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Applied functional analysis has an extensive history. In the last century, this field has often been used in physical sciences, such as wave and heat phenomena. In recent decades, with the development of nonlinear functional analysis, this field has been used to model a variety of engineering, medical, and computer sciences. Two of the most significant issues in this area are modeling and optimization. Thus, we consider some recently published works on fixed point, variational inequalities, and optimization problems. These works could lead readers to obtain new novelties and familiarize them with some applications of this area.

Research & information: general --- Mathematics & science --- vector variational-like inequalities --- vector optimization problems --- limiting (p,r)-α-(η,θ)-invexity --- Lipschitz continuity --- Fan-KKM theorem --- set-valued optimization problems --- higher-order weak adjacent epiderivatives --- higher-order mond-weir type dual --- benson proper efficiency --- fractional calculus --- ψ-fractional integrals --- fractional differential equations --- contraction --- hybrid contractions --- volterra fractional integral equations --- fixed point --- inertial-like subgradient-like extragradient method with line-search process --- pseudomonotone variational inequality problem --- asymptotically nonexpansive mapping --- strictly pseudocontractive mapping --- sequentially weak continuity --- method with line-search process --- pseudomonotone variational inequality --- strictly pseudocontractive mappings --- common fixed point --- hyperspace --- informal open sets --- informal norms --- null set --- open balls --- modified implicit iterative methods with perturbed mapping --- pseudocontractive mapping --- strongly pseudocontractive mapping --- nonexpansive mapping --- weakly continuous duality mapping --- set optimization --- set relations --- nonlinear scalarizing functional --- algebraic interior --- vector closure --- conjugate gradient method --- steepest descent method --- hybrid projection --- shrinking projection --- inertial Mann --- strongly convergence --- strict pseudo-contraction --- variational inequality problem --- inclusion problem --- signal processing

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Interfaces play an essential role in determining the mechanical properties and the structural integrity of a wide variety of technological materials. As new manufacturing methods become available, interface engineering and architecture at multiscale length levels in multi-physics materials open up to applications with high innovation potential. This Special Issue is dedicated to recent advances in fundamental and applications of solid material interfaces.

Technology: general issues --- Kirchhoff-Love plate --- composite material --- thin inclusion --- asymptotic analysis --- equivalent cylinder of finite length --- Steigmann–Ogden surface model --- anisotropic properties --- contact problem --- unilateral constraint --- variational inequality --- Tykhonov triple --- Tykhonov well-posedness --- approximating sequence --- multiphase fiber-reinforced composites --- asymptotic homogenization method --- effective complex properties --- elastic composite --- interfaces --- coupled thermoelasticity --- adhesive layer --- butt joint --- mode-I --- mixed-mode --- damage evolution --- analytical solution --- n/a --- Steigmann-Ogden surface model

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Interfaces play an essential role in determining the mechanical properties and the structural integrity of a wide variety of technological materials. As new manufacturing methods become available, interface engineering and architecture at multiscale length levels in multi-physics materials open up to applications with high innovation potential. This Special Issue is dedicated to recent advances in fundamental and applications of solid material interfaces.

Kirchhoff-Love plate --- composite material --- thin inclusion --- asymptotic analysis --- equivalent cylinder of finite length --- Steigmann–Ogden surface model --- anisotropic properties --- contact problem --- unilateral constraint --- variational inequality --- Tykhonov triple --- Tykhonov well-posedness --- approximating sequence --- multiphase fiber-reinforced composites --- asymptotic homogenization method --- effective complex properties --- elastic composite --- interfaces --- coupled thermoelasticity --- adhesive layer --- butt joint --- mode-I --- mixed-mode --- damage evolution --- analytical solution --- n/a --- Steigmann-Ogden surface model

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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Orthogonal polynomials --- Asymptotic theory --- Orthogonal polynomials -- Asymptotic theory. --- Polynomials. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Asymptotic theory. --- Asymptotic theory of orthogonal polynomials --- Algebra --- Airy function. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation error. --- Approximation theory. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Beta function. --- Boundary value problem. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Complex number. --- Complex plane. --- Correlation function. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Discrete measure. --- Distribution function. --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Fredholm determinant. --- Functional derivative. --- Gamma function. --- Gradient descent. --- Harmonic analysis. --- Hermitian matrix. --- Homotopy. --- Hypergeometric function. --- I0. --- Identity matrix. --- Inequality (mathematics). --- Integrable system. --- Invariant measure. --- Inverse scattering transform. --- Invertible matrix. --- Jacobi matrix. --- Joint probability distribution. --- Lagrange multiplier. --- Lax equivalence theorem. --- Limit (mathematics). --- Linear programming. --- Lipschitz continuity. --- Matrix function. --- Maxima and minima. --- Monic polynomial. --- Monotonic function. --- Morera's theorem. --- Neumann series. --- Number line. --- Orthogonal polynomials. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Parametrix. --- Pauli matrices. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Potential theory. --- Probability distribution. --- Probability measure. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random matrix. --- Random variable. --- Rate of convergence. --- Rectangle. --- Rhombus. --- Riemann surface. --- Special case. --- Spectral theory. --- Statistic. --- Subset. --- Theorem. --- Toda lattice. --- Trace (linear algebra). --- Trace class. --- Transition point. --- Triangular matrix. --- Trigonometric functions. --- Uniform continuity. --- Unit vector. --- Upper and lower bounds. --- Upper half-plane. --- Variational inequality. --- Weak solution. --- Weight function. --- Wishart distribution. --- Orthogonal polynomials - Asymptotic theory

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Solving nonlinear equations in Banach spaces (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others), is a non-trivial task that involves many areas of science and technology. Usually the solution is not directly affordable and require an approach using iterative algorithms. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots either with or without derivatives, iterative methods for approximating different generalized inverses, real or complex dynamics associated to the rational functions resulting from the application of an iterative method on a polynomial. Additionally, the analysis of the convergence has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, or partial differential equations, reveal the connection between iterative methods and other branches of science and engineering.

Lipschitz condition --- heston model --- rectangular matrices --- computational efficiency --- Hull–White --- order of convergence --- signal and image processing --- dynamics --- divided difference operator --- engineering applications --- smooth and nonsmooth operators --- Newton-HSS method --- higher order method --- Moore–Penrose --- asymptotic error constant --- multiple roots --- higher order --- efficiency index --- multiple-root finder --- computational efficiency index --- Potra–Pták method --- nonlinear equations --- system of nonlinear equations --- purely imaginary extraneous fixed point --- attractor basin --- point projection --- fixed point theorem --- convex constraints --- weight function --- radius of convergence --- Frédholm integral equation --- semi-local convergence --- nonlinear HSS-like method --- convexity --- accretive operators --- Newton-type methods --- multipoint iterations --- banach space --- Kantorovich hypothesis --- variational inequality problem --- Newton method --- semilocal convergence --- least square problem --- Fréchet derivative --- Newton’s method --- iterative process --- Newton-like method --- Banach space --- sixteenth-order optimal convergence --- nonlinear systems --- Chebyshev–Halley-type --- Jarratt method --- iteration scheme --- Newton’s iterative method --- basins of attraction --- drazin inverse --- option pricing --- higher order of convergence --- non-linear equation --- numerical experiment --- signal processing --- optimal methods --- rate of convergence --- n-dimensional Euclidean space --- non-differentiable operator --- projection method --- Newton’s second order method --- intersection --- planar algebraic curve --- Hilbert space --- conjugate gradient method --- sixteenth order convergence method --- Padé approximation --- optimal iterative methods --- error bound --- high order --- Fredholm integral equation --- global convergence --- iterative method --- integral equation --- ?-continuity condition --- systems of nonlinear equations --- generalized inverse --- local convergence --- iterative methods --- multi-valued quasi-nonexpasive mappings --- R-order --- finite difference (FD) --- nonlinear operator equation --- basin of attraction --- PDE --- King’s family --- Steffensen’s method --- nonlinear monotone equations --- Picard-HSS method --- nonlinear models --- the improved curvature circle algorithm --- split variational inclusion problem --- computational order of convergence --- with memory --- multipoint iterative methods --- Kung–Traub conjecture --- multiple zeros --- fourth order iterative methods --- parametric curve --- optimal order --- nonlinear equation