Renormalization of singular potentials and power counting

We use a toy model to illustrate how to build effective theories for singular potentials. We consider a central attractive 1/r(2) potential perturbed by a 1/r(4) correction. The power-counting rule, an important ingredient of effective theory, is established by seeking the minimum set of short-range counterterms that renormalize the scattering amplitude. We show that leading-order counterterms are needed in all partial waves where the potential overcomes the centrifugal barrier, and that the additional counterterms at next-to-leading order are the ones expected on the basis of dimensional analysis. (C) 2008 Elsevier B.V. All rights reserved.

A singular approach to discontinuous vector fields on the plane

In this paper we deal with discontinuous vector fields on R-2 and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation. The regularization process developed by Sotomayor and Teixeira is crucial for the development of this work. (c) 2006 Elsevier B.V. All rights reserved.

Regularization of discontinuous vector fields on R-3 via singular perturbation

Singular perturbations problems in dimension three which are approximations of discontinuous vector fields are studied in this paper. The main result states that the regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Moreover, the definition of' sliding vector field coincides with the reduced problem of the corresponding singular problem for a class of vector fields.

Hamilton-Jacobi formulation for singular systems with second-order Lagrangians

Recently, the Hamilton-Jacobi formulation for first-order constrained systems has been developed. In such formalism the equations of motion are written as total differential equations in many variables. We generalize the Hamilton-Jacobi formulation for singular systems with second-order Lagrangians and apply this new formulation to Podolsky electrodynamics, comparing with the results obtained through Dirac's method.

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in R-n without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

Analytical attitude propagation with non-singular variables and gravity gradient torque for spin stabilized satellite

An analytical approach for the spin stabilized satellite attitude propagation is presented using the non-singular canonical variables to describe the rotational motion. Two sets of variables were introduced for Fukushima in 1994 by a canonical transformation and they are useful when the angle between z-satellite axis of a coordinate system fixed in artificial satellite and the rotational angular momentum vector is zero or when the angle between Z-equatorial axis and rotation angular momentum vector is zero. Analytical solutions for rotational motion equations and torque-free motion are discussed in terms of the elliptic functions and by the application of some simplification to get an approximated solution. These solutions are compared with a numerical solution and the results show a good agreement for many rotation periods. When the mean Hamiltonian associated with the gravity gradient torque is included, an analytical solution is obtained by the application of the successive approximations' method for the satellite in an elliptical orbit. These solutions show that the magnitude of the rotation angular moment is not affected by the gravity gradient torque but this torque causes linear and periodic variations in the angular variables, long and short periodic variations in Z-equatorial component of the rotation angular moment and short periodic variations in x-satellite component of the rotation angular moment. The goal of this analysis is to emphasize the geometrical and physical meaning of the non-singular variables and to validate the approximated analytical solution for the rotational motion without elliptic functions for a non-symmetrical satellite. The analysis can be applied for spin stabilized satellite and in this case the general solution and the approximated solution are coincidence. Then the results can be used in analysis of the space mission of the Brazilian Satellites. (C) 2007 COSPAR. Published by Elsevier Ltd. All rights reserved.

GROUND-STATE-ENERGY OF SINGULAR POTENTIALS

It is shown that for singular potentials of the form lambda/r(alpha),the asymptotic form of the wave function both at r --> infinity and r --> 0 plays an important role. Using a wave function having the correct asymptotic behavior for the potential lambda/r(4), it is, shown that it gives the exact ground-state energy for this potential when lambda --> 0, as given earlier by Harrell [Ann. Phys. (NY) 105, 379 (1977)]. For other values of the coupling parameter X, a trial basis;set of wave functions which also satisfy the correct boundary conditions at r --> infinity and r --> 0 are used to find the ground-state energy of the singular potential lambda/r(4) It is shown that the obtained eigenvalues are in excellent agreement with their exact ones for a very large range of lambda values.

Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R(l), l >= 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

VECTOR FIELDS TANGENT TO FOLIATIONS

We investigate in this paper the topological stability of pairs (omega, X), where w is a germ of an integrable 1-form and X is a germ of a vector field tangent to the foliation determined by omega.

The Dirac equation in a non-Riemannian manifold. I. An analysis using the complex algebra

The Dirac wave equation is obtained in the non-Riemannian manifold of the Einstein-Schrödinger nonsymmetric theory. A new internal connection is determined in terms of complex vierbeins, which shows the coupling of the electromagnetic potential with gravity in the presence of a spin-1/2 field. © 1988 American Institute of Physics.

Riemannian and teleparallel descriptions of the scalar field gravitational interaction

A comparative study between the metric and the teleparallel descriptions of gravitation is made for the case of a scalar field. In contrast to the current belief that only spin matter could detect the teleparallel geometry, scalar matter being able to feel the metric geometry only, we show that a scalar field is able not only to feel anyone of these geometries, but also to produce torsion. Furthermore, both descriptions are found to be completely equivalent, which means that in fact, besides coupling to curvature, a scalar field couples also to torsion.

Construction of exact Riemannian instanton solutions

We present the exact construction of Riemannian (or stringy) instantons, which are classical solutions of 2D Yang-Mills theories that interpolate between initial and final string configurations. They satisfy the Hitchin equations with special boundary conditions. For the case of U(2) gauge group those equations can be written as the sinh-Gordon equation with a delta-function source. Using the techniques of integrable theories based on the zero curvature conditions, we show that the solution is a condensate of an infinite number of one-solitons with the same topological charge and with all possible rapidities.

Analysis of numeric conditioning of Hessian matrix of Lagrangian via singular values

This paper presents an analyze of numeric conditioning of the Hessian matrix of Lagrangian of modified barrier function Lagrangian method (MBFL) and primal-dual logarithmic barrier method (PDLB), which are obtained in the process of solution of an optimal power flow problem (OPF). This analyze is done by a comparative study through the singular values decomposition (SVD) of those matrixes. In the MBLF method the inequality constraints are treated by the modified barrier and PDLB methods. The inequality constraints are transformed into equalities by introducing positive auxiliary variables and are perturbed by the barrier parameter. The first-order necessary conditions of the Lagrangian function are solved by Newton's method. The perturbation of the auxiliary variables results in an expansion of the feasible set of the original problem, allowing the limits of the inequality constraints to be reached. The electric systems IEEE 14, 162 and 300 buses were used in the comparative analysis. ©2007 IEEE.

Limit cycles for singular perturbation problems via inverse integrating factor

In this paper singularly perturbed vector fields Xε defined in ℝ2 are discussed. The main results use the solutions of the linear partial differential equation XεV = div(Xε)V to give conditions for the existence of limit cycles converging to a singular orbit with respect to the Hausdorff distance. © SPM.

Stochastic exponential stability of singular linear systems with Markov jump parameters

This paper deals with exponential stability of discrete-time singular systems with Markov jump parameters. We propose a set of coupled generalized Lyapunov equations (CGLE) that provides sufficient conditions to check this property for this class of systems. A method for solving the obtained CGLE is also presented, based on iterations of standard singular Lyapunov equations. We present also a numerical example to illustrate the effectiveness of the approach we are proposing.