Browsing by Browse by FOR 2008 "019999 Mathematical Sciences not elsewhere classified"

We consider the semilinear elliptic equation Δu=h(u) in Ω {0}, where Ω is an open subset of ℝ^N (N≥2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q ∈ (1,CN)(that is, limu→∞h(λu)/h(u)=λ^q, for every λ>0), where CN denotes either N/(N-2) if N≥3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=u^q.

Due to changes to Australia's economic landscape (e.g., falling productivity and the end of the mining boom) and the emergence of disruptive digital technologies, the shape of the Australian workforce is rapidly changing and the development of STEM skills is an imperative. There has been a decline, however, in the number of students studying STEM subjects in senior secondary school, and the underrepresentation of females in many STEM disciplines further compounds the problem. The University of New England is a regional Australian university where a large proportion of students are from rural and regional areas, are mature-aged, and come from low SES backgrounds. Many commence their tertiary studies in STEM with diverse backgrounds, often without the necessary assumed knowledge. A mentoring program was designed to assist female students develop STEM-related study and career goals. Important components of the program included: face-to-face and online training and professional development for participants, two mentors (one academic and one industry-based) per student, accessibility for students studying at a distance, guest speakers, and outreach activities promoting STEM to the wider community. This program could be readily adapted for other cohorts of students (e.g., indigenous students) and expanded (e.g., for all students embarking on STEM studies). The program helped students recognise and address potential roadblocks to a sustained and successful STEM-based career, build confidence in pursuing study and career goals, and develop sound decision-making skills in career planning. For mentors, the program offered STEM-related professional development opportunities. Furthermore, academic mentors reported a positive impact on their approach to STEM teaching as a result of participation in the program.

We show that for small ε > 0, the boundary blow-up problem −ε2∆u = u(u − a(x))(1 - u) in Ω, u|∂Ω = ∞ has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

This article seeks to address a pedagogical theory of introducing the classicist and the frequentist approach to probability, by investigating important elements in 9th grade students' learning process while working with a TinkerPlots2 combinatorial problem. Results from this research study indicate that, after the students had seen the systematic construction of the event space via combinatorial analysis, they viewed the sample space as an essential property that regulated the results of the distribution of each sum's theoretical frequency.

We study the positive steady state of a quasi-linear reaction-diffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. The system has nonlocal dependence on the biomass function, and it has a biomass-dependent drifting term describing the active movement of the biomass towards the location of the optimal growth condition. We obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Our theoretical results reveal four critical numbers for the model not observed before and offer several further insights into the problem being modelled. This is Part I of a two-part series, where we obtain nearly optimal existence and nonexistence results. The asymptotic profile of the solutions is studied in the separate Part II.

This is Part II of our study on the positive steady state of a quasi-linear reaction-diffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. In Part I, we proved nearly optimal existence and nonexistence results. In Part II, we obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Moreover, we reveal four critical numbers for the model and provide further insights to the problem being modelled.

Agent based models of human systems, such as those found in behavioural economics, implement their agents intelligence through the use of simple rules. These models have proved to be worthwhile research tools. However, to improve the outputs of these models will require a greater emphasis on the development of intelligence within the model agents. To achieve a deeper understanding of the cognitive process of human decision making is needed. In this talk a conceptual model of the cognitive process of human decision making is put forward. This conceptual model is developed around the work of Lonergan and Polanyi. Further, the conceptual model utilises the fuzzy logic of Zadeh for its implementation. The resulting conceptual model can be embedded within agent's that model human systems.

This article investigates how 14- to 16- year-old students interpret representations of multivariate data generated by data visualisation tools and how they then seek to construct their own meaningful data visualizations that highlight emerging important aspects of data. Students were asked a single question - about where they would like to live - that involved reasoning about a complex data set with many different variables that they were able to explore using a dynamic visualization tool that allowed them to easily generate multiple visualizations of the relevant data set. Findings show the diverse inferences that students articulated to reason about covariation between multiple variables while using the cycle of inquiry and visual analysis. Students revisited their specific kinds of inferences while using complex data visualisation tools, inventing and revising their visual representations of data. Once they obtained some necessary insight, they readily made an informed decision.

In this paper, we study the existence and limiting behavior of the mountain pass solutions of the elliptic problem —Δu = λf(u — q(x)) in Ω C ℝ²; u 0 on ∂Ω, where q is a positive harmonic function. We show that the 'vortex core' Aλ = {x ∈ Ω: uλ (x) > q(x)} of the solution uλ, shrinks to a global minimum point of q on the boundary ∂Ω as λ → + ∞. Furthermore, we show that for each strict local minimum x₀ point of q(x) on the boundary ∂Ω, there exists a solution uλ whose vortex core shrinks to this strict local minimum point x₀ as λ → +∞.

This article discusses the special opportunities for teaching statistics that technology offers teachers who aim to provide rich learning experiences for their students. These opportunities involve automation of many activities such as quickly organising data, computing measures, and generating graphs. By automating the tasks of computing statistics or generating data, technology facilitates students' ability to visualise abstract concepts, affording an opportunity to focus on conceptual understandings and data analysis. This article also examines how GeoGebra can be integrated into the curriculum and learning process of introductory statistics to engage college students in cycles of investigation including (a) managing data (b) developing students' knowledge for understanding specific statistical concepts, (c) performing data analysis and inference, and (d) exploring probability models. Recommendations are included for ways mathematics educators can engage college learners in developing their knowledge for exploring data concepts and doing statistics with GeoGebra.

Let u∊ be a single layered radially symmetric unstable solution of the Allen-Cahn equation -∈²Δu=u(u-a(|x|))(1-u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u∊ when ∈ is small, we construct solutions of the form u∊ + v∊, with v∊ non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.

Penalized REML estimation can substantially reduce sampling variation in estimates of covariance matrices, and yield estimates of genetic parameters closer to population values than standard analyses. A number of suitable penalties based on prior distributions of correlation matrices from the Bayesian literature are described, and a simulation study is presented demonstrating their efficacy. Results show that reductions of 'loss' in estimates of the genetic covariance matrix, a conglomerate of sampling variance and bias, well over 50% are readily obtained for multivariate analyses of small samples. Default settings for a mild degree of penalization are proposed, which make such analyses suitable for routine use without increasing computational requirements.

The article investigates how 14- to 15- year-olds build informal conceptions of inferential statistics as they engage in a modelling process and build their own computer simulations with dynamic statistical software. This study proposes four primary phases of informal inferential reasoning for the students in the statistical modeling and simulation process. Findings show shifts in the conceptual structures across the four phases and point to the potential of all of these phases for fostering the development of students' robust knowledge of the logic of inference when using computer based simulations to model and investigate statistical questions.

We consider the Dirichlet problem for an elliptic system of FitzHugh-Nagumo type. We prove that the problem has a solution with a sharp peak inside the domain.

Recordings of lessons for the investigation of students’ understandings of Linear Relationships when using mathematical software as an exploration tool

This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝN with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on... We assume that f(u) behaves like u(In u)α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some uniqueness results.

We consider an elliptic problem of Ambrosetti–Prodi type involving critical Sobolev exponent. We prove that this problem has solutions blowing up near the boundary of the domain.

Let u... be a single layered radially symmetric unstable solution of the Allen-Cahn equation -∈²Δu=u(u-a(|x|))(1-u) over the unit ball with Neumann boundary conditions. We estimate the small eigenvalues of the linearized eigenvalue problem at u... when ∈ is small. As a consequence, we prove that the Morse index of u... is asymptotically given [μ*+o(1)]∈... with μ* a certain positive constant expressed in terms of parameters determined by the Allen-Cahn equation. Our estimates on the small eigenvalues have many other applications. For example, they may be used in the search of other non-radially symmetric solutions, which will be considered in forthcoming papers.

We consider the Dirichlet problem for an elliptic system of FitzHugh-Nagumo type. We prove that the problem has solutions with arbitrary many sharp peaks near the boundary but not on the boundary.

In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem...

For type I superconductors with penetration parameter λ below a certain critical value, the existence of multiple local minimizers of the energy functional is shown. The behavior of these minimizers is examined when the Ginzburg–Landau parameter is small, and a hysteresis phenomenon is demonstrated. In contrast to the case investigated by X. B. Pan [J. Math. Phys. 44, 2639 (2003)] where λ is above the critical value, the results presented in this paper show that type I superconductors with λ below the critical value have higher critical magnetic field Hc, and exhibit more complicated superconducting behavior.

We establish two general nonlinear Liouville theorems for equations of the type -Δu = h(x₁)f(u), u ≥ 0 in R^N, sup u < +∞. R^N We then show how these Liouville theorems can be used to obtain a priori estimates for positive solutions of indefinite superlinear elliptic equations for several new cases left open in previous research.

In this paper, using variational inequality techniques, we completely solve the existence and uniqueness question for a free boundary problem arising from the studies of certain population models. We also confirm a conjecture of ours in an earlier paper that this free boundary problem is the correct limiting problem of a degenerate predator-prey model.

We prove the Lazer-McKenna conjecture for an elliptic problem of Ambrosetti-Prodi type with critical and supercritical nonlinearities by constructing solutions concentrating on higher dimensional manifolds, under some partially symmetric assumption on the domain.

A basic goal of theoretical ecology is to understand how the interactions of individual organizations with each other under a certain inhabiting environment determine the spatiotemporal structure of distribution of populations. The diffusive logistic equation, which describes the spatial and temporal distribution of the population density of a single species, is one of the fundamental reaction-diffusion equation models in population biology. In this thesis, we are concerned with the periodic logistic equation with homogeneous Neumann boundary conditions. The theory for this basic case forms the foundation for further investigation of multispecies problems. The thesis consists of four chapters.

In this paper, we propose a diffusive prey-predator model with stage structure for the predator. We first analyze the stability of the nonnegative steady states for the reduced ODE system and then study the same question for the corresponding reaction-diffusion system with homogeneous Neumann boundary conditions. We find that a Hopf bifurcation occurs in the ODE system, but no Turing pattern happens in the reaction-diffusion system. However, when a natural cross diffusion term is included in the model, we can prove the emergence of stationary patterns (i.e., nonconstant positive stationary solutions) for this system; moreover, these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion.

We study the existence of peak solutions for the Neumann problem of an elliptic system of FitzHugh-Nagumo type. The solutions we construct have arbitrary many peaks on the boundary and arbitrary many peaks inside the domain, and all the peaks of the solutions approach some local minimum points of the mean curvature function of the boundary.

We study positive solutions of the equation... where n = 3, 4.5 and ∊ > 0 is small, with Neumann boundary condition in a unit ball B. We prove the existence of solutions with an interior bubble at the center and a boundary layer at the boundary ∂ B. To cite this article: J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The data revolution has given citizens access to enormous large-scale open databases. In order to take into account the full complexity of data, we have to change the way we think in terms of the nature of data and its availability, the ways in which it is displayed and used, and the skills that are required for its interpretation. Substantial changes in the content and processes involved in statistics education are needed. This paper calls for the introduction of new pedagogical constructs and principles needed in the age of the data revolution. The paper deals with a new construct of statistical literacy. We describe principles and dispositions that will become the building blocks of our pedagogical model. Our model suggests that effective engagement with large-scale data, modelling and interpretation situations requires the presence of knowledge-bases as well as supporting dispositions.