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Complex systems with symmetry arise in many fields, at various length scales, including financial markets, social, transportation, telecommunication and power grid networks, world and country economies, ecosystems, molecular dynamics, immunology, living organisms, computational systems, and celestial and continuum mechanics. The emergence of new orders and structures in complex systems means symmetry breaking and transitions from unstable to stable states. Modeling complexity has attracted many researchers from different areas, dealing both with theoretical concepts and practical applications. This Special Issue fills the gap between the theory of symmetry-based dynamics and its application to model and analyze complex systems.

History of engineering & technology --- multi-agent system (MAS) --- reinforcement learning (RL) --- mobile robots --- function approximation --- Opportunistic complex social network --- cooperative --- neighbor node --- probability model --- social relationship --- adapted PageRank algorithm --- PageRank vector --- networks centrality --- multiplex networks --- biplex networks --- divided difference --- radius of convergence --- Kung–Traub method --- local convergence --- Lipschitz constant --- Banach space --- fractional calculus --- Caputo derivative --- generalized Fourier law --- Laplace transform --- Fourier transform --- Mittag–Leffler function --- non-Fourier heat conduction --- Mei symmetry --- conserved quantity --- adiabatic invariant --- quasi-fractional dynamical system --- non-standard Lagrangians --- complex systems --- symmetry-breaking --- bifurcation theory --- complex networks --- nonlinear dynamical systems

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A plethora of problems from diverse disciplines such as Mathematics, Mathematical: Biology, Chemistry, Economics, Physics, Scientific Computing and also Engineering can be formulated as an equation defined in abstract spaces using Mathematical Modelling. The solutions of these equations can be found in closed form only in special case. That is why researchers and practitioners utilize iterative procedures from which a sequence is being generated approximating the solution under some conditions on the initial data. This type of research is considered most interesting and challenging. This is our motivation for the introduction of this special issue on Iterative Procedures.

Lipschitz condition --- order of convergence --- Scalar equations --- local and semilocal convergence --- multiple roots --- Nondifferentiable operator --- optimal iterative methods --- Order of convergence --- convergence order --- fast algorithms --- iterative method --- computational convergence order --- generalized mixed equilibrium problem --- nonlinear equations --- systems of nonlinear equations --- Chebyshev’s iterative method --- local convergence --- iterative methods --- divided difference --- Multiple roots --- semi-local convergence --- scalar equations --- left Bregman asymptotically nonexpansive mapping --- basin of attraction --- maximal monotone operator --- Newton–HSS method --- general means --- Steffensen’s method --- derivative-free method --- simple roots --- fixed point problem --- split variational inclusion problem --- weighted-Newton method --- ball radius of convergence --- Traub–Steffensen method --- Newton’s method --- fractional derivative --- Banach space --- multiple-root solvers --- uniformly convex and uniformly smooth Banach space --- Fréchet-derivative --- optimal convergence --- Optimal iterative methods --- basins of attraction --- nonlinear equation

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We use addition on a daily basis-yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series-long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms-the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

Number theory. --- Mathematics --- Number study --- Numbers, Theory of --- Algebra --- Absolute value. --- Addition. --- Analytic continuation. --- Analytic function. --- Automorphic form. --- Axiom. --- Bernoulli number. --- Big O notation. --- Binomial coefficient. --- Binomial theorem. --- Book. --- Calculation. --- Chain rule. --- Coefficient. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Congruence subgroup. --- Conjecture. --- Constant function. --- Constant term. --- Convergent series. --- Coprime integers. --- Counting. --- Cusp form. --- Determinant. --- Diagram (category theory). --- Dirichlet series. --- Division by zero. --- Divisor. --- Elementary proof. --- Elliptic curve. --- Equation. --- Euclidean geometry. --- Existential quantification. --- Exponential function. --- Factorization. --- Fourier series. --- Function composition. --- Fundamental domain. --- Gaussian integer. --- Generating function. --- Geometric series. --- Geometry. --- Group theory. --- Hecke operator. --- Hexagonal number. --- Hyperbolic geometry. --- Integer factorization. --- Integer. --- Line segment. --- Linear combination. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Matrix group. --- Modular form. --- Modular group. --- Natural number. --- Non-Euclidean geometry. --- Parity (mathematics). --- Pentagonal number. --- Periodic function. --- Polynomial. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Pythagorean theorem. --- Quadratic residue. --- Quantity. --- Radius of convergence. --- Rational number. --- Real number. --- Remainder. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Semicircle. --- Series (mathematics). --- Sign (mathematics). --- Square number. --- Square root. --- Subgroup. --- Subset. --- Sum of squares. --- Summation. --- Taylor series. --- Theorem. --- Theory. --- Transfinite number. --- Triangular number. --- Two-dimensional space. --- Unique factorization domain. --- Upper half-plane. --- Variable (mathematics). --- Vector space.

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The description for this book, Existence Theorems in Partial Differential Equations. (AM-23), Volume 23, will be forthcoming.

Differential equations, Partial. --- Existence theorems. --- 0O. --- 3N. --- Addition. --- Analytic function. --- Applied mathematics. --- Big O notation. --- Biharmonic equation. --- Boundary value problem. --- C0. --- Calculation. --- Cartesian coordinate system. --- Cauchy problem. --- Characteristic equation. --- Closed-form expression. --- Coefficient. --- Computation. --- Computational problem. --- Constructive proof. --- Continuous function (set theory). --- Continuous function. --- Convex set. --- Coordinate system. --- Derivative. --- Determination. --- Differential equation. --- Dirichlet problem. --- Elliptic partial differential equation. --- Empty set. --- Equation. --- Existence theorem. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Finite difference. --- Flattening. --- Formal scheme. --- Fourier transform. --- Fundamental solution. --- Geometry. --- Green's function. --- Harmonic function. --- Implicit function theorem. --- Implicit function. --- Improper integral. --- Initial value problem. --- Integral equation. --- Interval (mathematics). --- Laplace transform. --- Limit of a sequence. --- Linear combination. --- Linear differential equation. --- Linear equation. --- Mathematician. --- Method of characteristics. --- Nonlinear system. --- Numerical analysis. --- Ordinary differential equation. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pessimism. --- Plane curve. --- Power series. --- Probability of success. --- Probability. --- Pure mathematics. --- Radius of convergence. --- Real number. --- Real variable. --- Requirement. --- Scientific notation. --- Second derivative. --- Series (mathematics). --- Simultaneous equations. --- Special case. --- Terminology. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Truncation error. --- Uniform convergence. --- Upper and lower bounds. --- Variable (mathematics).

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The leading scientists who gave these papers under the sponsorship of the Royal Society in early 1987 provide reviews of facets of the subject of chaos ranging from the practical aspects of mirror machines for fusion power to the pure mathematics of geodesics on surfaces of negative curvature. The papers deal with systems in which chaotic conditions arise from initial value problems with unique solutions, as opposed to those where chaos is produced by the introduction of noise from an external source. Table of Contents Diagnosis of Dynamical Systems with Fluctuating Parameters D. Ruelle Nonlinear Dynamics, Chaos, and Complex Cardiac Arrhythmias L. Glass, A. L. Goldberger, M. Courtemanche, and A. Shrier Chaos and the Dynamics of Biological Populations R. M. May Fractal Bifurcation Sets, Renormalization Strange Sets, and Their Universal Invariants D. A. Rand From Chaos to Turbulence in Bnard Convection A. Libchaber Dynamics of Convection N. O. Weiss Chaos: A Mixed Metaphor for Turbulence E. A. Spiegel Arithmetical Theory of Anosov Diffeomorphisms F. Vivaldi Chaotic Behavior in the Solar System J. Wisdom Chaos in Hamiltonian Systems I. C. Percival Semi-Classical Quantization, Adiabatic Invariants, and Classical Chaos W. P. Reinhardt and I. Dana Particle Confinement and Adiabatic Invariance B. V. Chirikov Some Geometrical Models of Chaotic Dynamics C. Series The Bakerian Lecture: Quantum Chaology M. V. BerryOriginally published in 1989.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Chaotic behavior in systems --- Accuracy and precision. --- Action potential. --- Adiabatic invariant. --- Adiabatic theorem. --- Amplitude. --- Approximation. --- Attractor. --- Belousov–Zhabotinsky reaction. --- Bifurcation diagram. --- Bifurcation theory. --- Big O notation. --- Boolean function. --- Boundary value problem. --- Calculation. --- Cardiac arrhythmia. --- Chaos theory. --- Characteristic exponent. --- Classical limit. --- Computation. --- Convection. --- Correlation dimension. --- Degrees of freedom (statistics). --- Diagram (category theory). --- Differential equation. --- Dimension. --- Dirac delta function. --- Dynamical system. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Equation. --- Equations of motion. --- Estimation. --- Exponential growth. --- Factorization. --- Fractal dimension. --- Hamiltonian mechanics. --- Hamiltonian system. --- Heteroclinic bifurcation. --- Homoclinic bifurcation. --- Homoclinic orbit. --- Hopf bifurcation. --- Information dimension. --- Initial condition. --- Instability. --- Integrable system. --- Internal heating. --- Kirkwood gap. --- Libration. --- Limit cycle. --- Lorenz system. --- Mathematics. --- Non-Euclidean geometry. --- Nonlinear resonance. --- Nonlinear system. --- Orbital eccentricity. --- Orbital resonance. --- Orrery. --- Parameter. --- Parasystole. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Phase space. --- Pitchfork bifurcation. --- Plane wave. --- Poisson point process. --- Power series. --- Probability. --- Proportionality (mathematics). --- Quantum chaos. --- Quantum mechanics. --- Quasiperiodic motion. --- Radius of convergence. --- Rate of convergence. --- Rayleigh number. --- Regime. --- Renormalization. --- Resonance. --- Rotation around a fixed axis. --- Rotation number. --- Saddle-node bifurcation. --- Scattering. --- Separatrix (mathematics). --- Sinus rhythm. --- Soliton. --- Special case. --- Stable manifold. --- Standard map. --- Statistic. --- Statistical mechanics. --- Stochastic. --- Symbolic dynamics. --- Test particle. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Tidal locking. --- Time evolution. --- Two-dimensional space. --- Universality class. --- Winding number.