Ion-acoustic wave propagation in plasmas with ion beams having afinite cross section

The propagation of a low-density-modulated ion beam with finite cross section in a homogeneous plasma is considered. Analytical expressions describing a Cerenkov-like radiation pattern are obtained. An experimental setup is described that is suitable for investigating these phenomena. The results are in qualitative agreement with the analytical expressions     

Isotopic abundance measurements on solid nuclear-type samples by glow discharge mass spectrometry

A double-focusing Glow Discharge Mass Spectrometer (GDMS) installed in a glovebox for nuclear sample screening has been employed for isotopic measurements. Isotopic compositions of zirconium, silicon, lithium, boron, uranium and plutonium which are elements of nuclear concern have been determined. Interferences arising from the matrix sample and the discharge gas (Ar) for each of these elements are discussed. The GDMS results are compared with those from Thermal Ionization Mass Spectrometry (TIMS). For boron and lithium at g/g-ng/g levels, the two methods gave results in good agreement. In samples containing uranium the isotopic composition obtained by GDMS was in agreement with those from TIMS independently of the enrichment. Attempts for the determination of plutonium isotopic composition were also made. In this case, due to the interferences of uranium at mass 238 and americium at mass 241, the GDMS raw data are complementary with those values obtained from physical non-destructive techniques.     

Relaxation oscillations in spruce–budworm interactions

We review and complement the study of the mathematical properties of the predator–prey dynamical system for spruce–budworm interactions studied in Ludwig et al. [D. Ludwig, D. Jones, C.S. Holling, Qualitative analysis of insect outbreak systems: the spruce budworm and forest, Journal of Animal Ecology 47 (1978), 315–332]. We use the singular perturbation method to identify parameter regimes which permit relaxation oscillations. The leading order contribution to the period of these oscillations is computed by means of this method and compared with the outcome of direct numerical simulations.     

The early exercise region for Bermudan options on two underlyings

This paper investigates the early exercise region for Bermudan options on two underlying assets. We present a set of analytical validation results for the early exercise region which can be used as a means of validating pricing techniques. When all strike prices are identical we show the existence of an intersection point such that for any asset price pair below this point early exercise is always optimal. We develop an approximation to this point in the two asset put case. When the strike prices are not all equal, we show that three separate cases exist for the early exercise region. For a Bermudan put on two assets we present these cases and show that there exists a critical point in which the boundaries of the two asset early exercise region bifurcate. Comparisons are drawn between the Bermudan results presented and the corresponding American option results.     

Research on calculus: what do we know and where do we need to go?

In this introductory paper we take partial stock of the current state of field on calculus research, exemplifying both the promise of research advances as well as the limitations. We identify four trends in the calculus research literature, starting with identifying misconceptions to investigations of the processes by which students learn particular concepts, evolving into classroom studies, and, more recently research on teacher knowledge, beliefs, and practices. These trends are related to a model for the cycle of research and development aimed at improving learning and teaching. We then make use of these four trends and the model for the cycle of research and development to highlight the contributions of the papers in this issue. We conclude with some reflections on the gaps in literature and what new areas of calculus research are needed.     

Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.     

Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

The purpose of this paper is to further the notion of defining as a mathematical activity by elaborating a framework that structures the role of defining in student progress from informal to more formal ways of reasoning. The framework is the result of a retrospective account of a significant learning experience that occurred in an undergraduate geometry course. The framework integrates the instructional design theory of Realistic Mathematics Education (RME) and distinctions between concept image and concept definition and offers other researchers and instructional designers a structured way to analyze or plan for the role of defining in students’ mathematical progress.     

On the torsion of Jacobians of principal modular curves of level 3<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3ⁿ) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation. Received: 30 September 2005     

Computation of nonautonomous invariant and inertial manifolds

We derive a numerical scheme to compute invariant manifolds for time-variant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov–Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts. Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler–Bubnov–Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.     

Approximating the permanent: A simple approach

In this article, we present an efficient and very simple unbiased estimator for the Permanent of square (0–1) matrices. As the main result, we prove that our estimator, even though its worst case behavior is provably very bad, runs in time polynomial in the size of the input matrix for almost all matrices. We also generalize our technique and apply it to another enumeration problem, that of counting Hamilton cycles in directed graphs. The ease with which this was done suggests that the basic technique may have wide applicability. © 1994 John Wiley & Sons, Inc.