P-, I-, g-, and D-Angles in Normed Spaces

The notion of angles is known in a vector space equipped with an inner product, but not well established in a vector space equipped only with a norm. In this note, we shall develop some notions of angles between two vectors in a normed space and discuss their properties

Using the History of Circle and Parabolic Segment Areas as Learning Alternatives in Integral

This article will present some classic problems in the Ancient Greece period: the ratio of the areas of two circles problem solved by Eudoxus and the area of a parabola segment problem solved by Archimedes. These problems can be used as alternative teaching resources to give the students an early understanding of the integral concept. This article focuses on finding alternatives for teaching integral material through theorems and historical understanding without calculus knowledge. This study used a systematic literature review method to analyze the mathematical content and the historical influences on their problem-solving methods. The literature sources were indirect sources such as journals, books, and other written literature. The results show that Eudoxus' principle has been a special limit problem since the period, helping solve the ratio of the areas of two circles problem, and there has been a special case of infinite geometric series solving the area of parabolic segment problem. This article gives some recommendations for the teachers at the end of the article, on how to give a representation of the propositions discussed in this article to the students so the students can understand the connections between the prior area problem (in which the area is bounded by its line segments) and the integral concept which will be learned

Crossed products by endomorphisms and reduction of relations in relative Cuntz-Pimsner algebras

Starting from an arbitrary endomorphism \alpha of a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,\alpha) but also on the choice of an ideal orthogonal to kernel of \alpha. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C*-correspondence to the 'smallest' C*-correspondence yielding the same relative Cuntz-Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to the reduction of C*-dynamical systems and allow us to establish a coincidence between relative Cuntz-Pimsner algebras and crossed products introduced.Comment: The article is based on papers arXiv:math.OA/0703801 and arXiv:math.OA/0704.3811, and in essence forms their unification, refinement and developmen

The boundedness of Bessel-Riesz operators on Morrey spaces

In this paper, we shall discuss about Bessel-Riesz operators. Kurata et al. have investigated their boundedness on generalized Morrey spaces with weight. The boundedness of these operators on Lebesgue spaces and Morrey spaces will be reproved using a different approach. Moreover, we also find the norm of the operators are bounded by the norm of the kernel