1,076 research outputs found

    Sarnak's Conjecture for nilsequences on arbitrary number fields and applications

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    We formulate the generalized Sarnak's M\"obius disjointness conjecture for an arbitrary number field KK, and prove a quantitative disjointness result between polynomial nilsequences (Φ(g(n)Γ))nZD(\Phi(g(n)\Gamma))_{n\in\mathbb{Z}^{D}} and aperiodic multiplicative functions on OK\mathcal{O}_{K}, the ring of integers of KK. Here D=[K ⁣:Q]D=[K\colon\mathbb{Q}], X=G/ΓX=G/\Gamma is a nilmanifold, g ⁣:ZDGg\colon\mathbb{Z}^{D}\to G is a polynomial sequence, and Φ ⁣:XC\Phi\colon X\to \mathbb{C} is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of K\'atai in OK\mathcal{O}_{K}. We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on KK, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in OK\mathcal{O}_{K} to be zero; (3) we provide partition regularity results over KK for a large class of homogeneous equations in three variables. For example, for a,bZ\{0}a,b\in\mathbb{Z}\backslash\{0\}, we show that for every partition of OK\mathcal{O}_{K} into finitely many cells, where K=Q(a,b,a+b)K=\mathbb{Q}(\sqrt{a},\sqrt{b},\sqrt{a+b}), there exist distinct and non-zero x,yx,y belonging to the same cell and zOKz\in\mathcal{O}_{K} such that ax2+by2=z2ax^{2}+by^{2}=z^{2}.Comment: 65 page

    A pointwise cubic average for two commuting transformations

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    Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system (X,μ,S,T)(X,\mu,S,T) with commuting transformations SS and TT, the average 1N2i,j=0N1f0(Six)f1(Tjx)f2(SiTjx)\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x) converges a.e. as NN goes to infinity for any f1,f2,f3L(μ)f_1,f_2,f_3\in L^{\infty}(\mu)

    Evaluación de algoritmos de Machine Learning para conducción

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    Trabajo de Fin de Grado en Ingeniería Informática, Facultad de Informática UCM, Departamento de Arquitectura de Computadores y Automática, Curso 2020/2021.In this research and development project, our main purpose is to study four deep learning architectures for real-time object detection of people and bicycles encountered in front of driving. We use 4 different algorithms for the same data set, and compare the mAPs obtained after training. And discuss which method is the most accurate, but also consider the time it takes to get what is suitable for what kind of scene. The project I came up with would like to be used in a driving assistance system. The system uses camera sensors to get input, and then uses algorithms to assist, so that the safety of the car is guaranteed when driving. At the same time, it can run on a lowperformance version of the machine and compare the fps of different algorithms.Depto. de Arquitectura de Computadores y AutomáticaFac. de InformáticaTRUEunpu

    Sampling and Counting Crossing-Free Matchings

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    Sampling of combinatorial structures is an important statistical tool used in applications in a number of areas ranging from statistical physics, data mining, to biological sciences. Of comparable importance is the computation of the cor- responding partition function, which, in the case of the uniform distribution, is equivalent to the problem of counting all such structures. For self-reducible combinatorial structures, once we can produce an almost uniform sample from them, then we can approximately count them. Using a Markov chain Monte Carlo method, this thesis presents polynomial-time algorithms to approximately count and almost uniformly sample crossing-free matchings for certain input classes of graphs. Since the problem in its generality appears to be difficult, we made natural restrictions on the in- put graphs. Namely, we consider vertices arranged in a grid in the plane, where edges are line segments connecting the vertices and a matching is crossing-free if no two matching edges intersect. For appropriate bounds on the dimensions of the grid and the edge lengths, we show that a natural Markov chain is rapidly mixing and that the problem is self-reducible
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