S-expression: Difference between revisions

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As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply in terms of [[tempering out|tempering]]. The linked sections provide deeper information into each comma family.

* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form (''k''2)/(''k''2 - 1). Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).

* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form (''k''2)/(''k''2 - 1). Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).

* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', superparticular fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).

* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', superparticular fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).

* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')2 * S(''k''+1)''' and '''(S''k'')2 * S(''k''-1)'''. Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))2, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')2.

* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')2 * S(''k''+1)''' and '''(S''k'')2 * S(''k''-1)'''. Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))2, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')2.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: '''S''k''/S(''k''+1)'''. Tempering this out equates (''k''+2)/(''k''-1) with ((''k''+1)/''k'')3.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: '''S''k''/S(''k''+2)'''. Tempering this out equates (''k''+3)/(''k''-1) with ((''k''+2)/''k'')2.

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 As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply in terms of [[tempering out|tempering]].  The linked sections provide deeper information into each comma family.
-* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form (''k''<sup>2</sup>)/(''k''<sup>2</sup> - 1). Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).
+* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form (''k''<sup>2</sup>)/(''k''<sup>2</sup> - 1). <br /> Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).
-* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', superparticular fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).
+* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', superparticular fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). <br /> Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).
-* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')<sup>2</sup> * S(''k''+1)''' and '''(S''k'')<sup>2</sup> * S(''k''-1)'''. Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))<sup>2</sup>, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')<sup>2</sup>.
+* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')<sup>2</sup> * S(''k''+1)''' and '''(S''k'')<sup>2</sup> * S(''k''-1)'''. <br /> Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))<sup>2</sup>, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')<sup>2</sup>.
 * [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: '''S''k''/S(''k''+1)'''. Tempering this out equates (''k''+2)/(''k''-1) with ((''k''+1)/''k'')<sup>3</sup>.
 * [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: '''S''k''/S(''k''+2)'''. Tempering this out equates (''k''+3)/(''k''-1) with ((''k''+2)/''k'')<sup>2</sup>.