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n = 1 that yield a minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {( a n , b n )} N n = 1 for many variations of the exponential approximations and bounds up to N = 25 have been released to open access as a comprehensive dataset.
with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge (,) (,) into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
In the mathematical discipline of graph theory, the 2-factor theorem, discovered by Julius Petersen, is one of the earliest works in graph theory. It can be stated as follows: [ 1 ] Let G {\displaystyle G} be a regular graph whose degree is an even number, 2 k {\displaystyle 2k} .
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization.
For each integer n > 2, the function n x is defined and increasing for x ≥ 1, and n 1 = 1, so that the n th super-root of x, , exists for x ≥ 1. However, if the linear approximation above is used, then = + if −1 < y ≤ 0, so + cannot exist.
double x = 1.000000000000001; // rounded to 1 + 5*2^{-52} double y = 1.000000000000002; // rounded to 1 + 9*2^{-52} double z = y-x; // difference is exactly 4*2^{-52} The difference 1.000000000000002 − 1.000000000000001 {\displaystyle 1.000000000000002-1.000000000000001} is 0.000000000000001 = 1.0 × 10 − 15 {\displaystyle 0.000000000000001 ...
In graph theory, the trivial graph is a graph which has only 1 vertex and no edge. Database theory has a concept called functional dependency , written X → Y {\displaystyle X\to Y} . The dependence X → Y {\displaystyle X\to Y} is true if Y is a subset of X , so this type of dependence is called "trivial".