formulasearchengine - User contributions [en]

Squeeze theorem

2015-01-12T06:49:10Z

174.3.213.121: Undid revision 641455051 by 81.13.201.152 (talk) rv vandalism. what the hell do you think you're doing?i'll slap you, punk.

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible. Here are some common dental emergencies: Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached. At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN. Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours. Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary. Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital. Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital. Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief. In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off. If you liked this article as well as you wish to obtain guidance with regards to [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] kindly check out the page.

Rankin–Selberg method

2015-01-05T08:28:32Z

174.3.125.23: removed excessive formating

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Limit superior and limit inferior

2015-01-05T08:13:07Z

174.3.125.23: /* References */ unnecessary

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Ludwig Boltzmann

2015-01-03T06:27:34Z

174.3.213.121: Undid revision 640762606 by 174.3.213.121 (talk) LOL ADHD fail my bad

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Number theory

2015-01-02T22:32:18Z

174.3.125.23: /* Indian school: Āryabhaṭa, Brahmagupta, Bhāskara */ they did not collaborate; thus they were not part of a school

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Set-builder notation

2014-11-17T11:04:21Z

174.3.125.23: /* Formal set builder notation sets */ article needs links. readers such as me who are not mathematically inclined need a simple way to understand articles when they are reading!

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Naive set theory

2014-10-19T11:07:04Z

174.3.125.23: /* Setting of naive set theory */ this is pedagogical versus formal set language

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Yuktibhāṣā

2014-10-09T10:06:44Z

174.3.125.23: already linked

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Power rule

2014-10-09T06:24:15Z

174.3.125.23: needs more links

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Bernoulli process

2014-09-29T04:43:25Z

174.3.125.23: fmting

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Logic alphabet

2014-09-25T06:50:17Z

174.3.125.23: /* The logic alphabet */ WP:HEAD

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Poisson limit theorem

2014-09-23T07:05:57Z

174.3.125.23: /* The theorem */ WP:HEAD

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Latin hypercube sampling

2014-09-20T10:41:05Z

174.3.125.23: removed unnecessary spaces

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174.3.125.23: unrelated offtopic tanget

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2014-02-09T10:15:50Z

174.3.125.23: cleanup

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2014-01-18T04:18:12Z

174.3.125.23: WP:HEAD

'''Affine arithmetic''' ('''AA''') is a model for [[self-validated computation|self-validated]] [[numerical analysis]]. In AA, the quantities of interest are represented as [[affine combination]]s ('''affine forms''') of certain primitive variables, which stand for sources of uncertainty in the data or approximations made during the computation.

Affine arithmetic is meant to be an improvement on [[interval arithmetic]] (IA), and is similar to [[generalized interval arithmetic]], first-order [[Taylor arithmetic]], the [[center-slope model]], and [[ellipsoid calculus]] — in the sense that it is an automatic method to derive first-order guaranteed approximations to general formulas.

Affine arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving [[equation system|system]]s of non-linear equations, analyzing [[dynamical system]]s, [[integral|integrating]] functions [[differential equation]]s, etc. Applications include [[ray tracing (graphics)|ray tracing]], [[2D computer graphics|plotting]] [[curve]]s, intersecting [[implicit surface|implicit]] and [[parametric surface]]s, [[error analysis]], [[process control]], worst-case analysis of [[electric circuit]]s, and more.

==Definition==
In affine arithmetic, each input or computed quantity ''x'' is represented by a formula
<math>x = x_0 + x_1 \epsilon_1 + x_2 \epsilon_2 + {}</math><math>\cdots</math><math>{} + x_n \epsilon_n</math>
where <math>x_0, x_1, x_2,</math><math>\dots,</math><math> x_n </math> are known floating-point numbers, and <math>\epsilon_1, \epsilon_2,\epsilon_n</math> are symbolic variables whose values are only known to lie in the range [-1,+1].

Thus, for example, a quantity ''X'' which is known to lie in the range [3,7] can be represented by the affine form <math>x = 5 + 2 \epsilon_k</math>, for some ''k''. Conversely, the form <math>x = 10 + 2 \epsilon_3 - 5 \epsilon_8</math> implies that the corresponding quantity ''X'' lies in the range [3,17].

The sharing of a symbol <math>\epsilon_j</math> among two affine forms <math>x</math>, <math>y</math> implies that the corresponding quantities ''X'', ''Y'' are partially dependent, in the sense that their joint range is smaller than the [[Cartesian product]] of their separate ranges. For example, if
<math>x = 10 + 2 \epsilon_3 - 6 \epsilon_8</math> and
<math>y = 20 + 3 \epsilon_4 + 4 \epsilon_8</math>,
then the individual ranges of ''X'' and ''Y'' are [2,18] and [13,27], but the joint range of the pair (''X'',''Y'') is the [[hexagon]] with corners (2,27), (6,27), (18,19), (18,13), (14,13), (2,21) — which is a proper subset of the [[rectangle]] [2,18]×[13,27].

==Affine arithmetic operations==
Affine forms can be combined with the standard arithmetic operations or elementary functions, to obtain guaranteed approximations to formulas.

===Affine operations===
For example, given affine forms <math>x,y</math> for ''X'' and ''Y'', one can obtain an affine form <math>z</math> for ''Z'' = ''X'' + ''Y'' simply by adding the forms — that is, setting <math>z_j</math> <math>\gets</math> <math>x_j + y_j</math> for every ''j''. Similarly, one can compute an affine form <math>z</math> for ''Z'' = <math>\alpha</math>''X'', where <math>\alpha</math> is a known constant, by setting <math>z_j</math> <math>\gets</math> <math>\alpha x_j</math> for every ''j''. This generalizes to arbitrary affine operations like ''Z'' = <math>\alpha</math>''X'' + <math>\beta</math>''Y'' + <math>\gamma</math>.

===Non-affine operations===
A non-affine operation <math>Z</math> <math>\gets</math> <math>F(X,Y,</math><math>\dots</math><math>)</math>, like multiplication <math>Z</math> <math>\gets</math> <math>X Y</math> or <math>Z</math> <math>\gets</math> <math>\sin(X)</math>, cannot be performed exactly, since the result would not be an affine form of the <math>\epsilon_i</math>. In that case, one should take a suitable affine function ''G'' that approximates ''F'' to first order, in the ranges implied by <math>x</math> and <math>y</math>; and compute <math>z</math> <math>\gets</math> <math>G(x,y,</math><math>\dots</math><math>) + z_k\epsilon_k</math>, where <math>z_k</math> is an upper bound for the absolute error <math>|F-G|</math> in that range, and <math>\epsilon_k</math> is a new symbolic variable not occurring in any previous form.

The form <math>z</math> then gives a guaranteed enclosure for the quantity ''Z''; moreover, the affine forms <math>x,y,</math><math>\dots</math><math>,z</math> jointly provide a guaranteed enclosure for the point (''X'',''Y'',...,''Z''), which is often much smaller than the Cartesian product of the ranges of the individual forms.

===Chaining operations===
Systematic use of this method allows arbitrary computations on given quantities to be replaced by equivalent computations on their affine forms, while preserving first-order correlations between the input and output and guaranteeing the complete enclosure of the joint range. One simply replaces each arithmetic operation or elementary function call in the formula by a call to the corresponding AA library routine.

For smooth functions, the approximation errors made at each step are proportional to the square ''h''2 of the width ''h'' of the input intervals. For this reason, affine arithmetic will often yield much tighter bounds than standard interval arithmetic (whose errors are proportional to ''h'').

===Roundoff errors===
In order to provide guaranteed enclosure, affine arithmetic operations must account for the roundoff errors in the computation of the resulting coefficients <math>z_j</math>. This cannot be done by rounding each <math>z_j</math> in a specific direction, because any such rounding would falsify the dependencies between affine forms that share the symbol <math>\epsilon_j</math>. Instead, one must compute an upper bound <math>\delta_j</math> to the roundoff error of each <math>z_j</math>, and add all those <math>\delta_j</math> to the coefficient <math>z_k</math> of the new symbol <math>\epsilon_k</math> (rounding up). Thus, because of roundoff errors, even affine operations like ''Z'' = <math>\alpha</math>''X'' and ''Z'' = ''X'' + ''Y'' will add the extra term <math>z_k\epsilon_k</math>.

The handling of roundoff errors increases the code complexity and execution time of AA operations. In applications where those errors are known to be unimportant (because they are dominated by uncertainties in the input data and/or by the linearization errors), one may use a simplified AA library that does not implement roundoff error control.

==Affine projection model==
Affine arithmetic can be viewed in matrix form as follows. Let <math>X_1,X_2,</math><math>\dots,</math><math>X_m</math> be all input and computed quantities in use at some point during a computation. The affine forms for those quantities can be represented by a single coefficient matrix ''A'' and a vector ''b'', where element <math>A_{i,j}</math> is the coefficient of symbol <math>\epsilon_j</math> in the affine form of ''<math>X_i</math>''; and <math>b_i</math> is the independent term of that form. Then the joint range of the quantities — that is, the range of the point <math>(X_1,X_2,</math><math>\dots,</math><math>X_m)</math> — is the image of the hypercube <math>U^n = [-1,+1]^n</math> by the affine map from <math>U^n</math> to <math>R^m</math> defined by <math>\epsilon</math> <math>\to</math> <math>A \epsilon + b</math>.

The range of this affine map is a [[zonotope]] bounding the joint range of the quantities <math>X_1,X_2,</math><math>\dots,</math><math>X_m</math>. Thus one could say that AA is a "zonotope arithmetic". Each step of AA usually entails adding one more row and one more column to the matrix ''A''.

==Affine form simplification==
Since each AA operation generally creates a new symbol <math>\epsilon_k</math>, the number of terms in an affine form may be proportional to the number of operations used to compute it. Thus, it is often necessary to apply "symbol condensation" steps, where two or more symbols <math>\epsilon_k</math> are replaced by a smaller set of new symbols. Geometrically, this means replacing a complicated zonotope ''P'' by a simpler zonotope ''Q'' that encloses it. This operation can be done without destroying the first-order approximation property of the final zonotope.

== Implementation ==

===Matrix implementation===
Affine arithmetic can be implemented by a global array ''A'' and a global vector ''b'', as described above. This approach is reasonably adequate when the set of quantities to be computed is small and known in advance. In this approach, the programmer must maintain externally the correspondence between the row indices and the quantities of interest. Global variables hold the number ''m'' of affine forms (rows) computed so far, and the number ''n'' of symbols (columns) used so far; these are automatically updated at each AA operation.

===Vector implementation===
Alternatively, each affine form can be implemented as a separate vector of coefficients. This approach is more convenient for programming, especially when there are calls to library procedures that may use AA internally. Each affine form can be given a mnemonic name; it can be allocated when needed, be passed to procedures, and reclaimed when no longer needed. The AA code then looks much closer to the original formula. A global variable holds the number ''n'' of symbols used so far.

===Sparse vector implementation===
On fairly long computations, the set of "live" quantities (that will be used in future computations) is much smaller than the set of all computed quantities; and ditto for the set of "live" symbols <math>\epsilon_j</math>. In this situation, the matrix and vector implementations are too wasteful of time and space.

In such situations, one should use a [[sparse array|sparse]] implementation. Namely, each affine form is stored as a list of pairs (j,<math>x_j</math>), containing only the terms with non-zero coefficient <math>x_j</math>. For efficiency, the terms should be sorted in order of ''j''. This representation makes the AA operations somewhat more complicated; however, the cost of each operation becomes proportional to the number of nonzero terms appearing in the operands, instead of the number of total symbols used so far.

This is the representation used by LibAffa.

== References ==

*L. H. de Figueiredo and J. Stolfi (2004) "Affine arithmetic: concepts and applications." ''Numerical Algorithms'' '''37''' (1–4), 147–158.
* J. L. D. Comba and J. Stolfi (1993), "Affine arithmetic and its applications to computer graphics". ''Proc. SIBGRAPI'93 — VI Simpósio Brasileiro de Computação Gráfica e Processamento de Imagens (Recife, BR)'', 9–18.
* L. H. de Figueiredo and J. Stolfi (1996), "Adaptive enumeration of implicit surfaces with affine arithmetic". ''Computer Graphics Forum'', '''15''' ''5'', 287–296.
* W. Heidrich (1997), "A compilation of affine arithmetic versions of common math library functions". Technical Report 1997-3, Universität Erlangen-Nürnberg.
* M. Kashiwagi (1998), "An all solution algorithm using affine arithmetic". ''NOLTA'98 — 1998 International Symposium on Nonlinear Theory and its Applications (Crans-Montana, Switzerland)'', 14–17.
* L. Egiziano, N. Femia, and G. Spagnuolo (1998), "New approaches to the true worst-case evaluation in circuit tolerance and sensitivity analysis — Part II: Calculation of the outer solution using affine arithmetic". ''Proc. COMPEL'98 — 6th Workshop on Computer in Power Electronics (Villa Erba, Italy)'', 19–22.
* W. Heidrich, Ph. Slusallek, and H.-P. Seidel (1998), "Sampling procedural shaders using affine arithmetic". ''ACM Transactions on Graphics (TOG)'', '''17''' ''3'', 158–176.
* F. Messine and A. Mahfoudi (1998), "Use of affine arithmetic in interval optimization algorithms to solve multidimensional scaling problems". ''Proc. SCAN'98 — IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Budapest, Hungary)'', 22–25.
* A. de Cusatis Jr., L. H. Figueiredo, and M. Gattass (1999), "Interval methods for ray casting surfaces with affine arithmetic". ''Proc. SIBGRAPI'99 — 12th Brazilian Symposium on Computer Graphics and Image Processing'', 65–71. 
* K. Bühler and W. Barth (2000), "A new intersection algorithm for parametric surfaces based on linear interval estimations". ''Proc. SCAN 2000 / Interval 2000 — 9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics'', ???–???. 
* I. Voiculescu, J. Berchtold, A. Bowyer, R. R. Martin, and Q. Zhang (2000), "Interval and affine arithmetic for surface location of power- and Bernstein-form polynomials". ''Proc. Mathematics of Surfaces IX'', 410–423. Springer, ISBN 1-85233-358-8.
* Q. Zhang and R. R. Martin (2000), "Polynomial evaluation using affine arithmetic for curve drawing". ''Proc. of Eurographics UK 2000 Conference'', 49–56. ISBN 0-9521097-9-4.
* D. Michelucci (2000), "Reliable computations for dynamic systems". ''Proc. SCAN 2000 / Interval 2000 — 9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics'', ???–???.
* N. Femia and G. Spagnuolo (2000), "True worst-case circuit tolerance analysis using genetic algorithm and affine arithmetic — Part I". ''IEEE Transactions on Circuits and Systems'', '''47''' ''9'', 1285–1296.
* R. Martin, H. Shou, I. Voiculescu, and G. Wang (2001), "A comparison of Bernstein hull and affine arithmetic methods for algebraic curve drawing". ''Proc. Uncertainty in Geometric Computations'', 143–154. Kluwer Academic Publishers, ISBN 0-7923-7309-X.
* A. Bowyer, R. Martin, H. Shou, and I. Voiculescu (2001), "Affine intervals in a CSG geometric modeller". ''Proc. Uncertainty in Geometric Computations'', 1–14. Kluwer Academic Publishers, ISBN 0-7923-7309-X.
* T. Kikuchi and M. Kashiwagi (2001), "Elimination of non-existence regions of the solution of nonlinear equations using affine arithmetic". ''Proc. NOLTA'01 — 2001 International Symposium on Nonlinear Theory and its Applications''.
* T. Miyata and M. Kashiwagi (2001), "On range evaluation of polynomials of affine arithmetic". ''Proc. NOLTA'01 - 2001 International Symposium on Nonlinear Theory and its Applications''.
* Y. Kanazawa and S. Oishi (2002), "A numerical method of proving the existence of solutions for nonlinear ODEs using affine arithmetic". ''Proc. SCAN'02 — 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics''. 
* H. Shou, R. R.Martin, I. Voiculescu, A. Bowyer, and G. Wang (2002), "Affine arithmetic in matrix form for polynomial evaluation and algebraic curve drawing". ''Progress in Natural Science'', '''12''' ''1'', 77–81.
* A. Lemke, L. Hedrich, and E. Barke (2002), "Analog circuit sizing based on formal methods using affine arithmetic". ''Proc. ICCAD-2002 — International Conference on Computer Aided Design'', 486–489.
* F. Messine (2002), "Extensions of affine arithmetic: Application to unconstrained global optimization". ''Journal of Universal Computer Science'', '''8''' ''11'', 992–1015.
* K. Bühler (2002), "Implicit linear interval estimations". ''Proc. 18th Spring Conference on Computer Graphics (Budmerice, Slovakia)'', 123–132. ACM Press, ISBN 1-58113-608-0.
* L. H. de Figueiredo, J. Stolfi, and L. Velho (2003), "Approximating parametric curves with strip trees using affine arithmetic". ''Computer Graphics Forum'', '''22''' ''2'', 171–179.
* C. F. Fang, T. Chen, and R. Rutenbar (2003), "Floating-point error analysis based on affine arithmetic". ''Proc. 2003 International Conf. on Acoustic, Speech and Signal Processing''.
* A. Paiva, L. H. de Figueiredo, and J. Stolfi (2006), "Robust visualization of strange attractors using affine arithmetic". ''Computers & Graphics'', '''30''' ''6'', 1020– 1026.

== External links ==
*[http://www.dcc.unicamp.br/~stolfi/EXPORT/projects/affine-arith/Welcome.html] Stolfi's page on AA.
*[http://savannah.nongnu.org/projects/libaffa] LibAffa, an LGPL implementation of affine arithmetic.
*[http://sourceforge.net/projects/asol/] ASOL, a branch-and-prune method to find all solutions to systems of nonlinear equations using affine arithmetic

[[Category:Numerical analysis]]
[[Category:Affine geometry]]

Optical coating

2013-12-30T18:34:05Z

174.3.125.23: /* Types of coating */ WP:CAPTION

In [[mathematics]], more specifically in the area of [[Abstract algebra|modern algebra]] known as [[group theory]], a [[Group (mathematics)|group]] is said to be '''perfect''' if it equals its own [[commutator subgroup]], or equivalently, if the group has no nontrivial [[abelian group|abelian]] [[quotient group|quotients]] (equivalently, its [[abelianization]], which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that ''G''(1) = ''G'' (the commutator subgroup equals the group), or equivalently one such that ''G''ab = {1} (its abelianization is trivial).

== Examples ==
The smallest (non-trivial) perfect group is the [[alternating group]] ''A''5. More generally, any non-[[abelian group|abelian]] [[simple group]] is perfect since the commutator subgroup is a [[normal subgroup]] with abelian quotient. Conversely, a perfect group need not be simple; for example, the [[special linear group]] SL(2,5) (or the [[binary icosahedral group]] which is isomorphic to it) is perfect but not simple (it has a non-trivial [[center (group)|center]] containing <math>\left(\begin{smallmatrix}-1 & 0 \\ 0 & -1\end{smallmatrix}\right) = \left(\begin{smallmatrix}4 & 0 \\ 0 & 4\end{smallmatrix}\right)</math>).

More generally, a [[quasisimple group]] (a perfect [[Central extension (mathematics)|central extension]] of a simple group) which is a non-trivial extension (i.e., not a simple group itself) is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL(''n'',''q'') as extensions of the [[projective special linear group]] PSL(''n'',''q'') (SL(2,5) is an extension of PSL(2,5), which is isomorphic to ''A''5). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over '''F'''2, where it equals the special linear group), as the [[determinant]] gives a non-trivial abelianization and indeed the commutator subgroup is SL.

A non-trivial perfect group, however, is necessarily not [[solvable group|solvable]].

Every [[acyclic group]] is perfect, but the converse is not true: ''A''5 is perfect but not acyclic (in fact, not even [[Superperfect group|superperfect]]), see {{harv|Berrick|Hillman|2003}}. In fact, for ''n'' ≥ 5 the alternating group ''An'' is perfect but not superperfect, with ''H''2(''An'', '''Z''') = '''Z'''/2 for ''n'' ≥ 8.

Every perfect group ''G'' determines another perfect group ''E'' (its [[universal central extension]]) together with a surjection ''f:E'' → ''G'' whose kernel is in the center of ''E,''
such that ''f'' is universal with this property. The kernel of ''f'' is called the [[Schur multiplier]] of ''G'' because it was first studied by [[Schur]] in 1904; it is isomorphic to the
homology group ''H2(G)''.

==Grün's lemma==
A basic fact about perfect groups is '''Grün's lemma''' from {{harv|Grün|1935|loc=Satz 4,<ref group="note">''[[wikt:Satz#German|Satz]]'' is German for "theorem".</ref> p. 3}}: the [[quotient group|quotient]] of a perfect group by its [[center (group theory)|center]] is centerless (has trivial center).

<blockquote>'''Proof:''' If ''G'' is a perfect group, let ''Z''1 and ''Z''2 denote the first two terms of the [[Central_series#Upper_central_series|upper central series]] of ''G'' (i.e., ''Z''1 is the center of ''G'', and ''Z''2/''Z''1 is the center of ''G''/''Z''1). If ''H'' and ''K'' are subgroups of ''G'', denote the [[commutator]] of ''H'' and ''K'' by [''H'', ''K''] and note that [''Z''1, ''G''] = 1 and [''Z''2, ''G''] ⊆ ''Z''1, and consequently (the convention that [''X'', ''Y'', ''Z''] = [[''X'', ''Y''], ''Z''] is followed):

:<math>[Z_2,G,G]=[[Z_2,G],G]\subseteq [Z_1,G]=1</math>
:<math>[G,Z_2,G]=[[G,Z_2],G]=[[Z_2,G],G]\subseteq [Z_1,G]=1.</math>

By the [[three subgroups lemma]] (or equivalently, by the [[Commutator#Identities|Hall-Witt identity]]), it follows that [''G'', ''Z''2] = [[''G'', ''G''], ''Z''2] = [''G'', ''G'', ''Z''2] = {1}. Therefore, ''Z''2 ⊆ ''Z''1 = ''Z''(''G''), and the center of the quotient group ''G'' ⁄ ''Z''(''G'') is the [[trivial group]].</blockquote>

As a consequence, all [[Center (group theory)#Higher centers|higher centers]] (that is, higher terms in the [[upper central series]]) of a perfect group equal the center.

==Group homology==
In terms of [[group homology]], a perfect group is precisely one whose first homology group vanishes: ''H''1(''G'', '''Z''') = 0, as the first homology group of a group is exactly the abelianization of the group, and perfect means trivial abelianization. An advantage of this definition is that it admits strengthening:
* A [[superperfect group]] is one whose first two homology groups vanish: ''H''1(''G'', '''Z''') = ''H''2(''G'', '''Z''') = 0.
* An [[acyclic group]] is one ''all'' of whose (reduced) homology groups vanish <math>\tilde H_i(G;\mathbf{Z}) = 0.</math> (This is equivalent to all homology groups other than ''H''0 vanishing.)

==Quasi-perfect group==
Especially in the field of [[algebraic K-theory]], a group is said to be '''quasi-perfect''' if its commutator subgroup is perfect; in symbols, a quasi-perfect group is one such that ''G''(1) = ''G''(2) (the commutator of the commutator subgroup is the commutator subgroup), while a perfect group is one such that ''G''(1) = ''G'' (the commutator subgroup is the whole group). See {{harv|Karoubi|1973|pp=301–411}} and {{harv| Inassaridze | 1995 | p=76}}.

==Notes==
{{reflist | group = note }}

==References==
{{reflist}}
{{refbegin}}
* A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683–698. {{MR|2009444}}
* {{Citation | last1=Grün | first1=Otto | title=Beiträge zur Gruppentheorie. I. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002173409 | language=German | zbl=0012.34102 | year=1935 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=174 | pages=1–14}}
*{{Citation | last1=Inassaridze | first1=Hvedri | title=Algebraic K-theory | url=http://books.google.com/?id=rnSE3aoNVY0C | publisher=Kluwer Academic Publishers Group | location=Dordrecht | series=Mathematics and its Applications | isbn=978-0-7923-3185-8 | mr=1368402 | year=1995 | volume=311}}
* Karoubi, M.: Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications, Lecture Notes in Math. 343, Springer-Verlag, 1973
*{{Citation
| last = Rose
| first = John S.
| title = A Course in Group Theory
| publisher = Dover Publications, Inc.
| location = New York
| pages = 61
| year = 1994
| isbn = 0-486-68194-7
| mr = 1298629
}}
{{refend}}

==External links==
* {{MathWorld|urlname=PerfectGroup|title=Perfect Group}}
* {{MathWorld|urlname=GruensLemma|title=Grün's lemma}}

[[Category:Group theory]]
[[Category:Properties of groups]]
[[Category:Lemmas]]