potential benefits and challenges of this turn are also discussed.

Effects of white noise in multistable dynamics

We study the invariant measure of multistable dynamics under the influence of white noise. We show that the invariant measure exists and in the limit of vanishing white noise, the invariant measure approaches a Dirac type measure concentrated at the most stable equilibria if fluctuations are uniform; however, a lesser stable equilibrium may be selected by the fluctuation if its ability to fluctuate is sufficiently smaller than other stable equilibria. Certain related mathematical issues are also addressed.

Wide-angle vision for road views

AbstractThe field-of-view of a wide-angle image is greater than (say) 90 degrees, and so contains more information than available in a standard image. A wide field-of-view is more advantageous than standard input for understanding the geometry of 3D scenes, and for estimating the poses of panoramic sensors within such scenes. Thus, wide-angle imaging sensors and methodologies are commonly used in various road-safety, street surveillance, street virtual touring, or street 3D modelling applications. The paper reviews related wide-angle vision technologies by focusing on mathematical issues rather than on hardware.

Problems of mathematics subject integration into module studies

This article deals with relevant problems of mathematics’ subject studies in College. There are presented the development and essential issues of mathematics‘ subject integration into economics module studies. The presented analysis reveals further directions of mathematics quality improvement search.

Reply to “Comment on ‘Accurate and fast numerical solution of Poisson's equation for arbitrary, space-filling Voronoi polyhedra: Near-field corrections revisited’ ”

This is a Reply to the Comment by Gonis and Zhang1 on our recent paper.2 They discuss supposed issues with our “accurate and fast numerical solution of Poisson’s equation for arbitrary Voronoi polyhedra” (VP). The method of Gonis et al.3 and Vitos and Koll´ar4 correctly addresses the mathematical issues related to the solution of Poisson’s equation [and near-field corrections (NFCs)] using a spherical harmonic basis (L’s). However, as discussed in Ref. 2, their method is not numerically efficient and suffers from severe convergence issues from the multiply nested (conditionally convergent) multipole expansions to address r problems. Hence, the abstract in Ref. 2 clearly states that we provide a method to address a “fast but accurate numerical solution of the Poisson equation” for site-centered methods.

FORMALIZATION, PRIMITIVE CONCEPTS, AND PURITY

AbstractWe emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem.Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity.

Probabilistic Aspects of Finance

In the past decades, advanced probabilistic methods have had significant impact on the field of finance, both in academia and in the financial industry. Conversely, financial questions have stimulated new research directions in probability. In this survey paper, we review some of these developments and point to some areas that might deserve further investigation. We start by reviewing the basics of arbitrage pricing theory, with special emphasis on incomplete markets and on the different roles played by the 'real-world' probability measure and its equivalent martingale measures. We then focus on the issue of model ambiguity, also called Knightian uncertainty. We present two case studies in which it is possible to deal with Knightian uncertainty in mathematical terms. The first case study concerns the hedging of derivatives, such as variance swaps, in a strictly path wise sense. The second one deals with capital requirements and preferences specified by convex and coherent risk measures. In the final two sections we discuss mathematical issues arising from the dramatic increase of algorithmic trading in modern financial markets.

Editorial: Mathematics Teacher Educator: An Opportunity to Share, Verify, and Improve

Mathematics Teacher Educator is the first journal dedicated specifically to issues in mathematics teacher education, providing a much-needed forum for supporting and improving the practice of educating teachers of mathematics. As the Editorial Panel articulated in the call for manuscripts (http://www.amte.net/publications/mte), the mission of Mathematics Teacher Educator is “to contribute to building a professional knowledge base for mathematics teacher educators that stems from, develops, and strengthens practitioner knowledge. The journal provides a means for practitioner knowledge related to the preparation and support of teachers of mathematics to be not only shared but also verified and improved over time. The journal is a tool to build the personal knowledge that mathematics educators gain from their practice into a trustworthy knowledge base that can be shared with the profession.”

Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency

This paper discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation–dissipation theorems (FDTs) and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable ‘perfect’ system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long-range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e. the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the FDT opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.

Mathematics and The Hunger Games

The Hunger Games plot features a dystopian future in which twelve outer districts are oppressed by a centralized capital. The story focuses on the heroism of a sixteen-year-old girl named Katniss and how she tries to rise above the oppression that she experiences. It also features a special lottery and other twists that are sources of mathematical interest. This essay focuses on some of the mathematical issues raised by The Hunger Games in an effort to show that this story can be used to teach students (as well as other interested parties) some important concepts from mathematics.

An Enhanced On-Line Service for 3D Model Construction from Photographs

We present a web service for image based 3D reconstruction. The system allows a cultural heritage professional to easily create a 3D model of a scene or object out of images taken from different viewpoints. The user uploads the images to our server on which all processing takes place, and the final result can be downloaded upon completion. Any consumer-class digital camera can be used, and the system is free to use for non-commercial purposes. The service includes a number of innovations to greatly simplify the process of taking pictures suitable for reconstruction. In particular, we are able to construct models of planar scenes and from photographs shot using a turntable, and at varying zoom levels. Although the first two may seem like particularly simple cases, they cause some mathematical issues with traditional self-calibration techniques. We handle these cases by taking advantage of a new automatic camera calibration method that uses meta-data stored with the images. For fixed-lens camera setups, we can also reuse previously computed calibrations to support otherwise degenerate scenes. Furthermore, we can automatically compute the relative scale and transformation between two reconstructions of the same scene, merging two reconstructions into one. We demonstrate the capabilities of the system by two case studies: turntable reconstruction of various objects and the reconstruction of a cave, with walls and roof integrated into a complete model.

Risk in complex genetics: “All models are wrong but some are useful”

Although genome-wide association studies (GWAS) have proven remarkably effective at identifying reliably associated genetic variants, the biology underlying these discoveries is rarely immediately apparent and in most cases seems bound to require extensive fine mapping and functional analysis before it is revealed. In this context, it is logical and appropriate to try to interrogate available genetic data for biological insights. However, because such efforts invariably depend upon mathematical modeling, misperceptions can easily arise if the relevant mathematical properties are overlooked or forgotten. In this report, we will examine these mathematical issues, highlight some of the more common misconceptions, and hopefully help to clarify the somewhat blurry distinction between biology and mathematics that can so easily undermine and obscure the value of GWAS discoveries.

A fast Monte–Carlo method with a reduced basis of control variates applied to uncertainty propagation and Bayesian estimation

The reduced-basis control-variate Monte-Carlo method was introduced recently in [S. Boyaval, T. Lelièvre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm, Commun. Math. Sci. 8 (2010) 735–762 (Special issue “Mathematical Issues on Complex Fluids”)] as an improved Monte-Carlo method, for the fast estimation of many parametrized expected values at many parameter values. We provide here a more complete analysis of the method including precise error estimates and convergence results. We also numerically demonstrate that it can be useful to some parametric frameworks in Uncertainty Quantification, in particular (i) the case where the parametrized expectation is a scalar output of the solution to a Partial Differential Equation (PDE) with stochastic coefficients (an Uncertainty Propagation problem), and (ii) the case where the parametrized expectation is the Bayesian estimator of a scalar output in a similar PDE context. Moreover, in each case, a PDE has to be solved many times for many values of its coefficients. This is costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C. Le Bris, Y. Maday, N. Nguyen, A. Patera, A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable robin coefficient, Comput. Methods Appl. Mech. Eng. 198 (2009) 3187–3206]. To our knowledge, this is the first combination of various reduced-basis ideas, with a view to reducing as much as possible the computational cost of a simple versatile Monte-Carlo approach to Uncertainty Quantification.

Selected topics in computational mathematics

Computational mathematics is a discipline of mathematical research in areas of science where computation plays a central and essential role. Our everyday lives have been transformed by the impact of computation and communication. This impact can be observed in the world economy, where information technology has contributed in significant and numerous ways to growth in productivity and the development of a plethora of new products and services. For example, advances in computational techniques have led to improvements in healthcare diagnoses, access, and delivery; in education where learning is increasingly medicated by information technologies; and in science and engineering where increasingly pervasive and sophisticated cyberinfrastruture are revolutionalizing research methodology. Applying computation to problemsolving has led to the development of new ways of conceptualizing, analyzing, and solving problems that are otherwise difficult to tackle. This special issue consists of a few selected topics of research in computational mathematics from a prominent group of young researchers. The contributing papers address some of the key issues in computational mathematics: from model development to design, analysis, and implementation of numerical methods and computational algorithms. All papers are invited, but all have been carefully peer-reviewed. I would like to thank all the contributors for their diligent, high-quality work, and all the anonymous referees who have made constructive suggestions and comments that led to a much improved collection of papers.

The complexity of mathematics teaching and learning in mathematics teacher education and research

Coping with the complexity of mathematics teaching and learning in the practice ofteaching is an everyday task for a mathematics teacher, teacher educator, and researcher.Krainer (2003) considers that the field of mathematics education has moved toward a betterunderstanding of this complexity: ‘‘The growth of mathematics education as a scientificfield can be regarded as a continuous process of having a deeper and deeper understandingof the complexity of learning and teaching’’ (p. 93). In mathematics teacher educationresearch, as reported in a number of articles of the Journal of Mathematics TeacherEducation (JMTE), this complexity is tackled in different ways and contexts. Some studiesare exploratory with a focus on how teachers cope with the classroom complexity. Othersare developmental in the sense of creating contexts for teachers, which would promotetheir awareness of the complexity of teaching and learning mathematics and support themto cope with this in the classroom.At the exploratory level, the complexity lies in the classroom interactions and the waysthat the teacher is balancing mathematical goals, students’ reasoning and thinking, andclassroom management. As Wood (2002) argues, the alternate forms of teaching and theresearch on them have brought under consideration situations such as classroom norms,patterns of interaction, supporting students’ mathematical thinking, and reasoning asimportant issues in mathematics teaching that pose new challenges for the teacher, theteacher educator, and the researcher. Moving beyond this micro level of the mathematicsclassroom and addressing wider social factors that frame mathematics learning andteaching, Valero (2010) addresses another layer of complexity involving a network ofdifferent practices that seem to frame the didactical triangle and emphasizes the need formathematics education research to consider the connections between these differentpractices and mathematics teaching and learning. Sullivan (2006), in his editorial, alsooffers specific examples to indicate that ambiguities that exist in curriculum goals and inresearch findings create dilemmas to teachers and teacher educators who try to balancedifferent and often dichotomous goals of mathematics teaching. These complexities raised

ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS

Kumaraswamy Vela Velupillai74 presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser51 in his extension of Gödel's incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

Gender Issues in Mathematics: An Ontario Perspective

In many developed countries, including Canada, it is often perceived that gender issues in mathematics have been ‘solved’ and are no longer a relevant and timely issue. In this article, I challenge this perception by providing an overview of gender issues in mathematics in three domains – achievement, attitude, and participation – ranging from the elementary school level to the university level. My analysis of several sources of data from Ontario is compared to a meta-analysis of research involving data from culturally similar countries to Canada (e.g., Australia, United Kingdom). The data primarily arise from large-scale mathematics assessments (e.g., PISA, EQAO) and national statistics databases (e.g., Statistics Canada, National Center for Education Statistics). Counter to the aforementioned perception, this analysis indicates that gender issues still exist in mathematics in developed countries, including Canada. The gender gap is particularly wide in terms of students’ attitudes and participation: Males have substantially more positive attitudes toward mathematics and higher levels of participation in non-mandatory levels of mathematics than do females. The article concludes with a discussion of the implications of the findings and suggestions of possible steps that may be taken to help ameliorate the current situation.

FEYNMAN DIAGRAMS FOR DISPERSION INTERACTIONS OUT OF EQUILIBRIUM — TWO-BODY POTENTIALS FOR ATOMS WITH INITIAL EXCITATION

Diagrammatic techniques are well-known in the calculation of dispersion interactions between atoms or molecules. The multipolar coupling scheme combined with Feynman ordered diagrams significantly reduces the number of graphs compared to elementary stationary perturbation theory. We review calculations of van der Waals-Casimir-Polder forces, focusing on two atoms or molecules, one of which is excited. In this case, calculations of the corresponding force are notorious for mathematical issues connected with the spontaneous decay of the excitation. Treating such unstable states in a full non-equilibrium theory provides a physical interpretation of apparent contradictions in previous results and underlines the importance of decay processes for the intermolecular potential. This may have important implications on reactions in biological systems, where excited states may be relatively long-lived and the resonant intermolecular force may result in directed Brownian motion.

Mathematical Issues in the Inference of Causal Interactions among Multichannel Neural Signals

Within the last few decades, attempts have been made to characterize the underlying mechanisms of brain activity by analyzing neural signals recorded, directly or indirectly, from the human brain. Accordingly, inference of functional connectivity among neural signals has become an indispensable research tool in modern neuroscience studies aiming to explore how different brain areas are interacting with each other. Indeed, remarkable advances in computational sciences and applied mathematics even allow the estimation of causal interactions among multichannel neural signals. Here, we introduce the brief mathematical background of the use of causality inference in neuroscience and discuss the relevant mathematical issues, with the ultimate goal of providing applied mathematicians with the current state‐of‐the‐art knowledge on this promising multidisciplinary topic.

Le cœur numérique personnalisé

Progress in medical image analysis, biomathematics and biophysics has led to the development of the first personalized digital cardiac models that reproduce the anatomy and physiology of individual patients, permitting quantitative analysis of organ function and prediction of therapeutic effects. This article describes recent research in this field by our Asclepios project team at Inria, in collaboration with other Inria teams (Macs, Reo and Sisyphe) and external academic, clinical and industrial partners. While a number of mathematical and computational issues have to be solved before such personalized digital cardiac models can be used in routine clinical practice, these first results announce a new generation of tools in digital medicine which will help to improve preventive and predictive personalized medicine.