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Text The aim of this paper is to present a method to calculate the rate of growth of divergent series. In the case of the harmonic series, we study an alternate of other classical methods presented by L. Comtet, S.M. Zeyman, W. DeTemple and S.-H. Wang. Moreover, the method presented in this paper is suitable for obtaining similar results on other divergent series, as we will show in the final part of our investigation. Video For a video summary of this paper, please visit http://youtu.be/SFR7zY4pvG4 . [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to characterize QM-abelian surfaces which has a model of -type over . The “special” involutions on a corresponding indefinite quaternion algebra (which is defined in Section 2) play an essential role in this paper. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to determine all free separable quadratic algebras over the rings of integers of quadratic fields in terms of the properties of the fundamental unit in the real case. The paper corrects some earlier published results on the subject. [Copyright &y& Elsevier]

In this paper, we prove that the expected number of points in F q 2 of multiplicity m , for 0 ≤ m ≤ q + 1 , with respect to a randomly chosen arrangement of q + 1 lines with different slopes, is ( 1 / ( m ! e ) ) q 2 + O ( q ) , as q → ∞ . We further state that the distance between the number of such points in a randomly chosen arrangement and ( 1 / ( m ! e ) ) q 2 is lower than q ln ⁡ q with probability close to 1 for large q . [ABSTRACT FROM AUTHOR]

Abstract: We say the sets of nonnegative integers and are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set by using analytic tools. [Copyright &y& Elsevier]

Abstract: Let be positive integers. In this paper, we prove that the equation has no solutions in integers and k with , and . [Copyright &y& Elsevier]

Abstract: In this paper, we determine all sets A of integers such that, for any integral-valued polynomial which has no fixed divisor, for all integers and n, there are infinitely many integers and a choice of such that . The earlier result shows that is such a set. [Copyright &y& Elsevier]

Abstract: In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius–Euler polynomials and Frobenius–Euler polynomials of higher order. [Copyright &y& Elsevier]

Abstract: In this paper, we show that there is a frame of norm k in the odd Leech lattice for every . [Copyright &y& Elsevier]

Abstract: In this paper, we construct for arbitrary reductive group a full eigenvariety, which parameterizes all p-adic overconvergent cohomological eigenforms of the group in the sense of Ash–Stevens and Urban. We also see that the eigenvariety constructed by Eric Urban (2011) is the union of those irreducible components of codimension zero in the eigenvariety we constructed, therefore we can formulate Urbanʼs conjectures. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices. [Copyright &y& Elsevier]

Abstract: Text: We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field,” while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=az_0pxm1jrI. [Copyright &y& Elsevier]

Abstract: This paper continues the study of beta-expansions of 1 where β is a Pisot or Salem number. Special attention is given to regular Pisot numbers, and the Salem numbers that approach these Pisot numbers. [Copyright &y& Elsevier]

Abstract: In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums , , . The proof of this result is completely combinatorial and elementary. [Copyright &y& Elsevier]

Abstract: In this paper we give an alternative review over the subject of strata and simple characters as more generally already considered in the references [P. Broussous, M. Grabitz, Pure elements and intertwining classes of simple strata in local central simple algebras, Comm. Algebra 28 (11) (2000) 5405–5442] and [V. Sécherre, Représentations lisses de , I: charactères simples, Bull. Soc. Math. France 132 (3) (2004) 327–396]. Finally, we prove the intertwining implies conjugacy property of simple characters as known in the split case [C.J. Bushnell, P.C. Kutzko, The Admissible Dual of via Compact Open Subgroups, Ann. of Math. Stud., vol. 129, Princeton Univ. Press, 1993]. [Copyright &y& Elsevier]

Abstract: Let be a Galois extension of number fields and let be an abelian variety defined over . In this paper we establish the relation between the irreducible characters of the Galois group and the simple factors of the restriction of scalars of from to . Then we derive some equivalences of Birch and Swinnerton–Dyer conjectures. [Copyright &y& Elsevier]