1,721,015 research outputs found
Orthogonal-polynomial-based spectral methods for fractional differential equations: theories and applications
No full textFractional calculus is a well-established field whose practical applications have gained significant momentum in recent years, though solving fractional equations remains a substantial challenge. This project advances orthogonal-polynomial-related spectral methods for fractional integro-differential equations (FIE/FDEs) through several contributions. The Jacobi-fractional-polynomial spectral method \cite{zayernouri2014fractional} for one-sided FIE/FDEs is extended with additional operators, improved flexibility, and a stable procedure for generating operational matrices, demonstrating superior performance in challenging examples. For the half-order fractional Laplacian , three new approaches are developed: (1) a recurrence-based computation of half-order Riesz potentials of Legendre polynomials, potentially enabling approximation in ; (2) the discovery of an orthogonal basis related to the finite Hilbert transform, with associated operational matrices and a spectral method for on ; and (3) an enhancement of a sum-space spectral method via a novel complete basis, addressing limitations in earlier work. These developments collectively broaden the toolkit for fractional spectral methods and lay the groundwork for continued research in this area.Open Acces
Sparse spectral methods for integral equations and equilibrium measures
Full text linkIn this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems.
These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost.
This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels.
The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Numerics for classical applications of Riemann-Hilbert problems
No full textNon UBCUnreviewedAuthor affiliation: The University of SydneyFacult
Representations of the symmetric group are decomposable in polynomial time
No full textWe introduce an algorithm to decompose matrix representations of the symmetric group
over the reals into irreducible representations, which as a by-product also computes the mul tiplicities of the irreducible representations. The algorithm applied to a d-dimensional repre sentation of Sn is shown to have a complexity of O(n
2d
3
) operations for determining which
irreducible representations are present and their corresponding multiplicities and a further
O(nd4
) operations to fully decompose representations with non-trivial multiplicities. These
complexity bounds are pessimistic and in a practical implementation using floating point arith metic and exploiting sparsity we observe better complexity. We demonstrate this algorithm on
the problem of computing multiplicities of two tensor products of irreducible representations
(the Kronecker coefficients problem) as well as higher order tensor products. For hook and
hook-like irreducible representations the algorithm has polynomial complexity as n increases.
We also demonstrate an application to constructing a basis of homogeneous polynomials so
that applying a permutation of variables induces an irreducible representation
A general framework for solving Riemann-Hilbert problems\ud numerically
Full text linkA new, numerical framework for the approximation of solutions to matrix-valued Riemann-Hilbert problems is developed, based on a recent method for the homogeneous Painlev\'e II Riemann- Hilbert problem. We demonstrate its effectiveness by computing solutions to other Painlev\'e transcendents.\ud
\ud
An implementation in MATHEMATICA is made available online
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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