Variable Exponent Spaces of Analytic Functions

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

Inverse Scattering Source Problems

The purpose of this chapter is to discuss some of the highlights of the mathematical theory of direct and inverse scattering and inverse source scattering problem for acoustic, elastic and electromagnetic waves. We also briefly explain the uniqueness of the external source for acoustic, elastic and electromagnetic waves equation. However, we must first issue a caveat to the reader. We will also present the recent results for inverse source problems. The resents results including a logarithmic estimate consists of two parts: the Lipschitz part data discrepancy and the high frequency tail of the source function. In general, it is known that due to the existence of non-radiation source, there is no uniqueness for the inverse source problems at a fixed frequency.

Interference Pattern Representation on the Complex s-Plane

In this work, the normalized interference pattern produced by a coherence interferometer system was represented as a complex function. The Laplace transform was applied for the transformation. Poles and zeros were determined from this complex function, and then, its pole-zero map and its Bode diagram were proposed. Both graphical representations were implemented numerically. From our numerical results, pole location and zero location depend on the optical path difference (OPD), while the Bode diagram gives us information about the OPD parameter. Based on the results obtained from the graphical representations, the coherence interferometer systems, the low-coherence interferometer systems, the interferometric sensing systems, and the fiber optic sensors can be analyze on the complex s-plane.

Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves for the Structured Media

We have proven that the long wave with finite amplitude responds to the structure of the medium. The heterogeneity in a medium structure always introduces additional nonlinearity in comparison with the homogeneous medium. At the same time, a question appears on the inverse problem, namely, is there sufficient information in the wave field to reconstruct the structure of the medium? It turns out that the knowledge on the evolution of nonlinear waves enables us to form the theoretical fundamentals of the diagnostic method to define the characteristics of a heterogeneous medium using the long waves of finite amplitudes (inverse problem). The mass contents of the particular components can be denoted with specified accuracy by this diagnostic method.

Solution Methods of Large Complex-Valued Nonlinear System of Equations

Nonlinear systems of equations in complex plane are frequently encountered in applied mathematics, e.g., power systems, signal processing, control theory, neural networks, and biomedicine, to name a few. The solution of these problems often requires a first- or second-order approximation of nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to functions of complex and complex conjugate variables because they are necessarily nonanalytic. To overcome this problem, the Wirtinger calculus allows an expansion of nonlinear functions in its original complex and complex conjugate variables once they are analytic in their argument as a whole. Thus, the goal is to apply this methodology for solving nonlinear systems of equations emerged from applications in the industry. For instances, the complex-valued Jacobian matrix emerged from the power flow analysis model which is solved by Newton-Raphson method can be exactly determined. Similarly, overdetermined Jacobian matrices can be dealt, e.g., through the Gauss-Newton method in complex plane aimed to solve power system state estimation problems. Finally, the factorization method of the aforementioned Jacobian matrices is addressed through the fast Givens transformation algorithm which means the square root-free Givens rotations method in complex plane.

An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application to an AC/DC Flyback Converter

Flyback converters have been widely used in low- and high-power applications because of their simplicity and low cost. However, they incur electromagnetic compatibility problems which are more difficult to control. The present chapter proposes an efficient modeling method based on the near-field technique to solve real-world radiation problems of the power electronics circuits. Firstly, for the characterization of an AC/DC flyback converter, several experimental measurements of the magnetic near field are performed in the time domain over the converter. Subsequently, we have applied the time domain electromagnetic inverse method based on the genetic algorithms on the measured signals to find the equivalent radiating sources of the studied circuit. The accuracy and the efficiency of the proposed approach have been demonstrated by the good agreement between cartographies of the near magnetic field components calculated using the developed model and those measured. Finally, the developed equivalent model has been used to predict cartographies of other components of the magnetic field which will be compared to measured cartographies. This confirms that the identified equivalent sources can represent real sources in the studied structure. The proposed method could be used for diagnosis and fault location in power electronics systems.

Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature R μν , of Einstein’s equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for the reinterpretation of vector fields.

Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis

To analyze the properties of the coherent structures in near-wall turbulence, an extraction method based on wavelet transform (WT) and a verification procedure based on correlation analysis are proposed in this work. The flow field of the turbulent boundary layer is measured using the hot-film anemometer in a gravitational low-speed water tunnel. The obtained velocity profile and turbulence intensity are validated with traditional boundary layer theory. The fluctuating velocities at three testing positions are analyzed. Using the power spectrum density (PSD) and WT, coherent and incoherent parts of the near-wall turbulence are extracted and analyzed. The probability density functions (PDFs) of the extracted signals indicate that the incoherent structures of turbulence obey the Gaussian distribution, while the coherent structures deviate from it. The PDFs of coherent structures and original turbulence signals are similar, which means that coherent structures make the most contributions to the turbulence entrainment. A correlation parameter is defined at last to prove the validity of our extraction procedure.