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This monograph has arisen out of a number of attempts spanning almost five decades to understand how one might examine the evolution of densities in systems whose dynamics are described by differential delay equations. Though the authors have no definitive solution to the problem, they offer this contribution in an attempt to define the problem as they see it, and to sketch out several obvious attempts that have been suggested to solve the problem and which seem to have failed. They hope that by being available to the general mathematical community, they will inspire others to consider–and hopefully solve–the problem. Serious attempts have been made by all of the authors over the years and they have made reference to these where appropriate. .

Mathematical analysis. --- Analysis (Mathematics). --- Measure theory. --- Dynamics. --- Ergodic theory. --- Vibration. --- Dynamical systems. --- Probabilities. --- Analysis. --- Measure and Integration. --- Dynamical Systems and Ergodic Theory. --- Vibration, Dynamical Systems, Control. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Cycles --- Sound --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- 517.1 Mathematical analysis --- Mathematical analysis --- Delay differential equations. --- Delay equations (Differential equations) --- Delay functional differential equations --- Differential delay equations --- Differential equations --- Differential equations with lag --- Functional differential equations --- Retarded argument (Differential equations) --- Retarded differential equations --- Retarded functional differential equations --- Time-lag systems (Differential equations) --- Delay equations --- Retarded argument --- Time-lag equations

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This book presents collective works published in the recent Special Issue (SI) entitled "Multivariate Approximation for Solving ODE and PDE". These papers describe the different approaches and related objectives in the field of multivariate approximation. The articles in fact present specific contents of numerical methods for the analysis of the approximation, as well as the study of ordinary differential equations (for example oscillating with delay) or that of partial differential equations of the fractional order, but all linked by the objective to present analytical or numerical techniques for the simplification of the study of problems involving relationships that are not immediately computable, thus allowing to establish a connection between different fields of mathematical analysis and numerical analysis through different points of view and investigation. The present contents, therefore, describe the multivariate approximation theory, which is today an increasingly active research area that deals with a multitude of problems in a wide field of research. This book brings together a collection of inter-/multi-disciplinary works applied to many areas of applied mathematics in a coherent manner.

Research & information: general --- Mathematics & science --- nonlinear equations --- iteration methods --- one-point methods --- order of convergence --- oscillatory solutions --- nonoscillatory solutions --- second-order --- neutral differential equations --- multiple roots --- optimal convergence --- bivariate function --- divided difference --- inverse difference --- blending difference --- continued fraction --- Thiele–Newton’s expansion --- Viscovatov-like algorithm --- symmetric duality --- non-differentiable --- (G,αf)-invexity/(G,αf)-pseudoinvexity --- (G,αf)-bonvexity/(G,αf)-pseudobonvexity --- duality --- support function --- nondifferentiable --- strictly pseudo (V,α,ρ,d)-type-I --- unified dual --- efficient solutions --- Iyengar inequality --- right and left generalized fractional derivatives --- iterated generalized fractional derivatives --- generalized fractional Taylor’s formulae --- poisson equation --- domain decomposition --- asymmetric iterative schemes --- group explicit --- parallel computation --- even-order differential equations --- neutral delay --- oscillation --- Hilbert transform --- Hadamard transform --- hypersingular integral --- Bernstein polynomials --- Boolean sum --- simultaneous approximation --- equidistant nodes --- fourth-order --- delay differential equations --- riccati transformation --- parameter estimation --- physical modelling --- oblique decomposition --- least-squares

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This volume consists of a collection of 14 accepted submissions (including several invited feature articles) to the Special Issue of MDPI's journal Symmetry on the general subject area of integral transformations, operational calculus and their applications from many different parts around the world. The main objective of the Special Issue was to gather review, expository, and original research articles dealing with the state-of-the-art advances in integral transformations and operational calculus as well as their multidisciplinary applications, together with some relevance to the aspect of symmetry. Various families of fractional-order integrals and derivatives have been found to be remarkably important and fruitful, mainly due to their demonstrated applications in numerous diverse and widespread areas of mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional-order operators provide potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations; fractional-calculus analogues and extensions of each of these equations; and various other problems involving special functions of mathematical physics and applied mathematics, as well as their extensions and generalizations in one or more variables.

History of engineering & technology --- Stancu-type Bernstein operators --- Bézier bases --- Voronovskaja-type theorems --- modulus of continuity --- rate of convergence --- bivariate operators --- approximation properties --- statistical convergence --- P-convergent --- statistically and relatively modular deferred-weighted summability --- relatively modular deferred-weighted statistical convergence --- Korovkin-type approximation theorem --- modular space --- convex space --- N-quasi convex modular --- N-quasi semi-convex modular --- vehicle collaborative content downloading --- fuzzy comprehensive evaluation --- VANET --- delay differential equations --- integral operator --- periodic solutions --- subordinations --- exponential function --- Hankel determinant --- fractional differential equations with input --- Mittag-Leffler stability --- left generalized fractional derivative --- ρ-Laplace transforms --- functional integral equations --- Banach algebra --- fixed point theorem --- measure of noncompactness --- Geometric Function Theory --- q-integral operator --- q-starlike functions of complex order --- q-convex functions of complex order --- (δ,q)-neighborhood --- meromorphic multivalent starlike functions --- subordination --- univalent function --- symmetric differential operator --- unit disk --- analytic function --- analytic functions --- conic region --- Hadamard product --- differential subordination --- differential superordination --- generalized fractional differintegral operator --- Convex function --- Simpson’s rule --- differentiable function --- weights --- positive integral operators --- convolution operators --- n/a --- Bézier bases --- Simpson's rule

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This volume consists of a collection of 14 accepted submissions (including several invited feature articles) to the Special Issue of MDPI's journal Symmetry on the general subject area of integral transformations, operational calculus and their applications from many different parts around the world. The main objective of the Special Issue was to gather review, expository, and original research articles dealing with the state-of-the-art advances in integral transformations and operational calculus as well as their multidisciplinary applications, together with some relevance to the aspect of symmetry. Various families of fractional-order integrals and derivatives have been found to be remarkably important and fruitful, mainly due to their demonstrated applications in numerous diverse and widespread areas of mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional-order operators provide potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations; fractional-calculus analogues and extensions of each of these equations; and various other problems involving special functions of mathematical physics and applied mathematics, as well as their extensions and generalizations in one or more variables.

Stancu-type Bernstein operators --- Bézier bases --- Voronovskaja-type theorems --- modulus of continuity --- rate of convergence --- bivariate operators --- approximation properties --- statistical convergence --- P-convergent --- statistically and relatively modular deferred-weighted summability --- relatively modular deferred-weighted statistical convergence --- Korovkin-type approximation theorem --- modular space --- convex space --- N-quasi convex modular --- N-quasi semi-convex modular --- vehicle collaborative content downloading --- fuzzy comprehensive evaluation --- VANET --- delay differential equations --- integral operator --- periodic solutions --- subordinations --- exponential function --- Hankel determinant --- fractional differential equations with input --- Mittag-Leffler stability --- left generalized fractional derivative --- ρ-Laplace transforms --- functional integral equations --- Banach algebra --- fixed point theorem --- measure of noncompactness --- Geometric Function Theory --- q-integral operator --- q-starlike functions of complex order --- q-convex functions of complex order --- (δ,q)-neighborhood --- meromorphic multivalent starlike functions --- subordination --- univalent function --- symmetric differential operator --- unit disk --- analytic function --- analytic functions --- conic region --- Hadamard product --- differential subordination --- differential superordination --- generalized fractional differintegral operator --- Convex function --- Simpson’s rule --- differentiable function --- weights --- positive integral operators --- convolution operators --- n/a --- Bézier bases --- Simpson's rule