Summation

Summary

In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very o

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Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

Sabine Burgdorf

This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several examples are given illustrating the results.

Springer2013

Détection des aberrations chromosomiques basée sur les profiles d'expression des gènes et les données aCGH

The detection of chromosomal aberrations constitutes an important stage in the comprehension of the pathogenesis of several diseases, as cancer. In this case, the amplification or deletion (suppression) of some chromosomal regions, was identified as being an essential mechanism of tumour development. Several methods were proposed for the detection of regions, which copy number is altered, from aCGH data. However, it has been reported that 40-60 % of genes show changes in RNA abundance, when amplified [27, 38]. For this reason, the detection of chromosomal aberration from RNA abundance seems solvable. To achieve that, we propose, initially, a simple method, based on cumulated sums (CUMSUM), then more elaborated method, based on the hidden Markov models "HMM" [43]. Within the framework of the HMM, we propose a model HMM with two states (HMM2s) for the detection of only one type of aberration (amplification or deletion), a model HMM with three states (HMM3s) for the simultaneous detection of the amplifications and deletions. We also propose a model HMM with 2 states and double observations by state (HMM2s2o) for the simultaneous analysis of the expression and aCGH data. We tested the various models on simulated and real data, and finally we compared the detected regions of aberration from expression data to those detected from aCGH data of the Glioblastoma.

EPFL2006

Some applications of smooth bilinear forms with Kloosterman sums

Philippe Michel

We revisit a recent bound of I. Shparlinski and T. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet L-functions.

Maik Nauka/Interperiodica/Springer2017