formulasearchengine - User contributions [en]

Laplace–Beltrami operator

2015-01-01T22:23:29Z

128.6.62.214:

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Charles Fefferman

2014-11-21T20:39:27Z

128.6.62.89:

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Parrondo's paradox

2013-09-20T13:58:33Z

128.6.229.87: /* The saw-tooth example */ Filled in an answer 'why' question. Not perfectly satisfying, but it's an answer.

{{multiple issues|
{{Refimprove|date=December 2008}}
{{Technical|date=October 2009}}
{{original research|date=September 2013}}
}}
[[File:Hydrometer in action.jpg|thumb|right|A floating [[hydrometer]] in use to test the specific gravity of [[beer]]]]
'''Gravity''', in the [[Context (language use)|context]] of fermenting [[alcoholic beverage]]s, refers to the [[specific gravity]], or relative density compared to water, of the [[wort]] or [[must]] at various stages in the fermentation. The concept is used in [[brewing]] and wine making industry. Specific gravity is measured by a [[hydrometer]], [[refractometer]], [[pycnometer]] or [[oscillating U-tube]] electronic meter.

The density of a wort is largely dependent on the sugar content of the wort. During alcohol [[Ethanol fermentation|fermentation]], yeast converts sugars into carbon dioxide and alcohol. The decline in the sugar content ''and'' the presence of ethanol (which is appreciably less dense than water) drop the density of the wort. The percentage of alcohol can be calculated from the difference between the original gravity (abbreviated OG) of the wort and the current specific gravity (abbreviated SG) of wort. By monitoring the decline in SG over time the brewer obtains information about the health and progress of the fermentation and determines that it is complete when gravity stops declining. If the fermentation is finished, the specific gravity is called the final gravity (abbreviated FG). For example, for a typical strength beer, OG could be 1.050 and FG could be 1.010.

Several different scales have been used for measuring the original gravity. For historical reasons, the brewing industry largely uses the [[Plato scale]] (°P), which is essentially the same as the [[Brix]] scale used by the wine industry. For example, OG 1.050 is roughly equivalent to 12°P.

By considering the original gravity, the brewer or vintner obtains an indication as to the probable ultimate alcoholic content of his product. The OE (Original Extract) is often referred to as the "size" of the beer and is, in Europe, often printed on the label as Stammwürze or sometimes just as a percent. In the Czech Republic, for example, they speak of "10 degree beers", "12 degree beers" which refers to the gravity in Plato of the wort before the fermentation.

==Terms related to gravity==
===Specific gravity===

Specific gravity is the ratio of the density of a sample to the density of water. The ratio depends on the temperature and pressure of both the sample and water. The pressure is always considered (in brewing) to be 1 atmosphere (1013.25 hPa) and the temperature is usually 20 °C for both sample and water but in some parts of the world different temperatures may be used and there are hydrometers sold calibrated to, for example, {{convert|60|F|C}}. It is important, where any conversion to °P is involved, that the proper pair of temperatures be used for the conversion table or formula being employed. The current [[American Society of Brewing Chemists|ASBC]] table is (20°C/20°C) meaning that the density is measured at 20 °C and referenced to the density of water at 20 °C (0.998203 g/cm3). Mathematically

:<math> SG_\text{true} = {\rho_\text{sample} \over \rho_\text{water}}</math>

This formula gives the true specific gravity i.e. based on densities. Brewers cannot (unless using a [[Oscillating U-tube|U-tube meter]]) measure density directly and so must use a hydrometer, whose stem is bathed in air, or pycnometer weighings which are also done in air. Hydrometer readings and the ratio of pycnometer weights are influenced by air (see article [[Specific gravity|Specific Gravity]] for details) and are called "apparent" readings. True readings are easily obtained from apparent readings by

:<math>SG_\text{true} = SG_\text{apparent} - {\rho_\text{air} \over \rho_\text{water} }(SG_\text{apparent}-1)</math>

However, the ASBC table uses apparent specific gravities, so many electronic density meters will produce the correct °P numbers automatically.

=== Original Gravity (OG); Original Extract (OE) ===

The Original Gravity is the specific gravity measured before fermentation. From it the analyst can compute the Original Extract which is the mass (grams) of sugar in 100 grams of wort (°P) by use of the [[Plato scale]]. The symbol <math> p </math> will denote OE in the formulas which follow.

=== Final Gravity (FG); Apparent Extract (AE) ===

The Final Gravity is the specific gravity measured at the completion of fermentation. The Apparent Extract, denoted <math> m </math>, is the °P obtained by inserting the FG into the formulas or tables in the [[Plato scale|Plato]] article. The use of "apparent" here is not to be confused with the use of that term to describe specific gravity readings which have not been corrected for the effects of air.

=== True extract (TE) ===

The amount of extract which was not converted to yeast biomass, carbon dioxide or ethanol can be estimated by removing the alcohol from beer which has been degassed and clarified by filtration or other means. This is often done a part of a distillation in which the alcohol is collected for quantitative analysis but can also be done by evaporation in a water bath. If the residue is made back up to the original volume of beer which was subject to the evaporation process, the specific gravity of that reconstituted beer measured and converted to Plato using the tables and formulas in the [[Plato scale|Plato]] article then the TE is

:<math> n = P_{recon}{SG_{recon} \over SG_{beer}}</math>

See the [[Plato scale|Plato]] article for details. TE is denoted by the symbol <math> n </math>. This is the number of grams of extract remaining in 100 grams of beer at the completion of fermentation.

===Alcohol content===

Knowing the amount of extract in 100 grams of wort before fermentation and the number of grams of extract in 100 grams of beer at its completion, the amount alcohol (in grams) formed during the fermentation can be determined. The formula follows, attributed to [[Brix#Brix, Balling, Plato|Balling]]<ref name="DeClerck1958">De Clerck, J. A textbook of Brewing, Translated by Kathleen Barton-Wright, Chapman & Hall,, London 1958 Vol. II p427</ref>{{rp|427}}

:<math> A_w = {(p - n) \over (2.0665 - 1.0665p/100)} = f_{pn}(p - n)</math>

where <math>f_{pn} = {1 \over (2.0665 - 1.0665p/100) }</math>,gives the number of grams of alcohol per 100 grams of beer i.e. the ABW. Note that the alcohol content depends not only on the diminution of extract <math> (p - n) </math> but also on the multiplicative factor <math> f_{pn}</math> which depends on the OE. De Clerck<ref name="DeClerck1958" />{{rp|428}} tabulated Ballings values for <math> f_{pn}</math> but they can be calculated simply from p

:<math> f_{pn} = {1 \over (2.0665 - 1.0665p/100)} = 0.48394 + 0.0024688p + 0.00001561p^2 </math>

This formula is fine for those who wish to go to the trouble to compute TE (whose real value lies in determining attenuation) which is only a small fraction of brewers. Others want a simpler, quicker route to determining alcoholic strength. This lies in Tabarie's Principle<ref name="DeClerck1958" />{{rp|428}} which states that the depression of specific gravity in beer to which ethanol is added is the same as the depression of water to which an equal amount of alcohol (on a w/w basis) has been added. Use of Tabarie's principle lets us calculate the true extract of a beer with apparent extract <math> m </math> as

:<math> n =P(P^{-1}(m) + 1 - \frac {\rho_\text{EtOH}(A_w)} { \rho_\text{water}})</math>

where <math> P </math> is a function that converts SG to °P (see [[Plato scale|Plato]]) and <math> P^{-1} </math> (see [[Plato scale|Plato]]) its inverse and <math>\rho_\text{EtOH}(A_w)</math> is the density of an aqueous ethanol solution of strength <math> A_w </math>by weight at 20 °C. Inserting this into the alcohol formula the result, after rearrangement, is

:<math> {\left[p - P\left(P^{-1}(m) + 1 - \frac {\rho_\text{EtOH}(A_w)} { \rho_\text{water}}\right)\right] \over (2.0665 - 1.0665p/100)} - A_w = 0</math>

Which can be solved, albeit iteratively, for <math> A_w </math> as a function of OE and AE. It is again possible to come up with a relationship of the form

:<math> A_w = f_{pm}(p - m)\,</math>

De Clerk also tabulates values for <math> f_{pm} = 0.39661 + 0.001709p + 0.000010788p^2</math>.

Most brewers and consumers are used to having alcohol content reported by volume (ABV) rather than weight. Interconversion is simple but the specific gravity of the beer must be known:

:<math> A_v = A_w{SG_\text{beer} \over 0.79661}</math>

This is the number of cc of ethanol in 100 cc of beer.

Because ABV depends on multiplicative factors (one of which depends on the original extract and one on the final) as well as the difference between OE and AE it is impossible to come up with a formula of the form

:<math> A_v = k(p-m)\, </math>

where <math> k </math> is a simple constant. Because of the near linear relationship between extract and (SG-1) (see [[specific gravity]]) in particular because <math> p \approx 1000(SG-1)/4 </math> the ABV formula is written as

:<math> A_v = 250f_{pm}(OG -FG){SG_\text{beer} \over 0.79661}</math>

If the value given above for <math> f_{pm} </math> corresponds to an OE of 12°P which is 0.4187, and 1.010 can be taken as a typical FG then this simplifies to

:<math> A_v = 132.715(OG - FG) = (OG - FG)/0.00753\, </math>

With typical values of 1.050 and 1.010 for, respectively, OG and FG this simplified formula gives an ABV of 5.31% as opposed to 5.23% for the more accurate formula. Formulas for alcohol similar to this last simple one abound in the brewing literature and are very popular among home brewers. Formulas such as this one make it possible to mark hydrometers with "potential alcohol" scales based on the assumption that the FG will be close to 1 which is more likely to be the case in wine making than in brewing and it is to vintners that these are usually sold.

===Attenuation===

The drop in extract during the fermentation divided by the OE represents the percentage of sugar which has been consumed. The Real Degree of Attenuation (RDF) is based on TE

:<math> RDF = 100 {(p-n)\over p} </math>

and the Apparent Degree of Fermentation (ADF) is based on AE

:<math> ADF = 100 {(p-m) \over p} \approx 100 {(OG-FG) \over (OG-1)}</math>

Because of the near linear relationship between (SG-1) and °P specific gravities can be used in the ADF formula as shown.

=== Brewer's points ===

Many brewers like to exploit the near linear relationship between (SG -1) and °P to simplify calculations considerably. They define
:<math> p_t := 1000(SG-1)\, </math>,
call it "points" or "brewer's points" or "excess gravity" and use it as if it were extract. The Plato degree is thus approximately the points divided by 4:
:<math> p \approx p_t/4 = 1000(SG-1)/4.</math>

As an example, a wort of SG 1.050 would be said to have 1000(1.050-1)=50 points, and have Plato degree of approximately 50/4=12.5 °P.

Points can be used in the ADF and RDF formulas. Thus a beer with OG 1.050 which fermented to 1.010 would be said to have attenuated 100*(50 - 10)/50 = 80%. Points can also be used in the SG versions of the alcohol formulas. It is simply necessary to multiply by 1000 as points are 1000 times (SG-1).

==See also==
{{Portal box|Beer|Wine}}
*[[Plato scale]]
*[[Gravity (chemistry)]]

==References==
{{Reflist}}

{{DEFAULTSORT:Gravity (Alcoholic Beverage)}}
[[Category:Equations]]
[[Category:Brewing]]

Perfect group

2013-06-17T19:20:14Z

128.6.62.64: /* Examples */ added links to Schur multiplier and universal extensions

In [[mathematics]], an '''integer-valued polynomial''' (also known as a '''numerical polynomial''') ''P''(''t'') is a [[polynomial]] whose value ''P''(''n'') is an [[integer]] for every integer ''n''. Every polynomial with integer [[coefficient]]s is integer-valued, but the converse is not true. For example, the polynomial

:<math> \frac{1}{2} t^2 + \frac{1}{2} t=\frac{1}{2}(t)(t+1)</math>

takes on integer values whenever ''t'' is an integer. That is because one of ''t'' and ''t'' + 1 must be an [[even number]]. (The values this polynomial takes are the [[triangular number]]s.)

Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in [[algebraic topology]].{{citation needed|date=March 2012}}

==Classification==
The class of integer-valued polynomials was described fully by {{harvtxt|Pólya|1915}}. Inside the [[polynomial ring]] ''Q''[''t''] of polynomials with [[rational number]] coefficients, the [[subring]] of integer-valued polynomials is a [[free abelian group]]. It has as [[basis (linear algebra)|basis]] the polynomials

:''Pk''(''t'') = ''t''(''t'' − 1)...(''t'' − ''k'' + 1)/''k''!

for ''k'' = 0,1,2, ..., i.e., the [[binomial coefficient]]s. In other words, every integer-valued polynomial can be written as an integer [[linear combination]] of binomial coefficients in exactly one way. The proof is by the method of [[Difference operator|discrete Taylor series]]: binomial coefficients are integer-valued polynomials, and conversely, the discrete difference of an integer series is an integer series, so the discrete Taylor series of an integer series generated by a polynomial has integer coefficients (and is a finite series).

==Fixed prime divisors==
Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials ''P'' with integer coefficients that always take on even number values are just those such that ''P''/2 is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients.

In questions of prime number theory, such as [[Schinzel's hypothesis H]] and the [[Bateman–Horn conjecture]], it is a matter of basic importance to understand the case when ''P'' has no fixed prime divisor (this has been called ''Bunyakovsky's property''{{Citation needed|date=January 2013}}, for [[Viktor Bunyakovsky]]). By writing ''P'' in terms of the binomial coefficients, we see the highest fixed prime divisor is also the highest prime [[common factor]] of the coefficients in such a representation. So Bunyakovsky's property is equivalent to coprime coefficients.

As an example, the pair of polynomials ''n'' and ''n''2 + 2 violates this condition at ''p'' = 3: for every ''n'' the product

:''n''(''n''2 + 2)

is divisible by 3. Consequently there cannot be infinitely many prime pairs ''n'' and ''n''2 + 2. The divisibility is attributable to the alternate representation

:''n''(''n'' + 1)(''n'' − 1) + 3''n''.

==Other rings==
Numerical polynomials can be defined over other rings and fields, in which case the integer-valued polynomials above are referred to as '''classical numerical polynomials'''.{{citation needed|date=April 2012}}

==Applications==
The [[topological K-theory|K-theory]] of [[Classifying space for U(n)|BU(''n'')]] is numerical (symmetric) polynomials.

The [[Hilbert polynomial]] of a polynomial ring in ''k'' + 1 variables is the numerical polynomial <math>\binom{t+k}{k}</math>.

==References==

===Algebra===
* {{citation
|last1=Cahen
|first1=P-J.
|last2=Chabert
|first2=J-L.
|title=Integer-valued polynomials
|series=Mathematical Surveys and Monographs
|volume=48
|publisher=American Mathematical Society
|location=Providence, RI
|year=1997
}}
* {{citation | last=Pólya | first=G. | authorlink=George Pólya | title=Über ganzwertige ganze Funktionen | language=German | journal=Palermo Rend. | volume=40 | pages=1–16 | year=1915 | issn=0009-725X | jfm=45.0655.02 }}

===Algebraic topology===
* {{citation
|author=A. Baker, F. Clarke, N. Ray, L. Schwartz
|title=On the Kummer congruences and the stable homotopy of ''BU''
|journal=Trans. Amer. Math. Soc.
|volume=316
|issue=2
|year=1989
|pages=385–432
|doi=10.2307/2001355
|publisher=Transactions of the American Mathematical Society, Vol. 316, No. 2
|jstor=2001355
}}

* {{citation
|author=T. Ekedahl
|title=On minimal models in integral homotopy theory
|journal=Homology Homotopy Appl.
|volume=4
|issue=2
|year=2002
|pages=191–218
|url=http://projecteuclid.org/euclid.hha/1139852462
}}

* {{citation
|author=J. Hubbuck
|title=Numerical forms
|journal=J. London Math. Soc. (2)
|volume=55
|year=1997
|issue=1
|pages=65–75
|doi=10.1112/S0024610796004395
}}

==Further reading==
* {{cite book | last=Narkiewicz | first=Władysław | title=Polynomial mappings | series=Lecture Notes in Mathematics | volume=1600 | location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | isbn=3-540-59435-3 | issn=0075-8434 | zbl=0829.11002 }}

{{DEFAULTSORT:Integer-Valued Polynomial}}
[[Category:Polynomials]]
[[Category:Number theory]]
[[Category:Commutative algebra]]
[[Category:Ring theory]]

Confidence distribution

2013-05-01T13:34:45Z

128.6.168.245:

In [[mathematics]], the '''Hopf lemma''', named after [[Eberhard Hopf]], states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward pointing normal is strictly positive. The lemma is an important tool in the proof of the [[maximum principle]] and in the theory of [[partial differential equation]]s. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained.
==Statement for harmonic functions==
Let Ω be a bounded domain in '''R'''''n'' with smooth boundary. Let ''f'' be a real-valued function continuous on the closure of Ω and [[harmonic function|harmonic]] on Ω. If ''x'' is a boundary point such that ''f''(''x'') > ''f''(''y'') for all ''y'' in Ω sufficiently close to ''x'', then the (one-sided) [[directional derivative]] of ''f'' in the direction of the outward pointing normal to the boundary at ''x'' is strictly positive.

==Proof for harmonic functions==
Subtracting a constant, it can be assumed that ''f''(''x'') = 0 and ''f'' is strictly negative at interior points near ''x''. Since the boundary of Ω is smooth there is a small ball contained in Ω the closure of which is tangent to the boundary at ''x'' and intersects the boundary only at ''x''. It is then sufficient to check the result with Ω replaced by this ball. Scaling and translating, it is enough to check the result for the unit ball in '''R'''''n'', assuming ''f''(''x'') is zero for some unit vector ''x'' and ''f''(''y'') < 0 if |''y''| < 1.

By [[Harnack's inequality]] applied to −''f''

:<math>\displaystyle{-f(rx)\ge -{1-r\over (1+r)^{n-1}} f(0),}</math>

for ''r'' < 1. Hence

:<math>\displaystyle{{f(x)-f(rx)\over 1-r}={-f(rx)\over 1-r} \ge -{1\over (1+r)^{n-1}} f(0)> -{ f(0)\over 2^{n-1}} >0.}</math>

Hence the directional derivative at ''x'' is bounded below by the strictly positive constant on the right hand side.

==General discussion==
Consider a second order, uniformly [[elliptic operator]] of the form
:<math> Lu = a_{ij}(x)\frac{\partial^2 u}{\partial x_i \partial x_j} + b_i(x) \frac{\partial u}{\partial x_i} + c(x)u, \qquad x \in \Omega.</math>
Here <math>\Omega</math> is an open, bounded subset of <math>\mathbb{R}^n</math>.

The Weak Maximum Principle states that a solution of the equation <math>Lu=0</math> in <math>\Omega</math> attains its maximum value on the closure <math>\overline{\Omega}</math> at some point on the boundary <math>\partial\Omega</math>. Let <math>x_0 \in \partial\Omega</math> be such a point, then necessarily
:<math> \frac{\partial u}{\partial \nu}(x_0) \geq 0,</math>
where <math>\partial/\partial\nu</math> denotes the [[outer normal]] [[derivative]]. This is simply a consequence of the fact that <math>u(x)</math> must be nondecreasing as <math>x</math> approach <math>x_0</math>. The Hopf Lemma strengthens this observation by proving that, under mild assumptions on <math>\Omega</math> and <math>L</math>, we have

:<math> \frac{\partial u}{\partial \nu}(x_0) > 0.</math>

A precise statement of the Lemma is as follows. Suppose that <math>\Omega</math> is a bounded region in <math>\mathbb{R}^2</math> and let <math>L</math> be the operator described above. Let <math>u</math> be of class <math>C^2(\Omega) \cap C^1(\overline{\Omega})</math> and satisfy the differential inequality
:<math> Lu \geq 0, \qquad \textrm{in~} \Omega. </math>
Let <math>x_0 \in \partial\Omega</math> be given so that <math> 0\leq u(x_0) = \max_{x \in \overline{\Omega}} u(x)</math>.
If (i) <math>\Omega</math> is <math>C^2</math> at <math>x_0</math>, and (ii) <math>c \leq 0</math>, then either <math>u</math> is a constant, or <math> \frac{\partial u}{\partial \nu}(x_0) > 0</math>, where <math>\nu</math> is the outward pointing unit normal, as above.

The above result can be generalized in several respects. The regularity assumption on <math>\Omega</math> can be replaced with an interior ball condition: the lemma holds provided that there exists an open ball <math>B \subset \Omega</math> with <math>x_0 \in \partial B</math>. It is also possible to consider functions <math>c</math> that take positive values, provided that <math>u(x_0) = 0</math>. For the proof and other discussion, see the references below.

==See also==
*[[Hopf maximum principle]]
==References==

* {{citation
| last = Evans
| first = Lawrence
| year = 2000
| title = Partial Differential Equations
| publisher = [[American Mathematical Society]]
| isbn = 0-8218-0772-2
}}
* {{citation
| first = L. E.
| last = Fraenkel
| year = 2000
| title = An Introduction to Maximum Principles and Symmetry in Elliptic Problems
| publisher = Cambridge University Press
| isbn = 978-0-521-46195
}}
*{{citation|title=Geometric Function Theory: Explorations in Complex Analysis|first=Steven G.|last= Krantz|publisher=Springer|year= 2005|id=ISBN 0817643397|p=127-128}}
*{{citation| last=Taylor|first= Michael E.|authorlink=Michael E. Taylor|title= Partial differential equations I. Basic theory|edition=2nd |series= Applied Mathematical Sciences|volume= 115|publisher=Springer|year=2011|id= ISBN 978-1-4419-70}} (The Hopf lemma is referred to as "Zaremba's principle" by Taylor.)

[[Category:Partial differential equations]]
[[Category:Mathematical principles]]

Hypercone

2011-12-29T20:46:00Z

128.6.65.157:

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