formulasearchengine - User contributions [en]

Landweber exact functor theorem

2014-10-23T19:02:49Z

24.7.112.170:

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Probability amplitude

2014-02-17T08:51:43Z

24.7.123.82: /* In the context of the double-slit experiment */

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External sorting

2014-02-01T00:03:02Z

24.7.64.61: as i understand it, external sorts can be distribution or merge sorts as those terms are usually used

[[File:Menelaus' theorem 1.svg|right|thumb|Menelaus' theorem, case 1: line DEF passes inside triangle ABC]]

'''Menelaus' theorem''', named for [[Menelaus of Alexandria]], is a theorem about [[triangle]]s in [[plane geometry]]. Given a triangle ''ABC'', and a [[Transversal (geometry)|transversal]] line that crosses ''BC'', ''AC'' and ''AB'' at points ''D'', ''E'' and ''F'' respectively, with ''D'', ''E'', and ''F'' distinct from ''A'', ''B'' and ''C'', then
: <math>\frac{AF}{FB} \times \frac{BD}{DC} \times \frac{CE}{EA} = -1.</math>

or simply

: <math>AF \times BD \times CE= - FB \times DC \times EA .</math>

This equation uses signed lengths of segments, in other words the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line. For example, ''AF''/''FB'' is defined as having positive value when ''F'' is between ''A'' and ''B'' and negative otherwise.

The [[Theorem#Converse|converse]] is also true: If points ''D'', ''E'' and ''F'' are chosen on ''BC'', ''AC'' and ''AB'' respectively so that
: <math>\frac{AF}{FB} \times \frac{BD}{DC} \times \frac{CE}{EA} = -1,</math>
then ''D'', ''E'' and ''F'' are collinear. The converse is often included as part of the theorem.

The theorem is very similar to [[Ceva's theorem]] in that their equations differ only in sign.

== Proof ==
[[File:Menelaos' theorem 2.svg|right|thumb|Menelaus' theorem, case 2: line DEF is entirely outside triangle ABC]]

A standard proof is as follows:<ref>Follows Russel</ref>

First, the sign of the [[left-hand side]] will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (lower diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle. (See [[Pasch's axiom]].)

To check the magnitude, construct perpendiculars from ''A'', ''B'', and ''C'' to the line ''DEF'' and let their lengths be ''a, b,'' and ''c'' respectively. Then by [[Similarity (geometry)|similar]] triangles it follows that |''AF''/''FB''| = |''a''/''b''|, |''BD''/''DC''| = |''b''/''c''|, and |''CE''/''EA''| = ''c''/''a''. So
: <math>\left|\frac{AF}{FB}\right| \cdot \left|\frac{BD}{DC}\right| \cdot \left|\frac{CE}{EA}\right| = \left| \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} \right| = 1.</math>

For a simpler, if less symmetrical way to check the magnitude,<ref>Follows {{cite book |title=Inductive Plane Geometry|first=George Irving|last=Hopkins|publisher=D.C. Heath & Co.|year=1902|chapter=Art. 983}}</ref> draw ''CK'' parallel to ''AB'' where ''DEF'' meets ''CK'' at ''K''. Then by similar triangles
: <math>\left|\frac{BD}{DC}\right| = \left|\frac{BF}{CK}\right|,\,\left|\frac{AE}{EC}\right| = \left|\frac{AF}{CK}\right|</math>
and the result follows by eliminating ''CK'' from these equations.

The converse follows as a corollary.<ref>Follows Russel with some simplification</ref> Let ''D'', ''E'' and ''F'' be given on the lines ''BC'', ''AC'' and ''AB'' so that the equation holds. Let ''F''′ be the point where ''DE'' crosses ''AB''. Then by the theorem, the equation also holds for ''D'', ''E'' and ''F''′. Comparing the two,
: <math>\frac{AF}{FB} = \frac{AF'}{F'B}.</math>
But at most one point can cut a segment in a given ratio so ''F''=''F''′.

=== A non-computational proof using homothecies ===
The following proof<ref>See Michèle Audin, Géométrie, éditions BELIN, Paris 1998: indication for exercice 1.37, p. 273</ref> uses only notions of [[affine geometry]], notably [[Homothetic transformation|homothecies]].
Whether or not ''D'', ''E'', ''F'' are collinear, there are three homothecies with centers ''D'', ''E'', ''F'' that respectively send ''B'' to ''C'', ''C'' to ''A'', and ''A'' to ''B''. The composition of the three then is an element of the group of homothecy-translations that fixes ''B'', so it is a homothecy with center ''B'', possibly with ratio 1 (in which case it is the identity). This composition fixes the line ''DE'' if and only if ''F'' is collinear with ''D'' and ''E'' (since the first two homothecies certainly fix ''DE'', and the third does so only if ''F'' lies on ''DE''). Therefore ''D'', ''E'', ''F'' are collinear if and only if this composition is the identity, which means that the product of the three ratios is 1:
: <math>\frac{\overrightarrow{DC}}{\overrightarrow{DB}} \times
\frac{\overrightarrow{EA}}{\overrightarrow{EC}} \times
\frac{\overrightarrow{FB}}{\overrightarrow{FA}} = 1,</math>
which is equivalent to the given equation.

==History==
It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in ''Spherics'' by Menelaus. In this book, the plane version of the theorem is used as a lemma to prove a spherical version of the theorem.<ref>{{cite book|
last=Smith|first=D.E.|title=History of Mathematics|isbn=0-486-20430-8
|publisher=Courier Dover Publications|year=1958|volume=II|page=607}}</ref>

In [[Almagest]], [[Ptolemy]] applies the theorem on a number of problems in spherical astronomy.<ref name="rashed">{{cite book|last=Rashed|first=Roshdi|title=Encyclopedia of the history of Arabic science|volume=2|page=483|year=1996|publisher=Routledge|location=London|isbn=0-415-02063-8}}</ref> During the [[Islamic Golden Age]], Muslim scholars devoted a number of works that engaged in the study of Menelaus' theorem, which they referred to as "the proposition on the secants" (''shakl al-qatta'''). The [[complete quadrilateral]] was called the "figure of secants" in their terminology.<ref name="rashed" /> [[Al-Biruni]]'s work, ''The Keys of Astronomy'', lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's ''Almagest'' as in the works of [[al-Nayrizi]] and [[al-Khazin]] where each demonstrated particular cases of Menelaus' theorem that led to the [[sine rule]],<ref name="musa">{{cite journal|last=Moussa|first=Ali|title=Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations|journal=Arabic Sciences and Philosophy|year=2011|volume=21|issue=1|publisher=[[Cambridge University Press]]|doi=10.1017/S095742391000007X}}</ref> or works composed as independent treatises such as:

* The "Treatise on the Figure of Secants" (''Risala fi shakl al-qatta''') by [[Thabit ibn Qurra]].<ref name="rashed" />
* [[Husam al-DIn al-Salar]]'s ''Removing the Veil from the Mysteries of the Figure of Secants'' (Kashf al-qina' 'an asrar al-shakl al-qatta'), also known as "The Book on the Figure of Secants" (''Kitab al-shakl al-qatta''') or in Europe as ''The Treatise on the Complete Quadrilateral''. The lost treatise was referred to by [[Al-Tusi]] and [[Nasir al-Din al-Tusi]].<ref name="rashed" />
* Work by [[al-Sijzi]].<ref name="musa" />
* ''Tahdhib'' by [[Abu Nasr ibn Iraq]].<ref name="musa" />

==References==
{{Reflist}}
* {{cite book |title=Pure Geometry
|first=John Wellesley|last=Russell|publisher=Clarendon Press|year=1905
|chapter= Ch. 1 §6 "Menelaus' Theorem"
|url=http://books.google.com/books?id=r3ILAAAAYAAJ}}

== External links ==
* [http://planetmath.org/?op=getobj&from=objects&id=3092 Alternate proof] of Menelaus' theorem, from [[PlanetMath]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/MenelausFromCeva.shtml Menelaus From Ceva]
* [http://www.cut-the-knot.org/4travelers/CevaAndMenelaus.shtml Ceva and Menelaus Meet on the Roads]
* [http://www.mathpages.com/home/kmath442/kmath442.htm Menelaus and Ceva] at MathPages
* [http://demonstrations.wolfram.com/MenelausTheorem/ Menelaus' Theorem] by Jay Warendorff. [[The Wolfram Demonstrations Project]].
* {{MathWorld |title=Menelaus' Theorem |urlname=MenelausTheorem}}

[[Category:Affine geometry]]
[[Category:Triangle geometry]]
[[Category:Articles containing proofs]]
[[Category:Theorems in plane geometry]]

Simula

2013-12-29T03:47:21Z

24.7.101.21:

In [[mathematics]], a '''square-free''', or '''quadratfrei''', [[integer]] is one [[divisor|divisible]] by no [[square number|perfect square]], except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The smallest positive square-free numbers are

:1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ... {{OEIS|id=A005117}}

==Equivalent characterizations==
The positive integer ''n'' is square-free if and only if in the [[canonical representation of a positive integer|prime factorization]] of ''n'', no [[prime number]] occurs more than once. Another way of stating the same is that for every prime [[divisor|factor]] ''p'' of ''n'', the prime ''p'' does not evenly divide ''n'' / ''p''. Yet another formulation: ''n'' is square-free if and only if in every factorization ''n'' = ''ab'', the factors ''a'' and ''b'' are [[coprime]]. An immediate result of this definition is that all prime numbers are square-free.

The positive integer ''n'' is square-free [[if and only if]] μ(''n'') ≠ 0, where μ denotes the [[Möbius function]].

The positive integer ''n'' is square-free if and only if all [[abelian group]]s of [[order (group theory)|order]] ''n'' are [[group isomorphism|isomorphic]], which is the case if and only if all of them are [[cyclic group|cyclic]]. This follows from the classification of [[finitely generated abelian group]]s.

The integer ''n'' is square-free if and only if the [[factor ring]] '''Z''' / ''n'''''Z''' (see [[modular arithmetic]]) is a [[product of rings|product]] of [[field (mathematics)|field]]s. This follows from the [[Chinese remainder theorem]] and the fact that a ring of the form '''Z''' / ''k'''''Z''' is a field if and only if ''k'' is a prime.

For every positive integer ''n'', the set of all positive divisors of ''n'' becomes a [[partially ordered set]] if we use [[divisor|divisibility]] as the order relation. This partially ordered set is always a [[distributive lattice]]. It is a [[Boolean algebra (structure)|Boolean algebra]] if and only if ''n'' is square-free.

The [[radical of an integer]] is always square-free: an integer is square-free if it is equal to its radical.

==Dirichlet generating function==
The [[Dirichlet generating function]] for the square-free numbers is

:<math> \frac{\zeta(s)}{\zeta(2s) } = \sum_{n=1}^{\infty}\frac{ |\mu(n)|}{n^{s}} </math> where ζ(''s'') is the [[Riemann zeta function]].

This is easily seen from the [[Euler product]]
:<math> \frac{\zeta(s)}{\zeta(2s) } =\prod_p \frac{(1-p^{-2s})}{(1-p^{-s})}=\prod_p (1+p^{-s}). </math>

==Distribution==
Let ''Q''(''x'') denote the number of square-free (quadratfrei) integers between 1 and ''x''. For large ''n'', 3/4 of the positive integers less than ''n'' are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these events are independent, we obtain the approximation:

:<math>Q(x) \approx x\prod_{p\ \text{prime}} \left(1-\frac{1}{p^2}\right) = x\prod_{p\ \text{prime}} \frac{1}{(1-\frac{1}{p^2})^{-1}} </math>

:<math>Q(x) \approx x\prod_{p\ \text{prime}} \frac{1}{1+\frac{1}{p^2}+\frac{1}{p^4}+\cdots} = \frac{x}{\sum_{k=1}^\infty \frac{1}{k^2}} = \frac{x}{\zeta(2)} </math>

This argument can be made rigorous, and a very elementary estimate yields

:<math>Q(x) = \frac{x}{\zeta(2)} + O\left(\sqrt{x}\right) = \frac{6x}{\pi^2} + O\left(\sqrt{x}\right)</math>

(see [[pi]] and [[big O notation]]). By exploiting the largest known zero-free region of the Riemann zeta function, due to [[Ivan Matveyevich Vinogradov]], [[:ru:Коробов, Николай Михайлович|M.N. Korobov]] and [[Hans-Egon Richert]], the maximal size of the error term has been reduced
by [[Arnold Walfisz]]<ref>A. Walfisz. "Weylsche Exponentialsummen in der neueren Zahlentheorie" (VEB deutscher Verlag der Wissenschaften, Berlin 1963.</ref> and we have
:<math>Q(x) = \frac{6x}{\pi^2} + O\left(x^{1/2}\exp\left(-c\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right).</math>
for some positive constant ''c''. Under the [[Riemann hypothesis]], the error term can be further reduced<ref>Jia, Chao Hua. "The distribution of square-free numbers", ''Science in China Series A: Mathematics'' '''36''':2 (1993), pp. 154–169. Cited in Pappalardi 2003, [http://www.mat.uniroma3.it/users/pappa/papers/allahabad2003.pdf A Survey on ''k''-freeness]; also see Kaneenika Sinha, "[http://www.math.ualberta.ca/~kansinha/maxnrevfinal.pdf Average orders of certain arithmetical functions]", ''Journal of the Ramanujan Mathematical Society'' '''21''':3 (2006), pp. 267–277.</ref> to yield
:<math>Q(x) = \frac{x}{\zeta(2)} + O\left(x^{17/54+\varepsilon}\right) = \frac{6x}{\pi^2} + O\left(x^{17/54+\varepsilon}\right).</math>

See the race between the number of square-free numbers up to ''n'' and round(''n''/ζ(2)) on the OEIS:

{{OEIS2C|A158819}} – (Number of square-free numbers ≤ ''n'') minus round(''n''/ζ(2)). ]

The asymptotic/[[natural density]] of square-free numbers is therefore

:<math>\lim_{x\to\infty} \frac{Q(x)}{x} = \frac{6}{\pi^2} = \frac{1}{\zeta(2)}</math>

where ζ is the [[Riemann zeta function]] and 1/ζ(2) is approximately 0.6079 (over 3/5 of the integers are square-free).

Likewise, if ''Q''(''x'',''n'') denotes the number of ''n''-free integers (e.g. 3-free integers being cube-free integers) between 1 and ''x'', one can show
:<math>Q(x,n) = \frac{x}{\sum_{k=1}^\infty \frac{1}{k^n}} + O\left(\sqrt[n]{x}\right) = \frac{x}{\zeta(n)} + O\left(\sqrt[n]{x}\right).</math>

==Encoding as binary numbers==
If we represent a square-free number as the infinite product:

:<math>\prod_{n=0}^\infty {p_{n+1}}^{a_n}, a_n \in \lbrace 0, 1 \rbrace,\text{ and }p_n\text{ is the }n\text{th prime}. </math>

then we may take those <math>a_n</math> and use them as bits in a binary number, i.e. with the encoding:

:<math>\sum_{n=0}^\infty {a_n}\cdot 2^n</math>

e.g. The square-free number 42 has factorisation 2 × 3 × 7, or as an infinite product: 21 · 31  · 50 · 71 · 110 · 130 · ...; Thus the number 42 may be encoded as the binary sequence <tt>...001011</tt> or 11 decimal. (Note that the binary digits are reversed from the ordering in the infinite product.)

Since the prime factorization of every number is unique, so also is every binary encoding of the square-free integers.

The converse is also true. Since every positive integer has a unique binary representation it is possible to reverse this encoding so that they may be 'decoded' into a unique square-free integer.

Again, for example if we begin with the number 42, this time as simply a positive integer, we have its binary representation <tt>101010</tt>. This 'decodes' to become 20 · 31 · 50 · 71 · 110 · 131 = 3 × 7 × 13 = 273.

Among other things, this implies that the set of all square-free integers has the same [[cardinality]] as the set of all integers. In turn that leads to the fact that the in-order encodings of the square-free integers are a permutation of the set of all integers.

See sequences [[OEIS:A048672|A048672]] and [[OEIS:A064273|A064273]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]

==Erdős squarefree conjecture==
The [[central binomial coefficient]]

<math>{2n \choose n}</math>

is never squarefree for ''n'' > 4. This was proven in in 1985 for all sufficiently large integers by [[András Sárközy]],<ref>András Sárközy. On divisors of binomial coefficients, I.
J. Number Theory 20 (1985), no. 1, 70–80.</ref> and for all integers in 1996 by [[Olivier Ramaré]] and [[Andrew Granville]].<ref>Olivier Ramaré and Andrew Granville. Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43 (1996), no. 1, 73–107</ref>

==Squarefree core==
The [[multiplicative function]] <math>\mathrm{core}_t(n)</math> is defined
to map positive integers ''n'' to ''t''-free numbers by reducing the
exponents in the prime power representation modulo ''t'':
: <math>\mathrm{core}_t(p^e) = p^{e\mod t}.</math>
The value set of <math>\mathrm{core}_2</math>, in particular, are the
square-free integers. Their [[Dirichlet generating function]]s are
: <math>\sum_{n\ge 1}\frac{\mathrm{core}_t(n)}{n^s}
= \frac{\zeta(ts)\zeta(s-1)}{\zeta(ts-t)}</math>.

[[OEIS]] representatives are {{OEIS2C|A007913}} (''t''=2), {{OEIS2C|A050985}} (''t''=3) and {{OEIS2C|A053165}} (''t''=4).

== Notes ==
<references/>

== References ==
*{{cite journal | first1=Andrew | last1=Granville | first2=Olivier | last2=Ramaré | title=Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients | mr=1401709 | zbl=0868.11009 | year=1996 | journal=Mathematika | volume=43 | pages=73–107 | doi=10.1112/S0025579300011608 }}
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 }}

{{Divisor classes}}

{{Use dmy dates|date=September 2010}}

{{DEFAULTSORT:Square-Free Integer}}
[[Category:Number theory]]
[[Category:Integer sequences]]

SUHA (computer science)

2013-12-14T17:53:49Z

24.7.25.79: removed incorrect statement that SUHA is same as uniform hashing assumption (see talk page)

'''Galvani potential''' (also called Galvani potential difference, or inner potential difference, Δφ, delta phi) in [[electrochemistry]], is the electric potential difference between two points in the bulk of two phases.<ref>[http://www.iupac.org/goldbook/G02574.pdf IUPAC Gold Book, definition of Galvani potential difference.]</ref> These phases can be two different solids (e.g., two metals joint together), or a solid and a liquid (e.g., a metal [[electrode]] submerged in an [[electrolyte]]).

Generally, the Galvani potential difference is measurable only when the two phases have identical chemical composition.<ref>"Collected Works of J. Willard Gibbs, Vol. 1 Thermodynamics" (New Haven: Yale University Press, 1906) p. 429.</ref>

The Galvani potential is named after [[Luigi Galvani]].

==Galvani potential between two metals==

First, consider the Galvani potential between two metals. When two metals are electrically isolated from each other, an arbitrary voltage difference may exist between them. However, when two different metals are brought into electronic contact, electrons will flow from the metal with a lower voltage to the metal with the higher voltage until the [[Fermi level]] of the electrons in the bulk of both phases are equal. The actual numbers of electrons that passes between the two phases is small (it depends on the capacitance between the objects), and the occupancies of the electron bands are practically unaffected. Rather, this small increase or decrease in charge results in a shift in all the energy levels in the metals. An electrical [[Double layer (interfacial)|double layer]] is formed at the interface between the two phases.<ref>V.S. Bagotsky, "Fundamentals of Electrochemistry", Willey Interscience, 2006.</ref>

The equality of the electrochemical potential between the two different phases in contact can be written as:

:<math>\overline{\mu}_j^{(1)} = \overline{\mu}_j^{(2)}</math>
where:
* <math>\overline{\mu}</math> is the electrochemical potential
* j denotes the species which are the carrier of electrical current in the system (which are electrons in metals)
* (1) and (2) denote phase 1 and phase 2, respectively.

Now, the electrochemical potential of a species is defined as a sum of its chemical potential and the local electrostatic potential:

:<math>\overline{\mu}_j = \mu_j + z_j F \phi</math>
where:
* μ is the [[chemical potential]]
* z is the electrical charge carried by a single charge carrier (unity for electrons)
* F is the [[Faraday constant]]
* Φ is the electrostatic potential

From the two equations above:
:<math>\phi^{(2)} - \phi^{(1)} = \frac {\mu_j^{(1)} - \mu_j^{(2)}} {z_j F}</math>

where the difference on the left-hand side is the Galvani potential difference between the phases (1) and (2). Thus, the Galvani potential difference is determined entirely by chemical difference of the two phases; specifically by the difference of the chemical potential of the charge carriers in the two phases.

The Galvani potential difference between an electrode and electrolyte (or between other two electrically conductive phases) forms in an analogous fashion, although the chemical potentials in the equation above may need to include all species involved in the electrochemical reaction at the interface.

==Relation to measured cell potential==
The Galvani potential difference is not measurable. The measured potential difference between two metal electrodes assembled into a cell does not equal the difference of the Galvani potentials of the two metals (or their combination with the solution Galvani potential) because the cell needs to contain another metal-metal interface, as in the following schematic of a [[galvanic cell]]:

:M(1) | S | M(2) | M(1)'
where:
* M(1) and M(2) are the two different metals,
* S denotes the electrolyte,
* M(1)' is the additional metal (here assumed to be the metal (1)) that must be inserted into the circuit to close it,
* the vertical bar, |, denotes a phase boundary.

Instead, the measured cell potential can be written as:<ref name ="Trasatti">Sergio Trasatti, "The Absolute Electrode Potential: an Explanatory Note (Recommendations 1986)", International Union of Pure and Applied Chemistry, Pure & AppL Chem., Vol. 58, No.7, pp. 955—966, 1986. http://www.iupac.org/publications/pac/1986/pdf/5807x0955.pdf (pdf)</ref>

:<math>E^{(2)} - E^{(1)} = \left(\phi^{(2)} - \phi^{(S)} - \frac {\mu_j^{(2)}} {z_j F}\right) - \left(\phi^{(1)} - \phi^{(S)} - \frac {\mu_j^{(1)}} {z_j F}\right) =
\left(\phi^{(2)} - \phi^{(1)}\right) - \left(\frac {\mu_j^{(2)} - \mu_j^{(1)}} {z_j F}\right)
</math>
where:
* E is the potential of a single electrode,
* (S) denotes the electrolyte solution.

From the above equation, two metals in electronic contact (i.e., under electronic equilibrium) must have the same electrode potential.<ref name="Trasatti"/> Also, the electrochemical potentials of the electrons within the two metals will be the same. However, their Galvani potentials will be different (unless the metals are identical).

==See also==
* [[Electrode potential]]
* [[Absolute electrode potential]]
* [[Volta potential]]
* [[ITIES]]

==References==
{{reflist}}

{{DEFAULTSORT:Galvani Potential}}
[[Category:Electrochemistry]]
[[Category:Potentials]]

Bach tensor

2013-12-02T17:12:56Z

24.7.159.77:

[[Image:ChebyshevPsi.png|thumb|right|The Chebyshev function ''ψ''(''x''), with ''x'' < 50]]
[[Image:Chebyshev.svg|thumb|right|The function ''ψ''(''x'') − ''x'', for ''x'' < 10,000]]
[[Image:Chebyshev-big.svg|thumb|right|The function ''ψ''(''x'') − ''x'', for ''x'' < 10 million]]

In [[mathematics]], the '''Chebyshev function''' is either of two related functions. The '''first Chebyshev function''' ''&thetasym;''(''x'') or ''θ''(''x'') is given by

:<math>\vartheta(x)=\sum_{p\le x} \log p</math>

with the sum extending over all [[prime number]]s ''p'' that are less than or equal to ''x''.

The '''second Chebyshev function''' ''ψ''(''x'') is defined similarly, with the sum extending over all prime powers not exceeding ''x'':

:<math> \psi(x) = \sum_{p^k\le x}\log p=\sum_{n \leq x} \Lambda(n) = \sum_{p\le x}\lfloor\log_p x\rfloor\log p, </math>

where <math>\Lambda</math> is the [[von Mangoldt function]]. The Chebyshev functions, especially the second one ''ψ''(''x''), are often used in proofs related to [[prime numbers]], because it is typically simpler to work with them than with the [[prime-counting function]], ''π''(''x'') (See [[#The exact formula|the exact formula]], below.) Both Chebyshev functions are asymptotic to ''x'', a statement equivalent to the [[prime number theorem]].

Both functions are named in honour of [[Pafnuty Chebyshev]].

==Relationships==
The second Chebyshev function can be seen to be related to the first by writing it as

:<math>\psi(x)=\sum_{p\le x} k \log p</math>

where ''k'' is the unique integer such that ''p''''k'' ≤ ''x'' and ''x'' < ''p''''k''+1. The values ''k'' of are given in {{OEIS2C|id=A206722}}. A more direct relationship is given by

:<math>\psi(x)=\sum_{n=1}^\infty \vartheta \left(x^{1/n}\right).</math>

Note that this last sum has only a finite number of non-vanishing terms, as

:<math>\vartheta \left(x^{1/n}\right) = 0\text{ for }n>\log_2 x\,.</math>

The second Chebyshev function is the logarithm of the [[least common multiple]] of the integers from 1 to ''n''.

:<math>\operatorname{lcm}(1,2,\dots, n)=e^{\psi(n)}.</math>

Values of  <math>\operatorname{lcm}(1,2,\dots, n)</math>  for the integer variable ''n'' is given at {{OEIS2C|id=A003418}}.

==Asymptotics and bounds==
The following bounds are known for the Chebyshev functions:{{ref|Dusart1999}}{{ref|Dusart2010}} (in these formulas ''p''''k'' is the ''k''th prime number ''p''1 = 2, ''p''2 = 3, etc.)

:<math>\vartheta(p_k)\ge k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2.050735}{\ln k}\right)</math> for <math>k\ge10^{11},</math>

:<math>\vartheta(p_k)\le k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2}{\ln k}\right)</math> for ''k'' ≥ 198,

:<math>|\vartheta(x)-x|\le0.006788\frac{x}{\ln x}</math> for ''x'' ≥ 10,544,111,

:<math>|\psi(x)-x|\le0.006409\frac{x}{\ln x}</math> for ''x'' ≥ exp(22),

:<math>0.9999\sqrt x<\psi(x)-\vartheta(x)<1.00007\sqrt x+1.78\sqrt[3]x</math> for <math>x\ge121.</math>

Further, under the [[Riemann hypothesis]],

:<math>|\vartheta(x)-x|=O(x^{1/2+\varepsilon})</math>
:<math>|\psi(x)-x|=O(x^{1/2+\varepsilon})</math>

for any <math>\varepsilon>0.</math>

Upper bounds exist for both <math>\vartheta(x)</math> and <math>\psi(x)</math> such that,<ref>{{Cite journal
| last = Rosser
| first = J. Barkley
| last2 = Schoenfeld
| first2 = Lowell
| title = Approximate formulas for some functions of prime numbers.
| journal = Illinois J. Math.
| year = 1962
| volume = 6
| pages = 64–94.
| url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255631807}}</ref>

:<math>\vartheta(x)<1.01624x</math>
:<math>\psi(x)<1.03883x</math>

for any <math>x>0.</math>

An explanation of the constant 1.03883 is given at {{OEIS2C|id=A206431}}.

==The exact formula==
In 1895, [[Hans Carl Friedrich von Mangoldt]] proved{{ref|Dav104}} an explicit expression for <math>\psi(x)</math> as a sum over the nontrivial zeros of the [[Riemann zeta function]]:

: <math> \psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}). </math>

(The numerical value of ζ'(0)/ζ(0) is log(2π).) Here <math>\rho</math> runs over the nontrivial zeros of the zeta function, and ψ0 is the same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:

: <math>
\psi_0(x)
= \frac12\left( \sum_{n \leq x} \Lambda(n)+\sum_{n < x} \Lambda(n)\right)
=\begin{cases} \psi(x) - \frac{1}{2} \Lambda(x) & x = 2,3,4,5,7,8,9,11,13,16,\dots \\
\psi(x) & \mbox{otherwise.} \end{cases}
</math>

From the [[Taylor series]] for the [[logarithm]], the last term in the explicit formula can be understood as a summation of <math>x^{\omega}/{\omega}</math> over the trivial zeros of the zeta function, <math>\omega = -2, -4, -6, \ldots</math>, i.e.

: <math> \sum_{k=1}^{\infty} \frac{x^{-2k}}{-2k} = \frac{1}{2} \log ( 1 - x^{-2} ). </math>

Similarly, the first term, ''x'' = ''x''1/1, corresponds to the simple [[pole (complex analysis)|pole]] of the zeta function at 1. Its being a pole rather than zero accounts for the opposite sign of the term.

==Properties==
A theorem due to [[Erhard Schmidt]] states that, for some explicit positive constant ''K'', there are infinitely many natural numbers ''x'' such that

:<math>\psi(x)-x < -K\sqrt{x}</math>

and infinitely many natural numbers ''x'' such that

:<math>\psi(x)-x > K\sqrt{x}.</math>{{ref|Sch03}}{{ref|Hard16}}

In [[big-O notation|little-o notation]], one may write the above as

:<math>\psi(x)-x \ne o\left(\sqrt{x}\right).</math>

[[G. H. Hardy|Hardy]] and [[J. E. Littlewood|Littlewood]]{{ref|Hard16}} prove the stronger result, that

:<math>\psi(x)-x \ne o\left(\sqrt{x}\log\log\log x\right).</math>

==Relation to primorials==

The first Chebyshev function is the logarithm of the [[primorial]] of ''x'', denoted ''x''#:

:<math>\vartheta(x)=\sum_{p\le x} \log p=\log \prod_{p\le x} p = \log (x\#).</math>

This proves that the primorial ''x''# is asymptotically equal to exp((1+o(1))''x''), where "o" is the little-o notation (see [[Big O notation]]) and together with the prime number theorem establishes the asymptotic behavior of ''p''''n''#.

==Relation to the prime-counting function==
The Chebyshev function can be related to the prime-counting function as follows. Define

: <math> \Pi(x) = \sum_{n \leq x} \frac{\Lambda(n)}{\log n}. </math>

Then

: <math> \Pi(x) = \sum_{n \leq x} \Lambda(n) \int_n^x \frac{dt}{t \log^2 t} + \frac{1}{\log x} \sum_{n \leq x} \Lambda(n) = \int_2^x \frac{\psi(t)\, dt}{t \log^2 t} + \frac{\psi(x)}{\log x}. </math>

The transition from <math>\Pi</math> to the [[prime-counting function]], <math>\pi</math>, is made through the equation

: <math> \Pi(x) = \pi(x) + \frac{1}{2} \pi(x^{1/2}) + \frac{1}{3} \pi(x^{1/3}) + \cdots. </math>

Certainly <math>\pi(x) \leq x</math>, so for the sake of approximation, this last relation can be recast in the form

: <math> \pi(x) = \Pi(x) + O(\sqrt x). </math>

==The Riemann hypothesis==
The [[Riemann hypothesis]] states that all nontrivial zeros of the zeta function have real part 1/2. In this case, <math>|x^{\rho}|=\sqrt x</math>, and it can be shown that

: <math>\sum_{\rho} \frac{x^{\rho}}{\rho} = O(\sqrt x \log^2 x).</math>

By the above, this implies

: <math> \pi(x) = \operatorname{li}(x) + O(\sqrt x \log x). </math>

Good evidence that RH could be true comes from the fact proposed by [[Alain Connes]] and others, that if we differentiate the von Mangoldt formula with respect to x make ''x'' = exp(''u''). Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

: <math> \zeta(1/2+i \hat H )|n \ge \zeta(1/2+iE_{n})=0, \, </math>

:<math> \sum_n e^{iu E_{n}}=Z(u) = e^{u/2}-e^{-u/2} \frac{d\psi _0}{du}-\frac{e^{u/2}}{e^{3u}-e^u} = \operatorname{Tr}(e^{iu\hat H }),</math>

where the "trigonometric sum" can be considered to be the trace of the operator ([[statistical mechanics]]) <math> e^{iu \hat H} </math>,which is only true if <math> \rho =1/2+iE(n). </math>

Using the [[semiclassical approach]] the potential of ''H'' = ''T'' + ''V'' satisfies:

:<math> \frac{Z(u)u^{1/2}}{\sqrt \pi }\sim \int_{-\infty}^\infty e^{i (uV(x)+ \pi /4 )}\,dx </math>

with ''Z''(''u'') → 0 as ''u'' → ∞.

solution to this nonlinear integral equation can be obtained (among others) by <math>V^{-1} (x) \approx \sqrt (4\pi) \frac{d^{1/2}N(x)}{dx^{1/2}} </math> in order to obtain the inverse of the potential : <math> \pi N(E) = Arg \xi (1/2+iE) </math>

==Smoothing function==
[[Image:Chebyshev-smooth.svg|thumb|right|The difference of the smoothed Chebyshev function and ''x''2/2
for ''x'' < 106]]

The '''smoothing function''' is defined as

:<math>\psi_1(x)=\int_0^x \psi(t)\,dt.</math>

It can be shown that

:<math>\psi_1(x) \sim \frac{x^2}{2}.</math>

==Variational formulation==

The Chebyshev function evaluated at ''x'' = exp(''t'') minimizes the functional

:<math> J[f]=\int_{0}^{\infty}\frac{f(s)\zeta' (s+c)}{\zeta(s+c)(s+c)}\,ds-\int_{0}^{\infty}\!\!\!\int_{0}^{\infty} e^{-st}f(s)f(t)\,ds\,dt, </math>

so

:<math> f(t)= \psi (e^t)e^{-ct},\, </math>

for ''c'' > 0.

==Notes==
<references/>
* {{note|Dusart2010}} Pierre Dusart, "Estimates of some functions over primes without R.H.". {{arxiv|1002.0442}}
* {{note|Dusart1999}} Pierre Dusart, "Sharper bounds for ψ, θ, π, ''p''''k''", Rapport de recherche n° 1998-06, Université de Limoges. An abbreviated version appeared as "The ''k''th prime is greater than ''k''(ln ''k'' + ln ln ''k'' − 1) for ''k'' ≥ 2", ''Mathematics of Computation'', Vol. 68, No. 225 (1999), pp. 411–415.
* {{note|Sch03}}Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", ''Mathematische Annalen'', '''57''' (1903), pp. 195–204.
* {{note|Hard16}}G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", ''Acta Mathematica'', '''41''' (1916) pp. 119–196.
* {{note|Dav104}}[[Harold Davenport|Davenport, Harold]] (2000). In ''[http://books.google.com/books?vid=ISBN0387950974&id=U91lsCaJJmsC&pg=PA104&lpg=PA104 Multiplicative Number Theory]''. Springer. p. 104. ISBN 0-387-95097-4. Google Book Search.

==References==
* {{Apostol IANT}}

==External links==
* {{mathworld|urlname=ChebyshevFunctions|title=Chebyshev functions}}
* {{planetmathref|id=4020|title=Mangoldt summatory function}}
* {{planetmathref|id=4573|title=Chebyshev functions}}
* [http://www.math.ucsb.edu/~stopple/explicit.html Riemann's Explicit Formula], with images and movies

[[Category:Arithmetic functions]]

Co-citation Proximity Analysis

2013-02-20T06:31:53Z

24.7.87.170: /* Performance */

In the [[mathematical]] field of [[potential theory]], '''Boggio's formula''' is an explicit formula for the [[Green's function]] for the polyharmonic [[Dirichlet problem]] on the ball of radius 1. It was discovered by the Italian mathematician [[Tommaso Boggio]].

The polyharmonic problem is to find a function ''u'' satisfying
:<math>(-\Delta)^m u(x) = f(x)</math>
where ''m'' is a positive integer, and <math>(-\Delta)</math> represents the [[Laplace operator]]. The Green's function is a function satisfying
:<math>(-\Delta)^m G(x,y) = \delta(x-y)</math>
where <math>\delta</math> represents the [[Dirac delta function|Dirac delta distribution]], and in addition is equal to 0 up to order ''m-1'' at the boundary.

Boggio found that the Green's function on the ball in ''n'' spatial dimensions, whenever ''n > 2m'', is
:<math>G_{m,n} (x,y) = C_{m,n} |x-y|^{2m-n} \int_1^{\frac{\left||x|y - \frac{x}{|x|}\right|}{|x-y|}} (v^2-1)^{m-1} v^{1-n} dv</math>
The constant <math>C_{m,n}</math> is given by
:<math>C_{m,n} =\frac{1}{n e_n 4^{m-1} ((m-1)!)^2},</math> where <math>e_n = \frac{\pi^{\frac{n}{2}}}{\Gamma(1+\frac{n}{2})} </math>

==Sources==
*{{Citation|last=Boggio|first=Tomas|date=1905|title=Sulle funzioni di Green d'ordine ''m''|periodical=[[Rendiconti del Circolo Matematico di Palermo]]|volume=20|pages=97–135|url=http://link.springer.com/article/10.1007/BF03014033|doi=10.1007/BF03014033}}
* {{Citation
| last1=Gazzola
| first1=Filippo
| author1-link=
| last2=Grunau
| first2=Hans-Christoph
| author2-link=
| last3=Sweers
| first3=Guido
| title=Polyharmonic Boundary Value Problems
| series=Lecture Notes in Mathematics
| volume=1991
| edition=
| publisher=Springer
| date=2010
| place=Berlin
| url=http://www1.mate.polimi.it/~gazzola/book_GGS.pdf
| isbn=978-3-642-12244-6
}}

[[Category:Elliptic partial differential equations]]
[[Category:Potential theory]]

Reversible computing

2012-08-19T04:34:37Z

24.7.114.112: /* Reversible circuits */

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Schrödinger equation

2007-10-15T22:07:45Z

24.7.158.49: Undid revision 164818225 by 148.78.249.33 (talk)

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