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Daniel Eceizabarrena - One of the best experts on this subject based on the ideXlab platform.
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on the hausdorff dimension of riemann s non Differentiable Function
Transactions of the American Mathematical Society, 2021Co-Authors: Daniel EceizabarrenaAbstract:Recent findings show that the classical Riemann's non-Differentiable Function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's Function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.
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geometric differentiability of riemann s non Differentiable Function
Advances in Mathematics, 2020Co-Authors: Daniel EceizabarrenaAbstract:Abstract Riemann's non-Differentiable Function is a classic example of a continuous Function which is almost nowhere Differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's Function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.
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some geometric properties of riemann s non Differentiable Function
Comptes Rendus Mathematique, 2019Co-Authors: Daniel EceizabarrenaAbstract:Abstract Riemann's non-Differentiable Function is a celebrated example of a continuous but almost nowhere Differentiable Function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in C . The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.
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riemann s non Differentiable Function is intermittent
arXiv: Classical Analysis and ODEs, 2019Co-Authors: Alexandre Boritchev, Daniel Eceizabarrena, Victor Vilaca Da RochaAbstract:Riemann's non-Differentiable Function, introduced in the middle of the 19th century as a purely mathematical pathological object, is relevant in the study of the binormal flow, as shown recently by De La Hoz and Vega. From this physical point of view, the Function is therefore related to turbulent phenomena. We rigorously study the fine intermittent nature of this Function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways: one in the physical space and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0. The regularity of Riemann's non-Differentiable Function is a classical subject, heavily linked to its small-scale behaviour. However, our subtle asymptotics for a classical hydrodynamical quantity are new and sharp.
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intermittency of riemann s non Differentiable Function through the fourth order flatness
arXiv: Classical Analysis and ODEs, 2019Co-Authors: Alexandre Boritchev, Daniel Eceizabarrena, Victor Vilaca Da RochaAbstract:Riemann's non-Differentiable Function is one of the most famous examples of continuous but nowhere Differentiable Functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure Functions in the physical space and with high-pass filters in the Fourier space.
E H Doha - One of the best experts on this subject based on the ideXlab platform.
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On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials
Journal of Physics A: Mathematical and General, 2004Co-Authors: E H DohaAbstract:Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an infinitely Differentiable Function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an infinitely Differentiable Function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi–Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre–Jacobi and Hermite–Jacobi are developed.
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The ultraspherical coefficients of the moments of a general-order derivative of an infinitely Differentiable Function
Journal of Computational and Applied Mathematics, 1998Co-Authors: E H DohaAbstract:Abstract A formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given. Formulae for the ultraspherical coefficients of the moments of a general-order derivative of an infinitely Differentiable Function in terms of its ultraspherical coefficients are also obtained. The corresponding formulae for the important special cases of Chebyshev polynomials of the first and second kinds and of Legendre polynomials are deduced. Two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients, in the sense of Lewanowicz, are discussed. Comparisons with the results obtained by optimal algorithm of Lewanowicz (1976) are also made.
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On the legendre coefficients of the moments of the general order derivative of an infinitely Differentiable Function
International Journal of Computer Mathematics, 1995Co-Authors: E H Doha, S. I. El-soubhyAbstract:Formulae for the Legendre coefficients of the moments of the general order derivative of an infinitely Differentiable Function in terms of its Legendre coefficients are derived. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the Legendre expansion coefficients are discussed. Comparisons with the results obtained by the optimal algorithm of Lewanowicz (1976) are made.
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The first and second kind chebyshev coefficients of the moments for the general order derivative on an infinitely Differentiable Function
International Journal of Computer Mathematics, 1994Co-Authors: E H DohaAbstract:Expressions for the first and second kinds Chebyshev coefficients of the moments of the general order derivative of an infinitely Differentiable Function in terms of its Chebyshev coefficients are given. Two numerical applications of how to use these expressions for solving ordinary differential equations with polynomial coefficients are described. Comparisons with the results obtained by Lewanowicz optimum algorithm (1976) are noted.
Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.
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the general jacobi matrix method for solving some nonlinear ordinary differential equations
Applied Mathematical Modelling, 2012Co-Authors: M R Eslahchi, Mehdi DehghanAbstract:Abstract In this paper, we obtain the approximate solutions for some nonlinear ordinary differential equations by using the general Jacobi matrix method. Explicit formulae which express the Jacobi expansion coefficients for the powers of derivatives and moments of any Differentiable Function in terms of the original expansion coefficients of the Function itself are given in the matrix form. Three test problems are discussed to illustrate the efficiency of the proposed method.
Eceizabarrena Daniel - One of the best experts on this subject based on the ideXlab platform.
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On the Hausdorff dimension of Riemann's non-Differentiable Function
'American Mathematical Society (AMS)', 2021Co-Authors: Eceizabarrena DanielAbstract:Recent findings show that the classical Riemann's non-Differentiable Function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's Function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.Comment: v3: Major revision. v4: Accepted manuscript - Transactions of the American Mathematical Societ
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Geometric differentiability of Riemann's non-Differentiable Function
'Elsevier BV', 2020Co-Authors: Eceizabarrena DanielAbstract:Riemann's non-Differentiable Function is a classic example of a continuous Function which is almost nowhere Differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's Function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.Comment: 27 pages, 6 figures. v2: Typos corrected, references added. v3: Accepted manuscrip
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Some geometric properties of Riemann's non-Differentiable Function
'Elsevier BV', 2019Co-Authors: Eceizabarrena DanielAbstract:Riemann's non-Differentiable Function is a celebrated example of a continuous but almost nowhere Differentiable Function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.Comment: 6 pages, 2 figures. v2: References updated and correcte
Vsevolod I Ivanov - One of the best experts on this subject based on the ideXlab platform.
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higher order optimality conditions with an arbitrary non Differentiable Function
Optimization, 2016Co-Authors: Vsevolod I IvanovAbstract:In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real Function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the Function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Frechet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex Functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing ...
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higher order optimality conditions with an arbitrary non Differentiable Function
arXiv: Optimization and Control, 2015Co-Authors: Vsevolod I IvanovAbstract:In this paper, we introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real Function. This derivative is harmonized with the classical higher-order Fr\'echet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order $n$ ($n$ is a positive integer) for a local minimum and isolated local minimum of order $n$ of the given Function in terms of these derivatives and subdifferentials. We do not require any restrictions on the Function in our results. A notion of a higher-order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order $n$. Higher-order invex Functions are defined. They are the largest class such that our necessary conditions for local minima are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order $n$.