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X-ray diffractometry, X-ray fluorescence, and ICP-MS have been used to study the phase and elemental composition of the thermal coal from the Pavlovsk brown coal deposit and their ash-forming compounds. The results of the spectral analysis methods were combined with the data of thermal gravimetry, elemental analysis of OM, and scanning electron microscopy. The metal-bearing potential of the thermal coal from the deposit and the prospects for the associated extraction of useful components have been estimated. A set of analytical methods is proposed for the rapid assessment of the content of valuable and significant elements in solid fossil fuels for the comprehensive development of the coal deposits. [ABSTRACT FROM AUTHOR]

In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results. [ABSTRACT FROM AUTHOR]

Empirical networks often exhibit different meso-scale structures, such as community and core–periphery structures. Core–periphery structure typically consists of a well-connected core and a periphery that is well connected to the core but sparsely connected internally. Most core–periphery studies focus on undirected networks. We propose a generalization of core–periphery structure to directed networks. Our approach yields a family of core–periphery block model formulations in which, contrary to many existing approaches, core and periphery sets are edge-direction dependent. We focus on a particular structure consisting of two core sets and two periphery sets, which we motivate empirically. We propose two measures to assess the statistical significance and quality of our novel structure in empirical data, where one often has no ground truth. To detect core–periphery structure in directed networks, we propose three methods adapted from two approaches in the literature, each with a different trade-off between computational complexity and accuracy. We assess the methods on benchmark networks where our methods match or outperform standard methods from the literature, with a likelihood approach achieving the highest accuracy. Applying our methods to three empirical networks—faculty hiring, a world trade dataset and political blogs—illustrates that our proposed structure provides novel insights in empirical networks. [ABSTRACT FROM AUTHOR]

Computational requirements often impose limitations on the spatial and temporal resolutions of atmospheric CO2 inversions, increasing aggregation and representation errors. This study enables higher spatial and temporal resolution inversions with spatial and temporal error structures similar to those used in other published inversions by representing the prior flux error covariances as a Kronecker product of spatial and temporal covariances and by using spectral methods for the spatial correlations. Compared to existing inversion systems that are forced to degrade the resolution of the problem in order to bring the dimensionality down to computationally tractable levels, this inversion framework is able to take advantage of mesoscale transport simulations and more of the complexity of spatial and temporal covariances in the surface CO2 fluxes. This approach was successfully implemented over one month with an identical‐twin observing system simulation experiment (OSSE) using a set of assumptions about the prior flux uncertainties compatible both with continental‐scale uncertainty estimates and with comparisons of vegetation models to flux towers. The demonstration illustrates the potential of the newly developed inversion system to use high‐temporal‐resolution information from the North American tower network, to extract high‐resolution information about CO2 fluxes that is inaccessible to coarser resolution inversion systems, and to simultaneously optimize an ensemble of prior estimates. This demonstration sets the stage for regional flux inversions that can take full advantage of the high‐resolution data available in tower CO2 records and mesoscale atmospheric transport reanalyses, include more realistic prior error structures, and explore specifying prior fluxes with ensembles. Plain Language Summary: This paper describes a new inverse framework for deriving surface carbon dioxide (CO2) fluxes at high spatial and temporal resolutions from atmospheric mole fraction measurements. We demonstrate the potential of the system over North America by inverting for subdaily biogenic CO2 fluxes while accounting for the complexity of spatio‐temporal structures in flux error covariances in a data‐based manner. Key Points: A new inversion system uses spectral methods for spatial correlations to do a 27 km subdaily inversion over North AmericaThe software package allows simultaneous optimization of multiple prior mean flux estimatesThe system is demonstrated over continental North America for a network of seven towers [ABSTRACT FROM AUTHOR]

Weak codon bias in short exons and separation by a short intron induces difficulty in extracting period-3 component that marks the presence of exonic regions. The annotation task of such short exons has been addressed in the proposed model independent signal processing based method with following features: (a) DNA sequences have been mapped using multiple mapping schemes, (b) period-3 spectrums corresponding to multiple mappings have been optimized to enhance short exon-short intron discrimination, and (c) spectrums corresponding to multiple mapping schemes have been subjected to Principal Component Analysis (PCA) for identifying greater number of such short exons. A comparative study with other methods indicates improved detection of contiguous short exons disunited by a short intron. Apart from the annotation of exonic and intronic regions, the proposed algorithm can also complement the methods for the detection of alternative splicing by intron retention, as one of the characteristic feature for intron retention is the presence of two short exons flanking a short intron. [ABSTRACT FROM AUTHOR]

In the presented study, we are presenting the approximate solutions of two important equations, the Riccati and Logistic equations; the presented technique is based on the rational Legendre function. Since all the studied models are nonlinear, we convert these nonlinear equations to a sequence of linear ordinary differential equations (ODEs), then by applying the quasi-linearization method (QLM) on these resulting ODEs at each iteration, these ODEs will be converted to a simple linear system of algebraic equations which can be solved. Two numerical examples are presented and we compared between the approximate and the exact solutions. [ABSTRACT FROM AUTHOR]

This paper aims to provide a rigorous analysis of exponential convergence of an adaptive spectral collocation method for a general nonlinear system of rational-order fractional initial value problems. The key idea of the proposed method is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions to a class of fractional initial value problems, leading to spectrally accurate approximation. Numerical examples are presented, which verify the theoretical predictions and demonstrate that the new formulation of the spectral method leads to better performance compared to other known numerical approaches with a relatively smaller number of degree of freedoms. [ABSTRACT FROM AUTHOR]

Herein, the solution of partial differential equations (PDEs) using spectral methods is developed for irregular domains, which preserves their accuracy. Previously, to solve these problems, the finite differences method or the embedded domains method was typically applied. The approach presented in this article can be used for any boundary described by a Jordan curve, and the solution behavior outside the domain need not to considered. The computational process has low cost and generality because the map constructions and changing variables are unnecessary. In addition, by using the presented parametrization process, boundary conditions (boundary bordering) can be implemented conveniently, where the rectangular domains can be considered as an asymptotical case. The structure is numerically oriented, which facilitates the application of any algorithm related to spectral methods. [ABSTRACT FROM AUTHOR]

The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrödinger equation in the semiclassical regime, where the Planck constant ε is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrödinger equation with a linear potential. The local discretization error of the two time-splitting methods constructed here is O (max ⁡ { Δ t 3 , Δ t 5 / ε }) , while the well-known Lie-Trotter splitting scheme and the Strang splitting scheme are O (Δ t 2 / ε) and O (Δ t 3 / ε) , respectively, where Δ t is the time step-size. The global error estimates of new exponential splitting schemes with spectral discretization suggests that larger time step-size is admissible for obtaining high accuracy approximation of the solutions. Numerical studies verify our theoretical results and reveal that the new methods are especially efficient for linear semiclassical Schrödinger equation with a quadratic potential. [ABSTRACT FROM AUTHOR]

We introduce and analyze a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kinds with or without known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind problems, where the former builds on connections with Toeplitz operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with the application of the spectral tau and collocation methods to delay multi-order fractional differential equations with vanishing delay r x (0 < r < 1) . The fractional derivatives are described in the Caputo sense. The model solution is expanded in terms of Chebyshev polynomials. The convergence of the proposed approaches is investigated in the weighted L 2 -norm. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account. [ABSTRACT FROM AUTHOR]

Ensemble clustering via building co-association matrix and combining multiple basic partitions from the same dataset into the consensus one has been widely used in spectral clustering and subspace clustering. However, with the ever-increasing cost of calculating the co-association matrix, the conventional ensemble clustering algorithm is no longer fit for dealing with the large-scale datasets due to its less scalability and time-consuming. In this paper, we propose a scalable spectral ensemble clustering method via building a representative co-association matrix to improve the ensemble clustering problem. Our method mainly includes constructing a sparse matrix to select the representative points and building the co-association matrix, and a robust and denoising representation for the co-association matrix can be learned through a low-rank constraint in a unified optimization framework. The experiments verify the high efficiency and scalability but less time cost of our method compared with state-of-art clustering methods in the six real-world datasets, especially in the large-scale datasets. [ABSTRACT FROM AUTHOR]

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line ℝ ∪ ∞ , except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible. [ABSTRACT FROM AUTHOR]

RS-fMRI data analysis for functional connectivity explorations is a challenging topic in computational neuroimaging. Several approaches have been investigated to discover whole-brain data features. Among these, clustering techniques based on Competitive Learning (CL) and Spectral Methods (SM) have been shown effective in providing useful information in various contexts. We selected three clustering algorithms and two spectral methods, i.e the clustering algorithm are Self-organising Maps (SOM), Neural Gas (NG) and Growing Neural Gas (GNG), whereas the spectral methods are the classic Principal Component Analysis (PCA) and the Nonlinear Robust Fuzzy Principal Component Analysis (NRFPCA). We validated clustering with Davies–Bouldin Index (DBI) and we selected informative principal components using Random Matrix Theory (RMT). tools. We adopted these techniques to study the intrinsic functional properties of images coming from a shared repository of resting state fMRI experiments (1000 Functional Connectome Project). [ABSTRACT FROM AUTHOR]

Summary: Microstructural patterns emerge ubiquitously during phase transformations, deformation twinning, or crystal plasticity. Challenges are the prediction of such microstructural patterns and the resulting effective material behavior. Mathematically, the experimentally observed patterns are energy‐minimizing sequences produced by an underlying non‐(quasi)convex strain energy. Therefore, identifying the microstructure and effective response is linked to finding the quasiconvex, relaxed energy. Due to its nonlocal nature, quasiconvexification has traditionally been limited to (semi‐)analytical techniques or has been dealt with by numerical techniques such as the finite element method (FEM). Both have been restricted to primarily simple material models. We here contrast three numerical techniques—FEM, a Fourier‐based spectral formulation, and a meshless maximum‐entropy (max‐ent) method. We demonstrate their performance by minimizing the energy of a representative volume element for both hyperelasticity and finite‐strain phase transformations. Unlike FEM, which fails to converge in most scenarios, the Fourier‐based spectral formulation (FFT) scheme captures microstructures of intriguingly high resolution, whereas max‐ent is superior at approximating the relaxed energy. None of the methods are capable of accurately predicting both microstructures and relaxed energy; yet, both FFT and max‐ent show significant advantages over FEM. Numerical errors are explained by the energy associated with microstructural interfaces in the numerical techniques compared here. [ABSTRACT FROM AUTHOR]

Recently, the numerical solution of stiffly/highly oscillatory Hamiltonian problems has been attacked by using Hamiltonian boundary value methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective even when applied to a wider range of problems. Here, we fill this gap by providing a thorough convergence analysis of the methods and confirm the theoretical results with the aid of a few numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, an optimal control problem governed by elliptic partial differential equations, whose objective functionals do not depend on the controls, with L 2 -norm constraint on control variable is considered, and Galerkin spectral approximation of the optimal control problem without penalty term is established. To analyze the optimal control problem, optimality conditions of the control problem are first derived in detail. By virtue of some lemmas and auxiliary equations, a priori error estimates of spectral approximation for optimal control problems are analyzed rigorously. Moreover, a posteriori error estimates of the optimal control problem are also established carefully. In the end, some numerical examples are carried out to confirm the analytical results that the errors decay exponentially fast as long as the data is smooth. [ABSTRACT FROM AUTHOR]

An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel rational filter for the right half-plane. If the differential operator is normal or self-adjoint, then the operator analogue preserves that structure and robustly computes eigenvalues to near machine precision accuracy. The algorithm is particularly adept at computing high-frequency modes of differential operators that possess self-adjoint structure with respect to weighted Hilbert spaces. [ABSTRACT FROM AUTHOR]

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decaying slowly and subject to certain power law. Their numerical solutions are underexplored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identities related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by precomputing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach. [ABSTRACT FROM AUTHOR]

The Benjamin-Ono (BO) equation is a nonlinear partial integro-differential equation that describes internal waves in deep stratified fluids. Here we present a numerical study of the blow-up phenomenon for the BO-generalized KP version with different types of nonlinearities. A dynamic rescaling is used to identify the type of the singularity and explore the investigation of observed blow-ups. For the numerical experimentations, an exponential time-differencing fourth- order Runge-Kutta method combined with a space-scheme based on the Fourier spectral method is used. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient numerical method based on two-dimensional hybrid of block-pulse functions and Legendre polynomials for numerically solving an inverse reaction diffusion equation. The main idea of the present method is based upon some of the important benefits of the hybrid functions such as high accuracy, wide applicability, and adjustability of the orders of the block-pulse functions and Legendre polynomials to achieve highly accurate numerical solutions. By using the spectral method, inverse reaction diffusion equation with initial and boundary conditions would reduce to a system of nonlinear algebraic equations. Due to the ill-posed system of nonlinear algebraic equations, a regularization scheme is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of this method. [ABSTRACT FROM AUTHOR]

The vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using second‐order finite difference matrices produces O(1) errors for all but the few lowest modes, and increasing resolution quickly becomes too slow as the computational complexity of eigenvalue algorithms increases as O(n3). Existing methods are therefore inadequate for computing a full spectrum of internal waves, such as needed for initializing a numerical model with a full internal wave spectrum. Here we show that rewriting the eigenvalue problem in stretched coordinates and projecting onto Chebyshev polynomials results in substantially more accurate modes than finite differencing at a fraction of the computational cost. We also compute the surface quasigeostrophic modes using the same methods. All spectral and finite difference algorithms are made available in a suite of Matlab classes that have been validated against known analytical solutions in constant and exponential stratification. Plain Language Summary: The ocean is filled with internal waves that primarily move fluid back and forth horizontally and, to a lesser degree, vertically. The amount of vertical motion for each wave depends on depth because the fluid density in the ocean also changes with depth. The density profile of the ocean varies both seasonally and geographically, which means that vertical structure of each wave is also a function of season and geographic location. In order to diagnose the sizes of the internal waves, scientists need to accurately compute the motion predicted for each wave from a given density profile. The standard methodology for computing the vertical structure of the fluid motion only works well for the biggest and slowest moving waves. This paper introduces a different methodology which is both computationally faster and more accurate than the traditional methods. All of the code for computing the vertical structure of waves is implemented in Matlab and is freely available. The paper also explores some of the most common errors that will be encountered when using ocean data to diagnose the vertical structure. Key Points: The nonhydrostatic internal wave modes are computed with substantially higher accuracy in significantly less time than traditional methodsSQG modes are computed with high accuracy near the boundariesWe provide a suite of Matlab classes validated against known analytical solutions [ABSTRACT FROM AUTHOR]

In this paper we explore orthogonal systems in L 2 (R) which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for L 2 (R) , have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this family of functions is essentially the only orthonormal basis possessing a differentiation matrix of the above form and whose coefficients are equal to classical Fourier coefficients of a modified function though a monotone, differentiable change of variables. Examples of other orthogonal bases with skew-Hermitian, tridiagonal differentiation matrices are discussed as well. [ABSTRACT FROM AUTHOR]

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in R 3 . In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm. [ABSTRACT FROM AUTHOR]

We study the shear-driven instability, of Kelvin–Helmholtz type, that forms at the interface between a cylindrical jet of flowing fluid and its surroundings. The results of an infinitesimal-amplitude theory based on linearising the system about the undisturbed jet are given. A novel numerical model based on the Galerkin spectral method is developed and employed to simulate the non-linear development of the jet in the weakly-compressible Boussinesq regime. Representative results demonstrating the viability of this model are given, and are shown to agree with the linear analysis for small disturbances, but to develop cylindrical roll-up in the severely non-linear regime. [ABSTRACT FROM AUTHOR]

In this work, a deterministic approach to the solution of the integral linear differential Boltzmann equation, in one and twodimensional media, is presented. Explicit solutions, in terms of the spatial variables, for the discrete ordinates approximation of the original model are obtained from a spectral formulation. A relevant feature of the methodology is the reduced order of the eigenvalue problems, which is given as half of the number of the discrete directions. Such aspect, along with the use of arbitrary quadrature schemes to represent the integral term of the equation, are fundamental to provide fast, concise and accurate solutions to several problems of interest. In particular, the solutions here are derived for models associated with two different applications: rarefied gas dynamics and neutron transport. [ABSTRACT FROM AUTHOR]

Spectral methods have long been used for time series analysis, but their use for exploring spatially-referenced data is relatively newer and less common. Within spatial statistics the spectral representation of the spatial covariance function, the spectral density function, offers an alternative perspective of the variation of the random field. We use the spectral representation of the covariance function to develop a new method to test the assumption of spatial isotropy. The spatial isotropy spectral (SIS) test is the first to use the spectral representation of the covariance function with the primary goal of testing isotropy for geostatistical data. The SIS test is nonparametric and overcomes several deficiencies of other nonparametric tests of isotropy, such as the need for the user to make a number of choices before implementing the test. We address several challenges in using the spectral domain to construct a test statistic and develop new theory on the asymptotic distribution of the periodogram. We use a simulation study and application to investigate the utility and limitations of the SIS test. For large data sets, the SIS test is particularly useful as the computational cost is low. • The spatial isotropy spectral test is a nonparametric test for spatial anisotropy. • We derive the asymptotic distribution of the periodogram under mixed spatial asymptotics. • The size and power of the spatial isotropy spectral test are competitive with other nonparametric tests for spatial anisotropy. • The spatial isotropy spectral test provides a computationally fast method for testing for spatial anisotropy. [ABSTRACT FROM AUTHOR]

We investigate a spectral Galerkin method for the fractional advection-diffusionreaction equations in one dimension. We first prove sharp regularity estimates of solutions in nonweighted and weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin methods for both fractional advection-diffusion and diffusion-reaction equations. We also present an iterative solver with a quasi-optimal complexity. Numerical results are presented to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

We present a more detailed numerical investigation of the head-on collision of a two-kink/two-antikink system. We identified the escape of oscillon-like configurations as a pair of kinks of the standard ϕ4 model moving apart from each other. New pieces of evidence support that the lump-like defects can emerge from the two-kink interaction to form metastable configurations. Moreover, these configurations signalize the windows of escape that have a fractal structure similar to the n-bounce sequence when the kinks of ϕ4 interact. As the last piece of the numerical experiment, we show that by perturbing conveniently a lump-like defect, it is possible to recover another lump-like configuration as a metastable configuration. [ABSTRACT FROM AUTHOR]

This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any C1(a,b)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {C}^1(a,b)$$\end{document} weight function such that w(a)=w(b)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w(a)=w(b)=0$$\end{document}, we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case a=-∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a=-\infty $$\end{document}, b=+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b=+\infty $$\end{document}, only few powers of that matrix are bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function xαe-x\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x^\alpha \textrm{e}^{-x}$$\end{document} for x>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x>0$$\end{document} and α>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >0$$\end{document} and the ultraspherical weight function (1-x2)α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(1-x^2)^\alpha $$\end{document}, x∈(-1,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x\in (-1,1)$$\end{document}, α>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >0$$\end{document}, and establish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sensitive to the choice of α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}, and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions. [ABSTRACT FROM AUTHOR]

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training. • Symbolic RONS uses symbolic computing to develop a scalable computational method for solving the RONS equations. • Collocation RONS introduces a collocation point approach to RONS. • Regularization of the RONS equations speeds up their time integration by addressing the stiffness issue. • Application to evolutional neural network is investigated and demonstrated on a numerical example. [ABSTRACT FROM AUTHOR]

We motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have poly-logarithmic worst-case guarantees under the new measure. The main result of the article is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worst-case guarantees; another finds a "good" clustering, if one exists. [ABSTRACT FROM AUTHOR]