01/2019, ISBN 9783039216215

.... Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials...

eBook

Journal of computational physics, ISSN 0021-9991, 2010, Volume 229, Issue 22, pp. 8333 - 8363

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration...

Time dependence | Stochastic differential equations | Polynomial chaos | Monte-Carlo simulation | SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | QUANTIFICATION | UNCERTAINTY | SIMULATIONS | WIENER-HERMITE EXPANSION | PHYSICS, MATHEMATICAL | FLOW | Monte Carlo method | Analysis | Aerospace engineering | Approximation | Chaos theory | Mathematical analysis | Exact solutions | Mathematical models | Stochasticity | Probability density functions | Convergence | DIFFERENTIAL EQUATIONS | STOCHASTIC PROCESSES | MONTE CARLO METHOD | APPROXIMATIONS | CALCULATION METHODS | EQUATIONS | FUNCTIONS | SIMULATION | CHAOS THEORY | ANALYTICAL SOLUTION | COMPUTERIZED SIMULATION | POLYNOMIALS | MATHEMATICS | MATHEMATICAL SOLUTIONS | PROBABILITY | CONVERGENCE | TIME DEPENDENCE | MATHEMATICAL METHODS AND COMPUTING

Time dependence | Stochastic differential equations | Polynomial chaos | Monte-Carlo simulation | SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | QUANTIFICATION | UNCERTAINTY | SIMULATIONS | WIENER-HERMITE EXPANSION | PHYSICS, MATHEMATICAL | FLOW | Monte Carlo method | Analysis | Aerospace engineering | Approximation | Chaos theory | Mathematical analysis | Exact solutions | Mathematical models | Stochasticity | Probability density functions | Convergence | DIFFERENTIAL EQUATIONS | STOCHASTIC PROCESSES | MONTE CARLO METHOD | APPROXIMATIONS | CALCULATION METHODS | EQUATIONS | FUNCTIONS | SIMULATION | CHAOS THEORY | ANALYTICAL SOLUTION | COMPUTERIZED SIMULATION | POLYNOMIALS | MATHEMATICS | MATHEMATICAL SOLUTIONS | PROBABILITY | CONVERGENCE | TIME DEPENDENCE | MATHEMATICAL METHODS AND COMPUTING

Journal Article

Computer methods in applied mechanics and engineering, ISSN 0045-7825, 2015, Volume 290, Issue C, pp. 73 - 97

Independent sampling of orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models using Polynomial Chaos (PC) expansions...

Uncertainty quantification | Hermite polynomials | Legendre polynomials | Markov chain Monte Carlo | Least squares regression | Polynomial Chaos | Markov chain monte carlo | Polynomial chaos | EXPANSIONS | STOCHASTIC COLLOCATION | PROJECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Monte Carlo method | Analysis | Markov processes | Differential equations | Aerospace engineering | Statistical analysis | Samples | Coherence | Mathematical models | Polynomials | Statistical methods | Sampling | Personal computers

Uncertainty quantification | Hermite polynomials | Legendre polynomials | Markov chain Monte Carlo | Least squares regression | Polynomial Chaos | Markov chain monte carlo | Polynomial chaos | EXPANSIONS | STOCHASTIC COLLOCATION | PROJECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Monte Carlo method | Analysis | Markov processes | Differential equations | Aerospace engineering | Statistical analysis | Samples | Coherence | Mathematical models | Polynomials | Statistical methods | Sampling | Personal computers

Journal Article

ESAIM. Mathematical modelling and numerical analysis, ISSN 1290-3841, 2011, Volume 46, Issue 2, pp. 317 - 339

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables...

Determinate measure | Wiener integral | Spectral elements | Generalized polynomial chaos | Stochastic Galerkin method | Moment problem | Equations with random data | Polynomial chaos | Wiener-Hermite expansion | moment problem | MULTIDIMENSIONAL MOMENT PROBLEM | MATHEMATICS, APPLIED | stochastic Galerkin method | ELEMENT-METHOD | determinate measure | REPRESENTATIONS | APPROXIMATIONS | generalized polynomial chaos | DIFFERENTIAL-EQUATIONS | polynomial chaos | FLOW | spectral elements | COEFFICIENTS | MODELING UNCERTAINTY | Studies | Polynomials | Stochastic models | Integrals | Mathematical analysis | Numerical analysis | Chaos theory | Mathematical models | Complement | Models | Random variables | Convergence

Determinate measure | Wiener integral | Spectral elements | Generalized polynomial chaos | Stochastic Galerkin method | Moment problem | Equations with random data | Polynomial chaos | Wiener-Hermite expansion | moment problem | MULTIDIMENSIONAL MOMENT PROBLEM | MATHEMATICS, APPLIED | stochastic Galerkin method | ELEMENT-METHOD | determinate measure | REPRESENTATIONS | APPROXIMATIONS | generalized polynomial chaos | DIFFERENTIAL-EQUATIONS | polynomial chaos | FLOW | spectral elements | COEFFICIENTS | MODELING UNCERTAINTY | Studies | Polynomials | Stochastic models | Integrals | Mathematical analysis | Numerical analysis | Chaos theory | Mathematical models | Complement | Models | Random variables | Convergence

Journal Article

Journal of computational physics, ISSN 0021-9991, 2015, Volume 280, Issue C, pp. 363 - 386

Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions...

Sparse approximation | [formula omitted]-minimization | Stochastic PDEs | Uncertainty quantification | Polynomial chaos | Markov Chain Monte Carlo | Hermite polynomials | Legendre polynomials | Compressive sampling | ℓ | minimization | ATOMIC DECOMPOSITION | EQUATIONS | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | l-minimization | Markov processes | Monte Carlo method | Analysis | Monte Carlo methods | Chaos theory | Polycarbonates | Coherence | Strategy | Mathematical models | Polynomials | Sampling | Mathematics - Probability

Sparse approximation | [formula omitted]-minimization | Stochastic PDEs | Uncertainty quantification | Polynomial chaos | Markov Chain Monte Carlo | Hermite polynomials | Legendre polynomials | Compressive sampling | ℓ | minimization | ATOMIC DECOMPOSITION | EQUATIONS | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | l-minimization | Markov processes | Monte Carlo method | Analysis | Monte Carlo methods | Chaos theory | Polycarbonates | Coherence | Strategy | Mathematical models | Polynomials | Sampling | Mathematics - Probability

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2017, Volume 454, Issue 1, pp. 303 - 334

This paper introduces a new, transformation-free, generalized polynomial chaos expansion (PCE...

Multivariate Hermite polynomials | Uncertainty quantification | Polynomial chaos | MATHEMATICS | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | MECHANICS | INTEGRATION | CHAOS EXPANSIONS | SYSTEMS

Multivariate Hermite polynomials | Uncertainty quantification | Polynomial chaos | MATHEMATICS | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | MECHANICS | INTEGRATION | CHAOS EXPANSIONS | SYSTEMS

Journal Article

Journal of computational physics, ISSN 0021-9991, 02/2016, Volume 307, pp. 94 - 109

.... Specifically, we consider rotation-based linear mappings which are determined iteratively for Hermite polynomial expansions...

Uncertainty quantification | Generalized polynomial chaos | Compressive sensing | High dimensions | Active subspace | Iterative rotations | BASIS PURSUIT | REWEIGHTED L MINIMIZATION | SIGNAL RECOVERY | DECOMPOSITION | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | COLLOCATION METHOD | L-MINIMIZATION | CHAOS EXPANSIONS | DERIVATIVE INFORMATION | Mechanical engineering | Differential equations | Uncertainty | Partial differential equations | Mapping | Representations | Hermite polynomials | Random variables | Stochasticity | Iterative methods

Uncertainty quantification | Generalized polynomial chaos | Compressive sensing | High dimensions | Active subspace | Iterative rotations | BASIS PURSUIT | REWEIGHTED L MINIMIZATION | SIGNAL RECOVERY | DECOMPOSITION | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | COLLOCATION METHOD | L-MINIMIZATION | CHAOS EXPANSIONS | DERIVATIVE INFORMATION | Mechanical engineering | Differential equations | Uncertainty | Partial differential equations | Mapping | Representations | Hermite polynomials | Random variables | Stochasticity | Iterative methods

Journal Article

Computer aided geometric design, ISSN 0167-8396, 02/2017, Volume 51, pp. 30 - 47

•Special classes of polynomial 2-surfaces possessing polynomial area element are investigated...

Polynomial area element | PN surfaces | Hermite interpolation | MOS surfaces | RATIONAL OFFSETS | MATHEMATICS, APPLIED | SPHERE | PATCHES | CURVES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | BOUNDARIES | PYTHAGOREAN HODOGRAPHS | TRANSFORMS

Polynomial area element | PN surfaces | Hermite interpolation | MOS surfaces | RATIONAL OFFSETS | MATHEMATICS, APPLIED | SPHERE | PATCHES | CURVES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | BOUNDARIES | PYTHAGOREAN HODOGRAPHS | TRANSFORMS

Journal Article

International journal for numerical methods in engineering, ISSN 0029-5981, 09/2017, Volume 111, Issue 12, pp. 1192 - 1200

.... Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect...

uncertainty quantification | linearization problem | polynomial chaos | orthogonal polynomials | LINEARIZATION | REPRESENTATIONS | DIFFERENTIAL-EQUATIONS | HERMITE-POLYNOMIALS | SOIL-STRUCTURE INTERACTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | PRODUCT | COEFFICIENTS | UNCERTAINTY | RULES | JACOBI-POLYNOMIALS | Thermal expansion | Uncertainty | Gaussian process | Computational fluid dynamics | Computer simulation | Fluid flow | Solvers | Polynomials | Mathematical models | Galerkin method | Velocity | Quadratures

uncertainty quantification | linearization problem | polynomial chaos | orthogonal polynomials | LINEARIZATION | REPRESENTATIONS | DIFFERENTIAL-EQUATIONS | HERMITE-POLYNOMIALS | SOIL-STRUCTURE INTERACTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | PRODUCT | COEFFICIENTS | UNCERTAINTY | RULES | JACOBI-POLYNOMIALS | Thermal expansion | Uncertainty | Gaussian process | Computational fluid dynamics | Computer simulation | Fluid flow | Solvers | Polynomials | Mathematical models | Galerkin method | Velocity | Quadratures

Journal Article

Nonlinear analysis, ISSN 0362-546X, 2019, Volume 187, pp. 18 - 48

We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to Hermite polynomials in the large-N limit...

Gaussian measures | Spherical harmonics | Spherical Laplacian | MATHEMATICS | MATHEMATICS, APPLIED | LIMIT | Hermite polynomials

Gaussian measures | Spherical harmonics | Spherical Laplacian | MATHEMATICS | MATHEMATICS, APPLIED | LIMIT | Hermite polynomials

Journal Article

Numerische Mathematik, ISSN 0029-599X, 5/2019, Volume 142, Issue 1, pp. 167 - 203

...Numerische Mathematik (2019) 142:167–203
https://doi.org/10.1007/s00211-018-0996-9
Numerische
Mathematik
Generalized Taylor operators and polynomial chains...

Mathematical Methods in Physics | Numerical Analysis | Theoretical, Mathematical and Computational Physics | 65D10 | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | MATHEMATICS, APPLIED

Mathematical Methods in Physics | Numerical Analysis | Theoretical, Mathematical and Computational Physics | 65D10 | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | MATHEMATICS, APPLIED

Journal Article

Pattern recognition, ISSN 0031-3203, 12/2016, Volume 60, pp. 921 - 935

... in the size of problem. In this paper, a new kernel function is proposed for SVM which is derived from Hermite orthogonal polynomials...

Hermite orthogonal polynomial kernel | Kernel function | Support Vector Machine (SVM) | Combined kernel | FEATURE-SELECTION | REGRESSION | REDUCTION | ALGORITHM | SVM | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | MOMENTS | ENGINEERING, ELECTRICAL & ELECTRONIC

Hermite orthogonal polynomial kernel | Kernel function | Support Vector Machine (SVM) | Combined kernel | FEATURE-SELECTION | REGRESSION | REDUCTION | ALGORITHM | SVM | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | MOMENTS | ENGINEERING, ELECTRICAL & ELECTRONIC

Journal Article

Advances in difference equations, ISSN 1687-1839, 12/2017, Volume 2017, Issue 1, pp. 1 - 17

...)-analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as (p, q).( 1, 1...

(p, q) -difference operators | (p, q) -Sturm-Liouville problems | (p, q) -classical orthogonal polynomials | (p, q) -integrals | (p, q) -Pearson difference equation | MATHEMATICS | MATHEMATICS, APPLIED | (p, q)-integrals | (p, q)-Pearson difference equation | (p, q)-Sturm-Liouville problems | (p, q)-difference operators | (p, q)-classical orthogonal polynomials | Theorems (Mathematics) | Usage | Polynomials | Difference equations | Tests, problems and exercises | Theorems | Hermite polynomials | ( p , q ) $(p,q)$ -integrals | ( p , q ) $(p,q)$ -classical orthogonal polynomials | ( p , q ) $(p,q)$ -Pearson difference equation | ( p , q ) $(p,q)$ -Sturm-Liouville problems | ( p , q ) $(p,q)$ -difference operators

(p, q) -difference operators | (p, q) -Sturm-Liouville problems | (p, q) -classical orthogonal polynomials | (p, q) -integrals | (p, q) -Pearson difference equation | MATHEMATICS | MATHEMATICS, APPLIED | (p, q)-integrals | (p, q)-Pearson difference equation | (p, q)-Sturm-Liouville problems | (p, q)-difference operators | (p, q)-classical orthogonal polynomials | Theorems (Mathematics) | Usage | Polynomials | Difference equations | Tests, problems and exercises | Theorems | Hermite polynomials | ( p , q ) $(p,q)$ -integrals | ( p , q ) $(p,q)$ -classical orthogonal polynomials | ( p , q ) $(p,q)$ -Pearson difference equation | ( p , q ) $(p,q)$ -Sturm-Liouville problems | ( p , q ) $(p,q)$ -difference operators

Journal Article

International Journal of Modern Physics E, ISSN 0218-3013, 11/2017, Volume 26, Issue 11, p. 1750078

.... Here, we modify the tail of the cut-off WS (CWS) and cut-off generalized WS (CGWS) potentials by attaching Hermite polynomial tails to them beyond the cut...

finite-range potential | Resonance | PHYSICS, NUCLEAR | PHYSICS, PARTICLES & FIELDS | Physics - Nuclear Theory

finite-range potential | Resonance | PHYSICS, NUCLEAR | PHYSICS, PARTICLES & FIELDS | Physics - Nuclear Theory

Journal Article

Annals of physics, ISSN 0003-4916, 2013, Volume 335, pp. 78 - 85

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects...

Group representation theory | Quantum mechanics | Coherent states | Orthogonal polynomials | PHYSICS, MULTIDISCIPLINARY | Algebra | Polynomials | Lie groups | Hilbert space | Coherence | Differential equations | DIFFERENTIAL EQUATIONS | INTEGRAL CALCULUS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ALGEBRA | EIGENSTATES | MATRIX ELEMENTS | QUANTUM MECHANICS | HILBERT SPACE | LEGENDRE POLYNOMIALS | RECURSION RELATIONS | HERMITE POLYNOMIALS | ANNIHILATION OPERATORS | LIE GROUPS

Group representation theory | Quantum mechanics | Coherent states | Orthogonal polynomials | PHYSICS, MULTIDISCIPLINARY | Algebra | Polynomials | Lie groups | Hilbert space | Coherence | Differential equations | DIFFERENTIAL EQUATIONS | INTEGRAL CALCULUS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ALGEBRA | EIGENSTATES | MATRIX ELEMENTS | QUANTUM MECHANICS | HILBERT SPACE | LEGENDRE POLYNOMIALS | RECURSION RELATIONS | HERMITE POLYNOMIALS | ANNIHILATION OPERATORS | LIE GROUPS

Journal Article

Russian mathematical surveys, ISSN 0036-0279, 06/2018, Volume 73, Issue 3, pp. 457 - 518

This paper considers the zero distribution of Hermite-Pade polynomials of the first kind associated with a vector function. (f) over right arrow = (f(1),...,f(s)) whose components f(k...

rational approximations | equilibrium problem | APPROXIMANTS | S-compact set | Hermite-Pade polynomials | MATHEMATICS | zero distribution | EQUILIBRIUM MEASURES | CONVERGENCE | SYSTEMS | ORTHOGONAL POLYNOMIALS | ASYMPTOTICS | BRANCH-POINTS | Riemann surfaces | Hermite polynomials | Orthogonality | Mathematical analysis

rational approximations | equilibrium problem | APPROXIMANTS | S-compact set | Hermite-Pade polynomials | MATHEMATICS | zero distribution | EQUILIBRIUM MEASURES | CONVERGENCE | SYSTEMS | ORTHOGONAL POLYNOMIALS | ASYMPTOTICS | BRANCH-POINTS | Riemann surfaces | Hermite polynomials | Orthogonality | Mathematical analysis

Journal Article

18.
Full Text
Unified Hermite Polynomial Model and Its Application in Estimating Non-Gaussian Processes

Journal of Engineering Mechanics, ISSN 0733-9399, 03/2019, Volume 145, Issue 3, p. 04019001

.... To estimate non-Gaussian processes, various third-order Hermite polynomial models have been proposed and widely applied...

Technical Papers | Hermite polynomial | Hardening process | Softening process | Unified model | Non-Gaussian process | LOAD | WIND PRESSURE | ENGINEERING, MECHANICAL | Damage assessment | Estimation | Gaussian distribution | Wind pressure | Structural engineering | Fatigue failure | Softening | Model accuracy | Crack propagation | Hardening | Gaussian process | Mathematical models | Hermite polynomials | Translations

Technical Papers | Hermite polynomial | Hardening process | Softening process | Unified model | Non-Gaussian process | LOAD | WIND PRESSURE | ENGINEERING, MECHANICAL | Damage assessment | Estimation | Gaussian distribution | Wind pressure | Structural engineering | Fatigue failure | Softening | Model accuracy | Crack propagation | Hardening | Gaussian process | Mathematical models | Hermite polynomials | Translations

Journal Article

1939, 4th ed., Colloquium publications - American Mathematical Society, ISBN 0821810235, Volume 23., xiii, 432

Book

Proceedings of the American Mathematical Society, ISSN 0002-9939, 08/2014, Volume 142, Issue 8, pp. 2581 - 2591

Let ℝ〈x〉 denote the ring of polynomials in g freely noncommuting variables x = (x
1
,..., x
g
). There is a natural involution...

Integers | Mathematical theorems | Algebra | Linear polynomials | Eigenvalues | Polynomials | Matrices | Convexity | Hermite polynomials | Symmetry | Free polynomials | Quasi-convex | Free real algebraic geometry | MATHEMATICS | MATHEMATICS, APPLIED | free real algebraic geometry | quasi-convex | SUMS

Integers | Mathematical theorems | Algebra | Linear polynomials | Eigenvalues | Polynomials | Matrices | Convexity | Hermite polynomials | Symmetry | Free polynomials | Quasi-convex | Free real algebraic geometry | MATHEMATICS | MATHEMATICS, APPLIED | free real algebraic geometry | quasi-convex | SUMS

Journal Article

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