Search results
Results from the WOW.Com Content Network
This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0 (because the definition demands F=0 at some t and first derivative =0 i.e. its value 0 and ...
The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable. Envelope for a modulated sine wave.
The upper envelope or pointwise maximum is defined symmetrically. For an infinite set of functions, the same notions may be defined using the infimum in place of the minimum, and the supremum in place of the maximum. [1] For continuous functions from a given class, the lower or upper envelope is a piecewise function whose pieces are from the ...
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space , or equivalently as the set of all convex combinations of points in the subset.
In mathematics, the lower convex envelope ˘ of a function defined on an interval [,] is defined at each point of the interval as the supremum of all ...
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. [1] As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in ...
If ω is a linear function of k, but not directly proportional (ω = ak + b), then the group velocity and phase velocity are different. The envelope of a wave packet (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity.
By Hopf–Lax formula, the Moreau envelope is a viscosity solution to a Hamilton–Jacobi equation. [3] Stanley Osher and co-authors used this property and Cole–Hopf transformation to derive an algorithm to compute approximations to the proximal operator of a function. [4]