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

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]

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]

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

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]

Abstract: Given a strictly increasing sequence s of non-negative integers, filtering a word by s consists in deleting the letters such that i is not in the set . By a natural generalization, denote by , where L is a language, the set of all words of L filtered by s. The filtering problem is to characterize the filters s such that, for every regular language L, is regular. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions. [Copyright &y& Elsevier]

We give a method to decide whether or not a given D0L Parikh set can be recognized in some base. In the positive case the bases can be computed effectively. [ABSTRACT FROM AUTHOR]

Abstract: We study the action of the mapping class group of an oriented genus g surface with n punctures and a disc removed on a Poisson algebra which arises in the combinatorial description of Chern–Simons gauge theory when the gauge group is a semidirect product . We prove that the mapping class group acts on this algebra via Poisson isomorphisms and express the action of Dehn twists in terms of an infinitesimally generated G-action. We construct a mapping class group representation on the representation spaces of the associated quantum algebra and show that Dehn twists can be implemented via the ribbon element of the quantum double and the exchange of punctures via its universal R-matrix. [Copyright &y& Elsevier]

The article defines and investigates the new quasi-Stone algebras (QSA's) class. It characterizes the simple QSA's class and the class of irreducible QSA's which forms a primary class and with various locally finite QSA's. It points out the generation of various Stone algebras and Boolean algebras through a three-element Stone algebra and two-element Boolean algebra.

Let G be a graph of order p, and let a and b and n be nonnegative integers with 1 ≤ a ≤ b, and let g and f be two integer-valued functions defined on V (G) such that a ≤ g(x) ≤ f(x) ≤ b for all x &sin; (G). A (g, f)-factor of graph G is defined as a spanning subgraph F of G such that g(x) ≤ dF (x) ≤ f(x) for each x . V (G). Then a graph G is called a (g, f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a (g, f)-factor. In this paper, we prove that every graph G is a (g, f, n)- critical graph if its minimum degree is greater than p+a+b-2√(a +1)p - bn + 1. Furthermore, it is showed that the result in this paper is best possible in some sense. [ABSTRACT FROM AUTHOR]

The article explores the proof of the infinitude of primes from mathematician Hillel Furstenberg. It avoids topological language and exhibits the real reason that Furstenberg's approach works as viewed by the author. It denotes that if m and r are integers, a set of integers congruent to r mod m exists which refers as arithmetic progression. The author proves that a finite intersection of AP is either empty or infinite and if S is any collection of sets, then a finite intersection of finite unions of sets in S is also a finite union of finite intersections of sets in S.