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.

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.

Moduli of flat conformal structures of hyperbolic type

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

Characteristic reflection patterns in the southeast Canadian cordillera, northern Appalachians and Swiss Alps

There are some striking similarities and some differences between the seismic reflection sections recorded across the fold and thrust belts of the southeast Canadian Cordillera, Quebec-Maine Appalachians and Swiss Alps. In the fold and thrust belts of all three mountain ranges, seismic reflection surveys have yielded high-quality images of. (1) nappes (thin thrust sheets) stacked on top of ancient continental margins; (2) ramp anticlines in the hanging walls of faults that have ramp-flat or listric geometries; (3) back thrusts and back folds that developed during the terminal phases of orogeny; and (4) tectonic wedges and regional decollements. A principal result of the Cordilleran and Appalachian deep crustal studies has been the recognition of master decollements along which continental margin strata have been transported long distances, whereas a principal result of the Swiss Alpine deep crustal program has been the identification of the Adriatic indenter, a crustal-scale wedge that caused delamination of the European lithosphere. Significant crustal roots are observed beneath the fold and thrust belts of the Alps, southeast Canadian Cordillera and parts of the southern Appalachians, but such structures beneath the northern Appalachians have probably been removed by post-orogenic collapse and/or crustal attenuation associated with the Mesozoic opening of the Atlantic Ocean.

Geometry and physics of knots

KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration(1-4). Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. For a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number-the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility(5).

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.

Contact melting of a three-dimensional phase change material on a flat substrate

In this paper a model is developed to describe the three dimensional contact melting process of a cuboid on a heated surface. The mathematical description involves two heat equations (one in the solid and one in the melt), the Navier-Stokes equations for the flow in the melt, a Stefan condition at the phase change interface and a force balance between the weight of the solid and the countering pressure in the melt. In the solid an optimised heat balance integral method is used to approximate the temperature. In the liquid the small aspect ratio allows the Navier-Stokes and heat equations to be simplified considerably so that the liquid pressure may be determined using an igenfunction expansion and finally the problem is reduced to solving three first order ordinary differential equations. Results are presented showing the evolution of the melting process. Further reductions to the system are made to provide simple guidelines concerning the process. Comparison of the solutions with experimental data on the melting of n-octadecane shows excellent agreement.

An approximate solution method for boundary layer flow of a power law fluid over a flat plate

The work in this paper deals with the development of momentum and thermal boundary layers when a power law fluid flows over a flat plate. At the plate we impose either constant temperature, constant flux or a Newton cooling condition. The problem is analysed using similarity solutions, integral momentum and energy equations and an approximation technique which is a form of the Heat Balance Integral Method. The fluid properties are assumed to be independent of temperature, hence the momentum equation uncouples from the thermal problem. We first derive the similarity equations for the velocity and present exact solutions for the case where the power law index n = 2. The similarity solutions are used to validate the new approximation method. This new technique is then applied to the thermal boundary layer, where a similarity solution can only be obtained for the case n = 1.

Wing geometry as a tool for studying the Lutzomyia longipalpis (Diptera: Psychodidae) complex

Toro Toro (T) and Yungas (Y) have been described as genetically well differentiated populations of the Lutzomyia longipalpis (Lutz & Neiva, 1912) complex in Bolivia. Here we use geometric morphometrics to compare samples from these populations and new populations (Bolivia and Nicaragua), representing distant geographical origins, qualitative morphological variation ("one-spot" or "two-spots" phenotypes), ecologically distinct traits (peridomestic and silvatic populations), and possibly different epidemiological roles (transmitting or nor transmitting Leishmania chagasi). The Nicaragua (N) (Somotillo) sample was "one-spot" phenotype and a possible peridomestic vector. The Bolivian sample of the Y was also "one-spot" phenotype and a demonstrated peridomestic vector of visceral leishmaniasis (VL). The three remaining samples were silvatic, "two-spots" phenotypes. Two of them (Uyuni and T) were collected in the highlands of Bolivian where VL never has been reported. The last one (Robore, R) came from the lowlands of Bolivia, where human cases of VL are sporadically reported. The decomposition of metric variation into size and shape by geometric morphometric techniques suggests the existence of two groups (N/Y/R, and U/T). Several arguments indicate that such subdivision of Lu. longipalpis could correspond to different evolutionary units.

Geometry and physics of catenanes applied to the study of DNA replication.

The concept of ideal geometric configurations was recently applied to the classification and characterization of various knots. Different knots in their ideal form (i.e., the one requiring the shortest length of a constant-diameter tube to form a given knot) were shown to have an overall compactness proportional to the time-averaged compactness of thermally agitated knotted polymers forming corresponding knots. This was useful for predicting the relative speed of electrophoretic migration of different DNA knots. Here we characterize the ideal geometric configurations of catenanes (called links by mathematicians), i.e., closed curves in space that are topologically linked to each other. We demonstrate that the ideal configurations of different catenanes show interrelations very similar to those observed in the ideal configurations of knots. By analyzing literature data on electrophoretic separations of the torus-type of DNA catenanes with increasing complexity, we observed that their electrophoretic migration is roughly proportional to the overall compactness of ideal representations of the corresponding catenanes. This correlation does not apply, however, to electrophoretic migration of certain replication intermediates, believed up to now to represent the simplest torus-type catenanes. We propose, therefore, that freshly replicated circular DNA molecules, in addition to forming regular catenanes, may also form hemicatenanes.

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

Temporal variation of wing geometry in Aedes albopictus

Although native to the tropical and subtropical areas of Southeast Asia, Aedes albopictus is now found on five continents, primarily due to its great capacity to adapt to different environments. This species is considered a secondary vector of dengue virus in several countries. Wing geometric morphometrics is widely used to furnish morphological markers for the characterisation and identification of species of medical importance and for the assessment of population dynamics. In this work, we investigated the metric differentiation of the wings of Ae. albopictus samples collected over a four-year period (2007-2010) in São Paulo, Brazil. Wing size significantly decreased during this period for both sexes and the wing shape also changed over time, with the wing shapes of males showing greater differences after 2008 and those of females differing more after 2009. Given that the wings play sex-specific roles, these findings suggest that the males and females could be affected by differential evolutionary pressures. Consistent with this hypothesis, a sexually dimorphic pattern was detected and quantified: the females were larger than the males (with respect to the mean) and had a distinct wing shape, regardless of allometric effects. In conclusion, wing alterations, particularly those involving shape, are a sensitive indicator of microevolutionary processes in this species.