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Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid solutions, and the other refers to interstitial solid solutions.
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 ...
The solutions in terms of the original variable are obtained by substituting x 3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested only in real solutions, the solutions of the original equation are
Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx,} where R {\displaystyle R} is a rational function of x {\displaystyle x} and a x 2 + b x + c {\textstyle {\sqrt {ax^{2}+bx+c}}} .
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Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts , and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
For example, using x,y,z as variables, and taking f to be an uninterpreted function, the singleton equation set { f(1,y) = f(x,2) } is a syntactic first-order unification problem that has the substitution { x ↦ 1, y ↦ 2 } as its only solution. Conventions differ on what values variables may assume and which expressions are considered ...
brings the Abel equation of the first kind to the canonical form u ′ = u 3 + ϕ ( ξ ) . {\displaystyle u'=u^{3}+\phi (\xi ).\,} Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation in an implicit form.