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"Scientists and engineers use computer simulations to study relationships between a model's input parameters and its outputs. However, thorough parameter studies are challenging, if not impossible, when the simulation is expensive and the model has several inputs. To enable studies in these instances, the engineer may attempt to reduce the dimension of the model's input parameter space. Active subspaces are an emerging set of dimension reduction tools that identify important directions in the parameter space. This book describes techniques for discovering a model's active subspace and proposes methods for exploiting the reduced dimension to enable otherwise infeasible parameter studies. Readers will find new ideas for dimension reduction; easy-to-implement algorithms; and several examples of active subspaces in action" -- From the publisher.

Functional analysis --- Shift operators (Operator theory) --- Hilbert space --- Linear topological spaces --- Function spaces

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The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra. The second part is devoted to applications. Standard spectral problems in Hilbert spaces are of the form A-λI for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered. The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail. Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader’s background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed.

Mathematics. --- Operator Theory. --- Ordinary Differential Equations. --- Mathematical Physics. --- Operator theory. --- Differential Equations. --- Mathématiques --- Théorie des opérateurs --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Spectral theory (Mathematics) --- Polynomial operator pencils. --- Operator pencils, Polynomial --- Operator polynomials --- Pencils, Polynomial operator --- Polynomial pencils --- Polynomials, Operator --- Differential equations. --- Mathematical physics. --- Linear operators --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- 517.91 Differential equations --- Differential equations --- Physical mathematics --- Physics

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The book describes the direct problems and the inverse problem of the multidimensional Schrödinger operator with a periodic potential. This concerns perturbation theory and constructive determination of the spectral invariants and finding the periodic potential from the given Bloch eigenvalues. The unique method of this book derives the asymptotic formulas for Bloch eigenvalues and Bloch functions for arbitrary dimension. Moreover, the measure of the iso-energetic surfaces in the high energy region is construct and estimated. It implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed in this book, the spectral invariants of the multidimensional operator from the given Bloch eigenvalues are determined. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential. This way the possibility to determine the potential constructively by using Bloch eigenvalues as input data is given. In the end an algorithm for the unique determination of the potential is given.

Physics. --- Quantum Physics. --- Solid State Physics. --- Mathematical Physics. --- Quantum theory. --- Physique --- Théorie quantique --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Mathematical physics. --- Quantum physics. --- Solid state physics. --- Solids --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Physical mathematics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematics --- Schrödinger operator. --- Perturbation (Mathematics) --- Spectral theory (Mathematics) --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Perturbation equations --- Perturbation theory --- Approximation theory --- Mathematical physics --- Operator, Schrödinger --- Differential operators --- Quantum theory --- Schrödinger equation

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Book 4 in the Princeton Mathematical Series.Originally published in 1941.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Abelian group. --- Additive group. --- Adjunction (field theory). --- Algebraic connectivity. --- Algebraic number. --- Annihilator (ring theory). --- Automorphism. --- Barycentric coordinate system. --- Barycentric subdivision. --- Big O notation. --- Boundary (topology). --- Cantor set. --- Cardinal number. --- Cartesian coordinate system. --- Cauchy sequence. --- Character group. --- Circumference. --- Cohomology. --- Combinatorics. --- Compact space. --- Complete metric space. --- Complex number. --- Computation. --- Continuous function (set theory). --- Continuous function. --- Contractible space. --- Cyclic group. --- Dense set. --- Diameter. --- Dimension (vector space). --- Dimension function. --- Dimension theory (algebra). --- Dimension. --- Dimensional analysis. --- Discrete group. --- Disjoint sets. --- Domain of a function. --- Equation. --- Euclidean space. --- Existential quantification. --- Exponentiation. --- Function (mathematics). --- Function space. --- Fundamental theorem. --- Geometry. --- Group theory. --- Hausdorff dimension. --- Hausdorff space. --- Hilbert cube. --- Hilbert space. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hyperplane. --- Integer. --- Interior (topology). --- Invariance of domain. --- Inverse system. --- Linear space (geometry). --- Linear subspace. --- Lp space. --- Mathematical induction. --- Mathematics. --- Metric space. --- Multiplicative group. --- N-sphere. --- Natural number. --- Natural transformation. --- Ordinal number. --- Orientability. --- Parity (mathematics). --- Partial function. --- Partially ordered set. --- Point (geometry). --- Polytope. --- Projection (linear algebra). --- Samuel Eilenberg. --- Separable space. --- Separated sets. --- Set (mathematics). --- Set theory. --- Sign (mathematics). --- Simplex. --- Special case. --- Subgroup. --- Subsequence. --- Subset. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topological group. --- Topological property. --- Topological space. --- Transfinite. --- Transitive relation. --- Unit sphere. --- Upper and lower bounds. --- Variable (mathematics).