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The vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π and its vanishing point. Some authors have used the phrase, "the image of a line includes its vanishing point".
Ramanujam vanishing theorem (algebraic geometry) Ramanujan–Skolem's theorem (Diophantine equations) Ramsey's theorem (graph theory, combinatorics) Rank–nullity theorem (linear algebra) Rao–Blackwell theorem ; Rashevsky–Chow theorem (control theory) Rational root theorem (algebra, polynomials) Rationality theorem
Note that for large m the line bundle K M ⊗ L ⊗m is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). Fujita conjecture provides an explicit bound on m, which is optimal for projective spaces.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.
In machine learning, the vanishing gradient problem is encountered when training neural networks with gradient-based learning methods and backpropagation. In such methods, during each training iteration, each neural network weight receives an update proportional to the partial derivative of the loss function with respect to the current weight ...
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. [7] Naturally also same function of one variable that depends on some complex parameter is a candidate.
The strange car-chase movie 'Vanishing Point' has had an equally strange afterlife, as detailed in this new book about the film and its star, a 1970 Dodge Challenger R/T 440.
The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a non-zero 3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task. [10]