Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer Verlag, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For us a versal decomposition will be an explicit decomposition of a perturbation into its tangential and normal components, and we will not derive any miniversal deformations that may have simpler forms, but hide the metric information. Versal deformations for function spaces are discussed in [18, 25, 4, 5]. The first application of these ideas for the matrix Jordan canonical form is due to Arnold [1] Further references closely related to Arnold s matrix approach are [30] and [6] The latter reference, 6] also includes applications to differential equations. Applications of the matrix idea ....

....written a wonderfully readable introduction to singularity theory emphasizing the elementary geometrical viewpoint. After reading this introduction, it is easy to be lulled into the belief that one has mastered the subject, but a whole further more advanced wealth of information may be found in [18, 25, 4]. Finally, what none of these references do very well is explain clearly that there is still much in this area that mankind does not yet fully understand. Singularity theory may be viewed as a branch of the study of curves and surfaces, but its crowning application is towards the topological ....

Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer Verlag, 1976.

....by Zeeman [74] to unify singularity theory, bifurcation theory and their applications and gained wide popularity. A thorough mathematical treatment on singularity theory can be found in the work of Arnol d [2, 3, 4, 5, 6, 7] More pragmatic introductions and applications are widely published, e.g. [9, 23, 24, 52, 60, 73, 74]. The catastrophe points are also called non Morse critical points, since a higher order Taylor expansion is essentially needed to describe the qualitative properties. Although the dimension of the variables is arbitrary, the Thom Splitting Lemma states that one can split up the function in a ....

Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer-Verlag, Berlin, second corrected printing edition, 1976.

....by the results in specific image analysis applications, an increasing interest has recently emerged in trying to establish a generic underpinning of deep structure. This may serve as a basis for a diversity of multiresolution schemes. Such bottom up approaches often rely on catastrophe theory [1, 14, 38, 44, 50, 51], which is now fairly well established in the context of the scale space paradigm. 1.3 Related Work The application of catastrophe theory in Gaussian scale space has been studied by Damon [8] probably the most comprehensive account on the subject as well as by many others [15, 16, 19, 20, ....

....and the intuitive forecast. A Appendix We clarify the theory presented in section 2 by discussing the appearance of scale space saddles at the generic catastrophe event in scale space describing an annihilation. This event, called a Fold catastrophe, is known from catastrophe theory (see e.g. [1, 2, 5, 13, 14, 38, 44, 50, 51]) and applied to and used in scale space (see e.g. 8, 9, 10, 12, 20, 21, 22, 23, 24, 31] Firstly, an example on one dimensional images is given, because scale space saddles coincide with the catastrophe points. Secondly, the results on a multi dimensional image is discussed. 26 1 0.8 0.6 ....

Y.-C. Lu, editor. Singularity Theory and an Introduction to Catastrophe Theory. Springer-Verlag, Berlin, second corrected printing edition, 1976.

....Zeeman [56] to unify singularity theory, bifurcation theory and their applications and gained wide popularity. A thoroughly mathematical treatment on singularity theory can be found in the work of Arnol d [1, 2, 3, 4, 5, 6] More practical introductions and applications are widely published, e.g. [7, 20, 22, 41, 45]. name nickname CG PT A 2 Fold x # 1 x A 3 Cusp A 4 Swallowtail x A 5 Butterfly D 4 Hyperbolic Umbillic x y y D 4 Elliptic Umbillic x D 5 Parabolic Umbillic x Table 1: The catastrophe germs for maximal 4 different perturbation parameters. ....

Y.-C. Lu, editor. Singularity Theory and an Introduction to Catastrophe Theory. Springer-Verlag, Berlin, second corrected printing edition, 1976.

....suggested by Zeeman [70] to unify singularity theory, bifurcation theory and their applications and gained wide popularity. A thorough mathematical treatment on singularity theory can be found in the work of Arnol d [2 7] More pragmatic introductions and applications are widely published, e.g. [9,24,26,52,59,70]. The catastrophe points are also called non Morse critical points, since a higher order Taylor expansion is essentially needed to describe the qualitative properties. Although the dimension of the variables is arbitrary, the Thom Splitting Lemma states that one can split up the function in a ....

Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer-Verlag, Berlin, second corrected printing edition, 1976.

....q 0 such that M ae fx 2 IR n j 1 2 kxk 2 b T m s 1 x qg; 1 2 kx 0 k 2 b T m s 1 x 0 q for all q q: 4.1) Let A ae IR 1 2 n(n 1) be the set of all non singular symmetric (n,n) Matrices. Then A is open in IR 1 2 n(n 1) and IR 1 2 n(n 1) nA has the Lebesgue mesure 0 ([23]) In the sequel almost all is always meant in the sense of the Lebesgue measure of the corresponding dimension. 36 R. Fandom Noubiap Theorem 4.1 Assume that f; h i ; g j 2 C 3 (IR n ; IR) i 2 I; j 2 J: Then, for almost all (A; x 0 ; b; q) 2 A Theta IR n Theta B Theta fq 2 IRjq ....

Lu, Y.-C; Singularity Theory and an Introduction to Catastrophe theory, universitext, Springer-Verlag,1976

....A) 0 B D 2 y [ Phi 0 (y;t; A;b) Gamma P k2K 1 u k Phi k (y;t;B) D T y Phi K 1 (y;t;B) Dy Phi K 1 (y;t;B) 0 1 C A 9 Let M(K 2 ) ae M s (fl) be the set of those symmetric matrices with rank jK 2 j whose columns with indices in K 2 are linearly independent. It is known (cf. [27]) that M(K 2 ) is a smooth manifold with codimension 0:5(fl Gamma jK 2 j) fl Gamma jK 2 j 1) We denote by Theta 2 C 3 (IR 0:5fl(fl 1) IR 0:5(fl GammajK 2 j) fl GammajK 2 j 1) a smooth mapping defining the manifold M(K 2 ) locally. An important observation about these mappings ....

Lu, Y. C., Singularity Theory and an Introduction to Catastrophe Theory, Universitext, Springerverlag, Berlin, Heidelberg, New York, 1976. 26

.... a general smooth function (R n R) into a small number of qualitatively distinct local structures, and attach distinct labels to each of the parts Fortunately, it is possible to give a precise form to these issues within the framework of Catastrophe or Singularity Theory (Thom, 1975) (Lu, 1976), Poston and I.N.Stewart, 1978) Gilmore, 1981) Catastrophe Theory (CT) deals with qualitative behavior of dynamic systems governed by smooth potentials . Naturally, the qualitative behavior of a system is determined by the qualitative structure of the potential, which the theory seeks to ....

....can be put into one of several canonical forms. It would be helpful, at this point, to give an intuitive picture of how a function can be molded into one of several canonical forms by manipulating the control parameters. For detailed derivations the interested reader may consult (Gilmore, 1981) (Lu, 1976). Example 2: a) Consider a member function of a family of functions, f 1 (x; c) x 2 R, c 2 R. Further, assuming that f 1 (x) has a CP at the origin, consider the Taylor series of f 1 (x) around the origin, in order to study the local structure of the function: f 1 (x; c) a 2 (c)x 2 a 3 ....

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Lu, Y. (1976). Singularity Theory and an introduction to catastrophe theory. Springer-Verlag, New York.

....in the dynamics of eqn. 6) which is brought forth by synchronous input presentation. An in depth mathematical study confirms this observation and explains how a non zero w has a stabilizing effect. However, the study involves powerful tools of catastrophe or singularity theory [Gil81] [Lu76], which is beyond the scope of the present paper. The details of this work will be presented in a forthcoming publication. 10 3 10 2 10 1 10 0 10 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Scale Figure 10: Effect of non zero w ( 0.07) on cluster tree for Data Set I Figure 10: Effect of ....

Y.C. Lu. Singularity Theory and an introduction to catastrophe theory. Springer-Verlag, New York, 1976.

....For us a versal decomposition will be an explicit decomposition of a perturbation into its tangential and normal components, and we will not derive any miniversal deformations that may have simpler forms, but hide the metric information. Versal deformations for function spaces are discussed in [17, 24, 4, 5]. The first application of these ideas for the matrix Jordan canonical form is due to Arnold [1] Further references closely related to Arnold s matrix approach are [28] and [6] The latter reference, 6] also includes applications to differential equations. Applications of the matrix idea ....

....written a wonderfully readable introduction to singularity theory emphasizing the elementary geometrical viewpoint. After reading this introduction, it is easy to be lulled into the belief that one has mastered the subject, but a whole further more advanced wealth of information may be found in [17, 24, 4]. Finally, what none of these 2 1 0 1 2 2 1 0 1 2 4 2 0 2 4 Figure 2: Manifold of singular matrices. The axis of the cylindrical stick is tangent to the manifold. references do very well is explain clearly that there is still much in this area that mankind does not yet fully understand. ....

Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer Verlag, 1976.

....For us a versal decomposition will be an explicit decomposition of a perturbation into its tangential and normal components, and we will not derive any miniversal deformations that may have simpler forms, but hide the metric information. Versal deformations for function spaces are discussed in [18, 25, 4, 5]. The first application of these ideas for the matrix Jordan canonical form is due to Arnold [1] Further references closely related to Arnold s matrix approach are [30] and [6] The latter reference, 6] also includes applications to differential equations. Applications of the matrix idea ....

....written a wonderfully readable introduction to singularity theory emphasizing the elementary geometrical viewpoint. After reading this introduction, it is easy to be lulled into the belief that one has mastered the subject, but a whole further more advanced wealth of information may be found in [18, 25, 4]. Finally, what none of these references do very well is explain clearly that there is still much in this area that mankind does not yet fully understand. Singularity theory may be viewed as a branch of the study of curves and surfaces, but its crowning application is towards the topological ....

Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer Verlag, 1976.

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Lu, Y.C. "Singularity Theory and an Introduction to Catastrophe Theory", SpringerVerlag, New York, 1976

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Lu, Y.C. "Singularity Theory and an Introduction to Catastrophe Theory", Springer-Verlag, New York, 1976

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Lu, Y.C., 1976, "Singularity Theory and an Introduction to Catastrophe Theory", Springer-Verlag, New York.

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Y.-C.Lu, Singularity Theory and an Introduction to Catastrophe Theory, SpringerVerlag 1976.

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