A note on the equivalence of shock manifold equations

The shock manifold equation is a first order nonlinear partial differential equation, which describes the kinematics of a shockfront in an ideal gas with constant specific heats. However, it was found that there was more than one of these shock manifold equations, and the shock surface could be embedded in a one parameter family of surfaces, obtained as a solution of any of these shock manifold equations. Associated with each shock manifold equation is a set of characteristic curves called lsquoshock raysrsquo. This paper investigates the nature of various associated shock ray equations.

Euclideanization, topological theories, higher dimensions and all that

It is shown that the euclideanized Yukawa theory, with the Dirac fermion belonging to an irreducible representation of the Lorentz group, is not bounded from below. A one parameter family of supersymmetric actions is presented which continuously interpolates between the N = 2 SSYM and the N = 2 supersymmetric topological theory. In order to obtain a theory which is bounded from below and satisfies Osterwalder-Schrader positivity, the Dirac fermion should belong to a reducible representation of the Lorentz group and the scalar fields have to be reinterpreted as the extra components of a higher dimensional vector field.

Some properties of boundary layer flow during the transition from laminar to turbulent motion

Transition in the boundary layer on a flat plate is examined from the point of view of intermittent production of turbulent spots. On the hypothesis of localized laminar breakdown, for which there is some expermental evidence, Emmons’ probability calculations can be extended to explain the observed statistical similarity of transition regions. Application of these ideas allows detailed calculations of the boundary layer parameters including mean velocity profiles and skin friction during transition. The mean velocity profiles belong to a universal one-parameter family with the intermittency factor as the parameter. From an examination of experimental data the probable existence of a relation between the transition Reynolds number and the rate of production of the turbulent spots is deduced. A simple new technique for the measurement of the intermittency factor by a Pitot tube is reported.

KCL, Ray Theories and Application to Sonic Boom

We first review a general formulation of ray theory and write down the conservation forms of the equations of a weakly nonlinear ray theory (WNLRT) and a shock ray theory (SRT) for a weak shock in a polytropic gas. Then we present a formulation of the problem of sonic boom by a maneuvering aerofoil as a one parameter family of Cauchy problems. The system of equations in conservation form is hyperbolic for a range of values of the parameter and has elliptic nature else where, showing that unlike the leading shock, the trailing shock is always smooth.

Calculation of the front part of the sonic boom signature for a Maneuvering aerofopil

The sonic boom at a large distance from its source consists of a leading shock, a trailing shock and a one parameter family of nonlinear wavefronts in between these shocks. A new ray theoretical method using a shock ray theory and a weakly nonlinear lay theory has been used to obtain the shock fronts and wavefronts respectively, for a maneuvering aerofoil in a homogeneous medium. This method introduces a one parameter family of Cauchy problems to calculate the shock and wave fronts emerging from the surface of the aerofoil. These problems are solved numerically to obtain the leading shock front and the nonlinear wavefronts emerging from the front portion of the aerofoil.

Static balancing of a four-bar linkage and its cognates

Motivated by the need to statically balance the inherent elastic forces in linkages, this paper presents three techniques to statically balance a four-bar linkage loaded by a zero-free-length spring attached between its coupler point and an anchor point on the ground. The number of auxiliary links and balancing springs required for the three techniques is less than or equal to that of the only technique currently in the literature. One of the three techniques does not require auxiliary links. In these techniques, the set of values for the spring constants and the ground-anchor point of the balancing springs can vary over a one-parameter family. Thrice as many balancing choices are available when the cognates are considered. The ensuing numerous options enable a user to choose the most practical solution. To facilitate the evaluation of the balancing choices for all the cognates, Roberts-Chebyshev cognate theorem is extended to statically balanced four-bar linkages. (C) 2011 Elsevier Ltd. All rights reserved.

Analyticity of the Schrodinger propagator on the Heisenberg group

We discuss the analytic extension property of the Schrodinger propagator for the Heisenberg sublaplacian and some related operators. The result for the sublaplacian is proved by interpreting the sublaplacian as a direct integral of an one parameter family of dilated special Hermite operators.

Minimization Problems Based on Relative alpha-Entropy I: Forward Projection

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative alpha-entropies (denoted I-alpha), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative alpha-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative alpha-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Renyi or Tsallis entropy principle. The minimizing probability distribution (termed forward I-alpha-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse I-alpha-projection is studied.

The Type Ia supernova 2004s, a clone of SN 2001el, and the optimal photometric bands for extinction estimation

We present optical (UBVRI) and near-IR (YJHK) photometry of the normal Type Ia supernova (SN) 2004S. We also present eight optical spectra and one near-IR spectrum of SN 2004S. The light curves and spectra are nearly identical to those of SN 2001el. This is the first time we have seen optical and IR light curves of two Type Ia SNe match so closely. Within the one parameter family of light curves for normal Type Ia SNe, that two objects should have such similar light curves implies that they had identical intrinsic colors and produced similar amounts of Ni-56. From the similarities of the light-curve shapes we obtain a set of extinctions as a function of wavelength that allows a simultaneous solution for the distance modulus difference of the two objects, the difference of the host galaxy extinctions, and RV. Since SN 2001el had roughly an order of magnitude more host galaxy extinction than SN 2004S, the value of R-V = 2.15(-0.22)(+0.24) pertains primarily to dust in the host galaxy of SN 2001el. We have also shown via Monte Carlo simulations that adding rest-frame J-band photometry to the complement of BVRI photometry of Type Ia SNe decreases the uncertainty in the distance modulus by a factor of 2.7. A combination of rest-frame optical and near-IR photometry clearly gives more accurate distances than using rest-frame optical photometry alone.

STABLE PIECEWISE POLYNOMIAL VECTOR FIELDS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Slow-fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4], Llibre et al. (2007, 2008) [12,13] concerning the dynamics of non-smooth systems. In those papers a piecewise C-k discontinuous vector field Z on R-n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F :U -> R a polynomial function defined on the open subset U subset of R-n. The set F-1 (0) divides U into subdomains U-1, U-2,...,U-k, with border F-1(0). These subdomains provide a Whitney stratification on U. We consider Z(i) :U-i -> R-n smooth vector fields and we get Z = (Z(1),...., Z(k)) a discontinuous vector field with discontinuities in F-1(0). Our approach combines several techniques such as epsilon-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an epsilon-regularization of Z (see Sotomayor and Teixeira, 1996 [18]; Llibre and Teixeira, 1997 [15]). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16]), in systems with hysteresis (Seidman, 2006 [17]) and in mechanical systems with impacts (di Bernardo et al., 2008 [5]). (C) 2011 Elsevier Masson SAS. All rights reserved.

Interference phenomenon for the Faddeevian regularization of 2D chiral fermionic determinants

The classification of the regularization ambiguity of a 2D fermionic determinant in three different classes according to the number of second-class constraints, including the new Faddeevian regularization, is examined and extended. We find a new and important result that the Faddeevian class, with three second-class constraints, possesses a free continuous one parameter family of elements. The criterion of unitarity restricts the parameter to the same range found earlier by Jackiw and Rajaraman for the two-constraint class. We studied the restriction imposed by the interference of right-left modes of the chiral Schwinger model (χQED2) using Stone's soldering formalism. The interference effects between right and left movers, producing the massive vectorial photon, are shown to constrain the regularization parameter to belong to the four-constraint class which is the only nonambiguous class with a unique regularization parameter. ©1999 The American Physical Society.

3-Dimensional hopf bifurcation via averaging theory

We consider the Lorenz system ẋ = σ(y - x), ẏ = rx - y - xz and ż = -bz + xy; and the Rössler system ẋ = -(y + z), ẏ = x + ay and ż = b - cz + xz. Here, we study the Hopf bifurcation which takes place at q± = (±√br - b,±√br - b, r - 1), in the Lorenz case, and at s± = (c+√c2-4ab/2, -c+√c2-4ab/2a, c±√c2-4ab/2a) in the Rössler case. As usual this Hopf bifurcation is in the sense that an one-parameter family in ε of limit cycles bifurcates from the singular point when ε = 0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two 2-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other 3- or n-dimensional differential systems.

On the consistency conditions to braneworlds in scalar-tensor gravity for arbitrary dimensions

We derive an one-parameter family of consistency conditions to braneworlds in the Brans-Dicke gravity. The General Relativity case is recovered by taking a correct limit of the Brans-Dicke parameter. We show that it is possible to build a multiple AdS brane scenario in a six-dimensional bulk only if the brane tensions are negative. Besides, in the five-dimensional case, it is showed that no fine tuning is necessary between the bulk cosmological constant and the brane tensions, in contrast to the Randall-Sundrum model. Copyright © owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial- ShareAlike Licence.

Variedade central para laços homoclínicos

Pós-graduação em Matemática - IBILCE