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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".
Kleene fixed-point theorem (order theory) Kleene's recursion theorem (recursion theory) Knaster–Tarski theorem (order theory) Kneser's theorem (combinatorics) Kneser's theorem (differential equations) Kochen–Specker theorem ; Kodaira embedding theorem (algebraic geometry) Kodaira vanishing theorem (complex manifold) Koebe 1/4 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.
In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish. Andreotti–Grauert vanishing theorem; Bogomolov–Sommese vanishing theorem; Grauert–Riemenschneider vanishing theorem; Kawamata–Viehweg vanishing theorem; Kodaira vanishing theorem; Le Potier's vanishing theorem; Mumford vanishing theorem
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 ...
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.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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.