aarjaneiro/planetmath huggingface dataset:
Hermitian dot product (finite fields)

Hermitian dot product (finite fields)

Let

q

be an

http://planetmath.org/node/4703

even

http://planetmath.org/node/438

prime power (in particular,

q

is a square) and

𝔽

q

the

finite field

with

q

elements. Then

𝔽

q

is a

subfield

of

𝔽

q

. The

k

¯

of an element

k

∈

𝔽

q

is defined by
the

q

-th power

Frobenius map

k

¯

:=

Frob

q

⁡

(

k

)

=

k

q

.

The has properties

similar

to the

complex conjugate

. Let

k

1

,

k

2

∈

𝔽

q

, then

1.

k

1

+

k

2

¯

=

k

1

¯

+

k

2

¯

,

2.

k

1

⁢

k

2

¯

=

k

1

¯

⁢

k

2

¯

,

3.

k

1

¯

¯

=

k

1

.

Properties 1 and 2 hold because the Frobenius map is a

http://planetmath.org/node/1011

homomorphism

.
Property 3 holds because of the

identity

k

q

=

k

which
holds for any

k

in any finite field with

q

elements.
See also

http://planetmath.org/node/2893

finite field

.

Now let

𝔽

q

n

be the

n

-dimensional

vector space

over

𝔽

q

, then the

Hermitian dot product

of two vectors

(

u

1

,

…

,

u

n

)

,

(

v

1

,

…

,

v

n

)

∈

𝔽

q

n

is

(

u

1

,

…

,

u

n

)

⋅

(

v

1

,

…

,

v

n

)

:=

∑

i

=

1

n

u

i

⁢

v

i

¯

.

Again, this kind of Hermitian dot product has properties similar to

Hermitian inner products

on

complex vector spaces

. Let

k

1

,

k

2

∈

𝔽

q

and

v

1

,

v

2

,

v

,

,

w

∈

𝔽

q

n

, then

1.

(

k

1

⁢

v

1

+

k

2

⁢

v

2

)

⋅

w

=

k

1

⁢

(

v

1

⋅

w

)

+

k

2

⁢

(

v

2

⋅

w

)

(linearity)

2.

v

⋅

w

=

w

⋅

v

¯

3.

v

⋅

v

∈

𝔽

q

Property 3 follows since

q

+

1

divides

q

-

1

(see

http://planetmath.org/node/2893

finite field

).

Title

Hermitian dot product (finite fields)

Canonical name

HermitianDotProductfiniteFields

Date of creation

2013-03-22 15:13:26

Last modified on

2013-03-22 15:13:26

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

7

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 11E39

Classification

msc 12E20

Synonym

Hermitian dot product

Related topic

FiniteField

Defines

conjugate

(finite fields)

Defines

conjugation

(finite fields)

Generated on Fri Feb 9 15:33:29 2018 by

LaTeXML
Hermitian function

Hermitian function

A

Hermitian function

is

complex

valued function whose

real part

is even and

imaginary part

is odd.

1

Definition

Let

f

⁢

(

x

)

be a complex valued function of a real variable

x

.

f

⁢

(

x

)

is a

Hermitian function

if

f

⁢

(

-

x

)

=

f

⁢

(

x

)

¯

for all

x

.

Title

Hermitian function

Canonical name

HermitianFunction

Date of creation

2013-03-22 17:56:02

Last modified on

2013-03-22 17:56:02

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

6

Author

stitch (17269)

Entry type

Definition

Classification

msc 11E39

Related topic

EvenoddFunction

Generated on Fri Feb 9 15:33:31 2018 by

LaTeXML
inner product

inner product

An

inner product

on a

vector space

V

over a field

K

(which must be either the field

ℝ

of

real numbers

or the field

ℂ

of

complex numbers

) is a

function

(

,

)

:

V

×

V

⟶

K

such that, for all

k

1

,

k

2

∈

K

and

𝐯

1

,

𝐯

2

,

𝐯

,

𝐰

∈

V

, the following properties hold:

1.

(

k

1

⁢

𝐯

1

+

k

2

⁢

𝐯

2

,

𝐰

)

=

k

1

⁢

(

𝐯

1

,

𝐰

)

+

k

2

⁢

(

𝐯

2

,

𝐰

)

(linearity

1

1

A small minority of authors impose linearity on the second

coordinate

instead of the first coordinate.

)

2.

(

𝐯

,

𝐰

)

=

(

𝐰

,

𝐯

)

¯

, where

¯

denotes

complex conjugation

(

conjugate

symmetry

)

3.

(

𝐯

,

𝐯

)

≥

0

, and

(

𝐯

,

𝐯

)

=

0

if and only if

𝐯

=

𝟎

(

positive definite

)

(Note: Rule 2 guarantees that

(

𝐯

,

𝐯

)

∈

ℝ

, so the

inequality

(

𝐯

,

𝐯

)

≥

0

in rule 3 makes sense even when

K

=

ℂ

.)

The standard example of an inner product is the

dot product

on

K

n

:

(

(

x

1

,

…

,

x

n

)

,

(

y

1

,

…

,

y

n

)

)

:=

∑

i

=

1

n

x

i

⁢

y

i

¯

Every

inner product space

is a

normed vector space

, with the norm being defined by

||

𝐯

||

:=

(

𝐯

,

𝐯

)

.

Title

inner product

Canonical name

InnerProduct

Date of creation

2013-03-22 12:13:39

Last modified on

2013-03-22 12:13:39

Owner

djao (24)

Last modified by

djao (24)

Numerical id

15

Author

djao (24)

Entry type

Definition

Classification

msc 11E39

Classification

msc 15A63

Synonym

Hermitian inner product

Related topic

InnerProductSpace

Related topic

HermitianForm

Related topic

EuclideanVectorSpace

Generated on Fri Feb 9 15:33:33 2018 by

LaTeXML
positive definite form

positive definite form

A

bilinear form

B

on a real or

complex vector space

V

is

positive definite

if

B

⁢

(

x

,

x

)

>

0

for all nonzero vectors

x

∈

V

. On the other hand, if

B

⁢

(

x

,

x

)

<

0

for all nonzero vectors

x

∈

V

, then we say

B

is

negative definite

. If

B

⁢

(

x

,

x

)

≥

0

for all vectors

x

∈

V

, then we say

B

is

nonnegative definite

. Likewise,
if

B

⁢

(

x

,

x

)

≤

0

for all vectors

x

∈

V

, then we say

B

is

nonpositive definite

.

A form which is neither positive definite nor negative definite is called

indefinite

.

Title

positive definite form

Canonical name

PositiveDefiniteForm

Date of creation

2013-03-22 12:25:50

Last modified on

2013-03-22 12:25:50

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 11E39

Classification

msc 15A63

Classification

msc 47A07

Synonym

positive definite

Synonym

negative definite form

Synonym

negative definite

Synonym

indefinite form

Synonym

indefinite

Synonym

nonnegative definite

Synonym

nonpositive definite

Generated on Fri Feb 9 15:33:34 2018 by

LaTeXML
semi-inner product

semi-inner product

0.0.1

Definition

Let

V

be a

vector space

over a field

𝕂

, where

𝕂

is

ℝ

or

ℂ

.

A

semi-inner product

on

V

is a function

⟨

⋅

,

⋅

⟩

:

V

×

V

⟶

𝕂

that the following conditions:

1.

⟨

λ

1

⁢

v

1

+

λ

2

⁢

v

2

,

w

⟩

=

λ

1

⁢

⟨

v

1

,

w

⟩

+

λ

2

⁢

⟨

v

2

,

w

⟩

for every

v

1

,

v

2

,

w

∈

V

and

λ

1

,

λ

2

∈

𝕂

.

2.

⟨

v

,

w

⟩

=

⟨

w

,

v

⟩

¯

for every

v

,

w

∈

V

, where the above means complex conjugation.

3.

⟨

v

,

v

⟩

≥

0

( semi definite).

Hence, a semi-inner product on a vector space is just like an

inner product

, but for which

⟨

v

,

v

⟩

can be zero ( if

v

≠

0

).

A

semi-inner product space

is just a vector space endowed with a semi-inner product.

0.0.2

Topology

Every semi-inner product space

V

can be given a

topology

associated with the semi-inner product. In fact, a

semi-norm

∥

⋅

∥

can be defined in

V

by

∥

v

∥

:=

⟨

v

,

v

⟩

0.0.3

Cauchy-Schwarz inequality

The

Cauchy-Schwarz inequality

is valid for semi-inner product spaces:

|

⟨

v

,

w

⟩

|

≤

⟨

v

,

v

⟩

⁢

⟨

w

,

w

⟩

0.0.4

Properties

Let

V

be a semi-inner product space and

W

:=

{

v

∈

V

:

⟨

v

,

v

⟩

=

0

}

. It is not difficult to see, using the Cauchy-Schwarz inequality, that

W

is a

vector subspace

.

The semi-inner product in

V

induces a well defined semi-inner product in the

quotient

(

http://planetmath.org/QuotientModule

)

V

/

W

which is, in fact, an inner product. Thus, the

V

/

W

is an

inner product space

.

Title

semi-inner product

Canonical

name

SemiinnerProduct

Date of creation

2013-03-22 17:47:10

Last modified on

2013-03-22 17:47:10

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Definition

Classification

msc 11E39

Classification

msc 15A63

Classification

msc 46C50

Synonym

positive semi-definite inner product

Synonym

semi inner product

Defines

semi-inner product space

Defines

Cauchy-Schwartz inequality for semi-inner products

Generated on Fri Feb 9 15:33:36 2018 by

LaTeXML
symmetric bilinear form

symmetric bilinear form

A

symmetric bilinear form

is a

bilinear form

B

which is

symmetric

in the two

coordinates

; that is,

B

⁢

(

x

,

y

)

=

B

⁢

(

y

,

x

)

for all vectors

x

and

y

.

Every

inner product

over a

real vector space

is a

positive definite

symmetric bilinear form.

Title

symmetric bilinear form

Canonical name

SymmetricBilinearForm

Date of creation

2013-03-22 12:25:45

Last modified on

2013-03-22 12:25:45

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 11E39

Classification

msc 15A63

Classification

msc 47A07

Synonym

symmetric form

Related topic

AntiSymmetric

Related topic

QuadraticForm

Related topic

SkewSymmetricBilinearForm

Generated on Fri Feb 9 15:33:38 2018 by

LaTeXML
orders and structure of classical groups

orders and structure of classical groups

Theorem 1

.

|

S

⁢

L

⁢

(

d

,

q

)

|

=

q

(

d

2

)

⁢

∏

i

=

2

d

(

q

i

-

1

)

,

|

P

⁢

S

⁢

L

⁢

(

d

,

q

)

|

=

|

S

⁢

L

⁢

(

d

,

q

)

|

(

d

,

q

-

1

)

,

|

G

⁢

L

⁢

(

d

,

q

)

|

=

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

,

|

P

⁢

G

⁢

L

⁢

(

d

,

q

)

|

=

|

G

⁢

L

⁢

(

d

,

q

)

|

q

-

1

,

|

Γ

⁢

L

⁢

(

d

,

q

)

|

=

(

q

-

1

)

⁢

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

,

|

P

⁢

Γ

⁢

L

⁢

(

d

,

q

)

|

=

|

Γ

⁢

L

⁢

(

d

,

q

)

|

q

-

1

.

A proof of this theorem can proceed in many

directions

which each explain more of the structure of the these groups so each approach is considered equally valid and presented here. We list them from most elementary to most theoretical.

1.

Elementary

linear algebra

proof (

http://planetmath.org/ElementaryProofOfOrders

)

2.

Proof from subgroups of

G

⁢

L

⁢

(

V

)

(

http://planetmath.org/TheoryFromOrdersOfClassicalGroups2

)

3.

Proof from Lie theory/Chevally groups. [PENDING.]

Title

orders and structure of classical groups

Canonical name

OrdersAndStructureOfClassicalGroups

Date of creation

2013-03-22 15:56:45

Last modified on

2013-03-22 15:56:45

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

10

Author

Algeboy (12884)

Entry type

Topic

Classification

msc 11E57

Classification

msc 05E15

Generated on Fri Feb 9 15:33:47 2018 by

LaTeXML
topic entry on axioms and foundations of homology and cohomology theories

topic entry on axioms and foundations of homology and cohomology theories

1

Axioms for Homology and Cohomology theories

1.

Axioms for

homology

theory and uniqueness theorems

2.

Cech types

3.

K

-theory

4.

Generalized cohomology

5.

Generalized (extraordinary) homology and

cohomology

theories

6.

Galois Cohomology

and Categorical Galois theories

7.

(Co)homology of commutative rings and

algebras

(e.g., Hochschild, André–Quillen, cyclic, dihedral, etc.)

8.

Bordism

and cobordism theories, formal group laws

9.

Homology with local coefficients,

equivariant cohomology

10.

Sheaf cohomology

11.

Cohomology in Noncommutative algebraic geometry

12.

Classifying spaces

for foliations; Gelfand-Fuks cohomology

13.

Intersection homology and cohomology

14.

Elliptic cohomology

15.

Equivariant homology and cohomology

16.

Homology and

homotopy

of

topological groups

and related structures

17.

Homotopy

Quantum Field Theories

and Axiomatic Quantum Field Theories

18.

Non-Abelian

Homological Algebra

19.

Grothendieck’s ‘Anabelian Geometry’

20.

Other homology theories–Your new

additions

References

1

Hatcher, A. 2001.

http://www.math.cornell.edu/ hatcher/AT/AT.pdf

Algebraic Topology

(textbook on line)., Cambridge University Press; Cambridge, UK., 405 pages.

2

Ronald Brown:

Topology and Groupoids

, BookSurge LLC (2006).

3

Ronald Brown R, P.J. Higgins, and R. Sivera.:

“Non-Abelian algebraic topology”

, (2008).

4

R. Brown and J.-L. Loday: Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc., 54:(3), 176-192, (1987).

5

R. Brown and J.-L. Loday:

Van Kampen Theorems

for

diagrams

of spaces,

Topology

, 26: 311-337 (1987).

6

R. Brown and C.B. Spencer:

Double groupoids

and crossed modules,

Cahiers Top. Géom. Diff.

, 17 (1976), 343-362.

7

Allain Connes:

Noncommutative Geometry

, Academic Press 1994.

8

May, J.P. 1999,

A Concise Course in Algebraic Topology.

, The University of Chicago Press: Chicago

Title

topic entry on axioms and foundations of homology and cohomology theories

Canonical name

TopicEntryOnAxiomsAndFoundationsOfHomologyAndCohomologyTheories

Date of creation

2013-03-22 18:20:43

Last modified on

2013-03-22 18:20:43

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

12

Author

bci1 (20947)

Entry type

Topic

Classification

msc 11E72

Classification

msc 55N99

Classification

msc 55N25

Classification

msc 55N30

Classification

msc 55N05

Classification

msc 12G05

Classification

msc 14A22

Classification

msc 11R34

Classification

msc 55N40

Synonym

fundamentals of homology and cohomology theories

Generated on Fri Feb 9 15:33:54 2018 by

LaTeXML
diagonal quadratic form

diagonal quadratic form

Let

Q

⁢

(

𝒙

)

∈

k

⁢

[

x

1

,

…

,

x

n

]

be a

quadratic form

over a field

k

(

char

⁡

(

k

)

≠

2

), where

𝒙

is the

column vector

(

x

1

,

…

,

x

n

)

T

. We write

Q

as

Q

⁢

(

𝒙

)

=

𝒙

T

⁢

M

⁢

(

Q

)

⁢

𝒙

,

where

M

⁢

(

Q

)

is the associated

n

×

n

symmetric matrix

over

k

. We say that

Q

is a

diagonal quadratic form

if

M

⁢

(

Q

)

is a

diagonal matrix

.

Let’s see what a diagonal quadratic form looks like. If

M

=

M

⁢

(

Q

)

is

diagonal

whose diagonal entry in cell

(

i

,

i

)

is

r

i

, then

Q

⁢

(

𝒙

)

=

𝒙

T

⁢

(

r

1

⋯

0

⋮

⋱

⋮

0

⋯

r

n

)

⁢

(

x

1

⋮

x

n

)

=

(

x

1

⋯

x

n

)

⁢

(

r

1

⁢

x

1

⋮

r

n

⁢

x

n

)

=

r

1

⁢

x

1

2

+

⋯

+

r

n

⁢

x

n

2

.

So the coefficients of

x

i

⁢

x

j

for

i

≠

j

are all

0

in a diagonal quadratic form. A diagonal quadratic form is completely determined by the diagonal entries of

M

⁢

(

Q

)

.

Remark.

Every quadratic form is

equivalent

(

http://planetmath.org/EquivalentQuadraticForms

) to a diagonal quadratic form. On the other hand, a quadratic form may be to more than one diagonal quadratic form.

Title

diagonal quadratic form

Canonical name

DiagonalQuadraticForm

Date of creation

2013-03-22 15:42:05

Last modified on

2013-03-22 15:42:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11E81

Classification

msc 15A63

Classification

msc 11H55

Synonym

canonical quadratic form

Related topic

DiagonalizationOfQuadraticForm

Generated on Fri Feb 9 15:33:56 2018 by

LaTeXML
quadratic function associated with a linear functional

quadratic function associated with a linear functional

Let

V

be a real

hilbert space

(and thus an

inner product space

), and
let

f

be a

continuous

linear functional

on

V

. Then

f

has an associated quadratic function

φ

:

V

→

ℝ

given by

φ

⁢

(

v

)

=

1

2

⁢

∥

v

∥

2

-

f

⁢

(

v

)

Title

quadratic function associated with a linear functional

Canonical name

QuadraticFunctionAssociatedWithALinearFunctional

Date of creation

2013-03-22 14:00:23

Last modified on

2013-03-22 14:00:23

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 11Exx

Classification

msc 46Exx

Related topic

HilbertSpace

Related topic

InnerProductSpace

Generated on Fri Feb 9 15:34:01 2018 by

LaTeXML
cusp form

cusp form

A

cusp form

is a

modular form

whose first coefficient in any

expansion

around a cusp is zero. Another more general way to define cusp forms is to consider the forms

orthogonal

to

Eisenstein series

with respect to the Petersson

scalar product

.

The Weierstrass

Δ

function

, also called

modular discriminant

is a weight 12 cusp form for the full modular group

SL

2

⁢

(

ℤ

)

Δ

⁢

(

z

)

=

q

⁢

∏

n

=

1

∞

⁢

(

1

-

q

n

)

24

(1)

The

vector space

of weight

k

cusp forms for the full modular group is finite dimensionnal and non-trivial for

k

integral

greater than 12 and not 14.

Title

cusp form

Canonical name

CuspForm

Date of creation

2013-03-22 14:07:43

Last modified on

2013-03-22 14:07:43

Owner

olivierfouquetx (2421)

Last modified by

olivierfouquetx (2421)

Numerical id

9

Author

olivierfouquetx (2421)

Entry type

Definition

Classification

msc 11F11

Generated on Fri Feb 9 15:34:04 2018 by

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Hecke algebra

Hecke algebra

Let

f

be a

modular form

for

Γ

a congruence subgroup of

SL

2

⁢

(

ℤ

)

.

f

⁢

(

z

)

=

∑

n

=

0

∞

⁢

a

n

⁢

q

n

(1)

where

q

=

e

2

⁢

i

⁢

π

⁢

τ

.

For

m

∈

ℕ

, let

T

m

⁢

f

⁢

(

z

)

=

∑

n

=

0

∞

⁢

b

n

⁢

q

n

with :

b

n

=

∑

d

|

gcd

(

m

,

n

)

⁢

d

k

-

1

⁢

a

m

⁢

n

/

d

2

(2)

In particular, for

p

a prime,

T

p

⁢

f

⁢

(

z

)

=

∑

n

=

0

∞

⁢

b

n

⁢

q

n

with;

b

n

=

a

p

⁢

n

+

p

k

-

1

⁢

a

n

/

p

(3)

where

a

n

/

p

=

0

if

n

is not

divisible

by

p

.

The operator

T

n

is a

linear operator

on the space of modular forms called a

Hecke operator

.

The Hecke operators leave the space of modular forms and

cusp forms

invariant

and turn out to be

self-adjoint

for a

scalar product

called the Petersson
scalar product. In particular they have real

eigenvalues

. Hecke operators
also satisfy

multiplicative

properties that are best summarized by the formal

identity

:

∑

n

=

1

∞

⁢

T

n

⁢

n

-

s

=

∏

𝑝

⁢

(

1

-

T

p

⁢

p

-

s

+

p

k

-

1

-

2

⁢

s

)

-

1

(4)

That

equation

in particular implies that

T

m

⁢

T

n

=

T

n

⁢

T

m

whenever

gcd

⁡

(

n

,

m

)

=

1

.

The set of all Hecke operators is usually denoted

𝕋

and is called the

Hecke algebra

.

0.1

Group algebra example

Definition 0.1

Let

G

l

⁢

c

⁢

d

be a locally compact totally disconnected group; then the

Hecke algebra

H

⁢

(

G

l

⁢

c

⁢

d

)

of the group

G

l

⁢

c

⁢

d

is defined as the

convolution algebra

of
locally constant complex-valued functions on

G

l

⁢

c

⁢

d

with compact support.

Such

ℋ

⁢

(

G

)

algebras

play an important role in the theory of

decomposition

of

group representations

into

tensor products

.

Title

Hecke algebra

Canonical name

HeckeAlgebra

Date of creation

2013-03-22 14:08:13

Last modified on

2013-03-22 14:08:13

Owner

olivierfouquetx (2421)

Last modified by

olivierfouquetx (2421)

Numerical id

13

Author

olivierfouquetx (2421)

Entry type

Definition

Classification

msc 11F11

Classification

msc 20C08

Related topic

ModularForms

Related topic

AlgebraicNumberTheory

Defines

Hecke operator

Defines

Hecke algebra

H

⁢

(

G

)

of the group G

Generated on Fri Feb 9 15:34:06 2018 by

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Fourier coefficients

Fourier coefficients

Let

𝕋

n

=

ℝ

n

/

(

2

⁢

π

⁢

ℤ

)

n

be the

n

-dimensional torus, let

{

ϕ

k

⁢

(

x

)

}

k

∈

ℤ

n

be an orthonormal basis for

L

2

⁢

(

𝕋

n

)

, and suppose that

f

⁢

(

x

)

∈

L

2

⁢

(

𝕋

n

)

.

We can expand

f

as a

Fourier series

∑

k

∈

ℤ

n

f

^

⁢

(

k

)

⁢

ϕ

k

,

and we call the numbers

f

^

⁢

(

k

)

the

Fourier coefficients

of

f

with respect to the given basis. In particular, the Fourier series for

f

converges to

f

in the

L

2

norm.

The most basic incarnation of this is finding the Fourier coefficients of a

Riemann integrable

function

with respect to the orthonormal basis given by the

trigonometric functions

:

Let

f

be a Riemann integrable function from

[

-

π

,

π

]

to

ℝ

. Then the numbers

a

0

=

1

2

⁢

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

𝑑

x

,

a

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

(

n

⁢

x

)

⁢

𝑑

x

,

b

n

=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

(

n

⁢

x

)

⁢

𝑑

x

.

are called the Fourier coefficients of the function

f

.

The above can be repeated for a Lebesgue-integrable function

f

if we use the

Lebesgue integral

in place of the Riemann integral. This is the usual setting for modern

Fourier analysis

.

The trigonometric series

a

0

+

∑

n

=

1

∞

(

a

n

⁢

cos

⁡

(

n

⁢

x

)

+

b

n

⁢

sin

⁡

(

n

⁢

x

)

)

is called the trigonometric series of the function

f

, or Fourier series of the function

f

.

Title

Fourier coefficients

Canonical name

FourierCoefficients

Date of creation

2013-03-22 13:57:07

Last modified on

2013-03-22 13:57:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

19

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11F30

Related topic

GeneralizedRiemannLebesgueLemma

Related topic

FourierSeriesOfFunctionOfBoundedVariation

Related topic

DirichletConditions

Defines

Fourier series

Defines

trigonometric series

Generated on Fri Feb 9 15:34:15 2018 by

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period

period

A

real number

x

is a

period

if it is

expressible

as the

integral

of an (with

algebraic

coefficients

) over an

algebraic domain

, and this integral is

absolutely convergent

. This is called the number’s

period representation

. An

algebraic domain

is a subset of

ℝ

n

given by

inequalities

with algebraic coefficients. A

complex number

is defined to be a period if both its real and

imaginary parts

are. The set of all complex periods is denoted by

𝒫

.

1

Examples

Example 1

.

The

transcendental number

π

is a period since we can write

π

=

∫

x

2

+

y

2

≤

1

𝑑

x

⁢

𝑑

y

.

Example 2

.

Any

algebraic number

α

is a period since we use the somewhat definition that integration over a 0-dimensional space is taken to mean evaluation:

α

=

∫

{

α

}

x

Example 3

.

The

logarithms

of algebraic numbers are periods:

log

⁡

α

=

∫

1

α

1

x

⁢

𝑑

x

2

Non-periods

It is by no means

trivial

to find complex non-periods, though their

existence

is clear by a counting

argument

: The set of complex numbers is

uncountable

, whereas the set of periods is

countable

, as there are only countably many algebraic domains to choose and countably many

algebraic functions

over which to integrate.

3

Inclusion

With the existence of a non-period, we have the following chain of

set inclusions

:

ℤ

⊊

ℚ

⊊

ℚ

¯

⊊

𝒫

⊊

ℂ

,

where

ℚ

¯

denotes the set of algebraic numbers. The periods promise to prove an interesting and important set of numbers in that nebulous between

ℚ

¯

and

ℂ

.

4

References

Kontsevich and Zagier.

Periods

. 2001. Available on line at

\url

http://www.ihes.fr/PREPRINTS/M01/M01-22.ps.gz.

Title

period

Canonical name

Period

Date of creation

2013-03-22 13:55:43

Last modified on

2013-03-22 13:55:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11F67

Defines

period representation

Defines

algebraic domain

Generated on Fri Feb 9 15:34:17 2018 by

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Schanuel’s conjecture

Schanuel’s conjecture

Let

x

1

,

x

2

,

…

,

x

n

be

complex numbers

linearly independent

over

ℚ

. Then the set

{

x

1

,

x

2

,

…

,

x

n

,

e

x

1

,

e

x

2

,

…

,

e

x

n

}

has

transcendence degree

(

http://planetmath.org/TransendenceDegree

) greater than or equal to

n

.

Though seemingly innocuous, a proof of Schanuel’s conjecture would prove hundreds of conjectures in

transcendental number

.

Title

Schanuel’s conjecture

Canonical name

SchanuelsConjecture

Date of creation

2013-03-22 13:58:08

Last modified on

2013-03-22 13:58:08

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Conjecture

Classification

msc 11F67

Synonym

Schanuel conjecture

Related topic

LindemannWeierstrassTheorem

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