Primitive root modulo n

2014-01-14T09:32:36Z

128.103.135.53: Link to multiplicative group of integers mod n.

{{Refimprove|date=October 2010}}
[[Image:Polynomialdeg2.svg|thumb|right|<center><math>x^2 - x - 2\!</math></center>]]
A '''quadratic function''', in [[mathematics]], is a [[polynomial function]] of the form

:<math>f(x)=ax^2+bx+c,\quad a \ne 0.</math><ref name="wolfram">{{cite web | url=http://mathworld.wolfram.com/QuadraticEquation.html | title=Quadratic Equation -- from Wolfram MathWorld | accessdate=January 6, 2013}}</ref>

The [[graph of a function|graph]] of a quadratic function is a [[parabola]] whose axis of symmetry is parallel to the {{math|''y''}}-axis.

The expression {{math|''ax''2 + ''bx'' + ''c''}} in the definition of a quadratic function is a '''polynomial of [[Degree of a polynomial|degree]] 2''' or second order, or a '''2nd degree polynomial''', because the highest exponent of {{math|''x''}} is 2. This expression is also called a '''quadratic polynomial''' or '''quadratic'''.

If the quadratic function is set equal to zero, then the result is a [[quadratic equation]]. The solutions to the equation are called the [[root of a function|root]]s of the equation.

==Origin of word==

The adjective ''quadratic'' comes from the [[Latin]] word ''[[wikt:en:quadratum#Latin|quadrātum]]'' ("[[square (geometry)|square]]"). A term like {{math|''x''2}} is called a [[square (algebra)|square]] in algebra because it is the area of a ''square'' with side {{math|''x''}}.

In general, a prefix [[quadr(i)-]] indicates the number {{math|[[4 (number)|4]]}}. Examples are quadrilateral and quadrant. ''Quadratum'' is the Latin word for square because a square has four sides.

==Roots==
{{Further2|[[Quadratic equation]]}}

The [[root of a function|roots]] (zeros) of the quadratic function
: <math>f(x) = ax^2+bx+c\,</math>
are the values of {{math|''x''}} for which {{math|''f''(''x'') {{=}} 0}}.

When the [[coefficient]]s {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, are [[real numbers|real]] or [[complex numbers|complex]], the roots are
:<math>x=\frac{-b \pm \sqrt{\Delta}}{2 a}, </math>
where the [[discriminant]] is defined as
:<math>\Delta = b^2 - 4 a c \, . </math>

==Forms of a quadratic function==
A quadratic function can be expressed in three formats:<ref>{{citation
|title=College Algebra
|first1=Deborah
|last1=Hughes-Hallett
|first2=Eric
|last2=Connally
|first3=William G.
|last3=McCallum
|publisher=John Wiley & Sons Inc
|year=2007
|isbn=0-471-27175-6, 9780471271758
|page=205
|url=http://books.google.be/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22}}, [http://books.google.be/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22 Search result]
</ref>

* <math>f(x) = a x^2 + b x + c \,\!</math> is called the '''standard form''',
* <math>f(x) = a(x - x_1)(x - x_2)\,\!</math> is called the '''factored form''', where {{math|''x''1}} and {{math|''x''2}} are the roots of the quadratic equation, it is used in [[logistic map]]
* <math>f(x) = a(x - h)^2 + k \,\!</math> is called the '''vertex form''', where {{math|''h''}} and {{math|''k''}} are the {{math|''x''}} and {{math|''y''}} coordinates of the vertex, respectively.

To convert the '''standard form''' to '''factored form''', one needs only the [[quadratic formula]] to determine the two roots {{math|''x''1}} and {{math|''x''2}}. To convert the '''standard form''' to '''vertex form''', one needs a process called [[completing the square]]. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

==Graph==
[[Image:Function ax^2.svg|thumb|350px|<math>f(x) = ax^2 |_{a=\{0.1,0.3,1,3\}} \!</math>]]
[[Image:Function x^2+bx.svg|thumb|350px|<math>f(x) = x^2 + bx |_{b=\{1,2,3,4\}} \!</math>]]
[[Image:Function x^2-bx.svg|thumb|350px|<math>f(x) = x^2 + bx |_{b=\{-1,-2,-3,-4\}} \!</math>]]
Regardless of the format, the graph of a quadratic function is a [[parabola]] (as shown above).
* If {{math|''a'' > 0}}, (or is a positive number), the parabola opens upward.
* If {{math|''a'' < 0}}, (or is a negative number), the parabola opens downward.

The coefficient {{math|''a''}} controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive {{math|''a''}} makes the function increase faster and the graph appear more closed.

The coefficients {{math|''b''}} and {{math|''a''}} together control the axis of symmetry of the parabola (also the {{math|''x''}}-coordinate of the vertex) which is at <math>x = -\frac{b}{2a}</math>.

The coefficient {{math|''b''}} alone is the declivity of the parabola as {{math|''y''}}-axis intercepts.

The coefficient {{math|''c''}} controls the height of the parabola, more specifically, it is the point where the parabola intercept the {{math|''y''}}-axis.

===Vertex===

The '''vertex''' of a parabola is the place where it turns, hence, it's also called the '''turning point'''. If the quadratic function is in vertex form, the vertex is {{math|(''h'', ''k'')}}. By the method of completing the square, one can turn the general form
:<math>f(x) = a x^2 + b x + c \,\!</math>
into
: <math> f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} ,</math>
so the vertex of the parabola in the vertex form is
: <math> \left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right). </math>
If the quadratic function is in factored form
:<math>f(x) = a(x - r_1)(x - r_2) \,\!</math>
the average of the two roots, i.e.,
: <math>\frac{r_1 + r_2}{2} \,\!</math>
is the {{math|''x''}}-coordinate of the vertex, and hence the vertex is
: <math> \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).\!</math>

The vertex is also the maximum point if {{math|''a'' < 0}}, or the minimum point if {{math|''a'' > 0}}.

The vertical line

: <math> x=h=-\frac{b}{2a} </math>

that passes through the vertex is also the '''axis of symmetry''' of the parabola.

====Maximum and minimum points====

Using [[calculus]], the vertex point, being a [[minima and maxima|maximum or minimum]] of the function, can be obtained by finding the roots of the [[derivative]]:
:<math>f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b \,\!,</math>
giving
:<math>x=-\frac{b}{2a}</math>
with the corresponding function value
:<math>f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = -\frac{(b^2-4ac)}{4a} = -\frac{\Delta}{4a} \,\!,</math>
so again the vertex point coordinates can be expressed as
:<math> \left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right). </math>

==The square root of a quadratic function==
The [[square root]] of a quadratic function gives rise to one of the four conic sections, [[almost always]] either to an [[ellipse]] or to a [[hyperbola]]. If <math>a>0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes a hyperbola. The axis of the hyperbola is determined by the [[ordinate]] of the [[minimum]] point of the corresponding parabola <math> y_p = a x^2 + b x + c \,\!</math>. If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is positive, then the hyperbola's axis is vertical. If <math>a<0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either an ellipse or nothing at all. If the ordinate of the [[maximum]] point of the corresponding parabola
<math> y_p = a x^2 + b x + c \,\!</math> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an [[Empty set|empty]] locus of points.

==Iteration==
Given an <math>f(x)=ax^2+bx+c</math>, one cannot always deduce the analytic form of <math>f^{(n)}(x)</math>, which means the ''nth'' iteration of <math>f(x)</math>. (The superscript can be extended to negative number referring to the iteration of the inverse of <math>f(x)</math> if the inverse exists.) But there is one easier case, in which <math>f(x)=a(x-x_0)^2+x_0</math>.

In such case, one has
:<math>f(x)=a(x-x_0)^2+x_0=h^{(-1)}(g(h(x)))\,\!</math>,
where
:<math>g(x)=ax^2\,\!</math> and <math>h(x)=x-x_0\,\!</math>.
So by induction,
:<math>f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!</math>
can be obtained, where <math>g^{(n)}(x)</math> can be easily computed as
:<math>g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}\,\!</math>.
Finally, we have
:<math>f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}+x_0\,\!</math>,
in the case of <math>f(x)=a(x-x_0)^2+x_0</math>.

See [[Topological conjugacy]] for more detail about such relationship between ''f'' and ''g''. And see [[Complex quadratic polynomial]] for the chaotic behavior in the general iteration.

==Bivariate (two variable) quadratic function==
{{see|Quadric|Quadratic form}}
A '''bivariate quadratic function''' is a second-degree polynomial of the form
:<math> f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F \,\!</math>
Such a function describes a quadratic [[surface]]. Setting <math>f(x,y)\,\!</math> equal to zero describes the intersection of the surface with the plane <math>z=0\,\!</math>, which is a [[locus (mathematics)|locus]] of points equivalent to a [[conic section]].

===Minimum/maximum===

If <math> 4AB-E^2 <0 \,</math> the function has no maximum or minimum, its graph forms an hyperbolic [[paraboloid]].

If <math> 4AB-E^2 >0 \,</math> the function has a minimum if ''A''>0, and a maximum if ''A''<0, its graph forms an elliptic paraboloid.

The minimum or maximum of a bivariate quadratic function is obtained at <math> (x_m, y_m) \,</math> where:

:<math>x_m = -\frac{2BC-DE}{4AB-E^2}</math>

:<math>y_m = -\frac{2AD-CE}{4AB-E^2}</math>

If <math> 4AB- E^2 =0 \,</math> and <math> DE-2CB=2AD-CE \ne 0 \,</math> the function has no maximum or minimum, its graph forms a parabolic [[cylinder (geometry)|cylinder]].

If <math> 4AB- E^2 =0 \,</math> and <math> DE-2CB=2AD-CE =0 \,</math> the function achieves the maximum/minimum at a line. Similarly, a minimum if ''A''>0 and a maximum if ''A''<0, its graph forms a parabolic cylinder.

==Quadratic polynomial==
In mathematics, a quadratic polynomial or quadratic is a [[polynomial]] of [[degree of a polynomial|degree]] two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. For example, <math>x^2 - 4x + 7</math> is a quadratic polynomial, while <math>x^3 - 4x + 7</math> is not.

===Coefficients===
The [[coefficients]] of a polynomial are often taken to be real or [[Complex quadratic polynomial|complex number]]s, but in fact, a polynomial may be defined over any [[ring (mathematics)|ring]].

===Degree===
When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "[[Degeneracy (mathematics)|degenerate case]]". Usually the context will establish which of the two is meant.

Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial.

===Variables===

A quadratic polynomial may involve a single [[Variable (mathematics)|variable]] ''x'', or multiple variables such as ''x'', ''y'', and ''z''.

====The one-variable case====

Any single-variable quadratic polynomial may be written as
:<math>ax^2 + bx + c,\,\!</math>
where ''x'' is the variable, and ''a'', ''b'', and ''c'' represent the [[coefficient]]s. In [[elementary algebra]], such polynomials often arise in the form of a [[quadratic equation]] <math>ax^2 + bx + c = 0</math>. The solutions to this equation are called the [[Root of a function|roots]] of the quadratic polynomial, and may be found through [[factorization]], [[completing the square]], [[Graph of a function|graphing]], [[Newton's method]], or through the use of the [[quadratic formula]]. Each quadratic polynomial has an associated quadratic function, whose [[graph of a function|graph]] is a [[parabola]].

If the polynomial is a polynomial in one [[Variable (mathematics)|variable]], it determines a quadratic function in one variable. An example is given by ''f''(''x'') = ''x''2 + ''x'' − 2;. The [[Graph of a function|graph]] of such a [[Function (mathematics)|function]] is a [[parabola]] (in degenerate cases a [[line (mathematics)|line]]), and its [[Root of a function|zero]]es can be found by solving the [[quadratic equation]] ''f''(''x'') = 0.

There are three main '''forms''' :
* general form, <math> f(x) = a x^2 + b x + c \,</math>.
* [[logistic map|logistic form]], <math>f_r(x) = r x ( 1-x ) \,</math>, used to study [[Euclidean space|1D]] [[Dynamical_system#Maps|discrete dynamics]],
* [[Complex quadratic polynomial|monic and centered form]], <math>f_c(x) = x^2 +c\,</math>, used to study [[complex dynamics]].

====Two variables case====

Any quadratic polynomial with two variables may be written as
:<math>ax^2 + bxy + cy^2 + dx + ey + f,\,\!</math>
where ''x'' and ''y'' are the variables and ''a'', ''b'', ''c'', ''d'', ''e'', and ''f'' are the coefficients. Such polynomials are fundamental to the study of [[conic section]]s.
Similarly, quadratic polynomials with three or more variables correspond to [[quadric]] surfaces and [[hypersurface]]s. In [[linear algebra]], quadratic polynomials can be generalized to the notion of a [[quadratic form]] on a [[vector space]].

====N variables case====
In the general case, a quadratic polynomial in ''n'' variables ''x''1, ..., ''x''''n'' can be written in the form
:<math>
\sum_{i, j = 1}^{n} Q_{i,j} x_i x_j + \sum_{i = 1}^{n} P_i x_i + R
</math>
where ''Q'' is a symmetric ''n''-dimensional [[matrix (mathematics)|matrix]], ''P'' is an ''n''-dimensional [[Vector (geometric)|vector]], and ''R'' a constant.

==See also==
* [[Quadratic form]]
* [[Quadratic equation]]
* [[Matrix representation of conic sections]]
* [[Quadric]]
* [[Periodic points of complex quadratic mappings]]
* [[List of mathematical functions]]

==References==
{{reflist}}
*Algebra 1, Glencoe, ISBN 0-07-825083-8
*Algebra 2, Saxon, ISBN 0-939798-62-X

==External links==
* {{MathWorld|title=Quadratic|urlname=Quadratic}}

{{Polynomials}}

{{DEFAULTSORT:Quadratic Function}}
[[Category:Polynomials]]
[[Category:Parabolas]]

The Product Space

2013-10-29T15:39:41Z

128.103.252.68: /* The concept of ‘proximity’ */

[[Image:Hemithioacetal.png|150px|thumb|Hemithioacetal functional group]]

'''Hemithioacetal''' is an organic [[functional group]] with the general formula RCH(OR)SR.<ref name=March>{{March6th}}</ref> They form in a spontaneous reaction between a [[thiol]] and an [[aldehyde]]. Since the formerly carbonyl carbon bears four different substituents, hemiacetals are chiral. Hemithioacetals are usually intermediates in the catalytic reactions and usually arise via acid or base [[catalysis]]. The hemithioacetal features [[vicinal (chemistry)|vicinal]] [[hydroxyl]] and [[thioether]] functionalities. Although they are important intermediates, hemithioacetals are usually not isolated since they exist in equilibrium with the thiol and aldehyde:

:RCHO + R’SH <math>\overrightarrow{\leftarrow}</math> RCH(OH)(SR’)

==Isolable hemithioacetal==
Hemithioacetals ordinarily readily dissociate into thiol and aldehyde. Some hemithioacetals have been isolated. The few isolable hemithioacetals are all cyclic, which disfavors dissociation. One example is 2-hydroxy[[tetrahydrothiophene]].<ref>Cox, J.M.; Owen, L.N., J. Chem. Soc. C, Cyclic hemithioacetals: Analogues of thiosugars with sulphur in the ring, 1967, 1130-1134. {{DOI|10.1039/J39670001130}}</ref> Another isolable hemithioacetal can be prepared by addition of thiol to methyl glyoxalate.<ref>Milton, J; Brand, S; Jones, M.F; Rayner, C.M., Tetrahedron Letters, Enantioselective Enzymatic Synthesis of the Anti-Viral Agent Lamivudine, 1995, volume 36, 6961-6964, {{DOI|10.1016/0040-4039(95)01380-Z}}</ref> The stability of hemithioacetal is enhanced in the presence of acid.<ref>Barnett, R. E.; Jencks, W. P, J. Am. Chem. Soc, Diffusion-controlled and concerted catalysis in the decomposition of hemithioacetals, 1969, volume 91, 6758-6765. {{DOI|10.1021/ja01052a038}}</ref> Another class of isolable hemithioacetals are derived from carbonyl groups that form stable hydrates. For example, thiols react with hexafluoroacetone trihydrate to give hemithioacetals, which can be isolated.<ref>Field, L.; Sweetman, B.J.; Bellas, M., Journal of Medicinal Chemistry, Biologically oriented organic sulfur chemistry. II. Formation of hemimercaptals or hemimercaptoles as a means of latentiating thiols, 1969, 12(4), 624-628. {{DOI|10.1021/jm00304a014}}</ref>
[[File:C4H7(OH)S.png|thumb|140px|left|2-Hydroxy[[tetrahydrothiophene]] is a rare example of a hemithioacetal that can be isolated.]]

==Hemithioacetals in nature==
[[Glyoxalase I]], which is part of the glyoxalase system present in the [[cytosol]], catalyzes the conversion of α-oxoaldehyde (RC(O)CHO) and the thiol [[glutathione]] (abbreviated GSH) to S-2-hydroxyacylglutathione derivatives [RCH(OH)CO-SG]. The catalytic mechanism involves an intermediate hemithioacetal adduct [RCOCH(OH)-SG]. The spontaneous reaction forms [[methylglyoxal]]-glutathione hemithioacetal and human glyoxalse I.<ref>Thornalley, P.J., Biochemical Society Transactions, Glyoxalase I - Structure, function and a critical role in the enzymatic defence against glycation, 2003, 31 (6), 1343-1348. ISSN: 03005127</ref>

A hemithioacetal is also invoked in the mechanism of prenylcysteine [[lyase]]. In catalytic mechanism, S-farnesylcysteine is oxidized by a [[flavin]] to a thiocarbenium ion. The thiocarbenium ion hydrolyzes to form the hemithioacetal:
: [(RS)C(R’)(H)]+ + H2O → (RS)C(R’)(H)OH + H+
After formation, the hemithioacetal breaks into [[hydrogen peroxide]], farnesal, and [[cysteine]].<ref>Digits, J.A.; Pyun, H.-J.; Coates, R.M.; Casey, P.J. Journal of biological chemistry, Stereospecificity and kinetic mechanism of human prenylcysteine lyase, an unusual thioether oxidase, 2002, volume 277, 41086-41093. {{DOI|10.1074/jbc.M208069200}}</ref>

==References==
{{Reflist}}

[[Category:Acetals]]
[[Category:Functional groups]]

Discretization

2012-07-11T19:34:52Z

128.103.40.237:

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formulasearchengine - User contributions [en]

Primitive root modulo n

The Product Space

Discretization