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© 2021 Functional Analysis with application in algorithm Edit:Prof. Dr. Wenfeng Wang, Bin Hu Tenured Professor, The IMT Institute, India ISBN:978-1-80430-000-8 |
Description
Functional analysis was born in the early 20th century. Functional analysis was developed from the study of variational problems, integral equations and theoretical physics. The theoretical value of this book lies in the comprehensive application of functional theory, geometry and modern mathematics to study the functional, operator and limit theory on infinite dimensional vector space. Functional analysis can be regarded as analytic geometry and mathematical analysis of infinite dimensional vector space. Therefore, the practical significance of this book lies in that functional analysis is applied in mathematical physical equations, probability theory, computational mathematics and other branches, and it is also a mathematical tool to study physical systems with infinite degrees of freedom.
With the development of artificial intelligence, functional analysis has been successfully applied in deep learning, width learning and optimization theory. This book describes the practical application of functional analysis in specific intelligent algorithms, and provides important mathematical knowledge and practical knowledge for various research fields.
Functional analysis is an important subject in our study, and also an important tool for us to solve problems. It is involved in many fields. In this book, some basic concepts and lemmas in functional analysis are introduced. In the first part, the definition and properties of several Spaces in functional field are introduced, as well as the narration and proof of related theorems and inferences. In the second part, the definition and properties of fully ordered set and partially ordered set are discussed. The third part describes the equivalence of norm; The last part of the book describes the application of functional analysis in different fields and mathematical knowledge, such as the application of functional analysis in economics, the application of Hopf branch of differential equation and the application of artificial intelligence and so on. Functional analysis is an important mathematical tool, and it is of great significance to study it.
About the authors
Wenfeng Wang is a full professor in Shanghai Institute of Technology. A key tallent in Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences (2018- ). A reviewer of many SCI journals,including some top journals - Water Research, Science China-Information Sciences, Science of the Total Environment, Environmental Pollution, IEEE Transactions on Automation Science and Engineering and etc. A keynote speaker of AMICR2019, IACICE2020, OAES2020, AICS2021, 3DIT-MSP&DL2020 and etc. A scientist in chief of RealMax, Shanghai Lingang Artificial Intelligence Laboratory and etc.
Bin Hu is an engineer and a counselor at Changsha Normal University, ACM Member, IEEE Member, GCDF, he is the guest editor of EI-indexed journals or book series: International Journal of Reasoning-based Intelligent Systems (IJRIS) , International Journal of Information Systems and Supply Chain Management (IJISSCM),International Journal of Information Systems in the Service Sector(IJISSS) and Smart Innovation, Systems and Technologies(SIST), the technical editor of International Journal of Computer and Communication Technology (IJCCT). He published many articles in high standard journals like Knowledge-based System(KS), Journal of Intelligent and Fuzzy Control (JIFS), Enterprise Information System(EIS), Financial Innovation(FI) and etc.
Table of Contents
Preference
Chapter1 Metric space and topological space
1.1 Introduction1.2 Complete order and partial order sets
1.3 Complete metric space
1.3.1 Metric Space
1.3.2 Hilbert Space
1.4 From measure space to a topological space
1.4.1 An important theorem in metric space and topological space.
1.4.2 the proposition proved
References
Chapter2 Infinite product and Tikhonov's theorem2.1 Introduction
2.2 Proof of the infinite product theorem and Tikhonov's theorem
2.1.1 Related properties:
2.3 The proof process of Tikhonov's theorem:
Chapter3 Normed space and Banach space
3.1 The basic concept of normed space
3.2 Normed linear Spaces
3.3 Examples of normed spaces
3.3.1 Continuous function space
3.3.2 space
3.4 Equivalent norm finite dimensional normed space
3.4.1 Equivalent norm
3.4.2 Equivalent norm examples
3.4.3 Equivalence of norm
3.5 Finite dimensional space
3.6 Geometric characteristics of finite dimensional normed space
3.7 Banach space
3.7.1 An example of Banach space
3.7.2 Prove is Banach space
3.7.3 Application of functional analysis to dynamical systems
References
Chapter4 Linear Operator and isomorphism relations
4.1 Linear operator
4.1.1The basic concepts of linear operators and linear functionals
4.1.2 normed space composed of bounded linear operators
4.1.3 Related propositions
4.2 Convergence and completion of bounded linear operator spaces
4.2.1 convergence in bounded linear operator spaces
4.2.3 completeness of bounded linear operator spaces
4.3 Compact operator
4.3.1 Space composed of compact operators
4.3.2 Spectrum of Compact Operators
4.3.3 Definition of Operator Spectrum
4.3.4 Structure of Zero Space of Compact Linear Operators
4.4The Sequence Space
References
Chapter5 Banach-Alaoglu theorem
5.1 Banach-Alaoglu theorem
Chapter6 Open mappings and isomorphism relations
6.1 Open mapping theorem
6.2 Parallel and some relative definition
6.3 Quantum detuning
Chapter7 Closed graph theorem and uniform boundedness
7.1 Closed graph theorem
Chapter8 Extension of functional analysis
8.1 Measurement theory
8.1.1 Definition of external measure
8.1.2 The basic properties of lateral degree
8.1.3 Definition of measurable set:
8.1.4 Summary
8.2 The connection with functional analysis in singular perturbation
8.2.1 Fixed point theorem
8.2.2 Brouwer's fixed point theorem (1910)
8.2.3 KakutaniFixed point theorem (1941)
8.2.4 Fredholm’s alternative lemma:
8.3 Baire Category Theorem
8.3.1 Application of Baire Category Theorem
8.3.2 BCT1 indicates that each of the following is a Bell space
8.3.3 Several Definitions
References
Chapter 9 Image edge detection and recognition and reinforcement learning expansion
9.1 Introduction
9.2 Median filter denoising
9.3 Automatic threshold segmentation of grayscale images(Otsu method)
9.4 Intelligent detection of occluded face images
9.4.1Basic principles of image edge detection
9.4.2 The basic ideas and steps of edge detection
9.4.3 Edge detection based on first derivative
9.4.4 Edge detection based on second derivative
9.3 Image edge detection algorithm
9.3.1 Roberts operator
9.3.2 Prewitt operator
9.3.3 Sobel operator
9.3.4 LoG operator
9.3.5 Canny operator
References
Chapter 10 Canny Operator and Back Propagation Algorithm in Neural Network
10.1 Canny operator
10.1.1 Three criteria of Canny operator
10.1.2 The specific algorithm of Canny operator to find edge points
10.2 Derivation of back propagation algorithm in neural network
10.2.1 Principle overview
10.2.2 Algorithm derivation
10.3 Backpropagation algorithm implementation
10.3.1 Experiment summary:
References
Chapter 11 A Brief Introduction to Several Algorithms of Image Processing
11.1 Image signal acquisition and reconstruction
11.1.1 Low-pass sampling formula
11.2 Reconstruction formula of sampling signal
Chapter12 Summary of image denoising and image fusion
12.1 Introduction
12.2 Image denoising
12.2.1 Mean filter denoising
12.3 Image fusion
12.3.1 Image pyramid
12.3.2Gaussian Pyramid
12.3.3 Pyramid of Laplace
12.3.4 Wavelet transform fusion
12.4 Image translation
12.4.1 Mirror transformation of image
12.5 Image cropping
12.5.1 Experimental results
12.6 Image rotation
12.6.1 Experimental results
12.7 Image zoom
12.8 deep learning
12.9 Summary
References
Chapter13 Explanation of several knowledge points about intelligent algorithms
13.1 Introduction
13.2 Nonlocal mean filtering
13.2.1 Detailed explanation of the principle and formula of non-local mean filtering
13.2.2 Detailed explanation of fast non-local mean filtering formula
13.3 Optimal threshold
13.4 Regional growth
13.4.1 principle
13.4.2 Residual attention network
13.4.3 Grey model
13.4.4 Convolutional network convolutional layer and pooling layer[1]
13.4.5 Pooling layer
13.5 Softmax Return[2]
13.6 Conclusion
References
Chapter 14 Wavelet soft hard threshold denoising and image fusion algorithm
14.1 Introduction
14.2 Wavelet soft and hard threshold denoising
14.2.1 The process of wavelet threshold denoising method
14.2.2 Wavelet base choice
14.2.3 The choice of decomposition scale
14.2.4 Selection of threshold function
14.2.5 Wavelet hard threshold denoising
14.2.6 Comparison of the advantages and disadvantages of soft and hard threshold functions
14.2.7 Threshold selection rules
14.3 Image fusion algorithm
14.3.1 Gray-scale weighted average, take the big, take the small fusion
14.3.2 PCA integration
References
Chapter 15 Full-text Interpretation of Object Recognition Based on Linux Embedded System
15.1 Introduction
15.2 Linux system
15.2.1 Embedded
15.2.2 Ubuntu
15.3 Install OpenCV Dependency
15.4 Download OpenCV
15.5 Configure and compile OpenCV
15.6 For different versions of Python
Chapter16 Overview and Analysis of Image Recognition Theory
16.1 Introduction
16.2 Traditional edge detection
16.2.1 Sobel operator
16.2.2 Sobel algorithm process
2.1.2 Evaluation of Sobel Algorithm
16.2.3 Robert operator
16.2.4 Robert algorithm process
16.2.5 Evaluation of Robert's Algorithm
16.3 Prewitt operator
16.3.1 Overview of Pprewitt operator
16.3.2 Prewitt algorithm process
16.3.3 Evaluation of Prewitt Algorithm
16.3.4 Comparison of several operators
16.4 CNN edge detection
16.4.1 Overview of Neural Networks
3.2 Application of Convolutional Neural Network and Edge Detection
16.4.2 Summary
References
Chapter17 Three steps to realize face recognition
17.1 Contour wave threshold denoising
17.2 Histogram
17.2.1 What is a histogram
17.2.2 Histogram equalization (normalization)
17.2.3 Balanced experimental results
17.2.4 Histogram matching (prescribed)
17.2.5 Histogram matching experiment results
17.3 Three steps to realize face recognition
17.3.1 Face recognition process
17.3.2 Final realization
17.3.3 Summary
References