Ebook: Intelligent Mathematics: Computational Analysis

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Knowledge can be modelled and computed using computational mathematical methods, then lead to real world conclusions. The strongly related to that Computational Analysis is a very large area with lots of applications. This monograph includes a great variety of topics of Computational Analysis. We present: probabilistic wavelet approximations, constrained abstract approximation theory, shape preserving weighted approximation, non positive approximations to definite integrals, discrete best approximation, approximation theory of general Picard singular operators including global smoothness preservation property, fractional singular operators. We also deal with non-isotropic general Picard singular multivariate operators and q-Gauss-Weierstrass singular q-integral operators.We talk about quantitative approximations by shift-invariant univariate and multivariate integral operators, nonlinear neural networks approximation, convergence with rates of positive linear operators, quantitative approximation by bounded linear operators, univariate and multivariate quantitative approximation by stochastic positive linear operators on univariate and multivariate stochastic processes. We further present right fractional calculus and give quantitative fractional Korovkin theory of positive linear operators. We also give analytical inequalities, fractional Opial inequalities, fractional identities and inequalities regarding fractional integrals.We further deal with semigroup operator approximation, simultaneous Feller probabilistic approximation. We also present Fuzzy singular operator approximations.We give transfers from real to fuzzy approximation and talk about fuzzy wavelet and fuzzy neural networks approximations, fuzzy fractional calculus and fuzzy Ostrowski inequality. We talk about discrete fractional calculus, nabla discrete fractional calculus and inequalities.We study the q-inequalities, and q-fractional inequalities. We further study time scales: delta and nabla approaches, duality principle and inequalities. We introduce delta and nabla time scales fractional calculus and inequalities.We finally study convergence with rates of approximate solutions to exact solution of multivariate Dirichlet problem and multivariate heat equation, and discuss the uniqueness of solution of general evolution partial differential equation in multivariate time. The exposed results are expected to find applications to: applied and computational mathematics, stochastics, engineering, artificial intelligence, vision, complexity and machine learning. This monograph is suitable for graduate students and researchers.

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PLEASE USE THE FILE BACK COVER!

Knowledge can be modelled and computed using computational mathematical methods, then lead to real world conclusions. The strongly related to that Computational Analysis is a very large area with lots of applications. This monograph includes a great variety of topics of Computational Analysis. We present: probabilistic wavelet approximations, constrained abstract approximation theory, shape preserving weighted approximation, non positive approximations to definite integrals, discrete best approximation, approximation theory of general Picard singular operators including global smoothness preservation property, fractional singular operators. We also deal with non-isotropic general Picard singular multivariate operators and q-Gauss-Weierstrass singular q-integral operators.We talk about quantitative approximations by shift-invariant univariate and multivariate integral operators, nonlinear neural networks approximation, convergence with rates of positive linear operators, quantitative approximation by bounded linear operators, univariate and multivariate quantitative approximation by stochastic positive linear operators on univariate and multivariate stochastic processes. We further present right fractional calculus and give quantitative fractional Korovkin theory of positive linear operators. We also give analytical inequalities, fractional Opial inequalities, fractional identities and inequalities regarding fractional integrals.We further deal with semigroup operator approximation, simultaneous Feller probabilistic approximation. We also present Fuzzy singular operator approximations.We give transfers from real to fuzzy approximation and talk about fuzzy wavelet and fuzzy neural networks approximations, fuzzy fractional calculus and fuzzy Ostrowski inequality. We talk about discrete fractional calculus, nabla discrete fractional calculus and inequalities.We study the q-inequalities, and q-fractional inequalities. We further study time scales: delta and nabla approaches, duality principle and inequalities. We introduce delta and nabla time scales fractional calculus and inequalities.We finally study convergence with rates of approximate solutions to exact solution of multivariate Dirichlet problem and multivariate heat equation, and discuss the uniqueness of solution of general evolution partial differential equation in multivariate time. The exposed results are expected to find applications to: applied and computational mathematics, stochastics, engineering, artificial intelligence, vision, complexity and machine learning. This monograph is suitable for graduate students and researchers.

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PLEASE USE THE FILE BACK COVER!

Knowledge can be modelled and computed using computational mathematical methods, then lead to real world conclusions. The strongly related to that Computational Analysis is a very large area with lots of applications. This monograph includes a great variety of topics of Computational Analysis. We present: probabilistic wavelet approximations, constrained abstract approximation theory, shape preserving weighted approximation, non positive approximations to definite integrals, discrete best approximation, approximation theory of general Picard singular operators including global smoothness preservation property, fractional singular operators. We also deal with non-isotropic general Picard singular multivariate operators and q-Gauss-Weierstrass singular q-integral operators.We talk about quantitative approximations by shift-invariant univariate and multivariate integral operators, nonlinear neural networks approximation, convergence with rates of positive linear operators, quantitative approximation by bounded linear operators, univariate and multivariate quantitative approximation by stochastic positive linear operators on univariate and multivariate stochastic processes. We further present right fractional calculus and give quantitative fractional Korovkin theory of positive linear operators. We also give analytical inequalities, fractional Opial inequalities, fractional identities and inequalities regarding fractional integrals.We further deal with semigroup operator approximation, simultaneous Feller probabilistic approximation. We also present Fuzzy singular operator approximations.We give transfers from real to fuzzy approximation and talk about fuzzy wavelet and fuzzy neural networks approximations, fuzzy fractional calculus and fuzzy Ostrowski inequality. We talk about discrete fractional calculus, nabla discrete fractional calculus and inequalities.We study the q-inequalities, and q-fractional inequalities. We further study time scales: delta and nabla approaches, duality principle and inequalities. We introduce delta and nabla time scales fractional calculus and inequalities.We finally study convergence with rates of approximate solutions to exact solution of multivariate Dirichlet problem and multivariate heat equation, and discuss the uniqueness of solution of general evolution partial differential equation in multivariate time. The exposed results are expected to find applications to: applied and computational mathematics, stochastics, engineering, artificial intelligence, vision, complexity and machine learning. This monograph is suitable for graduate students and researchers.

Content:
Front Matter....Pages -
Introduction....Pages 1-11
Convex Probabilistic Wavelet Like Approximation....Pages 13-27
Bidimensional Constrained Wavelet Like Approximation....Pages 29-40
Multidimensional Probabilistic Scale Approximation....Pages 41-56
Multidimensional Probabilistic Approximation in Wavelet Like Structure....Pages 57-67
About L-Positive Approximations....Pages 69-87
About Shape Preserving Weighted Uniform Approximation....Pages 89-91
Jackson-Type Nonpositive Approximations for Definite Integrals....Pages 93-98
Quantitative Uniform Convergence of Smooth Picard Singular Integral Operators....Pages 99-113
Global Smoothness and Simultaneous Approximation by Smooth Picard Singular Operators....Pages 115-136
Approximation with Rates by Fractional Smooth Picard Singular Operators....Pages 137-150
Multivariate Generalized Picard Singular Integral Operators....Pages 151-167
Approximation by q-Gauss-Weierstrass Singular Integral Operators....Pages 169-189
Quantitative Approximation by Univariate Shift-Invariant Integral Operators....Pages 191-205
Quantitative Approximation by Multivariate Shift-Invariant Convolution Operators....Pages 207-214
Approximation by a Nonlinear Cardaliaguet-Euvrard Neural Network Operator of Max-Product Kind....Pages 215-237
A Generalized Shisha - Mond Type Inequality....Pages 239-260
Quantitative Approximation by Bounded Linear Operators....Pages 261-271
Quantitative Stochastic Korovkin Theory....Pages 273-274
Quantitative Multidimensional Stochastic Korovkin Theory....Pages 275-280
About the Right Fractional Calculus....Pages 281-298
Fractional Convergence Theory of Positive Linear Operators....Pages 299-331
Fractional Trigonometric Convergence Theory of Positive Linear Operators....Pages 333-354
Extended Integral Inequalities....Pages 355-376
Balanced Fractional Opial Integral Inequalities....Pages 377-397
Montgomery Identities for Fractional Integrals and Fractional Inequalities....Pages 399-422
Simultaneous Approximation Using the Feller Probabilistic Operator....Pages 423-433
Global Smoothness Preservation and Uniform Convergence of Singular Integral Operators in the Fuzzy Sense....Pages 435-441
Real Approximations Transferred to Vectorial and Fuzzy Setting....Pages 443-467
High Order Multivariate Approximation by Multivariate Wavelet Type and Neural Network Operators in the Fuzzy Sense....Pages 469-485
Fuzzy Fractional Calculus and the Ostrowski Integral Inequality....Pages 487-501
About Discrete Fractional Calculus with Inequalities....Pages 503-522
Discrete Nabla Fractional Calculus with Inequalities....Pages 523-552
About q- Inequalities....Pages 553-574
About q- Fractional Inequalities....Pages 575-585
Inequalities on Time Scales....Pages 587-600
Nabla Inequalities on Time Scales....Pages 601-613
The Principle of Duality in Time Scales with Inequalities....Pages 615-625
Foundations of Delta Fractional Calculus on Time Scales with Inequalities....Pages 627-648
Principles of Nabla Fractional Calculus on Time Scales with Inequalities....Pages 649-672
Optimal Error Estimate for the Numerical Solution of Multidimensional Dirichlet Problem....Pages 673-694
Optimal Estimate for the Numerical Solution of Multidimensional Dirichlet Problem for the Heat Equation....Pages 695-710
Uniqueness of Solution in Evolution in Multivariate Time....Pages 711-729
Back Matter....Pages 731-747
....Pages 749-764