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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 ...
The envelope theorem is an important tool for comparative statics of optimization models. [ 2 ] The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions { f ( x , ⋅ ) } x ∈ X {\displaystyle \left\{f\left(x,\cdot \right)\right\}_{x\in X}} that are ...
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.
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 mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, ...
In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.