Search results for "second order"

showing 10 items of 33 documents

Indefinite integrals from Wronskians and related linear second-order differential equations

2021

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica.

Second order differential equationsApplied MathematicsMathematics::Metric GeometryApplied mathematicsVDP::Matematikk og Naturvitenskap: 400::Matematikk: 410AnalysisMathematicsIntegral Transforms and Special Functions
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AC Stark shift of the ground state of atomic hydrogen

2004

An analytical expression for the second-order AC Stark shift of the ground state of atomic hydrogen is derived, which is convergent for negative as well as for positive energies of intermediate states except for the resonances. To clarify the applicability of the second-order perturbation theory, we compared results with those which are obtained by us and other authors using nonperturbative methods. It appears that values obtained for the AC Stark shift using our simple formula agree on average with Floquet-method calculations up to the field strength F=0.12 (a.u.), which corresponds to I=1015 W/cm2.

PhysicsSecond order theoryQuantum opticsPhysics and Astronomy (miscellaneous)Condensed matter physicsHydrogenGeneral EngineeringGeneral Physics and Astronomychemistry.chemical_elementField strengthsymbols.namesakeStark effectchemistrysymbolsIntermediate statePhysics::Atomic PhysicsPerturbation theoryAtomic physicsGround stateApplied Physics B: Lasers and Optics
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Existence and multiplicity of periodic solutions for second order Hamiltonian systems depending on a parameter

2013

The existence of at least one nontrivial periodic solution for a class of second order Hamiltonian systems depending on a parameter is obtained, under an algebraic condition on the nonlinearity G and without requiring any asymptotic behavior neither at zero nor at infinity. The existence is still deduced in the particular case when G is subquadratic at zero. Finally, two multiplicity results are proved if G, in addition, is required to fulfill some different Ambrosetti-Rabinowitz type superquadratic conditions at infinity. The approach is fully variational. © Heldermann Verlag.

Critical points Periodic solutions Second order hamiltonian systemsPeriodic solutionsPeriodic solutionCritical pointsSecond order hamiltonian systemsCritical point
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Periodic solutions for a class of second-order Hamiltonian systems

2005

Multiplicity results for an eigenvalue second-order Hamiltonian system are investigated. Using suitable critical points arguments, the existence of an exactly determined open interval of positive eigenvalues for which the system admits at least three distinct periodic solutions is established. Moreover, when the energy functional related to the Hamiltonian system is not coercive, an existence result of two distinct periodic solutions is given.© 2005 Texas State University - San Marcos.

Second order Hamiltonian systemPeriodic solutioncritical pointslcsh:MathematicsMultiple solutioneigenvalue problemperiodic solutionslcsh:QA1-939Second order Hamiltonian systemsAnalysisCritical pointmultiple solutions.Electronic Journal of Differential Equations
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Geometric optimal control : homotopic methods and applications

2012

This work is about geometric optimal control applied to celestial and quantum mechanics. We first dealt with the minimum fuel consumption problem of transfering a satellite around the Earth. This brought to the creation of the code HamPath which permits first of all to solve optimal control problem for which the command law is smooth. It is based on the Pontryagin Maximum Principle (PMP) and on the notion of conjugate point. This program combines shooting method, differential homotopic methods and tools to compute second order optimality conditions. Then we are interested in quantum control. We study first a system which consists in two different particles of spin 1/2 having two different r…

[SPI.OTHER]Engineering Sciences [physics]/OtherMéthodes de tirHomotopie différentielle[ SPI.OTHER ] Engineering Sciences [physics]/OtherOrbital transferContrôle optimal géométrique[SPI.OTHER] Engineering Sciences [physics]/Other[ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM]Shooting methodsDifferential homotopyAutomatic differentiationContraste en RMNQuantum control[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM]Geometric optimal controlConditions du deuxième ordreTransfert orbitalLieux conjugués et de coupureDifférenciation automatiqueSecond order conditions[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM]Cut and conjugate lociContrast imaging in NMRContrôle quantique
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Global Non-monotonicity of Solutions to Nonlinear Second-Order Differential Equations

2018

We study behavior of solutions to two classes of nonlinear second-order differential equations with a damping term. Sufficient conditions for the first derivative of a solution x(t) to change sign at least once in a given interval (in a given infinite sequence of intervals) are provided. These conditions imply global non-monotone behavior of solutions.

Differential equationGeneral Mathematics010102 general mathematicsMonotonic functionInterval (mathematics)01 natural sciencesNonlinear differential equationsTerm (time)010101 applied mathematicsSecond order differential equationsNonlinear systemApplied mathematics0101 mathematicsNonlinear differential equations ; non-monotone behaviour ; second order ; damping term ; reciprocal equationSign (mathematics)MathematicsMediterranean Journal of Mathematics
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Blind Source Separation Based on Joint Diagonalization in R: The Packages JADE and BSSasymp

2017

Blind source separation (BSS) is a well-known signal processing tool which is used to solve practical data analysis problems in various fields of science. In BSS, we assume that the observed data consists of linear mixtures of latent variables. The mixing system and the distributions of the latent variables are unknown. The aim is to find an estimate of an unmixing matrix which then transforms the observed data back to latent sources. In this paper we present the R packages JADE and BSSasymp. The package JADE offers several BSS methods which are based on joint diagonalization. Package BSSasymp contains functions for computing the asymptotic covariance matrices as well as their data-based es…

Statistics and ProbabilityComputer scienceJADE (programming language)02 engineering and technologyLatent variableMachine learningcomputer.software_genre01 natural sciencesBlind signal separation010104 statistics & probabilityMatrix (mathematics)nonstationary source separationMixing (mathematics)0202 electrical engineering electronic engineering information engineeringsecond order source separation0101 mathematicslcsh:Statisticslcsh:HA1-4737computer.programming_languageta113Signal processingta112matematiikkamultivariate time seriesmathematicsbusiness.industryEstimator020206 networking & telecommunicationsriippumattomien komponenttien analyysiindependent component analysis; multivariate time series; nonstationary source separation; performance indices; second order source separationIndependent component analysisperformance indicesstatisticsindependent component analysisArtificial intelligenceStatistics Probability and UncertaintybusinesscomputerAlgorithmSoftwareJournal of Statistical Software
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Multiplicity results for systems of asymptotically linear second order equations

2002

Abstract We prove the existence and multiplicity of solutions, with prescribed nodal properties, for a BVP associated with a system of asymptotically linear second order equations. The applicability of an abstract continuation theorem is ensured by upper and lower bounds on the number of zeros of each component of a solution.

Asymptotically linearAsymptotically linear second order system continuation theoremGeneral MathematicsMultiplicity resultsMathematical analysisSecond order equationStatistical and Nonlinear PhysicsMathematicsAdvanced Nonlinear Studies
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Three periodic solutions for perturbed second order Hamiltonian systems

2009

AbstractIn this paper we study the existence of three distinct solutions for the following problem−u¨+A(t)u=∇F(t,u)+λ∇G(t,u)a.e. in [0,T],u(T)−u(0)=u˙(T)−u˙(0)=0, where λ∈R, T is a real positive number, A:[0,T]→RN×N is a continuous map from the interval [0,T] to the set of N-order symmetric matrices. We propose sufficient conditions only on the potential F. More precisely, we assume that G satisfies only a usual growth condition which allows us to use a variational approach.

Continuous mapPeriodic solutionsApplied MathematicsSecond order equationHamiltonian systemCritical pointCombinatoricssymbols.namesakesymbolsSymmetric matrixHamiltonian (quantum mechanics)Second order Hamiltonian systemsAnalysisMathematicsJournal of Mathematical Analysis and Applications
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Signal dimension estimation in BSS models with serial dependence

2022

Many modern multivariate time series datasets contain a large amount of noise, and the first step of the data analysis is to separate the noise channels from the signals of interest. A crucial part of this dimension reduction is determining the number of signals. In this paper we approach this problem by considering a noisy latent variable time series model which comprises many popular blind source separation models. We propose a general framework for the estimation of the signal dimension that is based on testing for sub-sphericity and give examples of different tests suitable for time series settings. In the inference we rely on bootstrap null distributions. Several simulation studies are…

nonstationary source separationdimension reductionsignaalinkäsittelyaikasarjatsub-sphericitysecond order source separationblock bootstrapaikasarja-analyysi2022 International Conference on Electrical, Computer, Communications and Mechatronics Engineering (ICECCME)
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