On the k-generalized Fibonacci numbers and high-order linear recurrence relations

We are investigating the distribution of the number of peptides for given masses, and especially the observation that peptide density reaches a local maximum approximately every 14 Daltons. This wave pattern exists across species (e.g. human or yeast) and enzyme digestion techniques. To analyze this phenomenon we have developed a mathematical method for computing the mass distributions of peptides, and we present both theoretical and empirical evidence that this 14-Dalton periodicity does not arise from species selection of peptides but from the number-theoretic properties of the masses of amino acid residues. We also describe other, more subtle periodic patterns in the distribution of peptide masses. We also show that these periodic patterns are robust under a variety of conditions, including the addition of amino acid modifications and selection of mass accuracy scale. The method used here is also applicable to any family of sequential molecules, such as linear hydrocarbons, RNA, single- and double-stranded DNA.

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“…Numerical sequences generated by recurrence relationships in the form of Equation 2 are known as k -generalized Fibonacci numbers [19]. …”

Section: Mathematical Modelmentioning

confidence: 99%

Periodic patterns in distributions of peptide masses

Hubler

Crăciun

2012

Biosystems

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“…and, using the theory of lower triangular Toeplitz matrices (see for example [13] or [16]), one can express the k-generalized Fibonacci numbers in terms of determinants of matrices with entries given by the coefficients of their recurrence equation, that is (2. 16)…”

Section: 3mentioning

confidence: 99%

On classification of discrete, scalar-valued Poisson Brackets

Parodi

2011

Preprint

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We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on target space of dimension 1. It is proved that these Poisson brackets (PBs) are in one-to-one correspondence with the intersection points of certain projective hypersurfaces. In addition, they can be reduced to cubic PB of standard Volterra lattice by discrete Miura-type transformations. Finally, improving a consolidation lattice procedure, we obtain new families of non-degenerate, vector-valued and first order dDGPBs, which can be considered in the framework of admissible Lie-Poisson group theory.

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“…As in computation of spline functions, time series analysis, signal and image processing, queueing theory, polynomial and power series computations and many other areas, typical problems modelled by Toeplitz matrices are the numerical solution of certain differential and integral equations [1,2,3]. The literature includes many papers dealing with the special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Mersenne, Fermat, Padovan and Perrin [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The books written by Koshy and Vajda collect and classify many results dealing with these number sequences, most of which have been obtained quite recently [16,18,19].…”

Section: Introductionmentioning

confidence: 99%