aarjaneiro/planetmath huggingface dataset:
Alexander trick

Alexander trick

Want to extend a

homeomorphism

of the circle

S

1

to the whole disk

D

2

?

Let

f

:

S

1

→

S

1

be a homeomorphism. Then the formula

F

⁢

(

x

)

=

||

x

||

⁢

f

⁢

(

x

/

||

x

||

)

allows you to define a map

F

:

D

2

→

D

2

which extends

f

, for if

x

∈

S

1

⊂

D

2

then

||

x

||

=

1

and

F

⁢

(

x

)

=

1

⋅

f

⁢

(

x

/

1

)

=

f

⁢

(

x

)

. Clearly this map is

continuous

, save (maybe) the origin, since this formula is undefined there. Nevertheless this is removable.

To check

continuity

at the origin use:

“A map

f

is continuous at a point

p

if and only if for each sequence

x

n

→

p

,

f

⁢

(

x

n

)

→

f

⁢

(

p

)

”

.

So take a sequence

u

n

∈

D

2

such that

u

n

→

0

(i.e. which tends to the origin). Then

F

⁢

(

u

n

)

=

||

u

n

||

⁢

f

⁢

(

u

n

/

||

u

n

||

)

and since

f

⁢

(

u

n

/

||

u

n

||

)

≠

0

, hence

||

u

n

||

→

0

implies

F

⁢

(

u

n

)

→

0

, that is

F

is also continuous at the origin.

The same method works for

f

-

1

.

In the same vein one can extend homeomorphisms

S

n

→

S

n

to

D

n

+

1

→

D

n

+

1

.

Title

Alexander trick

Canonical name

AlexanderTrick

Date of creation

2013-03-22 15:53:38

Last modified on

2013-03-22 15:53:38

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

7

Author

juanman (12619)

Entry type

Definition

Classification

msc 37E30

Classification

msc 57S05

Related topic

Homeomorphism

Generated on Fri Feb 9 20:47:04 2018 by

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Feigenbaum constant

Feigenbaum constant

\PMlinkescapephrase

generated by

The

Feigenbaum delta constant

has the value

δ

=

4.669201609102990671853203820466

⁢

…

It governs the structure and behavior of many types of

dynamical systems

.
It was discovered in the 1970s by

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Feigenbaum.html

Mitchell Feigenbaum,
while studying the

logistic map

y

′

=

μ

⋅

y

⁢

(

1

-

y

)

,

which produces the Feigenbaum tree:

Generated by GNU Octave and GNUPlot.

If the

bifurcations

in this tree (first few shown as dotted blue lines)
are at points

b

1

,

b

2

,

b

3

,

…

, then

lim

n

→

∞

⁡

b

n

-

b

n

-

1

b

n

+

1

-

b

n

=

δ

.

That is, the ratio of the intervals between the bifurcation points
approaches Feigenbaum’s constant.

However, this is only the beginning.
Feigenbaum discovered that this constant
arose in

any

dynamical system
that approaches

chaotic behavior

via period-doubling bifurcation,
and has a single quadratic maximum.
So in some sense, Feigenbaum’s constant
is a universal constant of chaos theory.

Feigenbaum’s constant appears in problems of fluid-flow turbulence,
electronic oscillators, chemical reactions, and even the

Mandelbrot set

(the “budding” of the Mandelbrot set along the negative real axis
occurs at intervals determined by Feigenbaum’s constant).

References

1

http://www.research.att.com/ njas/sequences/A006890

A006890, “Decimal expansion of Feigenbaum bifurcation velocity”, in the

On-Line Encyclopedia of Integer Sequences

(

http://planetmath.org/OnLineEncyclopediaOfIntegerSequences

)

2

“Bifurcations”:

http://mcasco.com/bifurcat.html

http://mcasco.com/bifurcat.html

Title

Feigenbaum constant

Canonical

name

FeigenbaumConstant

Date of creation

2013-03-22 12:34:25

Last modified on

2013-03-22 12:34:25

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 37G15

Synonym

Feigenbaum delta constant

Synonym

Feigenbaum bifurcation velocity constant

Synonym

Feigenbaum bifurcation velocity

Synonym

Feigenbaum number

Synonym

Feigenbaum’s number

Synonym

Feigenbaum’s constant

Generated on Fri Feb 9 20:47:06 2018 by

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Feigenbaum fractal

Feigenbaum fractal

A

Feigenbaum fractal

is any

bifurcation

fractal produced by a period-doubling

cascade

. The “canonical” Feigenbaum fractal is produced by the

logistic map

(a simple population model),

y

′

=

μ

⋅

y

⁢

(

1

-

y

)

where

μ

is varied smoothly along one

dimension

. The logistic iteration either terminates in a cycle (set of repeating values) or behaves chaotically. If one plots the points of this cycle versus the

μ

-value, a graph like the following is produced:

Note the distinct bifurcation (branching) points and the

chaotic behavior

as

μ

increases.

Many other iterations will generate this same type of plot, for example the iteration

p

′

=

r

⋅

sin

⁡

(

π

⋅

p

)

One of the most amazing things about this class of fractals is that the bifurcation

intervals

are always described by Feigenbaum’s constant.

Octave/Matlab Code to generate the above image is available

\PMlinktofile

hereoctave_feigen.zip.

References.

•

“Quadratic Iteration, bifurcation, and chaos”:

http://mathforum.org/advanced/robertd/bifurcation.html

http://mathforum.org/advanced/robertd/bifurcation.html

•

“Bifurcation”:

http://spanky.triumf.ca/www/fractint/bif_type.html

http://spanky.triumf.ca/www/fractint/bif_type.html

•

“Feigenbaum’s Constant”:

http://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html

http://fractals.iuta.u-bordeaux.fr/sci-faq/feigenbaum.html

Title

Feigenbaum fractal

Canonical name

FeigenbaumFractal

Date of creation

2013-03-22 12:34:18

Last modified on

2013-03-22 12:34:18

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 37G15

Synonym

Feigenbaum tree

Defines

logistic map

Generated on Fri Feb 9 20:47:08 2018 by

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bifurcation problem with symmetry group

bifurcation problem with symmetry group

Let

Γ

be a Lie group acting on a

vector space

V

and let the

system of ordinary differential equations

𝐱

˙

+

g

⁢

(

𝐱

,

λ

)

=

0

where

g

:

ℝ

n

×

ℝ

→

ℝ

n

is smooth. Then

g

is called a

bifurcation problem with symmetry group

Γ

if

g

∈

ℰ

→

x

,

λ

⁢

(

Γ

)

(where

E

→

⁢

(

Γ

)

is the space of

Γ

-

equivariant

germs, at the origin, of

C

∞

mappings of

V

into

V

)

satisfying

g

⁢

(

0

,

0

)

=

0

and

(

d

⁢

g

)

0

,

0

=

0

where

(

d

⁢

g

)

0

,

0

denotes the

Jacobian Matrix

evaluated at

(

0

,

0

)

.

[

GSS

]

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in

Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

Title

bifurcation problem with symmetry group

Canonical name

BifurcationProblemWithSymmetryGroup

Date of creation

2013-03-22 13:53:36

Last modified on

2013-03-22 13:53:36

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Definition

Classification

msc 37G40

Generated on Fri Feb 9 20:47:10 2018 by

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Poénaru (1976) theorem

Poénaru (1976) theorem

Let

Γ

be a

compact Lie group

and let

g

1

,

…

,

g

r

generate the module

𝒫

→

⁢

(

Γ

)

(the space of

Γ

-

equivariant

polynomial

mappings)

of

Γ

-equivariant polynomials over the ring

𝒫

⁢

(

Γ

)

(the ring of

Γ

-

invariant polynomial

)

. Then

g

1

,

…

,

g

r

generate the module

ℰ

→

⁢

(

Γ

)

(the space of

Γ

-equivariant germs at the origin of

C

∞

mappings)

over the ring

ℰ

⁢

(

Γ

)

(the ring of

Γ

-invariant germs)

.

[

GSS

]

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

PV

Poénaru, V.:Singularités

C

∞

en Présence de Symétrie. Lecture Notes in Mathematics 510, Springer-Verlag, Berlin, 1976.

Title

Poénaru (1976) theorem

Canonical name

Poenaru1976Theorem

Date of creation

2013-03-22 13:40:59

Last modified on

2013-03-22 13:40:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 37G40

Generated on Fri Feb 9 20:47:11 2018 by

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trace formula

trace formula

Let

Γ

be a compact Lie group acting on

V

and let

Σ

⊂

Γ

be a Lie subgroup. Then

dim

⁡

Fix

⁡

(

Σ

)

=

∫

σ

∈

Σ

trace

⁡

(

σ

)

where

∫

denotes the

normalized Haar integral

on

Σ

and

Fix

⁡

(

Σ

)

is the

fixed-point subspace

of

Σ

.

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in

Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

Title

trace formula

Canonical name

TraceFormula

Date of creation

2013-03-22 13:44:22

Last modified on

2013-03-22 13:44:22

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 37G40

Generated on Fri Feb 9 20:47:13 2018 by

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chaotic dynamical system

chaotic dynamical system

As Strogatz says in reference [1], “No

definition

of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition”.

Chaos is the aperiodic long-term in a deterministic system that exhibits sensitive dependence on

initial conditions

.

Aperiodic long-term means that there are

trajectories

which do not settle down to

fixed points

, periodic orbits (

http://planetmath.org/Orbit

), or quasiperiodic as

t

→

∞

. For the purposes of this definition, a trajectory which approaches a limit of

∞

as

t

→

∞

should be considered to have a fixed point at

∞

.

Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast; i.e. (

http://planetmath.org/Ie

), the system has a

positive

Liapunov exponent.

Strogatz notes that he favors additional constraints on the aperiodic long-term , but leaves open (

http://planetmath.org/OpenQuestion

) what form they may take. He suggests two alternatives to fulfill this:

1.

Requiring that there exists an open set of initial conditions having aperiodic trajectories, or

2.

If one picks a random initial condition

x

⁢

(

0

)

then there must be a nonzero chance of the associated trajectory

x

⁢

(

t

)

being aperiodic.

0.1

Further reading

1.

B. Codenotti and Luciano Margara. Chaos in Mathematics, Physics, and Computer Science:

Similarities

and Dissimilarities. http://pespmc1.vub.ac.be/Einmag_Abstr/BCodenotti.html

0.2

References

1.

Steven H. Strogatz, ”Nonlinear Dynamics and Chaos”. Westview Press, 1994.

Title

chaotic dynamical system

Canonical

name

ChaoticDynamicalSystem

Date of creation

2013-03-22 13:05:26

Last modified on

2013-03-22 13:05:26

Owner

bshanks (153)

Last modified by

bshanks (153)

Numerical id

15

Author

bshanks (153)

Entry type

Definition

Classification

msc 37G99

Synonym

chaotic system

Synonym

deterministic chaotic system

Synonym

chaotic behavior

Related topic

DynamicalSystem

Related topic

SystemDefinitions

Generated on Fri Feb 9 20:47:14 2018 by

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Oseledets multiplicative ergodic theorem

Oseledets multiplicative ergodic theorem

Oseledets multiplicative ergodic theorem

, or Oseledets decomposition, considerably extends the results of Furstenberg-Kesten

theorem

, under the same conditions.

Consider

μ

a

probability measure

, and

f

:

M

→

M

a

measure preserving

dynamical system

. Consider

A

:

M

→

G

⁢

L

⁢

(

d

,

𝐑

)

, a measurable

transformation

, where GL(d,

R

) is the space of

invertible

square matrices of size

d

.
Consider the multiplicative cocycle

(

ϕ

n

⁢

(

x

)

)

n

defined by the transformation

A

, and assume

log

+

⁡

|

|

A

|

|

and

log

+

⁡

|

|

A

-

1

|

|

are integrable.

Then,

μ

almost everywhere

x

∈

M

, one can find a

natural number

k

=

k

⁢

(

x

)

and real numbers

λ

1

⁢

(

x

)

>

⋯

>

λ

k

⁢

(

x

)

and a

filtration

𝐑

d

=

V

x

1

>

⋯

>

V

x

k

>

V

x

k

+

1

=

{

0

}

such that, for

μ

almost everywhere and for all

i

∈

{

1

,

…

,

k

}

1.

k

⁢

(

f

⁢

(

x

)

)

=

k

⁢

(

x

)

and

λ

i

⁢

(

f

⁢

(

x

)

)

=

λ

i

⁢

(

x

)

and

A

⁢

(

x

)

⋅

V

x

i

=

V

f

⁢

(

x

)

i

;

2.

lim

n

⁡

1

n

⁢

log

⁡

|

|

ϕ

n

⁢

(

x

)

⁢

v

|

|

=

λ

i

⁢

(

x

)

for all

v

∈

V

x

i

\

V

x

i

+

1

;

3.

lim

n

⁡

1

n

⁢

log

⁡

|

det

⁡

ϕ

n

⁢

(

x

)

|

=

∑

i

=

1

k

d

i

⁢

(

x

)

⁢

λ

i

⁢

(

x

)

where

d

i

⁢

(

x

)

=

dim

⁡

V

x

i

-

dim

⁡

V

x

i

+

1

Furthermore, the numbers

k

i

⁢

(

x

)

and the

subspaces

V

x

i

depend measurably on the point

x

.

The numbers

λ

i

⁢

(

x

)

are called

Lyapunov exponents

of

A

relatively to

f

at the point

x

. Each number

d

i

⁢

(

x

)

is called the

multiplicity

of the Lyapunov exponent

λ

i

⁢

(

x

)

. We also have that

λ

1

=

λ

max

and

λ

k

=

λ

min

, where

λ

m

⁢

a

⁢

x

and

λ

min

are as given by Furstenberg-Kesten theorem.

Title

Oseledets multiplicative ergodic theorem

Canonical name

OseledetsMultiplicativeErgodicTheorem

Date of creation

2014-03-26 14:21:35

Last modified on

2014-03-26 14:21:35

Owner

Filipe (28191)

Last modified by

Filipe (28191)

Numerical id

6

Author

Filipe (28191)

Entry type

Theorem

Classification

msc 37H15

Synonym

Oseledets decomposition

Related topic

Lyapunov exponent

Related topic

Furstenberg-Kesten Theorem

Generated on Fri Feb 9 20:47:17 2018 by

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divided difference

divided difference

Let

f

be a real (or complex)

function

. Given distinct real (or complex) numbers

x

0

,

x

1

,

x

2

,

…

, the

divided differences

of

f

are defined recursively as follows:

Δ

1

⁢

f

⁢

[

x

0

,

x

1

]

=

f

⁢

(

x

1

)

-

f

⁢

(

x

0

)

x

1

-

x

0

Δ

n

+

1

⁢

f

⁢

[

x

0

,

x

1

,

…

,

x

n

+

1

]

=

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

,

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

2

,

…

,

x

n

+

1

]

x

1

-

x

0

It is also convenient to define the zeroth divided difference of

f

to be

f

itself:

Δ

0

⁢

f

⁢

[

x

0

]

=

f

⁢

[

x

0

]

Some important

properties

of divided differences are:

1.

Divided differences are

invariant

under

permutations

of

x

0

,

x

1

,

x

2

,

…

2.

If

f

is a polynomial of order

m

and

m

<

n

, then the

n

-th divided differences of

f

vanish identically

3.

If

f

is a polynomial of order

m

+

n

, then

Δ

n

⁢

(

x

,

x

1

,

…

,

x

n

)

is a polynomial in

x

of order

m

.

Divided differences are useful for interpolating functions when the values are given for unequally spaced values of the

argument

.

Becuse of the first property listed above, it does not matter with
respect to which two arguments we compute the divided difference
when we compute the

n

+

1

-st divided difference from the

n

-th
divided difference. For

instance

, when computing the

divided
difference table

for tabulated values of a function, a convenient
choice is the following:

Δ

n

+

1

⁢

f

⁢

[

x

0

,

x

1

,

…

,

x

n

+

1

]

=

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

,

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

1

,

…

,

x

n

]

x

n

+

1

-

x

0

Title

divided difference

Canonical

name

DividedDifference

Date of creation

2013-03-22 14:40:59

Last modified on

2013-03-22 14:40:59

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 39A70

Generated on Fri Feb 9 20:47:35 2018 by

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divided difference interpolation formula

divided difference interpolation formula

Newton’s

divided difference interpolation formula

is the analogue
of the Gregory-Newton and

Taylor series

for

divided differences

.

If

f

is a

real function

and

x

0

,

x

1

,

…

is a

sequence

of distinct
real numbers, then we have, for any

integer

n

>

0

,

f

⁢

(

x

)

=

f

⁢

(

x

0

)

+

(

x

-

x

0

)

⁢

Δ

⁢

f

⁢

(

x

0

,

x

1

)

+

⋯

+

(

x

-

x

0

)

⁢

⋯

⁢

(

x

-

x

n

-

1

)

⁢

Δ

n

⁢

f

⁢

(

x

0

,

…

⁢

x

n

)

+

R

where the

remainder

can be expressed either as

R

=

(

x

-

x

0

)

⁢

⋯

⁢

(

x

-

x

n

)

⁢

Δ

n

+

1

⁢

f

⁢

(

x

,

x

1

,

…

,

x

n

)

or as

R

=

1

(

n

+

1

)

!

⁢

(

x

-

x

0

)

⁢

⋯

⁢

(

x

-

x

n

)

⁢

f

(

n

+

1

)

⁢

(

η

)

where

η

lies between the smallest and the largest of

x

,

x

0

,

…

,

x

n

.

Remark

. If

f

is a

polynomial

of degree

n

, then

R

vanishes.

Title

divided difference interpolation formula

Canonical name

DividedDifferenceInterpolationFormula

Date of creation

2013-03-22 16:19:13

Last modified on

2013-03-22 16:19:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 39A70

Generated on Fri Feb 9 20:47:37 2018 by

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divided differences of powers

divided differences of powers

In this entry, we will prove the claims about

divided differences

of polynomials. Because the divided difference is a

linear
operator

, we can focus our attention on powers.

Theorem 1

.

If

f

⁢

(

x

)

=

x

n

and

m

≤

n

, then

Δ

m

⁢

f

⁢

[

x

0

,

⋯

⁢

x

m

]

=

∑

k

0

+

⋯

+

k

m

=

n

-

m

x

0

k

0

⁢

⋯

⁢

x

m

k

m

.

If

m

>

n

, then

Δ

m

⁢

f

⁢

[

x

0

,

⋯

⁢

x

m

]

=

0

.

Proof.

We proceed by induction. The formula is trivially true
when

m

=

0

. Assume that the formula is true for a
certain value of

m

. Then we have

Δ

m

+

1

⁢

f

⁢

[

x

0

,

⋯

⁢

x

m

+

1

]

=

Δ

m

⁢

f

⁢

[

x

1

,

x

2

⁢

⋯

⁢

x

m

]

-

Δ

m

⁢

f

⁢

[

x

0

,

x

2

⁢

⋯

⁢

x

m

]

x

1

-

x

0

=

∑

k

+

k

2

+

⋯

+

k

m

=

n

-

m

(

x

1

k

-

x

0

k

)

⁢

x

2

k

2

⁢

⋯

⁢

x

m

k

m

x

1

-

x

0

Using the identity for the sum of a geometric series,

x

1

k

-

x

0

k

x

1

-

x

0

=

∑

k

0

+

k

1

=

k

-

1

x

0

k

0

⁢

x

1

k

1

,

this becomes

Δ

m

+

1

⁢

f

⁢

[

x

0

,

⋯

⁢

x

m

+

1

]

=

∑

k

+

k

2

+

⋯

+

k

m

=

n

-

m

∑

k

0

+

k

1

=

k

-

1

x

0

k

0

⁢

x

1

k

1

⁢

x

2

k

2

⁢

⋯

⁢

x

m

+

1

k

m

+

1

=

∑

k

0

+

k

1

+

k

2

+

⋯

+

k

m

=

n

-

(

m

+

1

)

x

0

k

0

⁢

x

1

k

1

⁢

x

2

k

2

⁢

⋯

⁢

x

m

+

1

k

m

+

1

.

Note that when

k

=

0

, we have

x

1

k

-

x

0

k

=

0

, which
is consistent with the formula given above because, in that
case, there are no solutions to

k

1

+

k

2

=

k

, so the
sum is empty and, by convention, equals zero. Likewise,
when

n

=

m

, then the only solution to

k

0

+

⋯

+

k

m

=

0

is

k

0

=

⋯

=

k

m

=

0

, so the sum only consists of
one term,

x

0

0

⁢

⋯

⁢

x

m

0

=

1

so

Δ

n

⁢

f

⁢

[

x

0

,

⋯

⁢

x

n

]

=

1

, hence taking
further differences produces zero.
∎

Title

divided differences of powers

Canonical name

DividedDifferencesOfPowers

Date of creation

2013-03-22 16:47:59

Last modified on

2013-03-22 16:47:59

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 39A70

Generated on Fri Feb 9 20:47:43 2018 by

LaTeXML
example of divided difference interpolaton

example of divided difference interpolaton

To illustrate how one interpolates a

function

using

divided differences

, we will interpolate

sin

⁡

40

∘

from the sines of

0

∘

,

30

∘

,

45

∘

,

60

∘

, and

90

∘

. To keep from having too
many zeros in our numbers, we will actually
interpolate

sin

⁡

(

10

⁢

x

)

instead.

We begin by making a

divided difference table

:

0.0

0.0000

0.1667

3.0

0.500

-

0.00636

0.1381

-

0.001786

4.5

0.7071

-

0.01071

-

0.0001445

0.1060

-

0.000486

6.0

0.8660

-

0.01362

0.0447

9.0

1.0000

Reading off the top numbers from each column, we may
form the following divided difference series:

sin

⁡

(

10

⁢

x

)

=

0.1667

⁢

x

-

0.00636

⁢

x

⁢

(

x

-

3

)

-

0.001786

⁢

x

⁢

(

x

-

3

)

⁢

(

x

-

4.5

)

-

0.0001445

⁢

x

⁢

(

x

-

3

)

⁢

(

x

-

4.5

)

⁢

(

x

-

6

)

+

R

Substituting

0.4

for

x

, we obtain

0.6502

as
an approximate value for

sin

⁡

40

∘

. When
compared with the actual value of

0.6428

, this
is a reasonable approximation —it is correct
to

1

%

.

Title

example of divided difference interpolaton

Canonical name

ExampleOfDividedDifferenceInterpolaton

Date of creation

2013-03-22 16:49:19

Last modified on

2013-03-22 16:49:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Example

Classification

msc 39A70

Generated on Fri Feb 9 20:47:45 2018 by

LaTeXML
functional equation

functional equation

A

functional equation

is an equation whose unknowns are

functions

.

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

+

f

⁢

(

y

)

,

f

⁢

(

x

⋅

y

)

=

f

⁢

(

x

)

⋅

f

⁢

(

y

)

are examples of such
equations. The systematic study of these didn’t begin before the 1960’s,
although various mathematicians have been studying them before, including
Euler and Cauchy just to mention a few.

Functional equations appear many , for example,
the

gamma function

and Riemann’s

zeta function

both satisfy functional equations.

Title

functional equation

Canonical name

FunctionalEquation

Date of creation

2013-03-22 13:06:53

Last modified on

2013-03-22 13:06:53

Owner

jgade (861)

Last modified by

jgade (861)

Numerical id

9

Author

jgade (861)

Entry type

Definition

Classification

msc 39B05

Related topic

Equation

Generated on Fri Feb 9 20:48:13 2018 by

LaTeXML
proof of arithmetic-geometric-harmonic means inequality

proof of arithmetic-geometric-harmonic means inequality

We can use the

Jensen inequality

for an easy proof of the

arithmetic-geometric-harmonic means inequality

.

Let

x

1

,

…

,

x

n

>

0

; we shall first prove that

x

1

⋅

…

⋅

x

n

n

≤

x

1

+

…

+

x

n

n

.

Note that

log

is a

concave function

. Applying it to the

arithmetic mean

of

x

1

,

…

,

x

n

and using Jensen’s

inequality

, we see that

log

⁡

(

x

1

+

…

+

x

n

n

)

≥

log

⁡

(

x

1

)

+

…

+

log

⁡

(

x

n

)

n

=

log

⁡

(

x

1

⋅

…

⋅

x

n

)

n

=

log

⁡

x

1

⋅

…

⋅

x

n

n

.

Since

log

is also a

monotone function

, it follows that the arithmetic mean is at least as large as the

geometric mean

.

The proof that the geometric mean is at least as large as the

harmonic mean

is the usual one (see “

proof of arithmetic-geometric-harmonic means inequality

”).

Title

proof of arithmetic-geometric-harmonic means inequality

Canonical

name

ProofOfArithmeticgeometricharmonicMeansInequality

Date of creation

2013-03-22 12:43:07

Last modified on

2013-03-22 12:43:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Example

Classification

msc 39B62

Classification

msc 26D15

Related topic

ArithmeticGeometricMeansInequality

Related topic

ProofOfArithmeticGeometricMeansInequalityUsingLagrangeMultipliers

Generated on Fri Feb 9 20:48:15 2018 by

LaTeXML