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| {{Expert-subject|mathematics|date=July 2013}}
| | The nature of this psycho-social poison from televised violence was highlighted for me by an experience in Tasmania within the late 1970’s that I have retold many instances. I had organized to purchase the22 calibre rifle that I still personal as a “licensed weapon” from Janet Mollison (Bill Mollison’s oldest daughter who was my own age). On the time, Janet lived in a flat above the Salvo’s Op shop in Elizabeth St, North Hobart. I had organized to choose up the rifle on a Saturday evening. The transaction was simple with no paper work being required for ownership of rim-hearth rifles in these instances.<br><br><br><br>Promotional pocket knives and multi-instruments from Leatherman and Swiss Military are nice recognition or appreciation gifts for palms-on audiences. Identify-model promotional knives will be carried day by day and used for a lot of functions due to fold-out features like bottle openers, mini scissors, nail information and more! 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So in the nineteen sixties switchblades that where made in America the place solely being soled to the military [http://www.thebestpocketknifereviews.com/the-best-survival-knife-top-rated-wild-knives-in-the-world/ best survival knife in the world]. Later knife makers found away round these laws by promoting switchblades in construct it your self kits. Now although this loophole is closed in lots of states. Milwaukee Fastback knife fans might be glad to know that the belt clips on these new knives are reversible. Smooth Pocket Fastback Knife forty eight-22-1990<br><br>I was reminded of how a lot I appreciate a fine pocket knife lately when my colleague Chris Bennett blogged about an Australian farmer who dug himself out of a life-threatening scenario with nothing however a chest full of heart and a pocket knife. 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They are used extensively in lots of tropical areas for clearing brush, and chopping by way of jungle growth. Many cultures depend on them for farming purposes. Having said that, there are those that do use them for weapons. I took the pocket knife with me as I took Tuck to the social gathering, and I quietly slipped it to Coach. Of course, he knew precisely learn how to close it, and he immediately engaged Tucker within the first rules of pocket knife safety. Equally essential, Tyler got to hold it, too. |
| {{main|Iterated function}} {{main|Infinite compositions of analytic functions}}
| |
| In mathematics, a '''superfunction''' is a nonstandard name for an [[iterated function]] for complexified continuous iteration index.
| |
| Roughly, for some function ''f'' and for some variable ''x'', the superfunction could be defined by the expression
| |
| :<math> S(z;x) =
| |
| \underbrace{f\Big(f\big(\dots f(x)\dots\big)\Big)}_{z \text{ evaluations of the function }f} .</math>
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| Then, ''S(z;x)'' can be interpreted as the superfunction of the function ''f(x)''.
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| Such a definition is valid only for a positive integer index ''z''. The variable ''x'' is often omitted.
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| Much study and many applications of superfunctions employ various ''extensions of these superfunctions to complex and continuous indices''; and the analysis of the existence, uniqueness and their evaluation. The [[Ackermann function]]s and [[tetration]] can be interpreted in terms of super-functions.
| |
| | |
| ==History==
| |
| Analysis of superfunctions arose from applications of the evaluation of fractional iterations of functions. Superfunctions and their inverses allow evaluation of not only the first negative power of a function (inverse function), but also of any real and even complex iterate of that function. Historically, an early function of this kind considered was <math>\sqrt{\exp}~</math>; the function <math>\sqrt{!~}~</math> has then been used as the logo of the Physics department of the [[Moscow State University]].
| |
| <ref name="logo">Logo of the physics department of Moscow State University. (In Russian);
| |
| [http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml]. V.P.Kandidov. About the time and myself. (In Russian) | |
| [http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf]. 250 anniversary of the Moscow State University. (In Russian) | |
| ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250! [http://nauka.relis.ru/11/0412/11412002.htm]</ref>
| |
| | |
| At that time, these investigators did not have computational access for the evaluation of such functions, but the function <math>\sqrt{\exp}</math> was luckier than <math>~\sqrt{!~}~~</math>: at the very least, the existence of the [[holomorphic function]]
| |
| <math>\varphi</math> such that <math>\varphi(\varphi(u))=\exp(u)</math> had been demonstrated in 1950 by [[Helmuth Kneser]].<ref name="kneser">
| |
| {{cite journal
| |
| |author=[[Helmuth Kneser|H.Kneser]]
| |
| |title=Reelle analytische L¨osungen der Gleichung <math>\varphi(\varphi(x)) = e^x </math> und verwandter Funktionalgleichungen
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| |journal=[[Journal fur die reine und angewandte Mathematik]]
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| |volume=187
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| |year=1950
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| |pages=56–67}}
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| </ref>
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| | |
| Relying on the elegant functional conjugacy theory of [[Schröder's equation]],<ref name="schr">{{cite journal |last=Schröder |first=Ernst |authorlink=Ernst Schröder |coauthors= |year=1870 |month= |title=Ueber iterirte Functionen|journal=[[Mathematische Annalen]] |volume=3 |issue= |pages=296–322 | doi=10.1007/BF01443992 |url=}}</ref> for his proof, Kneser had constructed the "superfunction" of the exponential map through the corresponding ''Abel function'' <math>\mathcal{X}</math>, satisfying the related [[Abel equation]]
| |
| : <math>\mathcal{X}(\exp(u))=\mathcal{X}(u)+1.\ </math>
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| so that <math>\mathcal{X}(S(z;u))=\mathcal{X}(u)+z\ </math>. The inverse function Kneser found,
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| :<math>S(z;u)=\mathcal{X}^{-1}(z+\mathcal{X}(u))</math>
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| is an [[entire function|entire]] super-exponential, although it is not real on the real axis; it cannot be interpreted as [[tetration]]al, because the condition <math>S(0;x)=x</math> cannot be realized for the entire super-exponential. The [[real function|real]] <math>\sqrt{\exp}</math> can be constructed with the [[tetration]]al (which is also a superexponential); while the real <math>\sqrt{!~}~</math> can be constructed with the [[superfactorial]].
| |
| | |
| ==Extensions==
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| The recurrence formula of the above preamble can be written as | |
| :<math>S(z + 1;x)=f(S(z;x)) ~~~~~~~~ \forall z\in \mathbb{N} : z>0</math>
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| :<math>S(1)=f(x).</math>
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| Instead of the last equation, one could write the identity function,
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| :<math>S(0)=x~,</math>
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| and extend the range of definition of the superfunction ''S'' to the non-negative integers. Then, one may posit
| |
| :<math>S(-1)=f^{-1}(x), </math>
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| and extend the range of validity to the integer values larger than −2.
| |
| | |
| The following extension, for example,
| |
| | |
| :<math>S(-2)=f^{-2}(x)</math>
| |
| | |
| is not trivial, because the inverse function may happen to be not defined for some values of <math>x</math>.
| |
| In particular, [[tetration]] can be interpreted as super-function of exponential for some real base <math>b</math>; in this case,
| |
| :<math>f=\exp_{b}.</math>
| |
| Then, at ''x''=1,
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| :<math>S(-1) = \log_b 1 = 0, </math>
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| but
| |
| :<math>S(-2) = \log_b 0 </math>
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| is not defined.
| |
| | |
| For extension to non-integer values of the argument, the superfunction should be defined in a different way.
| |
| | |
| For complex numbers <math>~a~</math> and <math>~b~</math>, such that <math>~a~</math> belongs to some connected domain <math>D\subseteq \mathbb{C}</math>,
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| the superfunction (from <math>a</math> to <math>b</math>) of a [[holomorphic function]] ''f'' on the domain <math>D</math> is
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| function <math> S </math>, [[holomorphic]] on domain <math>D</math>, such that
| |
| :<math>S(z\!+\!1)=f(S(z)) ~ \forall z\in D : z\!+\!1 \in D\ </math>
| |
| :<math>S(a)=b.\ </math>
| |
| | |
| ==Uniqueness==
| |
| In general, the superfunction is not unique.
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| For a given base function <math>f</math>, from a given <math>(a\mapsto d)</math> superfunction <math>S</math>, another <math>(a\mapsto d)</math> superfunction <math>G</math> could be constructed as
| |
| : <math>G(z)=S(z+\mu(z))\ </math>
| |
| where <math>\mu</math> is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that <math> \mu(a)=0 </math>.
| |
| | |
| The modified super-function may have a narrower range of holomorphy. | |
| The variety of possible super-functions is especially large in the limiting case, when the width of the range of holomorphy becomes zero; in this case, one deals with real-analytic superfunctions.<ref name="walker">
| |
| {{cite journal
| |
| |author=P.Walker
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| |title=Infinitely differentiable generalized logarithmic and exponential functions
| |
| |journal= [[Mathematics of computation]]
| |
| |volume=196
| |
| |year=1991
| |
| |pages=723–733
| |
| |jstor=2938713
| |
| }}
| |
| </ref>
| |
| | |
| If the range of holomorphy required is large enough, then, the super-function is expected to be unique,
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| at least in some specific base functions <math>H</math>. In particular, the <math>(C, 0\mapsto 1)</math> super-function of
| |
| <math>\exp_b</math>, for <math>b>1</math>, is called [[tetration]] and is believed to be unique, at least for
| |
| <math>C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \}</math>; for the case <math> b>\exp(1/\mathrm{e})</math>,<ref name="kouznetsov">
| |
| {{cite journal
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| |author=D.Kouznetsov.
| |
| |title=Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.
| |
| |journal=[[Mathematics of Computation]]
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| |year=2009
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| |volume=78
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| |pages=1647–1670
| |
| |url= http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
| |
| |doi=10.1090/S0025-5718-09-02188-7
| |
| }} preprint: [http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf PDF]</ref>
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| but up to 2009, the uniqueness was more [[conjecture]] than a theorem with a formal mathematical proof.
| |
| | |
| ==Examples==
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| This short collection of elementary superfunctions is illustrated in.<ref name="superfactorial">
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| D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. [[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12. (Preprint ILS UEC, 2009:
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| [http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf] )
| |
| </ref> Some superfunctions can be expressed through elementary functions;
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| they are used without mention that they are superfunctions.
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| For example, for the transfer function "++", which means unity increment,
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| the superfunction is just addition of a constant.
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| | |
| ===Addition===
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| Chose a [[complex number]] <math>c</math> and define the function <math>\mathrm{add}_c</math> as
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| <math>\mathrm{add}_c(x)=c +x, ~~~ ~ \forall x \in \mathbb{C}</math>. Further define the function <math>\mathrm{mul_c}</math> as
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| <math>\mathrm{mul_c}(x)=c\cdot x,~~~ ~ \forall x \in \mathbb{C}</math>.
| |
| | |
| Then, the function <math>S(z;x)=x+\mathrm{mul_c}(z)</math> is the superfunction (0 to ''c'')
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| of the function <math>~\mathrm{add_c}~</math> on '''''C'''''.
| |
| | |
| ===Multiplication===
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| Exponentiation <math>\exp_c</math> is superfunction (from 1 to <math>c</math>) of function <math>\mathrm{mul}_c </math>.
| |
| | |
| ===Quadratic polynomials===
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| The examples but the last one, below, are essentially from Schröder's pioneering 1870 paper.<ref name="schr"/>
| |
| | |
| Let <math>f(x)=2 x^2-1</math>.
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| Then,
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| ::<math>S(z;x)=\cos( 2^z \arccos (x)) </math>
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| is a <math>(\mathbb{C},~ 0\! \rightarrow\! 1)</math> superfunction (iteration orbit) of ''f''. | |
| | |
| Indeed,
| |
| :<math> S(z+1;x)=\cos(2 \cdot 2^z \arccos (x) )=2\cos( 2^z \arccos (x))^2 -1 =f(S(z;x))\ </math>
| |
| and <math>S(0;x)=x ~.\ </math> | |
| | |
| In this case, the superfunction <math>S</math> is periodic, with period
| |
| <math>T=\frac{2\pi}{\ln(2)} i\approx 9.0647202836543876194 \!~i </math>;
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| and the superfunction approaches unity in the negative direction of the real axis, | |
| :<math> \lim_{z\rightarrow -\infty} S(z)=1.\ </math>
| |
| | |
| ===Algebraic function===
| |
| Similarly,
| |
| :<math>f(x)=2x \sqrt{1-x^2}</math>
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| has an iteration orbit
| |
| :<math>S(z;x)=\sin( 2^z\arcsin (x)).</math>
| |
| | |
| ===Rational function===
| |
| In general, the transfer (step) function ''f(x)'' needs not be an [[entire function]]. An example involving a [[meromorphic function]] ''f'' reads,
| |
| :<math>f(x)=\frac{2x}{1-x^2} ~~~~~ \forall x\in D~</math>; <math>~ D=\mathbb{C} \backslash \{-1,1\}.</math> | |
| Its iteration orbit (superfunction) is
| |
| :<math> S(z;x)=\tan( 2^z \arctan(x))</math>
| |
| on '''''C''''', the set of complex numbers except for the singularities of the function ''S''.
| |
| To see this, recall the double angle trigonometric formula
| |
| :<math>\tan(2 \alpha)=\frac{2 \tan(\alpha)}{1-\tan(\alpha)^2}~~
| |
| \forall \alpha \in \mathbb{C} \backslash \{\alpha\in \mathbb{C} : \cos(\alpha)=0 || \sin(\alpha)=\pm \cos(\alpha) \}.
| |
| </math>
| |
| | |
| ===Exponentiation===
| |
| Let
| |
| <math>b>1</math>,
| |
| <math>f(u)= \exp_b(u)</math>,
| |
| <math> C= \{ z \in \mathbb{C} : \Re(u)>-2 \}</math>.
| |
| The [[tetration]] <math> \mathrm{tet}_b </math> is then a <math>(C,~ 0\! \rightarrow\! 1)</math> superfunction of <math>\exp_b</math>.
| |
| | |
| ==Abel function==
| |
| The inverse of a superfunction for a suitable argument ''x'' can be interpreted as the [[Abel function]], the solution of the [[Abel equation]],
| |
| : <math>\mathcal{X}(\exp(u))=\mathcal{X}(u)+1.\ </math>
| |
| and hence
| |
| : <math>\mathcal{X}(S(z;u))=\mathcal{X}(u)+z.\ </math>
| |
| The inverse function when defined, is
| |
| :<math>S(z;u)=\mathcal{X}^{-1}(z+\mathcal{X}(u)),</math>
| |
| for suitable domains and ranges, when they exist. The recursive property of ''S'' is then self-evident.
| |
|
| |
| The figure at left shows an example of transition from
| |
| <math>\exp^{1}\!=\!\exp </math> to
| |
| <math>\exp^{\!-1}\!=\!\ln </math>. | |
| The iterated function <math>\exp^z</math> versus real argument is plotted for
| |
| <math>z=2,1,0.9, 0.5, 0.1, -0.1,-0.5, -0.9, -1,-2</math>. The [[tetration]]al and ArcTetrational were used as superfunction
| |
| <math>S</math> and Abel function <math>A</math> of the exponential.
| |
| The figure at right shows these functions in the complex plane. | |
| At non-negative integer number of iteration, the iterated exponential is an [[entire function]]; at non-integer values, it has two [[branch points]], thich correspond to the [[Fixed point (mathematics)|fixed point]] <math>L</math> and
| |
| <math>L^*</math> of natural logarithm. At <math>z\!\ge\! 0</math>, function <math>\exp^z ~(x)</math> remains [[holomorphic function|holomorphic]] at least in the strip <math>|\Im(z)|<\Im(L)\approx 1.3 </math> along the real axis.
| |
| | |
| ==Applications of superfunctions and Abel functions==
| |
| | |
| Superfunctions, usually the [[tetration|superexponential]]s, are proposed as a fast-growing function for an
| |
| upgrade of the [[floating point]] representation of numbers in computers. Such an upgrade would greatly extend the
| |
| range of huge numbers which are still distinguishable from infinity.
| |
| | |
| Other applications refer to the calculation of fractional iterates
| |
| (or fractional power) of a function. Any holomorphic function can be declared as a "transfer function", then its superfunctions and
| |
| corresponding Abel functions can be considered.
| |
| | |
| ===Nonlinear optics===
| |
| In the investigation of the nonlinear response of optical materials, the sample is supposed to be optically thin, in such a way, that the intensity of the light does not change much as it goes through. Then one can consider, for example, the absorption as function of the intensity. However, at small variation of the intensity in the sample, the precision of measurement of the absorption as function of intensity is not good. The reconstruction of the superfunction from the transfer function allows to work with relatively thick samples, improving the precision of measurements. In particular, the transfer function of the similar sample, which is half thiner, could be interpreted as the square root (i.e. half-iteration) of the transfer function of the initial sample.
| |
| | |
| Similar example is suggested for a nonlinear optical fiber.<ref name="kouznetsov">
| |
| {{cite journal
| |
| |author=D.Kouznetsov.
| |
| |title=Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.
| |
| |journal=[[Mathematics of Computation]]
| |
| |year=2009
| |
| |volume=78
| |
| |pages=1647–1670
| |
| |url= http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
| |
| |preprint=[http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf]
| |
| | doi=10.1090/S0025-5718-09-02188-7
| |
| }}</ref>
| |
| | |
| ===Nonlinear acoustics===
| |
| It may make sense to characterize the nonlinearities in the attenuation of shock waves in a homogeneous tube. This could find an application in some advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves without to disturb the flux of the gas. Again, the analysis of the nonlinear response, i.e. the transfer function, may be boosted with the superfunction.
| |
| | |
| ===Evaporation and condensation===
| |
| In analysis of condensation, the growth (or vaporization) of a small drop of liquid can be considered,
| |
| as it diffuses down through a tube with some uniform concentration of vapor.
| |
| In the first approximation, at fixed concentration of the vapor,
| |
| the mass of the drop at the output end can be interpreted as the
| |
| Transfer Function of the input mass.
| |
| The square root of this Transfer Function will characterize the tube of half length.
| |
| | |
| ===Snow avalanche===
| |
| The mass of a snowball that rolls down a hill can be considered as a function of the path it has already passed. At fixed length of this path
| |
| (that can be determined by the altitude of the hill) this mass can be considered also as a Transfer Function of the input mass. The mass of the snowball could be measured at the top of the hill and at the bottom, giving the Transfer Function; then, the mass of the snowball, as a function of the length it passed, is a superfunction.
| |
| | |
| ===Operational element===
| |
| If one needs to build-up an operational element with some given transfer function <math>H</math>,
| |
| and wants to realize it as a sequential connection of a couple of identical operational elements, then, each of these two elements should have transfer function
| |
| <math> h=\sqrt{H}</math>. Such a function can be evaluated through the superfunction and the Abel function of the transfer function <math>H</math>.
| |
| | |
| The operational element may have any origin: it can be realized as an electronic microchip,
| |
| or a mechanical couple of curvilinear grains, or some asymmetric U-tube filled with different liquids, and so on.
| |
| | |
| ==References==
| |
| {{Citizendium}}
| |
| <references/>
| |
| | |
| [[Category:Functions and mappings]]
| |
| [[Category:Functional equations]]
| |