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...ically, a harmonic prime ''p'' is always a factor of the numerators of the partial harmonic sums at positions ''p-1'', ''p*(p-1)'', and ''(p-1)*(p+1)''. For example, the numerators of the fractions given by <math> \sum_{i=1}^{4} \frac{1}{i} </math>, <math> \sum_{i=1}^{20} ...

...quid. The equation gives the relation between changes in mole fraction and partial pressure of the components. Because the sum of mole fractions of all components in the mixture is unity, i.e., ...

==Fractions== In general, this technique may be used to evaluate any fractions involving trigonometric functions. For example, consider the integral ...

...ent of interest. Every extensive property of a mixture has a corresponding partial molar property. ...and ethanol always have negative excess volumes when mixed, indicating the partial molar volume of each component is less when mixed than its molar volume whe ...

...example of cubic power series in <math>\mathbb{F}_2((X^{-1}))</math> whose partial quotients are bounded. (The degree of the power series in Baum and Sweet's They go on to prove that this unique root has partial quotients of degree <math>\le 2</math>. Before doing so, they state (in the ...

The transport equations for the fuel and oxygen mass fractions are ...equation to calculate combustion flows rather than calculating individual partial differential equations for the mass fraction. ...

== Relation to continued fractions == ...y from zero. An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded.<ref name=Bug245>{{cite boo ...

...- D_2) \frac{\partial N_1}{\partial x} = (D_2 - D_1) \frac{\partial N_2}{\partial x}.</math> * <math>N_1</math> and <math>N_2</math> are the [[Atomic ratio|atomic fractions]] of the two components. ...

...=Wang|first=Paul S.|contribution=A p-adic algorithm for univariate partial fractions|doi=10.1145/800206.806398|isbn=0-89791-047-8|location=New York, NY, USA|pag ...

* The [[Partially ordered set|partial order]] relations of divisibility are [[Lattice (order)|lattices]]; ...[[Ore condition for multiplicative sets]] and hence embeds in its group of fractions: such a group is a '''Garside group'''. A Garside group is [[biautomatic g ...

As we can see this is a rational function which can be solved using partial fractions. ...ational functions of <math>t</math>. In principle, [[factorization]] and [[partial fraction decomposition]] can be employed to break the integral down into si ...

...ns are either 0 or 1. The ''size'' of a fractional matching is the sum of fractions of all hyperedges. ...led '''perfect''' if for every vertex {{mvar|v}} in {{mvar|V}}, the sum of fractions of hyperedges in {{mvar|M}} containing {{mvar|v}} is ''exactly'' 1. ...

If ''α'' is the [[golden ratio]], then all the partial quotients ''a''_''n'' are equal to 1, the denominators ''q''<sub> ...

...of a [[rational function]] in a new variable, which can be calculated by [[partial fraction decomposition]]. which can be integrated using partial fractions, since <math>\frac {1}{1-u^2} = \frac {1}{2} \left( \frac{1}{1+u}+\frac{1}{ ...

===By partial fractions and a substitution (Barrow's approach)=== ...uchinsky"/> Barrow's proof of the result was the earliest use of [[partial fractions]] in integration.<ref name="rickey-tuchinsky"/> Adapted to modern notation ...

...frac{\partial}{\partial x_1} + \cdots + f_n({\mathbf x})\,\frac{\partial}{\partial x_n} . </math> ...frac{\partial}{\partial x_1} + \cdots + f_n({\mathbf x})\,\frac{\partial}{\partial x_n} </math> ...

...a [[Preference (economics)|preference-relation]] on individual items, to a partial preference-relation of item-bundles. ...RS extension of the ranking <math>w \prec x \prec y \prec z</math> is a [[partial order]] on the bundles of items, that includes all relations that can be de ...

...{Cite journal |last=Hirschhorn |first=Michael D. |date=2000 |title=Partial Fractions and Four Classical Theorems of Number Theory |journal=The American Mathemat ...

...Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them. ...of this theorem are presented: the ''[[simple continued fraction|continued fractions]]'' version due to Vincent,<ref name="paper_18342">{{cite journal|last=Vinc ...

...ber|rationals]]; specifically, the [[Continued fraction#Infinite continued fractions and convergents|convergents of the continued fraction]] are the best [[Diop ...rüche (On the approximate representation of irrational numbers by rational fractions)|journal=[[Mathematische Annalen]]|language=de|url=https://gdz.sub.uni-goet ...