Dissertations / Theses on the topic 'Harmonic function'

This dissertation follows the history of functional ideas and their pedagogy, illuminates with many examples the implementation of my updated system of Functional Analysis, and discusses the pedagogical implications that this updated system implies. The main goal is to update a system of labeling to be as pedagogically friendly as possible, in order to assist students and teachers of harmony to more easily and enjoyably learn, teach, and engage with common-practice tonal harmonic practice. Example syllabi, assignments, classroom demonstrations, and long projects are also included, and each aspect of the labeling is carefully discussed as it is presented. By surveying the history of functional thinking in music theory, we find that desire to analyze for function is not a new idea, and has been a goal of many theorists and harmony teachers for centuries. However, the current methods for instructing in function still leave students confused or baffled, as they struggle to match functional concepts to labels that do not exemplify their analysis goals and methods that insist on starting from tiny detail instead of coming from a more complete musical perspective. The elaboration of each detail of my Functional Analysis system shows how each part of Functional Analysis has been designed to help make harmonic analysis quicker, easier, more intuitive, and more personalized. The greater pedagogical implications on a larger scale involving courses and curricula are also covered, informed by my experience both as a teacher of today’s standard system and from teaching Functional Analysis in the classroom.

In this thesis, an approach for robot motion control with collision avoidance and human-following is investigated. Using velocity potential fields approach in a modified, quasi-harmonic, solution, the navigation controller is developed. A quasi-harmonic function based controller uses harmonic solutions for collision avoidance and smoothly changes toward a non-harmonic solution which tends toward a zero velocity command only when approaching the goal. After the motion controller was created, human-following strategy was designed to let a non-holonomic robot have the ability to follow a human in an unknown environment with obstacles. The approach is based on velocity potential fields that permit to generate velocity vector commands that drive the robot at a safe distance with regard to the human while avoiding obstacles. The quasi-harmonic approach is investigated analytically using symbolic math solutions of MAPLETM as well as in simulations using MATLABTM. Motion simulations of both holonomic and non-holonomic robot motion illustrate how the proposed approach operates. Experiments are also made with LEGO MINDSTROMS NXT robot to test the algorithm in environment with simple and complex obstacles.

We consider the problem of characterising Besov-Lipshitz and Triebel-Lizorkin spaces using kernels with limited smoothness and decay. This extends the work of H.-Q. Bui et al in [4] and [5] from kernels in S to more general kernels, including the Poisson kernel. We overcome the difficulty of defining the convolution of a general kernel with a distribution by using the concept of a bounded distribution introduced by E. Stein [12]. The characterisations we obtain are valid for the full range of indices.

In this work we will study Hardy-Littlewood maximal function and maximal operator, basing on both classical and most up to date works. In the first chapter we will give definitions for different types of those objects and consider some of their most important properties. The second chapter is entirely devoted to an overview of the fundamental properties of Hardy-Littlewood maximal function, which are strong (p, p) and weak (1, 1) inequalities. Here we list the most actual results on this inequalities in correspondence to the way the maximal func-tion is defined. The third chapter presents the theorem on asymptotic behavior of the lower bound of the constant in the weak-type (1, 1) inequality for the maximal function associated with cubes of Rd, then the dimension d tends to infinity. In the last chapter a method forcomputing constant c, appearing in the main theorem of chapter 3, is given.

QC 20140527

This work uses the method of the Bellman function to show a new proof of Hardy's inequality for Carleson measures. Bellman functions come from the theory of stochastic optimal control and there is a method to prove theorems about inequalities over dyadic trees (which have applications in harmonic analysis) that takes inspiration from concepts from the theory of the Bellman functions. The work will display the important concepts of the theory of Bellman functions in stochastic analysis, will show how to use the method of the Bellman function to prove the estimate over dyadic trees for Carleson measures for Hardy spaces (while also showing the connections between the stochastic theory and the harmonic analysis) and will give a new proof of Hardy's inequality for dyadic trees (which is related to the characterization of Carleson measures in Besov spaces) using the Bellman function method.

Filters are integral components in all wireless communication systems, and their function is to permit predefined band of frequencies into the system and reject all other signals. The ever-growing demand in the use of the radio frequency (RF) spectrum for new applications has resulted in the need for high performance microwave filters with strict requirements on both inband and out-of-band characteristics. High selectivity, high rejection, low loss and extremely wide spurious-free performance are required for both transmitter and receiver channels. In addition, these devices need to be highly compact, easy to integrate within transceivers and should be amenable to low cost manufacturing. High selectivity is essential to enable the guard band between adjacent channels to be reduced thus improving the efficiency of the RF spectrum and hence increasing the capacity of the system. A low insertion-loss, high return-loss and small group-delay in the passband are necessary to minimize signal degradation. A wide stopband is necessary to suppress spurious passbands outside the filter’s bandwidth that may allow spurious emissions from modulation process (harmonic, parasitic, intermodulation and frequency conversion products) and interfere with other systems. The EMC Directive 89/336/EEC mandates that all electronic equipment must comply with the applicable EN specification for EMI. This thesis presents the research work that has resulted in the development of innovative and compact microstrip bandpass filters that fulfil the above stringent requirements for wireless communication systems. In fact, the proposed highly compact planar microstrip filters provide an alternative solution for existing and next generation of wireless communications systems. In particular, the proposed filters exhibit a low-loss and quasi-elliptic function response that is normally only possible with filter designs using waveguides and high temperature superconductors. The selectivity of the filters has been improved by inserting a pair of transmission zeros between the passband edges, and implementing notched rejection bands in the filter’s frequency response to widen its stopband performance. The filter structures have been analysed theoretically and modelled by using Keysight Technologies’ Advanced Design System (ADS™) and Momentum® software. The dissertation is essentially composed of four main sections. In the first section, several compact and quasi-elliptic function bandpass filter structures are proposed and theoretically analysed. Selectivity and stopband performance of these filters is enhanced by loading the input and output feed-lines with inductive stubs that introduce transmission zeros at specified frequencies in the filter’s frequency response. This technique is shown to provide a sharp 3-dB roll-off and steep selectivity skirt with high out-of-band rejection over a wide frequency span. In addition, the 3-dB fractional bandwidth of the filters is shown to be controllable by manipulating the filter’s geometric parameters. Traditional microwave bandpass filters are designed using quarter-wavelength distributed transmission-line resonators that are either end-coupled or side-coupled. The sharpness of the filter response is determined by the number of resonators employed which degrades the filter’s passband loss performance. This results in a filter with a significantly larger footprint which precludes miniaturization. To circumvent these drawbacks the second section describes the development of a novel and compact wideband bandpass filter with the desired characteristics. The quasi-elliptic function filter comprises open-loop resonators that are coupled to each other using a stub loaded resonator. The proposed filter is shown to achieve a wideband 3-dB fractional bandwidth of 23% with much better loss performance, sharp skirt selectivity and very wide rejection bandwidth. The third section describes the investigation of novel ultra-wideband (UWB) microstrip bandpass filter designs. Parametric study enabled the optimization of the filter’s performance which was verified through practical measurements. The proposed filters meet the stringent characteristics required by modern communications systems, i.e. the filters are highly compact and miniature even when fabricated on a low dielectric constant substrate, possess a sharp quasi-elliptic function bandpass response with low passband insertion-loss, and ultra-wide stopband performance. With the rapid development of multi-band operation in modern and next generation wireless communication systems, there is a great demand for single frequency discriminating devices that can operate over multiple frequency bands to facilitate miniaturization. These multi-band bandpass filters need to be physically small, have low insertion-loss, high return-loss, and excellent selectivity. In the fourth section two miniature microstrip dual-band and triple-band bandpass filter designs are explored. A detailed parametric study was conducted to fully understand how the geometric parameters of the filters affected their performance. The optimized filters were fabricated and measured to validate their performance.

A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\mathscr{H}$ of functions with the property that the values $f(x)$ for $f \in \mathscr{H}$ are reproduced from the inner product in $\mathscr{H}$. Recent applications are found in stochastic processes (Ito Calculus), harmonic analysis, complex analysis, learning theory, and machine learning algorithms. This research began with the study of RKHSs to areas such as learning theory, sampling theory, and harmonic analysis. From the Moore-Aronszajn theorem, we have an explicit correspondence between reproducing kernel Hilbert spaces (RKHS) and reproducing kernel functions—also called positive definite kernels or positive definite functions. The focus here is on the duality between positive definite functions and their boundary spaces; these boundary spaces often lead to the study of Gaussian processes or Brownian motion. It is known that every reproducing kernel Hilbert space has an associated generalized boundary probability space. The Arveson (reproducing) kernel is $K(z,w) = \frac{1}{1-_{\C^d}}, z,w \in \B_d$, and Arveson showed, \cite{Arveson}, that the Arveson kernel does not follow the boundary analysis we were finding in other RKHS. Thus, we were led to define a new reproducing kernel on the unit ball in complex $n$-space, and naturally this lead to the study of a new reproducing kernel Hilbert space. This reproducing kernel Hilbert space stems from boundary analysis of the Arveson kernel. The construction of the new RKHS resolves the problem we faced while researching “natural” boundary spaces (for the Drury-Arveson RKHS) that yield boundary factorizations: \[K(z,w) = \int_{\mathcal{B}} K^{\mathcal{B}}_z(b)\overline{K^{\mathcal{B}}_w(b)}d\mu(b), \;\;\; z,w \in \B_d \text{ and } b \in \mathcal{B} \tag*{\it{(Factorization of} $K$).}\] Results from classical harmonic analysis on the disk (the Hardy space) are generalized and extended to the new RKHS. Particularly, our main theorem proves that, relaxing the criteria to the contractive property, we can do the generalization that Arveson's paper showed (criteria being an isometry) is not possible.

The steady state response of thin-walled members subjected to harmonic forces is investigated in the present study. The governing differential equations of motion and associated boundary conditions are derived from the Hamilton variational principle. The harmonic form of the applied forces is exploited to eliminate the need to discretize the problem in the time domain, resulting in computational efficiency. The formulation is based on a generalization of the Timoshenko-Vlasov beam theory and accounts for warping effects, shear deformation effects due to bending and non-uniform warping, translational and rotary inertial effects and captures flexural-torsional coupling arising in asymmetric cross-sections. Six of the resulting seven field equations are observed to be fully coupled for asymmetric cross-sections while the equation of longitudinal motion is observed to be uncoupled. Separate closed form solutions are provided for the cases of (i) doubly symmetric cross sections, (ii) monosymmetric cross-sections, and (iii) asymmetric cross-sections. The closed-form solutions are provided for cantilever and simply-supported boundary conditions. A family of shape functions is then developed based on the exact solution of the homogeneous field equations and then used to formulate a series of super-convergent finite beam elements. The resulting two-noded beam elements are shown to successfully capture the static and dynamic responses of thin-walled members. The finite elements developed involve no special discretization errors normally encountered in other finite element formulations and provide results in excellent agreement with those based on other established finite elements with a minimal number of degrees of freedom. The formulation is also capable to predict the natural frequencies and mode-shapes of the structural members. Comparisons with non-shear deformable beam solutions demonstrate the importance of shear deformation effects within short-span members subjected to harmonic loads with higher exciting frequencies. Comparisons with shell element solution results demonstrate that distortional effects are more pronounced in cantilevers with short spans. A generalized stress extraction scheme from the finite element formulation is then developed. Also, a generalization of the analysis procedure to accommodate multiple loads with distinct exciting frequencies is established. The study is concluded with design examples which illustrate the applicability of the formulation, in conjunction with established principles of fatigue design, in determining the fatigue life of steel members subjected to multiple harmonic forces.

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A1 weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps A∞ weights into the A1 class. We then extend a multiparameter counterexample to the Coifman-Rochberg proposition proposed by Soria (1987) and discuss the difficulties in modifying it into an A∞ counterexample that would disapprove the boundedness of the strong maximal function.

With the rapid development in today&rsquo
s technology, reliability and performance requirements on components of various mechanical systems, which tend to be much lighter and work under much more severe working conditions, dramatically increased. In general, analysis techniques based on simplified model of structural components with linearity assumption may provide time saving for solutions with reasonable accuracy. However, since most engineering structures are often very complex and intrinsically nonlinear, in some cases they may behave in a different manner which cannot be fully described by linear mathematical models, or linear treatments may not be applicable at all. In fact, some studies revealed that deviations in the modal properties of dynamic structures gathered from measured data are due to nonlinearities in the structure. Hence, in problems where accuracy is the primary concern, taking the nonlinear effects into account becomes inevitable. In this thesis, it is aimed to analyze the harmonic response characteristics of multi degree of freedom nonlinear structures having different type of nonlinearities. The amplitude dependencies of nonlinearities are modelled by using describing function method. To increase the accuracy of the results, effect of the higher order harmonic terms will be considered by using multi harmonic describing function theory. Mathematical formulations are embedded in a computer program developed in MATLAB®
with graphical user interface. The program gets the system matricies from the file which is obtained by using substructuring analysis in ANSYS®
, and nonlinearities in the system can easily be defined through the graphical user interface of the MATLAB®
program.

The persistent thrust for a cleaner, greener environment has prompted air pollution regulations to be enforced with increased stringency by environmental protection bodies all over the world. This has prompted gas turbine manufacturers to move from non-premixed combustion to lean, premixed combustion. These lean premixed combustors operate quite fuel-lean compared to the stochiometric, in order to minimize CO and NOx productions, and are very susceptible to oscillations in any of the upstream flow variables. These oscillations cause the heat release rate of the flame to oscillate, which can engage one or more acoustic modes of the combustor or gas turbine components, and under certain conditions, lead to limit cycle oscillations. This phenomenon, called thermoacoustic instabilities, is characterized by very high pressure oscillations and increased heat fluxes at system walls, and can cause significant problems in the routine operability of these combustors, not to mention the occasional hardware damages that could occur, all of which cumulatively cost several millions of dollars. In a bid towards understanding this flow-flame interaction, this research works studies the heat release response of premixed flames to oscillations in reactant equivalence ratio, reactant velocity and pressure, under conditions where the flame preheat zone is convectively compact to these disturbances, using the G-equation. The heat release response is quantified by means of the flame transfer function and together with combustor acoustics, forms a critical component of the analytical models that can predict combustor dynamics. To this end, low excitation amplitude (linear) and high excitation amplitude (nonlinear) responses of the flame are studied in this work. The linear heat release response of lean, premixed flames are seen to be dominated by responses to velocity and equivalence ratio fluctuations at low frequencies, and to pressure fluctuations at high frequencies which are in the vicinity of typical screech frequencies in gas turbine combustors. The nonlinear response problem is exclusively studied in the case of equivalence ratio coupling. Various nonlinearity mechanisms are identified, amongst which the crossover mechanisms, viz., stoichiometric and flammability crossovers, are seen to be responsible in causing saturation in the overall heat release magnitude of the flame. The response physics remain the same across various preheat temperatures and reactant pressures. Finally, comparisons between the chemiluminescence transfer function obtained experimentally and the heat release transfer functions obtained from the reduced order model (ROM) are performed for lean, CH4/Air swirl-stabilized, axisymmetric V-flames. While the comparison between the phases of the experimental and theoretical transfer functions are encouraging, their magnitudes show disagreement at lower Strouhal number gains show disagreement.

Displacement Green's function is the building block for some semi-analytical methods like Boundary Element Method (BEM), Distributed Point Source Method (DPCM), etc. In this thesis, the displacement Green`s function in anisotropic media due to a time harmonic point force is studied. Unlike the isotropic media, the Green's function in anisotropic media does not have a closed form solution. The dynamic Green's function for an anisotropic medium can be written as a summation of singular and non-singular or regular parts. The singular part, being similar to the result of static Green's function, is in the form of an integral over an oblique circular path in 3D. This integral can be evaluated either by a numerical integration technique or can be converted to a summation of algebraic terms via the calculus of residue. The other part, which is the regular part, is in the form of an integral over the surface of a unit sphere. This integral needs to be evaluated numerically and its evaluation is considerably more time consuming than the singular part. Obtaining dynamic Green's function and its spatial derivatives involves calculation of these two types of integrals. The spatial derivatives of Green's function are important in calculating quantities like stress and stain tensors. The contribution of this thesis can be divided into two parts. In the first part, different integration techniques including Gauss Quadrature, Simpson's, Chebyshev, and Lebedev integration techniques are tried out and compared for evaluation of dynamic Green’s function. In addition the solution from the residue theorem is included for the singular part. The accuracy and performance of numerical implementation is studied in detail via different numerical examples. Convergence plots are used to analyze the numerical error for both Green's function and its derivatives. The second part of contribution of this thesis relates to the mathematical derivations. As mentioned above, the regular part of dynamic Green's function, being an integral over the surface of a unit sphere, is responsible for the majority of computational time. From symmetry properties, this integration domain can be reduced to a hemisphere, but no more simplification seems to be possible for a general anisotropic medium. In this thesis, the integration domain for regular part is further reduced to a quarter of a sphere for the particular case of transversely isotropic material. This reduction proposed for the first time in this thesis nearly halves the number of integration points for the evaluation of regular part of dynamic Green's function. It significantly reduces the computational time.

In this paper I give a description and derivation of Dirichlet's problem, a boundary value problem, for p-harmonic functions on graphs and study an iterative method for solving it.The method's convergence is proved and some preliminary results about its speed of convergence are presented.There is an implementation accompanying this thesis and a short description of the implementation is included. The implementation will be made available on the internet at http://www.mai.liu.se/~anbjo/pharmgraph/ for as long as possible.

The spectral zeta function, introduced by Minakshisundaram and Pleijel in [36] and denoted by ζg(s), encodes important spectral information for the Laplacian on Riemannian manifolds. For instance, the important notions of the determinant of the Laplacian and Casimir energy are defined via the spectral zeta function. On homogeneous manifolds, it is known that the spectral zeta function is critical with respect to conformal metric perturbations, (see e.g Richardson ([47]) and Okikiolu ([41])). In this thesis, we compute a second variation formula of ζg(s) on closed homogeneous Riemannian manifolds under conformal metric perturbations. It is well known that the quadratic form corresponding to this second variation is given by a certain pseudodifferential operator that depends meromorphically on s. The symbol of this operator was analysed by Okikiolu in ([42]). We analyse it in more detail on homogeneous spaces, in particular on the spheres Sn. The case n = 3 is treated in great detail. In order to describe the second variation we introduce a certain distributional integral kernel, analyse its meromorphic properties and the pole structure. The Casimir energy defined as the finite part of ζg(-½) on the n-sphere and other points of ζg(s) are used to illustrate our results. The techniques employed are heat kernel asymptotics on Riemannian manifolds, the associated meromorphic continuation of the zeta function, harmonic analysis on spheres, and asymptotic analysis.

A novel framework for quantitative analysis of shape and function in magnetic resonance imaging (MRI) of the brain is proposed. First, an efficient method to compute invariant spherical harmonics (SPHARM) based feature representation for real valued 3D functions was developed. This method addressed previous limitations of obtaining unique feature representations using a radial transform. The scale, rotation and translation invariance of these features enables direct comparisons across subjects. This eliminates need for spatial normalization or manually placed landmarks required in most conventional methods [1-6], thereby simplifying the analysis procedure while avoiding potential errors due to misregistration. The proposed approach was tested on synthetic data to evaluate its improved sensitivity. Application on real clinical data showed that this method was able to detect clinically relevant shape changes in the thalami and brain ventricles of Parkinson's disease patients. This framework was then extended to generate functional features that characterize 3D spatial activation patterns within ROIs in functional magnetic resonance imaging (fMRI). To tackle the issue of intersubject structural variability while performing group studies in functional data, current state-of-the-art methods use spatial normalization techniques to warp the brain to a common atlas, a practice criticized for its accuracy and reliability, especially when pathological or aged brains are involved [7-11]. To circumvent these issues, a novel principal component subspace was developed to reduce the influence of anatomical variations on the functional features. Synthetic data tests demonstrate the improved sensitivity of this approach over the conventional normalization approach in the presence of intersubject variability. Furthermore, application to real fMRI data collected from Parkinson's disease patients revealed significant differences in patterns of activation in regions undetected by conventional means. This heightened sensitivity of the proposed features would be very beneficial in performing group analysis in functional data, since potential false negatives can significantly alter the medical inference. The proposed framework for reducing effects of intersubject anatomical variations is not limited to functional analysis and can be extended to any quantitative observation in ROIs such as diffusion anisotropy in diffusion tensor imaging (DTI), thus providing researchers with a robust alternative to the controversial normalization approach.

In this work we study vertical, acoustic propagation in a non-homogeneous media for a spatially-compact, time-harmonic source. An analytical, 2-layer model is developed representing the acoustic pressure disturbance propagating in the atmosphere. The validity of the model spans the distance from the Earth's surface to 30,000 meters. This includes the troposphere (adiabatic), ozone layer (isothermal), and part of the stratosphere (isothermal). The results of the model derivation in the adiabatic region yield pressure solutions as Bessel functions of the First (J) and Second (Y) Kind of order $-\frac{7}{2}$ with an argument of $2 \Omega \tau$ (where $\Omega$ represents a dimensionless frequency and $\tau$ is a dimensionless vertical height in z (vertical coordinate)). For an added second layer (isothermal region), the pressure solution is a decaying sinusoidal, exponential function above the first layer. In particular, the vertical, acoustic propagation is examined for various configurations. These are divided into 2 basic classes. The first class consists of examining the pressure response function when the source is located on boundary interfaces, while the second class consists of situations where the source is arbitrarily located within a finite layer. In all instances, a time-harmonic, compact source is implicitly understood. However, each class requires a different method of solution. The first class conforms to a general boundary value problem, while the second requires the use of Green's functions method. In investigating problems of the first class, 3 different scenarios are examined. In the first case, we apply our model to a semi-infinite medium with a time-harmonic source ($e^{-i \omega t}$) located on the ground. In the next 2 cases, a semi-infinite medium is overlain on the previous medium at a height of z=13,000 meters. Thus, there exist two boundaries: the ground and the layer interface between the 2 media. Sources placed at these interfaces represent the 2nd and 3rd scenarios, respectively. The solutions to all 3 cases are of the form $A \frac{J_{-\frac{7}{2}}(2 \Omega \tau)}{{\tau}^{-\frac{7}{2}}} + B \frac{Y_{-\frac{7}{2}}(2 \Omega \tau)}{{\tau}^{-\frac{7}{2}}}$, where \textit{A} and \textit{B} are constants determined by the boundary conditions. For the 2nd class, we examine the application to a time-harmonic, compact source placed arbitrarily within the 1st layer. The method of Green's functions is used to obtain a particular solution for the model equations. This result is compared with a Fast Field Program (FFP) which was developed to test these solutions. The results show that the response given by the Green's function compares favorably with that of the FFP. Keywords: Linear Acoustics, Inhomogeneous Medium, Layered Atmosphere, Boundary Value Problem, Green's Function Method

We establish a strong connection between buildings and Hecke algebras through the study of two algebras of averaging operators on buildings. To each locally finite regular building we associate a natural algebra B of chamber set averaging operators, and when the building is affine we also define an algebra A of vertex set averaging operators. In the affine case, it is shown how the building gives rise to a combinatorial and geometric description of the Macdonald spherical functions, and of the centers of affine Hecke algebras. The algebra homomorphisms from A into the complex numbers are studied, and some associated spherical harmonic analysis is conducted. This generalises known results concerning spherical functions on groups of p-adic type. As an application of this spherical harmonic analysis we prove a local limit theorem for radial random walks on affine buildings.

The evolution of diatonic tonality into a more complex, multidimensional order is evident in Arnold Schoenberg's late tonal and early atonal compositions. The development is inclusive of the synthesis of a major key with the related tonic minor key: twenty-four major and minor diatonic keys integrate to form twelve major-minor keys. The effect of the major-minor merger, or, Modal Interchangeability, on the evolution of tonality is that the increased diversity of degree forms results in expanded degree function and metamorphosed harmonic progressions. Schoenberg implements the change in tonal parameters by focusing on degrees with more than one functional interpretation. The concept is codified as 'Multiple Degree Function'. Degrees function simultaneously to implicate two or more keys without explicitly referring to one key. Initially a localized phenomenon, multiplicity of harmonic intent is gradually structural in manifestation. Compositions that use more than one key area through the application of Multiple Degree Function are based on a Key complex. The Key complex is defined as two or more keys interacting to form the harmonic structure of a composition. There are eleven classes of bipartite Key Complexes, based on the interval between two modally interchangeable keys. Potential degree function is systematized according to the particular degree interface between the two key areas. Compound Key Complexes combine more than one class of key relationship. The Key Complex System and the associated concept of Multiple Degree Function comprise the foundation of this study. Analysis of Schoenberg's transitional works demonstrates an evolving harmonic practice in which potential degree function and harmonic progression can be categorized. It is shown through the method of Key Complex analysis that the transitional and 'atonal' compositions of Arnold Schoenberg are structured on a defined set of key areas.

The Lie symmetry groups of minimal surfaces by way of planar harmonic functions are determined. It is shown that a symmetry group acting on the minimal surfaces is isomorphic with H × H^2 — the analytic functions and the harmonic functions. A subgroup of this gives a generalization of the associated family which is examined.

Approved for public release, distribution unlimited
A new method of array modeling which will be used to predict the performance of low frequency active sonar arrays is being investigated. In support of this effort, a network representation of a spherical shell piezoelectric transducer was developed. The transducer was modeled using the finite-element code MARTSAM, from which a nodal description of the transducer was obtained. A procedure was developed to reduce and transform the nodal description of the transducer into a spherical harmonic description. The spherical harmonic description of the transducer was computed at two frequencies, 112.5 Hz and 1125.3 Hz, corresponding to values of ka of 0.1 and 1. 0, respectively where a is the radius of the sphere.

The thesis consists of an introduction and four appended papers. In the introduction we give an overview of pulse-modulated systems and provide a few examples of such systems. Furthermore, we introduce the so-called dynamic phasor model which is used as a basis for analysis in two of the appended papers. We also introduce the harmonic transfer function and finally we provide a summary of the appended papers. The first paper considers stability analysis of a class of pulse-width modulated systems based on a discrete time model. The systems considered typically have periodic solutions. Stability of a periodic solution is equivalent to stability of a fixed point of a discrete time model of the system dynamics. Conditions for global and local exponential stability of the discrete time model are derived using quadratic and piecewise quadratic Lyapunov functions. A griding procedure is used to develop a systematic method to search for the Lyapunov functions. The second paper considers the dynamic phasor model as a tool for stability analysis of a general class of pulse-modulated systems. The analysis covers both linear time periodic systems and systems where the pulse modulation is controlled by feedback. The dynamic phasor model provides an $\textbf{L}_2$-equivalent description of the system dynamics in terms of an infinite dimensional dynamic system. The infinite dimensional phasor system is approximated via a skew truncation. The truncated system is used to derive a systematic method to compute time periodic quadratic Lyapunov functions. The third paper considers the dynamic phasor model as a tool for harmonic analysis of a class of pulse-width modulated systems. The analysis covers both linear time periodic systems and non-periodic systems where the switching is controlled by feedback. As in the second paper of the thesis, we represent the switching system using the L_2-equivalent infinite dimensional system provided by the phasor model. It is shown that there is a connection between the dynamic phasor model and the harmonic transfer function of a linear time periodic system and this connection is used to extend the notion of harmonic transfer function to describe periodic solutions of non-periodic systems. The infinite dimensional phasor system is approximated via a square truncation. We assume that the response of the truncated system to a periodic disturbance is also periodic and we consider the corresponding harmonic balance equations. An approximate solution of these equations is stated in terms of a harmonic transfer function which is analogous to the harmonic transfer function of a linear time periodic system. The aforementioned assumption is proved to hold for small disturbances by proving the existence of a solution to a fixed point equation. The proof implies that for small disturbances, the approximation is good. Finally, the fourth paper considers control synthesis for switched mode DC-DC converters. The synthesis is based on a sampled data model of the system dynamics. The sampled data model gives an exact description of the converter state at the switching instances, but also includes a lifted signal which represents the inter-sampling behavior. Within the sampled data framework we consider H-infinity control design to achieve robustness to disturbances and load variations. The suggested controller is applied to two benchmark examples; a step-down and a step-up converter. Performance is verified in both simulations and in experiments.
QC 20100628

this thesis presents a new approach in dealing with characterize LPVT and proposes determining the impulse response of LPVT, purposing to find transfer function (h(t)) which contains most electrical characteristics of LPVTs as a dynamic system.

We introduce the notion of harmonic thin sets and establish a refinement of the Fatou-Naim-Doob theorem in the axiomatic system of Brelot. We also introduce in a Brelot harmonic space several notions of sets of determination for harmonic functions which were introduced by Bonsall (5), Hayman and Lyons (18), and Aikawa (1) for the classical Laplace operator on $ IR sp{n}.$ We then discuss the relation between our results on harmonic thinness in such an abstract setting and certain recent results of Bonsall (5), Hayman and Lyons (18), Gardiner (12), Essen (11), and Aikawa (1). We consider in particular the potential theory and the above problems for the Laplace-Beltrami operator associated with the Bergman metric on the unit ball of the n-dimensional complex space, i.e., in the context of Poisson-Szego integrals.

Thesis (Ph. D.)--University of California, San Diego, 2009.
Title from first page of PDF file (viewed June 25, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 82-83).

[EN] The Ph.D. thesis "Weighted Banach Spaces of harmonic functions" presented here, treats several topics of functional analysis such as weights, composition operators, Fréchet and Gâteaux differentiability of the norm and isomorphism classes. The work is divided into four chapters that are preceded by one in which we introduce the notation and the well-known properties that we use in the proofs in the rest of the chapters. In the first chapter we study Banach spaces of harmonic functions on open sets of R^d endowed with weighted supremun norms. We define the harmonic associated weight, we explain its properties, we compare it with the holomorphic associated weight introduced by Bierstedt, Bonet and Taskinen, and we find differences and conditions under which they are exactly the same and conditions under which they are equivalent. The second chapter is devoted to the analysis of composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions. We characterize the continuity, the compactness and the essential norm of composition operators among these spaces in terms of their weights, thus extending the results of Bonet, Taskinen, Lindström, Wolf, Contreras, Montes and others for composition operators between spaces of holomorphic functions. We prove that for each value of the interval [0,1] there is a composition operator between weighted spaces of harmonic functions such that its essential norm attains this value. Most of the contents of Chapters 1 and 2 have been published by E. Jordá and the author in [48]. The third chapter is related with the study of Gâteaux and Fréchet differentiability of the norm. The \v{S}mulyan criterion states that the norm of a real Banach space X is Gâteaux differentiable at x\inX if and only if there exists x^* in the unit ball of the dual of X weak^* exposed by x and the norm is Fréchet differentiable at x if and only if x^* is weak^* strongly exposed in the unit ball of the dual of X by x. We show that in this criterion the unit ball of the dual of X can be replaced by a smaller convenient set, and we apply this extended criterion to characterize the points of Gâteaux and Fréchet differentiability of the norm of some spaces of harmonic functions and continuous functions with vector values. Starting from these results we get an easy proof of the theorem about the Gâteaux differentiability of the norm for spaces of compact linear operators announced by Heinrich and published without proof. Moreover, these results allow us to obtain applications to classical Banach spaces as the space H^\infty of bounded holomorphic functions in the disc and the algebra A(\overline{\D}) of continuous functions on \overline{\D} which are holomorphic on \D. The content of this chapter has been included by E. Jordá and the author in [47]. Finally, in the forth chapter we show that for any open set U of R^d and weight v on U, the space hv0(U) of harmonic functions such that multiplied by the weight vanishes at the boundary on U is almost isometric to a closed subspace of c0, extending a theorem due to Bonet and Wolf for the spaces of holomorphic functions Hv0(U) on open sets U of C^d. Likewise, we also study the geometry of these weighted spaces inspired by a work of Boyd and Rueda, examining topics such as the v-boundary and v-peak points and we give the conditions that provide examples where hv0(U) cannot be isometric to c0. For a balanced open set U of R^d, some geometrical conditions in U and convexity in the weight v ensure that hv0(U) is not rotund. These results have been published by E. Jordá and the author [46].
[ES] La presente memoria, "Espacios de Banach ponderados de funciones armónicas ", trata diversos tópicos del análisis funcional, como son las funciones peso, los operadores de composición, la diferenciabilidad Fréchet y Gâteaux de la norma y las clases de isomorfismos. El trabajo está dividido en cuatro capítulos precedidos de uno inicial en el que introducimos la notación y las propiedades conocidas que usamos en las demostraciones del resto de capítulos. En el primer capítulo estudiamos espacios de Banach de funciones armónicas en conjuntos abiertos de R^d dotados de normas del supremo ponderadas. Definimos el peso asociado armónico, explicamos sus propiedades, lo comparamos con el peso asociado holomorfo introducido por Bierstedt, Bonet y Taskinen, y encontramos diferencias y condiciones para que sean exactamente iguales y condiciones para que sean equivalentes. El capítulo segundo está dedicado al análisis de los operadores de composición con símbolo holomorfo entre espacios de Banach ponderados de funciones pluriarmónicas. Caracterizamos la continuidad, la compacidad y la norma esencial de operadores de composición entre estos espacios en términos de los pesos, extendiendo los resultados de Bonet, Taskinen, Lindström, Wolf, Contreras, Montes y otros para operadores de composición entre espacios de funciones holomorfas. Probamos que para todo valor del intervalo [0,1] existe un operador de composición sobre espacios ponderados de funciones armónicas tal que su norma esencial alcanza dicho valor. La mayoría de los contenidos de los capítulos 1 y 2 han sido publicados por E. Jordá y la autora en [48]. El capítulo tercero está relacionado con el estudio de la diferenciabilidad Gâteaux y Fréchet de la norma. El criterio de \v{S}mulyan establece que la norma de un espacio de Banach real X es Gâteaux diferenciable en x\in X si y sólo si existe x^* en la bola unidad del dual de X débil expuesto por x y la norma es Fréchet diferenciable en x si y sólo si x^*es débil fuertemente expuesto en la bola unidad del dual de X por x. Mostramos que en este criterio la bola del dual de X puede ser reemplazada por un conjunto conveniente más pequeño, y aplicamos este criterio extendido para caracterizar los puntos de diferenciabilidad Gâteaux y Fréchet de la norma de algunos espacios de funciones armónicas y continuas con valores vectoriales. A partir de estos resultados conseguimos una prueba sencilla del teorema sobre la diferenciabilidad Gâteaux de la norma de espacios de operadores lineales compactos enunciado por Heinrich y publicado sin la prueba. Además, éstos nos permiten obtener aplicaciones para espacios de Banach clásicos como H^\infty de funciones holomorfas acotadas en el disco y A(\overline{\D}) de funciones continuas en \overline{\D} que son holomorfas en \D. Los contenidos de este capítulo han sido incluidos por E. Jordá y la autora en [47]. Finalmente, en el capítulo cuarto mostramos que para cualquier abierto U contenido en R^d y cualquier peso v en U, el espacio hv0(U), de funciones armónicas tales que multiplicadas por el peso desaparecen en el infinito de U, es casi isométrico a un subespacio cerrado de c0, extendiendo un teorema debido a Bonet y Wolf para los espacios de funciones holomorfas Hv0(U) en abiertos U de C^d. Así mismo, inspirados por un trabajo de Boyd y Rueda también estudiamos la geometría de estos espacios ponderados examinando tópicos como la v-frontera y los puntos v-peak y damos las condiciones que proporcionan ejemplos donde hv0(U) no puede ser isométrico a c0. Para un conjunto abierto equilibrado U de R^d, algunas condiciones geométricas en U y sobre convexidad en el peso v aseguran que hv0(U) no es rotundo. Estos resultados han sido publicados por E. Jordá y la autora en [46].
[CAT] La present memòria, "Espais de Banach ponderats de funcions harmòniques", tracta diversos tòpics de l'anàlisi funcional, com són les funcions pes, els operadors de composició, la diferenciabilitat Fréchet i Gâteaux de la norma i les clases d'isomorfismes. El treball està dividit en quatre capítols precedits d'un d'inicial en què introduïm la notació i les propietats conegudes que fem servir en les demostracions de la resta de capítols. En el primer capítol estudiem espais de Banach de funcions harmòniques en conjunts oberts de R^d dotats de normes del suprem ponderades. Definim el pes associat harmònic, expliquem les seues propietats, el comparem amb el pes associat holomorf introduït per Bierstedt, Bonet i Taskinen, i trobem diferències i condicions perquè siguen exactament iguals i condicions perquè siguen equivalents. El capítol segon està dedicat a l'anàlisi dels operadors de composició amb símbol holomorf entre espais de Banach ponderats de funcions pluriharmòniques. Caracteritzem la continuïtat, la compacitat i la norma essencial d'operadors de composició entre aquests espais en termes dels pesos, estenent els resultats de Bonet, Taskinen, Lindström, Wolf, Contreras, Montes i altres per a operadors de composició entre espais de funcions holomorfes. Provem que per a tot valor de l'interval [0,1] hi ha un operador de composició sobre espais ponderats de funcions harmòniques tal que la seua norma essencial arriba aquest valor. La majoria dels continguts dels capítols 1 i 2 han estat publicats per E. Jordá i l'autora en [48]. El capítol tercer està relacionat amb l'estudi de la diferenciabilitat Gâteaux y Fréchet de la norma. El criteri de \v{S}mulyan estableix que la norma d'un espai de Banach real X és Gâteaux diferenciable en x\inX si i només si existeix x^* a la bola unitat del dual de X feble exposat per x i la norma és Fréchet diferenciable en x si i només si x^* és feble fortament exposat a la bola unitat del dual de X per x. Mostrem que en aquest criteri la bola del dual de X pot ser substituïda per un conjunt convenient més petit, i apliquem aquest criteri estès per caracteritzar els punts de diferenciabilitat Gâteaux i Fréchet de la norma d'alguns espais de funcions harmòniques i contínues amb valors vectorials. A partir d'aquests resultats aconseguim una prova senzilla del teorema sobre la diferenciabilitat Gâteaux de la norma d'espais d'operadors lineals compactes enunciat per Heinrich i publicat sense la prova. A més, aquests ens permeten obtenir aplicacions per a espais de Banach clàssics com l'espai H^\infty de funcions holomorfes acotades en el disc i l'àlgebra A(\overline{\D}) de funcions contínues en \overline{\D} que són holomorfes en \D. Els continguts d'aquest capítol han estat inclosos per E. Jordá i l'autora en [47]. Finalment, en el capítol quart mostrem que per a qualsevol conjunt obert U de R^d i qualsevol pes v en U, l'espai hv0(U), de funcions harmòniques tals que multiplicades pel pes desapareixen en el infinit d'U, és gairebé isomètric a un subespai tancat de c0, estenent un teorema degut a Bonet y Wolf per als espais de funcions holomorfes Hv0(U) en oberts U de C^d. Així mateix, inspirats per un treball de Boyd i Rueda també estudiem la geometria d'aquests espais ponderats examinant tòpics com la v-frontera i els punts v-peak i donem les condicions que proporcionen exemples on hv0(U) no pot ser isomètric a c0. Per a un conjunt obert equilibrat U de R^d, algunes condicions geomètriques en U i sobre convexitat en el pes v asseguren que hv0(U) no és rotund. Aquests resultats han estat publicats per E. Jordá i l'autora en [46].
Zarco García, AM. (2015). Weighted Banach spaces of harmonic functions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/56461
TESIS

Au cours des dernières années, des changements importants dans le domaine des assurances et des finances attirent de plus en plus l’attention sur la nécessité d’élaborer un cadre normalisé pour la mesure des risques. Récemment, il y a eu un intérêt croissant de la part des experts en assurance sur l’utilisation de l’espérance conditionnelle des pertes (CTE) parce qu’elle partage des propriétés considérées comme souhaitables et applicables dans diverses situations. En particulier, il répond aux exigences d’une mesure de risque “cohérente”, selon Artzner [2]. Cette thèse représente des contributions à l’inférence statistique en développant des outils, basés sur la convergence des intégrales fonctionnelles, pour l’estimation de la CTE qui présentent un intérêt considérable pour la science actuarielle. Tout d’abord, nous développons un outil permettant l’estimation de la moyenne conditionnelle E[X|X > x], ensuite nous construisons des estimateurs de la CTE, développons la théorie asymptotique nécessaire pour ces estimateurs, puis utilisons la théorie pour construire des intervalles de confiance. Pour la première fois, l’approche de bootstrap non paramétrique est explorée dans cette thèse en développant des nouveaux résultats applicables à la valeur à risque (VaR) et à la CTE. Des études de simulation illustrent la performance de la technique de bootstrap.

Many results in real and complex analysis are the consequence of mean value properties and theorems. This is the case for harmonic and holomorphic functions as well. The mean value property builds the foundation for several properties of each set of functions. Using this property one can derive more properties like the maximum principle for harmonic functions and the maximum modulus principle for holomorphic functions. These results are then used to show other properties. The goal is to compare the theorems and proofs for harmonic and holomorphic functions and to understand why the results seem to be similar.

Some geometrical figures like balls, annuli and infinite strips are characterized by quadrature formulae involving mean values of certain harmonic functions.
Quite a number of research papers starting with the work of Kuran (1972) to the present with an, as yet, unpublished work of Armitage and Goldstein have been devoted to different aspects of the inverse mean value property. This thesis contains a unified exposition of results concerning the inverse mean value characterization.

Orientadores: Sergio Antonio Tozoni, Alexander Kushpel
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: O objetivo da dissertação e desenvolver um texto em português sobre Análise Harmônica na esfera d-dimensional real e aplicar os resultados deste texto no estudo de um teorema de multiplicadores. Nos dois primeiros capítulos e realizado um estudo sobre funções harmônicas em um domínio do espaço euclidiano Rd+1, harmônicos esféricos, representações de SO(d+1), harmônicos zonais, polinômios ultraesféricos e sobre o operador de Laplace Beltrami para a esfera. Finalmente, no terceiro capítulo é estudado um teorema de multiplicadores recente, o qual fornece condições suficientes para que um operador multiplicador seja limitado de Lp(Sd) em Lq(Sd), para quaisquer p e q, 1=p, q=8. Como aplicação deste teorema são obtidas estimativas superiores para n-larguras de Kolmogorov de classes de Sobolev nos espaços Lq(Sd), 1=p, q= 8, g > 0
Abstract: The purpose of this work is to develop a text in Portuguese about Harmonic Analysis on the d-dimensional real sphere Sd and to apply the results of the text to study a multiplier theorem. In the first two chapters it is made a study about harmonic functions in a domain of the euclidian space Rd+1, spherics harmonics, representations of SO(d+1), zonal harmonics, ultraspherics polynomials and about the Laplace Beltrami operator on the sphere. Finally, in the third chapter it is studied a recent multiplier theorem which gives sufficient conditions for a multiplier operator be bounded from Lp(Sd) to Lq(Sd), for 1=p, q=8. As application of this theorem are obtained upper bounds for n-widths of Kolmogorov type of Sobolev classes in the spaces Lq(Sd), 1=p, q= 8, g > 0
Mestrado
Matematica
Mestre em Matemática

A high spatial resolution acoustic directivity acquisition system (ADAS) has been developed to acquire anechoic measurements of the far field radiation of musical instruments that are either remote controlled or played by musicians. Building upon work performed by the BYU Acoustic Research Group in the characterization of loudspeaker directivity, one can rotate a musical instrument with sequential azimuthal angle increments under a fixed semicircular array of microphones while recording repeated notes or sequences of notes. This results in highly detailed and instructive directivity data presented in the form of high-resolution balloon plots. The directivity data and corresponding balloon plots may be shown to vary as functions of time or frequency. This thesis outlines the development of a prototype ADAS and its application to different sources including loudspeakers, a concert grand piano, trombone, flute, and violin. The development of a method of compensating for variations in the played amplitude at subsequent measurement positions using a near-field reference microphone and Frequency Response Functions (FRF) is presented along with the results of its experimental validation. This validation involves a loudspeaker, with known directivity, to simulate a live musician. It radiates both idealized signals and anechoic recordings of musical instruments with random variations in amplitude. The concept of coherence balloon maps and surface averaged coherence are introduced as tools to establish directivity confidence. The method of creating composite directivities for musical instruments is also introduced. A composite directivity comes from combining the directivities of all played partials to approximate what the equivalent directivity from a musical instrument would be if full spectral excitation could be used. The composite directivities are derived from an iterative averaging process that uses coherence as an inclusion criterion. Sample directivity results and discussions of experimental considerations of the piano, trombone, flute, and violin are presented. The research conducted is preliminary and will be further developed by future students to expand and refine the methods presented here.