Search results for "Cauchy–Schwarz inequality"

showing 4 items of 4 documents

Vectors and Vector Fields

2012

The purpose of this book is to explain in a rigorous way Stokes’s theorem and to facilitate the student’s use of this theorem in applications. Neither of these aims can be achieved without first agreeing on the notation and necessary background concepts of vector calculus, and therein lies the motivation for our introductory chapter.

AlgebraSolenoidal vector fieldStandard basisPhysics::Physics EducationVector fieldCross productDirection vectorVector calculusComplex lamellar vector fieldCauchy–Schwarz inequalityMathematics
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Hardy’s inequality and the boundary size

2002

We establish a self-improving property of the Hardy inequality and an estimate on the size of the boundary of a domain supporting a Hardy inequality.

Hölder's inequalityKantorovich inequalityMathematics::Functional AnalysisPure mathematicsInequalityMathematics::Complex VariablesApplied MathematicsGeneral Mathematicsmedia_common.quotation_subjectMathematical analysisMathematics::Classical Analysis and ODEsBoundary (topology)Mathematics::Spectral TheoryLog sum inequalityRearrangement inequalityCauchy–Schwarz inequalityHardy's inequalityMathematicsmedia_commonProceedings of the American Mathematical Society
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On improved fractional Sobolev–Poincaré inequalities

2016

We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

Hölder's inequalityKantorovich inequalityPure mathematicsYoung's inequalityBernoulli's inequalityGeneral Mathematics010102 general mathematicsMathematical analysisMinkowski inequality01 natural sciences010101 applied mathematicsLog sum inequalityRearrangement inequality0101 mathematicsCauchy–Schwarz inequalityMathematicsArkiv för Matematik
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An approximate technique for determining in closed-form the response transition probability density function of diverse nonlinear/hysteretic oscillat…

2019

An approximate analytical technique is developed for determining, in closed form, the transition probability density function (PDF) of a general class of first-order stochastic differential equations (SDEs) with nonlinearities both in the drift and in the diffusion coefficients. Specifically, first, resorting to the Wiener path integral most probable path approximation and utilizing the Cauchy–Schwarz inequality yields a closed-form expression for the system response PDF, at practically zero computational cost. Next, the accuracy of this approximation is enhanced by proposing a more general PDF form with additional parameters to be determined. This is done by relying on the associated Fokke…

Nonlinear stochastic dynamics Path integral Cauchy–Schwarz inequalityFokker–Planck equationStochastic differential equationsSettore ICAR/08 - Scienza Delle Costruzioni
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