Crosscap number: Difference between revisions
en>Sun Creator Typo fixing and General, typos fixed: , → , using AWB |
en>Hyacinth ==External links== *"[http://www.indiana.edu/~knotinfo/descriptions/crosscap_number.html Crosscap Number]", ''KnotInfo''. |
||
| Line 1: | Line 1: | ||
In [[mathematical analysis]], '''nullclines''', sometimes called zero-growth [[isocline]]s, are encountered in a system of [[ordinary differential equation]]s | |||
:<math>x_1'=f_1(x_1, \ldots, x_n)</math> | |||
:<math>x_2'=f_2(x_1, \ldots, x_n)</math> | |||
::<math>\vdots</math> | |||
:<math>x_n'=f_n(x_1, \ldots, x_n)</math> | |||
where <math>x'</math> here represents a [[derivative]] of <math>x</math> with respect to another parameter, such as time <math>t</math>. The <math>j</math>'th nullcline is the geometric shape for which <math>x_j'=0</math>. The [[Fixed_point_(mathematics)|fixed point]]s of the system are located where all of the nullclines intersect. | |||
In a two-dimensional [[linear system]], the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves. | |||
== History == | |||
The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi<sup>1</sup>. This article also defined 'directivity vector' as | |||
<math>\mathbf{w} = \mathrm{sign}(P)\mathbf{i} + \mathrm{sign}(Q)\mathbf{j}</math>, | |||
where P and Q are the dx/dt and dy/dt differential equations, and i and j are the x and y direction unit vectors. | |||
Simonyi developed a new stability test method from these new definitions, and with it he studied differential equations. This method, beyond the usual stability examinations, provided semiquantative results. | |||
==References== | |||
{{reflist}} | |||
1.E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967 | |||
2. E. Simonyi – M. Kaszás: Method for the Dynamic Analysis of Nonlinear Systems, Periodica Polytechnica Chemical Engineering – Chemisches Ingenieurwesen, Polytechnical University Budapest, 1969 | |||
==External links== | |||
* {{planetmath reference|id= 5972|title=Nullcline}} | |||
* [http://ocw.mit.edu/NR/rdonlyres/D369E47F-064F-48E0-83CD-3B83725B5555/0/r02sol.pdf Notes] from [[MIT OpenCourseWare]] | |||
* [http://www.sosmath.com/diffeq/system/qualitative/qualitative.html SOS Mathematics: Qualitative Analysis] | |||
[[Category:Differential equations]] | |||
{{mathanalysis-stub}} | |||
[[de:Isokline]] | |||
[[nl:nullcline]] | |||
Latest revision as of 14:11, 26 May 2013
In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations
where here represents a derivative of with respect to another parameter, such as time . The 'th nullcline is the geometric shape for which . The fixed points of the system are located where all of the nullclines intersect. In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves.
History
The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi1. This article also defined 'directivity vector' as , where P and Q are the dx/dt and dy/dt differential equations, and i and j are the x and y direction unit vectors.
Simonyi developed a new stability test method from these new definitions, and with it he studied differential equations. This method, beyond the usual stability examinations, provided semiquantative results.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
1.E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967
2. E. Simonyi – M. Kaszás: Method for the Dynamic Analysis of Nonlinear Systems, Periodica Polytechnica Chemical Engineering – Chemisches Ingenieurwesen, Polytechnical University Budapest, 1969