Pre-service Teachers’ Developing Conceptions about the Nature and Pedagogy of Mathematical Modeling in the Context of a Mathematical Modeling Course

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers’ developing conceptions about (a) the nature of mathematical modeling in simulations of “real life” problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students’ ways of thinking. The researchers’ conceptual development, on the other hand, involved realizing the critical aspect of the “teacher role” played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.     

An Extension of Yuan’s Lemma and Its Applications in Optimization

We prove an extension of Yuan’s lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of Baccari and Trad (SIAM J Optim 15(2):394–408, 2005), where the classical necessary second-order optimality condition is proved, under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the Hessian of the Lagrangian, evaluated at the vertices of the Lagrange multiplier set, is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the Jacobian matrix.     

A Dynamic Analysis of Schelling’s Binary Corruption Model: A Competitive Equilibrium Approach

Schelling (in Micromotives and Macrobehavior, Norton, New York, 1978) suggested a simple binary choice model to explain the variation of corruption levels across societies. His basic idea was that the expected profitability of engaging in corruption depends on its prevalence. The key result of the so-called Schelling diagram is the existence of multiple equilibria and a tipping point. The present paper puts Schelling’s essentially static approach into an intertemporal setting. We show how the existence of an unstable interior steady state leads to thresholds such that history alone or history in addition to expectations (or coordination) is necessary to determine the long-run outcome. In contrast to the related literature, which classifies these two cases according to whether the unstable equilibrium is a node or a focus, the actual differentiation is more subtle because even a node can lead to an overlap of solution paths such that the initial conditions alone are insufficient to uniquely determine the competitive equilibrium. Another insight is that a (transiently) cycling competitive equilibrium can dominate the direct and monotonic route to a steady state, even if the direct route is feasible.     

On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

Journal of Nonlinear Science - A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a...     

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.     

Forecasting a Cyclical Downturn (Recession) in the US Economy Using a Mathematical Model of Hyman Minsky’s Theory of Financial Instability

Doklady Mathematics - By using the US economy as an example, the paper shows how the COVID-19 pandemic has changed its short-term dynamics, causing a deep crisis recession in 2020 rather than the...     

The Evolution of Jupiter’s Orbit in the Case of a Stellar Approach to the Solar System

The spatial motion of a passively gravitating body is investigated within the restricted threebody problem. The exact expression of the force function without expansion in series is used. The influence of the perturbing star as it approaches the Sun on the orbit of Jupiter is investigated. It is shown that a star of one to five solar masses that approaches the Solar System in a hyperbolic orbit within a minimum distance of 50 to 100 AU significantly affects the size and shape of Jupiter’s orbit only in the case when the sample star is at the perihelion, and Jupiter is in conjunction or in opposition to it. The results are shown in the form of figures and tables.     

Modeling the interaction of an electromagnetic field with a two-dimensional surface in the framework of symanzik’s approach

Symanzik’s approach to the construction of models for the interaction between quantum fields and macroobjects allows finding the general form of the action functional with a Chern-Simons potential up to an arbitrary dimensionless constant; it is used to describe the interaction of a material surface with the electromagnetic field. We discuss results obtained in static models of such a type. We also consider a simple dynamic model of the interaction of a massless scalar field with moving planes.     

Emergence of Condensation in Kingman’s Model of Selection and Mutation

We describe the onset of condensation in the simple model for the balance between selection and mutation given by Kingman in terms of a scaling limit theorem. Loosely speaking, this shows that the wave moving towards genes of maximal fitness has the shape of a gamma distribution. We conjecture that this wave shape is a universal phenomenon that can also be found in a variety of more complex models, well beyond the genetics context, and provide some further evidence for this.     

Initiation of Cracks in Griffith’s Theory: An Argument of Continuity in Favor of Global Minimization

The initiation of a crack in a sound body is a real issue in the setting of Griffith’s theory of brittle fracture. If one uses the concept of critical energy release rate (Griffith’s criterion), it is in general impossible to initiate a crack. On the other hand, if we replace it by a least energy principle (Francfort–Marigo’s criterion), it becomes possible to predict the onset of cracking in any circumstance. However this latter criterion can appear too strong. We propose here to reinforce its interest by an argument of continuity. Specifically, we consider the issue of the initiation of a crack at a notch whose angle ω is considered as a parameter. The result predicted by the Griffith criterion is not continuous with respect to ω, since no initiation occurs when ω>0 while a crack initiates when ω=0. In contrast, the Francfort–Marigo’s criterion delivers a response which is continuous with respect to ω, even though the onset of cracking is necessarily brutal when ω>0. The theoretical analysis is illustrated by numerical computations.     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

Efficient Computation of First Passage Times in Kou’s Jump-diffusion Model

S. G. Kou and H. Wang [First passage times of a jump diffusion process, Adv. Appl. Probab. 35 (2003) 504–531] give expressions of both the real Laplace transform of the distribution of first passage time and the real Laplace transform of the joint distribution of the first passage time and the running maxima of a jump-diffusion model called Kou model. These authors invert the former Laplace transform by using Gaver-Stehfest algorithm, and for the latter they need a large computing time with an algebra computer system. In the present paper, we give a much simpler expression of the Laplace transform of the joint distribution, and we also show, using Complex Analysis techniques, that both Laplace transforms can be extended to the complex plane. Hence, we can use inversion methods based on the complex inversion formula or Bromwich integral which are very efficent. The improvement in the computing times and accuracy is remarkable.     

An Extension of Polyak’s Theorem in a Hilbert Space

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅) _H . Let us consider three linear bounded operators,