Characterizations of the symmetrized polydisc via another family of domains

We find new characterizations for the points in the symmetrized polydisc [Formula: see text], a family of domains associated with the spectral interpolation, defined by [Formula: see text] We introduce a new family of domains which we call the extended symmetrized polydisc [Formula: see text], and define in the following way: [Formula: see text] [Formula: see text] We show that [Formula: see text] for [Formula: see text] and that [Formula: see text] for [Formula: see text]. We first obtain a variety of characterizations for the points in [Formula: see text] and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in [Formula: see text]. Also, we obtain similar characterizations for the points in [Formula: see text], where [Formula: see text]. A set of [Formula: see text] fractional linear transformations plays central role in the entire program. We also show that for [Formula: see text], [Formula: see text] is nonconvex but polynomially convex and is starlike about the origin but not circled.

The space spectral interpolation collocation method for reaction-diffusion systems

A space spectral interpolation collocation method is proposed to study nonlinear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus.

Numerical modeling of mechanical wave propagation

The numerical modeling of mechanical waves is currently a fundamental tool for the study and investigation of their propagation in media with heterogeneous physical properties and/or complex geometry, as, in these cases, analytical methods are usually not applicable. These techniques are used in geophysics (geophysical interpretation, subsoil imaging, development of new methods of exploration), seismology (study of earthquakes, regional and global seismology, accurate calculation of synthetic seismograms), in the development of new methods for ultrasonic diagnostics in materials science (non-destructive methods) and medicine (acoustic tomography). In this paper we present a review of numerical methods that have been developed and are currently used. In particular we review the key concepts and pioneering ideas behind finite-difference methods, pseudospectral methods, finite-volume methods, Galerkin continuous and discontinuous finite-element methods (classical or based on spectral interpolation), and still others such as physics-compatible, and multiscale methods. We focus on their formulations in time domain along with the main temporal discretization schemes. We present the theory and implementation for some of these methods. Moreover, their computational characteristics are evaluated in order to aid the choice of the method for each practical situation.

Stellar spectral interpolation using machine learning

ABSTRACT Theoretical stellar spectra rely on model stellar atmospheres computed based on our understanding of the physical laws at play in the stellar interiors. These models, coupled with atomic and molecular line databases, are used to generate theoretical stellar spectral libraries (SSLs) comprising of stellar spectra over a regular grid of atmospheric parameters (temperature, surface gravity, abundances) at any desired resolution. Another class of SSLs is referred to as empirical spectral libraries; these contain observed spectra at limited resolution. SSLs play an essential role in deriving the properties of stars and stellar populations. Both theoretical and empirical libraries suffer from limited coverage over the parameter space. This limitation is overcome to some extent by generating spectra for specific sets of atmospheric parameters by interpolating within the grid of available parameter space. In this work, we present a method for spectral interpolation in the optical region using machine learning algorithms that are generic, easily adaptable for any SSL without much change in the model parameters, and computationally inexpensive. We use two machine learning techniques, Random Forest (RF) and Artificial Neural Networks (ANN), and train the models on the MILES library. We apply the trained models to spectra from the CFLIB for testing and show that the performance of the two models is comparable. We show that both the models achieve better accuracy than the existing methods of polynomial based interpolation and the Gaussian radial basis function (RBF) interpolation.

Spectral and spatial interpolation of precipitation for daily and hourly time series

<p>Generating synthetic precipitation for weather generators were always a challenging issue in hydro-climate simulations because of its high variability in time and space. We present a spectral method for generating the synthetic precipitation time series which is in accordance with the observed precipitation statistical characteristics not only for the observed points, but also for any desired location by interpolating the time series spectrum. In this regard, time series spectra derived from the observed signal converting from its time domain to the corresponding frequency domain using the Fourier transform.</p><p>The main problem for spectral interpolation of precipitation time series is highly occurrence of non-rainy days which can be even more inaccurate for the finer resolutions such as hourly and sub-hourly data. In order to overcome the highly frequent occurrence of non-rainy days, transformation between indicator and normal correlation has been taken into account.</p><p>This method enables us to generate synthetic time series with same statistical characteristics for the observed points and also for any point of interests rather than the observed points. The introduced so called spectral and spatial interpolation method applied for daily and hourly precipitation time series for the selected stations in state Baden-Württemberg, Germany.</p>

A computational study of solitary wave solutions of Kawahara-type equations by meshless spectral interpolation method

In this paper, a meshless spectral radial point interpolation (MSRPI) method using weighted [Formula: see text]-scheme is formulated for the numerical solutions of a class of nonlinear Kawahara-type evolutionary equations. The formulated method is applied for simulation of single and double solitary waves motion, wave generation and oscillatory shock waves propagation. Quality of approximation is measured via discrete [Formula: see text], [Formula: see text] and [Formula: see text] error norms. Three invariant quantities corresponding to mass, momentum and energy are also computed for the method validation. Stability analysis of the proposed method is briefly discussed and verified computationally. Comparison of the obtained results are made with other existing results in the literature revealing the method superiority.