Reticulados em problemas de comunicação

Orientadores: Sueli Irene Rodrigues Costa, Vinay Anant VaishampayanTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O estudo de códigos no contexto de reticulados e outras constelações discretas para aplicações em comunicações é um tópico de interesse na área de teoria da informação. Certas construções de reticulados, como é o caso das Construções A e D, e de outras constelações que não são reticulados, como a Construção C, são utilizadas na decodificação multi-estágio e para quantização vetorial eficiente. Isso motiva a primeira contribuição deste trabalho, que consiste em investigar características da Construção C e propor uma nova construção baseada em códigos lineares, que chamamos de Construção $C^\star,$ analisando suas propriedades (condições para ser reticulado, uniformidade geométrica e distância mínima) e relação com a Construção C. Problemas na área de comunicações envolvendo reticulados podem ser computacionalmente difíceis à medida que a dimensão aumenta, como é o caso de, dado um vetor no espaço real $n-$dimensional, determinar o ponto do reticulado mais próximo a este. A segunda contribuição deste trabalho é a análise desse problema restrito a um sistema distribuído, ou seja, onde o vetor a ser decodificado possui cada uma de suas coordenadas disponíveis em um nó distinto desse sistema. Nessa investigação, encontramos uma solução aproximada para duas e três dimensões considerando a partição de Babai e também estudamos o custo de comunicação envolvidoAbstract: The study of codes in the context of lattices and other discrete constellations for applications in communications is a topic of interest in the area of information theory. Some lattice constructions, such as the known Constructions A and D, and other special nonlattice constellations, as Construction C, are used in multi-stage decoding and efficient vector quantization. This motivates the first contribution of this work, which is to investigate characteristics of Construction C and to propose a new construction based on linear codes that we called Construction $C^\star,$ analyzing its properties (latticeness, geometric uniformity and minimum distance) and relations with Construction C. Communication problems related to lattices can be computationally hard when the dimension increases, as it is the case of, given a real vector in the $n-$dimensional space, determine the closest lattice point to it. The second contribution of this work is the analysis of this problem restricted to a distributed system, i.e., where the vector to be decoded has each coordinate available in a separated node in this system. In this investigation, we find the approximate solution for two and three dimensions considering the Babai partition and study the communication cost involvedDoutoradoMatematica AplicadaDoutora em Matemática Aplicada140797/2017-3CNPQCAPE

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer is given to the Hilbert basis question raised by Billera, Jia, and Reiner.Comment: 25 pages; exposition tightened, typos corrected; to appear in the Journal of Combinatorial Theory, Series

Combinatorial and Additive Number Theory Problem Sessions: '09--'19

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019, fixed a few issues from some presenters 6/29/201

Geodesics on Flat Surfaces

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.Comment: (25 pages, 5 figures) Based on the talk at ICM 2006 at Madrid; see Proceedings of the ICM, Madrid, Spain, 2006, EMS, 121-146 for the final version. For a more detailed survey see the paper "Flat Surfaces", arXiv.math.DS/060939

Typicality, Black Hole Microstates and Superconformal Field Theories

We analyze the structure of heavy multitrace BPS operators in N = 1 superconformal quiver gauge theories that arise on the worldvolume of D3-branes on an affine toric cone. We exhibit a geometric procedure for counting heavy mesonic operators with given U(1) charges. We show that for any fixed linear combination of the U(1) charges, the entropy is maximized when the charges are in certain ratios. This selects preferred directions in the charge space that can be determined with the help of a piece of string. We show that almost all heavy mesonic operators of fixed U(1) charges share a universal structure. This universality reflects the properties of the dual extremal black holes whose microstates they create. We also interpret our results in terms of typical configurations of dual giant gravitons in AdS space.Comment: 40 pages + 3 appendices, 11 figure

Composite QDrift-Product Formulas for Quantum and Classical Simulations in Real and Imaginary Time

Recent work has shown that it can be advantageous to implement a composite channel that partitions the Hamiltonian $H$ for a given simulation problem into subsets $A$ and $B$ such that $H=A+B$, where the terms in $A$ are simulated with a Trotter-Suzuki channel and the $B$ terms are randomly sampled via the QDrift algorithm. Here we show that this approach holds in imaginary time, making it a candidate classical algorithm for quantum Monte-Carlo calculations. We upper-bound the induced Schatten-$1 \to 1$ norm on both imaginary-time QDrift and Composite channels. Another recent result demonstrated that simulations of Hamiltonians containing geometrically-local interactions for systems defined on finite lattices can be improved by decomposing $H$ into subsets that contain only terms supported on that subset of the lattice using a Lieb-Robinson argument. Here, we provide a quantum algorithm by unifying this result with the composite approach into ``local composite channels" and we upper bound the diamond distance. We provide exact numerical simulations of algorithmic cost by counting the number of gates of the form $e^{-iH_j t}$ and $e^{-H_j \beta}$ to meet a certain error tolerance $\epsilon$. We show constant factor advantages for a variety of interesting Hamiltonians, the maximum of which is a $\approx 20$ fold speedup that occurs for a simulation of Jellium.Comment: 49 pages, 13 figure

James' Conjecture for Hecke algebras of exceptional type, I

In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' Conjecture for Iwahori--Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are: - the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham--Lehrer, and - the explicit determination of $W$-graphs for the irreducible (generic) representations of Hecke algebras of type $E_7$ and $E_8$ by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's {\sf MeatAxe} and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.Comment: 24 pages; corrected some misprints, added Remark 4.1