Pascal triangles and Sierpiński gasket extended to binomial coefficients of words

The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead from October 2015 on an extension of the Pascal triangles to base-2 expansions of integers. In a first part, I define two new objects that both generalize the classical Pascal triangle and the Sierpinski gasket. In a second part, I define a new sequence extracted from the Pascal triangle in base 2 and study its regularity. In a third part, I exhibit an exact formula for the behavior of the summatory function of the latter sequence

Adjoining a universal inner inverse to a ring element

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ We obtain normal forms for elements of $R',$ and for elements of $R'$-modules arising by extension of scalars from $R$-modules. The details depend on where in the chain $pR\cap Rp \subseteq pR\cup Rp \subseteq pR + Rp \subseteq R$ the unit $1$ of $R$ first appears. This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant. We end with a normal form result for the algebra obtained by tying together a $k$-algebra $R$ given with a nonzero element $p$ satisfying $1\notin pR+Rp$ and a $k$-algebra $S$ given with a nonzero $q$ satisfying $1\notin qS+Sq,$ via the pair of relations $p=pqp,$ $q=qpq.$Comment: 28 pages. Results on mutual inner inverses added at end of earlier version, and much clarification of wording etc.. After publication, any updates, errata, related references etc. found will be recorded at http://math.berkeley.edu/~gbergman/paper

Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences

We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those n for which an automatic sequence x has a closed (resp., palindromic, privileged, rich, trape- zoidal, balanced) factor of length n is automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in a prefix of length n of a k-automatic sequence is not k-synchronized

Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems

We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand property, i.e., we can add or remove trailing zeroes to valid representations. It is a folklore fact that the Sierpi\'{n}ski gasket is the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from the classical Pascal triangle modulo $2$. In a similar way, we describe and study the subset of $[0, 1] \times [0, 1]$ associated with the latter generalization of the Pascal triangle modulo a prime number.Comment: 30 pages; 32 figure

Overlap-Free Words and Generalizations

The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. This thesis will consider several different generalizations of Thue's work. In particular we shall study the properties of infinite words avoiding various types of repetitions. In Chapter 1 we introduce the theory of combinatorics on words. We present the basic definitions and give an historical survey of the area. In Chapter 2 we consider the work of Thue in more detail. We present various well-known properties of the Thue-Morse word and give some generalizations. We examine Fife's characterization of the infinite overlap-free words and give a simpler proof of this result. We also present some applications to transcendental number theory, generalizing a classical result of Mahler. In Chapter 3 we generalize a result of Seebold by showing that the only infinite 7/3-power-free binary words that can be obtained by iterating a morphism are the Thue-Morse word and its complement. In Chapter 4 we continue our study of overlap-free and 7/3-power-free words. We discuss the squares that can appear as subwords of these words. We also show that it is possible to construct infinite 7/3-power-free binary words containing infinitely many overlaps. In Chapter 5 we consider certain questions of language theory. In particular, we examine the context-freeness of the set of words containing overlaps. We show that over a three-letter alphabet, this set is not context-free, and over a two-letter alphabet, we show that this set cannot be unambiguously context-free. In Chapter 6 we construct infinite words over a four-letter alphabet that avoid squares in any arithmetic progression of odd difference. Our constructions are based on properties of the paperfolding words. We use these infinite words to construct non-repetitive tilings of the integer lattice. In Chapter 7 we consider approximate squares rather than squares. We give constructions of infinite words that avoid such approximate squares. In Chapter 8 we conclude the work and present some open problems

Spoken content retrieval: A survey of techniques and technologies

Speech media, that is, digital audio and video containing spoken content, has blossomed in recent years. Large collections are accruing on the Internet as well as in private and enterprise settings. This growth has motivated extensive research on techniques and technologies that facilitate reliable indexing and retrieval. Spoken content retrieval (SCR) requires the combination of audio and speech processing technologies with methods from information retrieval (IR). SCR research initially investigated planned speech structured in document-like units, but has subsequently shifted focus to more informal spoken content produced spontaneously, outside of the studio and in conversational settings. This survey provides an overview of the field of SCR encompassing component technologies, the relationship of SCR to text IR and automatic speech recognition and user interaction issues. It is aimed at researchers with backgrounds in speech technology or IR who are seeking deeper insight on how these fields are integrated to support research and development, thus addressing the core challenges of SCR