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Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side. A significant development in the study of such equations occurred in the 1980's. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary. Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.

Calculus of variations. --- Differential equations, Nonlinear. --- Geometry, Riemannian. --- Riemann geometry --- Riemannian geometry --- Generalized spaces --- Geometry, Non-Euclidean --- Semi-Riemannian geometry --- Nonlinear differential equations --- Nonlinear theories --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Asymptotic analysis. --- Cayley–Hamilton theorem. --- Contradiction. --- Curvature. --- Diffeomorphism. --- Differentiable manifold. --- Equation. --- Estimation. --- Euclidean space. --- Laplace's equation. --- Maximum principle. --- Nonlinear system. --- Polynomial. --- Princeton University Press. --- Result. --- Ricci curvature. --- Riemannian geometry. --- Riemannian manifold. --- Simply connected space. --- Sphere theorem (3-manifolds). --- Stone's theorem. --- Submanifold. --- Subsequence. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Unit sphere.

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This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present.

independent domination --- stable graph --- tree --- unicyclic graph --- graph --- automorphism group --- determining number --- cograph --- unit interval graph --- simultaneous strong resolving set --- simultaneous strong metric dimension --- simultaneous strong resolving graph --- total outer k-independent domination --- total domination --- k-independence --- total Roman domination --- Roman domination --- semitotal domination --- domination --- efficient open domination --- digraphs --- products of digraphs --- secure total domination --- rooted product graph --- Roman domination number --- perfect Roman domination number --- strong resolving graph --- strong metric dimension --- strong resolving set --- cactus graphs --- unicyclic graphs --- network science --- network curvature --- discrete Ricci curvature --- earth-mover’s distance

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The present book contains 14 papers published in the Special Issue “Differential Geometry” of the journal Mathematics. They represent a selection of the 30 submissions. This book covers a variety of both classical and modern topics in differential geometry. We mention properties of both rectifying and affine curves, the geometry of hypersurfaces, angles in Minkowski planes, Euclidean submanifolds, differential operators and harmonic forms on Riemannian manifolds, complex manifolds, contact manifolds (in particular, Sasakian and trans-Sasakian manifolds), curvature invariants, and statistical manifolds and their submanifolds (in particular, Hessian manifolds). We wish to mention that among the authors, there are both well-known geometers and young researchers. The authors are from countries with a tradition in differential geometry: Belgium, China, Greece, Japan, Korea, Poland, Romania, Spain, Turkey, and United States of America. Many of these papers were already cited by other researchers in their articles. This book is useful for specialists in differential geometry, operator theory, physics, and information geometry as well as graduate students in mathematics.

statistical structure --- constant ratio submanifolds --- Euclidean submanifold --- framed helices --- Sasakian statistical manifold --- L2-harmonic forms --- Hodge–Laplacian --- complete connection --- concircular vector field --- cylindrical hypersurface --- k-th generalized Tanaka–Webster connection --- Casorati curvature --- symplectic curves --- generalized 1-type Gauss map --- rectifying submanifold --- manifold with singularity --- ruled surface --- Minkowski plane --- compact complex surfaces --- conjugate connection --- T-submanifolds --- L2-Stokes theorem --- inextensible flow --- shape operator --- generalized normalized ?-Casorati curvature --- Sasakian manifold --- centrodes --- circular helices --- non-flat complex space form --- invariant --- Frenet frame --- Darboux frame --- trans-Sasakian 3-manifold --- singular points --- symplectic curvatures --- Kähler–Einstein metrics --- conjugate symmetric statistical structure --- sectional ?-curvature --- circular rectifying curves --- developable surface --- capacity --- Ricci soliton --- Reeb flow symmetry --- Minkowskian pseudo-angle --- conical surface --- lie derivative --- position vector field --- pinching of the curvatures --- Hessian manifolds --- Minkowskian angle --- Hessian sectional curvature --- Minkowskian length --- lightlike surface --- affine sphere --- concurrent vector field --- slant --- affine hypersurface --- anti-invariant --- statistical manifolds --- Ricci operator --- C-Bochner tensor --- Ricci curvature --- real hypersurface --- scalar curvature --- framed rectifying curves

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The present monograph grew out of the fifth set of Hermann Weyl Lectures, given by Professor Griffiths at the Institute for Advanced Study, Princeton, in fall 1974.In Chapter 1 the author discusses Emile Borel's proof and the classical Jensen theorem, order of growth of entire analytic sets, order functions for entire holomorphic mappings, classical indicators of orders of growth, and entire functions and varieties of finite order.Chapter 2 is devoted to the appearance of curvature, and Chapter 3 considers the defect relations. The author considers the lemma on the logarithmic derivative, R. Nevanlinna's proof of the defect relation, and refinements of the classical case.

Complex analysis --- Holomorphic mappings --- Applications holomorphes --- 517.53 --- Mappings, Holomorphic --- Functions of several complex variables --- Mappings (Mathematics) --- Functions of a complex variable --- Holomorphic mappings. --- 517.53 Functions of a complex variable --- Fonctions de plusieurs variables complexes --- Fonctions entières --- Functions, Entire --- Algebraic variety. --- Analytic function. --- Analytic set. --- Armand Borel. --- Big O notation. --- Canonical bundle. --- Cartesian coordinate system. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Chern class. --- Compact Riemann surface. --- Compact space. --- Complex analysis. --- Complex manifold. --- Complex projective space. --- Corollary. --- Counting. --- Curvature. --- Degeneracy (mathematics). --- Derivative. --- Differential form. --- Dimension. --- Divisor. --- Elementary proof. --- Entire function. --- Equation. --- Exponential growth. --- Gaussian curvature. --- Hermann Weyl. --- Hodge theory. --- Holomorphic function. --- Hyperplane. --- Hypersurface. --- Infinite product. --- Integral geometry. --- Invariant measure. --- Inverse problem. --- Jacobian matrix and determinant. --- Kähler manifold. --- Line bundle. --- Linear equation. --- Logarithmic derivative. --- Manifold. --- Meromorphic function. --- Modular form. --- Monograph. --- Nevanlinna theory. --- Nonlinear system. --- Phillip Griffiths. --- Picard theorem. --- Polynomial. --- Projective space. --- Q.E.D. --- Quantity. --- Ricci curvature. --- Riemann sphere. --- Scientific notation. --- Several complex variables. --- Special case. --- Stokes' theorem. --- Subset. --- Summation. --- Theorem. --- Theory. --- Uniformization theorem. --- Unit square. --- Volume form. --- Fonctions entières

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This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion. The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained. The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions.

Differential equations, Hypoelliptic. --- Laplacian operator. --- Metric spaces. --- Spaces, Metric --- Operator, Laplacian --- Hypoelliptic differential equations --- Generalized spaces --- Set theory --- Topology --- Differential equations, Partial --- Alexander Grothendieck. --- Analytic function. --- Asymptote. --- Asymptotic expansion. --- Berezin integral. --- Bijection. --- Brownian dynamics. --- Brownian motion. --- Chaos theory. --- Chern class. --- Classical Wiener space. --- Clifford algebra. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Connection form. --- Coordinate system. --- Cotangent bundle. --- Covariance matrix. --- Curvature tensor. --- Curvature. --- De Rham cohomology. --- Derivative. --- Determinant. --- Differentiable manifold. --- Differential operator. --- Dirac operator. --- Direct proof. --- Eigenform. --- Eigenvalues and eigenvectors. --- Ellipse. --- Embedding. --- Equation. --- Estimation. --- Euclidean space. --- Explicit formula. --- Explicit formulae (L-function). --- Feynman–Kac formula. --- Fiber bundle. --- Fokker–Planck equation. --- Formal power series. --- Fourier series. --- Fourier transform. --- Fredholm determinant. --- Function space. --- Girsanov theorem. --- Ground state. --- Heat kernel. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Hypoelliptic operator. --- Integration by parts. --- Invertible matrix. --- Logarithm. --- Malliavin calculus. --- Martingale (probability theory). --- Matrix calculus. --- Mellin transform. --- Morse theory. --- Notation. --- Parameter. --- Parametrix. --- Parity (mathematics). --- Polynomial. --- Principal bundle. --- Probabilistic method. --- Projection (linear algebra). --- Rectangle. --- Resolvent set. --- Ricci curvature. --- Riemann–Roch theorem. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Sign convention. --- Smoothness. --- Sobolev space. --- Spectral theory. --- Square root. --- Stochastic calculus. --- Stochastic process. --- Summation. --- Supertrace. --- Symmetric space. --- Tangent space. --- Taylor series. --- Theorem. --- Theory. --- Torus. --- Trace class. --- Translational symmetry. --- Transversality (mathematics). --- Uniform convergence. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Wave equation.