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Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x 4 ≡ p (mod q) to that of x 4 ≡ q (mod p).
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), [1] states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).
At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. [12] [6] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. [13]
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. [1]
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Euler class of the bundle. The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking.
In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division: