aarjaneiro/planetmath huggingface dataset:
arithmetic-geometric-harmonic means inequality

arithmetic-geometric-harmonic means inequality

Let

x

1

,

x

2

,

…

,

x

n

be positive numbers.
Then

max

⁡

{

x

1

,

x

2

,

…

,

x

n

}

≥

x

1

+

x

2

+

⋯

+

x

n

n

≥

x

1

⁢

x

2

⁢

⋯

⁢

x

n

n

≥

n

1

x

1

+

1

x

2

+

⋯

+

1

x

n

≥

min

⁡

{

x

1

,

x

2

,

…

,

x

n

}

The equality is obtained if and only if

x

1

=

x

2

=

⋯

=

x

n

.

There are several

generalizations

to this

inequality

using

power means

and

weighted power means

.

Title

arithmetic-geometric-harmonic means inequality

Canonical

name

ArithmeticgeometricharmonicMeansInequality

Date of creation

2013-03-22 11:42:32

Last modified on

2013-03-22 11:42:32

Owner

drini (3)

Last modified by

drini (3)

Numerical id

22

Author

drini (3)

Entry type

Theorem

Classification

msc 00A05

Classification

msc 20-XX

Classification

msc 26D15

Synonym

harmonic-geometric-arithmetic means inequality

Synonym

arithmetic-geometric means inequality

Synonym

AGM inequality

Synonym

AGMH inequality

Related topic

ArithmeticMean

Related topic

GeometricMean

Related topic

HarmonicMean

Related topic

GeneralMeansInequality

Related topic

WeightedPowerMean

Related topic

PowerMean

Related topic

RootMeanSquare3

Related topic

ProofOfGeneralMeansInequality

Related topic

JensensInequality

Related topic

DerivationOfHarmonicMeanAsTheLimitOfThePowerMean

Related topic

MinimalAndMaximalNumber

Related topic

ProofOfArithm

Generated on Tue Feb 6 22:17:32 2018 by

LaTeXML
arity

arity

The

arity

of something is the number of arguments it takes. This is usually applied to functions: an

n

-ary function is one that takes

n

arguments.

Unary

is a synonym for

1

-ary, and binary for

2

-ary.

Title

arity

Canonical

name

Arity

Date of creation

2013-03-22 13:00:01

Last modified on

2013-03-22 13:00:01

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 00A05

Synonym

ary

Synonym

-arity

Related topic

BinaryOperation

Defines

unary

Defines

binary

Generated on Tue Feb 6 22:17:34 2018 by

LaTeXML
biogroupoids: mathematical models of species evolution

biogroupoids: mathematical models of species evolution

\xyoption

curve

0.1

Introduction

Biogroupoids

,

𝒢

ℬ

, were introduced as

mathematical representations

of
evolving biological species (

[

1

,

2

]

) generated by

weakly equivalent

classes of living organisms

,

E

O

, specified by inter-breeding organisms;in this case, a

weak equivalence

relation

,

∼

w

, is defined on the set of evolving organisms modeled in terms of

functional

,

isomorphic

genome networks

,

G

i

⁢

s

⁢

o

N

, such as those described by

L

⁢

M

n

-logic networks in Łukasiewicz-Moisil,

ℒ

⁢

M

n

topoi (

[

1

]

).

0.2

AT-Formulation

This

concept

allows an

algebraic topology

formulation of the origin of species and biological evolution
both at organismal/organismic and biomolecular levels; it

represents

a new approach to biological evolution
from the standpoint of

super-complex systems

biology.

References

1

Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in

Categories

,

Higher Dimensional Algebra

and Łukasiewicz-Moisil Topos:

Transformations

of Neuronal, Genetic and Neoplastic Networks.,

Axiomathes

,

16

Nos. 1-2: 65-122.

2

Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007,

Categorical Ontology

of Complex Spacetime

Structures

: The Emergence of Life and Human Consciousness, Axiomathes,

17

: 35-168.

Title

biogroupoids: mathematical models of species evolution

Canonical name

BiogroupoidsMathematicalModelsOfSpeciesEvolution

Date of creation

2013-03-22 18:11:47

Last modified on

2013-03-22 18:11:47

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

48

Author

bci1 (20947)

Entry type

Topic

Classification

msc 00A05

Classification

msc 92B05

Classification

msc 03G20

Classification

msc 92D15

Synonym

mathematical model of species

Synonym

origin of species

Related topic

SupercategoriesOfComplexSystems

Related topic

AlgebraicCategoryOfLMnLogicAlgebras

Related topic

RosettaGroupoids

Defines

biogroupoid

Generated on Tue Feb 6 22:17:41 2018 by

LaTeXML
calculator

calculator

A

calculator

is an electronic, electrical or mechanical device (hardware) or a

computer

program

(software) designed to perform a predefined set of

arithmetic

computations

on numbers entered by the user and display the results.

The abacus is sometimes given as an example of an early calculator; however, the completion of a calculation requires the active participation of the user and is easily subject to user error even when the device passes a diagnostic with flying colors, whereas a modern calculator in working order

calculates

and displays the correct result regardless of whether or not the user sticks around to see it (this is subject to certain caveats regarding precision, however). Of course it has always been the case in the history of calculators that users may enter incorrect inputs or misunderstand the

operation

of the device and thus get irrelevant results.

One of the first mechanical calculators was designed by

Blaise Pascal

in the 17th Century. It used a gear for each digit and the gears were

connected

, it took up a little space on a desk. It was limited to

addition

and

subtraction

. By the 1990s, electronic calculators were ubiquitous and small enough to put in a pocket.

A basic calculator has about twenty keys: the digits 0 to 9, a

decimal point

, sign change, addition, subtraction,

multiplication

,

division

, equal or Enter key and a C key to clear error exceptions. Occasionally such calculators also have keys for

percentage

and square root. Another option sometimes found on basic calculators is a memory register and associated keys M+, MR and MC (for adding to memory register, recalling memory and clearing memory, respectively).

More advanced calculators include

scientific calculators

, graphing calculators, programmable calculators. Most calculators use standard

infix notation

, but there are

reverse Polish notation

calculators also available on the market.

Title

calculator

Canonical

name

Calculator

Date of creation

2013-03-22 16:39:16

Last modified on

2013-03-22 16:39:16

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Classification

msc 01A07

Related topic

CalculatorAndCASSupportForVariousPositionalBases

Generated on Tue Feb 6 22:17:43 2018 by

LaTeXML
characterisation

characterisation

In mathematics,

characterisation

usually means a property or a condition to define a certain notion.  A notion may, under some presumptions, have different ways to define it.

For example, let

R

be a

commutative ring

with

non-zero unity

(the presumption).  Then the following are

equivalent

:

(1) All

finitely generated

regular ideals

of

R

are

invertible

.

(2) The

(

a

,

b

)

⁢

(

c

,

d

)

=

(

a

⁢

c

,

b

⁢

d

,

(

a

+

b

)

⁢

(

c

+

d

)

)

for multiplying ideals of

R

is valid always when at least one of the

elements

a

,

b

,

c

,

d

of

R

is not zero-divisor.

(3) Every

overring

of

R

is

integrally closed

.

Each of these conditions is

sufficient

(and necessary) for characterising and defining the Prüfer ring.

Title

characterisation

Canonical name

Characterisation

Date of creation

2013-03-22 14:22:28

Last modified on

2013-03-22 14:22:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A05

Synonym

characterization

Synonym

defining property

Related topic

AlternativeDefinitionOfGroup

Related topic

EquivalentFormulationsForContinuity

Related topic

MultiplicationRuleGivesInverseIdeal

Generated on Tue Feb 6 22:17:44 2018 by

LaTeXML
Conway’s chained arrow notation

Conway’s chained arrow notation

Conway’s

chained arrow notation

is a way of writing numbers even larger than those provided by the up arrow notation. We define

m

→

n

→

p

=

m

(

p

+

2

)

⁢

n

=

m

⁢

↑

⋯

↑

⏟

p

⁢

n

and

m

→

n

=

m

→

n

→

1

=

m

n

. Longer chains are evaluated by

m

→

⋯

→

n

→

p

→

1

=

m

→

⋯

→

n

→

p

m

→

⋯

→

n

→

1

→

q

=

m

→

⋯

→

n

and

m

→

⋯

→

n

→

p

+

1

→

q

+

1

=

m

→

⋯

→

n

→

(

m

→

⋯

→

n

→

p

→

q

+

1

)

→

q

For example:

3

→

3

→

2

=

3

→

(

3

→

2

→

2

)

→

1

=

3

→

(

3

→

2

→

2

)

=

3

→

(

3

→

(

3

→

1

→

2

)

→

1

)

=

3

→

(

3

→

3

→

1

)

=

3

3

3

=

3

27

=

7625597484987

A much larger example is:

3

→

2

→

4

→

4

=

3

→

2

→

(

3

→

2

→

3

→

4

)

→

3

=

3

→

2

→

(

3

→

2

→

(

3

→

2

→

2

→

4

)

→

3

)

→

3

=

3

→

2

→

(

3

→

2

→

(

3

→

2

→

(

3

→

2

→

1

→

4

)

→

3

)

→

3

)

→

3

=

3

→

2

→

(

3

→

2

→

(

3

→

2

→

(

3

→

2

)

→

3

)

→

3

)

→

3

=

3

→

2

→

(

3

→

2

→

(

3

→

2

→

9

→

3

)

→

3

)

→

3

Clearly this is going to be a very large number. Note that, as large as it is, it is proceeding towards an eventual final evaluation, as evidenced by the fact that the final number in the chain is getting smaller.

Title

Conway’s chained arrow notation

Canonical

name

ConwaysChainedArrowNotation

Date of creation

2013-03-22 12:58:46

Last modified on

2013-03-22 12:58:46

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 00A05

Synonym

chained arrow notation

Synonym

chained arrow

Synonym

chained-arrow

Synonym

chained-arrow notation

Synonym

Conway notation

Related topic

KnuthsUpArrowNotation

Generated on Tue Feb 6 22:17:47 2018 by

LaTeXML
decimal point

decimal point

A

decimal point

is a symbol separating those digits representing

integer

powers of a base (usually base 10) on the left, and those representing fractional powers of a base (the base raised to a

negative number

) on the right. For example, in

π

≈

3.14

, the 3 to the left of the decimal point corresponds to

3

×

10

0

, while the 1 to the right of the decimal point corresponds to

1

×

10

-

1

.

Most

scientific calculators

capable of displaying binary, octal and

hexadecimal

limit numbers in those bases to integers, making moot the issue of what to call the decimal point in those bases.

The decimal point is generally omitted for integers. However,

Mathematica

will use the decimal point at the end of an integer to indicate the value has been computed using

floating-point arithmetic

and loss of precision is possible. For example,

(1/2)^(-1)

gives “2” as an answer but

.5^(-1)

gives “2.” for the answer. Even more pointedly,

1 + 1

gives “2” as the answer but

1. + 1.

gives “2.” as the answer.

In the

United States

, the decimal point is usually aligned with the bottom of the digit glyphs, while in the United Kingdom it is usually centered (and is distinguished from the central dot

multiplication operator

purely on spacing). In Europe, a comma is used instead, so our example would be written

π

≈

3

,

14

.

Title

decimal point

Canonical

name

DecimalPoint

Date of creation

2013-03-22 17:23:33

Last modified on

2013-03-22 17:23:33

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Generated on Tue Feb 6 22:17:58 2018 by

LaTeXML
division

division

Division

is the

operation

which assigns to every two numbers (or more generally, elements of a field)

a

and

b

their

quotient

or ratio, provided that the latter,

b

, is distinct from zero.

The

quotient

(or

ratio

)

a

b

of

a

and

b

may be defined as such a number (or element of the field)

x

that

b

⋅

x

=

a

.  Thus,

b

⋅

a

b

=

a

,

which is the “fundamental property of quotient”.

The quotient of the numbers

a

and

b

(

≠

0

) is a
uniquely determined number, since if one had

a

b

=

x

≠

y

=

a

b

,

then we could write

b

⁢

(

x

-

y

)

=

b

⁢

x

-

b

⁢

y

=

a

-

a

=

0

from which the supposition

b

≠

0

would imply

x

-

y

=

0

, i.e.

x

=

y

.

The explicit general

expression

for

a

b

is

a

b

=

b

-

1

⋅

a

where

b

-

1

is the

inverse number

(the

multiplicative inverse

) of

a

, because

b

⁢

(

b

-

1

⁢

a

)

=

(

b

⁢

b

-

1

)

⁢

a

=

1

⁢

a

=

a

.

•

For positive numbers the quotient may be obtained by performing the

division algorithm

with

a

and

b

.  If

a

>

b

>

0

,  then

a

b

indicates how many times

b

fits in

a

.

•

The quotient of

a

and

b

does not change if both numbers (

elements

) are multiplied (or divided, which is called

reduction

) by any

k

≠

0

:

k

⁢

a

k

⁢

b

=

(

k

⁢

b

)

-

1

⁢

(

k

⁢

a

)

=

b

-

1

⁢

k

-

1

⁢

k

⁢

a

=

b

-

1

⁢

a

=

a

b

So we have the method for getting the quotient of

complex numbers

,

a

b

=

b

¯

⁢

a

b

¯

⁢

b

,

where

b

¯

is the

complex conjugate

of

b

, and the quotient of

http://planetmath.org/SquareRootOfSquareRootBinomial

square root polynomials

, e.g.

1

5

+

2

⁢

2

=

5

-

2

⁢

2

(

5

-

2

⁢

2

)

⁢

(

5

+

2

⁢

2

)

=

5

-

2

⁢

2

25

-

8

=

5

-

2

⁢

2

17

;

in the first case one aspires after a real and in the second case after a

rational

denominator

.

•

The division is neither

associative

nor

commutative

, but it is

right distributive

over

addition

:

a

+

b

c

=

a

c

+

b

c

Title

division

Canonical

name

Division

Date of creation

2014-08-08 17:51:29

Last modified on

2014-08-08 17:51:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

29

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A05

Classification

msc 12E99

Related topic

InverseFormingInProportionToGroupOperation

Related topic

DivisionInGroup

Related topic

ConjugationMnemonic

Related topic

Difference2

Related topic

UniquenessOfDivisionAlgorithmInEuclideanDomain

Defines

quotient

Defines

ratio

Defines

fundamental property of quotient

Defines

reduction

Generated on Tue Feb 6 22:18:02 2018 by

LaTeXML
fundamental theorems in mathematics

fundamental theorems in mathematics

In different areas of mathematics, there are certain

theorems

called

fundamental

.  We list a number of them below.

•

fundamental cohomology theorem

•

fundamental homomorphism theorem

•

fundamental theorem of algebra

•

fundamental theorem of arithmetics

•

fundamental theorem of ideal theory

•

fundamental theorem of calculus

•

fundamental theorem of calculus for Kurzweil-Henstock integral

•

fundamental theorem of calculus for Lebesgue integration

•

fundamental theorem of calculus for Riemann integration

•

http://planetmath.org/node/6745

fundamental theorem of calculus of variations

•

fundamental theorem of complex analysis

•

fundamental theorem of demography

•

fundamental theorem of finitely generated Abelian groups

•

fundamental theorem of Galois theory

•

fundamental theorem of Gaussian quadrature

•

fundamental theorem of integral calculus

•

fundamental theorem on isogonal lines

•

fundamental theorem of linear algebra

•

fundamental theorem of linear programming

•

http://planetmath.org/node/12096

fundamental theorem of line integrals

•

fundamental theorem of plane curves

•

fundamental theorem of projective geometry

•

fundamental theorem of Riemannian geometry

•

fundamental theorem of space curves

•

fundamental theorem of symmetric polynomials

•

fundamental theorem of

topology

•

fundamental theorem of topos theory

•

fundamental theorem of transcendence

•

http://planetmath.org/node/10506

fundamental theorem of vector calculus

Title

fundamental theorems in mathematics

Canonical

name

FundamentalTheoremsInMathematics

Date of creation

2013-03-22 19:11:13

Last modified on

2013-03-22 19:11:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Topic

Classification

msc 00A05

Generated on Tue Feb 6 22:18:04 2018 by

LaTeXML
Graham’s number

Graham’s number

Graham’s number

G

is an

upper bound

in a problem in

Ramsey theory

, first mentioned in a paper by

Ronald Graham

and B. Rothschild. The

Guinness Book of World Records

calls it the largest number ever used in a mathematical proof. Graham’s number is too difficult to write in

scientific notation

, so it is generally written using Knuth’s arrow-up notation. In Graham’s paper, the bound is given as

6

≤

N

⁢

(

2

)

≤

A

⁢

(

A

⁢

(

A

⁢

(

A

⁢

(

A

⁢

(

A

⁢

(

A

⁢

(

12

,

3

)

,

3

)

,

3

)

,

3

)

,

3

)

,

3

)

,

3

)

,

where

N

⁢

(

2

)

is the least

natural number

such that

n

≥

N

⁢

(

2

)

implies that given any arbitrary

2

-

coloring

of the

line segments

between pairs of vertices of an

n

-dimensional box, there must exist a monochromatic

rectangle

in the box.

Here

A

is the

function

defined by (this is a direct quote):

A

⁢

(

1

,

n

)

=

2

n

,

A

⁢

(

m

,

2

)

=

4

,

m

≥

1

,

n

≥

2

,

A

⁢

(

m

,

n

)

=

A

⁢

(

m

-

1

,

A

⁢

(

m

,

n

-

1

)

)

,

m

≥

2

,

n

≥

3

.

In the earlier paper Graham and Rothschild called this function

F

instead of

A

, and commented: “Clearly, there is some room for improvement here.”

In Knuth’s arrow-up notation, Graham’s number is still cumbersome to write: we define the

recurrence relation

g

1

=

3

↑

↑

↑

↑

3

and

g

n

=

3

↑

g

n

-

1

3

. Graham’s number is then

G

=

g

64

.

To help understand Graham’s number from the more familiar viewpoint of standard

exponentiation

,

Wikipedia

offers the following chart:

g

1

=

3

↑

↑

↑

↑

3

=

3

↑

↑

↑

(

3

↑

↑

↑

3

)

=

3

↑

↑

↑

(

3

3

⋅

⋅

⋅

3

⏟

3

3

3

threes

)

=

3

3

⋅

⋅

⋅

3

⏟

3

3

⋅

⋅

⋅

3

⏟

threes

⋮

⋮

3

3

⋅

⋅

⋅

3

⏟

threes

3

3

3

threes

}

3

3

⋅

⋅

⋅

3

⏟

layers

3

3

3

threes

We don’t know what the most significant base 10 digits of Graham’s number are, but we do know that the least significant digit is 7 (and of course 0 in base 3).

Graham’s number has its own entry in Wells’s

Dictionary of Curious and Interesting Numbers

, it is the very last entry, right after Skewes’ number, which it significantly dwarfs, and which was once also said to be the largest number ever used in a serious mathematical proof.

References

1

M. P. Slone,

PlanetMath

message, March 19, 2007.

2

R. L. Graham and B. L. Rothschild, “Ramsey’s

theorem

for

n

-

parameter

sets”,

Trans. Amer. Math. Soc.

,

159

(1971): 257 - 292

3

R. L. Graham and B. L. Rothschild. Ramsey theory.

Studies in

combinatorics

, ed. G.-C. Rota,

Mathematical Association of America

, 1978.

Title

Graham’s number

Canonical

name

GrahamsNumber

Date of creation

2013-03-22 17:15:15

Last modified on

2013-03-22 17:15:15

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Classification

msc 05A05

Classification

msc 68P30

Generated on Tue Feb 6 22:18:07 2018 by

LaTeXML
Grapher

Grapher

Grapher

is a software graphing

calculator

that comes bundled with the Apple Mac OS Xoperating system. It can graph 2-dimensional equations like

y

=

sin

⁡

x

, as well as 3-dimensional equations like

z

=

y

3

x

2

+

y

2

(Cartan’s umbrella, illustrated below).

Equations to be graphed can be entered in Cartesian form or in

parametric form

,

polar coordinates

, logarithmic, etc. The program automatically “typesets” the formulas as they are entered, for example, converting

x^2

to

x

2

or

1/x

to

1

x

. The program can create animations, and one can also move a graph or rotate it (just by dragging it, for 3D graphs; for 2D graphs one must first select the Move Tool).

In a default installation, the program is in the Utilities subfolder of the Applications folder. Mac OS 9 had a

similar

program, called Graphing Calculator, which was also capable of graphing 2D and 3D equations as well as animations.

References

1

Maria Langer,

Visual Quickstart Guide: Mac OS 9.1

. New York: Peachpit Press (2001): 107

2

David Pogue & Adam Goldstein,

The Missing Manual: Switching to the Mac, Tiger Edition

. Sebastopol: O’Reilly Media, Inc. (2005): 452 - 453

Title

Grapher

Canonical

name

Grapher

Date of creation

2013-03-22 17:46:13

Last modified on

2013-03-22 17:46:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Classification

msc 01A07

Generated on Tue Feb 6 22:18:09 2018 by

LaTeXML
hypothesis

hypothesis

In mathematics, a

hypothesis

is an unproven statement which is supported by all the available data and by many weaker results. An unproven mathematical statement is usually called a “

conjecture

”, and while experimentation can sometimes produce millions of examples to

support

a conjecture, usually nothing short of a proof can convince experts in the field. But when a conjecture is supported not only but all the available data but also by numerous weaker results, it is upgraded in label to a hypothesis. The most famous conjecture in mathematics is the

Riemann hypothesis

, which despite many attempts at a proof, is supported by many related results. The

convexity conjecture

, on the other hand, is considered “incompatible” with the

n

-tuples conjecture and more results appear to support the latter, thus neither is upgraded to hypothesis.

References

1

R. Crandall & C. Pomerance,

Prime Numbers

: A Computational Perspective

, Springer, NY, 2001: 1.2.4

Title

hypothesis

Canonical

name

Hypothesis

Date of creation

2013-03-22 17:15:18

Last modified on

2013-03-22 17:15:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Generated on Tue Feb 6 22:18:10 2018 by

LaTeXML
indeterminate

indeterminate

An

indeterminate

is simply a

variable

that is not known or solvable.  It is usually denoted by a mathematical alphabet (

x

,

y

,

z

, or

α

,

β

, etc…).  It is important to distinguish between a variable and an indeterminate in that a variable is solvable, at least conditionally.  To make this more precise, let’s see two examples:

1.

Let

x

be a variable such that

2

+

3

⁢

x

=

a

+

b

⁢

x

, where

a

,

b

∈

ℚ

. Then

x

=

(

a

-

2

)

/

(

3

-

b

)

.  Here

x

is solvable conditioned on the equation given. Any values of

a

and

b

(

≠

3

)

will yield a value for

x

.

2.

Let

x

be an indeterminate such that

2

+

3

⁢

x

=

a

+

b

⁢

x

, where

a

,

b

∈

ℚ

.  Since

x

can not be solved, we have

2

=

a

and

3

=

b

.  Note that if

a

and

b

are previously assigned to be values other than 2 and 3 respectively, then

x

is no longer an indeterminate.

Title

indeterminate

Canonical

name

Indeterminate

Date of creation

2013-03-22 14:47:33

Last modified on

2013-03-22 14:47:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 00A05

Related topic

Parameter

Generated on Tue Feb 6 22:18:12 2018 by

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