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  2. Substitution (logic) - Wikipedia

    en.wikipedia.org/wiki/Substitution_(logic)

    A substitution σ is called a flat substitution if xσ is a variable for every variable x. A substitution σ is called a renaming substitution if it is a permutation on the set of all variables. Like every permutation, a renaming substitution σ always has an inverse substitution σ −1, such that tσσ −1 = t = tσ −1 σ for every term t ...

  3. Change of variables - Wikipedia

    en.wikipedia.org/wiki/Change_of_variables

    In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.

  4. Millennium Prize Problems - Wikipedia

    en.wikipedia.org/wiki/Millennium_Prize_Problems

    The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are ...

  5. Extraneous and missing solutions - Wikipedia

    en.wikipedia.org/wiki/Extraneous_and_missing...

    One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions. However, strictly speaking, this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before. For example, consider the following ...

  6. Integration by substitution - Wikipedia

    en.wikipedia.org/wiki/Integration_by_substitution

    In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."

  7. Marginal rate of technical substitution - Wikipedia

    en.wikipedia.org/wiki/Marginal_rate_of_technical...

    where and are the marginal products of input 1 and input 2, respectively. Along an isoquant, the MRTS shows the rate at which one input (e.g. capital or labor) may be substituted for another, while maintaining the same level of output. Thus the MRTS is the absolute value of the slope of an isoquant at the point in question.

  8. Tangent half-angle substitution - Wikipedia

    en.wikipedia.org/.../Tangent_half-angle_substitution

    The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution, [6] and also known by variant names such as half-tangent substitution or half-angle substitution.

  9. Imputation (statistics) - Wikipedia

    en.wikipedia.org/wiki/Imputation_(statistics)

    In statistics, imputation is the process of replacing missing data with substituted values. When substituting for a data point, it is known as "unit imputation"; when substituting for a component of a data point, it is known as "item imputation".

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