In search of mathematical primitives for deriving universal projective hash families

We provide some guidelines for deriving new projective hash families of cryptographic interest. Our main building blocks are so called group action systems; we explore what properties of this mathematical primitives may lead to the construction of cryptographically useful projective hash families. We point out different directions towards new constructions, deviating from known proposals arising from Cramer and Shoup's seminal work.

Mixed Hodge structures and vector bundles on the projective Plane I

We describe an equivalence of categories between the category of mixed Hodge structures and a category of vector bundles on the toric complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalises the notion of R-split mixed Hodge structure and compute extensions in the category of mixed Hodge structures in terms of extensions of the corresponding vector bundles. We also give a relative version of this correspondence and apply it to define stratifications of the bases of the variations of mixed Hodge structure.

Diophantine approximation on projective varieties I: algebraic distance and metric Bézout theorem

"Vegeu el resum a l'inici del document del fitxer adjunt."

Projective modules over the Gerasimov-Sakhaev counterexample

We investigate the structure of the so-called Gerasimov- Sakhaev counterexample, which is a particular example of a universal localization, and classify (both finitely and infinitely generated) projective modules over it.

Elements of algebraic geometry and the positive theory of partially commutative groups

The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Counting polygon dissections in the projective plane

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Horospheres in hyperbolic geometry

In this paper we investigate the role of horospheres in Integral Geometry and Differential Geometry. In particular we study envelopes of families of horocycles by means of “support maps”. We define invariant “linear combinations” of support maps or curves. Finally we obtain Gauss-Bonnet type formulas and Chern-Lashof type inequalities.

Geometria integral i convexitat en espais de curvatura holomorfa constant

En aquest treball es tracten qüestions de la geometria integral clàssica a l'espai hiperbòlic i projectiu complex i a l'espai hermític estàndard, els anomenats espais de curvatura holomorfa constant. La geometria integral clàssica estudia, entre d'altres, l'expressió en termes geomètrics de la mesura de plans que tallen un domini convex fixat de l'espai euclidià. Aquesta expressió es dóna en termes de les integrals de curvatura mitja. Un dels resultats principals d'aquest treball expressa la mesura de plans complexos que tallen un domini fixat a l'espai hiperbòlic complex, en termes del que definim com volums intrínsecs hermítics, que generalitzen les integrals de curvatura mitja. Una altra de les preguntes que tracta la geometria integral clàssica és: donat un domini convex i l'espai de plans, com s'expressa la integral de la s-èssima integral de curvatura mitja del convex intersecció entre un pla i el convex fixat? A l'espai euclidià, a l'espai projectiu i hiperbòlic reals, aquesta integral correspon amb la s-èssima integral de curvatura mitja del convex inicial: se satisfà una propietat de reproductibitat, que no es té en els espais de curvatura holomorfa constant. En el treball donem l'expressió explícita de la integral de la curvatura mitja quan integrem sobre l'espai de plans complexos. L'expressem en termes de la integral de curvatura mitja del domini inicial i de la integral de la curvatura normal en una direcció especial: l'obtinguda en aplicar l'estructura complexa al vector normal. La motivació per estudiar els espais de curvatura holomorfa constant i, en particular, l'espai hiperbòlic complex, es troba en l'estudi del següent problema clàssic en geometria. Quin valor pren el quocient entre l'àrea i el perímetre per a successions de figures convexes del pla que creixen tendint a omplir-lo? Fins ara es coneixia el comportament d'aquest quocient en els espais de curvatura seccional negativa i que a l'espai hiperbòlic real les fites obtingudes són òptimes. Aquí provem que a l'espai hiperbòlic complex, les cotes generals no són òptimes i optimitzem la superior.

Continuity of set-valued maps revisited in the light of tame geometry

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result –depending on stratification techniques– holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.

Rigid flat webs on the projective plane

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Legendre transform, is an avatar of classical projective duality in the realm of differential equations. We show that the Legendre transform of what we call reduced convex foliations are webs with zero curvature, and we exhibit a countable infinity family of convex foliations which give rise to a family of webs with zero curvature not admitting non-trivial deformations with zero curvature.

Revisiting the compositional data. Some fundamental questions and new prospects in Archaeometry and Archaeology

In this paper we examine the problem of compositional data from a different startingpoint. Chemical compositional data, as used in provenance studies on archaeologicalmaterials, will be approached from the measurement theory. The results will show, in avery intuitive way that chemical data can only be treated by using the approachdeveloped for compositional data. It will be shown that compositional data analysis is aparticular case in projective geometry, when the projective coordinates are in thepositive orthant, and they have the properties of logarithmic interval metrics. Moreover,it will be shown that this approach can be extended to a very large number ofapplications, including shape analysis. This will be exemplified with a case study inarchitecture of Early Christian churches dated back to the 5th-7th centuries AD

Vertebrates Limb Geometry in the Simplex space

A novel metric comparison of the appendicular skeleton (fore and hind limb) ofdifferent vertebrates using the Compositional Data Analysis (CDA) methodologicalapproach it’s presented.355 specimens belonging in various taxa of Dinosauria (Sauropodomorpha, Theropoda,Ornithischia and Aves) and Mammalia (Prothotheria, Metatheria and Eutheria) wereanalyzed with CDA.A special focus has been put on Sauropodomorpha dinosaurs and the Aitchinsondistance has been used as a measure of disparity in limb elements proportions to infersome aspects of functional morphology

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities