Characterization of the unfolding of a weak focus and modulus of analytic classification

La thèse présente une description géométrique d’un germe de famille générique déployant un champ de vecteurs réel analytique avec un foyer faible à l’origine et son complexifié : le feuilletage holomorphe singulier associé. On montre que deux germes de telles familles sont orbitalement analytiquement équivalents si et seulement si les germes de familles de difféomorphismes déployant la complexification de leurs fonctions de retour de Poincaré sont conjuguées par une conjugaison analytique réelle. Le “caractère réel” de la famille correspond à sa Z2-équivariance dans R^4, et cela s’exprime comme l’invariance du plan réel sous le flot du système laquelle, à son tour, entraîne que l’expansion asymptotique de la fonction de Poincaré est réelle quand le paramètre est réel. Le pullback du plan réel après éclatement par la projection monoidal standard intersecte le feuilletage en une bande de Möbius réelle. La technique d’éclatement des singularités permet aussi de donner une réponse à la question de la “réalisation” d’un germe de famille déployant un germe de difféomorphisme avec un point fixe de multiplicateur égal à −1 et de codimension un comme application de semi-monodromie d’une famille générique déployant un foyer faible d’ordre un. Afin d’étudier l’espace des orbites de l’application de Poincaré, nous utilisons le point de vue de Glutsyuk, puisque la dynamique est linéarisable auprès des points singuliers : pour les valeurs réels du paramètre, notre démarche, classique, utilise une méthode géométrique, soit un changement de coordonée (coordonée “déroulante”) dans lequel la dynamique devient beaucoup plus simple. Mais le prix à payer est que la géométrie locale du plan complexe ambiante devient une surface de Riemann, sur laquelle deux notions de translation sont définies. Après avoir pris le quotient par le relèvement de la dynamique nous obtenons l’espace des orbites, ce qui s’avère être l’union de trois tores complexes plus les points singuliers (l’espace résultant est non-Hausdorff). Les translations, le caractère réel de l’application de Poincaré et le fait que cette application est un carré relient les différentes composantes du “module de Glutsyuk”. Cette propriété implique donc le fait qu’une seule composante de l’invariant Glutsyuk est indépendante.

Desingularization of singular Riemannian foliation

Let F be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F we construct a regular Riemannian foliation (F) over cap on a compact Riemannian manifold (M) over cap and a desingularization map (rho) over cap : (M) over cap -> M that projects leaves of (F) over cap into leaves of F. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F are compact, then, for each small epsilon > 0, we can find (M) over cap and (F) over cap so that the desingularization map induces an epsilon-isometry between M/F and (M) over cap/(F) over cap. This implies in particular that the space of leaves M/F is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {((M) over cap (n)/(F) over cap (n))}.

Generic Bifurcations of Planar Filippov Systems via Geometric Singular Perturbations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sliding Vector Fields via Slow-Fast Systems

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R(l), l >= 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

A NOTE ON THE UNIQUENESS OF SOLUTIONS FOR THE YAMABE PROBLEM

Using recent results on the compactness of the space of solutions of the Yamabe problem, we show that in conformal classes of metrics near the class of a nondegenerate solution which is unique (up to scaling) the Yamabe problem has a unique solution as well. This provides examples of a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.

Formation of Singularities for Weakly Non-Linear N×N Hyperbolic Systems

We present some results on the formation of singularities for C^1 - solutions of the quasi-linear N × N strictly hyperbolic system Ut + A(U )Ux = 0 in [0, +∞) × Rx . Under certain weak non-linearity conditions (weaker than genuine non-linearity), we prove that the first order derivative of the solution blows-up in finite time.

Some new examples for nonuniqueness of the evolution problem of harmonic maps

We find some new examples to show nonuniquence for the heat flow of harmonic maps where weak solutions satisfy the same monotonicity property.

Synthesis, characterization and electrochemical behavior of the vanadium pentoxide/cetyl pyridinium chloride hybrid material

In situ and ex situ studies concerning the new hybrid material vanadium pentoxide xerogel in the presence of the cationic surfactant cetyl pyridinium chloride (V(2)O(5)/CPC) are presented. The in situ characterization studies revealed the presence of a lamellar structure for the V(2)O(5)/CPC hybrid material. The intercalation reaction was evidenced on the basis of the increase in the d-spacing as well as the displacement of the infrared bands toward lower energy levels. Electrochemical studies comprising the cyclic voltammetry and the electrochemical impedance spectroscopy techniques showed that the behavior of the hybrid material is considerably influenced by the electrolyte composition. The ion insertion/de-insertion into the V(2)O(5) xerogel structure accompanying the charge transfer process is influenced by the solid-state diffusion process modeled by using the finite-space Warburg element.

Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model

Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recover the recent result about the global in time existence of weak-solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the sub-critical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudo-inverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of mesh-refinement the blow-up of solutions for super-critical masses.

Pure motives, mixed motives and extensions of motives associated to singular surfaces

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface X and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor D in a non-singular blow-up of X. If all geometric irreducible components of D are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical subquotients of the motive with compact support of the smooth part of X. Specializing to Hilbert-Blumenthal surfaces, we recover a motivic interpretation of a recent construction of A. Caspar.

Bounds on multisecant lines

The purpose of this paper is two fold. First, we give an upper bound on the orderof a multisecant line to an integral arithmetically Cohen-Macaulay subscheme in Pn of codimension two in terms of the Hilbert function. Secondly, we givean explicit description of the singular locus of the blow up of an arbitrary local ring at a complete intersection ideal. This description is used to refine standardlinking theorem. These results are tied together by the construction of sharp examples for the bound, which uses the linking theorems.

Nitrous oxide and methane fluxes in south Brazilian gleysol as affected by nitrogen fertilizers

Nitrogen fertilizers increase the nitrous oxide (N2O) emission and can reduce the methane (CH4) oxidation from agricultural soils. However, the magnitude of this effect is unknown in Southern Brazilian edaphoclimatic conditions, as well as the potential of different sources of mineral N fertilizers in such an effect. The aim of this study was to investigate the effects of different mineral N sources (urea, ammonium sulphate, calcium nitrate, ammonium nitrate, Uran, controlled- release N fertilizer, and urea with urease inhibitor) on N2O and CH4 fluxes from Gleysol in the South of Brazil (Porto Alegre, RS), in comparison to a control treatment without a N application. The experiment was arranged in a randomized block with three replications, and the N fertilizer was applied to corn at the V5 growth stage. Air samples were collected from a static chambers for 15 days after the N application and the N2O and CH4 concentration were determined by gas chromatography. The topmost emissions occurred three days after the N fertilizer application and ranged from 187.8 to 8587.4 µg m-2 h-1 N. The greatest emissions were observed for N-nitric based fertilizers, while N sources with a urease inhibitor and controlled release N presented the smallest values and the N-ammonium and amidic were intermediate. This peak of N2O emissions was related to soil NO3--N (R² = 0.56, p < 0.08) when the soil water-filled pore space was up to 70 % and it indicated that N2O was predominantly produced by a denitrification process in the soil. Soil CH4 fluxes ranged from -30.1 µg m-2 h-1 C (absorption) to +32.5 µg m-2 h-1 C (emission), and the accumulated emission in the period was related to the soil NH4+-N concentration (R² = 0.82, p < 0.001), probably due to enzymatic competition between nitrification and metanotrophy processes. Despite both of the gas fluxes being affected by N fertilizers, in the average of the treatments, the impact on CH4 emission (0.2 kg ha-1 equivalent CO2-C ) was a hundredfold minor than for N2O (132.8 kg ha-1 equivalent CO2-C). Accounting for the N2O and CH4 emissions plus energetic costs of N fertilizers of 1.3 kg CO2-C kg-1 N regarding the manufacture, transport and application, we estimated an environmental impact of N sources ranging from 220.4 to 664.5 kg ha-1 CO2 -C , which can only be partially offset by C sequestration in the soil, as no study in South Brazil reported an annual net soil C accumulation rate larger than 160 kg ha-1 C due to N fertilization. The N2O mitigation can be obtained by the replacement of N-nitric sources by ammonium and amidic fertilizers. Controlled release N fertilizers and urea with urease inhibitor are also potential alternatives to N2O emission mitigation to atmospheric and systematic studies are necessary to quantify their potential in Brazilian agroecosystems.

Rummet i vårdandets värld

This study is made in the context of basic research within the field ofcaring science. The aim is to make a theoretical and ontological investigation of what the space is in the world of caring. The basic proposition is that the space, as a fundamental dimension, has an impact on how the appreciation of one's mental health and suffering is shaped, and vice versa. The overall purpose is to develop a theoretical model of space from the caring science point of view andalso to offer an ideal concept of space to caring science. Guided by a theoretical horizon (Eriksson 1993, Eriksson 1995, Eriksson 2001) and methodological approach grounded in Gadamer's philosophic and existential hermeneutics a three-stage analysis and interpretation is conducted. The hermeneutic spiral of this investigation starts through a procedure in accordance with Eriksson's model (1997) of concept definition. The goal is to clarify the etymology of the concept as well as semantic differences between synonymous concepts, i.e. to identify the different extents of the concept of `space` (`rum`) in order to bring these closer for an exploration. The second phase is to analyse and interpret a sample of narratives in order to explicate the ontological nature and meaning of the space. The material used here is literary texts. The goal is to clarify the characteristics of the very inside of the space when it is shaped in relation to the human being in encountering suffering. In the third phase an interview study is taken place. The focus of the study is directed towards the phenomenon of space as it is known by a patient in a landscape of psychiatric care, i.e. what the space is in a contextual meaning. Then, a gradual hermeneutic understanding of the space is attempted by using theories from the field of caring science as well as additional theories from other disciplines. Metaphors are used as they are vivid and expressive tools for generating meaning. Different metaphoric space formations depict here a variety of purports that, although not quite the same, share extensive elements. Six metaphorically summarized entities of meaning emerged. The comprehensive form of space is pointed out as the Mobile-Immobile Room. Furthermore, the Standby, the Asylum, the Wall and the Place. In the further dialogue with the texts the understanding has deepened ontologically. The theoretical model ofthe space sums up the vertical, horizontal and the inward extent of deepness inthe movement of mental health. Three entities of ontological meaning have emerged as three significant rooms: the Common Land emerges as the ideal concept of mutual creation in the freedom of doing, being and becoming health. On the interpersonal level it means freedom, which includes sovereignty, choice and dignity of the human being. The Ice World signifies, ultimately, the space as a kind of frozenness of despair which "wallpapers" the person's entire being in the world in the drama of suffering. The Spiritual Home is shaped when the human being has acquired the very core of his/her inner and outer placeness as a kind of "at-homeness" and rootedness. Time is a central element and the inward extent of deepness of this trialectic space. Each of the metaphors is then the human being's unique, although even paradoxical, way of conceiving reality, and mastering spiritual suffering. They condense characteristic structures and patterns of dynamic scenery, which take place within the movement of health. The space encloses a contradictory spatiality constituted through the dynamic field of meaningfulness and meaninglessness. Anyway, it is not through a purging of these contradictions but through bringing them together in a drama of suffering that the space is shaped as ontologically good and meaningful in the world of caring.