Fixed point: Difference between revisions

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imported>Dmitrii Kouznetsov
imported>Dmitrii Kouznetsov
Line 15: Line 15:
[[Exponential]] if fixed point or [[operator of differentiation]] D,
[[Exponential]] if fixed point or [[operator of differentiation]] D,
because
because
<math>{\rm D} \exp(x) =\exp'(x)=\exp(x)</math>
<math>{\rm D}~ \exp(x) =\exp'(x)=\exp(x)</math>


The [[Gaussian exponential]]
The [[Gaussian exponential]]
:(2) <math>L(x)=\exp(-x^2/2)</math>, <math>x \in</math>[[real number|reals]]  
:(2) <math>L(x)=\exp(-x^2/2)</math> , <math>~x \in</math>[[real number|reals]]  
is fixed point of the [[Fourier operator]], defined with its action on a function <math>g</math>:
is fixed point of the [[Fourier operator]], defined with its action on a function <math>g</math>:
:(3) <math>F(g)(p)=\frac{1}{\sqrt{2\pi}\int_{-\infty}^{\infty}
:(3) <math>F(g)(p)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}
g(x)\exp(-{\rm i}px) {\rm d}p </math>
g(x)\exp(-{\rm i}px) {\rm d}p </math>
in general, functors have no need to be linear, so, there is no [[associativity]]
in general, functors have no need to be linear, so, there is no [[associativity]]

Revision as of 02:46, 31 May 2008

Fixed point of a functor is solution of equation

(1)

Simple examples

Elementary functions

In particular, functor can be elementaty function. For example, 0 and 1 are fixed points of function sqrt, because and .

In similar way, 0 is fixed point of sine function, because .

Operators

Functor in the equation (1) can be a linear operator. In this case, the fixed point of functor is its eigenfunction with eigenvalue equal to unity.

Exponential if fixed point or operator of differentiation D, because

The Gaussian exponential

(2) , reals

is fixed point of the Fourier operator, defined with its action on a function :

(3)

in general, functors have no need to be linear, so, there is no associativity at application of several functiors in row, and parenthesis are necessary in the left hand side of eapression (3). [1]

==Fixed points of exponential and fixed points of logarithm

Notes

  1. Note that that there is certain ambiguity in commonly uused writing of mathematical formulas, omiting sign * of multiplication; in equaiton (3), expression does not mean that ; it means that result of action of operator on function , whith is function, is evaluated at arcument .