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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 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]

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: 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: 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]

This article provides a survey of some basic results in algebraic number theory and applies this material to prove that the cyclotomic integers generated by a seventh root of unity are a unique factorization domain. Part of the proof uses the computer algebra system Maple to find and verify factorizations. The proofs use a combination of historic and modern techniques and some attempt has been made to discuss the history of this material. [ABSTRACT FROM AUTHOR]

We prove that for almost all real primitive characters χd of modulus |d|, the least positive integer nχd at which χd takes a value not equal to 0 and 1 satisfies nχd ≪ log|d|, and give a quite precise estimate on the size of the exceptional set. [ABSTRACT FROM AUTHOR]

We investigate the class of ±1 polynomials evaluated at q defined as: A(q) = {ϵ0 + ϵ1q + … + ϵmqm : ϵi Є {-1, 1}} and usually called spectrum, and show that, if q is the root of the polynomial xn - xn-1 - … - xk+1 + xk + xk-1 + … + x + 1 between 1 and 2, and n > 2k + 3, then A(q) is discrete, which means that it does not have any accumulation points. [ABSTRACT FROM AUTHOR]

Let T 2 k+1 be the set of trees on 2 k+1 vertices with nearly perfect matchings and α( T) be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T 2 k+1. Specifically, 10 trees T 2, T 3,⋯, T 11 and two classes of trees T(1) and T(12) in T 2 k+1 are introduced. It is shown in this paper that for each tree T , T ∈ T(1) and T , T ∈ T(12) and each i, j with 2 ≤ i < j ≤ 11, α( T ) = α( T ) > α( T i ) > α( T j ) > α( T ) = α( T ). It is also shown that for each tree T with T ∈ T 2 k+1 ( T(1) ∪ { T 2,T 3,⋯, T 11} ∪ T(12)), α( T ) > α( T). [ABSTRACT FROM AUTHOR]

The article traces the contributions of Pierre Hérigone in making mathematics algebraic rather than geometric. It discusses the algebraic concepts and notation of Hérigone which can be found in his volume on algebra. It introduces his universal symbolic language applicable to all branches of mathematics. It presents his algebraic procedures and subsections on identities, equations, geometric constructions and ambiguous equations. His influence on later mathematicians including Isaac Barrow and Gottfried Wilhelm von Leibniz is also discussed.