Search results for "Jacob"

Hamilton–Jacobi semi-groups in infinite dimensional spaces

2006

AbstractLet (X,ρ) be a Polish space endowed with a probability measure μ. Assume that we can do Malliavin Calculus on (X,μ). Let d:X×X→[0,+∞] be a pseudo-distance. Consider QtF(x)=infy∈X{F(y)+d2(x,y)/2t}. We shall prove that QtF satisfies the Hamilton–Jacobi inequality under suitable conditions. This result will be applied to establish transportation cost inequalities on path groups and loop groups in the spirit of Bobkov, Gentil and Ledoux.

On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method

2017

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method. …

Scheduled Relaxation Jacobi method: improvements and applications

2016

Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficien…

Optical Bistability and Switching in Oppositely Directed Coupler

2016

We report the optical bistability in two core oppositely directed coupler with negative index material channel. Using Langrangian variational method and Jacobi elliptic functions, we construct the solutions of the coupled nonlinear Schrodinger equations. The bistability arises due to the effective feedback mechanism as a result of opposite directionality of the phase velocity and energy flow in the negative index material channel. We report the various ways to control and manipulate the bistability threshold and hysteresis loop, which could be useful in the design and development of fast and low-threshold optical switches.

Relativistic positioning: errors due to uncertainties in the satellite world lines

2014

Global navigation satellite systems use appropriate satellite constellations to get the coordinates of an user -close to Earth- in an almost inertial reference system. We have simulated both GPS and GALILEO constellations. Uncertainties in the satellite world lines lead to dominant positioning errors. In this paper, a detailed analysis of these errors is developed inside a great region surrounding Earth. This analysis is performed in the framework of the so-called relativistic positioning systems. Our study is based on the Jacobian, J, of the transformation giving the emission coordinates in terms of the inertial ones. Around points of vanishing J, positioning errors are too large. We show …

A simple method for limiting the quadrature oscillator amplitude

1981

How to perform QCD analysis of DIS in Analytic Perturbation Theory

2015

We apply (Fractional) Analytic Perturbation Theory (FAPT) to the QCD analysis of the nonsinglet nucleon structure function $F_2(x,Q^2)$ in deep inelastic scattering up to the next leading order and compare the results with ones obtained within the standard perturbation QCD. Based on a popular parameterization of the corresponding parton distribution we perform the analysis within the Jacobi Polynomial formalism and under the control of the numerical inverse Mellin transform. To reveal the main features of the FAPT two-loop approach, we consider a wide range of momentum transfer from high $Q^2\sim 100 {\rm GeV}^2$ to low $Q^2\sim 0.3 {\rm GeV}^2$ where the approach still works.

The Numerical Simulation of Relativistic Fluid Flow with Strong Shocks

2001

In this review we present and analyze the performance of a Go-dunov type method applied to relativistic fluid flow. Our model equations are the corresponding Euler equations for special relativistic hydrodynamics. By choosing an appropriate vector of unknowns, the equations of special relativistic fluid dynamics (RFD) can be written as a hyperbolic system of conservation laws. We give a complete description of the spectral decomposition of the Jacobian matrices associated to the fluxes in each spatial direction, (see (Donat et al., 1998), for details), which is the essential ingredient of the Godunov-type numerical method we propose in this paper. We also review a numerical flux formula tha…

Computing Strong Shocks in Ultrarelativistic Flows: A Robust Alternative

1999

In recent years, shock capturing methods have started to be used in numerical simulations in Relativistic Fluid Dynamics (RFD). These techniques lead to explicit numerical codes that are able to successfully simulate the extreme conditions of the ultrarelativistic regime. After [2], an explicit, ready-to-use description of the full spectral decomposition of the Jacobian matrices of the RFD system is available, and this allows us to implement Marquina’s scheme [3] in RFD. The scheme is seen to maintain the good behavior shown in [3] with respect to certain numerical pathologies.

The Lie algebra of polynomial vector fields and the Jacobian conjecture

1998

The Jacobian conjecture for polynomial maps ϕ:Kn→Kn is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra\(\mathbb{D}\) of formal vector fields inn variables. To be precise, let\(\mathbb{D}_0 \) be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of\(\mathbb{D}_0 \),Hλ the root spaces corresponding to linear forms λ onH and\(A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda \). Then every polynomial map ϕ:Kn→Kn with invertible Jacobian matrix is an automorphism if and only if every automorphism Φ of\(\mathbb{D}\) with Φ(A)\( \subseteq A\) satisfies Φ(A)=A.