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The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations and stochastic processes.In 1947, when Kac and Feynman were both faculty members at Cornell University, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. [1]
For a parametric equation of a parabola in general position see § As the affine image of the unit parabola. The implicit equation of a parabola is defined by an irreducible polynomial of degree two: + + + + + =, such that =, or, equivalently, such that + + is the square of a linear polynomial.
In this equation, the origin is the midpoint of the horizontal range of the projectile, and if the ground is flat, the parabolic arc is plotted in the range . This expression can be obtained by transforming the Cartesian equation as stated above by y = r sin ϕ {\displaystyle y=r\sin \phi } and x = r cos ϕ {\displaystyle x=r\cos \phi } .
Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, Hamilton (1993a) proved the following "Li–Yau inequality". [ 5 ] Let M {\displaystyle M} be a smooth manifold, and let g t {\displaystyle g_{t}} be a solution of the Ricci flow with t ∈ ( 0 , T ) {\displaystyle t ...
Fermat's spiral: a>0, one branch = + Fermat's spiral, both branches. A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant.
Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. J. Serrin , and J. Moser (1961, 1964) generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions.
If this transformation is performed on each conic in an orthogonal net of confocal ellipses and hyperbolas, the limit is an orthogonal net of confocal parabolas facing opposite directions. Every parabola with focus at the origin and x-axis as its axis of symmetry is the locus of points satisfying the equation
The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R 3 certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ...