Perturbations of embedded eigenvalues for the planar bilaplacian

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues is linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials

Nonlinear stability of source defects in oscillatory media

In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects are important as organizing centers of more complicated flows. Our analysis uses spatial dynamics combined with an instantaneous phase-tracking technique to obtain detailed pointwise estimates describing perturbations to lowest order as a phase-shift radiating outward at a linear rate plus a pair of localized approximately Gaussian excitations along the phase-shift boundaries; we show that in the wake of these outgoing waves the perturbed solution converges time-exponentially to a space-time translate of the original source pattern.https://arxiv.org/abs/1802.07676First author draf

Nonlinear stability of source defects in the complex Ginzburg-Landau equation

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for the Green's function, which allow one to close a nonlinear iteration scheme.Comment: 53 pages, 5 figure

A hybrid particle–ensemble Kalman filter for Lagrangian data assimilation

Author Posting. © American Meteorological Society, 2015. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Monthly Weather Review 143 (2015): 195–211, doi:10.1175/MWR-D-14-00051.1.Lagrangian measurements from passive ocean instruments provide a useful source of data for estimating and forecasting the ocean’s state (velocity field, salinity field, etc.). However, trajectories from these instruments are often highly nonlinear, leading to difficulties with widely used data assimilation algorithms such as the ensemble Kalman filter (EnKF). Additionally, the velocity field is often modeled as a high-dimensional variable, which precludes the use of more accurate methods such as the particle filter (PF). Here, a hybrid particle–ensemble Kalman filter is developed that applies the EnKF update to the potentially high-dimensional velocity variables, and the PF update to the relatively low-dimensional, highly nonlinear drifter position variable. This algorithm is tested with twin experiments on the linear shallow water equations. In experiments with infrequent observations, the hybrid filter consistently outperformed the EnKF, both by better capturing the Bayesian posterior and by better tracking the truth.The work of Apte benefited from the support of the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems established in ICTS-TIFR. Spiller would like to acknowledge support by NSF Grant DMS-1228265 and ONR Grant N00014-11-1-0087. Sandstede gratefully acknowledges support by the NSF through Grant DMS-0907904. Slivinski was supported by the NSF through Grants DMS-0907904 and DMS-1148284.2015-07-0

Localized radial roll patterns in higher space dimensions

Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves (“isolas”), or the length increases to infinity so that branches are unbounded in function space (“snaking”). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyse the structure of branches of localized radial roll solutions in dimension 1+ε, with 0 < ε 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.http://math.bu.edu/people/mabeck/Bramburgeretal18.pdfFirst author draf

Crowdsourcing Classroom Observations to Identify Misconceptions in Data Science

Web-browsing histories, online newspapers, streaming music, and stock prices all show that we live in an age of data. Extracting meaning from data is necessary in many fields to comprehend the information flow. This need has fueled rapid growth in data science education aiming to serve the next generation of policy makers, data science researchers, and global citizens. Initially, teaching practices have been drawn from data science\u27s parent disciplines (e.g., computer science and mathematics). This project addresses the early stages of developing a concept inventory of student difficulty within the newly emerging field of data science. In particular this project will address three primary research objectives: (1) identify student misconceptions in data science courses; (2) document students’ prior knowledge and identify courses that teach early data science concepts; and (3) confirm expert identification of data science concepts, and their importance for introductory-level data science curricula. During the first year of this grant, we have collected approximately 200 responses for a survey to confirm concepts from an existing body of knowledge presented by the Edison Project. Survey respondents are comprised of faculty and industry practitioners within data science and closely related fields. Preliminary analysis of these results will be presented with respect to our third research objective. In addition, we developed and launched a pilot assessment for identifying student difficulties within data science courses. The protocol includes regular responses to reflective questions by faculty, teaching assistants, and students from selected data science courses offered at the three participating institutions. Preliminary analyses will be presented along with implications for future data collection in year two of the project. In addition to the anticipated results, we expect that the data collection and analysis methodologies will be of interest to many scholars who have or will engage in discipline-based educational research

A Mixed-Method Approach to Investigating Difficulty in Data Science Education

The purpose of this study was to define a methodology to identify any disconnect between students and instructors in data science classrooms through analyzing qualitative data. A combined qualitative and quantitative approach was used for analysis of survey data from students, faculty/instructors, and teaching assistants across three institutions. Using a manual content analysis paired with a TF-IDF analysis, researchers were able to pull out frequently used terms within responses and encode them into categories and subcategories. Trends were identified from these categories and subcategories to examine general areas of disconnect within the data science classroom. Additionally, a quality analysis was run to determine the effectiveness of the phrasing of the questions posed during the survey. As a whole, the methods used throughout this research process provide direction for researchers in interpretation and analysis of the survey data in an efficient and time-sensitive manner. Furthermore, it allows researchers to analyze the quality of responses to give insight towards rephrasing of survey questions in future analyses. Although the research was applied to data science classrooms, this method has the potential to be applied into other fields and areas of study when performed with coordination between a field expert and a data scientist