| 14/2009 |
Tempered Generalized Functions Algebra, Hermite Expansions and Itô Formula Pedro J. Catuogno, Christian Olivera The space of tempered distributions $\mathcal{S}^{\prime}$ can be realized as a sequence spaces by means of the Hermite representation theorems (see \cite{Miku}). In this work we introduce and study a new tempered generalized functions algebra $\mathcal{H}$, in this algebra the tempered distributions are embedding via its Hermite expansion. We study the Fourier transform, point value of generalized tempered functions and the relation of the product of generalized tempered functions with the Hermite product of tempered distributions. Furthermore, we give a generalized It\^o formula for elements of $\mathcal{H}$ and finally we show some applications to stochastic analysis.  rp-2009-14.pdf |
| 13/2009 |
A Girsanov Theorem in Manifolds and Projective Maps Simão Stelmastchuk Our purpose is to show a version of Girsanov theorem in smooth manifolds. After, we will use it theorem to give stochastic characterization for strongly projective maps. This stochastic characterization yields a proof that projective maps of rank $\geq 2$ between Riemannian manifolds, with connected domain, are affine maps. In particular, the groups of affine and projective transformations, in connected Riemannian manifold, are equal. rp-2009-13.pdf |
| 12/2009 |
Self-similarity and Uniqueness of Solutions for Semilinear Reaction-Diffusion Systems Lucas C. F. Ferreira, Eder Mateus We study the well-posedness of the initial value problem for a coupled semilinear reaction-diffusion system in Marcinkiewicz spaces $L^{(p_{1},\infty)}(\Omega)\times L^{(p_{2},\infty)}(\Omega)$. The exponents $p_{1},p_{2}$ of the initial value space are chosen to allow the existence of self-similar solutions (when $\Omega=\mathbb{R}^{n}$). As a nontrivial consequence of our coupling-term estimates, we prove the uniqueness of solutions in the scaling invariant class $C([0,\infty);L^{p_{1}}(\Omega)\times L^{p_{2}}(\Omega))$ regardless of their size and sign. We also analyze theasymptotic stability of the solutions, show the existence of a basin of attraction for each self-similar solution and that solutions in $L^{p_{1}}\times L^{p_{2}}$ \ present a simple long time behavior. rp-2009-12.pdf |
| 11/2009 |
On the Stability Problem for the Boussinesq Equations in Weak-L$^P$ Spaces Lucas C. F. Ferreira, Elder J. Villamizar-Roa We consider the Boussinesq equations in either an exterior domain in $\mathbb{R}^{n}$, the whole space $\mathbb{R}^{n}$, the half space $\mathbb{R}_{+}^{n}$ or a bounded domain in $\mathbb{R}^{n}$, where the space dimension $n$ satisfies $n\geq3$. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-$L^{p}$ spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class $L_{\sigma}^{(n,\infty)}$. Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done. rp-2009-11.pdf |
| 10/2009 |
Navier-Stokes Equations with Fractional Dissipation in Sum of Pseudomeasure Spaces Lucas C. F. Ferreira, Elder J. Villamizar-Roa In this paper is studied the local well-posedness of the Navier-Stokes system with initial data belonging to a sum of two pseudomeasure-type spaces denoted by $PM^{a,b}:=PM^{a}+PM^{b}.$ New results about local well-posedness and regularity of mild solutions of Navier-Stokes system are obtained. The proof requires to show an interesting H\"{o}lder-type inequality in $PM^{a,b}$, as well as to establish estimates of the semigroup generated by fractional power of \ Laplacian $(-\Delta)^{\gamma}$ on these spaces. rp-2009-10.pdf |
| 9/2009 |
A Combinatorial Proof for an Identity Involving Partitions with Distinct Odd Parts José Plínio O. Santos, Robson da Silva We follow Santos idea to present a combinatorial interpretation for a sum and provide a bijective proof for an identity involving the number of partitions of an integer in which the odd parts are distinctand greater than 1. |
| 8/2009 |
New Two-Line Arrays Representing Partitions José Plínio O. Santos, Paulo Mondek, Andréia C. Ribeiro We present combinatorial interpretations for sums into two parameters from which we have, as special cases, combinatorial interpretations for many identities of Slater´s list including Rogers-Ramanujan identities, unrestricted partitions and Lebesgues´ partition identity. In this work we are representing a number as a "vector" and providing representation of this "vector" as a sum of "vectors". It is possible to write this representation as a two-line matrix which can be interpreted as lattice paths. We provide three distinct representations for unrestricted partitions. One of them has the property of giving also a complete description for the conjugate partition. |
| 7/2009 |
Limit Cycles of Resonant Four-Dimensional Polynomial Systems Jaume LLibre, Ana Cristina Mereu, Marco A. Teixeira We study the bifurcation of limit cycles from four-dimensional centers inside a class of polynomial differential systems. Our results establish an upper bounded for the number of limit cycles which can be prolonged in function of the degree of the polynomial perturbation considered, up to first-order expansion of the displacement function with respect to small parameter. The main tool for proving such results is the averaging theory. rp-2009-7.pdf |
| 6/2009 |
Invariant Tori Fulfilled of Periodic Orbits for 4-Dimensional C$^2$ Differential Systems in Presence of Resonance Jaume LLibre, Ana Cristina Mereu, Marco A. Teixeira We provide an algorithm for studying invariant tori fulfilled of periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in $p:q$ resonance.We illustrate the algorithm with an application. rp-2009-6.pdf |
| 5/2009 |
Weak KAM Methods and Ergodic Optimal Problems for Countable Markov Shifts Rodrigo Bissacot, Eduardo Garibaldi Let \(\sigma\colon \Sigma\to \Sigma\) be the left shift acting on \(\Sigma\), a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of \(\sigma\)-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential \(A\colon \Sigma\to \mathbb{R}\). Under certain conditions, we are able to show not only that \(A\)-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions). |
