The Finite Fourier Transform

We present "one-dimensional" Fourier theory on commutative groups T hH , 0 h h < X , 0< H h X within the framework of the so-called calculus on measure chains (or time scales). Depending on certain values of the graininess h and length H of the group the four classical types of Fourier transform are covered: Fourier integral ( T 0 X = R ), Fourier series ( T 1 X = Z ), Fourier analysis of periodic functions ( T 0,2 ~ = S 1 (0) unit circle) and discrete Fourier transform ( T 1 N = Z N ). We will present Fourier theory on these groups in a unified manner. This also allows to closely track the roles of the graininess h and length H of the group--especially for h M 0 and H M X . In the final part of the paper, we investigate the solution of a fundamental equation on T hH , which can be considered as a generalization of the Gauss function. It finally leads to a version of the Heisenberg uncertainty principle, which extends the classical one, valid for T 0 X = R , to the case T hH , where either h >0 or H < X .

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e&gt;0 there exists a d&gt;0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

In this work, we propose a method for the derivation of prolate spheroidal wave functions (PSWFs) and Slepian functions on continuous and disjoint intervals on the real number line. The proposed method uses Fourier series to obtain a closed-form approximation for Slepian functions on the real line. With this closed-form expression, Slepian functions can be evaluated at arbitrary points in the region of interest with high accuracy. The conventional method uses properties of the Slepian concentration problem to evaluate PSWFs on finite number of points in an interval. The conventional method is computationally expensive and does not allow for easy storage. By approximating an interval containing regions of interest as periodic, we express the Slepian concentration problem as a finite dimensional problem using the Fourier series domain. Solutions to the Slepian concentration problem in this form are Fourier series coefficients corresponding to the Slepian functions. Reconstruction in Fourier series basis, scaling and subsequent truncation provides the closed-form expression for the Slepian problem. Upon comparison with PSWFs obtained by the conventional method, we find negligible difference.

In our research, we explore a coupled mKdV system within the Yijima–Oikawa long-wave–short-wave hierarchy, involving both real and complex dynamical variables. By establishing an explicit Lax pair, Darboux–Bäcklund transformation, we proved the generalized Bianchi’s permutability theorem for the coupled system utilizing Riccati representation of the Lax pair. We then uncover new solutions in unique closed forms, showcasing the intricate relationship between the real and complex variables. These solutions encompass diverse forms such as one-soliton, stationary and moving breathers, and two-soliton solutions, each demonstrating distinct properties. Notably, the dynamics reveal intriguing behavior where the complex potential oscillates precisely on the unit circle, contingent on the initial seed solution, thus, a novel dynamical Pythagorean triple associated is detected.

This book considers various spaces and algebras made up of functions, measures, and other objects-situated always on one or another locally compact abelian group, and studied in the light of the Fourier transform. The emphasis is on the objects themselves, and on the structure-in-detail of the spaces and algebras. A mathematician needs to know only a little about Fourier analysis on the commutative groups, and then may go many ways within the large subject of harmonic analysis-into the beautiful theory of Lie group representations, for example. But this book represents the tendency to linger on the line, and the other abelian groups, and to keep asking questions about the structures thereupon. That tendency, pursued since the early days of analysis, has defined a field of study that can boast of some impressive results, and in which there still remain unanswered questions of compelling interest. We were influenced early in our careers by the mathematicians Jean-Pierre Kahane, Yitzhak Katznelson, Paul Malliavin, Yves Meyer, Joseph Taylor, and Nicholas Varopoulos. They are among the many who have made the field a productive meeting ground of probabilistic methods, number theory, diophantine approximation, and functional analysis. Since the academic year 1967-1968, when we were visitors in Paris and Orsay, the field has continued to see interesting developments. Let us name a few. Sam Drury and Nicholas Varopoulos solved the union problem for Helson sets, by proving a remarkable theorem (2.1.3) which has surely not seen its last use.

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

This study proposes a tone experiment using multiple smartphones or tablets to assist Fourier series teaching combined with a pedagogical strategy of the Visible Thinking framework. Fourier analysis is widely applied in many fields; therefore, it is an important subject in many professional courses of colleges, especially in engineering. While Fourier analysis is the core theoretical foundation of various experimental techniques, undergraduate courses often teach Fourier analysis from a mathematical perspective, leaving students with little intuition about how Fourier analysis works. The proposed tone experiment requires teamwork for students to manipulate smartphones to synthesize harmonic tones to simulate a periodic wave so that students can intuitively understand the physical meaning of the Fourier series formula. In addition, we conducted a pedagogical study on this course to evaluate the effects. The proposed method successfully improved the learning outcomes of beginner students and reduced the anxiety associated with learning Fourier analysis. Compared with the class that did not use the proposed tone experiment, the quiz scores improved by an average of 3.77 points. The students gave positive feedback and comments on the proposed experiments and teaching methods.

The origin of structural optimization can be traced back several centuries (Ref. 1), but it is only during the last two decades or so, with the advent of modern computers, that it has evolved into a mature discipline in engineering. The literature published in this field is extensive and it can be reasonably discussed here only by referring to some recently written articles and textbooks found in Refs. 2–4. The major part of the articles deal with such numerical optimization techniques in finite-dimensional problems as optimality criteria or mathematical programming methods, but considerable efforts have also been made in applying the control theory approach to distributed parameter structural systems. The finite element method is commonly used in analyzing load supporting structures and there is usually a finite-dimensional optimization problem associated with it. In this chapter truss design problems, which by nature belong to this class, are considered. Various mainly nonlinear programming approaches have been developed to numerically solve scalar problems where the number of design variables and constraints is constantly increasing.

Generating Function Method for the solution of point reactor kinetic equations

Large space buildings play a significant role in modern society because of their environmental advantages and market value. While the zonal model is promising for the efficient and rapid evaluation of the stratified thermal environment, there is a lack of a reasonable and convenient zoning strategy with the advent of modern computing. This paper presents a universal and practical zonal model, in which a simplified momentum equation is applied to consider air momentum preservation, transformation, and dissipation. Hence, the zoning structure is generalized and flexible. Moreover, limiting the dimensionless temperature constraint between adjacent zones establishes the connection between thermal nonuniformity and zoning results automatically. Simultaneously, the dimension and number of zones should be restricted within reasonable ranges to satisfy the characteristics of zonal simulation and reach the criteria of convergence. To further explore and validate the zonal model, a reduced-scale experimental model was constructed to replicate the thermal stratification in a mechanically ventilated large space by considering many crucial realistic factors. A particle image velocimetry (PIV) measurement was then conducted to visualize the airflow pattern and support the partition of zones. The results showed that the zonal simulation with adaptive zoning method can realize a similar accuracy with fewer zones and exhibit a better tolerance for zoning results compared to the conventional empirical zoning method. Furthermore, a case study of an atrium was performed to demonstrate the practicality and efficiency of the method for long-term dynamic simulations of complex thermal environments and building energy use.

The field of Geomorphology covers the essential link between climate and geological processes such as tectonics. Because both of these processes operate on planetary scales, and over million year periods, landscape evolution models must, by necessity, do the same. With the advent of modern computing and the reduction in computational complexity of the Stream Power Law algorithm (SPL), it has become much easier to conduct investigations of landscape evolution on these scales. By doing so we can test model interactions between Earth system processes such as geodynamics, weathering, sediment flux, and erosion. In this work we aim to conduct landscape evolution modelling with the SPL algorithm on pre-industrial Earth, using high resolution climate models (CMIP &amp;#8211; Coupled Model Intercomparison Project) and topographic maps with uplift histories as input. This model has already been used for planetary scale modelling on ancient Mars, and now we aim to use it to conduct a broad sensitivity analysis of the landscape evolution of pre-industrial Earth. We will compare the model outputs to established datasets and to other landscape evolution studies to best constrain the input parameters of the model (e.g., incision coefficient) to reproduce known water and sediment fluxes for the period. Once the model is calibrated, we aim to use it to look at periods of deep time where landscape evolution was perturbed by tectonic and climate excursions such as supercontinent assembly, and transitions to and from icehouse climate states. As with the pre-industrial study, this work would also include coupling to climate models but furthermore would be coupled to a global geodynamic model to produce topography, reducing reliance on paleo-topographical maps and allowing for comparison to previous studies that used those maps as topographic input.

Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.

Pressure buildup/annular pressure buildup in subsea oil and gas equipment occurs primarily due to the thermal expansion of trapped liquids. With the advent of modern computers, it has become increasingly possible to numerically analyze such problems with commercial codes available in the market. The objective of the present study is to propose a methodology for numerical prediction of structural damage in subsea oil and gas equipment due to pressure buildup. A judicious combination of computational fluid dynamics (CFD) with structural finite element analysis code has been used to perform a sample numerical analysis that is truly representative of a wide class of problems encountered in subsea oil and gas applications. The mitigation of trapped pressure is one among the prime areas of concern in the subsea oil and gas industry. In the present study, CFD analysis is used to determine the maximum pressure buildup due to the thermal expansion of trapped liquids in small leak tight enclosed volumes with rigid walls and the pressure obtained is used as a boundary condition for the structural analysis. In a nutshell, the analysis has been split into three steps (1) a steady-state CFD analysis to determine the temperature distribution within the oil and gas equipment under consideration, (2) the temperature contours obtained from the steady-state analysis are imposed as a boundary condition for the transient analysis to calculate the trapped pressure in the small volumes of interest and finally and (3) a structural analysis is used to determine the damage to the oil and gas equipment. The methodology adapted is similar to a one-way coupled fluid structure interaction analysis, but provides the added advantage of a significant reduction in computational cost. In the present study, the proposed methodology has been extended to a subsea Christmas tree (XT) and the pressure buildup in the hydraulic lines has been calculated. The results obtained using the present technique has been compared with analytical solution. The proposed numerical technique can be applied to any subsea or surface oil and gas equipment where pressure buildup due to trapped volume is a major issue. The findings of this study can help for better understanding of pressure buildup in trapped volumes within subsea/surface oil and gas equipment. This study can be applied to predict the thermal expansion of trapped volumes in subsea XTs, manifolds, pipe line end manifolds (PLEM) and pipe line end termination (PLET) units.

It is now almost 25 years since Edward Lorenz, thinking about weather prediction, reported chaotic numerical solutions of a deterministic differential equation, and Ruelle and Takens speculated on the relevance of deterministic chaos to fluid turbulence. Stimulated in part by the advent of modern computers much has happened since then. An indication of the explosion of scientific interest is the proportionately larger explosion in popular interest: the chaologist in Jurassic Park now lines up alongside the likes of Superman and Indiana Jones as a Hollywood “good guy” and it is no longer a major faux pas to admit at parties that you study “chaos theory”.

The use for mathematical models of natural phenomena has underpinned science and engineering for centuries, but until the advent of modern computers and computational methods, the full utility of most of these models remained outside the reach of the engineering communities. Since World War II, advances in computational methods have transformed the way engineering and science is undertaken throughout the world. Today, theories of mechanics of solids and fluids, electromagnetism, heat transfer, plasma physics, and other scientific disciplines are implemented through computational methods in engineering analysis, design, manufacturing, and in studying broad classes of physical phenomena. The discipline concerned with the application of computational methods is now a key area of research, education, and application throughout the world. In the early 1980's, the International Association for Computational Mechanics (IACM) was founded to promote activities related to computational mechanics and has made impressive progress. The most important scientific event of IACM is the World Congress on Computational Mechanics. The first was held in Austin (USA) in 1986 and then in Stuttgart (Germany) in 1990, Chiba (Japan) in 1994, Buenos Aires (Argentina) in 1998, Vienna (Austria) in 2002, Beijing (China) in 2004, Los Angeles (USA) in 2006 and Venice, Italy; in 2008. The 9th World Congress on Computational Mechanics is held in conjunction with the 4th Asian Pacific Congress on Computational Mechanics under the auspices of Australian Association for Computational Mechanics (AACM), Asian Pacific Association for Computational Mechanics (APACM) and International Association for Computational Mechanics (IACM). The 1st Asian Pacific Congress was in Sydney (Australia) in 2001, then in Beijing (China) in 2004 and Kyoto (Japan) in 2007.The WCCM/APCOM 2010 publications consist of a printed book of abstracts given to delegates, along with 247 full length peer reviewed papers published with free access online in IOP Conference Series: Materials Science and Engineering. The editors acknowledge the help of the paper reviewers in maintaining a high standard of assessment and the co-operation of the authors in complying with the requirements of the editors and the reviewers. We also would like to take this opportunity to thank the members of the Local Organising Committee and the International Scientific Committee for helping make WCCM/APCOM 2010 a successful event. We also thank The University of New South Wales, The University of Newcastle, the Centre for Infrastructure Engineering and Safety (CIES), IACM, APCAM, AACM for their financial support, along with the United States Association for Computational Mechanics for the Travel Awards made available. N. Khalili S. Valliappan Q. Li A. Russell 19 July 2010 Sydney, Australia