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For example, one might say that: "Given an arbitrary integer, multiplying it by two will result in an even number." Even further, the implication of the use of "arbitrary" is that generality will hold—even if an opponent were to choose the item in question. In which case, arbitrary can be regarded as synonymous to worst-case. [5]
Arbitrary inference is a classic tenet of cognitive therapy created by Aaron T. Beck in 1979. [1] He defines the act of making an arbitrary inference as the process of drawing a conclusion without sufficient evidence, or without any evidence at all.
The statement " is non-negative for arbitrarily large ." is a shorthand for: "For every real number , () is non-negative for some value of greater than .". In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers.
For example, some unicellular organisms have genomes much larger than that of humans. Cole's paradox: Even a tiny fecundity advantage of one additional offspring would favor the evolution of semelparity. Gray's paradox: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large accelerations.
For example, oxygen is necessary for fire. But one cannot assume that everywhere there is oxygen, there is fire. A condition X is sufficient for Y if X, by itself, is enough to bring about Y. For example, riding the bus is a sufficient mode of transportation to get to work.
The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).
For example, any irrational number x, such as x = √ 2, is a "gap" in the rationals Q in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers p/q, in the sense that distance of x and p/q given by the absolute value | x − p/q | is as small as desired. The following table lists some examples of ...
The first examples were the arbitrary width case.George Cybenko in 1989 proved it for sigmoid activation functions. [3] Kurt Hornik [], Maxwell Stinchcombe, and Halbert White showed in 1989 that multilayer feed-forward networks with as few as one hidden layer are universal approximators. [1]