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Distributional properties of powers of matrices

Fernando Chamizo, Dulcinea Raboso (2014)

Czechoslovak Mathematical Journal

We apply the larger sieve to bound the number of 2 × 2 matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar...

Dynamique des nombres et physique des oscillateurs

Jacky Cresson (2008)

Journal de Théorie des Nombres de Bordeaux

Nous présentons un modèle mathématique permettant de reproduire le spectre expérimental des fréquences dans un composant électronique appelé boucle ouverte. Le spectre semble s’organiser suivant une contrainte de nature diophantienne sur les fréquences. Sa structure peut donc se comprendre via une étude de l’ensemble des fractions continues en fonction de leur longueur et de la taille des quotients partiels.

From Planck to Ramanujan : a quantum 1 / f noise in equilibrium

Michel Planat (2002)

Journal de théorie des nombres de Bordeaux

We describe a new model of massless thermal bosons which predicts an hyperbolic fluctuation spectrum at low frequencies. It is found that the partition function per mode is the Euler generating function for unrestricted partitions p ( n ). Thermodynamical quantities carry a strong arithmetical structure : they are given by series with Fourier coefficients equal to summatory functions σ k ( n ) of the power of divisors, with k = - 1 for the free energy, k = 0 for the number of particles and k = 1 for the internal energy. Low...

Gosset polytopes in integral octonions

Woo-Nyoung Chang, Jae-Hyouk Lee, Sung Hwan Lee, Young Jun Lee (2014)

Czechoslovak Mathematical Journal

We study the integral quaternions and the integral octonions along the combinatorics of the 24 -cell, a uniform polytope with the symmetry D 4 , and the Gosset polytope 4 21 with the symmetry E 8 . We identify the set of the unit integral octonions or quaternions as a Gosset polytope 4 21 or a 24 -cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the E 8 or D 4 actions on the 4 21 or the 24 -cell, respectively. Moreover, we show that each...

Gravity, strings, modular and quasimodular forms

P. Marios Petropoulos, Pierre Vanhove (2012)

Annales mathématiques Blaise Pascal

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ...The latter often appear as gravitational instantons i.e. as special solutions of Einstein’s equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by...

On q-orders in primitive modular groups

Jacek Pomykała (2014)

Acta Arithmetica

We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that * p has no generator in the interval [1,B]. As a consequence we prove that if Q > e x p [ c ( l o g p ) / ( l o g B ) ( l o g l o g p ) ] with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that ν q ( o r d p b ) = ν q ( p - 1 ) for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.

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