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In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number.
The content ranges from extremely difficult algebra and pre-calculus problems to problems in branches of mathematics not conventionally covered in secondary or high school and often not at university level either, such as projective and complex geometry, functional equations, combinatorics, and well-grounded number theory, of which extensive knowledge of theorems is required.
[17] [20] If, as usual in modern mathematics, the axiom of choice is assumed, then f is surjective if and only if there exists a function : such that =, that is, if f has a right inverse. [20] The axiom of choice is needed, because, if f is surjective, one defines g by g ( y ) = x , {\displaystyle g(y)=x,} where x {\displaystyle x} is an ...
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin.
Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O (2 p ( n ) ) time, where p ( n ) is a polynomial function of n .
There are also two SL only courses: a transdisciplinary course, Environmental Systems and Societies, that satisfies Diploma requirements for Groups 3 and 4, [2] and Sports, Exercise and Health Science (previously, for last examinations in 2013, a pilot subject [3]). Astronomy also exists as a school-based syllabus.
Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous. [6] A chain complex is said to be exact if the image of the (n+1)th map is always equal to the kernel of the nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being ...
Klein's j-invariant in the complex plane. In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers.