1,720,972 research outputs found
Invariance principle and hydrodynamic limits on Riemannian manifolds
No full textIn this report we study Markov processes on compact and connected Riemannian manifolds. We define a random walk on such manifolds and give a direct proof of the invariance principle. This principle says that under some conditions on the jumping distributions (i.e. the distributions of single steps), the random walk converges to Brownian motion when space is scaled by 1/N, time by N^2 and N tends to infinity (which has been shown with more general methods by Jorgensen and by Blum). To prove this, we show convergence of the generators on the set of smooth functions and we apply the Trotter-Kurtz theorem (as has been done by Blum, in a rather sketchy way and in a slightly different setting). We also show convergence of the corresponding Dirichlet forms. Then we show that the conditions on the jumping distributions are satisfied if they are compactly supported and have mean 0 and a covariance matrix which is invariant under orthogonal transformations.Next, we define random grids on a Riemannian manifold and we define random walks on them. We show that their Dirichlet forms converge to the Dirichlet form of Brownian motion, using the results above. We also prove a result that is a bit weaker than convergence of the generators in this case.Finally, these grids allow us to define the Symmetric Exclusion Process (SEP) on a manifold. Using the convergence results above, we follow the line of a proof of Seppäläinen to show that the hydrodynamic limit of the SEP satisfies the heat equation. Some details still need to be filled in, but we believe that this method will allow us to study interacting particle systems and their hydrodynamic limits on Riemannian manifolds.Before all of this we start with an introduction to Markov processes, their semigroups and generators. In particular we focus on time-reversible (or symmetric) processes and the Dirichlet form with its properties. We also give an introduction to Riemannian manifolds and related notions
Orthogonality relations of q-Meixner polynomials: with the use of spectral analysis
No full textOrthogonality relations of q-Meixner polynomials, polynomials in terms of basic hypergeometric series, will be proved by using spectral measures and a difference operator
Quantum gradient estimation and its application to quantum reinforcement learning
No full textIn 2005, Jordan showed how to estimate the gradient of a real-valued function with a high-dimensional domain on a quantum computer. Subsequently, in 2017, it was shown by Gilyén et al. how to do this with a different input model. They also proved optimality of their algorithm for \ell^\infty -approximations of functions satisfying some smoothness conditions.In this text, we expand the ideas of Gilyén et al., and extend their algorithm such that functions with fewer regularity constraints can be used as input. Moreover, we show that their algorithm is essentially optimal in the query complexity to the phase oracle even for classes of functions that have more stringent smoothness conditions. Finally, we also prove that their algorithm is optimal for approximating gradients with respectto general \ell^p -norms, where p \in [1,\infty].Furthermore, we investigate how Gilyén et al.’s algorithm can be used to do reinforcement learning on a quantum computer. We elaborate on Montanaro’s ideas for quantum Monte-Carlo simulation, and show how they can be used to implement quantum value estimation of Markov reward processes. We also show essential optimality of this algorithm in the query complexity of all its oracles. Next, we show how we can construct a quantum policy evaluation algorithm, and how we can use these algorithms as subroutines in Gilyén et al.’s quantum gradient estimation algorithm to perform quantum policy optimization.The most important open questions remain whether it is possible to improve the query complexity of the extension of Gilyén et al.’s algorithm, when function classes containing functions of Gevrey-type 1 are used as input, as at the moment for this specific parameter setting the algorithm is not better than a very simple classical gradient estimation procedure. Improvement of this result would automatically improve the quantum policy optimization routine as well
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
Binomial formulas for Macdonald polynomials
No full textSymmetric and nonsymmetric Macdonald polynomials associated to root systems are very general families of orthogonal polynomials in multiple variables. Their definition is quite complex, but in certain cases one can define so-called interpolation polynomials that have a surprisingly simple definition and are related to the Macdonald polynomials by a binomial formula. In this thesis we will discuss such formulas for two kinds of root systems: type A and type (C∨,C). For the latter case, there are still some open questions that remain unanswered.Applied Mathematic
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
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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