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Walter Rudin called it "the most important function in mathematics". [1] It is therefore useful to have multiple ways to define (or characterize) it. Each of the characterizations below may be more or less useful depending on context. The "product limit" characterization of the exponential function was discovered by Leonhard Euler. [2]
A cut is maximum if the size of the cut is not smaller than the size of any other cut. The illustration on the right shows a maximum cut: the size of the cut is equal to 5, and there is no cut of size 6, or |E| (the number of edges), because the graph is not bipartite (there is an odd cycle).
Eric W. Weisstein conjectured, [12] and McKay et al. (2004) proved, that the same numbers count the (0,1) matrices for which all eigenvalues are positive real numbers. The proof is bijective: a matrix A is an adjacency matrix of a DAG if and only if A + I is a (0,1) matrix with all eigenvalues positive, where I denotes the identity matrix.
Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1, [17] + + + + + + =. Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start.
The distribution of X 1 + ⋯ + X n / √ n need not be approximately normal (in fact, it can be uniform). [38] However, the distribution of c 1 X 1 + ⋯ + c n X n is close to (,) (in the total variation distance) for most vectors (c 1, ..., c n) according to the uniform distribution on the sphere c 2 1 + ⋯ + c 2 n = 1.
For example, instead of testing whether x equals 1.1, one might test whether (x <= 1.0), or (x < 1.1), either of which would be certain to exit after a finite number of iterations. Another way to fix this particular example would be to use an integer as a loop index , counting the number of iterations that have been performed.
Sometimes γ(G) is used, since χ(G) is also used to denote the Euler characteristic of a graph. [5] A graph that can be assigned a (proper) k-coloring is k-colorable, and it is k-chromatic if its chromatic number is exactly k. A subset of vertices assigned to the same color is called a color class, every such class forms an independent set.
In open circuit scuba the upper limit for oxygen partial pressure is generally accepted as 1.6 bar, [40] equivalent to a depth of 6 msw (metres of sea water), but in-water and surface decompression at higher partial pressures is routinely used in surface supplied diving operation, both by the military and civilian contractors, as the ...