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Michael Danos and Johann Rafelski edited the Pocketbook of Mathematical Functions, published by Verlag Harri Deutsch in 1984. [14] [15] The book is an abridged version of Abramowitz's and Stegun's Handbook, retaining most of the formulas (except for the first and the two last original chapters, which were dropped), but reducing the numerical tables to a minimum, [14] which, by this time, could ...
The notion of a "many-one" functional relation": Russell first discusses the notion of "identity", then defines a descriptive function (pages 30ff) as the unique value ιx that satisfies the (2-variable) propositional function (i.e., "relation") φลท. N.B. The reader should be warned here that the order of the variables are reversed!
The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker : X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f : X → X to its ...
A specific element x of X is a value of the variable, and the corresponding element of Y is the value of the function at x, or the image of x under the function. The image of a function, sometimes called its range, is the set of the images of all elements in the domain. [6] [7] [8] [9]
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The converse of this implication leads to functions that are order-reflecting, i.e. functions f as above for which f(a) ≤ f(b) implies a ≤ b. On the other hand, a function may also be order-reversing or antitone, if a ≤ b implies f(a) ≥ f(b). An order-embedding is a function f between orders that is both order-preserving and order ...
The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function.
(Note - the relation between pressure, volume, temperature, and particle number which is commonly called "the equation of state" is just one of many possible equations of state.) If we know all k+2 of the above equations of state, we may reconstitute the fundamental equation and recover all thermodynamic properties of the system.