Search results for "Mathematical analysis"
showing 10 items of 2409 documents
The fixed angle scattering problem with a first order perturbation
2021
We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by $2n$ measurements up to a natural gauge. We also show that one can recover the full first order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and M. Salo to Hamiltonians with first order perturbations, and it is based on wave equation methods and Carleman estimates.
The unequal mass sunrise integral expressed through iterated integrals on M‾1,3
2020
Abstract We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter e. In order to do so, we transform the system of differential equations for the master integrals to an e-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M 1 , 3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M ‾ 1 , 3 . On the hypersurface τ = const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.
EXTRACTION OF INFRARED DIVERGENCES IN THE DIMENSIONAL REGULARIZED TWO-LOOP LADDER GRAPH
1994
We consider the evaluation of the fundamental scalar integral in the on-shell two-loop ladder graph with different external masses and arbitrary transfer momentum. A method for cleanly extracting the infrared divergences in the Feynman parameter integrals using dimensional regularization is presented, and we analyze one of the finite part contributions to this integral.
Electrical analogous in viscoelasticity
2014
In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behavior are firstly approached by using fractional mathematical operators. Viscoelastic models have elastic and viscous components which are obtained by combining springs and dashpots. Various arrangements of these elements can be used, and all of these viscoelastic models can be equivalently modeled as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models are validated by using modal analysis. Moreover, a comparison with numerical expe…
Efficient finite difference formulation of a geometrically nonlinear beam element
2021
The article is focused on a two-dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element lev…
Two-dimensional quantum scattering by non-isotropic interactions localized on a circle, applications to open billiards
2018
Two-dimensional quantum scattering by isotropic and non-isotropic interactions localized on a circle is considered. The expansion of the interaction on the circle in a Fourier series allows us to express basic objects of scattering theory (resolvent, T operator, differential cross length, cross length, and cross length averaged over all orientations of the incident particle), in terms of operations on matrices. For numerical applications, these matrices are truncated to a given order. If the interaction is isotropic, this general formulation reduces to the usual one, and the resonances in the isotropic cases are studied because they allow us to interpret resonances in some non-isotropic cas…
Comparison of two non-primitive methods for path integral simulations: Higher-order corrections vs. an effective propagator approach
2002
Two methods are compared that are used in path integral simulations. Both methods aim to achieve faster convergence to the quantum limit than the so-called primitive algorithm (PA). One method, originally proposed by Takahashi and Imada, is based on a higher-order approximation (HOA) of the quantum mechanical density operator. The other method is based upon an effective propagator (EPr). This propagator is constructed such that it produces correctly one and two-particle imaginary time correlation functions in the limit of small densities even for finite Trotter numbers P. We discuss the conceptual differences between both methods and compare the convergence rate of both approaches. While th…
An exact soliton solution for an averaged dispersion-managed fibre system equation
2001
We consider the nonlinear wave propagation in an averaged dispersion-managed (DM) fibre system. We present the explicit Lax pair with a variable spectral parameter and derive the exact soliton solution using the Backlund transformation. A similar study is also carried out for simultaneous propagation of N nonlinear pulses in the averaged DM fibre system.
Collective variable theory for optical solitons in fibers
2000
We present a projection-operator method to express the generalized nonlinear Schrödinger equation for pulse propagation in optical fibers, in terms of the pulse parameters, called collective variables, such as the pulse width, amplitude, chirp, and frequency. The collective variable (CV) equations of motion are derived by imposing a set of constraints on the CVs to minimize the soliton dressing during its propagation. The lowest-order approximation of this CV approach is shown to be equivalent to the variational Lagrangian method. Finally, we demonstrate the application of this CV theory for pulse propagation in dispersion-managed optical fiber links.
Analytical design of densely dispersion-managed optical fiber transmission systems with Gaussian and raised cosine return-to-zero Ansätze
2004
We propose an easy and efficient way to analytically design densely dispersion-managed fiber systems for ultrafast optical communications. This analytical design is based on the exact solution of the variational equations derived from the nonlinear Schrodinger equation by use of either a Gaussian or a raised-cosine (RC) Ansatz. For the input pulses of dispersion-managed optical fiber transmission systems we consider a RC profile and show that RC return-to-zero pulses are as effective as Gaussian pulses in high-speed (160-Gbits/s) long-distance transmissions.