Discrepancy of Fractions with Divisibility Constraints

We provide bounds for the absolute discrepancy of sequences of fractions with denominators streaming in a given arithmetic progression and satisfying divisibility constraints. Supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

Lipscomb’s L(A) Space Fractalized in l p (A)

In this paper, by using some of our new results concerning the shift space for an infinite IFS (see A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. Bucur. 11 (2009), 21?C32), we show that, for an infinite set A, the embedded version of the Lipscomb space L(A) in l ^p (A), ${p \in [1,\infty)}$ , with the metric induced from l ^p (A), denoted by ${\omega_p^A}$ , is the attractor of an infinite iterated function system comprising affine transformations of l ^p (A). In this way we provide a generalization of the positive answer that we gave to an open problem of J.C. Perry (see Lipscomb??s universal space is the attractor of an infinite iterated function system, Proc. Amer. Math. Soc. 124 (1996), 2479?C2489) in one of our previous works (see R. Miculescu and A. Mihail, Lipscomb space ?? ^A is the attractor of an infinite IFS containing affine transformations of l ²(A), Proc. Amer. Math. Soc. 136 (2008), 587?C592). Moreover, as a byproduct, we provide a generalization of Corollary 15 from Perry??s paper by proving that ${\omega_p^A}$ is a closed subset of l ^p (A).     

On a Problem of Brocard

It is proved that, if P is a polynomial with integer coefficients,having degree 2, and 1 > > 0, then n(n – 1) ...(n – k + 1) = P(m) has only finitely many natural solutions(m,n,k), n k > n, provided that the abc conjecture is assumedto hold under Szpiro's formulation. 2000 Mathematics SubjectClassification 11D75, 11J25, 11N13.     

Arithmetic differential equations on $$GL_n$$, III Galois groups

Differential equations have arithmetic analogues (Buium in Arithmetic differential equations, Mathematical Surveys and Monographs, vol 118. American Mathematical Society, Providence 2005) in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations, and the present paper is concerned with the “linear” ones. The equations themselves were introduced in a previous paper (Buium and Dupuy, in Arithmetic differential equations on \(GL_{n}\), II: arithmetic Lie–Cartan theory, arXiv:1308.0744). In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.     

A finiteness theorem for a class of exponential congruences

For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.

     

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\).     

Quasistationary distributions for one-dimensional diffusions with singular boundary points

In the present work we characterize the existence of quasistationary distributions for diffusions on $(0,\infty )$ allowing singular behavior at 0 and $\infty$. If absorption at 0 is certain, we show that there exists a quasistationary distribution as soon as the spectrum of the generator is strictly positive. This complements results of Cattiaux et al. (2009) and Kolb and Steinsaltz (2012) for 0 being a regular boundary point and extends results by Cattiaux et al. (2009) on singular diffusions.     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.