Testing non-uniform k-wise independent distributions over product spaces (extended abstract)

A distribution D over Σ1× ⋯ ×Σ n is called (non-uniform) k-wise independent if for any set of k indices {i 1, ..., i k } and for any z1zki1ik, PrXD[Xi1Xik=z1zk]=PrXD[Xi1=z1]PrXD[Xik=zk]. We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al. [1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.National Science Foundation (U.S.) (NSF grant 0514771)National Science Foundation (U.S.) (grant 0728645)National Science Foundation (U.S.) (Grant 0732334)Marie Curie International Reintegration Grants (Grant PIRG03-GA-2008-231077)Israel Science Foundation (Grant 1147/09)Israel Science Foundation (Grant 1675/09)Massachusetts Institute of Technology (Akamai Presidential Fellowship

Topologies and Laplacian spectra of a deterministic uniform recursive tree

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.Comment: 7 pages, 1 figures, definitive version accepted for publication in EPJ

Headland-bay beach planform stability of Santa Catarina State and of the Northern Coast of São Paulo State

This paper presents the results of the planform stability classification for the headland-bay beaches of the State of Santa Catarina and of the Northern Coast of São Paulo, based on the application of the Parabolic Bay-Shape Equation (PBSE) to aerial images of the beaches, using the software MEPBAY®. For this purpose, georeferenced mosaics of the QuickBird2® satellite imagery (for the State of Santa Catarina) and vertical aerial photographs (for the northern coast of São Paulo State) were used. Headland-bay beach planform stability can be classified as: (1) in static equilibrium, (2) in dynamic equilibrium, (3) unstable or (4) in a state of natural beach reshaping. Static equilibrium beaches are the most frequent along the coast of the State of Santa Catarina and the Northern Shore of São Paulo, notably along the most rugged sectors of the coast and those with experiencing lower fluvial discharge. By comparison, dynamic equilibrium beaches occur primarily on the less rugged sectors of the coast and along regions with higher fluvial discharge. Beaches in a state of natural beach reshaping have only been found in SC, associated with stabilized estuarine inlets or port breakwaters. However, it is not possible to classify any of these beaches as unstable because only one set of images was used. No clear relation was observed between a beach's planform stability and other classification factors, such as morphodynamics or orientation.<br>Este trabalho apresenta resultados da classificação da estabilidade da forma em planta das praias de enseada do Estado de Santa Catarina e do Litoral Norte de São Paulo, realizada através da aplicação do modelo parabólico a imagens aéreas utilizando o software MEPBAY®. Para isso foram construídos mosaicos georeferenciados com imagens do satélite QuickBird2® (para o Estado de Santa Catarina) e com fotografias aéreas verticais (para o Litoral Norte de São Paulo). Quanto à estabilidade de sua forma em planta, as praias de enseada podem ser classificadas como: (1) equilíbrio estático, (2) equilíbrio dinâmico, (3) instável e (4) remodelamento natural. No Estado de Santa Catarina e no Litoral Norte de São Paulo foi observado o domínio de praias em equilíbrio estático, predominantemente nos setores do litoral que são mais recortados e apresentam menor aporte fluvial. As praias em equilíbrio dinâmico, por outro lado, são mais freqüentes nos setores menos recortados e com maior influências do aporte sedimentar por via fluvial. Praias em remodelamento natural ocorrem apenas em SC, associadas à desembocadura de rios retificados ou a quebra-mares de portos. Como foi utilizada apenas uma série de imagens, não foi possível enquadrar praias no estado instável. Não foi observada uma clara relação entre o estado de equilíbrio da forma em planta com outras classificações de praias, como morfodinâmica e orientação