Liouville number (7281 bytes)
1: ...er theory]], a '''Liouville number''' is a [[real number]] ''x'' with the property that, for any positive ...
5: ... positive integer ''n'', there exists an infinite number of pairs of integers (''p'',''q'') obeying the ab...
7: ...'x'' is a Liouville number, ''x'' is [[irrational number | irrational]]. Assume otherwise; then there exis...
13: ...de the first example of a provably transcendental number,
19: ... constant'''. Liouville's constant is a Liouville number; if we define ''p''_''n'' and ''q''<sub...

Liouville function (1458 bytes)
1: [[sv:Liouvilles lambda-funktion]]
2: ...mportant [[function (mathematics)|function]] in [[number theory]].
8: where &Omega;(''n'') is the number of [[prime number|prime]] [[divisor|factors]] of ''n'', counted wit...
10: ...ave &Omega;(1)=0 and therefore &lambda;(1)=1. The Liouville function satisfies the identity:
15: The Liouville function is related to the [[Riemann zeta functio...

Liouville (crater) (1008 bytes)
20: |[[Joseph Liouville]]
22: '''Liouville''' is a small [[lunar]] impact [[crater]] that is...

Joseph Liouville (2129 bytes)
1: '''Joseph Liouville''' (born [[March 24]] [[1809]], died [[September ...
3: Liouville graduated from the [[École Polytechnique]] in 182...
5: ...e Galois]] which appeared in his journal in 1846. Liouville was also involved in politics for some time, and ...
7: ... is a second "[[Liouville's theorem (Hamiltonian)|Liouville's theorem]]" in Hamiltonian dynamics.
9: [[Category:1809 births|Liouville, Joseph]]

Liouville equation (1829 bytes)
1: The '''Liouville equation''' is the most important equation of [[S...
3: ...tem in the 6N-dim [[phase space]], where N is the number of particles.
14: (See [[Liouville's theorem (Hamiltonian)]] for further discussion ...
31: Another way to write down the Liouville Equation is
39: The Liouville Equation is a [[continuity equation]] for the [[p...

Liouville's theorem (380 bytes)
1: '''Liouville's theorem''' has various meanings:
3: *In [[complex analysis]], see [[Liouville's theorem (complex analysis)]].
4: *In [[Hamiltonian mechanics]], see [[Liouville's theorem (Hamiltonian)]].
5: ...that any [[Liouville number]] is [[Transcendental number|transcendental]], ''or'' the lemma involved on [[...

Liouville-Neumann series (990 bytes)
1: In [[mathematics]], the '''[[Liouville]]-Neumann series''' is an [[infinite series]] def...

Riemann-Liouville differintegral (2486 bytes)
13: The '''Riemann-Liouville differintegral''' (RL) is the simplest and easies...
15: === Constructing the Riemann-Liouville differintegral ===
16: We first introduce the ''Riemann-Liouville fractional integral'', which is a straightforward...

Sturm-Liouville theory (10705 bytes)
1: ...Charles Francois Sturm]] (1803-1855) and [[Joseph Liouville]] (1809-1882), is a second-order linear [[differe...
9: '''Sturm-Liouville theory''' is important in applied mathematics, wh...
11: == Sturm-Liouville form ==
14: is said to be in Sturm-Liouville form. The function ''w'' is known as the weight f...
19: can easily be put into Sturm-Liouville form, since ''D''(1-''x''²)=-2''x'', s...

Liouville's theorem (Hamiltonian) (6261 bytes)
1: ...ouville's theorem''' (also sometimes known as the Liouville equation) is a key theorem in [[statistical mecha...
4: ...her being in the field of complex analysis (see [[Liouville's theorem (complex analysis)]].
31: ... ''D'' is given by ''N''/''V'' where ''N'' is the number of points in the cloud, of volume ''V''.

Liouville's theorem (complex analysis) (2134 bytes)
2: ...s that every bounded (i.e., there exists a [[real number]] ''M'' such that |''f''(''z'')| &le; ''M'' for a...
4: Liouville's theorem can be used to give an elegant short pr...
6: ...e function whose image omits at least two complex numbers must be constant.
21: See also [[Joseph Liouville]].
23: [[de:Satz von Liouville]]

50 (number) (3306 bytes)
1: [[Category:Integers]] [[Category:Composite numbers]]
2: ... following [[49 (number)|49]] and preceding [[51 (number)|51]].
5: <tr><td colspan=2 align=center>{{numbers_50s}}
6: <tr><td>[[Cardinal number|Cardinal]] <td>fifty
7: <tr><td>[[Ordinal number|Ordinal]]<td>50th (fiftieth)

45 (number) (3601 bytes)
1: [[Category:Integers]] [[Category:Composite numbers]]
2: ...ollowing [[44 (number)|44]] and followed by [[46 (number)|46]].
5: <tr><td colspan=2 align=center>{{numbers_40s}}
6: <tr><td>[[Cardinal number|Cardinal]] <td>forty-five
7: <tr><td>[[Ordinal number|Ordinal]]<td>45th (forty-fifth)

33 (number) (3339 bytes)
1: [[Category:Integers]] [[Category:Composite numbers]]
2: '''33''' is the [[natural number]] following [[thirty-two|32]] and preceding [[thi...
5: <tr><td colspan=2>{{numbers_30s}}
6: <tr><td>[[Cardinal number|Cardinal]]<td>thirty-three
7: <tr><td>[[Ordinal number|Ordinal]]<td>33rd (thirty-third)

78 (number) (2653 bytes)
1: ...ollowing [[77 (number)|77]] and followed by [[79 (number)|79]].
4: <tr><td colspan=2>{{Numbers_70s}}
5: <tr><td>[[Cardinal number|Cardinal]] <td>seventy-eight
6: <tr><td>[[Ordinal number|Ordinal]]<td>78th (seventy-eighth)
15: ...es it a [[composite number#Sphenic number|sphenic number]].

-1 (number) (5662 bytes)
3: | colspan="2" | {{numbers (integers)}}
18: ...r than negative two (&minus;2) and less than [[0 (number)|0]].
20: ...ut slightly different properties to positive [[1 (number)|one]]. Negative one would be a multiplicative id...
24: .../sup> = 1/''x'', meaning that we define raising a number to the [[exponentiation|power]] &minus;1 to have ...
26: The two square roots of the real number negative one are the [[imaginary unit|imaginary u...

30 (number) (3491 bytes)
1: '''30''' ('''thirty''') is the [[natural number]] following [[twenty-nine|29]] and preceding [[th...
4: <tr><td colspan=2>{{numbers_30s}}
5: <tr><td>[[Cardinal number|Cardinal]]<td>thirty
6: <tr><td>[[Ordinal number|Ordinal]]<td>thirtieth
16: ...four squares, which makes it a [[square pyramidal number]].

28 (number) (3851 bytes)
1: [[Category:Composite numbers|2 8]] [[Category:Integers|2 8]]
2: [[Category:Perfect numbers|28]]
3: ... following [[27 (number)|27]] and preceding [[29 (number)|29]].
6: <tr><td colspan=2>{{numbers_20s}}
7: <tr><td>[[Cardinal number|Cardinal]] <td>twenty-eight

38 (number) (2347 bytes)
1: ... following [[37 (number)|37]] and preceding [[39 (number)|39]].
4: <tr><td colspan=2>{{numbers_30s}}
5: <tr><td>[[Cardinal number|Cardinal]] <td>thirty-eight
6: <tr><td>[[Ordinal number|Ordinal]]<td>38th (thirty-eighth)
22: *The [[atomic number]] of [[strontium]]

27 (number) (3296 bytes)
1: [[Category:Integers|2 7]] [[Category:Composite numbers|2 7]]
2: ... following [[26 (number)|26]] and preceding [[28 (number)|28]]. Twenty-seven is the smallest positive inte...
5: <tr><td colspan=2>{{numbers_20s}}
6: <tr><td>[[Cardinal number|Cardinal]] <td>twenty-seven
7: <tr><td>[[Ordinal number|Ordinal]]<td>twenty-seventh

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