DIOPHANTINE ANALYSIS AND RELATED FIELDS—2010: DARF—2010

Generators for the elliptic curve
View Description Hide DescriptionLet be an elliptic curve over the rationals with a positive integer n. Mordell’s theorem asserts that the group of rational points on E is finitely generated. Our interest is in the generators for its free part. Duquesne (2007) showed that if is square‐free, then certain two points of infinite order can always be in a system of generators. We generalize this result and show that the same is true for “infinitely many” infinite families with two variables.

Algebraic independence of real numbers with low density of nonzero digits (survey)
View Description Hide DescriptionWe study transcendence and algebraic independence of the value where z is an algebraic number with 0<z<1 and w(n) is a strictly increasing sequence of nonnegative integers. First we survey the case where w(n) is lacunary. In our main results, we give criteria for algebraic independence of the values f(z;w(n)) in the case where for some integer α greater than 1 and w(n) is not lacunary. In particular, using our criteria, we deduce that the uncountable set is algebraically independent, where

On the complexity of the binary expansions of algebraic irrational numbers (survey)
View Description Hide DescriptionBorel conjectured that all irrational numbers are normal in any integral base α. For each positive number ξ and integer α greater than 1, ξ is normal in base α if and only if the sequence is uniformly distributed modulo 1. In this paper we survey not only the digit of algebraic irrational numbers in integral base but also the fractional parts of geometric progressions whose common ratios are algebraic numbers greater than 1. In our main results, we give new lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.

Some recognizable forms of simple continued fractions
View Description Hide DescriptionAn explicit form of Hurwitzian continued fraction whose quasi‐period is determined by polynomials of degree larger than 1 had been unknown. Here, we show a reduction formula that yields some recognizable expressions given the sequence of partial quotients in the simple continued fraction expansion. As an illustrations, we can obtain and a number of other interesting results.

Generalizations of classical results on Jeśmanowicz’ conjecture concerning Pythagorean triples
View Description Hide DescriptionIn 1956, Leon Jeśmanowicz conjectured that, for any primitive Pythagorean triple (a,b,c) with the equation in positive integers has only the solution There are some classical and celebrated results on this conjecture. In this paper, we broadly generalize many of them. As a corollary, we can verify that the conjecture is true when

Mahler measure of the Horie unit and Weber’s Class Number Problem in the Cyclotomic ‐extension of
View Description Hide DescriptionLet p be a prime number. It is an interesting problem to consider whether a prime number ℓ divides the class numbers of the intermediate fields of the cyclotomic ‐extension of Q. In the case R. Okazaki developed a theory for this problem by using Mahler measure. In this paper, we focus on the case and show that a prime number ℓ does not divide the class numbers of the intermediate fields of the cyclotomic 3‐extension of if ℓ satisfies mod 27.

On Siegel’s lemma and linear spaces of random variables
View Description Hide DescriptionUsing a linear space of random variables, we present a statement that covers some types of Siegel’s lemma in a certain sense.

A congruence modulo p of zeta polynomial for cyclotomic function fields
View Description Hide DescriptionA cyclotomic function field is defined as adding torsion points of Carlitz module to the rational function field of characteristic p. Properties of these fields are remarkably similar to those of cyclotomic number fields. In 1999, L. Guo and L. Shu gave congruence relations for class numbers of cyclotomic function fields. In this paper, we will generalize these results from the view point of congruence zeta function.

Preserving log‐concavity and generalized triangles
View Description Hide DescriptionWe introduce generalized triangles called s‐triangles for s given positive integer, as a bi‐indexed sequence of non negative numbers satisfying for k<0 or k>ns. A such s‐triangle is LC‐positive if for each r, the sequence of polynomials is q‐log‐concave. We extend some results of Wang and Yeh, Log‐concavity and LC‐positivity, J. Combin. Theory Ser. A (2007), and show that if is LC‐positive then the log‐concavity of the sequence implies the log‐concavity of the sequence defined by Applications related to ordinary multinomials are given.

Characterization of linear recurrences associated to rays in Pascal’s triangle
View Description Hide DescriptionOur purpose is to describe the recurrence relations associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We also answer Horadam’s question posed in his paper entitled Chebyshev and Pell connections, Fibonacci Quart., (2005). Further, using Morgan‐Voyce sequence, we establish the nice identity of Fibonacci numbers, where Finally, connections to continued fractions, bivariate polynomials and finite differences are given.

Asymptotic expansions for certain multiple q‐integrals and q‐differentials of Thomae‐Jackson type
View Description Hide DescriptionThis is a summarized version of the forthcoming paper [11]. Let q be a complex parameter with q<1. We shall study in this paper asymptotic aspects when q→1 of certain general classes of q‐integral and q‐differential operations given in (1.5) and (1.6) below respectively; this leads us to establish complete asymptotic expansions for their iterated extensions (Theorems 1 and 2) under fairly generic situations (Theorem 3). Several applications of of our main formulae (2.4) and (2.9) are further given for the generalized Lerch zeta‐function defined by (3.3) (Theorems 4–6 and Corollary 6.1), the q‐factorials (Corollary 4.1), q‐analogues of the exponential (Corollary 4.2), the binomial (Corollary 4.3), and the poly‐logarithmic functions (Corollaries 4.4 and 5.1).

Open Problems on Densities II
View Description Hide DescriptionThis is a collection of open questions and problems concerning various density concepts on subsets of It is a continuation of paper [10].

On Generalized Lipschitz‐type Formulae and Applications
View Description Hide DescriptionThe purpose of this paper is to present a certain Lipschitz‐type formula and its generalization via Mellin‐Barnes type integrals. We further introduce a class of double Eisenstein series for defined on the pair of the upper (or lower) half planes, and show their transformation properties by using the generalized Lipschitz‐type formulae.

Algebraic Jacobi‐Perron algorithm for biquadratic numbers
View Description Hide DescriptionWe introduced a new algorithm [6] which is something like the modified Jacobi‐Perron algorithm and we conjectured that the expansion obtained by our algorithm for (with some natural conditions on ) becomes periodic for any real number field K as far as We announce Theorem and some computer experiments for certain biquadratic real number fields which support our conjecture.

Back Matter for Volume 1264
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