On the Sequence of Pedal Triangles

Although geometers have studied the properties of triangles for over two thousand years, there still remain problems of interest involving operations performed infinitely often. A given triangle T_0 generates a sequence of triangles T_n where T_(n+1) is the pedal triangle of T_n. This sequence was discussed by Hobson (1897, 1925) but, while his formulae for the transition from T_n to T_(n+1) are correct, those for T_n in terms of T_0 are not. Lacking correct formulae, we experimented numerically, taking the angles of T_0 to be integers in degrees. To our surprise the angles in the pedal sequence became periodic with periods of 12 steps. The explanation of this curious fact led to a general investigation of pedal sequences, revealing that (a) the sequence may stop by degeneration of the triangle to a straight segment, (b) the angles may develop any periodicity, or (c) the sequence may proceed to infinity without any periodicity. We give necessary and sufficient conditions on the angles of T_0 corresponding to these options, and discuss the periodic case in some detail

Tomography of atomic number and density of materials using dual-energy imaging and the Alvarez and Macovski attenuation model

Dual-energy computed tomography and the Alvarez and Macovski [Phys. Med. Biol. 21, 733 (1976)] transmitted intensity (AMTI) model were used in this study to estimate the maps of density (ρ) and atomic number (Z) of mineralogical samples. In this method, the attenuation coefficients are represented [Alvarez and Macovski, Phys. Med. Biol. 21, 733 (1976)] in the form of the two most important interactions of X-rays with atoms that is, photoelectric absorption (PE) and Compton scattering (CS). This enables material discrimination as PE and CS are, respectively, dependent on the atomic number (Z) and density (ρ) of materials [Alvarez and Macovski, Phys. Med. Biol. 21, 733 (1976)]. Dual-energy imaging is able to identify sample materials even if the materials have similar attenuation coefficients at single-energy spectrum. We use the full model rather than applying one of several applied simplified forms [Alvarez and Macovski, Phys. Med. Biol. 21, 733 (1976); Siddiqui et al., SPE Annual Technical Conference and Exhibition (Society of Petroleum Engineers, 2004); Derzhi, U.S. patent application 13/527,660 (2012); Heismann et al., J. Appl. Phys. 94, 2073–2079 (2003); Park and Kim, J. Korean Phys. Soc. 59, 2709 (2011); Abudurexiti et al., Radiol. Phys. Technol. 3, 127–135 (2010); and Kaewkhao et al., J. Quant. Spectrosc. Radiat. Transfer 109, 1260–1265 (2008)]. This paper describes the tomographic reconstruction of ρ and Z maps of mineralogical samples using the AMTI model. The full model requires precise knowledge of the X-ray energy spectra and calibration of PE and CS constants and exponents of atomic number and energy that were estimated based on fits to simulations and calibration measurements. The estimated ρ and Z images of the samples used in this paper yield average relative errors of 2.62% and 1.19% and maximum relative errors of 2.64% and 7.85%, respectively. Furthermore, we demonstrate that the method accounts for the beam hardening effect in density (ρ) and atomic number (Z) reconstructions to a significant extent.S.J.L., G.R.M., and A.M.K. acknowledge funding through the DigiCore consortium and the support of a linkage grant (LP150101040) from the Australian Research Council and FEI Company

Anomalously small wave tails in higher dimensions

We consider the late-time tails of spherical waves propagating on even-dimensional Minkowski spacetime under the influence of a long range radial potential. We show that in six and higher even dimensions there exist exceptional potentials for which the tail has an anomalously small amplitude and fast decay. Along the way we clarify and amend some confounding arguments and statements in the literature of the subject.Comment: 13 page

Effect of crumb rubber on mechanical properties of multi-phase syntactic foams

Syntactic foam is a lightweight and strong material which can be used in marine and aeronautical applications. However, the brittleness of the material limits its application to a broader range. Adding crumb rubber to the syntactic foam can increase its energy absorption capacity. The effect of crumb rubber on the fracture toughness and energy absorption capacity of 2-phase and 3-phase syntactic foam is evaluated under both static and impact loads. The experimental results have shown that the fracture toughness of the 2-phase rubberized syntactic foam increased by 8% while an increase of 22% of its fracture energy was observed. Under quasi-static loads, the 3-phase rubberized syntactic foam showed decreases in the compressive strength and elastic modulus but an increase in the energy absorption capacity as compared to the syntactic foam without crumb rubber. In addition, the impact energy absorption of the 3-phase rubberized syntactic foam increased by 24% as compared to that of the 3-phase syntactic foam without crumb rubber

Solid State Physics

Contains reports on three research projects

New results on group classification of nonlinear diffusion-convection equations

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page

Experiences of using mobile technologies and virtual field tours in Physical Geography: implications for hydrology education

Education in hydrology is changing rapidly due to diversification of students, emergent major scientific and practical challenges that our discipline must engage with, shifting pedagogic ideas and higher education environments, the need for students to develop new discipline specific and transferrable skills, and the advent of innovative technologies for learning and teaching. This paper focuses on new technologies in the context of learning and teaching in Physical Geography and reflects on the implications of our experiences for education in hydrology. We evaluate the experience of designing and trialling novel mobile technology-based field exercises and a virtual field tour for a Year 1 undergraduate Physical Geography module at a UK university. The new exercises are based on using and obtaining spatial data, operation of meteorological equipment (explained using an interactive DVD), and include introductions to global positioning systems (GPS) and geographical information systems (GIS). The technology and exercises were well received in a pilot study and subsequent rolling-out to the full student cohort (∼150 students). A statistically significant improvement in marks was observed following the redesign. Although the students enjoyed using mobile technology, the increased interactivity and opportunity for peer learning were considered to be the primary benefits by students. This is reinforced further by student preference for the new interactive virtual field tour over the previous "show-and-tell" field exercise. Despite the new exercises having many advantages, exercise development was not trivial due to the high start-up costs, the need for provision of sufficient technical support and the relative difficulty of making year-to-year changes (to the virtual field tour in particular). Our experiences are highly relevant to the implementation of novel learning and teaching technologies in hydrology education