A Primer of Real Analytic Functions

Krantz, Steven G.; Parks, Harold R.

doi:10.1007/978-3-0348-7644-5

Cited by 333 publications

(493 citation statements)

References 12 publications

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“…We follow this direction and consider h l as an analytical function. According to the definition of analytical functions [15], h l can be expressed explicitly by the converged power series as…”

Section: Analytical Dependency Modelingmentioning

confidence: 99%

“…According to Abel's Lemma [15], the series in (10) is uniformly converged. Thus, it can be integrated term by term, and equation (10) becomes…”

Section: Analytical Dependency Modelingmentioning

confidence: 99%

“…Similarly, since the series defined by (6) is converged, h l can be approximated by the first R + 1 terms in (6) as follows, (15) where R is a positive integer. According to the analysis in Section 3.1, the approximation in (15) can model dependency by setting solutions to the first R equations in (13) as non-trivial.…”

Section: Reduced Form Of the Adm Modelmentioning

confidence: 99%

“…Therefore, the reduced form of ADM can model dependency. Equations (14) and (15) indicate that the highest order of each term and each variable are K and R, respectively. Combining these two equations, the reduced analytical dependency model (RADM) is given by…”

Section: Reduced Form Of the Adm Modelmentioning

confidence: 99%

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Reduced Analytical Dependency Modeling for Classifier Fusion

Andy

Yuen

2012

Computer Vision – ECCV 2012

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Abstract. This paper addresses the independent assumption issue in classifier fusion process. In the last decade, dependency modeling techniques were developed under some specific assumptions which may not be valid in practical applications. In this paper, using analytical functions on posterior probabilities of each feature, we propose a new framework to model dependency without those assumptions. With the analytical dependency model (ADM), we give an equivalent condition to the independent assumption from the properties of marginal distributions, and show that the proposed ADM can model dependency. Since ADM may contain infinite number of undetermined coefficients, we further propose a reduced form of ADM, based on the convergent properties of analytical functions. Finally, under the regularized least square criterion, an optimal Reduced Analytical Dependency Model (RADM) is learned by approximating posterior probabilities such that all training samples are correctly classified. Experimental results show that the proposed RADM outperforms existing classifier fusion methods on Digit, Flower, Face and Human Action databases.

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Section: Analytical Dependency Modelingmentioning

confidence: 99%

“…According to Abel's Lemma [15], the series in (10) is uniformly converged. Thus, it can be integrated term by term, and equation (10) becomes…”

Section: Analytical Dependency Modelingmentioning

confidence: 99%

Section: Reduced Form Of the Adm Modelmentioning

confidence: 99%

Section: Reduced Form Of the Adm Modelmentioning

confidence: 99%

See 2 more Smart Citations

Reduced Analytical Dependency Modeling for Classifier Fusion

Andy

Yuen

2012

Computer Vision – ECCV 2012

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“…The local structure of these sets is well known (see [13]): they can contain isolated points and points with a finite number of branches emanating from them. In the second case the branches are described by Puiseux series.…”

Section: Fix(h)mentioning

confidence: 99%

Retracts, fixed point index and differential equations

Ortega

2008

Rev. R. Acad. Cien. Serie A. Mat.

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Abstract. Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions. Retractos,índice de punto fijo y ecuaciones diferencialesResumen. Algunos problemas de las ecuaciones diferenciales evolucionan hacia la Topología desde un origen analítico. Se discutirán dos problemas de este tipo: la existencia de soluciones asintóticas al equilibrio y la estabilidad de lasórbitas cerradas de los sistemas Hamiltonianos. La teoría de retractos y elíndice de punto fijo se han convertido en herramientas muyútiles para el estudio de estas cuestiones.

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Convergence of Adaptive Compression Methods for Hartree‐Fock‐Like Equations

Lin

Lindsey

2018

Comm Pure Appl Math

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The adaptively compressed exchange (ACE) method provides an efficient way to solve Hartree‐Fock‐like equations in quantum physics, chemistry, and materials science. The key step of the ACE method is to adaptively compress an operator that is possibly dense and full‐rank. In this paper, we present a detailed study of the adaptive compression operation and establish rigorous convergence properties of the adaptive compression method in the context of solving linear eigenvalue problems. Our analysis also elucidates the potential use of the adaptive compression method in a wide range of problems. © 2018 Wiley Periodicals, Inc.

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A Primer of Real Analytic Functions

Cited by 333 publications

References 12 publications

Reduced Analytical Dependency Modeling for Classifier Fusion

Reduced Analytical Dependency Modeling for Classifier Fusion

Retracts, fixed point index and differential equations

Convergence of Adaptive Compression Methods for Hartree‐Fock‐Like Equations

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