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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

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]

A commutative hypercomplex system L1(Q,m) is, roughly speaking, a space which is defined by a structure measure (c(A,B, r), (A,B ϵ β(Q)). Such space has been studied by Berezanskii and Krein. Our main purpose is to establish a generalization of convolution semigroups and to discuss the role of the Lévy measure in the Lévy-Khinchin representation in terms of continuous negative definite functions on the dual hypercomplex system. [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]

Abstract: In 1988 Manickam and Singhi conjectured that for every positive integer and every , every set of real numbers whose sum is non-negative contains at least subsets of size whose sums are non-negative. In this paper we make use of Hall’s matching theorem in order to study some numbers related to this conjecture. [Copyright &y& Elsevier]

Abstract: In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums , , . The proof of this result is completely combinatorial and elementary. [Copyright &y& Elsevier]

In this paper we look at linear codes over the Galois ring $$GR(p^{\ell}, m)$$ with the homogeneous weight and we prove that the number of codewords with homogenous weights in a particular residue class modulo p e are divisible by high powers of p. We also state a result for a more generalized weight on linear codes over Galois rings. We obtain similar results for the Lee weights of linear codes over $$\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$$ and we prove that the results we obtain are best possible. The results that we obtain are an improvement to Wilson’s results in [Wilson RM (2003) In: Proceedings of international workshop on Cambridge linear algebra and graph coloring] [ABSTRACT FROM AUTHOR]

Abstract: Let and be sets of nonnegative integers with . Let be a family of subsets of with for each and for any . We prove that when we have the conditions that and ’s are consecutive. We also prove the same bound under the condition instead of the above conditions. Finally, an observation gives us a bound of on when . [Copyright &y& Elsevier]

The article presents several proposed mathematical problems and solutions. Raimond Struble of Santa Monica, California proposed a sequence of complex numbers while Manuel Kauers of the Research Institute for Symbolic Computation offers a problem on Fibonnaci number. Nobuhisa Abe of NBU Attached Senior High School, Saiki, Japan proposes a problem related to integers. Solutions to other mathematical problems are also discussed including secondary inradii, polynomials with three distinct zeros summing to zero, a cousin of Gronwall's inequality and partial fractions.

Presents the mathematical theory in the lesser-known delta-function in number theory. Number of contexts required in analytic number theory; Versions of the classical sieve of Eratorsthenes that produces primes; Presentation of a number of results concerning the delta function; Expressions of the two values of delta function.

The article presents a curious proof of Fermat's little theorem. Any formal power series can be represented in a unique way as a formal product where the coefficients being used are integers. A binomial theorem is not used in the proof of Fermat's little theorem. The author proves that any infinite sequence of integers has an inverse sequence of integers wherein all such sequences split into mutually inverse pairs. An inverse of a certain sequence for which the infinite product has a rich mathematical content is also expressed.

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.

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.