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Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations. This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.

Burgers equation. --- Heat equation.

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Measurements, Mechanisms, and Models of Heat Transport offers an interdisciplinary approach to the dynamic response of matter to energy input. Using a combination of fundamental principles of physics, recent developments in measuring time-dependent heat conduction, and analytical mathematics, this timely reference summarizes the relative advantages of currently used methods, and remediates flaws in modern models and their historical precursors. Geophysicists, physical chemists, and engineers will find the book to be a valuable resource for its discussions of radiative transfer models and the kinetic theory of gas, amended to account for atomic collisions being inelastic. This book is a prelude to a companion volume on the thermal state, formation, and evolution of planets. Covering both microscopic and mesoscopic phenomena of heat transport, Measurements, Mechanisms, and Models of Heat Transport offers both the fundamental knowledge and up-to-date measurements and models to encourage further improvements.--

Terrestrial heat flow. --- Terrestrial heat transfer --- Burgers equation --- Earth temperature --- Geophysics --- Heat --- Heat budget (Geophysics) --- Heat equation --- Transmission

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Mathematical analysis --- Reaction-diffusion equations. --- Approximation theory. --- Burgers equation. --- Equations de réaction-diffusion --- Théorie de l'approximation --- Burgers, Equation de --- 51 <082.1> --- Mathematics--Series --- Équations de réaction-diffusion. --- Approximation, Théorie de l'. --- Equations de réaction-diffusion --- Théorie de l'approximation --- Approximation theory --- Burgers equation --- Reaction-diffusion equations --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Diffusion equation, Nonlinear --- Heat flow equation, Nonlinear --- Nonlinear diffusion equation --- Nonlinear heat flow equation --- Heat equation --- Navier-Stokes equations --- Turbulence --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems

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These lecture notes are woven around the subject of Burgers' turbulence/KPZ model of interface growth, a study of the nonlinear parabolic equation with random initial data. The analysis is conducted mostly in the space-time domain, with less attention paid to the frequency-domain picture. However, the bibliography contains a more complete information about other directions in the field which over the last decade enjoyed a vigorous expansion. The notes are addressed to a diverse audience, including mathematicians, statisticians, physicists, fluid dynamicists and engineers, and contain both rigorous and heuristic arguments. Because of the multidisciplinary audience, the notes also include a concise exposition of some classical topics in probability theory, such as Brownian motion, Wiener polynomial chaos, etc.

Partial differential equations --- Interfaces (Physical sciences) --- Turbulence --- Burgers equation --- Differential equations, Parabolic --- Mathematical models --- Mathematical Theory --- Atomic Physics --- Physics --- Mathematics --- Physical Sciences & Mathematics --- Differentiaalvergelijkingen [Parabolische ] --- Differential equations [Parabolic] --- Diffusion equation [Nonlinear ] --- Equations differentielles paraboliques --- Heat flow equation [Nonlinear ] --- Interface (Physical sciences) --- Partial differential equations. --- Probabilities. --- Partial Differential Equations. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Burgers equation. --- Mathematical models. --- Diffusion equation, Nonlinear --- Heat flow equation, Nonlinear --- Nonlinear diffusion equation --- Nonlinear heat flow equation --- Heat equation --- Navier-Stokes equations --- Interfaces (Physical sciences) - Mathematical models --- Turbulence - Mathematical models

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Numerical methods are a specific form of mathematics that involve creating and use of algorithms to map out the mathematical core of a practical problem. Numerical methods naturally find application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, and even arts. The common uses of numerical methods include approximation, simulation, and estimation, and there is almost no scientific field in which numerical methods do not find a use. Results communicated here include topics ranging from statistics (Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions) and Statistical software packages (dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS) to new approaches for numerical solutions (Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1; On q-Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems; Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method; On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence; Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations) to the use of wavelets (Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron) and methods for visualization (A Simple Method for Network Visualization).

Research & information: general --- Mathematics & science --- Clenshaw–Curtis–Filon --- high oscillation --- singular integral equations --- boundary singularities --- local convergence --- nonlinear equations --- Banach space --- Fréchet-derivative --- finite integration method --- shifted Chebyshev polynomial --- Caputo fractional derivative --- Burgers’ equation --- coupled Burgers’ equation --- maxmin --- supporting vector --- matrix norm --- TMS coil --- optimal geolocation --- probability computing --- Monte Carlo simulation --- order statistics --- extreme values --- outliers --- multiobjective programming --- methods of quasi-Newton type --- Pareto optimality --- q-calculus --- rate of convergence --- wavelets on 3D ball --- uniform 3D grid --- volume preserving map --- Network --- graph drawing --- planar visualizations --- multiple root solvers --- composite method --- weight-function --- derivative-free method --- optimal convergence --- multivariate polynomial regression designs --- G-optimality --- D-optimality --- multiplicative algorithms --- G-efficiency --- Caratheodory-Tchakaloff discrete measure compression --- Non-Negative Least Squares --- accelerated Lawson-Hanson solver