Energy minimization, periodic sets and spherical designs

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.Comment: 16 pages; incorporated referee comment

Spherical Designs and Heights of Euclidean Lattices

We study the connection between the theory of spherical designs and the question of extrema of the height function of lattices. More precisely, we show that a full-rank n-dimensional Euclidean lattice, all layers of which hold a spherical 2-design, realises a stationary point for the height function, which is defined as the first derivative at 0 of the spectral zeta function of the associated flat torus. Moreover, in order to find out the lattices for which this 2-design property holds, a strategy is described which makes use of theta functions with spherical coefficients, viewed as elements of some space of modular forms. Explicit computations in dimension up to 7, performed with Pari/GP and Magma, are reported.Comment: 22 page

Why cultural attitudes and lifestyles remains so stratified and differentiated in the context of cultural globalisation?

A well-established tradition of research in the social sciences insists on the symbolic dimensions of social stratification and social classes’ relations in contemporary societies that are not reducible to their relations in the sphere of production. Fist and foremost, many authors, from Halbwachs and Veblen to Bourdieu, have intended to challenge the predominantly “production-based” conception of social classes, inherited form the Marxian tradition, insisting on what could be designed as a “consumption-based” theory of social classes, in which people tend to differentiate themselves from each other on the base of their patterns of consumption expenditures, lifestyles, tastes and habits in such diverse areas than clothing, food or cultural activities, but also on the base of their moral or political values, and even their religious or spiritual beliefs. Theses two conceptions – production-based and lifestyles-based theories of social classes – are in fact far form being mutually exclusive, to the extent that the social stratification of lifestyles often extends end expresses the social cleavages that originate in the sphere of production. But the lifestyles-based theory nonetheless suggests the relative autonomy of people lifestyles as to the purely economic factors. In particular, in the sphere of consumption, the behaviours patterns are merely considered as an exclusive matter of price/earning optimization, but entail a cultural and social framing of people choices and preferences. In this paper, we will try to evaluate the relevance of this “consumption-based” or “lifestylesbased” orientation and, perhaps more generally, the relevance of all the class-schemes that focus on the symbolic dimensions of social interactions, be they religious, spiritual or cultural, in a broader sense

Reading and Television-Watching: Changes in the Cultural Role of the School in the Age of Mass Education

What are the cultural effects of generalized access to upper secondary education, which became effective in France in the 1980s and 1990s? Data on reading and television-watching suggest that reading is in decline, a decline concentrated among high school and higher education graduates, and that television occupies a greater place in the lives of generations who attended school in the 1980s and 1990s than in those of the preceding generations. These two developments result from a complex combination of the “net” and structural effects of mass education. Mass education has slowed the decline of reading and mitigated the increase in television-watching: the proportional weight in the population at large of categories most likely to read heavily and least likely to be intensive TV watchers – i.e., high school and higher education graduates– has grown, as reading and television-watching levels among the highest educated have gradually, over the generations, come to resemble those for the less educated. This combined development may be interpreted in two ways that are not necessarily mutually exclusive: 1) both the decline of reading and the increase in television-watching reflect a change in the cultural function of education, a change in turn due to changes in teaching methods and curriculum content reinforced by the sharp rise in influence of the mass media; but 2) these changes also reflect morphological changes in the world of upper secondary and higher education students: with the generalizing of access to the baccalauréat [French high school-leaving degree], the proportion of “inheritors” has considerably fallen. The changes can therefore be read persuasively both as a sign that schooling in France has lost some of its cultural authority and that cultural gaps within the generations who experienced mass education have narrowed

Testing the “Omnivore/Univore” Hypothesis in a Cross-National Perspective. On the Social Meaning of Ecletism in Musical Tastes:Paper presented at the Summer meeting of the ISA RC28, UCLA – 19/08/2005

Are musical tastes still strongly and universally correlated with social class, as asserted by Pierre Bourdieu in a suggestive formulation (Bourdieu, 1984, p. 158) . A lot of evidences, based on the sociological literature on tastes and cultural consumption, support the robustness of this correlation, while less straightforwardly defined than stated by Bourdieu. Indeed, this correlation appears to be slightly defined as a correspondence between social stratification and cultural legitimacy scale (i.e. highbrow arts and culture for the upper middle classes vs. lowbrow arts and culture for the lower classes. see Gans, 1974; Levine, 1988), and it tends to be more adequately described as a matter of scope of tastes and cultural habits, as stated by Richard Peterson, basically opposing the quite ‘omnivorous’ upper-middle classes, listening as much higbrow music as middlebrow or lowbrow music, to the more ‘univorous’ popular classes, narrowly restricted to a limited number of lowbrow musical genres (Peterson & Simkus, 1992; Peterson &Kern, 1996) (...)

Local Energy Optimality of Periodic Sets

We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\geq 9$ we can hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$.Comment: 27 pages, 2 figure

Cultural and Sports Participation in France: Choices, Diversity, and Accumulation

The French 2003 survey of household living conditions includes a cultural and sports participation module that covers radio and television consumption, reading, cultural outings, amateur activities, and sports, but also hunting and fishing. The social differentiation of attitudes to recreational activities is primarily determined by an accumulation criterion, which categorises individuals on the basis of the number and frequency of the activities in which they engage. Most activities do not seem mutually exclusive. People are all the more likely to engage in one if they already engage in another. An individuals ranking on the activity-accumulation scale seems more correlated with educational attainment than with income level. Differences in attitudes do not appear solely determined by the accumulation criterion. There is a contrast between persons who are not significantly engaged and two profiles of persons substantially more involved: individuals participating chiefly in sports activities, and individuals participating mainly in cultural activities.Cultural Consumption, Social Stratification, Leisure, Time Use

Codes and designs in Grassmannian spaces

AbstractThe notion of t-design in a Grassmannian space Gm,n was introduced by the first and last authors and G. Nebe in a previous paper. In the present work, we give a general lower bound for the size of such designs. The method is inspired by Delsarte, Goethals and Seidel work in the case of spherical designs. This leads us to introduce a notion of f-code in Grassmannian spaces, for which we obtain upper bounds, as well as a kind of duality tight-designs/tight-codes. The bounds are in terms of the dimensions of the irreducible representations of the orthogonal group O(n) occurring in the decomposition of the space L2(Gm,n°) of square integrable functions on Gm,n°, the set of oriented Grassmanianns

L’équipement automobile, entre contrainte et distinction sociale

Emblématique de la société de consommation et de production de masse, l'automobile connaît en France une diffusion qui, tant du point de vue de l'accès à la motorisation que de la distribution des différentes catégories de véhicules (marque, modèle, puissance, ancienneté et statut d'acquisition), fait apparaître des différences importantes entre les groupes sociaux. Entre ces derniers s'insinuent de nouveaux types de clivages (multi-motorisation, progression de la place des marques étrangères, notamment). L'analyse conjointe des habitudes de déplacements et des caractéristiques d'équipement automobile des ménages motorisés en 2008 met en évidence l'articulation d'effets propres aux contraintes de mobilité et aux habitudes de déplacement, qui n'entament pas toutefois la robustesse des écarts entre groupes sociaux. Le statut d'acquisition des véhicules apparaît comme un marqueur social soumis à un cycle de diffusion et de banalisation que révèle l'accès socialement différencié au marché du neuf et de l'occasion. Il se combine au type de véhicule. Les voitures allemandes, en particulier les plus puissantes, apparaissent ainsi comme un marqueur spécifique de l'appartenance aux classes supérieures, en particulier chez les indépendants. Ces écarts soulignent les éléments de compétition statutaire et symbolique qui continuent d'entourer l'acquisition et les usages de l'automobile