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The Beta distribution on [0,1], a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution , a straight-forward generalization of the Beta distribution to arbitrary bounded intervals [ a , b ...
Conversely, it is possible to 2-colour a K 5 without creating any monochromatic K 3, showing that R(3, 3) > 5. The unique [b] colouring is shown to the right. Thus R(3, 3) = 6. The task of proving that R(3, 3) ≤ 6 was one of the problems of William Lowell Putnam Mathematical Competition in 1953, as well as in the Hungarian Math Olympiad in 1947.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
An R 2 of 1 indicates that the regression predictions perfectly fit the data. Values of R 2 outside the range 0 to 1 occur when the model fits the data worse than the worst possible least-squares predictor (equivalent to a horizontal hyperplane at a height equal to the mean of the observed data). This occurs when a wrong model was chosen, or ...
Ramsey defines a proposition as "ethically neutral" when two possible outcomes have an equal value. In other words, if the probability can be defined as a preference, each proposition should have 1 / 2 to be indifferent between both options. [5] Ramsey shows that = (()) (() ()) [6]
Consider the ring A = R[x,y,z] of polynomials in three variables over the real numbers and its field of fractions M = R(x,y,z). The projective dimension of M as A-module is either 2 or 3, but it is independent of ZFC whether it is equal to 2; it is equal to 2 if and only if CH holds. [14]
If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative. On the other hand, r 1 – r 2 and r 1 – r 3 are complex conjugates, and their product is real and positive. [23]
Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are