On the Computation of the Topology of a Non-Reduced Implicit Space Curve

An algorithm is presented for the computation of the topology of a non-reduced space curve defined as the intersection of two implicit algebraic surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the original space curve. The algorithm is designed to provide the exact result for all inputs. It's a symbolic-numeric algorithm based on subresultant computation. Simple algebraic criteria are given to certify the output of the algorithm. The algorithm uses only one projection of the non-reduced space curve augmented with adjacency information around some "particular points" of the space curve. The algorithm is implemented with the Mathemagix Computer Algebra System (CAS) using the SYNAPS library as a backend

Computation of the degree of rational surface parametrizations

A rational affine parametrization of an algebraic surface establishes a rational correspondence of the affine plane with the surface. We consider the problem of computing the degree of such a rational map. In general, determining the degree of a rational map can be achieved by means of elimination theoretic methods. For curves, it is shown that the degree can be computed by gcd computations. In this paper, we show that the degree of a rational map induced by a surface parametrization can be computed by means of gcd and univariate resultant computations. The basic idea is to express the elements of a generic fibre as the finitely many intersection points of certain curves directly constructed from the parametrization, and defined over the algebraic closure of a field of rational functions

Inversion mechanism for the transport current in type-II superconductors

The longitudinal transport problem (the current is applied parallel to some bias magnetic field) in type-II superconductors is analyzed theoretically. Based on analytical results for simplified configurations, and relying on numerical studies for general scenarios, it is shown that a remarkable inversion of the current flow in a surface layer may be predicted under a wide set of experimental conditions. Strongly inhomogeneous current density profiles, characterized by enhanced transport toward the center and reduced, or even negative, values at the periphery of the conductor, are expected when the physical mechanisms of flux depinning and consumption (via line cutting) are recalled. A number of striking collateral effects, such as local and global paramagnetic behavior, are predicted. Our geometrical description of the macroscopic material laws allows a pictorial interpretation of the physical phenomena underlying the transport backflow.Comment: 8 pages, 6 figures (Best quality pictures are available by author's contact

Polymer-mediated entropic forces between scale-free objects

The number of configurations of a polymer is reduced in the presence of a barrier or an obstacle. The resulting loss of entropy adds a repulsive component to other forces generated by interaction potentials. When the obstructions are scale invariant shapes (such as cones, wedges, lines or planes) the only relevant length scales are the polymer size R_0 and characteristic separations, severely constraining the functional form of entropic forces. Specifically, we consider a polymer (single strand or star) attached to the tip of a cone, at a separation h from a surface (or another cone). At close proximity, such that h<<R_0, separation is the only remaining relevant scale and the entropic force must take the form F=AkT/h. The amplitude A is universal, and can be related to exponents \eta governing the anomalous scaling of polymer correlations in the presence of obstacles. We use analytical, numerical and epsilon-expansion techniques to compute the exponent \eta for a polymer attached to the tip of the cone (with or without an additional plate or cone) for ideal and self-avoiding polymers. The entropic force is of the order of 0.1 pN at 0.1 micron for a single polymer, and can be increased for a star polymer.Comment: LaTeX, 15 pages, 4 eps figure

Computational Methods for Nonlinear Systems Analysis With Applications in Mathematics and Engineering

An investigation into current methods and new approaches for solving systems of nonlinear equations was performed. Nontraditional methods for implementing arc-length type solvers were developed in search of a more robust capability for solving general systems of nonlinear algebraic equations. Processes for construction of parameterized curves representing the many possible solutions to systems of equations versus finding single or point solutions were established. A procedure based on these methods was then developed to identify static equilibrium states for solutions to multi-body-dynamic systems. This methodology provided for a pictorial of the overall solution to a given system, which demonstrated the possibility of multiple candidate equilibrium states for which a procedure for selection of the proper state was proposed. Arc-length solvers were found to identify and more readily trace solution curves as compared to other solvers making such an approach practical. Comparison of proposed methods was made to existing methods found in the literature and commercial software with favorable results. Finally, means for parallel processing of the Jacobian matrix inherent to the arc-length and other nonlinear solvers were investigated, and an efficient approach for implementation was identified. Several case studies were performed to substantiate results. Commercial software was also used in some instances for additional results verification

Theoretical study of corrugated plates: Shearing of a corrugated plate with curvilinear corrugations

A theoretical elastic analysis is presented for the in-plane shear of a corrugated plate with curvilinear corrugations and with discrete attachments between the ends of the corrugations and the surrounding members. The purpose of the analysis is to obtain information about the effective shear stiffness of the plate and the flexural strains that develop in it during the shearing process as a result of the attendant cross-sectional deformations. The crests and troughs of the corrugation cross section are assumed to be identical circular arcs. The following four kinds of discrete attachment at the corrugation ends are considered in the analysis: (1) point attachments in the troughs, (2) point attachments at the crests and in the troughs, (3) point attachments at mid-height, and (4) point attachments at mid-height, at the crests, and in the troughs