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Universal generalization / instantiation. Existential generalization / instantiation. In propositional logic and Boolean algebra, De Morgan's laws, [ 1][ 2][ 3] also known as De Morgan's theorem, [ 4] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British ...
Mathematical fallacy. In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known ...
The proposition in probability theory known as the law of total expectation, [ 1] the law of iterated expectations[ 2] ( LIE ), Adam's law, [ 3] the tower rule, [ 4] and the smoothing theorem, [ 5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then.
Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
Group theory. The commutator of two elements, g and h, of a group G, is the element. [g, h] = g−1h−1gh. This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by ...
Fundamental theorem of algebra. The fundamental theorem of algebra, also called d'Alembert's theorem[ 1] or the d'Alembert–Gauss theorem, [ 2] states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a ...
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product is a product distribution .
The Lie bracket is an R - bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra . The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory ...
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