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This book studies the construction methods for solving one-dimensional and multidimensional inverse dynamical problems for hyperbolic equations with memory. The theorems of uniqueness, stability and existence of solutions of these inverse problems are obtained. This book discusses the processes, by using generalized solutions, the spread of elastic or electromagnetic waves arising from sources of the type of pulsed directional “impacts” or “explosions”. This book presents new results in the study of local and global solvability of kernel determination problems for a half-space. It describes the problems of reconstructing the coefficients of differential equations and the convolution kernel of hyperbolic integro-differential equations by the method of Dirichlet-to-Neumann. The book will be useful for researchers and students specializing in the field of inverse problems of mathematical physics.

Differential equations. --- Differential Equations. --- 517.91 Differential equations --- Differential equations --- Equacions diferencials hiperbòliques --- Funcions de Kernel --- Problemes inversos (Equacions diferencials)

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R (Computer program language) --- Kernel functions. --- Functions, Kernel --- Functions of complex variables --- Geometric function theory --- GNU-S (Computer program language) --- Domain-specific programming languages --- Funcions de Kernel --- Aprenentatge automàtic --- R (Llenguatge de programació)

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The most crucial ability for machine learning and data science is mathematical logic for grasping their essence rather than relying on knowledge or experience. This textbook addresses the fundamentals of kernel methods for machine learning by considering relevant math problems and building Python programs. The book's main features are as follows: The content is written in an easy-to-follow and self-contained style. The book includes 100 exercises, which have been carefully selected and refined. As their solutions are provided in the main text, readers can solve all of the exercises by reading the book. The mathematical premises of kernels are proven and the correct conclusions are provided, helping readers to understand the nature of kernels. Source programs and running examples are presented to help readers acquire a deeper understanding of the mathematics used. Once readers have a basic understanding of the functional analysis topics covered in Chapter 2, the applications are discussed in the subsequent chapters. Here, no prior knowledge of mathematics is assumed. This book considers both the kernel for reproducing kernel Hilbert space (RKHS) and the kernel for the Gaussian process; a clear distinction is made between the two.

Programming --- Artificial intelligence. Robotics. Simulation. Graphics --- neuronale netwerken --- fuzzy logic --- cybernetica --- programmeren (informatica) --- KI (kunstmatige intelligentie) --- Artificial intelligence. --- Artificial intelligence --- Data processing. --- Funcions de Kernel --- Aprenentatge automàtic --- Python (Llenguatge de programació) --- AI (artificiële intelligentie)

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This book contains select chapters on support vector algorithms from different perspectives, including mathematical background, properties of various kernel functions, and several applications. The main focus of this book is on orthogonal kernel functions, and the properties of the classical kernel functions—Chebyshev, Legendre, Gegenbauer, and Jacobi—are reviewed in some chapters. Moreover, the fractional form of these kernel functions is introduced in the same chapters, and for ease of use for these kernel functions, a tutorial on a Python package named ORSVM is presented. The book also exhibits a variety of applications for support vector algorithms, and in addition to the classification, these algorithms along with the introduced kernel functions are utilized for solving ordinary, partial, integro, and fractional differential equations. On the other hand, nowadays, the real-time and big data applications of support vector algorithms are growing. Consequently, the Compute Unified Device Architecture (CUDA) parallelizing the procedure of support vector algorithms based on orthogonal kernel functions is presented. The book sheds light on how to use support vector algorithms based on orthogonal kernel functions in different situations and gives a significant perspective to all machine learning and scientific machine learning researchers all around the world to utilize fractional orthogonal kernel functions in their pattern recognition or scientific computing problems.

Algebraic fields. --- Polynomials. --- Mathematical optimization. --- Quantitative research. --- Machine learning. --- Pattern recognition systems. --- Python (Computer program language). --- Field Theory and Polynomials. --- Optimization. --- Data Analysis and Big Data. --- Machine Learning. --- Automated Pattern Recognition. --- Python. --- Scripting languages (Computer science) --- Pattern classification systems --- Pattern recognition computers --- Pattern perception --- Computer vision --- Learning, Machine --- Artificial intelligence --- Machine theory --- Data analysis (Quantitative research) --- Exploratory data analysis (Quantitative research) --- Quantitative analysis (Research) --- Quantitative methods (Research) --- Research --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Algebra --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Aprenentatge automàtic --- Algorismes --- Funcions de Kernel --- Python (Llenguatge de programació) --- Python (Computer program language)