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In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence. necessary and sufficient
Also acid ionization constant or acidity constant. A quantitative measure of the strength of an acid in solution expressed as an equilibrium constant for a chemical dissociation reaction in the context of acid-base reactions. It is often given as its base-10 cologarithm, p K a. acid–base extraction A chemical reaction in which chemical species are separated from other acids and bases. acid ...
Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.
Say (,) is equipped with its usual topology. Then the essential range of f is given by . = { >: < {: | | <}}. [7]: Definition 4.36 [8] [9]: cf. Exercise 6.11 In other words: The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.
This can be generalized to restrict the range of values in the dataset between any arbitrary points and , using for example ′ = + (). Note that some other ratios, such as the variance-to-mean ratio ( σ 2 μ ) {\textstyle \left({\frac {\sigma ^{2}}{\mu }}\right)} , are also done for normalization, but are not nondimensional: the units do not ...
For example, saying "the absolute value is denoted by | · |" is perhaps clearer than saying that it is denoted as | |. ± (plus–minus sign) 1. Denotes either a plus sign or a minus sign. 2. Denotes the range of values that a measured quantity may have; for example, 10 ± 2 denotes an unknown value that lies between 8 and 12.
There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science.
The term "scientific law" is traditionally associated with the natural sciences, though the social sciences also contain laws. [11] For example, Zipf's law is a law in the social sciences which is based on mathematical statistics. In these cases, laws may describe general trends or expected behaviors rather than being absolutes.