On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

INDEFINITE QUASILINEAR ELLIPTIC EQUATIONS IN EXTERIOR DOMAINS WITH EXPONENTIAL CRITICAL GROWTH

This paper is concerned with the existence of solutions for the quasilinear problem {-div(vertical bar del u vertical bar(N-2) del u) + vertical bar u vertical bar(N-2) u = a(x)g(u) in Omega u = 0 on partial derivative Omega, where Omega subset of R(N) (N >= 2) is an exterior domain; that is, Omega = R(N)\omega, where omega subset of R(N) is a bounded domain, the nonlinearity g(u) has an exponential critical growth at infinity and a(x) is a continuous function and changes sign in Omega. A variational method is applied to establish the existence of a nontrivial solution for the above problem.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.

The signature of dark energy perturbations in galaxy cluster surveys

Models of dynamical dark energy unavoidably possess fluctuations in the energy density and pressure of that new component. In this paper we estimate the impact of dark energy fluctuations on the number of galaxy clusters in the Universe using a generalization of the spherical collapse model and the Press-Schechter formalism. The observations we consider are several hypothetical Sunyaev-Zel`dovich and weak lensing (shear maps) cluster surveys, with limiting masses similar to ongoing (SPT, DES) as well as future (LSST, Euclid) surveys. Our statistical analysis is performed in a 7-dimensional cosmological parameter space using the Fisher matrix method. We find that, in some scenarios, the impact of these fluctuations is large enough that their effect could already be detected by existing instruments such as the South Pole Telescope, when priors from other standard cosmological probes are included. We also show how dark energy fluctuations can be a nuisance for constraining cosmological parameters with cluster counts, and point to a degeneracy between the parameter that describes dark energy pressure on small scales (the effective sound speed) and the parameters describing its equation of state.

Parallax

Parallax is a live contemporary dance work that incorporates 3D animation, stereoscopic illusions and dance. This work was performed within the Melbourne Fringe Festival at the Substation, Newport. Within this work the stereoscopic illusion creates a new choreographic palette that can be used to manipulate human physicality via animated bodies that appear within the performance space. The stereoscopic image is released from the wall and placed within the dancing environment the image becomes another body within the dance space that can be manipulated in ways that would be impossible for a real physical body. In turn, the dancing body is positioned within the digital environment. The performer’s abilities have not changed, but the space around the dancer can be manipulated with imagery that transforms the place of the dancer within time and space. The stereoscopic illusion and live dance are melded creating a new experience of choreography one that takes the infinite possibilities of 3D animation and places them directly within choreography. Thematically this performance draws on the historical events revolving around the development of the stereoscope in the 1830s and the seminal ideas of the virtual that surfaced at this time. In the early 1830s Charles Wheatstone drew on the ideas and writings of Euclid and Leonardo da Vinci and discovered binocular vision through the use of his stereoscope box. It was this box that became the entertainment sensation of this time becoming a standard parlour entertainment. Unlike now where imagery of people are everywhere in the 1830s these types of imagery were novel. The stereoscopic pictures often showed content of people doing ordinary tasks such as chopping wood, doing the washing or simply standing in front of their house. In Parallax a Victorian woman is transported from her hallway to virtual worlds where she encounters, Euclid’s ancient Greek column, a di Vinci sphere and one of the first stereoscopic images drawn by Charles Wheatstone’s a stick figure cube.

Geometrias não-eucludianas como anomalias: implicações para o ensino de geometria e medidas

This present research the aim to show to the reader the Geometry non-Euclidean while anomaly indicating the pedagogical implications and then propose a sequence of activities, divided into three blocks which show the relationship of Euclidean geometry with non-Euclidean, taking the Euclidean with respect to analysis of the anomaly in non-Euclidean. PPGECNM is tied to the line of research of History, Philosophy and Sociology of Science in the Teaching of Natural Sciences and Mathematics. Treat so on Euclid of Alexandria, his most famous work The Elements and moreover, emphasize the Fifth Postulate of Euclid, particularly the difficulties (which lasted several centuries) that mathematicians have to understand him. Until the eighteenth century, three mathematicians: Lobachevsky (1793 - 1856), Bolyai (1775 - 1856) and Gauss (1777-1855) was convinced that this axiom was correct and that there was another geometry (anomalous) as consistent as the Euclid, but that did not adapt into their parameters. It is attributed to the emergence of these three non-Euclidean geometry. For the course methodology we started with some bibliographical definitions about anomalies, after we ve featured so that our definition are better understood by the readers and then only deal geometries non-Euclidean (Hyperbolic Geometry, Spherical Geometry and Taxicab Geometry) confronting them with the Euclidean to analyze the anomalies existing in non-Euclidean geometries and observe its importance to the teaching. After this characterization follows the empirical part of the proposal which consisted the application of three blocks of activities in search of pedagogical implications of anomaly. The first on parallel lines, the second on study of triangles and the third on the shortest distance between two points. These blocks offer a work with basic elements of geometry from a historical and investigative study of geometries non-Euclidean while anomaly so the concept is understood along with it s properties without necessarily be linked to the image of the geometric elements and thus expanding or adapting to other references. For example, the block applied on the second day of activities that provides extend the result of the sum of the internal angles of any triangle, to realize that is not always 180° (only when Euclid is a reference that this conclusion can be drawn)

Search schemes for random optimization algorithms that preserve the asymptotic distribution

Markovian algorithms for estimating the global maximum or minimum of real valued functions defined on some domain Omega subset of R-d are presented. Conditions on the search schemes that preserve the asymptotic distribution are derived. Global and local search schemes satisfying these conditions are analysed and shown to yield sharper confidence intervals when compared to the i.i.d. case.

Generic Bifurcations of Planar Filippov Systems via Geometric Singular Perturbations

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

SINGULARITY THEORY and FORCED SYMMETRY BREAKING IN EQUATIONS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Finiteness theorem for topological contact equivalence of map germs

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The signature of dark energy perturbations in galaxy cluster surveys

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

The immune system of insects and the control of some parasitary human diseases

Parasitic diseases in humans, transmitted by insects, affect about 500 million people living mainly in countries of low economic power, the control of these diseases is difficult to carry out, mainly die to social and political problems, enhanced bg the capacity of these organisms to develop resistance to insecticides used to for their destruction.Some recent advances in the area of insect immunology have open the possibility for abetter epidemiological control of these diseases.The immune system of these insects, as well as that of other organisms, have the ability to recognize the infecting parasites and liberate a series of reactions which stop the infection. These reactions involve the circulating cells (hemocytes) against the parasite. These cells have the ability of phagocytize and liberate the production of various humoral factors, neutralizing the infection.Some promising results, obtained by the study of the immune system of malaria-transmitting insects, the sleeping disease, and dengue, are an example of this new sanitary strategy.

On the hardy space H¹ on products of half-spaces

We show that the Hardy space H¹ anal (R2+ x R2+) can be identified with the class of functions f such that f and all its double and partial Hubert transforms Hk f belong to L¹ (R2). A basic tool used in the proof is the bisubharmonicity of |F|q, where F is a vector field that satisfies a generalized conjugate system of Cauchy-Riemann type.

Combinatorial and geometrical properties of a class of tilings

In this paper, we consider a tiling generated by a Pisot unit number of degree d >= 3 which has a finite expansible property. We compute the states of a finite automaton which recognizes the boundary of the central tile. We also prove in the case d = 3 that the interior of each tile is simply connected.

Sliding Vector Fields via Slow-Fast Systems

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)