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Chinese remainder theorem. Sunzi's original formulation: x ≡ 2 (mod 3) ≡ 3 (mod 5) ≡ 2 (mod 7) with the solution x = 23 + 105k, with k an integer. In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the ...
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
Theorem — For any table of nodes there is a continuous function f(x) on an interval [a, b] for which the sequence of interpolating polynomials diverges on [a,b]. [ 14 ] The proof essentially uses the lower bound estimation of the Lebesgue constant, which we defined above to be the operator norm of X n (where X n is the projection operator on ...
Using reconstruction techniques (Chinese remainder theorem, rational reconstruction, etc.) one can recover the GCD of f and g from its image modulo a number of ideals I. One can prove [ 3 ] that this works provided that one discards modular images with non-minimal degrees, and avoids ideals I modulo which a leading coefficient vanishes.
Mathematics. Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." [1]
Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form . (When one needs only the remainder, the polynomial remainder theorem provides a simpler method.) A typical example, where one needs the quotient, is the factorization of a polynomial p ( x ) {\displaystyle p(x)} for which one knows a root r :
Hasse principle. 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. This is handled by examining the equation in the ...
Polynomial remainder theorem. In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) [1] is an application of Euclidean division of polynomials. It states that, for every number any polynomial is the sum of and the product by of a polynomial in of degree less than the degree of In particular, is ...