Symposium: Brown v. Board of Education and Its Legacy: A Tribute to Justice Thurgood Marshall, Learning Together: Justice Marshall\u27s Desegregation Opinions

In this Article, Professor Marcus examines the influence of Justice Thurgood Marshall on the Supreme Court\u27s current school desegregation agenda. Justice Marshall was part of the majority in desegregation cases during his earlier years on the high Court subsequently, however, his role became one of dissenter. Professor Marcus analyzes the divisive issues facing the Court in desegregation litigation, Marshall\u27s positions on such issues, and his legacy to the Court in this area. Finally, the Article assesses the vitality of this legacy in light of two Supreme Court decisions issued after Justice Marshall\u27s retirement

Adaptive ACMS: A robust localized Approximated Component Mode Synthesis Method

We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follows the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, however it deteriorates with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$

Hybrid Localized Spectral Decomposition for multiscale problems

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity

Collisional broadening and spectral line shape of an entire rotational band

The impact approximation is applied to the classical binary collision operator making it possible to derive an expression for the dipole correlation function for real systems in a form which is computationally tractable and contains no adjustable parameters. Trajectory calculations are performed (in order to evaluate the microscopic expression for the relaxation parameter in the correlation function) for the system CO in dense Ar gas. Comparison is made with experimental data and excellent agreement is found for certain densities when a quantum correction is included. At higher densities (i.e., ρ^(−1/3)< "the range of the potential") one approximation is not valid and comparison with experiment illustrates this point

A Jang Equation Approach to the Penrose Inequality

We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the nonspherical setting would yield a proof of the full Penrose Conjecture. Thus it remains as an important and challenging open problem to determine whether this system does indeed admit the desired solutions.Comment: 31 page

Laser extensometer

A drift compensated and intensity averaged extensometer for measuring the diameter or other properties of a substantially cylindrical sample based upon the shadow of the sample is described. A beam of laser light is shaped to provide a beam with a uniform intensity along an axis normal to the sample. After passing the sample, the portion of the beam not striking said sample is divided by a beam splitter into a reference signal and a measurement signal. Both of these beams are then chopped by a light chopper to fall upon two photodiode detectors. The resulting ac currents are rectified and then divided into one another, with the final output being proportional to the size of the sample shadow

Design of a Solid Freeform Fabrication Diamond Reactor

Solid Freeform Fabrication (SFF) has progressed from the visualization aided stage of computer aided designs (CAD) to rapid prototyping of structural parts. Among the promising techniques for producing structural prototypes is the technology ofchemical vapor deposition (CVD) ofpolycrystalline diamond. This paper discusses the thermodynamic and kinetic theories that suggest that structural diamond may be rapidly deposited at rates approaching 1 mmJhr from the vapor phase at metastable thermodynamic conditions. The design of a reactor that will produce structural diamond prototypes is discussed. This reactor combines downstream microwave plasma enhanced chemical vapor deposition (DMWPECVD) with a scanned CO2 laser that locally heats the substrate to diamond deposition temperatures. The input:Fases are H2, 02' CH4, and Ar. The operating pressure range of the reactor is 1 x 10- to 7 x 102 Torr. The reactor is designed for in situ determination of deposit thickness while deposition occurs as well as having the capacity of fitting on an existing resonance enhanced multiphoton ionization time of flight mass spectroscopy (REMPITOFMS) apparatus that will allow for plasma diagnostics immediately above the heated substrate. Plasma diagnostics will be employed to determine the active metastable species that results in diamond deposition so that optimization can be made ofthe operating parameters to maximize diamond selectivity and deposition rate.Mechanical Engineerin