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Let a , b , c be fixed coprime positive integers with min ⁡ { a , b , c } > 1 . In this paper, combining the Gel'fond–Baker method with an elementary approach, we prove that if max ⁡ { a , b , c } > 5 × 10 27 , then the equation a x + b y = c z has at most three positive integer solutions ( x , y , z ) . [ABSTRACT FROM AUTHOR]

Two decades ago, Steven Roman, Daniel E. Loeb and Gian-Carlo Rota introduced a family of harmonic numbers in their study of harmonic logarithms. We propose to refer to those numbers as Roman harmonic numbers . With the purpose of revitalizing the study of these mathematical objects, we recall here their known properties and unveil additional ones. An integral representation, several generating relations, and a collection of sum rules involving those numbers are presented. It is also shown that higher derivatives of the Pochhammer and reciprocal Pochhammer symbols are easily expressed in terms of Roman harmonic numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study multizeta values over function fields in characteristic p . For each d ≥ 2 , we show that when the constant field has cardinality >2, the field generated by all multizeta values of depth d is of infinite transcendence degree over the field generated by all single zeta values. [ABSTRACT FROM AUTHOR]

We prove new estimates on the number of algebraic points of fixed degree and bounded height on projective spaces over a given number field. These results extend previous works of Wolfgang Schmidt [13] , Gao Xia [3] and Martin Widmer [18] . Our approach, based on zeta functions, also gives a new proof of Schanuel's theorem. [ABSTRACT FROM AUTHOR]

Possible transcendental nature of Euler's constant γ has been the focus of study for sometime now. One possible approach is to consider γ not in isolation, but as an element of the infinite family of generalized Euler–Briggs constants. In a recent work [6] , it is shown that the infinite list of generalized Euler–Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler–Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see Theorem 5 and Theorem 6 ) on the dimension of these spaces and consequently establish the infinite dimensionality of the space spanned. Further, we study linear and algebraic independence of these constants over the field of all algebraic numbers. [ABSTRACT FROM AUTHOR]

Some congruences on conjectures of van Hamme are established. These results confirm some conjectures of Swisher. [ABSTRACT FROM AUTHOR]

For each 1 < β < 2 , let r n ( β ) be the maximal length of consecutive 0 digits in the first n digits of 1's β -expansions. We prove that for Lebesgue almost all 1 < β < 2 , lim n → ∞ ⁡ r n ( β ) / log β ⁡ n = 1 and r n ( β ) ≥ log β ⁡ n for infinitely many n , which answers two questions raised by Erdös, Joó and Komornik. [ABSTRACT FROM AUTHOR]

Text In 2014, Herzog, Kaplan and Lev introduced the critical numbers cr ( r , n ) of natural intervals [ r , n ] and determined the values of cr ( 1 , n ) and cr ( 2 , n ) for all n , and for r ≥ 3 , they determined the values of cr ( r , n ) for n ≤ ( 3 r 2 + 3 r ) / 2 and n ≥ ( 5 r 2 + r ) / 2 . In this note, we determine the remaining values of cr ( r , n ) for r ≥ 3 . Video For a video summary of this paper, please visit https://youtu.be/hZSI-Ul3K94 . [ABSTRACT FROM AUTHOR]

We define the analogue of q-additivity for the canonical number systems in the Gaussian ring of integers. We characterize all those functions f : Z[i] → C which are θ = -A + i-additive and completely multiplicative (Theorem 1). For θ = -1 + i we give all functions which are θ- and θ¯-additive (Theorem 2). [ABSTRACT FROM AUTHOR]

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham. [ABSTRACT FROM AUTHOR]

In this paper we work on indivisibility of the class numbers of real quadratic function fields. We find an explicit expression for a lower bound of the density of real quadratic function fields (with constant field F ) whose class numbers are not divisible by a given prime ℓ . We point out that the explicit lower bound of such a density we found only depends on the prime ℓ , the degrees of the discriminants of real quadratic function fields, and the condition: either | F | ≡ 1 ( mod ℓ ) or not. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field of finite degree over the rational field Q, and a( n) the number of integral ideals in K with norm n. When K is a Galois extension over Q, many authors contribute to the integral power sums of a( n), . This paper is interested in the distribution of integral ideals concerning different number fields. The author is able to establish asymptotic formulae for the convolution sum , where K and K are two different quadratic fields. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case. [ABSTRACT FROM AUTHOR]

Let m be a fifth power free integer greater than one. Let K be an algebraic number field generated by a fifth root of m over the rational number field. If m has a prime factor p congruent to −1 modulo five, the class number of K is a multiple of five. [ABSTRACT FROM AUTHOR]

The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions I0(x) and K0(x) for positive real x. [ABSTRACT FROM AUTHOR]

Let L = L1 ∪ L2 ∪ ⋯ ∪ Ln be a link in ℝ3 such that Li is a trivial link for each i, 1 ≤ i ≤ n. Let P1, P2,...,Pn be mutually distinct flat planes in ℝ3 such that no two of them are parallel. Then there is a link in ℝ3 such that L is ambient isotopic to L′ and for each i, 1 ≤ i ≤ n. [ABSTRACT FROM AUTHOR]

In this paper we study the class numbers in the finite layers of certain non-cyclotomic $\mathbb{Z}$p-extensions of the imaginary quadratic field $\mathbb{Q}(\sqrt{-1})$, for all primes p ≡ 1 modulo 4. By studying the Mahler measure of elliptic units, we are able to show that there are only finitely many primes ℓ congruent to a primitive root modulo p2 that divide any of the class numbers in the $\mathbb{Z}$p-extension. [ABSTRACT FROM AUTHOR]

Abstract: Text: Let K be an algebraic number field and the ring of integers in K. Let be the set of all elements which are sums of squares in and the minimal number of squares necessary to represent −1 in . Let be the smallest positive integer t such that every element in is a sum of t squares in . In this note, for , where are two distinct positive square-free integers, we show that and if , then . Video: For a video summary of this paper, please click here or visit http://youtu.be/F6UHHgzfdzA. [Copyright &y& Elsevier]

A ring R is called pseudopolar if for every a ∈ R there exists p2 = p ∈ R such that p ∈ 2(a), a + p ∈ U(R) and akp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M2(R) is pseudopolar. [ABSTRACT FROM AUTHOR]

The graphs in which every minimal vertex separator induces a complete or an edgeless subgraph are shown to be the edge sums of graphs of the two extreme types: (i) when every minimal vertex separator induces a complete subgraph - these are precisely the well-studied chordal graphs - and (ii) when every minimal vertex separator induces an edgeless subgraph - these have appeared in several recent papers. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, the author establishes several integral representations and properties of Stirling numbers of the first kind. [Copyright &y& Elsevier]

Abstract: Let K be a perfect field of characteristic not equal to two, an algebraic closure of K and let be the Galois group of the extension . Let T be an n-point set in . The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this paper we show that point sets of odd cardinality in with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality . Finally we prove that the following local-to-global principle holds for point sets of cardinality : if an n-point set T in is defined over for each prime p (including the prime at ∞), then it is defined over . From this result we derive an analogue local-to-global principle for hyperelliptic curves. [Copyright &y& Elsevier]

Abstract: We use Artin–Schreier base change to construct counterexamples to a Kummer–Vandiver type question for function fields. [Copyright &y& Elsevier]

Abstract: Let C be an algebraically closed field of positive characteristic p and complete with respect to a non-archimedean absolute value and a discrete -submodule. Suppose there exists an -basis of Λ such that and . For define the meromorphic function on C. We show that all the zeroes x of satisfy for some i. Furthermore, the number (counted with multiplicities) of zeroes for which (⁎) holds depends only on i and the p-adic expansion coefficients of k, but not on Λ. [Copyright &y& Elsevier]

Abstract: Following methods of Hayes, we state a conjectural product formula for ratios of Brumer–Stark units over real quadratic fields. [Copyright &y& Elsevier]

Let (K, v) be a complete rank-1 valued field. In this paper, we extend classical Hensel's Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind's theorem regarding splitting of primes in algebraic number fields. We also deduce an irreducibility criterion for polynomials over rank-1 valued fields which extends already known generalizations of Schönemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in Khanduja and Khassa [ Manuscripta Math. 134(1-2) (2010) 215-224] is also obtained as a corollary of the main result. [ABSTRACT FROM AUTHOR]

Abstract: Text: The isomorphism number of an F-crystal over an algebraically closed field of positive characteristic is the smallest non-negative integer such that the -th level truncation of determines the isomorphism class of . When is isoclinic, namely it has a unique Newton slope λ, we provide an efficiently computable upper bound for in terms of λ and the Hodge slopes of . This is achieved by providing an upper bound for the level torsion of introduced by Vasiu. We also check that this upper bound is optimal for many families of isoclinic F-crystals that are of special interest (such as isoclinic F-crystals of K3 type). Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=gVObAZZ1DKE. [Copyright &y& Elsevier]

In this paper, we study the notion of norm-Euclidean ideal class, which was defined by Lenstra [Euclidean ideal classes, Astérisque 61 (1979) 121-131]. Using a slight modification of an algorithm determining among other properties the Euclidean minimum described in [P. Lezowski, Computation of the Euclidean minimum of algebraic number fields, preprint (2011); http://hal.archives-ouvertes.fr/hal-00632997/en/], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari [The Euclidean condition in pure cubic and complex quartic fields, Math. Comput. 33 (1979) 389-398], we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class. Finally, we show that ${\mathbf{Q}}(\sqrt{2}, \sqrt{35})$, which is known to admit a non-principal Euclidean ideal class, has no norm-Euclidean ideal class. [ABSTRACT FROM AUTHOR]

The article focuses on divisibility of polynomials and their boundedness. Instead of going for conventional algebraic theory, a straightforward concept for finding divisibility of polynomials is investigated in the article. The second part of the article presents a theorem for finding out boundedness of polynomials.

Abstract: We prove an extension of the Ocone–Karatzas integral representation, valid for all functions on the classical Wiener space. We also establish an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of composite random variables. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime number. For the cyclotomic -extension of a finite algebraic number field F, we denote by the maximal unramified pro-p-extension of . In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for to be abelian, and give sufficient conditions in certain special cases. [Copyright &y& Elsevier]

Abstract: The number , appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many partial results about the minimal polynomial of this algebraic number. Here we obtain the general formula and it is Möbius inversion for this minimal polynomial by means of the Dickson polynomials and the Möbius inversion theory. Moreover, we investigate the homogeneous cyclotomic, Chebychev and Dickson polynomials in two variables and we show that our main results in one variable case nicely extend to this situation. In this paper, the deep results concerning these polynomials are proved by elementary arguments. [Copyright &y& Elsevier]

We study a $ \mathbb{Z}G $-module A such that $ \mathbb{Z} $ is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C( A) = 1, A is not a minimax $ \mathbb{Z} $-module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/ C( H) is a minimax $ \mathbb{Z} $-module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any positive fundamental discriminant D > 1596, there is always at least one prime p ≤ D0.45 such that the Kronecker symbol (D/p) = -1. This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime p in a real quadratic field of discriminant D > 3705 is at most √D/2. We use a "smoothed" version of the P´olya-Vinogradov inequality, which is very useful for numerically explicit estimates. [ABSTRACT FROM AUTHOR]

We obtain an integral representation of even functions of two variables for which the kernel [ k( x + y) + k( x − y)], x, y ∈ R, is positive definite. [ABSTRACT FROM AUTHOR]

Let F be a finite extension of ℚ. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over $\overline{ \mathbb{F}}_{p}$ to be supersingular. We then give the classification of irreducible admissible smooth GL( F)-representations over $\overline{ \mathbb{F}}_{p}$ in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses. [ABSTRACT FROM AUTHOR]

Abstract: The Hadamard product of two power series and is the power series . We define the (Hadamard) grade of a power series A to be the least number (finite or infinite) of algebraic power series, the Hadamard product of which equals A. We study and discuss this notion. [Copyright &y& Elsevier]

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted $\mathscr{N}$). When K is an imaginary quadratic extension, $\mathscr{N}$ is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in $\mathscr{N}$ can be written as $\alpha/\overline{\alpha}$ for some $\alpha \in \mathscr{O}_K$, which yields another ordering of $\mathscr{N}$ given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that $\mathscr{N}$ is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that $\mathscr{N}$ is equidistributed with respect to norm, under the map β ↦ log|β|(mod log|ϵ2|) where ϵ is a fundamental unit of $\mathscr{O}_K$. [ABSTRACT FROM AUTHOR]

Abstract: The family of semiabelian p-groups is the minimal family that contains {1} and is closed under quotients and semidirect products with finite abelian p-groups. Kisilevsky and Sonn have solved the minimal ramification problem for a certain subfamily of the family of semiabelian p-groups. We show that is in fact the entire family of semiabelian p-groups and by this complete their solution to all semiabelian p-groups. [Copyright &y& Elsevier]

Abstract: The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [Copyright &y& Elsevier]

Abstract: This paper gives the answer to a question of R.M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers such that . Then there exists an integer such that there exists a group divisible design of group type with block size k and index one for any integer satisfying the necessary arithmetic conditions [1.] , [2.] . This paper also presents a large-index asymptotic existence theorem for group divisible t-designs with a fixed number of groups, fixed group size and fixed block size. [Copyright &y& Elsevier]

family of tenth algebraic order eight-step methods is constructed in this paper. For this family of methods, we require the phase-lag and its first, second and third derivatives to be vanished. Three alternative methods are proposed which satisfy the above requirements. An error analysis and a stability analysis is also investigated in this paper and a comparison with other methods is also studied. The new proposed methods are applied for the numerical solution of the one dimensional Schrödinger equation. The efficiency of the new methodology is proved via theoretical analysis and numerical applications. [ABSTRACT FROM AUTHOR]

Abstract: Vector space partitions of an -dimensional vector space over a finite field are considered in Bu (1980) , Heden (1984) , and more recently in a series of papers Blinco et al. (2008) , El-Zanati et al. (2008, 2009) . In this paper, we consider the generalization of a vector space partition which we call a -fold partition (or simply a -partition). In particular, for a given positive integer, , we define a -fold partition of to be a multiset of subspaces of such that every nonzero vector in is contained in exactly subspaces in the given multiset. A -fold spread as defined in Hirschfeld (1998) is one example of a -fold partition. After establishing some definitions in the introduction, we state some necessary conditions for a -fold partition of to exist, then introduce some general ways to construct such partitions. We also introduce the construction of a dual -partition as a way of generating -partitions from a given -partition. One application of this construction is that the dual of a vector space partition will, in general, be a -partition for some . In the last section, we discuss a connection between -partitions and some designs over finite fields. [Copyright &y& Elsevier]

Abstract: All continuous and affinely contravariant matrix-valued valuations on the Sobolev space are completely classified. It is shown that there is a unique such valuation. This valuation turns out to be the Fisher information matrix. [ABSTRACT FROM AUTHOR]

It is a well-known consequence of the Baker-Pixley-Theorem that any clone containing a near-unanimity operation is finitely generated, leading to the question what arity the generating functions must have. In this paper, we show that, for arbitrary d ≥ 2 and large enough n, ( n − 1) − 1 is the smallest integer k such that, for every clone C on an n-element set that contains a ( d + 1)-ary near-unanimity operation, C generates C. [ABSTRACT FROM AUTHOR]

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted M∞(α) for an algebraic number α. We show that the computation of M∞(α) can be reduced to a certain search through a finite set. Although it is an open problem to record the points of this set in general, we provide some examples where it is reasonable to compute and our result can be used to determine M∞(α). [ABSTRACT FROM AUTHOR]

We study a card game called Mousetrap, together with its generalization He Loves Me, He Loves Me Not. We first present some results for the latter game, based, on one hand, on theoretical considerations and, on the other one, on Monte Carlo trials. Furthermore, we introduce a combinatorial algorithm, which allows us to obtain the best result at least for French card decks (52 cards with 4 suits). We then apply the algorithm to the study of Mousetrap and Modular Mousetrap, improving recent results. Finally, by means of our algorithm, we study the reformed permutations in Mousetrap, Modular Mousetrap and He Loves Me, He Loves Me Not, attaining new results which give some answers to several questions posed by Cayley and by Guy and Nowakowski in their papers. [ABSTRACT FROM AUTHOR]

Let C be a graph and ƒ ∈ C0(G). It is proved that P(ƒ)¯ = G if R(ƒ)¯ = G and P(ƒ) ≠ ∅. This result generalizes several corresponding results given in [3] and [10]. [ABSTRACT FROM AUTHOR]

In this work, we focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs on a given algebra. In our exposition, we try to survey general results by presenting some meaningful examples; moreover, where possible, we outline the proofs in order to give an insight of the methods and the techniques. We recall some recent results for the problem PosSLP, consisting of deciding if the integer defined by a straight-line program on the ring Z is greater than zero; we discuss some implications in the areas of numerical analysis and strategic games. Furthermore, we propose some methods for reducing Compressed Word Problem from an algebra to another; reductions from trace monoids to the semiring of nonnegative integers are exhibited and polynomial time algorithms for compressed equivalence in monoids related to Dyck reductions are shown. Finally, we consider inclusion problems for context-free languages, proving how in some cases efficient algorithms for these problems benefit from the ability to work with compressed data. [ABSTRACT FROM AUTHOR]

The field scattered by a plane composed of several homogenous layers on a perfectly reflecting plane is generally given by its plane wave expansion (Fourier representation). We here develop another approach to express the field, more suitable for point source illumination. For this, we sum the contributions of modes corresponding to zeros gj (or poles – gj) of the reflection coefficient of the imperfectly reflecting surface, developing an original integral representation and its expansions, valid for arbitrary mode which can be passive or active. Particular attention has been paid to active modes with Regj < 0, and to the vicinity of the mode with gj = – 1. We then obtain novel exact expressions for the field in acoustics and for potentials in electromagnetism. [ABSTRACT FROM PUBLISHER]