A Quaternionic Version Theory related to Spheroidal Functions

In dieser Arbeit wird eine neue Theorie der quaternionischen Funktionen vorgestellt, welche das Problem der Bestapproximation von Familien prolater und oblater sphäroidalen Funktionen im Hilberträumen behandelt. Die allgemeine Theorie beginnt mit der expliziten Konstruktion von orthogonalen Basen für Räume, definiert auf sphäroidalen Gebieten mit beliebiger Exzentrizität, deren Elemente harmonische, monogene und kontragene Funktionen sind und durch die Form der Gebiete parametrisiert werden. Eine detaillierte Studie dieser grundlegenden Elemente wird in dieser Arbeit durchgeführt. Der Begriff der kontragenen Funktion hängt vom Definitionsbereich ab und ist daher keine lokale Eigenschaft, während die Begriffe der harmonischen und monogenen Funktionen lokal sind. Es werden verschiedene Umwandlungsformeln vorgestellt, die Systeme harmonischer, monogener und kontragener Funktionen auf Sphäroiden unterschiedlicher Exzentrizität in Beziehung setzen. Darüber hinaus wird die Existenz gemeinsamer nichttrivialer kontragener Funktionen für Sphäroide jeglicher Exzentrizität gezeigt. Der zweite wichtige Beitrag dieser Arbeit betrifft eine quaternionische Raumfrequenztheorie für bandbegrenzte quaternionische Funktionen. Es wird eine neue Art von quaternionischen Signalen vorgeschlagen, deren Energiekonzentration im Raum und in den Frequenzbereichen unter der quaternionischen Fourier-Transformation maximal ist. Darüber hinaus werden diese Signale im Kontext der Spektralkonzentration als Eigenfunktionen eines kompakten und selbstadjungierteren quaternionischen Integraloperators untersucht und die grundlegenden Eigenschaften ihrer zugehörigen Eigenwerte werden detailliert beschrieben. Wenn die Konzentrationsgebiete beider Räume kugelförmig sind, kann der Winkelanteil dieser Signale explizit gefunden werden, was zur Lösung von mehreren eindimensionalen radialen Integralgleichungen führt. Wir nutzen die theoretischen Ergebnisse und harmonische Konjugierten um Klassen monogener Funktionen in verschiedenen Räumen zu konstruieren. Zur Charakterisierung der monogenen gewichteten Hardy- und Bergman-Räume in der Einheitskugel werden zwei konstruktive Algorithmen vorgeschlagen. Für eine reelle harmonische Funktion, die zu einem gewichteten Hardy- und Bergman-Raum gehört, werden die harmonischen Konjugiert in den gleichen Räumen gefunden. Die Beschränktheit der zugrundeliegenden harmonischen Konjugationsoperatoren wird in den angegebenen gewichteten Räumen bewiesen. Zusätzlich wird ein quaternionisches Gegenstück zum Satz von Bloch für monogene Funktionen bewiesen.This work presents a novel Quaternionic Function Theory associated with the best approximation problem in the setting of Hilbert spaces concerning families of prolate and oblate spheroidal functions. The general theory begins with the explicit construction of orthogonal bases for the spaces of harmonic, monogenic, and contragenic functions defined in spheroidal domains of arbitrary eccentricity, whose elements are parametrized by the shape of the corresponding spheroids. A detailed study regarding the elements that constitute these bases is carried out in this thesis. The notion of a contragenic function depends on the domain, and, therefore, it is not a local property in contrast to the concepts of harmonic and monogenic functions. Various conversion formulas that relate systems of harmonic, monogenic, and contragenic functions associated with spheroids of differing eccentricity are presented. Furthermore, the existence of standard nontrivial contragenic functions is shown for spheroids of any eccentricity. The second significant contribution presented in this work pertains to a quaternionic space-frequency theory for band-limited quaternionic functions. A new class of quaternionic signals is proposed, whose energy concentration in the space and the frequency domains are maximal under the quaternion Fourier transform. These signals are studied in the context of spatial-frequency concentration as eigenfunctions of a compact and self-adjoint quaternion integral operator. The fundamental properties of their associated eigenvalues are described in detail. When the concentration domains are spherical in both spaces, the angular part of these signals can be found explicitly, leading to a set of one-dimensional radial integral equations. The theoretical framework described in this work is applied to the construction of classes of monogenic functions in different spaces via harmonic conjugates. Two constructive algorithms are proposed to characterize the monogenic weighted Hardy and Bergman spaces in the Euclidean unit ball. For a real-valued harmonic function belonging to a Hardy and a weighted Bergman space, the harmonic conjugates in the same spaces are found. The boundedness of the underlying harmonic conjugation operators is proven in the given weighted spaces. Additionally, a quaternionic counterpart of Bloch’s Theorem is established for monogenic functions

A spectral representation solution for electromagnetic scattering from complex structures

Significant effort has been directed towards improving computational efficiency in calculating radiated or scattered fields from a complex structure over a broad frequency band. The formulation and solution of boundary integral equation methods in commercial and scientific software has seen considerable attention; methods presented in the literature are often abstract, “curve-fits” or lacking a sound foundation in the underlying physics of the problem. Anomalous results are often characterized incorrectly, or require user expertise for analysis, a clear disadvantage in a computer-aided design tool. This dissertation documents an investigation into the motivating theory, limitations and integration into SuperNEC of a technique for the analytical, continuous, wideband description of the response of a complex conducting body to an electromagnetic excitation. The method, referred to by the author as Transfer Function Estimation (TFE) has its foundations in the Singularity Expansion Method (SEM). For scattering and radiation from a perfect electric conductor, the Electric-Field Integral Equation (EFIE) and Magnetic-Field Integral Equation (MFIE) formulations in their Stratton-Chu form are used. Solution by spectral representation methods including the Singular Value Decomposition (SVD), the Singular Value Expansion (SVE), the Singular Function Method (SFM), Singularity Expansion Method (SEM), the Eigenmode Expansion Method (EEM) and Model-Based Parameter Estimation (MBPE) are evaluated for applicability to the perfect electric conductor. The relationships between them and applicability to the scattering problem are reviewed. A common theoretical basis is derived. The EFIE and MFIE are known to have challenges due to ill-posedness and uniqueness considerations. Known preconditioners present possible solutions. The Modified EFIE (MEFIE) and Modified Combined Integral Equation (MCFIE) preconditioner is shown to be consistent with the fundamental derivations of the SEM. Prony’s method applied to the SEM poleresidue approximation enables a flexible implementation of a reduced-order method to be defined, for integration into SuperNEC. The computational expense inherent to the calculation of the impedance matrix in SuperNEC is substantially reduced by a physically-motivated approximation based on the TFE method. iv Using an adaptive approach and relative error measures, SuperNEC iteratively calculates the best continuous-function approximation to the response of a conducting body over a frequency band of interest. The responses of structures with different degrees of resonant behaviour were evaluated: these included an attack helicopter, a log-periodic dipole array and a simple dipole. Remarkable agreement was achieved

Interpretation of equatorial current meter data as internal waves

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution January 1987Garrett and Munk use linear dynamics to synthesize frequency-wavenumber energy spectra for internal waves (GM72, GM75, GM79). The GM internal wave models are horizontally isotropic, vertically symmetric, purely propagating, and universal in both time and space. This set of properties effectively eliminates all the interesting physics, since such models do not allow localized sources and sinks of energy. Thus an important step in understanding internal wave dynamics is to make measurements of deviations from the simple GM models. This thesis continues the search for deviations from the GM models. It has three advantages over earlier work: extensive data from an equatorial region, long time series (2 years), and relatively sophisticated linear internal wave models. Since the GM models are based on mid-latitude data, having data from an equatorial region which has a strong mean current system offers an opportunity to examine a region with a distinctly different basic state. The longer time series mean there is a larger statistical ensemble of realizations, making it possible to detect smaller internal wave signals. The internal wave models include several important extensions to the GM models: horizontal anisotropy and vertical asymmetry, resolution between standing modes and propagating waves, general vertical structure, and kinematic effects of mean shear flow. Also investigated are the effects of scattering on internal waves, effects that are especially strong on the equator because the buoyancy frequency variability is a factor of ten higher than at mid-latitudes. In the high frequency internal wave field considered (frequencies between .125 cph and .458 cph), several features are found that are not included in the GM models. Both the kinematic effects of a mean shear flow and the phase-locking that distinguishes standing modes from propagating waves are observed. There is a seasonal dependence in energy level of roughly 10% of the mean level. At times the wave field is zonally and vertically asymmetric, with resulting energy fluxes that are a small (4% to 10%) fraction of the maximum energy flux the internal wave field could support. The fluxes are, however, as big as many of the postulated sources of energy for the internal wave field.This work has been supported under grants from the National Science Foundation and the Office of Naval Research, grants numbered NSF-89076, ONR-88914, NSF-9l002, NSF-94971, and NSF-93661

Theory and Applications of Aperiodic (Random) Phased Arrays

A need for network centric topologies using mobile wireless communications makes it important to investigate new distributed beamforming techniques. Platforms such as micro air vehicles (MAVs), unattended ground sensors (UGSs), and unpiloted aerial vehicles (UAVs) can all benefit from advances in this area utilizing advantages in stealth, enhanced survivability and maximum maneuverability. Moreover, in this dissertation, electromagnetic radiation is investigated such that the signal power of each element is coherently added in the far-field region of a specified target direction with net destructive interference occurring in all other regions to suppress sidelobe behavior. This provides superior range and resolution characteristics for a variety of applications including; early warning radar, ballistic missile defense and search and rescue efforts. A wide variety of topologies can be used to confine geometrically these mobile random arrays for analysis. The distribution function for these topologies must be able to generalize the randomness within the geometry. By this means it is feasible to assume the random element distribution of a very large volumetric space will yield either a normal or Gaussian distribution. Therefore the underlying assumption stands that the statistically averaged beam pattern develops from an arrangement of uniformly or Gaussian distrusted elements; both confined to a variety of geometry of radius A and is further generalized using a simple theory based upon the Fourier Transform. Hence, this theory will be derived and serve as the foundation for advanced performance characteristics of these arrays such as its ability for sidelobe tapering, adaptive nulling and multi beam control. In addition it will be shown that for the most ideal of conditions a steerable beam pattern free of sidelobe behavior (better known as a Gaussian distribution) is quite possible. As well these random array structures will be shown to provide superior bandwidth capability over tradiational array structures since they are frequency independent. Last of all a summary of the random array analysis and its results concludes this dissertation

Proceedings of the 1968 Summer Institute on Symbolic Mathematical Computation

Investigating symbolic mathematical computation using PL/1 FORMAC batch system and Scope FORMAC interactive syste

Electromagnetic waves in loaded cylindrical structures : a radial transmission line approach

The motivation for developing the computational electromagnetic methods presented in this thesis is to model the radiation of leaky slotted coaxial cables (LCXs), which are used as distributed antennas in environments that are not readily accessible via conventional antenna substations, and to model ring cavities that act as circular waveguide ??lters. We employ circuit-based electromagnetic wave theory in the solution of guided-wave scattering problems. Here the term ¿guided wave¿ is actually to be interpreted loosely, since even free space can be viewed as a waveguide. Propagation in usual rectilinear waveguides is often phrased in literature in the language of transmission line theory. The theory of equivalent transmission lines has been contrived as a way to give physical insight into the mathematical method of separation of variables. This opens the way to the use of unconventional equivalent transmission lines, such as radial or angular ones. In this thesis we have focused on the concept of radial waveguide, a structure that has the radial direction as the direction of propagation, and that is possibly bounded by metal plates parallel to the coordinate surfaces. Unlike the traditional vector mode functions encountered in conventional waveguides, the radial transmission line concept is introduced in a component basis. Radial lines are peculiar, because they have an absolute origin and hence is not shift invariant. Nevertheless, using a suitable vector formalism, the usual circuit theory concepts can be still applied, including the de??nition of voltages and currents, impedances, propagators, scattering matrices, etc. The LCXs are standard coaxial cables from which, on the outer conductor, slots are cut in order to induce energy exchange between the interior of the cable and the surrounding external domain. These kinds of antennas are usually employed for indoor communications in places where the traditional antenna systems fail or their application and installation are problematic, such as in subways and tunnels. They are also used for security reasons, e.g., in outstations and airports, in order to con??ne the communications inside speci??c places. In particular, nowadays, there is an increasing interest in the application of this technology in the GSM and UMTS frequency bands. LCXs have been studied by several researchers in the past. The analysis techniques employed in these studies produce solutions, to a varying degree of accuracy, for the particular problem of the in??nite periodically slotted cable. The problem of junctions between closed and slotted cables has so far not been addressed. The periodically slotted LCXs considered in the literature suffers from poor ef??ciency in terms of percentage of incident power used for the radiation. Indeed, since the decay of the power inside the cable is exponential and the radiated ??eld decays along the cable length with the same law, the standard periodically slotted LCX requires a compromise between an almost constant level of power along the slotted cable length and minimum power at the end of the cable that is not employed for radiation. In the present thesis we have developed accurate and ef??cient modeling techniques, enabling us to analyze both periodic and aperiodic LCXs, as well as transitions between open and closed cables. The second type of devices of interest is a particular category of stop-band ??lters commonly used in antenna systems to isolate receivers from the signals produced by transmitters, internal or external to the system, and operating in adjacent frequency bands. The structure that we have analyzed presents advantages in terms of the radial and longitudinal dimensions, which allows for the high level of integration that is often essential for space applications. Due to the resonance behavior of the device, the commercial numerical codes require long computational times before suf??ciently accurate ??eld solutions are obtained. Our dedicated modeling method is much more ef??cient in attaining the required results, which has made it possible to produce several design examples. Our modeling techniques are based on the magnetic ??eld integral equation. The associated kernel is the Green's function of the structure, which is been computed in the spectral domain, using radial transmission line theory. The solution of the corresponding integral equation is obtained, for both problems, by the method of moments in the Galerkin form, using a suitable set of basis functions. The computation of the moments requires particular care. We have developed dedicated numerical techniques by which the numerical convergence is improved and the computation of the integrals is accelerated considerably. For LCXs, we have developed a design procedure based on tapering the geometrical dimensions of the slots in order to obtain an uniform radiation and to maximize the radiated power. Since a typical LCX consists of thousands of slots, one approaches practical limitations of integral equation techniques, as the dimension of the linear system resulting from the discretization of the integral equation increases with the number of slots. For this reason, we have augmented our approach to analyze LCXs in two alternative directions. One is based on the application of the Bloch wave approach, the other comprises an extension, for the electromagnetic problem under consideration, of the so-called eigencurrent approach, that was originally developed for linear arrays of patches. First, the Bloch wave approach is not standard in this case since the structure consists of two different regions, one is closed (the interior of the coaxial cable) the other is open (the unbounded exterior domain). We have employed a particular mathematical formalism to overcome this problem, viz., we have solved the junction problem between an closed cable and a slotted one using the mode matching technique. In the Bloch wave approach a LCX with any number of slots, all equal and equally spaced, can ef??ciently be analyzed. Second, the eigencurrent approach is a versatile two-step technique for modelling large compound structures. The ??rst step is to evaluate the eigenvalues and current eigenfunctions of the integral operator associated with a single slot. Subsequently, the pertaining eigencurrents act as global-domain basis functions for the slotted array. In the resulting equivalent linear system, the interaction between the slots is adequately described in terms of very few of these eigencurrents. We have applied this method for LCXs with slots of different geometric dimensions, and have observed a substantial reduction of computation times. For a LCX with a large but ??nite number of identical slots, it turns out that the dominant Bloch wave is the same as the one excited in the semi-in??nite case. When this so-called Forward wave reaches the junction between the slotted and unslotted cable, it gives rise to several re??ected Bloch waves that, upon scattering at the ??rst junction, couple only with the Forward wave. Further, we have observed that all the regressive Bloch waves have globally a negligible effect on the magnetic currents on the slots. Hence the ??eld propagating in the slotted region of the ??nite slotted cable is essentially a progressive wave. As regards the radiation properties of an in??nite LCX, a paradox arises. In practical LCX applications the receiver is always in the near-??eld region of the array, but in the far-??eld region of the majority of the slots. This is related to the in??nite length of a LCX. Application of the Poisson sum formula to the expression for the radiated ??eld emanating from a LCX converts that expression into a linear superposition of spatial harmonics, in line with the Bloch-wave de scription. As a consequence, cables with different slot spacings are perfectly explained in terms of the various modes of operation resulting from the Bloch-wave description, i.e., surface-wave, mono-radiation and multi-radiation operation

Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation

Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo