Universal resonant ultracold molecular scattering

The elastic scattering amplitudes of indistinguishable, bosonic, strongly-polar molecules possess universal properties at the coldest temperatures due to wave propagation in the long-range dipole-dipole field. Universal scattering cross sections and anisotropic threshold angular distributions, independent of molecular species, result from careful tuning of the dipole moment with an applied electric field. Three distinct families of threshold resonances also occur for specific field strengths, and can be both qualitatively and quantitatively predicted using elementary adiabatic and semi-classical techniques. The temperatures and densities of heteronuclear molecular gases required to observe these univeral characteristics are predicted. PACS numbers: 34.50.Cx, 31.15.ap, 33.15.-e, 34.20.-bComment: 4 pages, 5 figure

Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonlinear equations are expressed as the logarithmic norm of the Jacobian of the right-hand side of the system of equations. Sufficient conditions for the stability of solutions of systems of differential equations expressed in terms of logarithmic norms of the right-hand sides of equations (for systems of linear equations) and the Jacobian of right-hand sides (for nonlinear equations) have the following advantages: (1) in investigating stability in different metrics from the same standpoints, we have obtained a set of sufficient conditions; (2) sufficient conditions are easily expressed; (3) robustness areas of systems are easily determined with respect to the variation of their parameters; (4) in case of impulse action, information on moments of impact distribution is not required; (5) a method to obtain sufficient conditions of stability is extended to other definitions of stability (in particular, to p-moment stability). The obtained sufficient conditions are used to study Hopfield neural networks with discontinuous synapses and discontinuous activation functions

Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonlinear equations are expressed as the logarithmic norm of the Jacobian of the right-hand side of the system of equations. Sufficient conditions for the stability of solutions of systems of differential equations expressed in terms of logarithmic norms of the right-hand sides of equations (for systems of linear equations) and the Jacobian of right-hand sides (for nonlinear equations) have the following advantages: (1) in investigating stability in different metrics from the same standpoints, we have obtained a set of sufficient conditions; (2) sufficient conditions are easily expressed; (3) robustness areas of systems are easily determined with respect to the variation of their parameters; (4) in case of impulse action, information on moments of impact distribution is not required; (5) a method to obtain sufficient conditions of stability is extended to other definitions of stability (in particular, to p-moment stability). The obtained sufficient conditions are used to study Hopfield neural networks with discontinuous synapses and discontinuous activation functions

Regular, Singular and Hypersingular Integrals over Fractal Contours

The paper is devoted to the approximate calculation of Riemann definite integrals, singular and hypersingular integrals over closed and open non-rectifiable curves and fractals. The conditions of existence for the Riemann definite integrals over non-rectifiable curves and fractals are provided. We give a definition of a singular integral over non-rectifiable curves and fractals which generalizes the known one. We define hypersingular integrals over non-rectifiable curves and fractals. We construct quadratures for the calculation of Riemann definite integrals, singular and hypersingular integrals over non-rectifiable curves and fractals and the corresponding error estimates for various classes of functions. Singular and hypersingular integrals are defined up to an additive constant (or a combination of constants) that are subject to a convention that depends on the actual problem being solved. We illustrate our theoretical results with numerical examples for Riemann definite integrals, singular integrals and hypersingular integrals over fractals

Hypersingular Integral Equations of Prandtl’s Type: Theory, Numerical Methods, and Applications

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations

Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations

Hypersingular Integral Equations of Prandtl’s Type: Theory, Numerical Methods, and Applications

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations

Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations

High-Performance Optimization of Algorithms Used in the BM@N Experiment of the NICA Project

Results of high-performance optimization of BmnRoot software modules are presented. The BmnRoot package used in the BM@N experiment of the NICA project plays a crucial role in the simulation and event reconstruction so its performance should be maximized to make the data processing efficient. Results of performance analysis on representative testcases are given and bottlenecks are localized. Most suitable approaches to BmnRoot optimization are chosen and numerical estimates of the scalability of the parallelized modules for event reconstruction are presented