Finite-Repetition threshold for infinite ternary words

The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at most r(a). This notion was introduced in 1972 by Dejean who gave the exact values of r(a) for every alphabet size a as it has been eventually proved in 2009. The finite-repetition threshold for an a-letter alphabet refines the above notion. It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contains a finite number of factors with exponent r(a). It is known from Shallit (2008) that FRt(2)=7/3. With each finite-repetition threshold is associated the smallest number of r(a)-exponent factors that can be found in the corresponding infinite word. It has been proved by Badkobeh and Crochemore (2010) that this number is 12 for infinite binary words whose maximal exponent is 7/3. We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite word containing only two 7/4-exponent words, the smallest number. Based on deep experiments we conjecture that FRt(4)=r(4)=7/5. The question remains open for alphabets with more than four letters. Keywords: combinatorics on words, repetition, repeat, word powers, word exponent, repetition threshold, pattern avoidability, word morphisms.Comment: In Proceedings WORDS 2011, arXiv:1108.341

The ⋆\star-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of \W^\star. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure

Repetitions in partial words

El objeto de esta tesis está representado por las repeticiones de palabras parciales, palabras que, además de las letras regulares, pueden tener un número de símbolos desconocidos,llamados símbolos "agujeros" o "no sé qué". Más concretamente, se presenta y se resuelve una extensión de la noción de repetición establecida por Axel Thue. Investigamos las palabras parciales con un número infinito de agujeros que cumplen estas propiedades y, también las palabras parciales que conservan las propiedades después de la inserción de un número arbitrario de agujeros, posiblemente infinito. Luego, hacemos un recuento del número máximo de 2-repeticiones distintas compatibles con los factores de una palabra parcial. Se demuestra que el problema en el caso general es difícil, y estudiamos el problema en el caso de un agujero. Al final, se estudian algunas propiedades de las palabras parciales sin fronteras y primitivas (palabras sin repeticiones) y se da una caracterización del lenguaje de palabras parciales con una factorización crítica

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization

Quantum XOR Games

We introduce quantum XOR games, a model of two-player one-round games that extends the model of XOR games by allowing the referee's questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck's inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance players can obtain in a given game, both in case they have no shared entanglement and in case they share unlimited entanglement. As a byproduct of the algorithm we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.Comment: 43 page

Feasible approach for the computer implementation of parametric visual calculating

Thesis (S.M. in Architecture Studies)--Massachusetts Institute of Technology, Dept. of Architecture, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 62-66).Computational design tools in architecture currently fall into two broad categories: Tools for representation and tools for generative design, including scripting. However, both categories address only relatively methodical aspects of designing, and do little to support the design freedom and serendipitous creativity that, for example, is afforded by iterative sketching. Calculating with visual rules provides an explicit notation for such artistic processes of seeing and drawing. Shape grammars have validated this approach by formalizing many existing designs and styles as visual rule-sets. In this way, visual rules store and transfer design knowledge. Visual calculating in a more general sense supports creativity by allowing a designer to apply any rule she wants, and to capriciously see and re-see the design. In contrast to other explicit design methodologies, visual calculating defines a decomposition into parts only after the design is calculated, thus allowing formalization without impeding design freedom. Located at the intersection between design and computation, the computer implementation of visual calculating presents an opportunity for more designerly computational design tools. Since parametric visual calculating affords the largest set of design possibilities, the computer implementation of parametric visual calculating will allow flexible, rule-based design tools that intelligently combine design freedom with computational processing power. In order to compute with shapes, a symbolic representation for shapes is necessary. This thesis examines several symbolic representations for shapes, including graphs. Especially close attention is given to graph-based representations, since graphs are well suited to represent parametric shapes. Based on this analysis, this thesis proposes a new graph for parametric shapes that is clearer, more compact and closer the original formulation of visual calculating than existing approaches, while also strongly supporting design freedom. The thesis provides algorithms and heuristics to construct this "inverted" graph, for connected and unconnected shapes.by Thomas Alois Wortmann.S.M.in Architecture Studie