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Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, ngeqslant 2, and let L^k be the k-th tensor power of a CR complex line bundle L over X. Given qin {0,1,ldots ,n-1}, let Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For lambda geq 0, let Pi ^{(q)}_{k,leq lambda} :=E((-infty ,lambda ]), where E denotes the spectral measure of Box ^{(q)}_{b,k}. In this work, the author proves that Pi ^{(q)}_{k,leq k^{-N_0}}F^*_k, F_kPi ^{(q)}_{k,leq k^{-N_0}}F^*_k, N_0geq 1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of Box ^{(q)}_{b,k}, where F_k is some kind of microlocal cut-off function. Moreover, we show that F_kPi ^{(q)}_{k,leq 0}F^*_k admits a full asymptotic expansion with respect to k if Box ^{(q)}_{b,k} has small spectral gap property with respect to F_k and Pi^{(q)}_{k,leq 0} is k-negligible away the diagonal with respect to F_k. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S^1 action.

Embedding theorems. --- CR submanifolds. --- Manifolds (Mathematics) --- Embeddings (Mathematics) --- Kernel functions. --- Asymptotic expansions.

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This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development.Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws-PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs-mixed type, free boundaries, and corner singularities-that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities.

Shock waves --- Von Neumann algebras. --- MATHEMATICS / Differential Equations / Partial. --- Algebras, Von Neumann --- Algebras, W --- Neumann algebras --- Rings of operators --- W*-algebras --- C*-algebras --- Hilbert space --- Shock (Mechanics) --- Waves --- Diffraction --- Diffraction. --- Mathematics. --- A priori estimate. --- Accuracy and precision. --- Algorithm. --- Andrew Majda. --- Attractor. --- Banach space. --- Bernhard Riemann. --- Big O notation. --- Boundary value problem. --- Bounded set (topological vector space). --- C0. --- Calculation. --- Cauchy problem. --- Coefficient. --- Computation. --- Computational fluid dynamics. --- Conjecture. --- Conservation law. --- Continuum mechanics. --- Convex function. --- Degeneracy (mathematics). --- Demetrios Christodoulou. --- Derivative. --- Dimension. --- Directional derivative. --- Dirichlet boundary condition. --- Dirichlet problem. --- Dissipation. --- Ellipse. --- Elliptic curve. --- Elliptic partial differential equation. --- Embedding problem. --- Equation solving. --- Equation. --- Estimation. --- Euler equations (fluid dynamics). --- Existential quantification. --- Fixed point (mathematics). --- Flow network. --- Fluid dynamics. --- Fluid mechanics. --- Free boundary problem. --- Function (mathematics). --- Function space. --- Fundamental class. --- Fundamental solution. --- Fundamental theorem. --- Hyperbolic partial differential equation. --- Initial value problem. --- Iteration. --- Laplace's equation. --- Linear equation. --- Linear programming. --- Linear space (geometry). --- Mach reflection. --- Mathematical analysis. --- Mathematical optimization. --- Mathematical physics. --- Mathematical problem. --- Mathematical proof. --- Mathematical theory. --- Mathematician. --- Melting. --- Monotonic function. --- Neumann boundary condition. --- Nonlinear system. --- Numerical analysis. --- Parameter space. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Phase boundary. --- Phase transition. --- Potential flow. --- Pressure gradient. --- Quadratic function. --- Regularity theorem. --- Riemann problem. --- Scientific notation. --- Self-similarity. --- Special case. --- Specular reflection. --- Stefan problem. --- Structural stability. --- Subspace topology. --- Symmetrization. --- Theorem. --- Theory. --- Truncation error (numerical integration). --- Two-dimensional space. --- Unification (computer science). --- Variable (mathematics). --- Velocity potential. --- Vortex sheet. --- Vorticity. --- Wave equation. --- Weak convergence (Hilbert space). --- Weak solution.