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The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field .
The Pythagorean theorem has at least 370 known proofs. [1]In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. [a] [2] [3] The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
The Euler class, in turn, relates to all other characteristic classes of vector bundles. For closed Riemannian manifolds , the Euler characteristic can also be found by integrating the curvature; see the Gauss–Bonnet theorem for the two-dimensional case and the generalized Gauss–Bonnet theorem for the general case.
Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation + = if n is an integer greater than two (n > 2).. Over time, this simple assertion became one of the most famous unproved claims in mathematics.
(also written as 0. 9, 0.., or 0.(9)) is a repeating decimal that is an alternative way of writing the number 1. Following the standard rules for representing numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, ... .
The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé. [4]: 98 The theorem was first proved by Bienaymé in 1853 [5] and more generally proved by Chebyshev in 1867. [6] [7] His student Andrey Markov provided another proof in his 1884 Ph.D. thesis ...
The Helmholtz decomposition in three dimensions was first described in 1849 [9] by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, [10] [11] which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. [11]
A related theorem is often called Fubini's theorem for infinite series, [1] although it is due to Alfred Pringsheim. [2] It states that if { a m , n } m = 1 , n = 1 ∞ {\textstyle \{a_{m,n}\}_{m=1,n=1}^{\infty }} is a double-indexed sequence of real numbers, and if ∑ ( m , n ) ∈ N × N a m , n {\textstyle \displaystyle \sum _{(m,n)\in ...