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This book celebrates the 80 years of the Professor Eugene P. Wigner paper “On Unitary Representations of the Inhomogeneous Lorentz Group", published in The Annals of Mathematics in 1939. We have collected several contributions divided into Research articles and Reviews. All contributions are technical, but the papers also bring a health element of didactic. Practitioners from several areas, from Gravity to Quantum Field Theory and Quantum Mechanics, as well as students, shall find a rich material in this Volume.

Research & information: general --- spinors in 4d --- regularization --- anomalies --- quantum gravity --- quantum mechanics --- symmetry --- quantum cosmology --- special relativity --- combination of velocities --- wigner angle --- quaternions --- gauge field theory --- Yang-Mills fields --- modified gravity --- non-Riemannian geometry --- spacetime symmetries --- gauge field theories --- gauge anomalies --- nonperturbative techniques --- ray representation --- strongly continuous --- continuity --- Hilbert space --- entanglement --- bispinors --- chirality --- spinors in 4d --- regularization --- anomalies --- quantum gravity --- quantum mechanics --- symmetry --- quantum cosmology --- special relativity --- combination of velocities --- wigner angle --- quaternions --- gauge field theory --- Yang-Mills fields --- modified gravity --- non-Riemannian geometry --- spacetime symmetries --- gauge field theories --- gauge anomalies --- nonperturbative techniques --- ray representation --- strongly continuous --- continuity --- Hilbert space --- entanglement --- bispinors --- chirality

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This book celebrates the 80 years of the Professor Eugene P. Wigner paper “On Unitary Representations of the Inhomogeneous Lorentz Group", published in The Annals of Mathematics in 1939. We have collected several contributions divided into Research articles and Reviews. All contributions are technical, but the papers also bring a health element of didactic. Practitioners from several areas, from Gravity to Quantum Field Theory and Quantum Mechanics, as well as students, shall find a rich material in this Volume.

Research & information: general --- spinors in 4d --- regularization --- anomalies --- quantum gravity --- quantum mechanics --- symmetry --- quantum cosmology --- special relativity --- combination of velocities --- wigner angle --- quaternions --- gauge field theory --- Yang-Mills fields --- modified gravity --- non-Riemannian geometry --- spacetime symmetries --- gauge field theories --- gauge anomalies --- nonperturbative techniques --- ray representation --- strongly continuous --- continuity --- Hilbert space --- entanglement --- bispinors --- chirality --- n/a

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There is significant interest in the Philosophy of Science community to understand the role that "effective theories" have in the work of forefront science. The ideas of effective theories have been implicit in science for a long time, but have only been articulated well in the last few decades. Since Wilson's renormalization group revolution in the early 1970's, the science community has come to more fully understand its power, and by the mid-1990's it had gained its apotheosis. It is still one of the most powerful concepts in science, which has direct impact in how one thinks about and formulates theories of nature. It is this power that this Brief sets out to emphasize through historical analysis and current examples.

Biomedical engineering -- Philosophy. --- Physics -- Philosophy. --- Physics. --- Engineering & Applied Sciences --- Applied Physics --- Science --- Philosophy. --- Normal science --- Philosophy of science --- Philosophy and science. --- Theoretical, Mathematical and Computational Physics. --- History and Philosophical Foundations of Physics. --- Philosophy of Science. --- Mathematical physics. --- Science and philosophy --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Physical mathematics --- Physics --- Mathematics --- Effectiv field theory --- Effective action --- Effective theories --- Naturalness and fine-tuning in theoretical physics --- Phenomenology --- Renormalization group --- Symmetries in Physics

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This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects.

adjoint-symmetry --- one-form --- symmetry --- vector field --- geometrical formulation --- nonlocal conservation laws --- differential coverings --- polynomial and rational invariants --- syzygy --- free resolution --- discretization --- differential invariants --- invariant derivations --- symplectic --- contact spaces --- Euler equations --- shockwaves --- phase transitions --- symmetries --- integrable systems --- Darboux-Bäcklund transformation --- isothermic immersions --- Spin groups --- Clifford algebras --- Euler equation --- quotient equation --- contact symmetry --- optimal investment theory --- linearization --- exact solutions --- Korteweg–de Vries–Burgers equation --- cylindrical and spherical waves --- saw-tooth solutions --- periodic boundary conditions --- head shock wave --- Navier–Stokes equations --- media with inner structures --- plane molecules --- water --- Levi–Civita connections --- Lagrangian curve flows --- KdV type hierarchies --- Darboux transforms --- Sturm–Liouville --- clamped --- hinged boundary condition --- spectral collocation --- Chebfun --- chebop --- eigenpairs --- preconditioning --- drift --- error control

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This volume was conceived as a Special Issue of the MDPI journal Mathematics to illustrate and show relevant applications of differential equations in different fields, coherently with the latest trends in applied mathematics research. All the articles that were submitted for publication are valuable, interesting, and original. The readers will certainly appreciate the heterogeneity of the 10 papers included in this book and will discover how helpful all the kinds of differential equations are in a wide range of disciplines. We are confident that this book will be inspirational for young scholars as well.

q-Hermite polynomials --- zeros of q-Hermite polynomials --- differential equation --- splitted separation --- Lie symmetries --- gauss hypergeometric functions --- initial value problem --- Kepler-type orbits --- Runge–Kutta --- differential evolution --- dynamical systems --- stability --- economics --- relationships --- networks --- oscillatory problems --- SEIR ODE model --- COVID-19 transmission --- convalescent plasma transfusion (CPT) --- degeneracy --- elliptic PDE --- ladder operator --- commuting operator --- eigenvalues --- mixing process --- simultaneous differential equations --- variable production rate --- simulated annealing --- financial markets --- investment style --- border collision bifurcation --- fundamental analysis --- technical analysis --- market maker --- differential equations with discontinuous right-hand sides --- Hopfield artificial neural networks --- n/a --- Runge-Kutta