| 9/2006 |
Canonical Lyapunov Graphs and the Morse Polytope Ricardo N. Cruz, Margarida P. Mello, Ketty A. de Rezende In this article we will show that, in general, for each integral point $(\gamma_0, \ldots, \gamma_n)$ in the Morse polytope, $\mathcal{P}_{\kappa}(h_0, \ldots, h_n)$, one can associate an abstract Lyapunov graph $L(h_0, \ldots, h_n,\kappa)$ with $ntd$-labelling and realize a corresponding flow on $M^n$, where the Betti numbers of $M^n$ satisfy $\beta_j(M^n)= \beta_{n-j}(M^n)=\gamma_j$, for all $0  rp-2006-9.pdf |
| 8/2006 |
Decomposability of High-Dimensional Diversity Measures: Quasi U-Statistics, Martingales and Nonstandard Asymptotics Aluísio Pinheiro, Pranab Kumar Sen, Hildete P. Pinheiro In complex diversity analysis, specially arising in genetics, genomics, ecology and other high-dimensional (and sometimes low sample size) data models, typically sub\-group-decomposability (analogous to ANOVA decomposability) arises. In group-divergence of diversity measures in a high-dimension low sample size scenario, it is shown that Hamming distance-type statistics lead to a general class of quasi U-statistics having a martingale (array) property, providing key to the study of general (nonstandard) asymptotics. Neither the stochastic independence nor homogeneity of the marginal probability laws play a basic role. A genomic MANOVA model is presented as an illustration. rp-2006-8.pdf |
| 7/2006 |
Generalized Second-Order Complementarity Problems: Theory and Numerical Experiments Roberto Andreani, Ana Friedlander, Margarida P. Mello, Sandra A. Santos The generalized second-order cone complementarity problem (GSOCCP) is reformulated via bound-constrained minimization, preserving differentiability of the original data. Four reformulations are proposed, which are tested in five low dimensional instances. A thorough presentation and discussion of the numerical experiments is provided, to illustrate the performance of the reformulations. In a companion paper, equivalence results relating global minimizers of two of the reformulations with zero objective function value and solutions to GSOCCP are proved, together with sufficient conditions for ensuring correspondence between stationary points of one of the reformulations and solutions to GSOCCP. Reference to these results are included, with additional theoretical insight into the second reformulation that has beenpreviously addressed. rp-2006-7.pdf |
| 6/2006 |
Multiplicity Results for a Superlinear Elliptic System with Partial Interference with the Spectrum Eugenio Massa In this work, we consider an elliptic system of two equations in dimension greater than one, with nonlinearities which are linear at $-\infty$ and superlinear at $+\infty$.We prove, by variational techniques which involve a strongly indefinite functional, the existence of two solutions for suitable forcing terms, under a condition on the linear part; which prevents resonance with the eigenvalues of the operator. rp-2006-6.pdf |
| 5/2006 |
Recurrence, Short Correlations and the Golden Number Miguel Abadi We consider a stochastic process with the weakest mixing condition: the so called $\alpha$. For \emph{any} fixed $n$-string we prove:(1) The hitting time has approximately exponential law.(2) The return time has approximately a convex combination between a Dirac measure at the origin and an exponential law.In both cases the parameter of the exponential law is $\lambda(A)IP(A)$ where $IP(A)$ is the measure of the string and $\lambda(A)$ is the short correlation function of the string with itself. Also, we show that the weight of the convex combination is a approximately $\lambda(A)$. We describe the autocorrelation function. Our results hold when the rate $\alpha$ decays polinomially fast with power larger than the golden number. rp-2006-5.pdf |
| 4/2006 |
Sharp Error Terms for Poisson Statistics Under Mixing Conditions: A New Approach Miguel Abadi, Nicolas Vergne We describe the statistics of the number of occurrences of a string of symbols in a stochastic process: Chosen a string $A$ of length $n$, we prove that the number of visits to $A$ up to time $t$, denoted by $N_t$, has approximately a Poisson distribution. We provide a sharp error for this approximation. Contrarily to previous works who presente uniform error terms based on the total variation distance, our error is point-wise. As a byproduct we obtain approximations for all the moments of $N_t$. Our result holds for processes that verify the \phi$-mixing condition. The error term is explcitely expressed as function of $\phi$ and then easily computable. We breafly extend our result to the weaker $\alpha$-mixing case. rp-2006-4.pdf |
| 3/2006 |
Sharp Error Terms for Return Time Statistics Under Mixing Conditions Miguel Abadi, Nicolas Vergne We describe the statistics of repetition times of a string of symbols in a stochastic process. We consider a string $A$ of length $n$ and prove:1) The time elapsed until the process starting with $A$ repeats $A$, denoted by $\ta$, has a distribution which can be well approximated by a degenerated law at the origin and an exponential law.2) The number of consecutive repetitions of $A$, denoted by $S_A$, has a distribution which is approximately a geometric law.We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of $\ta$ and $S_A$. Our results hold for processes that verify the $\phi$ mixing condition. rp-2006-3.pdf |
| 2/2006 |
Poisson Approximation in Biological Context Nicolas Vergne, Miguel Abadi Using recent results on the occurrence times of a string of symbols in a stochastic process with mixing properties, we present a new method for the search of rare words in biological sequences generally modelled by a Markov chain. We obtain a bound of the error between the law of the number of occurrences of a word in a sequence (under a Markov model) and its Poisson approximation. A global bound is already given by a Chen-Stein method. Our method, the $\psi$-mixing method, gives local bounds. We search a number of occurrences from which we can regard the studied word as a rare word. If the word appears more often than this number in the biological sequence, we conclude that it is an overrepresented word and then we suppose a biological role. Our method always give a limit number, while it was impossible with the Chen-Stein method.Comparing the methods, we observe a better accuracy for the $\psi$-mixing method for the bound of the tails of distribution. We also present the software PANOW 1 dedicated to the computation of the error term and the limit number of occurrences for a studied word. rp-2006-2.pdf |
| 1/2006 |
Integral Representation for the Dirac Delta Function Edmundo Capelas de Oliveira, Ary O. Chiacchio Using the concept of Green's function associated with an ordinary differential equation and the residue theorem, we discuss the connection between afinite number of poles and a branch cut and we obtain an integral representation for the Dirac delta function, which is interpreted as a spectralrepresentation associated with the Fourier sine transform. rp-2006-1.pdf |
| 61/2005 |
Morse Decompositions of Semiflows on Topological Spaces Mauro Patrão This paper studies Morse decompositions of discrete and continuous-time semiflows on compact Hausdorff topological spaces. We extend two classical results which are well known facts for flows on compact metric spaces: the characterization of the Morse decompositions through increasing sequences of attractors and the existence of Lyapunov functions. rp-2005-61.pdf |
