Spherical sets avoiding a prescribed set of angles

Let $X$ be any subset of the interval $[-1,1]$. A subset $I$ of the unit sphere in $R^n$ will be called \emph{$X$-avoiding} if $\notin X$ for any $u,v \in I$. The problem of determining the maximum surface measure of a $\{ 0 \}$-avoiding set was first stated in a 1974 note by Witsenhausen; there the upper bound of $1/n$ times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl-Wilson theorem is that this fraction decreases exponentially, but until now the $1/3$ upper bound for the case $n=3$ has not moved. We improve this bound to $0.313$ using an approach inspired by Delsarte's linear programming bounds for codes, combined with some combinatorial reasoning. In the second part of the paper, we use harmonic analysis to show that for $n\geq 3$ there always exists an $X$-avoiding set of maximum measure. We also show with an example that a maximiser need not exist when $n=2$.Comment: 21 pages, 3 figure

Fourier analysis on finite groups and the Lov\'asz theta-number of Cayley graphs

We apply Fourier analysis on finite groups to obtain simplified formulations for the Lov\'asz theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a recent article proving a version of the Erd\H{o}s-Ko-Rado theorem for $k$-intersecting families of permutations. We also introduce a $q$-analog of the notion of $k$-intersecting families of permutations, and we verify a few cases of the corresponding Erd\H{o}s-Ko-Rado assertion by computer.Comment: 9 pages, 0 figure

Sine dolo malo

This is the author accepted manuscript. The final version is available from Brill via http://dx.doi.org/10.1163/1568525x-12341822The Oscan law of the Tabula Bantina (tbo), the longest Oscan text found to this date, is the product of a politically significant time in Roman history in which the Italian socii revolted against, and tried to assert independence from, Rome. In spite of this, the tbo largely reads like a Roman law, particularly in vocabulary and phrasing. The aim of this article is to reveal the extent to which the tbo was influenced by Latin legal language. It will identify several remarkable syntactic, stylistic and epigraphic aspects of this Oscan law and compare them to conventions in the Latin legal register.</jats:p

Les effets adverses des politiques officielles en matière de drogue sur les mécanismes d’autorégulation des consommateurs de drogues illicites

Les interventions d’ordre politique et juridique entourant la consommation de drogues découlent principalement de la croyance répandue voulant que les drogues illicites génèrent une dépendance qui, elle, engendre le chaos. Cela étant dit, nos connaissances sur les divers schémas de consommation de substances licites et une foule d’études scientifiques sur l’usage contrôlé de drogues illicites et l’arrêt spontané de consommation de celles-ci remettent en question l’opinion généralisée voulant que l’usage mène inexorablement à une perte de contrôle. Pour comprendre comment et pourquoi certains usagers perdent le contrôle de leur consommation, nous devons chercher à savoir comment et pourquoi tant d’autres arrivent à acquérir ce contrôle et à le garder.Le contexte social peut créer de nouvelles sanctions sociales informelles (comme définies par N.E. Zinberg) et des rituels (contrôles informels). En diffusant de l’information selon plusieurs mécanismes informels, le contexte social est un facteur essentiel de l’usage contrôlé des substances intoxicantes. Contrairement à ce qui se passe avec les substances licites, les occasions d’apprendre à contrôler l’usage de drogues illicites sont extrêmement limitées.L’un des principaux stimuli externes influant sur les processus naturels d’autorégulation informelle est la définition sociale des drogues et des consommateurs véhiculée par la politique formelle en matière de drogues. La politique actuelle – axée particulièrement sur la répression – ne réussit pas à encourager une consommation sécuritaire et même paralyse dans une large mesure le développement et la transmission de normes sécuritaires. Les mécanismes de contrôle formel actuels encouragent la formation de sous-cultures déviantes d’usage de drogues illicites, engendrent et amplifient les stéréotypes négatifs (le junkie, le criminel, l’individu violent) et entravent la diffusion des rituels de contrôle et des sanctions sociales concernant la consommation dans son ensemble.Political and legal responses to drug use are largely underpinned with the ubiquitous belief that illicit drugs generate dependence which, in turn, creates chaos. However, our common knowledge about varying use patterns of licit drugs, and an extensive array of scientific studies on controlled use of illicit drugs and natural cessation of illicit drug use challenge the widely held belief that drug use inexorably tends towards loss of control. In order to understand how and why certain users have lost control over the drug or drugs they are using, we need to tackle the all-important question of how and why many others manage to achieve control and maintain it. The social setting, with its capacity to develop new informal social sanctions and rituals (informal controls), and its transmission of information in numerous informal ways, is a crucial factor in the controlled use of any intoxicant. Unlike the situation with licit drugs, the opportunities for learning how to control illicit drug consumption remain extremely limited. One of the weightiest external stimuli that affect the natural processes of informal self-regulation is the social definition of drugs and their users, embodied in formal drug policy. The current drug policy, with its mainly repressive characteristics, fails to reinforce safe use and even to a large extent obstructs the development and communication of safe standards. The present forms of formal control create the preconditions for deviant subcultures of illicit drug use, generates and reinforces the stereotypical negative image (junkie, criminal, violent), and impedes the dissemination of controlling rituals and social sanctions about drug use in general.Las intervenciones de carácter político y jurídico que rodean el consumo de drogas se originan principalmente en la creencia difundida de que las drogas ilícitas generan una dependencia, la que, a su vez, engendra el caos. Sin embargo, nuestros conocimientos sobre los diversos esquemas de consumo de sustancias ilícitas, así como una cantidad de estudios científicos sobre el uso controlado de drogas ilícitas y el cese espontáneo del consumo de las mismas, cuestionan la opinión generalizada que sostiene que su uso lleva inexorablemente a una pérdida de control. Para comprender por qué razón ciertos usuarios pierden el control del consumo debemos tratar de saber de qué manera y porqué tantos otros llegan a tener dicho control y a mantenerlo.El contexto social puede crear nuevas sanciones sociales informales (tal como las define N. E. Zinberg) y rituales (controles informales). El contexto social, al difundir la información a través de numerosos mecanismos informales, es un factor esencial para el uso controlado de sustancias intoxicantes. Contrariamente a lo que sucede con las sustancias lícitas, las ocasiones para aprender a controlar el uso de drogas ilícitas son extremadamente limitadas.Uno de los principales estímulos externos que influyen en los procesos naturales de autorregulación informal es la definición social de las drogas y de los consumidores transmitida por la política formal en materia de drogas. La política actual, centrada principalmente en la represión, no logra alentar un consumo seguro e incluso paraliza en gran medida el desarrollo y la transmisión de normas seguras. Los mecanismos de control formal actuales alientan la formación de subculturas desviadas de uso de drogas ilícitas, engendran y amplifican los estereotipos negativos (el junkie, el criminal, el individuo violento) y entorpecen la difusión de rituales de control y de las sanciones sociales referidas al consumo en su conjunto