1,721,017 research outputs found
Random splitting of point vortex flows
Full text linkWe consider a stochastic version of the point vortex system, in which the fluid velocity advects single vortices intermittently for small random times. Such system converges to the deterministic point vortex dynamics as the rate at which single components of the vector field are randomly switched diverges, and therefore it provides an alternative discretization of 2D Euler equations. The random vortex system we introduce preserves microcanonical statistical ensembles of the point vortex system, hence constituting a simpler alternative to the latter in the statistical mechanics approach to 2D turbulence
Random Splitting of Fluid Models: Ergodicity and Convergence
Full text linkWe introduce a family of stochastic models motivated by the study of
nonequilibrium steady states of fluid equations. These models decompose the
deterministic dynamics of interest into fundamental building blocks, i.e.,
minimal vector fields preserving some fundamental aspects of the original
dynamics. Randomness is injected by sequentially following each vector field
for a random amount of time. We show under general assumptions that these
random dynamics possess a unique invariant measure and converge almost surely
to the original, deterministic model in the small noise limit. We apply our
construction to the Lorenz-96 equations, often used in studies of chaos and
data assimilation, and Galerkin approximations of the 2D Euler and
Navier-Stokes equations. An interesting feature of the models developed is that
they apply directly to the conservative dynamics and not just those with
excitation and dissipation
The Gaussian structure of the singular stochastic Burgers equation
Full text linkWe consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time t, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite-dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes, which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed, as we are beyond the regime of classical solutions for much of the paper
Sticky central limit theorems on open books
Full text linkGiven a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine.We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky). © Institute of Mathematical Statistics, 2013
Random Splitting of Fluid Models: Ergodicity, Convergence, and Chaos
No full textIn this thesis we study random splitting and apply our results to random splittings of fluid models. Random splitting is loosely defined as follows. Consider the differential equation where is a time derivative and the vector field on splits as the sum . In traditional operator splitting one approximates solutions of by composing solutions of over (typically small) deterministic time steps. Here we take these times to be independent and identically distributed random variables. This turns the aforementioned compositions into Markov chains, which we call \textit{random splittings of } or simply \textit{random splittings}. We prove under relatively mild conditions that these random splittings possess a unique invariant measure (ergodicity), that their trajectories converge on average and almost surely to trajectories of the original system (convergence), and that, in certain cases, their top Lyapunov exponent is positive (chaos). After proving these general results, we construct random splittings of four fluid models: the conservative Lorenz-96 and Lorenz-96 equations, and Galerkin approximations of the 2d Euler and 2d Navier-Stokes equations on the torus. We prove all these random splittings are ergodic and converge to their deterministic counterparts in a certain sense, and, for conservative Lorenz-96 and 2d Euler, that their top Lyapunov exponent is positive. </p
A Generalized Lyapunov Construction for Proving Stabilization by Noise
No full textNoise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.</p
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
Dimensionality Reduction and Learning on Networks
No full textMachine learning is a powerful branch of mathematics and statistics that allows the automation of tasks that would otherwise require humans a long time to perform. Two particular fields of machine learning that have been developing in the last two decades are dimensionality reduction and semi-supervised learning.Dimensionality reduction is a powerful tool in the analysis of high dimensional data by reducing the number of variables under consideration while approximately preserving some quantity of interest (usually pairwise distances). Methods such as Principal Component Analysis (PCA) or Isometric Feature Mapping (ISOMAP) do this do this by embedding the data, equipped with a nonnegative, symmetric, similarity kernel or adjacency matrix into Euclidean space and finding a linear subspace or low dimensional submanifold which best fits the data, respectively.When the data takes the form of network data, how to perform such dimensionality reduction intrinsically, without resorting to an embedding, that can be extended to the case of nonnegative, non-symmetric adjacency matrices remains an important open problem. In the first part of my dissertation, using current techniques in local spectral clustering to partition the network using a Markov process induced by the adjacency matrix, we deliver an intrinsic dimensionality reduction of the network in terms of a non-Markov process on a reduced state space that approximately preserves transitions of the original Markov process between clusters. By iterating the process, one obtains a family of non-Markov processes on successively finer state spaces representing the original network ands its diffusion at different scales, which can be used to approximate the law of the original process at a particular time scale. We give applications of this theory to a variety of synthetic data sets and evaluate its performance accordingly.Next, consider the case of detecting astronomical phenomenon solely in terms of the light intensities observed. There already exists a large database of prior recorded phenomena that has been categorized by humans as a function of the observed light intensity. Given these so-called class labels then, how can we automate the procedure of extending these class labels to the massive amount of data that is currently being observed? This is the problem of concern in semi-supervised learning.In the second part of my thesis, we consider data sets in which relations between data points are more complex than simply pairwise. Examples include gene networks where the the data points are random variables, and similarities of a subset are measured by non-independence of the corresponding random variables. Such data sets can be illustrated as a hypergraph, and the natural question for diagnosis becomes: how does one perform transductive inference (a particular form of semi-supervised learning)? Using the simple case of pairwise and threewise similarities, we construct a reversible random walk on undirected edges induced by threewise relations (faces). By pulling the random walk back to a random walk on the vertex set and mixing it with the random walk induced by pairwise similarities, we perform diffusive transductive inference. We present applications and results of this technique, any analyze its performance on a variety of data sets.</p
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
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