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The law of iterated logarithms operates "in between" the law of large numbers and the central limit theorem.There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums S n, scaled by n −1, converge to zero, respectively in probability and almost surely:
There are at least four proofs: Hermann Weyl's original proof from the compact group point of view, [10] based on the Weyl character formula and the Peter–Weyl theorem. The theory of Verma modules contains the highest weight theorem. This is the approach taken in many standard textbooks (e.g., Humphreys and Part II of Hall).
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as a hypothesis and then proceed to derive B.
By the Pythagorean theorem, the plane located units above the "equator" intersects the sphere in a circle of radius and area (). The area of the plane's intersection with the part of the cylinder that is outside of the cone is also π ( r 2 − y 2 ) {\displaystyle \pi \left(r^{2}-y^{2}\right)} .
This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it.
The line graph of a bipartite graph is perfect (see Kőnig's theorem), but need not be bipartite as the example of the claw graph shows. The line graphs of bipartite graphs form one of the key building blocks of perfect graphs, used in the proof of the strong perfect graph theorem. [15]
The mountain pass theorem is an existence theorem from the calculus of variations, ... For a proof, see section 8.5 of Evans. Weaker formulation. Let ...
The full 6,500 line formal proof of Jordan's curve theorem in Mizar. Collection of proofs of the Jordan curve theorem at Andrew Ranicki's homepage; A simple proof of Jordan curve theorem (PDF) by David B. Gauld; Brown, R.; Antolino-Camarena, O. (2014).