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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 it is min/max at that t).
In category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called refinement.
In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. [1] 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 ...
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope.
In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, ...
A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions.
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.
The Moreau envelope has important applications in mathematical optimization: minimizing over and minimizing over are equivalent problems in the sense that the sets of minimizers of and are the same. However, first-order optimization algorithms can be directly applied to M f {\displaystyle M_{f}} , since f {\displaystyle f} may be non ...