Preface

**Chapter 9 101**

**Chapter 10 113**

Use of Transforms in Biomedical Signal Processing and Analysis

Insights from Systematic DFT Calculations on Superconductors *by Ian D.R. Mackinnon, Alanoud Almutairi and Jose A. Alarco*

*by Ette Harikrishna and Komalla Ashoka Reddy*

**II**

This book is devoted to some of the current fundamental tools on physics, communications, economy and health, that is, Fourier transforms and the treatment of superconductivity under the vision of the electromagnetic properties of both type I and type II superconductors. We begin in Chapter 1, "On the Zap Integral Operators over Fourier Transforms," with the latest applications and mathematical properties of Fourier transforms like projection operators aimed at communications and electromagnetic emissions. The authors prove that under an appropriate space generated by the solutions of the generalized inhomogeneous Fredholm equation it is possible to operate in it with integral operators like those of the quantum mechanics theory, killing and creating solutions of homogeneous or inhomogeneous Fredholm equations, which can be used to improve communications and broadcasting by selecting the type of behavior we need. Chapter 2, "Quantum Fourier Operators and Their Application" revises the structure of the Quantum Fourier Operators (QFT) and its implementation putting these concepts in their proper perspective, the authors provide a brief overview of quantum computation and provide a permutation structure for putting the QFT in the context of universal computation. In Chapter 3, "A Fast Method for Numerical Realization of Fourier Tools," the author presents a new algorithm for fast summations of truncated Fourier series. Chapter 4, "Fourier Transform Infrared Spectroscopy of the Animal Tissues," examines how animal tissues are extensively used as scaffolds for tissue engineering and regenerative therapies. They are typically subjected to a decellularization process to obtain cell-free extracellular matrix (ECM) scaffolds. It is important to identify the chemical structure of the ECM scaffolds, and Fourier transform infrared spectroscopy (FTIR) appears to be the technique of choice. The chapter presents FTIR spectra of native and decellularized buffalo aortae, buffalo diaphragms, goat skin, and native bovine cortical bone. The transmittance peaks are that of organic collagen amide A, amide B, amide I, amide II, and amide III chemical functional groups in both native and decellularized aortae, diaphragms, and skin. In bone, the transmittance peaks are that of inorganic ν1, ν3 PO4 3−, OH− in addition to organic collagen amide A, amide B, amide I, amide II, and amide III chemical functional groups. In Chapter 5, "Medical Image Classification Using the Discriminant Power Analysis (DPA) of Discrete Cosine Transform (DCT) Coefficients," we see the relevance of medical imaging systems in medicine. These systems assist specialists in making the final decision about a patient's condition, and strongly help in early cancer detection. The classification of mammogram images represents a very important operation to identify whether breast cancer is benign or malignant. The authors propose a new computer-aided diagnostic (CAD) system composed of three steps. In the first step, the input image is pre-processed to remove the noise and artifacts and to separate the breast profile from the pectoral muscle. This operation is a difficult task that can affect the final decision. In the second step, we propose a features extraction method based on the discrete cosine transform (DCT), where the processed images of the breast profiles are transformed by the DCT where the part containing the highest energy value is selected. Then, in the features selection step, a new most discriminative power coefficients algorithm is proposed to select the most significant features. The obtained results show the effectiveness. In Chapter 6, "Path Integral Two Dimensional Models of P– and D–Wave Superconductors and Collective Modes," we return to superconductivity in which the main parameter that is the order parameter describes superfluids and superconductors and all their main properties.

**Chapter 1**

**Abstract**

On the Zap Integral Operators

*Jaime Granados Samaniego and Alicia Cid Reborido*

*Ricardo Teodoro Paez Hernandez, Alejandro Perez Ricardez,*

We devote the current chapter to describe a class of integral operators with properties equivalent to a killer operator of the quantum mechanics theory acting over a determined state, literally killing the state but now operating over some kind of Fourier integral transforms that satisfies a certain Fredholm integral equation, we call this operators Zap Integral Operators (ZIO). The result of this action is to eliminate the inhomogeneous term and recover a homogeneous integral equation. We show that thanks to this class of operators we can explain the presence of two extremely different solutions of the same Generalized Inhomogeneous Fredholm equation. So we can regard the Generalized Inhomogeneous Fredholm Equation as a Super-Equation with two kinds of solutions, the resonant and the conventional but coexisting simultaneously. Also, we remember the generalized projection operators and we show they are the precursors of the ZIO. We present simultaneous academic

**Keywords:** integral operators, generalized inhomogeneous Fredholm equations,

Recently a new question about the solutions of integral Fredholm emerges, that is the question about the type of equation each of them solve. If we follow the steps or the clue marked by the linear second order differential equations the solutions of the inhomogeneous equation do not solve de homogeneous equation. But we have shown in a recent paper that both kind of solutions of the homogeneous and also the inhomogeneous Fredholm equations satisfy a third class of integral equation we named the Generalized Inhomogeneous Fredholm Equation (GIFE) which is only a bit different for the traditional inhomogeneous [1–3]. Even more, we can transform his appearance in a continuous form from homogeneous to inhomogeneous, but preserving his very extraordinary property: the two kinds of solutions are simultaneous solutions. This situation is quite different from differential equations but not the connection between eigenfunctions and solutions of inhomogeneous equations through the Green function [4–7]. And if we want to explain this behavior we find a founder: an integral operator which is hidden in the structure of the GIFE. There is no surprise in the fact that the new operator treats in different manner both kinds of

killer operators, evanescent waves, electromagnetic resonances

over Fourier Transforms

*Juan Manuel Velazquez Arcos,*

examples for both kinds of solutions.

**1. Introduction**

All properties of 2D–superconductors (for example, of CuO2 planes of HTSC) and, in particular, the collective excitations spectrum, are determined by these functionals. The authors consider all superconducting states, arising in symmetry classification of p-wave and d-wave 2D–superconductors, and calculate the full collective modes spectrum for each of these states. This will help to identify the type of pairing and the symmetry of the order parameter in HTSC and HFSC. In Chapter 7, "Periodogram Analysis under the Popper-Bayes Approach," the authors describe the Lomb-Scargle periodogram, its advantages, and pitfalls from a geometrical rather than statistical point of view. This means emphasizing more the transformation properties of the finite sampling (i.e., the available data) rather than the ensemble properties of the assumed model statistical distributions. The whole discussion is under the geophysical inverse theory point of view, the Tarantola's combination of information or the so-called Popper-Bayes approach. Finally, the authors discuss the Lomb-Scargle-Tarantola (LST) periodogram, which is an estimator of spectral content existing in irregularly sampled time series that implements these principles. Chapter 8, "Spread Option Pricing on Single-Core and Parallel Computing Architectures," introduces parallel computation for spread options using a two-dimensional Fourier transform. Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial securities. Their results indicate a significant increase in computational performance when the algorithm is performed on multiple CPU cores and GPU. Moreover, the literature on spread option pricing using FFT methods documents that the pricing accuracy increases with FFT grid size while the computational speed has opposite effect. By using the multi-core/GPU implementation, the tradeoff between pricing accuracy and speed is taken into account effectively. In Chapter 9, "Use of Transforms in Biomedical Signal Processing and Analysis," the authors write about low-frequency biomedical signals like electrocardiogram (ECG), photoplethysmographic (PPG), and blood pressure that need to be processed for further diagnosis and clinical monitoring. They use Fourier and wavelet transforms to reduce motion artifacts from PPG signals to produce correct blood oxygen saturation (SpO2) values. In an important contribution, FT is utilized for the generation of the reference signal for adaptive filter-based motion artifact reduction eliminating additional sensors for the acquisition of reference signals. In Chapter 10, "Insights from Systematic DFT Calculations on Superconductors," the authors present three systematic approaches to the use of Density Functional Theory (DFT) for interpretation and prediction of superconductivity in new or existing materials. Systematic calculations on conventional superconductors show that to attain a level of resolution comparable to the energy gap, two key parameters, Δk and the cut-off energy, must be optimized for a specific compound. The optimal level of resolution is achieved with k-grids smaller than the minimum reciprocal space separation between key parallel Fermi surfaces. These approaches enable estimates of superconducting properties including the transition temperature (Tc). They demonstrate these approaches for the conventional superconductors MgB2, metal substituted MgB2, and boron-doped diamond.

**Juan Manuel Velázquez Arcos**

Basic Sciences Department, Metropolitan Autonomous University, CDMX, México
