Mathematical Sciences

Inverse scattering series is the only nonlinear, direct inversion method for the multidimensional, acoustic or elastic equation. Recently developed techniques for inverse problems based on the inverse scattering series were shown to require two mappings, one associating nonperturbative description of seismic events with their forward scattering series description and a second relating the construction of events in the forward to their treatment in the inverse scattering series. This paper extends and further analyzes the first of these two mappings, introduced, for 1D normal incidence, in Matson [J. ]. It brings a new and more rigorous understanding of the mathematics and physics underlying the calculation of terms in the forward scattering series and the events in the seismic model. The convergence of the series for 1D acoustic models is examined, and the earlier precritical analysis is extended to critical and postcritical reflections. An explanation is proposed for the divergence of the series for postcritical incident planewaves.

The starting point for the derivation of a new set of approaches for predicting both the wavefield at depth in an unknown medium and transmission data from measured reflection data is the inverse scattering series. We present a selection of these maps that differ in order ͑i.e., linear or nonlin-ear͒, capability, and data requirements. They have their roots in the consideration of a data format known as the T-matrix and have direct applicability to the data construction techniques motivating this special issue. Of particular note, one of these, a construction of the wavefield at any depth ͑including the transmitted wavefield͒, order-by-order in the measured reflected wavefield, has an unusual set of capabilities ͑e.g., it does not involve an assumption regarding the minimum-phase nature of the data and is accomplished with processing in the simple reference medium only͒ and requirements ͑e.g., a suite of frequencies from surface data are required to compute a single frequency of the wavefield at depth when the subsurface is unknown͒. An alternative reflection-to-transmission data mapping ͑which does not require a knowledge of the wavelet, and in which the component of the unknown medium that is linear in the reflection data is used as a proxy for the component of the unknown medium that is linear in the transmission data͒ is also derivable from the inverse scattering series framework.

Inverse scattering series is the only nonlinear, direct inversion method for the multidimensional, acoustic or elastic equation. Recently developed techniques for inverse problems based on the inverse scattering series were shown to require two mappings, one associating nonperturbative description of seismic events with their forward scattering series description and a second relating the construction of events in the forward to their treatment in the inverse scattering series. This paper extends and further analyzes the first of these two mappings, introduced, for 1D normal incidence, in Matson [J. ]. It brings a new and more rigorous understanding of the mathematics and physics underlying the calculation of terms in the forward scattering series and the events in the seismic model. The convergence of the series for 1D acoustic models is examined, and the earlier precritical analysis is extended to critical and postcritical reflections. An explanation is proposed for the divergence of the series for postcritical incident planewaves.

This paper provides: (1) a review of the logic and promise behind the isolated task inverse-scattering sub-series concept for achieving all processing objectives directly in terms of data only, without knowing or determining or estimating the properties that govern wave propagation in the actual earth; (2) the recognition that an effective response to pressing seismic challenges requires understanding that those challenges arise when assumptions, and prerequisites behind current leading edge seismic processing are not satisfied and that those failures can be attributed to: (A) insufficient acquisition, and/or (B) compute power and (C) bumping into algorithmic limitations and assumptions, and (3) the status and plans of the inverse series campaign to address the fundamental algorithmic limitations of processing methods, that are not addressed by more complete acquisition and faster computers.

We analyze the nature of steady solutions of a sheared ferrofluid between two parallel boundaries and subject to an applied magnetic field H perpendicular to the boundaries. Making no a priori assumption about the magnitude of spin, we find solutions numerically for the velocity and spin fields under the combined pressure gradient and boundary flow forcing. The numerical technique is valid for arbitrary spin viscosity, and by approaching asymptotically small values we explore the impact of the spin boundary conditions on the flow. When the imposed magnetic field is time independent, its effect on the flow is dissipative, but spatially varying fields still permit control of the velocity profile, including the breaking of its midplane symmetry. Time dependent or rotating perpendicular fields can drive the flow and allow more complete flow control, as illustrated in a simple numerical experiment that approximates plug flow.

New type III and type N approximate solutions which are regular in the linear approximation are shown to exist. For that, we use complex transformations on self-dual Robinson-Trautman metrics rather then the classical approach. The regularity criterion is the boundedness and vanishing at infinity of a scalar obtained by saturating the Bel-Robinson tensor of the first approximation by a time-like vector which is constant with respect to the zeroth approximation.

This paper describes the design of Distributed Biography, an online tool for the co-construction of individual, event, location or experience-based narrative. As the application is currently in the later stages of alpha development, the focus herein centers on our original designs, the process of understanding the collaborative rhythms of the group and outlining the development trajectory.

This paper studies the behavior of a trainer-of-trainers (TofT) model of professional development using a finite-difference model. We built a computational environment using real data and a structure closely matching the model’s actual implementation that allowed for a type of controlled experimentation that is not otherwise possible. We found that over time a TofT model forces implementation only at the lowest level of the system where fidelity of implementation is less than appropriate and where buy-in is extremely low. We also found that when the number of “untrained” teachers declines over time and the program has excess capacity to train new teachers, the actual total number of teachers implementing the reform is a small portion of the total number of teachers in the system. These findings suggest that buy-in at the teacher level is critical and provoke an analysis of influences on teacher buy-in.

Children’s experiences in geometry are entirely Euclidean. But they also have non-metric ideas. Supported by a dynamic geometry environment I co-authored for this study, I conducted a teaching experiment with two children, ages six and seven. I found that these children developed significant and authentic forms of geometric reasoning. Qualitative geometry is the name I give to these findings.

In this paper we present case studies of student engagement in four classrooms selected from a randomized, experimental study that investigated scaling up the SimCalc approach to proportionality . The experimental study revealed significant learning gains in students' knowledge. These gains were robust across demographic and regional variation in teachers and students, and attributable to a combination of potential factors which the experiment did not attempt to disentangle, including professional development, the SimCalc curriculum replacement unit, and teacher knowledge and practices. Thus, the experimental study also generated new questions about what and how students learned. Noting that studies of curriculum interventions that found effects rarely studied how the effects were achieved, Stein, Remillard, and Smith (2007) called for more studies "designed to include both large-scale

Dynamic representational technologies (DRTs) in mathematics education are software environments that allow users to observe and interact with multiple and connected representations of concepts. This article makes a case for their use to teach rate concepts, which historically have been difficult for teachers to teach and for students to learn.

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as "qualitative geometry," that have implications for the teaching and learning of geometry and for research into students' mathematical reasoning.

This paper draws on findings from a prior study for which I designed and developed a dynamic geometry environment called Configure and used it in the episodes of a teaching experiment to engage children's informal and intuitive topological, or at least non-metric, conceptions and support their development. I found that "Amanda" developed significant and authentic forms of geometric reasoning and have given the name "qualitative geometry" to these findings. These newly identified forms of reasoning were seen to develop as Amanda engaged in mathematical activity mediated by the microworld of Configure. These findings are used to provide a perspective on mathematics thinking and learning as engagement in microworld-mediated mathematical activity. Essential elements of this particular form of mathematical activity are provided.

It is widely accepted in the mathematics education community that pedagogies oriented toward inquiry are aligned with a constructivist theory of learning, and that these pedagogies effectively support students' learning of mathematics. In order to promote such an orientation, we first separate the idea of inquiry from its conception as a collection of methods. Then, by grounding those methods in a generally accepted theory of learning, we construct an inquiry-oriented pedagogy from a constructivist 10 perspective. We then discuss the implications of this pedagogy for the design of mathematical tasks that democratize student access to inquiry. This work has implications for educators who wish to enact an inquiry-oriented pedagogy in their classroom in order to support their students' problem solving and problem posing.

We describe an experience within mathematics teacher preparation that engages pre-service teachers of mathematics (PMTs) in Making and design practices that we hypothesized would inform their conceptual, curricular, and pedagogical thinking. With a focus on the design of new tools that can generate new possibilities for mathematics teaching and learning, this Learning by Design experience has PMTs exploring at the intersection of content, pedagogy, and Making. We describe the forms of knowledge brought to bear on their experiences through a case study analysis of one pair of PMTs' Making experience. As the advancement of these forms of knowledge is essential to effective mathematics teaching, these findings suggest the promise of a Making experience within mathematics teacher preparation.

In this paper, we aim to explore how prospective elementary mathematics teachers (PMTs) learn to teach mathematics through their engagement in a pedagogically informative Making experience. Grounded in a commognitive perspective, we define learning to teach mathematics as changes in any of these four discourse activities: mathematizing and identifying (Heyd-Metzuyanim & Sfard, 2012), in addition to pedagogy and designing. We present our analysis of one PMT's discourse activity, exposing Making as an effective venue for provoking all four discourses and revealing their intertwined nature, further illustrating that one's identity is as central to learning to teach mathematics as is their learning of mathematics, pedagogy, and design. We conclude with a discussion of the implications of these findings for the research and practice of teacher education.