Examples. Let n ∈ N. Define functions τ, σ, π : N → N as follows: • τ(n) = the number of all natural divisors of n = #{d > 0 | d|n}; • σ(n) = the sum of all natural divisors of n = ∑ d|nd; • π(n) = the product of all natural divisors of n = ∏ d|nd. As we shall see below, τ and σ are multiplicative functions, while π is not. From now on we shall write the prime decomposition of n ∈ N as n = p1 1...

The present paper is part I of a series of three closely related papers in which the inverse measure (dt) of a given measure (dt) on 0; 1] is introduced. In the rst case discussed in detail, and are multifractal in the usual sense, that is, both are linearly self-similar and continuous but not diierentiable and both are non{zero for every interval of 0; 1]. Under these assumptions the HH older ...

We establish a pair of identities, which will provide a useful tool in the reconstruction of evolutionary trees in Kimura's 3-parameter model. The starting point of this paper was an attempt for a better understanding and generalization of an Hadamard inverse pair of formulae, which was used in statistics by Cooper [l], in image processing by Andrews [2, Chapters 6,7], and in information theory...

The commonly applied methods for seismic inversion are based on some drastic assumptions regarding the known background or macro-velocity model and the data acquisition, that limit their applicability in geologies of realistic complexity or to realistic, noisy and incomplete, data sets. The background is usually assumed smooth, often to such a degree that the wav e field can be described by sim...

A trace group (monoid) is the quotient of a free group (monoid) by relations of commutation between some pairs of generators. We prove an analog for the trace group of the Möbius inversion formula for the trace monoid (Cartier and Foata, 1969). c © 200? Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Une formule d’inversion de Möbius pour le groupe de traces Résumé. Un gr...

Tuy's cone-beam inversion formula was modified to develop a cone-beam reconstruction algorithm. The algorithm was implemented for a cone-beam vertex orbit consisting of a circle and two orthogonal lines. This orbit geometry satisfies the cone-beam data sufficiency condition and is easy to implement on commercial single photon emission computed tomography (PECT) systems. The algorithm, which con...

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra o...

In standard notation, let {H1,H2, · · · ,Hm} and {A1, A2, · · · , An} be two distinct partitions of the sample space, or equivalently two sets of events satisfying three properties: (i) each event is non-void, (ii) events in the same set are mutually exclusive (i.e. P (Hj ∪Hk) = P (Hj)+P (Hk) for j 6= k), and (iii) each set is collectively exhaustive, (i.e. P (∪i=1Hj) = 1). The Bayes formula in...

The Erlang B formula represents the steady-state blocking probability in the Erlang loss model or M/M/s/s queue. We derive asymptotic expansions for the offered load that matches, for a given number of servers, a certain blocking probability. In addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function. A similar inversion problem is invest...