Simulating non-linear coupled oscillators by an iterative differential quadrature method

An iterative method based on differential quadrature rules is proposed as a new unified frame of resolution for non-linear two-degree-of-freedom systems. Dynamical systems with Duffing-type non-linearity have been considered. Differential quadrature rules have been applied with a careful distribution of sampling points to reduce the governing equation of motion to two second-order non-linear, non-autonomous ordinary differential equations and to solve the time-domain problem. The time domain of the problem is discretized by means of time intervals, with the same distribution of sampling points used to discretize the space domain (which can be seen as a single interval). It will be shown that accurate solutions depend not only on the choice of the distribution of sampling points, but also on the length of the time interval one refers to in the computations. The numerical results, utilized to draw Poincaré maps, are successfully compared with those obtained using the Runge-Kutta method.     

Landscape of solutions in constraint satisfaction problems

We present a theoretical framework for characterizing the geometrical properties of the space of solutions in constraint satisfaction problems, together with practical algorithms for studying this structure on particular instances. We apply our method to the coloring problem, for which we obtain the total number of solutions and analyze in detail the distribution of distances between solutions.     

Using hysteresis for optimization

We propose a new optimization method based on a demagnetization procedure well known in magnetism. We show how this procedure can be applied as a general tool to search for optimal solutions in any system where the configuration space is endowed with a suitable "distance." We test the new algorithm on frustrated magnetic models and the traveling salesman problem. We find that the new method successfully competes with similar basic algorithms such as simulated annealing.     

On Solutions of the Integrable Boundary Value Problem for KdV Equation on the Semi-Axis

This paper is concerned with the Korteweg–de Vries (KdV) equation on the semi-axis. The boundary value problem with inhomogeneous integrable boundary conditions is studied. We establish some characteristic properties of solutions of the problem. Also we construct a wide class of solutions of the problem using the inverse spectral method.     

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.     

Application of the complex point source method to the Schrödinger equation

The paraxial wave equation is a reduced form of the Helmholtz equation. Its solutions can be directly obtained from the solutions of the Helmholtz equation by using the method of complex point source. We applied the same logic to quantum mechanics, because the Schrödinger equation is parabolic in nature as the paraxial wave equation. We defined a differential equation, which is analogous to the Helmholtz equation for quantum mechanics and derived the solutions of the Schrödinger equation by taking into account the solutions of this equation with the method of complex point source. The method is applied to the problem of diffraction of matter waves by a shutter.     

Sampling of fractional bandlimited signals associated with fractional Fourier transform

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of reconstructing a fractional bandlimited signal with FRFT. We propose a novel reconstruction method for fractional bandlimited signals using the fractional Fourier series (FRFS). The advantage is that the sampling expansion can be deduced directly not based on the Shannon theorem. By utilizing the generalized form of Parseval’s relation for complex FRFS, we obtain the sampling expansion for fractional bandlimited signals with FRFT. We show that the sampling expansion for fractional bandlimited signals with FRFT is a special case of Parseval’s relation for complex FRFS.     

Second harmonic generation: the solution for an amplitude-modulated initial pulse

We address the initial value problem for one-dimensional second harmonic generation starting from a purely amplitude-modulated fundamental wave. A general method to solve the problem in terms of a Schrödinger equation is presented, in which the initial pulse-shape is taken as a potential. Several examples with the complete solution given in analytical form are discussed. A much broader class of solutions can be found with the help of a single numerical integration. In particular, solutions with incident pulses approximating a sech -shape have been obtained.     

Consistent determination of photoluminescence quantum efficiency for phosphors in the form of solution, plate, thin film, and powder

We have developed an improved method for easily determining the photoluminescence quantum efficiencies of transparent materials, such as solutions, glass plates, and thin films on a substrate, having various absorbances from ca. 0.05 to 1.0 at the excitation wavelength when reabsorption is negligible. The estimated accuracy for emitting semiconductor nanocrystals is ±5% for solutions. The efficiencies of non-transparent material, such as powder, together with the above-mentioned transparent materials were measured using the traditional integrating sphere method. Comparison of the two values showed that the traditional integrating sphere method usually underestimates the efficiency of powder samples ca. 10-20% depending on the optical density of the powder. This is because the emissions from more than ca. 0.2 mm deep do not leave the powder sample due to internal scattering. We also developed a method to overcome this problem.     

Improving the sampling efficiency of Monte Carlo molecular simulations: an evolutionary approach

We present a new approach in order to improve the convergence of Monte Carlo (MC) simulations of molecular systems belonging to complex energetic landscapes: the problem is redefined in terms of the dynamic allocation of MC move frequencies depending on their past efficiency, measured with respect to a relevant sampling criterion. We introduce various empirical criteria with the aim of accounting for the proper convergence in phase space sampling. The dynamic allocation is performed over parallel simulations by means of a new evolutionary algorithm involving ‘immortal’ individuals. The method is bench marked with respect to conventional procedures on a model for melt linear polyethylene. We record significant improvement in sampling efficiencies, thus in computational load, while the optimal sets of move frequencies are liable to allow interesting physical insights into the particular systems simulated. This last aspect should provide a new tool for designing more efficient new MC moves.     

Ultrasonic approach to obtaining partial thermodynamic characteristics of solutions

We describe a method for evaluating the thermodynamic characteristics both of pure liquids and of solutes in solutions using data derived from ultrasonic velocity measurements. The principal possibility of using ultrasound velocity lies in the fact that the velocity of ultrasound is a simple function of the adiabatic compressibility. The problem is formulated as an initial value problem for the parabolic type differential equations in partial derivatives. The validity of the method is demonstrated by calculation of the thermodynamic parameters for water, glycine and alanine in aqueous solutions at infinite dilution.     

Virtual parallel computing and a search algorithm using matrix product States

We propose a form of parallel computing on classical computers that is based on matrix product states. The virtual parallelization is accomplished by representing bits with matrices and by evolving these matrices from an initial product state that encodes multiple inputs. Matrix evolution follows from the sequential application of gates, as in a logical circuit. The action by classical probabilistic one-bit and deterministic two-bit gates such as NAND are implemented in terms of matrix operations and, as opposed to quantum computing, it is possible to copy bits. We present a way to explore this method of computation to solve search problems and count the number of solutions. We argue that if the classical computational cost of testing solutions (witnesses) requires less than O(n^{2}) local two-bit gates acting on n bits, the search problem can be fully solved in subexponential time. Therefore, for this restricted type of search problem, the virtual parallelization scheme is faster than Grover's quantum algorithm.     

Efficient sampling of protein structures by model hopping

We introduce a novel simulation method, model hopping, that enhances sampling of low-energy configurations in complex systems. The approach is illustrated for a protein-folding problem. Thermodynamic quantities of proteins with up to 46 residues are evaluated from all-atom simulations with this method.     

Defining the sampling space in multidimensional NMR experiments: What should the maximum sampling time be?

Efficient sampling of signals is a key issue for multiple-dimensional NMR experiments to establish the best ratio between experiment time and spectral quality. Focussing on the most widely used sampling strategy using standard rectangular sampling and data analysis by Fourier transformation, a central question is concerned with determining the optimal maximum sampling time in the individual dimensions. The spectral resolution depends directly on this choice, as do the overall experiment times when addressing the indirect dimensions. We present a theoretical, numerical, and experimental analysis of the sampling space problem and propose approaches to efficient sampling for typical cases.     

A fast direct solver for the integral equations of scattering theory on planar curves with corners

We describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners. It is rather comprehensive in that it applies to a wide variety of boundary value problems; here, we treat the Neumann and Dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids. It achieves high accuracy, is applicable to large-scale problems and, perhaps most importantly, does not require asymptotic estimates for solutions. Instead, the singularities of solutions are resolved numerically. The approach is efficient, however, only in the low- and mid-frequency regimes. Once the scatterer becomes more than several hundred wavelengths in size, the performance of the algorithm of this paper deteriorates significantly. We illustrate our method with several numerical experiments, including the solution of a Neumann problem for the Helmholtz equation given on a domain with nearly 10000 corner points.     

On Decay Properties of Solutions of the k-Generalized KdV Equation

We prove special decay properties of solutions to the initial value problem associated to the k-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As an application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.     

Numerical solution of certain classes of transport equations in any dimension by Shannon sampling

A method is developed for computing solutions to some class of linear and nonlinear transport equations (hyperbolic partial differential equations with smooth solutions), in any dimension, which exploits Shannon sampling, widely used in information theory and signal processing. The method can be considered a spectral or a wavelet method, strictly related to the existence of characteristics, but allows, in addition, for some precise error estimates in the reconstruction of continuous profiles from discrete data. Non-dissipativity and (in some case) parallelizability are other features of this approach. Monotonicity-preserving cubic splines are used to handle nonuniform sampling. Several numerical examples, in dimension one or two, pertaining to single linear and nonlinear (integro-differential) equations, as well as to certain systems, are given.     

Compressive sampling photoacoustic tomography based on edge expander codes and TV regularization

A new photoacoustic （PA） signal sampling and image reconstruction method, called compressive sampling PA tomography （CSPAT）, is recently proposed to make low sampling rate and high-resolution PA tomogra- phy possible. A key problem within the CSPAT framework is the design of optic masks. We propose to use edge expander codes-based masks instead of the conventional random distribution masks, and efficient total variation （TV） regularization-based model to formulate the associated problem. The edge expander codesbased masks, corresponding to non-uniform sampling schemes, are validated by both theoretical analysis and results from computer simulations. The proposed method is expected to enhance the capability of CSPAT for reducing the number of measurements and fast data acquisition.     

奇异微分方程边值问题的数值解法

本文给出求解奇异微分方程边值问题的正则化方法。解在奇点邻域内展开成级数形式,在余下区间上推导出正则边值问题,应用差分方法求解,并给出收敛结果和数值算例。