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Hence, to get the solutions, we just make this substitution in the previous results. For x = 0, c 1 = 0 and c 2 = 1 − γ. Hence, in our case, c 1 = 0 while c 2 = γ − α − β. Let us now write the solutions. In the following we replaced each z by 1 - x.
The rows with minterm m(4,12) and m(10,11,14,15) can now be removed, together with all the columns they cover. The second prime implicant can be 'covered' by the third and fourth, and the third prime implicant can be 'covered' by the second and first, and neither is thus essential.
In mathematics, Appell series are a set of four hypergeometric series F 1, F 2, F 3, F 4 of two variables that were introduced by Paul Appell () and that generalize Gauss's hypergeometric series 2 F 1 of one variable.
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
Let α be a primitive element of GF(q m). For any positive integer i, let m i (x) be the minimal polynomial with coefficients in GF(q) of α i. The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m 1 (x),…,m d − 1 (x)). It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides x ...
According to the Erdős–Szekeres theorem, any sequence of + distinct integers has an increasing or a decreasing subsequence of length + [7] [8] For inputs in which each permutation of the input is equally likely, the expected length of the longest increasing subsequence is approximately . [9] [2]
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.