A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements

Numerical schemes using piecewise polynomial approximation are very popular for high order discretization of conservation laws. While the most widely used numerical scheme under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity of formulation and implementation. However, recently it has been shown that the scheme is not linearly stable on triangles. In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart–Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed. We demonstrate viability of the concept by showing linear stability both in the semi-discrete sense and for time stepping schemes of the SSP Runge–Kutta type. Furthermore, we present convergence studies, as well as case studies in compressible flow simulation using the Euler equations.     

Stable Algebraic Topology and Stable Topological Algebra

Algebraic topology is a young subject, and its foundations arenot yet firmly in place. I shall give some history, examplesand modern developments in that part of the subject called stablealgebraic topology, or stable homotopy theory. This is by farthe most calculationally accessible part of algebraic topology,although it is also the least intuitively grounded in visualizablegeometric objects. It has a great many applications to othersubjects such as algebraic geometry and geometric topology.Time will not allow me to say as much as I would like aboutthat. Rather, I shall emphasize some foundational issues thathave been central to this part of algebraic topology since theearly 1960s, but that have been satisfactorily resolved onlyin the last few years. 1991 Mathematics Subject Classification55P42, 55N20.     

A Functional Equation Whose Unknown is \mathcal{P}([0,1]) Valued

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.     

Students’ conceptual understanding of a function and its derivative in an experimental calculus course

Calculus has been witnessing fundamental changes in its curriculum, with an increased emphasis on visualization. This mode for representing mathematical concepts is gaining more strength due to the advances in computer technology and the development of dynamical mathematical software. This paper focuses on the understanding of the function and its derivative as viewed by students of a reformed Calculus 1 course offered in two experimental sections at the Lebanese American University in Beirut, Lebanon. Results have shown that the general approach adopted in the course proved to be unpopular for a great majority of the students, but rewarding for others. Interviews conducted with some students and a study of their performance on very specific exam questions reveal that for most students, the algebraic representation of a function still dominated their thinking; however, these students showed an almost complete understanding of the derivative, particularly the idea of the instantaneous rate of change and/or the slope of a curve at a given point. Furthermore, very few of these students referred to the mechanical methods for finding derivatives.     

Hypernuclear fine structure in (9)(lambda)Be

With a germanium detector array (Hyperball), we observed two gamma-ray peaks corresponding to the two transitions (5/2(+)-->1/2(+) and 3/2(+)-->1/2(+)) in the (9)(Lambda)Be hypernucleus which was produced by the 9Be(K-,pi(-)) reaction. The energies of the gamma rays are 3029 +/- 2 +/- 1 keV and 3060 +/- 2 +/- 1 keV. The energy difference was measured to be 31.4(+2.5)(-3.6) keV, which indicates a very small Lambda-spin-dependent spin-orbit force between a Lambda and a nucleon. This is the smallest level splitting by far ever measured in a hypernucleus.     

Surface reduction of freely floating smectic bubbles

Elongated freely floating smectic bubbles are observed during their relaxation to equilibrium sphere shape. Unlike soap bubbles that perform weakly damped oscillations into equilibrium, this relaxation is overdamped in smectics by internal structure reorganisation processes. The bubble area reduction of centimetre-sized freely floating bubbles with few nanometres film thickness is recorded with high-speed optical imaging in microgravity and analysed quantitatively. We find a nearly linear reduction of the film area with time, driven by capillary forces and inhibited by smectic layer reorganisations. Characteristic times are in the milliseconds range, with little correlation to the film thickness and bubble size. Instead, the homogeneity of the films and the number and sizes of islands of excess layers that spontaneously form on the films appear to have crucial influence on the dynamics. The efficiency of this process sets the time scale of the film area shrinkage. We discuss the limitations of a minimalistic model that captures smectic layer reorganisation processes.     

Rapid initiation of reactions in Al/Ni multilayers with nanoscale layering

Research into nanoenergetic materials is enabling new capabilities for controlling exothermic reaction rates and energy output, as well as new methods for integrating these materials with conventional electronics fabrication techniques. Many reactions produce primarily heat, and in some cases it is desirable to increase the rate of heat release beyond what is typically observed. Here we investigate the Al-Ni intermetallic reaction, which normally propagates across films or foils at rates lower than 10 m/s. However, models and experiments indicate that local heating rates can be very high (10⁷ K/s), and uniform heating of such a multilayer film can lead to a rapid, thermally explosive type of reaction. With the hopes of using a device to transduce electrical energy to kinetic energy of a flyer plate in the timescale of 100's of nanoseconds, we have incorporated a Ni/Al nanolayer film that locally heats upon application of a large electrical current. We observed flyer plate velocities in the 2-6 km/s range, corresponding to 4-36 kJ/g in terms of specific kinetic energy. Several samples containing Ni/Al films with different bilayer thicknesses were tested, and many produced additional kinetic energy in the 1.1-2.3 kJ/g range, as would be expected from the Ni-Al intermetallic reaction. These results provide evidence that nanoscale Ni/Al layers reacted in the timescale necessary to contribute to device output.     

On the cohomology of generalized homogeneous spaces

We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces , where is a finite loop space or a -compact group and is a ``subgroup' in the homotopical sense.

     

Correlation of speckle patterns generated by laser point source — Illuminated diffusers

In this paper it is shown that, when a ground-glass transilluminated by a diverging or converging beam is axially translated, there exists a particular plane in which the corresponding speckle pattern remains correlated. Consequently, under some experimental conditions on this incident beam and the geometry of the optical setup, a speckle pattern generated by a set of two parallel ground-glasses suffers no decorrelation when one of them is axially or laterally translated.