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Andrei Vieru

Publications and source records attributed to Andrei Vieru.

12 recordsLinked to original sources
arXiv2017

Analytic renormalization of multiple zeta functions. Geometry and combinatorics of the generalized Euler reflection formula for MZV

The renormalization of MZV was until now carried out by algebraic means. We show that renormalization in general, of the multiple zeta functions in particular, is more than mere convention. We show that simple calculus methods allow us to compute the renormalized values of multiple zeta functions in any dimension for arguments of the form (1,...,1), where the series do not converge. These values happen to be the coefficients of the asymptotic expansion of the inverse Gamma function. We focus on the geometric interpretation of these values, and on the combinatorics their closed form encodes, which happen to match the combinatorics of the generalized Euler reflection formula discovered by Michael E. Hoffman, which in turn is a kind of analogue of the Cayley-Hamilton theorem for matrices. By means of one single limit formula, we define a function on the positive open half-line which takes exactly the values of the Riemann zeta function, with the additional advantage that it equals the Euler constant when the argument is 1.

math.NT↗
arXiv2015

Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals. .

math.GM↗
arXiv2015

On iterating operators and on generalized periodic orbits

We try to define the more general form of iterative processes in which the Pomeau-Manneville and the Feigenbaum scenario may occur along with their specific scaling properties. Doing this we need to generalize other basic concepts. Thus, what we call a periodic carousel is a generalization of what is usually called a periodic orbit.

math.DS↗
arXiv2012

Short note on additive sequences and on recursive processes

Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver ratio, but a dense set of ratio limits that corresponds to an infinity of conceivable recursive additive rules. We shall show that some of these limits have nice properties. Identities involving Fibonacci and Lucas sequences will be viewed as special cases of more general identities. We shall show that some properties of the Pascal Triangle belong also to other similar objects. In Dynamical Systems and Chaos Theory we shall encounter weird orbits, whose order is higher than the number of its distinct elements and, beyond the chaos point, a rather unexpected belated convergence to 0, after a pseudo chaotic behaviour during as many terms as one may wish. Time and again, we shall find here the Feigenbaum constant. In Formal Grammars we shall see that recursive rules applied to concatenation are sometimes equivalent to formal grammars although generally more restrictive.

math.DS↗
arXiv2012

Agoh's conjecture: its generalizations, its analogues

In this paper we formulate two generalizations of Agoh's conjecture. We also formulate conjectures involving congruence modulo primes about hyperbolic secant, hyperbolic tangent, Nörlund numbers, as well as about coefficients of expansions in powers of other analytic functions. We formulate a thesis about combinatorial objects that do not produce fake primes.

math.NT↗
arXiv2012

Pisot Numbers and Primes

We define and study a transform whose iterates bring to the fore interesting relations between Pisot numbers and primes. Although the relations we describe are general, they take a particular form in the Pisot limit points. We give three elegant formulae, which permit to locate on the whole semi-line all limit points that are not integer powers of other Pisot numbers.

math.NT↗
arXiv2008

Bifurcations, Schwarzian derivatives and Feigenbaum constant revisited

The main purpose is to show that Feigenbaum delta constant is much more universal than believed. The paper is mainly devoted to period-doubling processes in families in one parameter of endomorphisms of the interval and consider generalizations of the Feigenbaum delta constant. We formulate the so-called parenthesis permeability hypothesis, a conjecture that holds for all types of bifurcation (i.e. for flip, fold, pitchfork and transcritical bifurcations, which states that under some conditions two or three different functions may have exactly the same bifurcation points. We propose a conjecture that considerably relaxes David Singer conditions for endomorphism families to generate at most one stable orbit, showing that Feigenbaum constant appears also in some classes of functions that have more than one maximum and have positive Schwarzian in at least one sub-interval. This version contains more arguments in favor of an even greater Feiganbaum constant universality.

math.DS↗
arXiv2008

General definitions of chaos for continuous and discrete-time processes

A precise definition of chaos for discrete processes based on iteration already exists. We shall first reformulate it in a more general frame, taking into account the fact that discrete chaotic behavior is neither necessarily based on iteration nor strictly related to compact metric spaces or to bounded functions. Then we shall apply the central idea of this definition to continuous processes. We shall try to see what chaos is, regardless of the way it is generated.

math.DS↗
arXiv2008

About Stable Periodic Helixes, L-iteration and Chaos Generated by Unbounded Functions

We consider stable periodic helixes as a generalization of stable periodic orbits. We see that in the studied class of iterated functions Chaos always arise suddenly. Therefore, we shall study the route from chaos to order rather than the route from order to chaos. We show that, paradoxically, genuine Chaos may look as much like Order and during as many iteration steps as one may wish. Then, we shall propose a generalization of the idea of map iteration that do not imply the existence of periodic orbits. We shall show that, within a strictly deterministic context, unpredictability, aperiodic order, sensitive dependence and chaos are completely different concepts and we shall try to show what this difference is made of. We shall also propose an example of non chaotic aperiodic order.

math.DS↗
arXiv2008

Generalized iteration, catastrophes and generalized Sharkovsky's ordering

We define iteration of functions that map n-dimensional vector spaces into m-dimensional vector spaces (m at most equal to n). It happens that usual iteration and Fibonacci iterative methods become special cases of this generalized iteration. Mathematical objects such as orbits, bifurcations, chaos, Feigenbaum constant, (generalized) Sharkovsky ordering, (generalized) Julia and Mandelbrot sets and a new kind of catastrophe can be found and studied in this enlarged context.

math.DS↗
arXiv2008

Lindenmayer systems and primes

We study the surprising discrepancy between the number of primes corresponding, respectively, to the two letters of an infinite word engendered by one of the simplest Lindenmayer systems. We formulate a conjecture concerning the rate of growth of this discrepancy, which seems to tend to e for every two sufficiently high consecutive even rank iterates of the Lindenmayer system.

math.NT↗