The Calculus of Variations in the Large

Morse, Marston

doi:10.1090/coll/018

Cited by 490 publications

(319 citation statements)

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“…According to the Morse theory [8], a continuous function µ defined on a surface can characterize the surface topology using its critical points. As we choose the integral of contact and merge frame15 frame16 Figure 2.…”

Section: Tip Detection and Correspondence Estimationmentioning

confidence: 99%

A 3D Shape Descriptor for Segmentation of Unstructured Meshes into Segment-Wise Coherent Mesh Series

Mukasa

Nobuhara

Tung

et al. 2014

2014 2nd International Conference on 3D Vision

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Section: Tip Detection and Correspondence Estimationmentioning

confidence: 99%

A 3D Shape Descriptor for Segmentation of Unstructured Meshes into Segment-Wise Coherent Mesh Series

Mukasa

Nobuhara

Tung

et al. 2014

2014 2nd International Conference on 3D Vision

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“…The simplest instance of this decomposition is the gradient flow of a Morse function ( [10], [12]). In this case the components of the chain recurrent set are the critical points of the function.…”

Section: Introductionmentioning

confidence: 99%

Morse decompositions and connection matrices

Moeckel

1988

Ergod. Th. Dynam. Sys.

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“…hold almost everywhere on a <t < A, and the transversality conditions We remark that [10] gives a more general theorem and includes the complex case. We also note that (8) is the integrated form of the 2/ith order inte-…”

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confidence: 99%

“…Both theorems are given in [8]. In Theorem 6 let a = 1,..., p and k -0,..., n -1 ; then we have We note that Kix) is Jix) except for the terms which give rise to the principal part of equation (10). Furthermore Dix) is elliptic (in the sense of Hestenes) and generates a norm on 8 equivalent to (1).…”

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An approximation theory for generalized Fredholm quadratic forms and integral-differential equations

Gregory¹,

López-Acedo²

1976

Trans. Amer. Math. Soc.

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ABSTRACT. An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integraldifferential equations by Lopez, and extended with applications for approximation problems by Gregory.The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas.In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order)ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the "Euler-Lagrange equations"for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations.In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.Received by the editors August 27, 1973 and, in revised form, April 8, 1975 AMS (MOS) subject classifications (1970). Primary 45J05, 49A30, 34C10.Key words and phrases. Approximation theory, conjugate points, oscillations, calculus of variations, Fredholm integral differential equations, spline approximations. 1. Introduction. The main purpose of this paper is to present an approximation theory of quadratic forms which is applicable to a very general class of quadratic forms and to linear selfadjoint operators of a generalized Fredholm type. That is 2wth order, integral-differential systems such as T»(r) = vß~l(t), or as the generalized system:

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The Calculus of Variations in the Large

Cited by 490 publications

References 0 publications

A 3D Shape Descriptor for Segmentation of Unstructured Meshes into Segment-Wise Coherent Mesh Series

A 3D Shape Descriptor for Segmentation of Unstructured Meshes into Segment-Wise Coherent Mesh Series

Morse decompositions and connection matrices

An approximation theory for generalized Fredholm quadratic forms and integral-differential equations

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