Functions and vector fields on C(CPn)-singular manifolds

Let M2n+1 be a C(CPn) -singular manifold. We study functions and vector fields with isolated singularities on M2n+1. A C(CPn) -singular manifold is obtained from a smooth manifold M2n+1 with boundary in the form of a disjoint union of complex projective spaces CPn boolean OR CPn boolean OR ... boolean OR CPn with subsequent capture of a cone over each component of the boundary. Let M2n+1 be a compact C(CPn) -singular manifold with k singular points. The Euler characteristic of M2n+1 is equal to chi(M2n+1) = k(1 - n)/2. Let M2n+1 be a C(CPn)-singular manifold with singular points m(1), ..., m(k). Suppose that, on M2n+1, there exists an almost smooth vector field V (x) with finite number of zeros m(1), ..., m(k), x(1), ..., x(1). Then chi(M2n+1) = Sigma(l)(i=1) ind(x(i)) + Sigma(k)(i=1) ind(m(i)).

THREE TIME SCALE SINGULAR PERTURBATION PROBLEMS AND NONSMOOTH DYNAMICAL SYSTEMS

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

On symmetries of singular implicit ODEs

We study implicit ODEs, cubic in derivative, with infinitesimal symmetry at singular points. Cartan showed that even at regular points the existence of nontrivial symmetry imposes restrictions on the ODE. Namely, this algebra has the maximal possible dimension 3 iff the web of solutions is flat. For cubic ODEs with flat 3-web of solutions we establish sufficient conditions for the existence of nontrivial symmetries at singular points and show that under natural assumptions such a symmetry is semi-simple, i.e. is a scaling is some coordinates. We use this symmetry to find first integrals of the ODE.

Local classification of singular hexagonal 3-webs with holomorphic Chern connection form and infinitesimal symmetries

Implicit ODE, cubic in derivative, generically has no infinitesimal symmetries even at regular points with distinct roots. Cartan showed that at regular points, ODEs with hexagonal 3-web of solutions have symmetry algebras of the maximal possible dimension 3. At singular points such a web can lose all its symmetries. In this paper we study hexagonal 3-webs having at least one infinitesimal symmetry at singular points. In particular, we establish sufficient conditions for the existence of non-trivial symmetries and show that under natural assumptions such a symmetry is semi-simple, i.e. is a scaling in some coordinates. Using the obtained results, we provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection form is holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal weighted homogeneous 3-webs is obtained.

Geometric singular perturbartion theory for non-smooth dynamical systems

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

A note on pairs of regular foliations on the solid torus

We discuss the geometry of the pair of foliations on a solid torus given by the Reeb foliation together with discs transverse to the boundary of the torus.

Singular value analyses of voltage stability on power system considering wind generation variability

Pós-graduação em Engenharia Elétrica - FEIS

Os desenhos da cidade observada: uma reflexão singular

How are teachers Cartoons observation Architecture Schools in several decades, the idea was to try and explain firsthand the look and fundamental elements of drawings to the Architect. We start looking through the design of: 1 A projective gaze: the invisible is made visible by interactivity and space and time are perceived by the distance that look place between them; 2nd In bidirectional contamination: the object fills the subject, double-hand, bringing the tactile qualities of architecture in continuous resonance, 3rd Strangeness by slow look: draw as perceptual expansion strategy; 4 Articulation of the structural elements of the design while similar language to architecture and city through the selection of appropriate signs and codes, exclusive and singular.

A dialética singular-particular-universal: implicações do método materialista dialético para a psicologia

Considering that Cultural-Historical theory has been gaining importance within Brazilian Psychology in the last decades, this essay aims at contributing to understanding its ontological and epistemological foundations, introducing the concepts of singularity, particularity and universality of the dialectics that exists between them. The analysis looks to explore the implications to Psychology, both as a science and as a professional practice, of the lukacsian indication about the need to apprehend the connections between singularity, particularity and universality as a condition for understanding the essence of phenomena. In that sense, this analysis brings light to the individual/society dynamics unity affirmed by historical-dialectical materialism, which contributes to overcoming of the dichotomies usually established between the poles of this relationship.

Autismo: aspectos cognitivos, educacionais e neurobiológicos com base na análise dos livros Mundo singular: entenda o autismo e o Os gatos nunca mentem sobre o amor e na literatura especializada

The autism is a severe mental illness that involves psychological, social, educational and neurobiological aspects. The Education plays an important role in recovering, preserving and increasing the cognitive task of people with autism. There are several ways to observe a person with autism. Through the studying of cases, it is also possible to address issues of everyday life of the person with the illness, its cognitive issues and its obstacles of psychosocial involvement, as well as psychological aspects such as illness acknowledgement. Through specialized literature, it is possible to identify characteristics referring to the learning process and neurobiological components involved in the condition, such as brain activity disorder and genetic factors. The objectives of this paper are: from the analysis of two books: Unique World: comprehend the Autism (SILVA, A. B. B; GAIATO, M.B; REVELES, L.T) and The cats never lie about love (DILLON, J.), it is intended to describe cognitive and psychosocial aspects of the autism. Based on the specialized literature, the goal is to identify cognitive and neurobiological components in the illness. The methods that were used: analysis of the books Unique World: comprehend the Autism and The cats never lie about love to describe psychosocial and educational aspects of the autism; analysis of the specialized literature to identify clinical and neurobiological components of the illness, such as changes in brain activity, genetic factors or clinical evolution; and also details of the Education role in preserving the person with autism and his cognitive and emotional development. The study had the documentary research as reference for the methodological design, including book analysis and research in specialized literature. It is intended to deepen discussions about the Education roles related to the cognitive and neurobiological aspects of the autism

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

A NOTE ON OPEN 3-MANIFOLDS SUPPORTING FOLIATIONS BY PLANES

We show that if N, an open connected n-manifold with finitely generated fundamental group, is C-2 foliated by closed planes, then pi(1)(N) is a free group. This implies that if pi(1)(N) has an abelian subgroup of rank greater than one, then F has at least a nonclosed leaf. Next, we show that if N is three dimensional with fundamental group abelian of rank greater than one, then N is homeomorphic to T-2 x R. Furthermore, in this case we give a complete description of the foliation.

Riemannian Submersions with Discrete Spectrum

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.

Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems

[EN] We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets.