Inhomogeneities in 3 dimensional oscillatory media

We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on heterogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at $\varepsilon=0$ the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.Comment: 3 figures, 15 pages. More accurate numerical results. Added a figure illustrating the decay of Amplitude of solution

Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction-diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when $\varepsilon$, the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in $\varepsilon$. The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in $\varepsilon$ does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be "correct" to all orders in $\varepsilon$. We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution, which obtained using matched asymptotics.Comment: 39 pages, 1 figur

Deformation of Striped Patterns by Inhomogeneities

We study the effects of adding a local perturbation in a pattern forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction which vanishes for a particular spatial shift only, which we interpret as a phase-selection mechanism through the inhomogeneity.Comment: 18 page

Pacemakers in large arrays of oscillators with nonlocal coupling

We model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation. We concentrate on the case of heterogeneities with positive average and show that steady solutions to the nonlocal problem exist. In particular, we show that these heterogeneities act as a wave source, sending out waves in the far field. This effect is not possible in 3 dimensional systems, such as the complex Ginzburg-Landau equation, where the wavenumber of weak sources decays at infinity. To obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation. We then use Fredholm properties of the Laplace operator in Kondratiev spaces to obtain solutions to the eikonal equation, and by extension to the nonlocal problem.Comment: 26 page

Apparent Horizons of N Black Holes and Black Hole Rings

We first study systems of N Schwarzschild black holes in a time-symmetric spacelike hypersurface with axial symmetry. Apparent horizons are found by numerically solving a non linear system of 3 coupled ODE\u27s using Mathematica. The location of the apparent horizon is calculated in each system in order to find the critical separation between the black holes that creates an encompassing apparent horizon. A method for approximating the critical separations of N black holes by representing them as an effective system of two black holes is developed. Next, we study black hole rings of different mass. The apparent horizon is used as an approximation to the event horizon in an effort to predict a critical ring radius that generates an event horizon of toroidal topology. We found that a good estimate for this ring radius is 20/(3 &pi) M

Existence of spiral waves in oscillatory media with nonlocal coupling

We prove existence of spiral waves in oscillatory media with nonlocal coupling. Our starting point is a nonlocal complex Ginzburg-Landau (cGL) equation, rigorously derived as an amplitude equation for integro-differential equations undergoing a Hopf bifurcation. Because this reduced equation includes higher order terms that are usually ignored in a formal derivation of the cGL, the solutions we find also correspond to solutions of the original nonlocal system. To prove existence of these patterns we use perturbation methods together with the implicit function theorem. Within appropriate parameter regions, we find that spiral wave patterns have wavenumbers, $\kappa$, with expansion $\kappa \sim C e^{-a/\varepsilon}$, where $a$ is a positive constant, $\varepsilon$ is the small bifurcation parameter, and the positive constant $C$ depends on the strength and spread of the nonlocal coupling. The main difficulty we face comes from the linear operators appearing in our system of equations. Due to the symmetries present in the system, and because the equations are posed on the plane, these maps have a zero eigenvalue embedded in their essential spectrum. Therefore, they are not invertible when defined between standard Sobolev spaces and a straightforward application of the implicit function theorem is not possible. We surpass this difficulty by redefining the domain of these operators using doubly weighted Sobolev spaces. These spaces encode algebraic decay/growth properties of functions, near the origin and in the far field, and allow us to recover Fredholm properties for these maps.Comment: 57 pages, 4 figure

A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions $u(x)$ which are nonconvex in the gradient $\nabla u$ and possibly also in $u$. To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: i) a semi-analytical method based on control systems theory, ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure.Comment: 34 pages, 10 figure

Mathematical literacy: A case study of pre-service teachers

This study addresses the question of whether or not pre-service teachers are ready and prepared to use and teach the highly-specialized language of each discipline. The disciplinary languages present teaching and learning challenges due to their lack of parallels in the daily language (Shanahan & Shanahan, 2008). Additionally, the languages of the disciplines are rarely taught and are commonly acquired through an isolated representation of words without a situated meaning within the theory (Gee, 2002). The knowledge of the particular ways of reading, writing, listening to, and talking in the content areas provides opportunities for students’ apprenticeship within the disciplines required for success in higher education contexts (Dobbs, Ippolito, and Charner, 2017). Moreover, this study addresses the question of how future teachers develop disciplinary knowledge and skills. The purpose of this case study was to investigate how mathematical literacy is shaped and defined by the experiences, language, and disciplinary practices of pre-service teachers and experts in mathematics. This overall aim was unfolded by three guiding research questions: 1) What do the Experiences of Pre-Service Teachers and Experts in Mathematics Reveal about their Understanding of Mathematical Literacy? 2) RQ 2. How do pre-service teachers and experts in mathematics use language when solving mathematical problems? and 3) What literacy practices do pre-service teachers and experts in mathematics utilize when presented with modules that require mathematics problem-solving? To structure the elements of analysis for the participants’ responses, I adopted the theoretical support from the emerging disciplinary literacy framework, the novice-expert paradigm, and the tenets of M. K. Halliday’s functional linguistic theory (i.e., Systemic Functional Linguistics; [SFL]). Four faculty in the Department of Mathematics and four pre-service teachers in the Department of Curriculum and Instruction at a large Midwest university agreed to participate in this case study. For the data collection, I asked the participants to participate in two sessions. In the first sessions, the participants responded to a semi-structured interview. Afterward, in a second session, the participants solved modules of mathematical problems following three protocols: a think-aloud, a silent-solving, and an oral-explanatory. The results of the participants’ responses to the semi-structured interview and the three protocols indicated that their experiences as learners and teachers of mathematics are tied to their definitions of literacy and disciplinary literacy. The SFL analysis showed that for the experts of mathematics, mathematical problem-solving is a more abstract and cognitive practice. The pre-service teachers’ registers indicated that mathematical problem-solving is experienced as more concrete and real practice. The unique literacy practices that these participants displayed showed the strong connection between language, literacy, and mathematical thought.The implications of this study are discussed in terms of the importance of language and disciplinary literacy in preparation for future teachers as they progress in their course of study within their teaching education programs

Analysis and Simulations of a Nonlocal Gray-Scott Model

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, $L^1$ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.Comment: 28 pages, 2 figure