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Potential functional approximations are an intriguing alternative to density functional approximations. The potential functional that is dual to the Lieb density functional is defined and its properties are reported. The relations...
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Potential functional approximations are an intriguing alternative to density functional approximations. The potential functional that is dual to the Lieb density functional is defined and its properties are reported. The relationship between the Thomas-Fermi theory as a density functional and the theory as a potential functional is derived. The properties of several recent semiclassical potential functionals are explored, especially regarding their approach to the large particle number and classical continuum limits. The lack of ambiguity in the energy density of potential functional approximations is demonstrated. The density-density response function of the semiclassical approximation is calculated and shown to violate a key symmetry condition.
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This paper describes a new approach to multivariate scattered data smoothing. It is assumed that the data are generated by a Lipschitz continuous function f, and include random noise to be filtered out. The proposed approach uses ...
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This paper describes a new approach to multivariate scattered data smoothing. It is assumed that the data are generated by a Lipschitz continuous function f, and include random noise to be filtered out. The proposed approach uses known, or estimated value of the Lipschitz constant of f, and forces the data to be consistent with the Lipschitz properties of f. Depending on the assumptions about the distribution of the random noise, smoothing is reduced to a standard quadratic or a linear programming problem. We discuss an efficient algorithm which eliminates the redundant inequality constraints. Numerical experiments illustrate applicability and efficiency of the method. This approach provides an efficient new tool of multivariate scattered data approximation.
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This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force f. These approximations are obtained by combining a fixed poi...
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This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force f. These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon's consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on f, the approximation method is always convergent. However, our error estimates show that the convergence's properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term f is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.
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We extend an approximate theory in order to examine the properties of the one-component plasma (OCP) of similarly charged rodlike polyelectrolytes system from the weak- to the strong-coupling regimes. In addition, Monte Carlo simu...
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We extend an approximate theory in order to examine the properties of the one-component plasma (OCP) of similarly charged rodlike polyelectrolytes system from the weak- to the strong-coupling regimes. In addition, Monte Carlo simulations are performed in order to verify the approximate theory. The electrostatic interactions are decomposed into short- and long-range components treated within different approximations. In addition to the Coulomb interactions present within the system, the excluded volume interactions between the charged rods are included in our analysis within a second order virial reference system. By theory and numerical simulation, we find that the isotropic phase becomes unstable with reference to the orientational ordering performed by both electrostatic and excluded volume interactions. Good agreement between theory and numerical simulation is obtained for the conditions investigated.
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We answer several open questions in the theory of approximate amenability for Banach algebras. First we give examples of Banach algebras which are boundedly approximately amenable but which do not have bounded approximate identiti...
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We answer several open questions in the theory of approximate amenability for Banach algebras. First we give examples of Banach algebras which are boundedly approximately amenable but which do not have bounded approximate identities. This answers a question open since the year 2000 when Ghahramani and Loy founded the notion of approximate amenability. We give a nice condition for a c _o-direct-sum of amenable Banach algebras to be approximately amenable, which gives us a reasonably large and varied class of such examples. Then we examine our examples in some detail, and thereby find answers to other open questions: the two notions of bounded approximate amenability and bounded approximate contractibility are not the same; the direct-sum of two approximately amenable Banach algebras does not have to be approximately amenable; and a 1-codimensional closed ideal in a boundedly approximately amenable Banach algebra need not be approximately amenable.
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The pointwise version of the Jackson theorem on the best approximation by trigonometric polynomials is proved. Some application of the result is also presented.
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In the subspace approximation problem, given m points in R^{n} and an integer k <= n, the goal is to find a k-dimension subspace of R^{n} that minimizes the l_{p}-norm of the Euclidean distances to the given points. This problem g...
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In the subspace approximation problem, given m points in R^{n} and an integer k <= n, the goal is to find a k-dimension subspace of R^{n} that minimizes the l_{p}-norm of the Euclidean distances to the given points. This problem generalizes several subspace approximation problems and has applications from statistics, machine learning, signal processing to biology. Deshpande et al. [Deshpande et al., 2011] gave a randomized O(sqrt{p})-approximation and this bound is proved to be tight assuming NP != P by Guruswami et al. [Guruswami et al., 2016]. It is an intriguing question of determining the performance guarantee of deterministic algorithms for the problem. In this paper, we present a simple deterministic O(sqrt{p})-approximation algorithm with also a simple analysis. That definitely settles the status of the problem in term of approximation up to a constant factor. Besides, the simplicity of the algorithm makes it practically appealing.
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We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in theLieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as theinteraction strength, f...
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We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in theLieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as theinteraction strength, frequency, and wave number. The suggested approximations are consistent with the exactresults known in limiting cases. In particular, we encompass exact edge exponents as well as the Luttinger-liquid and perturbation theoretic results. We further discuss derived approximations for the static structurefactor and the pair distribution function g(x). The approximate expressions show excellent agreement withnumerical results based on the algebraic Bethe ansatz.
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Many computer vision and human–computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation o...
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Many computer vision and human–computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation of these kind of functions often implies a very high computational cost, unacceptable in real-time applications. To alleviate this problem, functions are commonly approximated by simpler piecewise-polynomial representations. Following this idea, we propose a novel, efficient, and practical technique to evaluate complex and continuous functions using a nearly optimal design of two types of piecewise linear approximations in the case of a large budget of evaluation subintervals. To this end, we develop a thorough error analysis that yields asymptotically tight bounds to accurately quantify the approximation performance of both representations. It provides an improvement upon previous error estimates and allows the user to control the tradeoff between the approximation error and the number of evaluation subintervals. To guarantee real-time operation, the method is suitable for, but not limited to, an efficient implementation in modern graphics processing units, where it outperforms previous alternative approaches by exploiting the fixed-function interpolation routines present in their texture units. The proposed technique is a perfect match for any application requiring the evaluation of continuous functions; we have measured in detail its quality and efficiency on several functions, and, in particular, the Gaussian function because it is extensively used in many areas of computer vision and cybernetics, and it is expensive to evaluate.
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