HOPs and COPs: Room frames with partitionable transversals

In this paper, we construct Room frames with partitionable transversals. Direct and recursive constructions are used to find sets of disjoint complete ordered partitionable (COP) transversals and sets of disjoint holey ordered partitionable (HOP) transversals for Room frames. Our main results include upper and lower bounds on the number of disjoint COP transversals and the number of disjoint HOP transversals for Room frames of type 2n. This work is motivated by the large number of applications of these designs

Periodic time dependent Hamiltonian systems and applications

[eng] A dynamical system is one that evolves with time. This definition is so diffuse that seems to be completely useless, however, gives a good insight of the vast range of applicability of this field of Mathematics has. It is hard to track back in the history of science to find the origins of this discipline. The works by Fibonacci, in the twelfth century, concerning the population growth rate of rabbits can be already considered to belong to the above mentioned field. Newton's legacy changed the prism through the humankind watched the universe and established the starting shot of several areas of knowledge including the study of difierential equations. Newton's second law relates the acceleration, the second derivative of the position of a body with the net force acting upon it. The formulation of the law of universal gravitation settled the many body problem, the fundamental question around the field of celestial mechanics has grown. Newton itself solved the two body problem, providing an analytical proof of Kepler's laws. In the subsequent years a number of authors, among of them Euler and Lagrange, exhausted Newton's powerful ideas but none of them was able to find a closed solution of the many body problem. By the end of the nineteenth century, Poincaré changed again the point of view: The French mathematician realized that the many body problem could not be solved in the sense his predecessors expected, however, many other fundamental questions could be addressed by studying the solutions of not quantitatively but by means of their geometrical and topological properties. The ideas that bloomed in Poincaré's mind are nowadays a source of inspiration for modern scientist facing problems located along all the spectrum of human knowledge. Poincaré understood that invariant structures organize the long term behaviour of the solutions of the system. Invariant objects are, therefore, the skeleton of the dynamics. These invariant structures and their linear normal behaviour are to be analyzed carefully and this shall lead to a good insight on global aspects of the phase space. For nonintegrable systems the task of studying invariant objects and their stability is, in general, a problem which is hard to be handled rigorously. Usually, the hypotheses needed to prove specific statements on the solutions of the systems reduce the applicability of the results. This is especially relevant in physical problems: Indeed, we cannot, for instance, choose the mass of Sun to be suficiently small. The advent of the computers changed the way to undertake studies of dynamical systems. The task of writing programs for solving, numerically, problems related to specific examples is, at the present time, as important as theoretical studies. This has two main consequences: On the first hand, more involved models can be chosen to study real problems and this allow us to understand better the relation between abstract concepts and physical phenomena. Secondly, even when facing fundamental questions on dynamics, the numerical studies give us data from which build our theoretical developments. Nowadays, a large number of commercial (or public) software packages helps scientist to study simple problems avoiding the tedious work to master numerical algorithms and programming languages. These programs are coded to work in the largest possible number of different situations, therefore, they do not have the eficiency that programs written specifically for a certain purpose have. Some of the computations presented in this dissertation cannot be performed by using commercial software or, at least, not in a reasonable amount of time. For this reason, a large part of the work presented here has to do with coding and debugging programs to perform numerical computations. These programs are written to be highly eficient and adapted to each problem. At the same time, the design is done so that specific blocks of the code can be used for other computations, that is, there exist a commitment between eficiency and reusability which is hard to achieve without having full control on the code. Under these guiding principles we undertake the study of applied dynamical systems according to the following stages: From a particular problem we get a simple model, then perform a number of numerical experiments that permits us to understand the invariant objects of the system, with that information, we can isolate the relevant phenomena and identify the key elements playing a role on it. Next, we try to find an even simpler model in which we can develop theoretical arguments and produce theorems that, with more effort, can be generalized or related to other problems which, in principle, seem to be difierent to the original one. Paraphrasing Carles Simó, from a physical problem we can take the lift to the abstract world, use theoretical arguments, come out with conclusions and, finally, lift down to the real world and apply these conclusions to specific problems (maybe not only the original one). This methodology has been developed in the last decades over the world when it turned out to outstand among the most powerful approaches to cope with problems in applied mathematics. The group of Dynamical Systems from Barcelona has been one of the bulwarks of this development from the late seventies to the present days. Following the guidelines presented in the previous section, we concern with several problems, mostly from the field of celestial mechanics but we also deal with a phenomenon coming from high energy physics. All these situations can be modeled by means of periodically time dependent Hamiltonian systems. To cope with those investigations, we develop software which can be used to perform computations in any periodically perturbed Hamiltonian system. We split the contents of this dissertation in two parts. The first one is devoted to general tolos to handle periodically time dependent Hamiltonians, even though we fill this first part with a number of illustrating examples, the goal is to keep the exposition in the abstract setting. Most of the contents of Part I deal with the development of software used to be applied in the second part. Some of the software has not been applied to the specific contents of Part II, this is left for future work. We also devote a whole chapter to some theoretical issues that, while are motivated by physical problems, they fall out of the category of periodic time dependent Hamiltonians. This splitting of contents has the intention of reecting, somehow, the basic methodological principles presented in the previous paragraph, keeping separated the abstract and the physical world but keeping in mind the lift

Optics of polyhedra: from invisibility cloaks to curved spaces

Transformation optics is a new and highly active field of research, which employs the mathematics of differential geometry to design optical materials and devices with unusual properties.Probably the most exciting device proposed by transformation optics is the invisibility cloak. However, transformation optics can be employed in many other cases, for example when designing a setup mimicking a curved space-time phenomena in a lab. The purpose of this thesis is to establish a new concept of transformation optics: instead of designing complicated materials, we will design our devices using standard optical elements such as lenses or optical wedges. We will stretch the possibilities of geometrical optics by providing a novel description of imaging due to combinations of tilted lenses and the theory of invisibility with ideal thin lenses. This theory will be then applied to design novel transformation optics devices, namely the omnidirectional lens and a number of ideal lens invisibility cloaks. We also present a new approach of building optical systems that simulate light-field propagation in both 2D and 3D curved spaces. Instead of building the actual curved space, the light field is regarded to travel in the respective unfolded net, whose edges are optically identified, using the so-called space-cancelling wedges. By deriving a full analytical solution of the Schrodinger equation, we will also investigate a quantum motion in a number of two dimensional compact surfaces including the Klein bottle, Mobius strip and projective plane. We will show that the wavefunction exhibits perfect revivals on these surfaces and that quantum mechanics on many seemingly unphysical surfaces can be realised as simple diffraction experiments. Our work therefore offers a new concept of optical simulation of curved spaces, and potentially represents a new avenue for research of physics in curved spaces and simulating otherwise inaccessible phenomena in non-Euclidean geometries. We conclude with a summary of potential future projects which lead naturally from the results of this thesis

Statistical inference with paired observations and independent observations in two samples

A frequently asked question in quantitative research is how to compare two samples that include some combination of paired observations and unpaired observations. This scenario is referred to as `partially overlapping samples'. Most frequently the desired comparison is that of central location. Depending on the context, the research question could be a comparison of means, distributions, proportions or variances. Approaches that discard either the paired observations or the independent observations are customary. Existing approaches evoke much criticism. Approaches that make use of all available data are becoming more prominent. Traditional and modern approaches for the analyses for each of these research questions are reviewed. Novel solutions for each of the research questions are developed and explored using simulation. Results show that proposed tests which report a direct measurable difference between two groups provide the best solutions

Squared Law Algorithms: Theory and Applications.

This dissertation focuses on a new approach for a hardware implementation of the cyclic convolution operation. The cyclic convolution operation is the core of several functions used in applications related to digital signal processing and error control. Since the operation is multiplication intensive and the cost of a multiplication operation is very high, most of the present research effort attempts to reduce the number of multiplications. Our approach, however, aims at obtaining an efficient implementation by relying on the properties of the special case of multiplication, namely, the squaring operation. Due to the properties exhibited by the squaring operation the hardware cost and time delay of a squarer unit is both cheaper and faster than that of a multiplication unit. This is true for both memory and non-memory based implementations. In this dissertation we have developed all the necessary theory required to express the cyclic convolution of two n-point sequences, where n is a power of 2, in terms of the elementary arithmetic operations add, square, and subtract. Our algorithms require fewer squaring operations than multiplication operations required by a traditional implementation of the cyclic convolution operation, do not introduce any round-off errors, place no restriction on word length, and are valid when the number of points to be convolved is a power of two. We then clearly demonstrate that our algorithms are also more hardware efficient for both memory and non-memory based implementations. Further, schemes to multiply two numbers based on the cyclic convolution operation are presented. Finally, efficient ways of computing the squaring operation when arithmetic is performed in modular rings are developed

Applied Mathematics to Mechanisms and Machines

This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section “Engineering Mathematics”. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics