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Developing Workforce with Mathematical Modeling Skills
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| Pengarang | : | Ariel Cintrón-Arias, Ryan Andrew Nivens, Anant Godbole, Calvin B. Purvis |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 4) |
| Halaman | : | 778-792 |
| Abstrak | : | Mathematicians have traditionally been a select group of academics who produce high-impact ideas enabling substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant proportion of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow, and the list of required occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of data, computer simulation, and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates to pursue a double major that includes mathematics. Here we present our solution, some early results of its implementation, and a vision for possible nationwide adoption. |
Exploring Newton’s Second Law and Kinetic Friction Using the Accelerometer Sensor in Smartphones
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The Physics Teacher |
| Volume / Edisi | : | 61 (No. 6) |
| Halaman | : | 473–476 |
| Abstrak | : | Abstrak tidak tersedia. |
Sandpiles and Dunes: Mathematical Models for Granular Matter
| Pengarang | : | Piermarco Cannarsa, Stefano Finzi Vita |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 4) |
| Halaman | : | 751-777 |
| Abstrak | : | Granular materials are everywhere, in the environment but also in our pantry. Their properties are different from those of any solid material, due to the possibility of sudden phenomena such as avalanches or landslides. Here we present a brief survey on their characteristics and on what can be found (from the past thirty years) in the recent mathematics literature in order to reproduce their behavior. We discuss, in particular, differential models proposed for the growth of a sandpile on a table and, when wind comes into play, for the formation and dynamics of sand dunes. This field of research is still of great interest since there is no consolidated general model for the dynamics of granular matter, but rather only standalone models adapted to specific situations. |
A Tool for Exploring Our Fluid Earth
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The Physics Teacher |
| Volume / Edisi | : | 61 (No. 6) |
| Halaman | : | 432–435 |
| Abstrak | : | Five years ago, the Byrd Center’s Education and Outreach program recognized the need for a new weather and climate visualization for middle school youth and the public. Working across traditional disciplines, the team was able to create a tool that addressed challenges they had identified through delivery of outreach programs and allowed educators to create investigations appropriate for their audience and content. Inspiring the team were a series of “Challenges Inherent in Teaching Geosciences,” articulated by Kastens from the geosciences education research community, that were consistent with Cervenec’s science teaching experience over 11 years using the Modeling Instruction method.1–3 These challenges included (1) that the scale of phenomena were often well beyond the size of the classroom, (2) that there was a need for well-crafted experiences and models that could bridge the gap in scale, (3) that there was regularly a need to depend on data that students could not practically collect themselves, (4) that there was often a need to depend on nonexperimental ways of inquiry that deviated from the often-taught “scientific method,” (5) that spatial thinking was critically important, and (6) that understanding how phenomena emerged was aided by conceptualizing Earth systems and their interactions. |
A Bridge between Invariant Theory and Maximum Likelihood Estimation
| Pengarang | : | Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, Anna Seigal |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 4) |
| Halaman | : | 721-747 |
| Abstrak | : | We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous statistical models: we discuss log-linear models and Gaussian models, including matrix normal models and directed Gaussian graphical models. Our approach reveals promising consequences of the interplay between invariant theory and statistics. For instance, algorithms from statistics can be used in invariant theory, and vice versa. |
Feynmans Inverse Problem
| Pengarang | : | Adrian Kirkeby |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 4) |
| Halaman | : | 694-718 |
| Abstrak | : | We analyze an inverse problem for water waves posed by Richard Feynman in the BBC documentary Fun to Imagine. We show that the problem can be modeled as an inverse Cauchy problem for gravity-capillary waves, conduct a detailed analysis of the Cauchy problem, and give a uniqueness proof for the inverse problem. Somewhat surprisingly, this results in a positive answer to Feynman's question. In addition, we derive stability estimates for the inverse problem for both continuous and discrete measurements, propose a simple inversion method, and conduct numerical experiments to verify our results. |
Sigmoid Functions, Multiscale Resolution of Singularities, and Mesh Refinement
| Pengarang | : | Daan Huybrechs, Lloyd N. Trefethen |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 4) |
| Halaman | : | 683-693 |
| Abstrak | : | In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) ??-mesh refinement for solution of \pdes, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables ?=log?(?), and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of “linear tapering” near the singularity---of clustered poles in rational approximation and of polynomial orders in ??-mesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities. |
Oscillatory Networks: Insights from Piecewise-Linear Modeling
| Pengarang | : | Stephen Coombes, Mustafa ?ayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, Yi Ming Lai |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 4) |
| Halaman | : | 619-679 |
| Abstrak | : | There is enormous interest---both mathematically and in diverse applications---in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled-oscillator networks. We discuss phase reductions, phase--amplitude reductions, and the master stability function for smooth dynamical systems. We then focus, in particular, on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use saltation operators, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds. |
Neighborhood Watch in Mechanics: Nonlocal Models and Convolution
| Pengarang | : | Thomas Nagel, Tymofiy Gerasimov, Jere Remes, Dominik Kern |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 1) |
| Halaman | : | 176-193 |
| Abstrak | : | This paper is intended to serve as a low-hurdle introduction to nonlocality for graduate students and researchers with an engineering mechanics or physics background who did not have a formal introduction to the underlying mathematical basis. We depart from simple examples motivated by structural mechanics to form a physical intuition and demonstrate nonlocality using concepts familiar to most engineers. We then show how concepts of nonlocality are at the core of one of the most active current research fields in applied mechanics, namely, in phase-field modeling of fracture. From a mathematical perspective, these developments rest on the concept of convolution in both its discrete and its continuous forms. The previous mechanical examples may thus serve as an intuitive explanation of what convolution implies from a physical perspective. In the supplementary material we highlight a broader range of applications of the concepts of nonlocality and convolution in other branches of science and engineering by generalizing from the examples explained in detail in the main body of the article. |
Energy in Its Material and Social Context: Power Plants
| Pengarang | : | Rachel E. Scherr |
| Nama Majalah/Jurnal | : | The Physics Teacher |
| Volume / Edisi | : | 61 (No. 6) |
| Halaman | : | 428–431 |
| Abstrak | : | One way for science teaching to have significance beyond the classroom is for science education to be in the service of community organizing and ethical decision-making.1–3 Power plants have tremendous social significance both locally and globally. In what follows, we will consider the energy dynamics of two electrical power production facilities: (1) the largest coal-fired electrical power production facility in the United States and (2) one of the facilities that provides significant electrical power to the authors. Rather than analyzing power plants apart from their material and social context, we suggest an analysis that includes the relationships between the power plant and the surrounding human, plant, and animal communities, as well as lands, waters, and air. Our intent is to model an approach to energy learning that begins to prepare students to engage in ethical decision-making about energy resources. |