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In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J ( χ , ψ ) for Dirichlet characters χ , ψ modulo a prime number p , defined by
Given an integer a and a non-zero integer d, it can be shown that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < | d |. The number q is called the quotient, while r is called the remainder. (For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see Division algorithm.)
If m = n, then f is a function from R n to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
If the "numerator" is 1, rules 3 and 4 give a result of 1. If the "numerator" and "denominator" are not coprime, rule 3 gives a result of 0. Otherwise, the "numerator" and "denominator" are now odd positive coprime integers, so we can flip the symbol using rule 6, then return to step 1. In addition to the codes below, Riesel [4] has it in Pascal.
Sloping lines denote graphs of 2x+5y=n where n is the total in pence, and x and y are the non-negative number of 2p and 5p coins, respectively. A point on a line gives a combination of 2p and 5p for its given total (green). Multiple points on a line imply multiple possible combinations (blue). Only lines with n = 1 or 3 have no points (red).
The infinite series whose terms are the natural numbers 1 + 2 + 3 ... the sum of an infinite number of ... f decay quickly enough for the remainder terms in the ...
It was first conjectured in 1939 by Ott-Heinrich Keller, [1] and widely publicized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle ...