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This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes selfcontained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including SturmLiouville operators, canonical systems of differential equations, and multidimensional Schrödinger operators, where the abstract Weyl function appears as either the classical TitchmarshWeyl coefficient or the DirichlettoNeumann map. The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.
Mathematics  Operator theory  Differential equations  Partial differential equations  System theory
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This open access book features a selection of highquality papers from the presentations at the International Conference on Spectral and HighOrder Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions.
Partial Differential Equations  Numerical Analysis  Analysis  Highorder methods  Partial differential equations  Spectral methods  Isogeometric methods  Discontinuous Galerkin methods  Wave simulation  Uncertainty quantification  Open access  Differential calculus & equations  Numerical analysis
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This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is a beautiful area of mathematics with links to many other subjects. While describing the general theory of Hardy, Rellich, CaffarelliKohnNirenberg, Sobolev, and other inequalities in the setting of general homogeneous groups, the authors pay particular attention to the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes intricately intertwined with the properties of subLaplacians and subelliptic partial differential equations. These topics constitute the core of this book and they are complemented by additional, closely related topics such as uncertainty principles, function spaces on homogeneous groups, the potential theory for stratified groups, and the potential theory for general Hörmander's sums of squares and their fundamental solutions. This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. As can be attested as the winner of such an award, it is a vital contribution to literature of analysis not only because it presents a detailed account of the recent developments in the field, but also because the book is accessible to anyone with a basic level of understanding of analysis. Undergraduate and graduate students as well as researchers from any field of mathematical and physical sciences related to analysis involving functional inequalities or analysis of homogeneous groups will find the text beneficial to deepen their understanding.
Mathematics  Topological groups  Lie groups  Potential theory (Mathematics)  Partial differential equations  Harmonic analysis  Functional analysis  Differential geometry
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In Noether's original presentation of her celebrated theorem of 1918, allowances were made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the dependent variable(s), the socalled generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to only point transformations. In recent decades, this diminution of the power of Noether's Theorem has been partly countered, in particular, in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look at the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables.
wave equation  spherically symmetric spacetimes  lie symmetries  roots  optimal systems  invariant solutions  n/a  Noether symmetry approach  FLRW spacetime  action integral  variational principle  first integral  modified theories of gravity  GaussBonnet cosmology  Noether’s theorem  action integral  generalized symmetry  first integral  invariant  nonlocal transformation  boundary term  conservation laws  analytic mechanics  Noether’s theorem  generalized symmetry  energymomentum tensor  Lagrange anchor  viscoelasticity  KelvinVoigt equation  Lie symmetries  optimal system  groupinvariant solutions  conservation laws  multiplier method  continuous symmetry  symmetry reduction  integrable nonlocal partial differential equations  symmetries  conservation laws  Noether operator identity  quasiNoether systems  quasiLagrangians  Lie symmetry  conservation law  double dispersion equation  Boussinesq equation  systems of ODEs  Noether operators  Noether symmetries  first integrals  partial differential equations  approximate symmetry and solutions
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For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier–Plancherel formula for noncommutative harmonic analysis, applied to locallycompact, nonAbelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, subRiemannian manifolds, and Lie groups. In parallel, in geometric mechanics, JeanMarie Souriau interpreted the temperature vector of Planck as a spacetime vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring noncommutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.
WeylHeisenberg group  affine group  Weyl quantization  Wigner function  covariant integral quantization  Fourier analysis  special functions  rigged Hilbert spaces  quantum mechanics  signal processing  nonFourier heat conduction  thermal expansion  heat pulse experiments  pseudotemperature  GuyerKrumhansl equation  higher order thermodynamics  Lie groups thermodynamics  homogeneous manifold  polysymplectic manifold  dynamical systems  nonequivariant cohomology  Lie group machine learning  SouriauFisher metric  Born–Jordan quantization  shorttime propagators  timeslicing  Van Vleck determinant  thermodynamics  symplectization  metrics  nonequilibrium processes  interconnection  discrete multivariate sine transforms  orthogonal polynomials  cubature formulas  nonequilibrium thermodynamics  variational formulation  nonholonomic constraints  irreversible processes  discrete thermodynamic systems  continuum thermodynamic systems  fourier transform  rigid body motions  partial differential equations  Lévy processes  Lie Groups  homogeneous spaces  stochastic differential equations  harmonic analysis on abstract space  heat equation on manifolds and Lie Groups
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This Special Issue aims to be a compilation of new results in the areas of differential and difference Equations, covering boundary value problems, systems of differential and difference equations, as well as analytical and numerical methods. The objective is to provide an overview of techniques used in these different areas and to emphasize their applicability to reallife phenomena, by the inclusion of examples. These examples not only clarify the theoretical results presented, but also provide insight on how to apply, for future works, the techniques used.
Legendre wavelets  collocation method  threestep Taylor method  asymptotic stability  timedependent partial differential equations  noninstantaneous impulses  Caputo fractional derivative  differential equations  state dependent delays  lipschitz stability  limitperiodic solutions  difference equations  exponential dichotomy  strong nonlinearities  effective existence criteria  population dynamics  discrete Lyapunov equation  difference equations  Hilbert space  dichotomy  exponential stability  ?Laplacian operator  mean curvature operator  heteroclinic solutions  problems in the real line  lower and upper solutions  Nagumo condition on the real line  fixed point theory  coupled nonlinear systems  functional boundary conditions  Schauder’s fixed point theory  Arzèla Ascoli theorem  lower and upper solutions  first order periodic systems  SIRS epidemic model  mathematical modelling  Navier–Stokes equations  global solutions  regular solutions  a priori estimates  weak solutions  kinetic energy  dissipation  Bäcklund transformation  Clairin’s method  generalized Liouville equation  Miura transformation  Kortewegde Vries equation  secondorder differential/difference/qdifference equation of hypergeometric type  nonuniform lattices  divideddifference equations  polynomial solution  integrodifferentials  Sumudu decomposition method  dynamical system
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This Proceedings book presents papers from the 39th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MaxEnt 2019. The workshop took place at the Max Planck Institute for Plasma Physics in Garching near Munich, Germany, from 30 June to 5 July 2019, and invited contributions on all aspects of probabilistic inference, including novel techniques, applications, and work that sheds new light on the foundations of inference. Addressed are inverse and uncertainty quantification (UQ) and problems arising from a large variety of applications, such as earth science, astrophysics, material and plasma science, imaging in geophysics and medicine, nondestructive testing, density estimation, remote sensing, Gaussian process (GP) regression, optimal experimental design, data assimilation, and data mining.
hypothesis tests  precise hypotheses  pragmatic hypotheses  UAP  UAV  UFO  Nimitz  TicTac  image reconstruction  Bayesian Maximum a Posteriori approach  entropy prior probability  global statistical regularization  local statistical regularization  PET  SPECT  marginal likelihood  evidence  nested sampling  annealed importance sampling  Monte Carlo  stochastic gradients  SGHMC  model comparison  MCMC  thermodynamic Integration  HMC  uncertainty quantification  nonintrusive  spectral expansion  plasmawall interactions  Bayesian analysis  Deep Learning (DL)  Artificial Intelligence (AI)  Convolutional Neural Network (CNN)  classification  orthodontics  cervical vertebra maturation  machine learning  uncertainty quantification  multi fidelity  Gaussian processes  probability theory  Bayes  impedance cardiography  aortic dissection  Gaussian process regression  physicsinformed methods  kernel methods  field reconstruction  source localization  partial differential equations  meshless methods  nested sampling  cluster analysis  mean shift method  Bayesian evidence  model comparison  formant  steadystate  vowel  detrending  acoustic phonetics  sourcefilter theory  probability theory  uncertainty quantification  model averaging  nested sampling
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The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.
time fractional differential equations  mixedindex problems  analytical solution  asymptotic stability  conservative problems  Hamiltonian problems  energyconserving methods  Poisson problems  Hamiltonian Boundary Value Methods  HBVMs  line integral methods  constrained Hamiltonian problems  Hamiltonian PDEs  highly oscillatory problems  boundary element method  finite difference method  floating strike Asian options  continuous geometric average  barrier options  isogeometric analysis  adaptive methods  hierarchical splines  THBsplines  local refinement  linear systems  preconditioners  Cholesky factorization  limited memory  Volterra integral equations  Volterra integro–differential equations  collocation methods  multistep methods  convergence  Bspline  optimal basis  fractional derivative  Galerkin method  collocation method  spectral (eigenvalue) and singular value distributions  generalized locally Toeplitz sequences  discretization of systems of differential equations  higherorder finite element methods  discontinuous Galerkin methods  finite difference methods  isogeometric analysis  Bsplines  curl–curl operator  time harmonic Maxwell’s equations and magnetostatic problems  low rank completion  matrix ODEs  gradient system  ordinary differential equations  Runge–Kutta  tree  stump  order  elementary differential  edgehistogram  edgepreserving smoothing  histogram specification  initial value problems  onestep methods  Hermite–Obreshkov methods  symplecticity  Bsplines  BS methods  hyperbolic partial differential equations  high order discontinuous Galerkin finite element schemes  shock waves and discontinuities  vectorization and parallelization  high performance computing  generalized Schur algorithm  nullspace  displacement rank  structured matrices  stochastic differential equations  stochastic multistep methods  stochastic Volterra integral equations  meansquare stability  asymptotic stability  numerical analysis  numerical methods  scientific computing  initial value problems  onestep methods  Hermite–Obreshkov methods  symplecticity  Bsplines  BS methods
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