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This Special Issue of the journal Symmetry contains a collection of papers devoted to the use of symmetry in finite element approximation of partial differential equations. More specifically, applications ranging from mechanical engineering to electromagnetics and fluid dynamics are considered. Both theoretical and computational aspects are considered. The contributions were selected to ensure the widest variety of themes. In particular, we wanted to include both theoretical papers (well posedness, stability) and numerical computations.

Research & information: general --- Mathematics & science --- Oseen problem --- corner singularity --- weighted finite element method --- preconditioning --- symmetric boundary condition --- pattern formation --- computational design --- finite-element method --- three-layer composite shell --- Mindlin plate theory --- finite element method --- force vibration --- FGMshells --- edge-based smoothed finite element method (ES-FEM) --- mixed interpolation of tensorial components (MITC) --- electromagnetic scattering --- time-harmonic electromagnetic fields --- moving media --- rotating axisymmetric objects --- bianisotropic media --- variational formulation --- well posedness --- convergence of the approximation --- Oseen problem --- corner singularity --- weighted finite element method --- preconditioning --- symmetric boundary condition --- pattern formation --- computational design --- finite-element method --- three-layer composite shell --- Mindlin plate theory --- finite element method --- force vibration --- FGMshells --- edge-based smoothed finite element method (ES-FEM) --- mixed interpolation of tensorial components (MITC) --- electromagnetic scattering --- time-harmonic electromagnetic fields --- moving media --- rotating axisymmetric objects --- bianisotropic media --- variational formulation --- well posedness --- convergence of the approximation

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The topic of this book is the mathematical and numerical analysis of some recent frameworks based on differential equations and their application in the mathematical modeling of complex systems, especially of living matter. First, the recent new mathematical frameworks based on generalized kinetic theory, fractional calculus, inverse theory, Schrödinger equation, and Cahn–Hilliard systems are presented and mathematically analyzed. Specifically, the well-posedness of the related Cauchy problems is investigated, stability analysis is also performed (including the possibility to have Hopf bifurcations), and some optimal control problems are presented. Second, this book is concerned with the derivation of specific models within the previous mentioned frameworks and for complex systems in biology, epidemics, and engineering. This book is addressed to graduate students and applied mathematics researchers involved in the mathematical modeling of complex systems.

Research & information: general --- Mathematics & science --- boundedness --- delay --- Hopf bifurcation --- Lyapunov functional --- stability --- SEIQRS-V model --- kinetic theory --- integro-differential equations --- complex systems --- evolution equations --- thermostat --- nonequilibrium stationary states --- discrete Fourier transform --- discrete kinetic theory --- nonlinearity --- fractional operators --- Cahn–Hilliard systems --- well-posedness --- regularity --- optimal control --- necessary optimality conditions --- Schrödinger equation --- Davydov’s model --- partial differential equations --- exact solutions --- fractional derivative --- abstract Cauchy problem --- C0−semigroup --- inverse problem --- active particles --- autoimmune disease --- degenerate equations --- real activity variable --- Cauchy problem --- electric circuit equations --- wardoski contraction --- almost (s, q)—Jaggi-type --- b—metric-like spaces --- second-order differential equations --- dynamical systems --- compartment model --- epidemics --- basic reproduction number --- boundedness --- delay --- Hopf bifurcation --- Lyapunov functional --- stability --- SEIQRS-V model --- kinetic theory --- integro-differential equations --- complex systems --- evolution equations --- thermostat --- nonequilibrium stationary states --- discrete Fourier transform --- discrete kinetic theory --- nonlinearity --- fractional operators --- Cahn–Hilliard systems --- well-posedness --- regularity --- optimal control --- necessary optimality conditions --- Schrödinger equation --- Davydov’s model --- partial differential equations --- exact solutions --- fractional derivative --- abstract Cauchy problem --- C0−semigroup --- inverse problem --- active particles --- autoimmune disease --- degenerate equations --- real activity variable --- Cauchy problem --- electric circuit equations --- wardoski contraction --- almost (s, q)—Jaggi-type --- b—metric-like spaces --- second-order differential equations --- dynamical systems --- compartment model --- epidemics --- basic reproduction number

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The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics.

Research & information: general --- Mathematics & science --- well posedness --- constrained variational control problem --- monotonicity --- pseudomonotonicity --- hemicontinuity --- multiple integral functional --- lower semicontinuity --- fractional differential equations --- fractional derivative of Riemann-Liouville type --- integral boundary value problems --- Green's functions --- Guo-Krasnosel'skii fixed point theorem in cones --- sublinearity and superlinearity --- Arzelà-Ascoli Theorem --- multi-objective programming --- fractional transportation problem --- intuitionistic fuzzy set --- parametric programming --- convex function --- h-convex function --- Hermite-Hadamard inequality --- Caputo-Fabrizio fractional integral --- Jensen inequality --- Jensen-Mercer inequality --- multiobjective programs with vanishing constraints --- semidefinite programming --- convexificators --- nonsmooth analysis --- constraint qualifications --- interval-valued function --- Riemann integral --- LR-convex interval-valued function --- interval Hermite-Hadamard inequality --- interval Hermite-Hadamard-Fejér inequality --- Lieb concavity theorem --- deformed exponential --- Pick function --- convexity of matrix --- low carbon inventory --- discount --- payment in advance --- price-sensitive demand --- emission reduction --- advances of SDO --- applications of SDO --- metaheuristic optimization --- nature-inspired algorithms --- optimization problems --- spiral dynamics optimization --- spiral-inspired optimization algorithms --- spiral paths --- (p,s)-convex fuzzy-interval-valued function --- fuzzy Riemann integral --- Jensen type inequality --- Schur type inequality --- Hermite-Hadamard type inequality --- Hermite-Hadamard-Fejér type inequality --- inverse geometric problem --- Laplace equation --- method of fundamental solution --- least-square problem --- micro resonator --- fractal --- multistability --- safe jump --- hidden attractor --- chaos --- basin of attraction --- LR-Harmonically convexity --- fractional integral operator --- Hermite-Hadamard type inequalities --- multimodal multi-objective optimization --- manta ray foraging optimizer --- non-dominated solution --- crowing distance --- engineering design problem --- optimal power flow --- renewable energy sources --- improved chaos game optimization --- TD-TI controller --- load frequency control --- electrical vehicles --- well posedness --- constrained variational control problem --- monotonicity --- pseudomonotonicity --- hemicontinuity --- multiple integral functional --- lower semicontinuity --- fractional differential equations --- fractional derivative of Riemann-Liouville type --- integral boundary value problems --- Green's functions --- Guo-Krasnosel'skii fixed point theorem in cones --- sublinearity and superlinearity --- Arzelà-Ascoli Theorem --- multi-objective programming --- fractional transportation problem --- intuitionistic fuzzy set --- parametric programming --- convex function --- h-convex function --- Hermite-Hadamard inequality --- Caputo-Fabrizio fractional integral --- Jensen inequality --- Jensen-Mercer inequality --- multiobjective programs with vanishing constraints --- semidefinite programming --- convexificators --- nonsmooth analysis --- constraint qualifications --- interval-valued function --- Riemann integral --- LR-convex interval-valued function --- interval Hermite-Hadamard inequality --- interval Hermite-Hadamard-Fejér inequality --- Lieb concavity theorem --- deformed exponential --- Pick function --- convexity of matrix --- low carbon inventory --- discount --- payment in advance --- price-sensitive demand --- emission reduction --- advances of SDO --- applications of SDO --- metaheuristic optimization --- nature-inspired algorithms --- optimization problems --- spiral dynamics optimization --- spiral-inspired optimization algorithms --- spiral paths --- (p,s)-convex fuzzy-interval-valued function --- fuzzy Riemann integral --- Jensen type inequality --- Schur type inequality --- Hermite-Hadamard type inequality --- Hermite-Hadamard-Fejér type inequality --- inverse geometric problem --- Laplace equation --- method of fundamental solution --- least-square problem --- micro resonator --- fractal --- multistability --- safe jump --- hidden attractor --- chaos --- basin of attraction --- LR-Harmonically convexity --- fractional integral operator --- Hermite-Hadamard type inequalities --- multimodal multi-objective optimization --- manta ray foraging optimizer --- non-dominated solution --- crowing distance --- engineering design problem --- optimal power flow --- renewable energy sources --- improved chaos game optimization --- TD-TI controller --- load frequency control --- electrical vehicles