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Register a new userBypassing dynamical systems : A simple way to get the box-counting dimension of the graph of the Weierstrass function
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Authors
Claire David
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In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by~$ {cal W}(x)=displaystyle sum_{n=0}^{+infty} lambda^n,cos left ( 2, pi,N_b^n,x right) $, where~$lambda$ and~$N_b$ are two real numbers such that~mbox{$0 <lambda<1$},~mbox{$ N_b,in,N$} and~$ lambda,N_b > 1 $, using a sequence a graphs that approximate the studied one.
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Many large-scale machine learning problems involve estimating an unknown parameter $theta_{i}$ for each of many items. For example, a key problem in sponsored search is to estimate the click through rate (CTR) of each of billions of query-ad pairs. Most common methods, though, only give a point estimate of each $theta_{i}$. A posterior distribution for each $theta_{i}$ is usually more useful but harder to get. We present a simple post-processing technique that takes point estimates or scores $t_{i}$ (from any method) and estimates an approximate posterior for each $theta_{i}$. We build on the idea of calibration, a common post-processing technique that estimates $mathrm{E}left(theta_{i}!!bigm|!! t_{i}right)$. Our method, second order calibration, uses empirical Bayes methods to estimate the distribution of $theta_{i}!!bigm|!! t_{i}$ and uses the estimated distribution as an approximation to the posterior distribution of $theta_{i}$. We show that this can yield improved point estimates and useful accuracy estimates. The method scales to large problems - our motivating example is a CTR estimation problem involving tens of billions of query-ad pairs.
We investigate Weierstrass functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H={loggamma}/{log frac12}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough $gamma$ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.
In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $Lambda$. We begin by giving an alternative characterization of the $Lambda$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.
A class of two-dimensional linear differential systems is considered. The box-counting dimension of the graphs of solution curves is calculated. Criteria to obtain the box-counting dimension of spirals are also established.
We show that the graph of the classical Weierstrass function $sum_{n=0}^infty lambda^n cos (2pi b^n x)$ has Hausdorff dimension $2+loglambda/log b$, for every integer $bge 2$ and every $lambdain (1/b,1)$. Replacing $cos(2pi x)$ by a general non-constant $C^2$ periodic function, we obtain the same result under a further assumption that $lambda b$ is close to $1$.
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