By Robert Gulliver, Walter Littman, Roberto Triggiani
This quantity includes chosen papers that have been provided on the AMS-IMS-SIAM Joint summer season study convention on ""Differential Geometric equipment within the keep an eye on of Partial Differential Equations"", which was once held on the college of Colorado in Boulder in June 1999. the purpose of the convention used to be to discover the infusion of differential-geometric tools into the research of keep watch over thought of partial differential equations, really within the difficult case of variable coefficients, the place the actual features of the medium fluctuate from aspect to indicate. whereas a collectively ecocnomic hyperlink has been lengthy demonstrated, for no less than 30 years, among differential geometry and regulate of standard differential equations, a similar dating among differential geometry and regulate of partial differential equations (PDEs) is a brand new and promising topic.Very contemporary study, simply sooner than the Colorado convention, supported the expectancy that differential geometric equipment, while delivered to undergo on sessions of PDE modelling and regulate issues of variable coefficients, will yield major mathematical advances. The papers incorporated during this quantity - written via experts in PDEs and keep watch over of PDEs in addition to through geometers - jointly help the declare that the goals of the convention are being fulfilled. particularly, they advise the assumption that either matters - differential geometry and keep an eye on of PDEs - have a lot to achieve via nearer interplay with each other. therefore, extra learn actions during this region are sure to develop
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Additional info for Differential Geometric Methods in the Control of Partial Differential Equations: 1999 Ams-Ims-Siam Joint Summer Research Conference on Differential ... University of co
If r(so)is the position vector of a point P, then the position vector p(so) of the curvature center 0 of y at P is the sum of the vector r(so) and the vector PO. Thus we obtain the position vector of the evolute: From the formula evolute is 7: = kv it follows that v', = -kr. 2). Assume that y is the evolute of some planar curve 7; then 7is called the evolvent of y. Let us find the position vector T; = r(s) of the evolvent with respect to the arc length s of y. 2 The curve y is the evolute of 7,hence its tangent vector r is collinear to the principal normal vector to the evolvent.
Suppose that there exists a sequence of points M, E y converging to M such that for some arbitrary fixed positive number k. 8) it follows that C = 0, at the points M,, o ( A r 3 ) / A y 2-+0 as n contradicting the assumption that C # 0. e. the curve y is tangent to a straight line collinear to the x-axis. Let us rewrite the equation of y: © 2000 CRC Press SINGULAR POINTS OF PLANE CURVES Then from this equation we obtain: If k l # 0, then the expression under the square root is equivalent to -4CklAx3 as Ar 4 0.
Now we will consider families of curves such that different curves of a family can be mutually intersecting. The equation of any such circle is For different a we have equations of different circles of the family. Two arbitrary sufficiently near circles are intersecting. In general we will assume that an equation defines a family of plane curves. 2) represents some curve y,,of the family. 1 © 2000 CRC Press 52 DIFFERENTIAL GEOMETRY A N D TOPOLOGY OF CURVES On examination of a family of plane curves, it can be seen that some plane curves are singular with respect to the family.
Differential Geometric Methods in the Control of Partial Differential Equations: 1999 Ams-Ims-Siam Joint Summer Research Conference on Differential ... University of co by Robert Gulliver, Walter Littman, Roberto Triggiani