Hypergraph expanders of all uniformities from Cayley graphs

Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.Comment: 32 page

Bounding sequence extremal functions with formations

An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has no subsequence isomorphic to $u$. For every sequence $u$ define $\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which there exists $r$ such that there is a subsequence isomorphic to $u$ in every $(r, s)$-formation. We use $\mathit{fw}(u)$ to prove upper bounds on $\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternation with the same formation width as $u$. We generalize Nivasch's bounds on $\mathit{Ex}((ab)^{t}, n)$ by showing that $\mathit{fw}((12 \ldots l)^{t})=2t-1$ and $\mathit{Ex}((12\ldots l)^{t}, n) =n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for every $l \geq 2$ and $t\geq 3$, such that $\alpha(n)$ denotes the inverse Ackermann function. Upper bounds on $\mathit{Ex}((12 \ldots l)^{t} , n)$ have been used in other papers to bound the maximum number of edges in $k$-quasiplanar graphs on $n$ vertices with no pair of edges intersecting in more than $O(1)$ points. If $u$ is any sequence of the form $a v a v' a$ such that $a$ is a letter, $v$ is a nonempty sequence excluding $a$ with no repeated letters and $v'$ is obtained from $v$ by only moving the first letter of $v$ to another place in $v$, then we show that $\mathit{fw}(u)=4$ and $\mathit{Ex}(u, n) =\Theta(n\alpha(n))$. Furthermore we prove that $\mathit{fw}(abc(acb)^{t})=2t+1$ and $\mathit{Ex}(abc(acb)^{t}, n) = n2^{\frac{1}{(t-1)!}\alpha(n)^{t-1}\pm O(\alpha(n)^{t-2})}$ for every $t\geq 2$.Comment: 25 page

Joves i televisió: l'experiència de Televisió de Catalunya

El català mola? Si ens referim que el català pot ser una llengua d'ús habitual en els mitjans de comunicació quan es vol abordar un públic jove, podem dir que el català pot molar. Dependrà, però, de com juguem les nostres cartes. L'aparició del Canal 3XL,1 adreçat a públic jove, ha suposat un repte per a la creació de models lingüístics que equilibrin l'observació de la realitat, la creativitat i la tradició. L 'article exposa com cal treure profit de les possibilitats que obre el món actual i com es poden assolir models que permetin connectar amb el públic jove. S'ha procurat construir un model normatiu però alhora viu i fresc, que incorpori el llenguatge dels blocs, de Facebook o dels SMS (amb la fragmentació del discurs que això comporta), i que permeti afirmar, sense cap mena de dubte, que el català mola.Is Catalan “cool”? If this means that Catalan can be a language that is ordinarily used in the communication media when seeking to reach out to young people, it may be said that Catalan can be “cool”. It all depends, however, on how the media play their cards. The appearance of Channel 3XL, which is addressed to young people, has faced the challenge of creating language models that strike a balance between the observation of reality, creativity and tradition. This paper describes how advantage should be taken of the possibilities offered by today’s world and how to achieve models that will allow young people to relate to the media. It has been sought to build a model that is prescriptive as well as fresh and lively; that incorporates the language of blogs, Facebook or text messages (with the fragmentation of the discourse that this entails), and that will allow it to be affirmed, beyond the shadow of a doubt, that Catalan is “cool”

Quantitative bounds for the U4U^4-inverse theorem over low characteristic finite fields

This paper gives the first quantitative bounds for the inverse theorem for the Gowers $U^4$-norm over $\mathbb{F}_p^n$ when $p=2,3$. We build upon earlier work of Gowers and Mili\'cevi\'c who solved the corresponding problem for $p\geq 5$. Our proof has two main steps: symmetrization and integration of low-characteristic trilinear forms. We are able to solve the integration problem for all $k$-linear forms, but the symmetrization problem we are only able to solve for trilinear forms. We pose several open problems about symmetrization of low-characteristic $k$-linear forms whose resolution, combined with recent work of Gowers and Mili\'cevi\'c, would give quantitative bounds for the inverse theorem for the Gowers $U^{k+1}$-norm over $\mathbb{F}_p^n$ for all $k,p$.Comment: 17 page

Efficient Bayesian estimates for discrimination among topologically different systems biology models

A major effort in systems biology is the development of mathematical models that describe complex biological systems at multiple scales and levels of abstraction. Determining the topology—the set of interactions—of a biological system from observations of the system's behavior is an important and difficult problem. Here we present and demonstrate new methodology for efficiently computing the probability distribution over a set of topologies based on consistency with existing measurements. Key features of the new approach include derivation in a Bayesian framework, incorporation of prior probability distributions of topologies and parameters, and use of an analytically integrable linearization based on the Fisher information matrix that is responsible for large gains in efficiency. The new method was demonstrated on a collection of four biological topologies representing a kinase and phosphatase that operate in opposition to each other with either processive or distributive kinetics, giving 8–12 parameters for each topology. The linearization produced an approximate result very rapidly (CPU minutes) that was highly accurate on its own, as compared to a Monte Carlo method guaranteed to converge to the correct answer but at greater cost (CPU weeks). The Monte Carlo method developed and applied here used the linearization method as a starting point and importance sampling to approach the Bayesian answer in acceptable time. Other inexpensive methods to estimate probabilities produced poor approximations for this system, with likelihood estimation showing its well-known bias toward topologies with more parameters and the Akaike and Schwarz Information Criteria showing a strong bias toward topologies with fewer parameters. These results suggest that this linear approximation may be an effective compromise, providing an answer whose accuracy is near the true Bayesian answer, but at a cost near the common heuristics.National Cancer Institute (U.S.) (U54 CA112967)National University of Singapor