The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG '05), University of Windsor, Canada, 200

Finite Element and Differential Quadrature Methods for Heat Distribution in Rectangular Fins

Presently there are many numerical solution techniques such as finite element method (FEM), differential quadrature method (DQM), finite difference method (FDM), boundary element method (BEM), Raleigh-Ritz method (RRM), etc. These methods have their respective drawbacks. However, FEM and DQM are important techniques among those. The conventional FEM (CFEM) provides flexibility to model complex geometries than FDM and conventional DQM (CDQM) do in spite some of its own drawbacks. It has been widely used in solving structural, mechanical, heat transfer, and fluid dynamics problems as well as problems of other disciplines. It has the characteristic that the solution must be calculated with a large number of mesh points (uniformly distributed) in order to obtain moderately accurate results at the points of interest. Consequently, both the computing time and storage required often prohibit the calculation. Therefore, focus is given to optimize the CFEM. The Optimum FEM (OFEM) has been presented in this thesis to solve heat conduction problems in rectangular thin fins. This method is a simple and direct technique, which can be applied in a large number of cases to circumvent the computational time and complexity. The accuracy of the method depends mainly on the accuracy of the mesh generation (non-uniformly distributed) and stiffness matrix calculation, which is a key of the method. In this thesis, the algorithm for OFEM solution and the optimum mesh generation formula have been developed and presented. The technique has been illustrated with the solution of four heat conduction problems in fins for two types of mesh size distribution (uniformly distributed and non-uniformly distributed). The obtained OFEM results are of good accuracy with the exact solutions. It is also shown that the obtained OFEM results are at least 90% and 7% improved than those of similar published CFEM and ODQM results respectively. This method is a vital alternative to the conventional numerical methods, such as FDM, CFEM and DQM. On the other hand, DQM is suitable for simple geometry and not suitable for practical large-scale problems or on complex geometries. DQM is used efficiently to solve various one-dimensional heat transfer problems. For two-dimensional case, this technique is so far used to solve Poisson’s equation and some fluid flow problems but not the heat conduction problems in fins. Hence, in this thesis, a twodimensional heat conduction problem in a thin rectangular fin is solved using DQM by means of the accurate discretization (for uniformly distributed (CDQM) and non-uniformly distributed (ODQM) mesh size. DQM optimum discretization rule and mesh generation formula have been presented. The governing equations have been discretized according to DQM rule.The technique has been illustrated with the solution of two two-dimentional heat conduction problems in fins. The obtained results show that the DQM results are of good accuracy with the FEM results. Optimum DQM (ODQM) shows better accuracy and stability than CDQM and CFEM. But in some cases, OFEM shows better efficiency than ODQM

A parallel multigrid solver for multi-patch Isogeometric Analysis

Isogeometric Analysis (IgA) is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then. If large spline degrees are considered, one obtains the approximation power of a high-order method, but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers (like Gauss Seidel) does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree. In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF) for the financial support through the DK W1214-04, while the second author was supported by the FWF grant NFN S117-0

Extranoematic artifacts: neural systems in space and topology

During the past several decades, the evolution in architecture and engineering went through several stages of exploration of form. While the procedures of generating the form have varied from using physical analogous form-finding computation to engaging the form with simulated dynamic forces in digital environment, the self-generation and organization of form has always been the goal. this thesis further intend to contribute to self-organizational capacity in Architecture. The subject of investigation is the rationalizing of geometry from an unorganized point cloud by using learning neural networks. Furthermore, the focus is oriented upon aspects of efficient construction of generated topology. Neural network is connected with constraining properties, which adjust the members of the topology into predefined number of sizes while minimizing the error of deviation from the original form. The resulted algorithm is applied in several different scenarios of construction, highlighting the possibilities and versatility of this method