Fixed point: Difference between revisions

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imported>Dmitrii Kouznetsov
imported>Dmitrii Kouznetsov
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{{under construction}}
'''Fixed point''' <math> L </math> of a [[functor]] <math> F </math> is solution of equation
'''Fixed point''' <math> L </math> of a [[functor]] <math> F </math> is solution of equation
:(1) <math>F[L]=L</math>
:(1) <math>F[L]=L</math>
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<ref name="ambiguity">
<ref name="ambiguity">
Note that that there is certain [[ambiguity]] in commonly used writing of mathematical formulas, omiting sign * of multiplication; in equaiton (3), expression  
Note that that there is certain [[ambiguity]] in commonly used writing of mathematical formulas, omiting sign * of multiplication; in equaiton (3), expression  
<math>F(g)(p)</math> does not mean that <math>F(g)<math> is multiplied to value of <math>p</math>; it means that result <math>F(g)(p)</math> of action of operator <math>F </math> on function <math>g</math>, whith is function, is [[evaluate]]d at arcument <math>p</math>.
<math>F(g)(p)</math> does not mean that <math>F(g)</math> is multiplied to value of <math>p</math>; it means that result <math>F(g)(p)</math> of action of operator <math>F </math> on function <math>g</math>, whith is function, is [[evaluate]]d at arcument <math>p</math>.
</ref>
</ref>



Revision as of 02:50, 31 May 2008

Template:Under construction

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 functors 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 used writing of mathematical formulas, omiting sign * of multiplication; in equaiton (3), expression does not mean that is multiplied to value of ; it means that result of action of operator on function , whith is function, is evaluated at arcument .