We find all the inequivalent simple root systems for the complex reflection groups G12, G24, G25 and G26. Then we give all the non-congruent essential presentations of these groups by generators and relations. The methed used in the paper is applicable to any finite (complex) reflection groups. Introduction. Shephard and Todd classified all the finite complex reflection groups in paper [5]. Lat...

We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.

In this paper, the reflection of ultrasonic guided waves from simple dent in pipes has been investigated using finite element method and the relationship between reflection coefficient of these waves and deformation rate has been determined. Also, the effect of the parameters of wave generation source on the generated wave field has been investigated using normal modes expansion method. At firs...

Let s be a source point and t be a destination point inside an n-vertex simple polygon P . Euclidean shortest paths [Lee and Preparata, Networks, 1984; Guibas et al. , Algorithmica, 1987] and minimum-link paths [Suri, CVGIP, 1986; Ghosh, J. Algorithms, 1991] between s and t inside P have been well studied. Both these kinds of paths are simple and piecewise-convex. However, computing optimal pat...

We discuss the theory and practical issues behind creating reflection models to show the difficulty of the problem. We survey the current approaches towards reflection models for computer graphics to show that even for simple surfaces, the important issues are far from settled. We briefly discuss future directions for research. Finally, we present a case study of a particular type of light refl...

Models of spatially distributed populations and/or communities are still a matter of challenge for the students working in population biology, ecology, environmental sciences and mathematical modelling. An adequate model pattern to describe, model and predict the impact of spatial structure on a community dynamics, as well as the migration processes themselves, is a key problem here. Adequate m...