The approximate value of a given sum is obtained through the development of some results of inequalities. Inequalities are also used to show how the concepts of sum and integral are related. (MP)

Results are investigated when the axioms for a group have been altered. (DT)

Asserts that when college mathematics is taught as one of the liberal arts, Euclid's Elements of Geometry, which is one of the oldest math textbooks, may still be the best, because it helps students understand what is "lovely about mathematics". Suggests that "Elements of Geometry" promotes the mental exercise that comes from…

Described are some major developments in the history of ideas in which mathematics has played a central role. The discussion is organized around four key features of mathematics: certainty, applicability, more than one geometry, and opposition. (MNS)

Summarizes the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards document and suggests 10 implications for college mathematics. (MKR)

Offers an extended review of the book "Advanced Mathematical Thinking." Discusses each chapter, details connections and reactions, and responds in the context both of research in mathematics education and of writing about mathematics. (MKR)

Outlines a three-step approach to proof writing: exploration, explanation, and formalization. Provides a detailed example of its use. Offers reflections on this approach and adaptations to it. (Author/MKR)

Direct sum decomposition of Abelian groups appears in almost all textbooks on algebra for undergraduate students. This concept plays an important role in group theory. One simple example of this decomposition is obtained by using the kernel and range of a projection map on an Abelian group. The aim in this pedagogical note is to establish a direct…

A semigroup G is a group if it has a left identity and every element has a left inverse. The purpose of this note is to weaken this condition further in two different ways. A semigroup G with an identity e is a group if every element x in G has an inverse. It is well known that this statement can be weakened. A semigroup G with a left identity e…

Closed forms are computed for parametric integrals, generally using induction formulas. It is shown that these integrals can be core activities, mixing hand-work, computations with a computer algebra system and experimental mathematics with an interactive website.

A criterion (formula) for the termination of a continued fraction expansion leading to the solution of a standard differential eigenvalue problem from mathematical physics is presented. The criterion generates the eigenvalues in any specific case and is illustrated by elementary examples yielding well-known polynomial eigenfunctions. This…

In this paper the power of the [gamma] algorithm for obtaining the dual of a given cone and some of its multiple applications is discussed. The meaning of each sequential tableau appearing during the process is interpreted. It is shown that each tableau contains the generators of the dual cone of a given cone and that the algorithm updates the…

In this article, an alternative multiplication method for rhotrices is proposed. The method establishes some relationships between rhotrices and matrices. This article has discussed a modified multiplication method for rhotrices. The method has a direct relationship with matrix multiplication, and so rhotrices under this multiplication procedure…

The familiar property of integral of f (x) = 1/x led to the discovery of other functions with this property.

This article looks at how a particular curve associated with tangents of an exponential function is a copy of the exponential itself.