Suberitane sesterterpenoids from the Antarctic sponge Phorbas areolatus (Thiele, 1905)

Following ecological and chemical screenings, the Antarctic sponge Phorbas areolatus was chemically investigated for the first time. Three new suberitane derivatives named isosuberitenone B, 19-episuberitenone B, and isoxaspirosuberitenone were identified together with the known compounds suberitenones A and B, and oxaspirosuberitenone after a thorough inspection of their NMR data. These compounds were found to exhibit moderate cytotoxic activity and their presence in this sponge rules out their use as a chemotaxonomic marker for Suberites sponges.     

What Is a Fraction? Developing Fraction Understanding in Prospective Elementary Teachers

Classroom teachers need a well‐developed deep understanding of fractions and pedagogic practices so they can provide meaningful experiences for students to explore and construct ideas about fractions. This study sought to examine prospective elementary teachers' understandings of fraction by focusing specifically on their use of fractions meanings and interpretations. Results indicated that prospective elementary teachers bring with them to their final methods course a limited understanding of fractions and that experiences in methods courses resulted only in minor improvement of those limited understandings. The limited part‐whole understanding of fractions that prospective elementary teachers entered the course with was resilient. The implications of this study suggest a need for prospective elementary teachers to continue to develop their conceptual understanding of fractions and for changes to the content and instructional strategies of mathematics content courses designed for prospective elementary teachers.     

Efficient remainder rule

Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra.     

On an Inequality of Pual Turan Concerning Polynomials-Ⅱ

Let P(z) be a polynomial of degree n and for any complex number α, let D_αP(z) = nP(z)+(α- z) P′(z) denote the polar derivative of the polynomial P(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.     

An M/M/1 Driven Fluid Queue – Continued Fraction Approach

This paper presents complete solutions of the stationary distributions of buffer occupancy and buffer content of a fluid queue driven by an M/M/1 queue. We assume a general boundary condition when compared to the model discussed in Virtamo and Norros [Queueing Systems 16 (1994) 373–386] and Adan and Resing [Queueing Systems 22 (1996) 171–174]. We achieve the required solutions by transforming the underlying system of differential equations using Laplace transforms to a system of difference equations leading to a continued fraction. This continued fraction helps us to find complete solutions. We also obtain the buffer content distribution for this fluid model using the method of Sericola and Tuffin [Queueing Systems 31 (1999) 253–264].     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Hyperbolic Geometry and the Hillam–Thron Theorem

Every open ball within has an associated hyperbolic metric and Möbius transformations act as hyperbolic isometries from one ball to another. The Hillam–Thron Theorem is concerned with images of balls under Möbius transformation, yet existing proofs of the theorem do not make use of hyperbolic geometry. We exploit hyperbolic geometry in proving a generalisation of the Hillam–Thron Theorem and examine the precise configurations of points and balls that arise in that theorem.This work was supported by Science Foundation Ireland grant 05/RFP/MAT0003     

Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms

The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group. In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion. Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G . September 12, 2000. Final version received: April 16, 2001.