The umbilic bracelet and the double-cusp catastrophe

Zeeman, E. C.

doi:10.1007/bfb0077854

Cited by 25 publications

(16 citation statements)

References 6 publications

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“…According to the cusp catastrophe theory [41,42], one of the most important features of this region is that it has a cusp, at figure 3b.…”

Section: ð2:8þmentioning

confidence: 99%

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

Qian

2010

J. R. Soc. Interface.

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A theory for an non-equilibrium phase transition in a driven biochemical network is presented. The theory is based on the chemical master equation (CME) formulation of mesoscopic biochemical reactions and the mathematical method of large deviations. The large deviations theory provides an analytical tool connecting the macroscopic multi-stability of an open chemical system with the multi-scale dynamics of its mesoscopic counterpart. It shows a corresponding non-equilibrium phase transition among multiple stochastic attractors. As an example, in the canonical phosphorylation -dephosphorylation system with feedback that exhibits bistability, we show that the non-equilibrium steadystate (NESS) phase transition has all the characteristics of classic equilibrium phase transition: Maxwell construction, a discontinuous first-derivative of the 'free energy function', Lee-Yang's zero for a generating function and a critical point that matches the cusp in nonlinear bifurcation theory. To the biochemical system, the mathematical analysis suggests three distinct timescales and needed levels of description. They are (i) molecular signalling, (ii) biochemical network nonlinear dynamics, and (iii) cellular evolution. For finite mesoscopic systems such as a cell, motions associated with (i) and (iii) are stochastic while that with (ii) is deterministic. Both (ii) and (iii) are emergent properties of a dynamic biochemical network.

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“…According to the cusp catastrophe theory [41,42], one of the most important features of this region is that it has a cusp, at figure 3b.…”

Section: ð2:8þmentioning

confidence: 99%

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

Qian

2010

J. R. Soc. Interface.

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“…See also (Zeeman 1976), for the focal aspect, and (Darboux 1896), (Gutierrez and Sotomayor 1982, 1983, 1991 and (Bruce and Fidal 1989), for the Darbouxian types.…”

Section: At the Crossroads Of Geometry And Global Analysismentioning

confidence: 99%

A metric property of umbilic points

García

Sotomayor

2003

An. Acad. Bras. Ciênc.

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In the space of cubic forms of surfaces, regarded as a G-space and endowed with a natural invariant metric, the ratio of umbilic points with a negative index to those with a positive index is evaluated in terms of the asymmetry of the metric, defined here. This work contains also a section with references pertinent to umbilic points and one with a discussion of a connection of the above defined ratio with that reported in 1977 by Berry and Hannay in the domain of Statistical Physics.

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“…The range of its blossom is a 3-dimensional variety that spirals away from that space curve. The umbilic bracelet of catastrophe theory, neatly analyzed by E. C. Zeeman [7], is a double covering of this variety.…”

Section: This Algebraic Perspective Is Particularly Valuable When We mentioning

confidence: 99%

Béziers and B-splines as Multiaffine Maps

Ramshaw

1988

Theoretical Foundations of Computer Graphics and CAD

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It is a classical principle in mathematics that polynomials in a single variable of degree n are essentially equivalent to symmetric polynomials in n variables that are linear in each variable separately. We shall apply this principle to the Bezier and B-spline curves and surfaces that are used in computer aided geometric design. The main result is a method of labeling the Bezier points that control a curve segment or surface patch or the de Boor points that control a B-spline curve with symmetric, multivariate labels. The properties of these labels make it simple to understand or to reconstruct the basic algorithms in this area, such as the de Casteljau Algorithm and the de Boor Algorithm.

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The umbilic bracelet and the double-cusp catastrophe

Cited by 25 publications

References 6 publications

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

A metric property of umbilic points

Béziers and B-splines as Multiaffine Maps

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