It is commonly assumed that a signal bandlimited to 2 Hz cannot oscillate at frequencies higher than Hz. In fact, however, for any fixed bandwidth, there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients, called superoscillations, can only occur in signals that possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared 2 norm) as a function of the superoscillation's frequency, number, and maximum derivative. It briefly discusses some of the implications of superoscillating signals, in reference to information theory and time-frequency analysis, for example. It also shows, among other things, that the required energy grows exponentially with the number of superoscillations, and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations' period. . His research interests range from classical sampling theory to quantum information theory, quantum gravity, and inflationary cosmology. Prof. Kempf is a member of the Institute for Quantum Computing and an affiliated member of the Perimeter Institute for Theoretical Physics.

It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible -a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.

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Motivation: DNA sequences can be represented by sequences of four symbols, but it is often useful to convert the symbols into real or complex numbers for further analysis. Several mapping schemes have been used in the past, but they seem unrelated to any intrinsic characteristic of DNA. The objective of this work was to find a mapping scheme directly related to DNA characteristics and that would be useful in discriminating between different species. Mathematical models to explore DNA correlation structures may contribute to a better knowledge of the DNA and to find a concise DNA description.Results: We developed a methodology to process DNA sequences based on inter-nucleotide distances. Our main contribution is a method to obtain genomic signatures for complete genomes, based on the inter-nucleotide distances, that are able to discriminate between different species. Using these signatures and hierarchical clustering, it is possible to build phylogenetic trees. Phylogenetic trees lead to genome differentiation and allow the inference of phylogenetic relations. The phylogenetic trees generated in this work display related species close to each other, suggesting that the inter-nucleotide distances are able to capture essential information about the genomes. To create the genomic signature, we construct a vector which describes the inter-nucleotide distance distribution of a complete genome and compare it with the reference distance distribution, which is the distribution of a sequence where the nucleotides are placed randomly and independently. It is the residual or relative error between the data and the reference distribution that is used to compare the DNA sequences of different organisms.Contact: vera@ua.pt

Abstract-In this paper we study the eigenvalues of a matrix S which arises in the recovery of lost samples from oversampled band-limited signals. Emphasis is placed on the variation of the eigenvalues as a function of the distribution of the missing samples and as a function of the oversampling parameter. We present a number of results which help to understand the numerical difficulties that may occur in this class of problems, and ways to circumvent them.

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