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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]

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]

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]

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]

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]

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]

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]

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]

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: 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]

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]

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]

Abstract: For any odd prime , we present a reflection theorem which for is the Scholz Reflection Theorem. We obtain some formulas for the -rank and the -rank(), where is a number field. With the help of our reflection theorem, we establish relations between the -divisibility of the order of and the -divisibility of the class number of some algebraic number fields . [Copyright &y& Elsevier]

Abstract: We show how to calculate three low degree set-theoretic generators (i.e., algebraic surfaces) for all rational space curves of low degree (degree ) as well as for all higher degree rational space curves where at least one element of their -basis has degree 1 from a -basis of the parametrization. In addition to having low degree, at least two of these surface generators are always ruled surfaces. Whenever possible we also show how to compute two set-theoretic complete intersection generators for these rational space curves from a -basis of their parametrization. [Copyright &y& Elsevier]

Abstract: We give the values of the irreducible complex characters of the group where ℓ is an integer >1. [Copyright &y& Elsevier]

In the paper we consider deemed three mock theta functions introduced by Hikami. We have given their alternative expressions in double summation analogous to Hecke type expansion. In proving we also give a new Bailey pair relative to a². I presume they will be helpful in getting fundamental transformations. [ABSTRACT FROM AUTHOR]

Due to the complexity of Einstein's equations, it is often natural to study a question of interest in the framework of a restricted class of solutions. One way to impose a restriction is to consider solutions satisfying a given symmetry condition. There are many possible choices, but the present article is concerned with one particular choice, which we shall refer to as Gowdy symmetry. We begin by explaining the origin and meaning of this symmetry type, which has been used as a simplifying assumption in various contexts, some of which we shall mention. Nevertheless, the subject of interest here is strong cosmic censorship. Consequently, after having described what the Gowdy class of spacetimes is, we describe, as seen from the perspective of a mathematician, what is meant by strong cosmic censorship. The existing results on cosmic censorship are based on a detailed analysis of the asymptotic behavior of solutions. This analysis is in part motivated by conjectures, such as the BKL conjecture, which we shall therefore briefly describe. However, the emphasis of the article is on the mathematical analysis of the asymptotics, due to its central importance in the proof and in the hope that it might be of relevance more generally. The article ends with a description of the results that have been obtained concerning strong cosmic censorship in the class of Gowdy spacetimes. [ABSTRACT FROM AUTHOR]

It is proved that a ring R in which for every x ϵ R there exists a (and hence the smallest) natural number n(x) > 1 such that xn(x) = x is always a Smarandache Ring. Two examples are provided for justification. [ABSTRACT FROM AUTHOR]

For relatively prime positive integers $u_0$ and $r$, we consider the arithmetic progression ${u_k:=u_0+kr}_{k=0}^n$. par Define $L_n:=operatorname {lcm}{u_0, u_1, ..., u_n}$ and let $age 2$ be any integer. In this paper, we show that for integers $alpha , rgeq a$ and $ngeq 2alpha r$, we have $$L_ngeq u_0r^{alpha +a-2}(r+1)^n.$$ In particular, letting $a=2$ yields an improvement to the best previous lower bound on $L_n$ (obtained by Hong and Yang) for all but three choices of $alpha , rgeq 2$. [ABSTRACT FROM AUTHOR]

The $w$-index of a surface link $F$ is the minimal number of the triple points of surface braids representing $F$. In this paper, for a given $3$-cocycle, we consider the minimal number of the $w$-indices of surface links whose quandle cocycle invariants associated with $f$ are non-trivial, and denote it $omega (f)$. In particular, we show that $omega (theta _3)=6$ and $omega (theta _p)geq 7$, where $theta _n$ is Mochizuki's $3$-cocycle of the dihedral quandle of order $n$ and $p$ is an odd prime integer $not =3$. As a consequence, for a given non-negative integer $g$, there are surface knots with genus $g$ with the $w$-index $6$. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over and establish necessary conditions for the existence of this kind of irreducible pentanomials. Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties. [Copyright &y& Elsevier]

Abstract: In this paper, we study potential convergence of modulus patterns. A modulus pattern is convergent if all complex matrices in (i.e. all matrices with modulus pattern ) are convergent. A modulus pattern is potentially (absolutely) convergent if there exists a (nonnegative) convergent matrix in . We also introduce types of potential convergence that correspond to diagonal and D-convergence, studied in [E. Kaszkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhauser, 2000]. Convergent modulus patterns have been completely characterized by Kaszkurewicz and Bhaya [E. Kaszkurewicz, A. Bhaya, Qualitative stability of discrete-time systems, Linear Algebra Appl. 117 (1989) 65–71]. This paper presents some techniques that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for modulus patterns is given. Complete characterizations of all introduced types of potential convergence for modulus patterns are also presented. [Copyright &y& Elsevier]

Let F be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k + 1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in F, f (k) and g(k) share the set S = {a, b}, then F is normal in D. The condition that the zeros of functions in F have multiplicity at least k + 1 cannot be weakened. [ABSTRACT FROM AUTHOR]

We obtain the system of recurrence relations for rational approximations of the Euler constant generalizing the recurrence relations obtained earlier by Aptekarev with coauthors. The leading coefficient of the recurrence relations of this system is 1, which can be used to verify that the generated numbers are integers. [ABSTRACT FROM AUTHOR]

Abstract: For integers and with and , let denote the family of 2-edge-connected graphs such that for every bond with two or three edges, each component of has at least vertices. In this note we get: (1) If and , then is supereulerian if and only if cannot be contracted to some well classified special graphs. (2) If , and , then , the line graph of , is Hamilton-connected if and only if . Our results extend some earlier results in [P.A. Catlin, X.W. Li, Supereulerian graphs of minimum degree at least 4, J. Adv. Math. 28 (1999) 65–69], [H.J. Broersma, L.M. Xiong, A note on minimum degree conditions for supereulerian graphs, Discrete Appl. Math. 120 (2002) 35–43] and [D.X. Li, H.-J. Lai, M.Q. Zhan, Eulerian subgraphs and hamilton-connected line graphs, Discrete Appl. Math. 145 (2005) 422–428] by Catlin and Li, by Broersma and Xiong, and by Li, Lai and Zhan. [Copyright &y& Elsevier]

Abstract: Let A be a noetherian commutative -algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck–Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If , we show that the vanishing of its Euler class in the corresponding Grothendieck–Witt group is a necessary and sufficient condition for P to have a free factor of rank one. If d is odd, we also get some results in that direction. If A is regular, we show that the Chow–Witt groups defined by Morel and Barge appear naturally as some homology groups of a Gersten-type complex in Grothendieck–Witt theory. From this, we deduce that if then the vanishing of the Euler class of P in the corresponding Chow–Witt group is a necessary and sufficient condition for P to have a free factor of rank one. [Copyright &y& Elsevier]

Given a function f : E → R with E ⊂ &#x211D:n, we explain how to decide whether f extends to a ⊂m function F on ℝn. If E is finite, then one can efficiently compute an F as above, whose Cm norm has the least possible order of magnitude (joint work with B. Klartag). [ABSTRACT FROM AUTHOR]

The article presents several mathematical problems and solutions. These include works on positive integers as proposed by Erwin Just from Bronx Community College of the City University of New York, the usual gamma function as proposed by M. L. Glasser from Clarkson University, and partitioning values of a polynomial into sets of equal sum as proposed by Marian Tevita from the National College in Romania. Also explored are oddly triangulated polygons and the characterization of permutation matrices

Abstract: We give an alternative computation of the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field (Theorem 2), a particular case of results of Movahhedi–Nguyen Quang Do. This short Note is motivated by the paper [J. Jossey, Galois 2-extensions unramified outside 2, J. Number Theory 124 (2007) 42–76] and, at this occasion, we bring into focus some classical technics of abelian ℓ-ramification which, unfortunately, are often ignored, especially those developed by J.-F. Jaulent with the ℓ-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify the study of such subjects; for instance, our proof of Theorem 2 generalizes the purpose of Jossey''s paper in such a way using a result of Herfort–Zalesskii. This Note is mainly an attempt of clarification about ℓ-rationality. [Copyright &y& Elsevier]

A numerical algorithm for calculating the generalized Mittag-Leffler Eα,≈(z) function for arbitrary complex argument z and real parameters α > 0 and β ϵ ℝ is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm. [ABSTRACT FROM AUTHOR]

In this paper we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid (2006). Specifically, we show that generically there are no Cannon--Thurston maps from the Gromov boundary to Thurston's boundary of Teichmüller space. [ABSTRACT FROM AUTHOR]

We investigate models for programming and specifying in which higher-order functions and nondeterminacy (both demonic and angelic) coexist. The models are built using predicate transformers, binary multirelations, state transformers, and free lattices over a poset. We show there exist suitable models in each approach, and that they are isomorphic. The models support an algebra of nondeterminacy which we use to prove that the classical list fusion law holds even in the presence of nondeterminacy. [ABSTRACT FROM AUTHOR]