In the Manning equation the hydraulic radius can be defined as the cross-section dimension of the shape. In pipe flow the bed shear stress is assumed to be uniformly distributed along the wetted perimeter which cannot be true in open channel flow. Hence, three approximation of the true boundary shear-stress distribution are examined and more practical conveyance depth or resistance radius formu...

Let/j(z),... ,/m(z) be polynomials with complex coefficients, and let their product be of degree n. For any polynomial, let | |/ | | be the maximum of \J[z)\ on the unit circle. We determine constants Cm < 2 for which II/ill " ll/mll ^ C^H/j •••/„,|| for any n. The inequalities are asymptotically sharp as n ->oo. This improves earlier results of Gel'fond and Mahler, who gave the constants e and...

An edge tessellation of the plane is generated by the reflection of some polygon in its edges. We prove that a polygon generating an edge tessellation is exactly one of the following: An equilateral, 30-right, isoceles right, or 120-isoceles triangle, a rectangle, a 60-rhombus, a 60-90-120-kite, or a regular hexagon. The results in this paper were discovered by Millersville University students ...

The Virtual Math Teams (VMT) Project is exploring an approach to the teaching and learning of basic school geometry through a computer-supported collaborative-learning (CSCL) approach. As one phase of a design-based-research cycle of design/trial/analysis, two teams of three adults worked on a dynamic-geometry task in the VMT online environment. The case study presented here analyzes the progre...

In the first part of the paper we will explore the use of arguments from physics in mathematical proof and give some reasons why this approach might be worthwhile. In the second part we will relate this idea to Freudenthal’s concept of local organization. The third part of the paper will present the results of an empirical study conducted in Canada on the classroom use of arguments from physics...

We consider the functions of proof in mathematics from the perspective of the work of the mathematics teacher. The work of proving and the time spent on proving, what can a teacher account it to? How can he or she justify it? We frame that problem in a descriptive theory of teaching and place within that frame the work of scholars who have inquired on the function of proof in mathematics. We ar...

We studied the ultrafast laser-induced coherent phonon oscillation in prismatic shaped gold nanoparticles assembled in monolayer periodic arrays by using the nanosphere lithographic technique. The amplitude and phase of the oscillation observed by ultrafast pump−probe transient spectroscopy is monitored as the wavelength of the dipolar surface plasmon absorption decreases. At a certain waveleng...