Algebraic Boundary Research Articles

On Numerical Differential Algebraic Problems with Application to Semiconductor Device Simulation

This paper considers questions regarding conditioning of, and numerical methods, for certain differential algebraic equations subject to initial and boundary conditions. The approach taken is that of separating “differential” and “algebraic” solution components, at least theoretically. This yields conditioning results for differential algebraic boundary value problems in terms of “pure” differential problems, for which existing theory is well developed. The process is carried out for problems with (global) index 1 or 2. For semi-explicit boundary value problems of index 1 (where solution components are separated) a convergence theorem is given for a special class of collocation methods. For general index 1 problems advantages and disadvantages of certain symmetric difference schemes are discussed. For initial value problems with index 2 the use of BDF schemes is examined, with a summary of conditions for their successful and stable utilization. Finally, the present considerations and analysis are applied to two problems involving differential algebraic equations arising in semiconductor device simulation.

Tracking parameterized algebraic curves on raster displays

An algorithm is presented for the tracking of parameterized algebraic curve segments on raster displays. Such segments are generally obtained by projecting the edges of an algebraic surface boundary representation describing a solid object. We sub-divide the curve segments such that each resultant segment is properly bounded by a guiding triangle. The triangle is used to approximate the segment sequentially and unambiguously on the integer coordinates of the display. The algorithm is both efficient and robust as opposed to existing parametric and algebraic tracking techniques.

Quadrature identities and the schottky double

By using Riemann surface theory we obtain results on quadrature domains and identities for analytic functions, e.g., existence of multiply-connected quadrature domains, descriptions of their algebraic boundaries and results on the multitude of quadrature domains associated to a fixed quadrature identity. The main idea is to characterize quadrature domains in terms of meromorphic functions and differentials on Riemann surfaces conformally equivalent to the Schottky doubles of the domains.

Synthesis of fully rotatable R-S-S-R linkages

Algebraic-geometrical techniques to design fully rotatable R-S-S-R linkages with transmission angle control are presented. The concept of upper bounding Cos μ where μ is the transmission angle, is used to obtain feasible regions for linkages with full input-link rotatability. The proposed feasible regions have algebraic boundaries. This concept was developed earlier for the precision synthesis of planar four-bar linkage and it has now been extended to the precision synthesis of R-S-S-R linkage.

The k-space formulation of the scattering problem in the time domain

The arbitrary direct scattering problem is solved numerically in closed form in the time domain and spatial Fourier transform space. This solution consists of casting the general basic global laws (i.e., the second-order partial differential wave equation or its integral representation) as a local algebraic equation in the spatial Fourier transform space, and leaving the specific local constitutive equations (i.e., the algebraic boundary conditions, which specify a given structure, which are conventionally imposed on the differential or integral representation of the general basic global wave equation) as a local algebraic equation in real space, thereby reducing the scattering problem to a statement of two simultaneous local algebraic equations in two unknowns (the fields and the induced sources) in two spaces connected by the spatial Fourier transform. A temporally local representation in both spaces is obtained with the aid of an introduced auxiliary field and two propagators. By virtue of causality, a numerically efficient closed form solution to this set of equations is obtained that utilizes the fast Fourier transform algorithm as the transformations between the two spaces. By virtue of the numerically efficient fast Fourier transform algorithm and the local algebraic representations, the number of required complex multiply-add operations and storage allocation is of the order of N log2 N and N per temporal discretization, respectively (where N is the number of spatial cells into which the scattering problem is descretized). It is shown that the solution is only of the order of log2 N slower than an ideal solution. The solution is thus practical for very large one-, two-, and three-dimensional scattering problems. Numerico-experimental results for a variety of refrective index profiles, including perfect reflectors, are presented, thus verifying the presented algorithm.

Boundaries for polydisc algebras in infinite dimensions

AbstractLet AB be the algebra of all bounded continuous functions on the closed unit ball B of c0, analytic on the open unit ball, with sup norm, and let AU be the sub-algebra of AB of those functions which are uniformly continuous on B. Call a set S ⊂ B a boundary of AB (AU) iffor every f ∈ AB (f ∈AU, respectively). In the paper we study the boundaries of AB and AU. We give a complete description of the boundaries of AU and present some necessary and some sufficient conditions for a set to be a boundary of AB. We also give some examples of boundaries.

Effective boundary conditions for reflection and transmission by an absorbing shell of arbitrary shape

The reflection and transmission properties of an absorbing shell of arbitrary shape can be described by two algebraic boundary conditions relating the four tangential fields E_{+}, H_{+}, E_{-}, H_{-} at corresponding points on the outer surface S_{+} and inner surface S_{-} , respectively. The boundary conditions contain first-order curvature correction terms, but it is still necessary that the shell curvature be reasonably small compared to the attenuation per unit distance in the shell material. Especially simple conditions are obtained for shells thin compared to the attenuation length. The shell boundary conditions can also be used to determine an effective surface impedance for a conducting body coated by multiple layers of absorbing material.

IBM-Fortran computer calculations for analytical ultracentrifuge data

Weight-average sedimentation coefficients, S app, from the second moment of the total gradient curve (Goldberg equation), diffusion coefficients, D app, from σ 2 = 2 Dt, and molecular weight determination from approach to sedimentation equilibrium may be calculated on a computer. Gilbert-Jenkins algebraic boundary analysis is also programed and the validity of the conservation equation for a reversibly interacting system (A + B ⇆ C) has been tested. The results of computer calculations are shown, and the usefulness of the programing is demonstrated with the bovine serum albumin-pepsin interaction. Among the protein-protein interactions to which such a computer program can be applied are β-lactoglobulin-casein, casein-casein, trypsinovomucoid, conalbumin-lysozyme, DNA-bovine serum albumin, RNA-ovomucoid, RNA- or DNA-protein, antigen-antibody complex, and numerous others.