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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

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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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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.

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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Electromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. Because of their wide range of important applications, these materials have been intensely studied over the past twenty-five years, mainly from the perspectives of physics and engineering. But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory. Designed for researchers and advanced graduate students in applied mathematics, electrical engineering, and physics, this book introduces the electromagnetics of complex media through a systematic, state-of-the-art account of their mathematical theory. The book combines the study of well posedness, homogenization, and controllability of Maxwell equations complemented with constitutive relations describing complex media. The book treats deterministic and stochastic problems both in the frequency and time domains. It also covers computational aspects and scattering problems, among other important topics. Detailed appendices make the book self-contained in terms of mathematical prerequisites, and accessible to engineers and physicists as well as mathematicians.

Electromagnetism --- Stochastic control theory. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Control theory --- Stochastic processes --- Mathematics. --- AtkinsonЗilcox expansion theorem. --- Beltrami fields. --- Faedo-Galerkin approach. --- Herglotz wave functions. --- Hilbert Uniqueness method. --- Maxwell equations. --- Maxwell operator. --- PDEs. --- applied mathematics. --- auxiliary elliptic problems. --- boundary controllability. --- boundary integral equation. --- boundary value problem. --- chiral material. --- chiral media. --- chirality. --- compact embeddings. --- complex electromagnetic media. --- complex media. --- constitutive relations. --- controllability problem. --- controllability. --- decompositions. --- differential equations. --- dispersive media. --- dyadics. --- eigenvalue problems. --- electric flux density. --- electrical engineering. --- electromagnetic complex media. --- electromagnetic fields. --- electromagnetic media. --- electromagnetic wave scattering. --- electromagnetic waves. --- electromagnetics. --- evolution family approach. --- evolution operators. --- evolution problems. --- exterior problems. --- finite-dimensional space. --- fixed point approach. --- frequency. --- function spaces. --- general scattering theorem. --- generalised integral transforms. --- geometry. --- handedness. --- homogenisation problem. --- homogenisation. --- homogenised media. --- homogenised system. --- infinite Frchet differentiability. --- integrodifferential equations. --- integrodifferential evolution equation. --- interior domain problem. --- magnetic flux density. --- mathematical modelling. --- mathematical theory. --- nonlinear PDEs. --- nonlinear model. --- nonlinear phenomena. --- nonlinear problems. --- nonlinearity. --- operators. --- optical theorem. --- penetrable obstacle. --- perfectly conducting obstacle. --- periodic media. --- physics. --- plane electromagnetic waves. --- reciprocity principle. --- scattering problems. --- scattering process. --- scattering theories. --- scattering theory. --- semigroup approach. --- semigroup arguments. --- semigroup-based approach. --- solvability. --- spaces. --- spectral theory. --- standard differential. --- stochastic integrodifferential equations. --- time domain. --- time-harmonic electromagnetic wave. --- time-harmonic problems. --- time. --- trace operators. --- two-scale expansion. --- variational formulation. --- vector analysis. --- wave motions. --- wave operators. --- well posedness.