aarjaneiro/planetmath huggingface dataset:
presentations of algebraic objects

presentations of algebraic objects

Given an

algebraic category

with enough free objects one can use the general
description of the free object to provide a precise description of all other
objects in the

category

. The process is called a

presentation

.

Suppose

A

is an object generated by a subset

X

. Then if there exists
a free object on

X

,

F

, then there exists a unique

morphism

f

:

F

→

A

which matches the

embedding

of

X

in

F

to the embedding
of

X

in

A

.

As we are in an algebraic category we have a

fundamental homomorphism theorem

(we take this as our

definition

of an algebraic category in this

context

).
This means there is a notion of kernel

K

of

f

and

quotient

F

/

K

such that

F

/

K

is

isomorphic

to

A

.

Now

F

is generted by

X

so every element of

F

is expressed as an informal word over

X

. [By

informal word

we mean whatever process encodes
general

elements

as generated by

X

. For example, in groups and

semigroups

these are actual formal words, but in

algebras

these can be linear combinations
of words or

polynomials

with indeterminants in

X

, etc.] Hence a set of

generators

for the kernel

K

will be expressed as words over

X

.

Definition 1

.

A presentation of an object

A

is a pair of sets

⟨

X

|

R

⟩

where

X

generates

A

and

R

is a set of informal words over

X

such that
the free object

F

on

X

and the

normal subobject

K

of

F

generated by

R

has the

property

F

/

K

≅

A

.

Once again, normal refers to whatever property is required for

subobject

to allow quotients, so

normal subgroup

or ideals, etc.

Existence

of presentations is dependent on the category being considered.
The common categories: groups, rings, and modules all have presentations.

It is generally not possible to insist that a presentation is unique. First we have the

variable

choice of generators. Secondly, we may choose various

relations

. Indeed, it is possible that the relations will generate different subobjects

K

such that

F

/

K

≅

A

. In practice, presentations are a highly compactified description of an object which can hide many essential features of the object. Indeed, in the extreem case are the

theorems

of Boone which show that in the category of groups it is impossible to tell if an arbitrary presentation is a presentation of the

trivial group

.
For a detailed account of these theorems refer to

Joseph Rotman,

An

Introduction

to the Theory of Groups

, Springer, New York, Fourth edition, 1995.

Title

presentations of algebraic objects

Canonical

name

PresentationsOfAlgebraicObjects

Date of creation

2013-03-22 16:51:27

Last modified on

2013-03-22 16:51:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B20

Related topic

Presentationgroup

Defines

presentation

Generated on Fri Feb 9 14:05:43 2018 by

LaTeXML
restricted direct product of algebraic systems

restricted direct product of algebraic systems

Let

{

A

i

∣

i

∈

I

}

be a family of

algebraic systems

indexed by

a set

I

. Let

J

be a

Boolean ideal

in

P

⁢

(

I

)

, the

Boolean algebra

over the

power set

of

I

. A subset

B

of the

direct product

∏

{

A

i

∣

i

∈

I

}

is called a

restricted direct product

of

A

i

if

1.

B

is a

subalgebra

of

∏

{

A

i

∣

i

∈

I

}

, and

2.

given any

(

a

i

)

∈

B

, we have that

(

b

i

)

∈

B

iff

{

i

∈

I

∣

a

i

≠

b

i

}

∈

J

.

If it is necessary to distinguish the different restricted direct products of

A

i

, we often specify the “

restriction

”, hence we say that

B

is a

J

-restricted direct product of

A

i

, or that

B

is restricted to

J

.

Here are some special restricted direct products:

•

If

J

=

P

⁢

(

I

)

above, then

B

is the direct product

∏

A

i

, for if

(

b

i

)

∈

∏

A

i

, then clearly

{

i

∈

I

∣

a

i

≠

b

i

}

∈

P

⁢

(

I

)

, where

(

a

i

)

∈

B

(

B

is non-empty since it is a subalgebra). Therefore

(

b

i

)

∈

B

.

This justifies calling the direct product the “unrestricted direct product” by some people.

•

If

J

is the ideal consisting of all finite subsets of

I

, then

B

is called the

weak direct product

of

A

i

.

•

If

J

is the singleton

{

∅

}

, then

B

is also a singleton: pick

a

,

b

∈

B

, then

{

i

∣

a

i

≠

b

i

}

=

∅

, which is

equivalent

to saying that

(

a

i

)

=

(

b

i

)

.

Remark

. While the direct product of

A

i

always exists, restricted direct products may not. For example, in the last case above, A

∅

-restricted direct product exists only when there is an

element

a

∈

∏

A

i

that is fixed by all

operations

on it: that is, if

f

is an

n

-ary operation on

∏

A

i

, then

f

⁢

(

a

,

…

,

a

)

=

a

. In this case,

{

a

}

is a

∅

-restricted direct product of

∏

A

i

.

References

1

G. Grätzer:

Universal Algebra

, 2nd Edition, Springer, New York (1978).

Title

restricted direct product of algebraic systems

Canonical name

RestrictedDirectProductOfAlgebraicSystems

Date of creation

2013-03-22 17:05:57

Last modified on

2013-03-22 17:05:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B25

Defines

restricted direct product

Defines

weak direct product

Generated on Fri Feb 9 14:06:16 2018 by

LaTeXML
complex projective line

complex projective line

Let

ℂ

be the set of complex numbers. We define an

equivalence relation

on

ℂ

2

-

{

0

,

0

}

by

(

x

⁢

1

,

y

⁢

1

)

∼

(

x

⁢

2

,

y

⁢

2

)

⇔

∃

λ

∈

ℂ

*

⁢

λ

⁢

(

x

⁢

1

,

y

⁢

1

)

=

(

x

⁢

2

,

y

⁢

2

)

(1)

The set

ℂ

2

-

{

0

,

0

}

/

∼

is a projective variety called the

complex projective line

.

Title

complex projective line

Canonical name

ComplexProjectiveLine

Date of creation

2013-03-22 13:59:54

Last modified on

2013-03-22 13:59:54

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 08B30

Generated on Fri Feb 9 14:06:18 2018 by

LaTeXML
enumerating algebras

enumerating algebras

1

How many algebras are there?

Unlike categories of discrete objects, such as

simple graphs

with

n

vertices, (see article on

enumerating graphs

(

http://planetmath.org/EnumeratingGraphs

)) such a question is a little malposed as the quantity can be infinite. However the spirit of the question can be addressed by appealing to algebraic varieties and considering their

dimension

.

Let

A

be an

non-associative algebra

over a field

k

of dimension

n

. For example,

A

could be a

Lie algebra

, an

associative algebra

, or a

commutative algebra

.

From every basis

e

1

,

…

,

e

n

for

A

, the addition of the

algebra

is completely understood as all

n

-dimensional

k

-

vector spaces

are

isomorphic

. Thus we must consider only the multiplication. For this the

structure constants

of the algebra are considered. That is:

e

i

⁢

e

j

=

∑

k

=

1

n

c

i

⁢

j

k

⁢

e

k

for

c

i

⁢

j

k

∈

k

. These structure constants completely define the algebra

A

.

Due to the axioms of multiplication, the structure constants satisfy certain

relations

. For example, if

A

is a Lie algebra then multiplication is via the associated

Lie bracket

and we know

[

e

i

,

e

i

]

=

0

Hence we find

c

i

⁢

i

k

=

0

for all

1

≤

i

≤

n

. Likewise the

Jacobi identity

/

associativity

/

commutative

conditions each imply their particular relations. If one replaces the structure constants with

variables

x

i

⁢

j

⁢

k

we find that each algebra

A

of a given type (Lie/Associative/Commutative/etc.) is a

solution

to the

polynomial equations

given by the relations of the algebra. Thus the algebras themselves are parameterized by the algebraic variety, in

n

3

-dimensional affine space, of these equations.

Theorem 1

(Neretin, 1987).

The dimension of the algebraic variety for

n

-dimensional Lie algebras, associative
algebras, and commutative algebras is respectively

2

27

⁢

n

3

+

O

⁢

(

n

8

/

3

)

,

4

27

⁢

n

3

+

O

⁢

(

n

8

/

3

)

,

and

⁢

2

27

⁢

n

3

+

O

⁢

(

n

8

/

3

)

.

Lower bounds

of

2

27

⁢

n

3

+

O

⁢

(

n

2

)

(and/or

4

27

+

O

⁢

(

n

2

)

) are attainable by exhibiting large families of algebras. For example,

class 2

nilpotent Lie algebras

attain the lower bound.

As with the related problems for

p

-groups, it is also expected that the true upper bound has error term

O

⁢

(

n

2

)

[Neretin,Sims].

Neretin, Yu. A.,

An estimate for the number of parameters defining an

n

-dimensional algebra

, Izv. Akad. Nauk SSSR Ser. Mat., vol. 51,1987, no. 2, pp. 306–318, 447.

Mann, Avinoam,

Some questions about

p

-groups

,
J. Austral. Math. Soc. Ser. A, vol. 67, 1999, no. 3, pp. 356–379.

Title

enumerating algebras

Canonical name

EnumeratingAlgebras

Date of creation

2013-03-22 15:50:48

Last modified on

2013-03-22 15:50:48

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 08B99

Classification

msc 05A16

Related topic

EnumeratingGroups

Generated on Fri Feb 9 14:06:20 2018 by

LaTeXML
identity in a class

identity in a class

Let

K

be a class of

algebraic systems

of the same type. An

identity

on

K

is an expression of the form

p

=

q

, where

p

and

q

are

n

-ary

polynomial symbols

of

K

, such that, for every

algebra

A

∈

K

, we have

p

A

⁢

(

a

1

,

…

,

a

n

)

=

q

A

⁢

(

a

1

,

…

,

a

n

)

for all

⁢

a

1

,

…

,

a

n

∈

A

,

where

p

A

and

q

A

denote the induced polynomials of

A

by the corresponding polynomial symbols. An identity is also known sometimes as an

equation

.

Examples

.

•

Let

K

be a

class of algebras

of the type

{

e

,

-

1

,

⋅

}

, where

e

is nullary,

-

1

unary, and

⋅

binary. Then

(a)

x

⋅

e

=

x

,

(b)

e

⋅

x

=

e

,

(c)

(

x

⋅

y

)

⋅

z

=

x

⋅

(

y

⋅

z

)

,

(d)

x

⋅

x

-

1

=

e

,

(e)

x

-

1

⋅

x

=

e

, and

(f)

x

⋅

y

=

y

⋅

x

.

can all be considered identities on

K

. For example, in the fourth equation, the right hand side is the unary

polynomial

q

⁢

(

x

)

=

e

. Any algebraic system satisfying the first three identities is a monoid. If a monoid also satisfies identities 4 and 5, then it is a group. A group satisfying the last identity is an

abelian group

.

•

Let

L

be a class of algebras of the type

{

∨

,

∧

}

where

∨

and

∧

are both binary. Consider the following possible identities

(a)

x

∨

x

=

x

,

(b)

x

∨

y

=

y

∨

x

,

(c)

x

∨

(

y

∨

z

)

=

(

x

∨

y

)

∨

z

,

(d)

x

∧

x

=

x

,

(e)

x

∧

y

=

y

∧

x

,

(f)

x

∧

(

y

∧

z

)

=

(

x

∧

y

)

∧

z

,

(g)

x

∨

(

y

∧

x

)

=

x

,

(h)

x

∧

(

y

∨

x

)

=

x

,

(i)

x

∨

(

y

∧

(

x

∨

z

)

)

=

(

x

∨

y

)

∧

(

x

∨

z

)

,

(j)

x

∧

(

y

∨

(

x

∧

z

)

)

=

(

x

∧

y

)

∨

(

x

∧

z

)

,

(k)

x

∨

(

y

∧

z

)

=

(

x

∨

y

)

∧

(

x

∨

z

)

, and

(l)

x

∧

(

y

∨

z

)

=

(

x

∧

y

)

∨

(

x

∧

z

)

.

If algebras of

K

satisfy identities 1-8, then

K

is a class of lattices. If 9 and 10 are satisfied as well, then

K

is a class of modular lattices. If every identity is satisified by algebras of

K

, then

K

is a class of

distributive

lattices.

Title

identity in a class

Canonical name

IdentityInAClass

Date of creation

2013-03-22 16:48:05

Last modified on

2013-03-22 16:48:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B99

Defines

identity

Generated on Fri Feb 9 14:06:40 2018 by

LaTeXML
variation

variation

Variation and proportion are defined to be the relationship between two or more variables with regard to a constant of proportionality.

The traditional notation for direct proportionality is

x

∝

y

or, if using

regular

equality notation,

x

=

k

⁢

y

.

Here,

k

denotes the constant of proportionality.

Similarly, the traditional notation for inverse proportionality is

x

∝

1

/

y

or, with regular equality,

x

=

k

/

y

.

For direct proportionality, to find the value of an unknown

x

or

y

, you may use the formula:

y

1

/

x

1

=

y

2

/

x

2

Similarly, for inverse proportion it would be:

x

1

/

y

1

=

y

2

/

x

2

Title

variation

Canonical name

Variation

Date of creation

2013-03-22 14:53:49

Last modified on

2013-03-22 14:53:49

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Topic

Classification

msc 08C99

Synonym

Proportion

Related topic

HomogeneousEquation

Related topic

GraphOfEquationXyConstant

Related topic

ProportionalityOfNumbers

Defines

Relationships between two or more variables.

Generated on Fri Feb 9 14:06:51 2018 by

LaTeXML
abelian number field

abelian number field

Definition 1

.

An

abelian number field

is a

number field

K

such that

K

/

Q

is an

abelian extension

, i.e.

K

/

Q

is Galois and

Gal

⁡

(

K

/

Q

)

is an

abelian group

.

The abelian number fields are classified by the

Kronecker-Weber Theorem

.

Definition 2

.

A cyclic number field is an (

abelian

) number field

K

such that

K

/

Q

is a

Galois extension

and

Gal

⁡

(

K

/

Q

)

is a finite

cyclic group

(therefore abelian).

Title

abelian number field

Canonical name

AbelianNumberField

Date of creation

2013-03-22 16:01:24

Last modified on

2013-03-22 16:01:24

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 11-00

Related topic

GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ

Defines

cyclic number field

Generated on Fri Feb 9 15:06:32 2018 by

LaTeXML
Artin’s conjecture on primitive roots

Artin’s conjecture on primitive roots

Let

m

be a number in the list

2

,

4

,

p

k

or

2

⁢

p

k

for some

k

≥

1

. Then we know that

m

has a

primitive root

, but finding one can be a rather challenging problem (theoretically and computationally).

Gauss conjectured that the number

10

is a primitive root for infinitely many primes

p

. Much later, in

1927

, Emil Artin made the following

conjecture

:

Artin’s Conjecture

.

Let

n

be an

integer

not equal to

-

1

or a square. Then

n

is a primitive root for infinitely many primes

p

.

However, up to now, nobody has been able to show that a single integer

n

is a primitive root for infinitely many primes. It can be shown that the number

3

is a primitive root for every

Fermat prime

but, unfortunately, the existence of infinitely many Fermat primes is far from

obvious

, and in fact it is quite dubious (only five Fermat primes are known!).

Title

Artin’s conjecture on primitive roots

Canonical

name

ArtinsConjectureOnPrimitiveRoots

Date of creation

2013-03-22 16:21:04

Last modified on

2013-03-22 16:21:04

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Conjecture

Classification

msc 11-00

Synonym

Artin’s conjecture

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

LaTeXML
computation of powers using Fermat’s little theorem

computation of powers using Fermat’s little theorem

A straightforward application of Fermat’s theorem consists of rewriting the power of an

integer

mod

n

. Suppose we have

x

≡

a

b

(

mod

n

)

with

a

∈

𝒰

⁢

(

n

)

. Then, by Fermat’s theorem we have

a

ϕ

⁢

(

n

)

≡

1

(

mod

n

)

,

so

x

≡

a

b

⁢

(

1

)

k

≡

a

b

⁢

(

a

ϕ

⁢

(

n

)

)

k

≡

a

b

+

k

⁢

ϕ

⁢

(

n

)

(

mod

n

)

for any integer

k

. This means we can replace

b

by any integer

congruent

to it mod

ϕ

⁢

(

n

)

. In particular we have

x

≡

a

b

%

⁢

ϕ

⁢

(

n

)

(

mod

n

)

where

b

%

⁢

ϕ

⁢

(

n

)

denotes the

remainder

of

b

upon division by

ϕ

⁢

(

n

)

.

This can be used to make the computation of large powers easier. It also allows one to find an easy to compute inverse to

x

b

(

mod

n

)

whenever

b

∈

𝒰

⁢

(

n

)

. In fact, this is just

x

b

-

1

where

b

-

1

is an inverse to

b

mod

ϕ

⁢

(

n

)

. This forms the base of the

RSA cryptosystem

where a message

x

is encrypted by raising it to the

b

th power, giving

x

b

, and is decrypted by raising it to the

b

-

1

th power, giving

(

x

b

)

b

-

1

≡

x

b

⁢

b

-

1

,

which, by the above argument, is just

x

b

⁢

b

-

1

%

⁢

ϕ

⁢

(

n

)

≡

x

,

the original message!

Title

computation of powers using Fermat’s little theorem

Canonical name

ComputationOfPowersUsingFermatsLittleTheorem

Date of creation

2013-03-22 13:17:42

Last modified on

2013-03-22 13:17:42

Owner

basseykay (877)

Last modified by

basseykay (877)

Numerical id

5

Author

basseykay (877)

Entry type

Example

Classification

msc 11-00

Generated on Fri Feb 9 15:06:38 2018 by

LaTeXML
converse of Wilson’s theorem

converse of Wilson’s theorem

Theorem

. Given an

integer

n

>

1

, if

(

n

-

1

)

!

≡

-

1

mod

n

then

n

is prime.

To prove the

converse

of Wilson’s theorem it is enough to show that a

composite number

can’t

satisfy

the

congruence

. A number that does satisfy the congruence, then, would be not composite, and therefore prime.

Proof.

If

n

is composite, then its

greatest prime factor

is at most

n

2

, and

n

2

<

(

n

-

1

)

as long as

n

>

2

(and the smallest

positive

composite number is 4). Therefore,

(

n

-

1

)

!

being the

product

of the numbers from 1 to

n

-

1

includes among its

divisors

the greatest prime factor of

n

, and indeed all its

proper divisors

. In fact, for composite

n

>

4

, it is the case that

(

n

-

1

)

!

not only has all the same proper divisors of

n

as a subset of its own proper divisors, but has them with greater

multiplicity

than

n

does. For the special case of

n

=

4

, the congruence

(

n

-

1

)

!

≡

2

mod

n

is satisfied. For all larger composite

n

, the congruence

(

n

-

1

)

!

≡

0

mod

n

is satisfied instead of the congruence stated in the theorem.
∎

The special case of

n

=

4

deserves further special attention, as it is an exception which proves the rule. With any other

semiprime

n

=

p

⁢

q

, with either

p

or

q

being a prime greater than 2, the product

(

n

-

1

)

!

contains, in

addition

to

p

and

q

, both

(

p

-

1

)

⁢

q

and

p

⁢

(

q

-

1

)

(which are distinct numbers if

p

≠

q

). So if

p

and

q

are distinct, then

(

n

-

1

)

!

has both

prime factors

p

and

q

with a multiplicity of at least 2, which is greater than the multiplicity of 1 in the semiprime

p

⁢

q

. But with 4, the numbers

(

p

-

1

)

⁢

q

and

p

⁢

(

q

-

1

)

are both 2, and so 3! includes 2 as a factor with only a multiplicity of 1, which is less than that factor’s multiplicity in 4.

References

1

Thomas Kochy, ”

Elementary Number Theory

with Applications”, 2nd Edition. London: Elsevier (2007): 324 - 325

Title

converse of Wilson’s theorem

Canonical name

ConverseOfWilsonsTheorem

Date of creation

2013-03-22 17:58:55

Last modified on

2013-03-22 17:58:55

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11-00

Generated on Fri Feb 9 15:06:39 2018 by

LaTeXML
coprime

coprime

Two

integers

a

,

b

are

coprime

if their

greatest common divisor

is

1

. It is also said that

a

,

b

are

relatively prime

.

Title

coprime

Canonical name

Coprime

Date of creation

2013-03-22 11:46:13

Last modified on

2013-03-22 11:46:13

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 11-00

Classification

msc 53B20

Classification

msc 53C25

Classification

msc 53C17

Classification

msc 53B21

Synonym

relatively prime

Generated on Fri Feb 9 15:06:41 2018 by

LaTeXML
corollary of Euler-Fermat theorem

corollary of Euler-Fermat theorem

Corollary of Euler-Fermat theorem

(F. Smarandache):

Let

a

,

m

∈

ℕ

,

m

≠

0

, and

ϕ

be the

Euler totient function

. Then:

a

ϕ

⁢

(

m

s

)

+

s

≡

a

s

(

mod

m

)

where

s

and

m

s

depend on

a

and

m

, also

s

is one more than the number of steps in the

algorithm

, while

m

s

is a

divisor

of

m

, and they are both obtained from the following

integer

algorithm:

Step (0):

calculate

the gcd of

a

and

m

and denote it by

d

0

;

therefore

d

0

=

(

a

,

m

)

, and also denote

m

0

=

m

/

d

0

;

if

d

0

≠

1

go to the next step, otherwise stop;

Step (1):

calculate the gcd of

d

0

and

m

0

and denote it by

d

1

;

therefore

d

1

=

(

d

0

,

m

0

)

, and also denote

m

1

=

m

0

/

d

1

;

if

d

1

≠

1

go to the next step, otherwise stop;

…

⁢

…

⁢

…

⁢

…

⁢

…

⁢

…

⁢

…

Step (s-1):

calculate the gcd of

d

s

-

2

and

m

s

-

2

and denote it by

d

s

-

1

;

therefore

d

s

-

1

=

(

d

s

-

2

,

m

s

-

2

)

, and also denote

m

s

-

1

=

m

s

-

2

/

d

s

-

1

;

if

d

s

-

1

≠

1

go to the next step, otherwise stop;

Step (s):

calculate the gcd of

d

s

-

1

and

m

s

-

1

and denote it by

d

s

;

therefore

d

s

=

(

d

s

-

1

,

m

s

-

1

)

, and also denote

m

s

=

m

s

-

1

/

d

s

;

eventually

one arrives at a gcd

d

s

=

1

, stop the algorithm.

The algorithm ends when the gcd=1. Actually at each step the gcd decreases: from the maximum gcd=(a,m) at step (0) to the minimum gcd=1 at step (s). The algorithm is finite because the first gcd of (a,m) is finite and at each step one gets a smaller gcd.

For the particular case when

(

a

,

m

)

=

1

one has

s

=

0

(hence the algorithm finishes at step (0)) and

m

s

=

m

, which is

Euler-Fermat theorem

.

References

1

Florentin Smarandache,

A Generalization of Euler Theorem

, Bulet. Univ. Brasov, Series C, Vol. XXIII, 7-12, 1981;

http://xxx.lanl.gov/pdf/math.GM/0610607

online article in arXiv.

2

Florentin Smarandache,

Collected Papers

, Vol. I, 184-191 (in French), Tempus, Bucharest, 1996;

http://www.gallup.unm.edu/ smarandache/CP1.pdf

online book.

Title

corollary of Euler-Fermat theorem

Canonical name

CorollaryOfEulerFermatTheorem

Date of creation

2013-03-22 14:23:14

Last modified on

2013-03-22 14:23:14

Owner

kamala (5486)

Last modified by

kamala (5486)

Numerical id

9

Author

kamala (5486)

Entry type

Result

Classification

msc 11-00

Generated on Fri Feb 9 15:06:43 2018 by

LaTeXML