aarjaneiro/planetmath huggingface dataset:
ad hoc

ad hoc

The Latin phrase

ad hoc

translates as “toward this”, and is used in mathematics to describe anything that has been made up specifically for one particular purpose.

Examples

1.

an ad hoc

definition

is a temporary definition that might
be useful for the discussion at hand. For instance, let a

nice set

be a closed set with smooth boundary.

2.

If a proof makes use of an ad hoc construction, the proof
(typically) does not make use of a general theory for the problem.

See also

http://en.wikipedia.org/wiki/Ad_hoc

Article on ad hoc on Wikipedia

Title

ad hoc

Canonical

name

AdHoc

Date of creation

2013-03-22 14:45:20

Last modified on

2013-03-22 14:45:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 00-01

Generated on Tue Feb 6 22:16:54 2018 by

LaTeXML
Attachment to Textbook projects on PlanetMath

Attachment to Textbook projects on PlanetMath

1

List of current Textbook Projects on PlanetMath:

1.

http://planetmath.org/?op=getobj&from=books&id=289

Abelian and

nonabelian

mathematics applications in

quantum theories

and natural sciences

h

t

t

p

:

/

/

p

l

a

n

e

t

m

a

t

h

.

o

r

g

/

?

o

p

=

g

e

t

o

b

j

&

f

r

o

m

=

b

o

o

k

s

&

i

d

=

289

2.

http://planetmath.org/?op=getobj&from=books&id=288

Quantum Algebra

and

Symmetry

h

t

t

p

:

/

/

p

l

a

n

e

t

m

a

t

h

.

o

r

g

/

?

o

p

=

g

e

t

o

b

j

&

f

r

o

m

=

b

o

o

k

s

&

i

d

=

288

3.

http://planetphysics.us/?op=getobj&from=books&id=436

Higher dimensional algebroids and crossed modules

h

t

t

p

:

/

/

p

a

g

e

s

.

b

a

n

g

o

r

.

a

c

.

u

k

/

m

a

s

010

/

m

o

s

a

-

t

h

e

s

i

s

.

h

t

m

l

4.

http://planetmath.org/?op=getobj&from=objects&id=12478

Q-logics

of Quantum Automata
5a.

http://metameso.org/ joe/docs/book.pdf

PM Book Project in 2005
5b.

http://planetphysics.us/?op=getobj&from=books&id=435

PM Book Project in 2005, available at PlanetPhysics.org

h

t

t

p

:

/

/

p

l

a

n

e

t

p

h

y

s

i

c

s

.

u

s

/

?

o

p

=

g

e

t

o

b

j

&

f

r

o

m

=

b

o

o

k

s

&

i

d

=

435

…
N. The result would depend on how well that tag works at PM.

Title

Attachment to Textbook projects on PlanetMath

Canonical

name

AttachmentToTextbookProjectsOnPlanetMath

Date of creation

2013-03-22 19:33:35

Last modified on

2013-03-22 19:33:35

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Definition

Classification

msc 00-01

Generated on Tue Feb 6 22:16:56 2018 by

LaTeXML
expression

expression

An

expression

is a symbol or of symbols used to denote a quantity or value. Expressions consist of

constants

,

variables

,

operations

,

operators

,

functions

, and parentheses.

Note 1.

If an expression one or more operations to be performed in a certain

http://planetmath.org/OrderOfOperations

order, the expression may be named after the last (

http://planetmath.org/Ie

i.e. outermost) operation. For example, the expression

a

2

-

5

⁢

a

+

2

3

⁢

a

is a

sum expression

.

Note 2.

An equation is a denoted equality of two expressions.

References

1

Dictionary.com Unabridged

(version 1.1). Accessed on February 22, 2008. URL:

http://dictionary.reference.com/browse/expression

http://dictionary.reference.com/browse/expression

Title

expression

Canonical

name

Expression

Date of creation

2013-03-22 17:50:11

Last modified on

2013-03-22 17:50:11

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 00-01

Related topic

Equation

Defines

sum expression

Generated on Tue Feb 6 22:17:00 2018 by

LaTeXML
infix notation

infix notation

Infix notation

is how we usually read and write

arithmetic

expressions

. In this notation, the

operator

goes between the operands in the expression:

(operand1)

(operator)

(operand2)

E.g.,

3

+

2

, or

196

×

11

, etc.

Infix notation suffers from some ambiguity; e.g.

3

+

9

×

2

could mean

(

3

+

9

)

×

2

or

3

+

(

9

×

2

)

. Parentheses are needed to specify the

order of operations

unambiguously.

Postfix notation

(or reverse-Polish notation) does not suffer this ambiguity; but it is considered harder for humans to read (hence its primary use in

computer

applications).

The “usual” fix for the ambiguity problem described above is to provide a convention regarding precedence of operations. This is typically done for computer parsing of mathematical expressions rather than in math done by hand, because in the former case, the computer

must

have some standard rules to proceed. For example, it is typical to make

multiplication

“higher precedence” than

addition

, so in the above case,

9

×

2

would be performed before adding the result to 3.

The ambiguity problem only occurs when multiple operators are present in one expression, and thus, the associative law does not hold. E.g., there is no ambiguity in

1

+

2

+

3

, because

(

1

+

2

)

+

3

is the same as

1

+

(

2

+

3

)

, by the associative property of addition.

Title

infix notation

Canonical

name

InfixNotation

Date of creation

2013-03-22 16:21:20

Last modified on

2013-03-22 16:21:20

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 00-01

Synonym

infix

Related topic

GeneralAssociativity

Defines

infix arithmetic

Generated on Tue Feb 6 22:17:05 2018 by

LaTeXML
rigid

rigid

Suppose

C

is a

collection

of mathematical objects
(for

instance

, sets, or

functions

).
Then we say that

C

is

rigid

if every

c

∈

C

is uniquely determined by less information about

c

than
one would expect.

It should be emphasized that the above “definition” does not
define a

mathematical object

. Instead, it describes in what sense
the adjective rigid is typically used in mathematics,
by mathematicians.

Let us illustrate this by some examples:

1.

Harmonic functions

on the

unit disk

are rigid in the sense that
they are uniquely determined by their boundary values.

2.

By the

fundamental theorem of algebra

,

polynomials

in

ℂ

are rigid in the sense that any polynomial is completely determined by
its values on any

countably infinite

set, say

ℕ

, or the unit disk.

3.

Linear maps

ℒ

⁢

(

X

,

Y

)

between

vector spaces

X

,

Y

are rigid in the sense that any

L

∈

ℒ

⁢

(

X

,

Y

)

is completely
determined by its values on any set of basis vectors of

X

.

4.

Mostow’s rigidity theorem

Title

rigid

Canonical name

Rigid

Date of creation

2013-03-22 14:38:10

Last modified on

2013-03-22 14:38:10

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Definition

Classification

msc 00-01

Synonym

rigidity result

Synonym

rigidity theorem

Synonym

rigidity

Generated on Tue Feb 6 22:17:06 2018 by

LaTeXML
strict

strict

In mathematical writing, the adjective

strict

is used in to modify technical which have meanings. It indicates that the exclusive meaning of the term is to be understood. (More formally, one could say that this is the meaning which implies the other meanings.)

This term is commonly used in the context of

inequalities

— the phrases “

strictly

less than” and “strictly

greater

than” mean “less than and not equal to” and “greater than and not equal to”, respectively. A related use occurs when comparing numbers to zero — “strictly

positive

” and “strictly negative” mean “positive and not equal to zero” and “negative and not equal to zero”, respectively. Also, in the context of functions, the adverb “strictly ” is used to modify the terms “

monotonic

”, “increasing”, and “decreasing”.

On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons, one can use the phrases “non-negative”, “non-positive”, “non-increasing”, and “non-decreasing” to make it clear that the inclusive sense of the terms is intended.

Using such terminology helps avoid possible ambiguity and confusion. For instance, upon reading the phrase “x is negative”, it is not immediately clear whether x = 0 is possible, since some authors consider zero to be positive while others consider zero not to be positive. Therefore, it is prudent to write either “x is strictly negative” or “x is non-positive” unless the distinction is unimportant or the context makes

obvious

which meaning is intended or it has been explicitly stated that the term ”positive” is to be used in only one sense. (Here, in Planet Math, we have taken the third option by adopting the convention that zero is neither positive nor negative. Hence, the terms ”strictly positive” and ”positive” are synonyms here. However, the adverb ”strictly” may still be necessary in some of the other contexts described avove.)

Title

strict

Canonical

name

Strict

Date of creation

2013-03-22 14:45:23

Last modified on

2013-03-22 14:45:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 00-01

Defines

strict

Defines

strictly

Generated on Tue Feb 6 22:17:08 2018 by

LaTeXML
Textbook projects on PlanetMath

Textbook projects on PlanetMath

This will be a guide to textbook projects on PM. Since I don’t know
if there currently are any textbook projects on PM, this is just a
“stub” entry for now. If you create a textbook project, please add
a link to it here. Let me know what other things you would like to
see as part of this entry. (Eventually we will probably create a
documentation item for textbook writing.) I think the 00-01 category
can and should contain articles

about

instructional exposition,
just as well as examples of instructional exposition, so perhaps such
discussions can begin here as well, and then branch off into other
entries as that becomes convenient.

An example of the sort of textbook we might want to have would
be: a free book

comparable

to

Introduction

to

Algorithms

by
Cormen

et al.

.

•

Statistics on PlanetMath

Title

Textbook projects on PlanetMath

Canonical

name

TextbookProjectsOnPlanetMath

Date of creation

2013-03-22 14:46:20

Last modified on

2013-03-22 14:46:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00-01

Generated on Tue Feb 6 22:17:09 2018 by

LaTeXML
toy theorem

toy theorem

A

toy theorem

is a simplified version of a more general

theorem

.
For

instance

, by introducing some simplifying

assumptions

in a theorem,
one obtains a toy theorem.

Usually, a toy theorem is used to illustrate the claim of a theorem.
It can also be illustrative and insightful to study proofs of
a toy theorem derived from a non-trivial theorem.
Toy theorems also have a great education value.
After presenting a theorem (with, say, a highly non-trivial proof),
one can sometimes give some assurance that the theorem
really holds, by proving a toy version of the theorem.

For instance, a toy theorem of the

Brouwer fixed point theorem

is obtained by restricting the

dimension

to one.
In this case, the Brouwer fixed point theorem follows
almost immediately from the

intermediate value theorem

(see

http://planetmath.org/BrouwerFixedPointInOneDimension

this page).

Title

toy theorem

Canonical name

ToyTheorem

Date of creation

2013-03-22 13:55:35

Last modified on

2013-03-22 13:55:35

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 00-01

Generated on Tue Feb 6 22:17:11 2018 by

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failure function tests

failure function tests

1

FAILURE FUNCTION

An abstract

definition

is:

Let

ϕ

⁢

(

x

)

be a

function

of

x

. Then,

x

=

ψ

⁢

(

x

0

)

is a failure function if the values of

x

generated by

ψ

⁢

(

x

0

)

, when substituted in

ϕ

⁢

(

x

)

, generate only failures in accordance with our definition of a failure. Here

x

0

is a specific value of

x

.

1.1

Examples

•

(i) Let the mother function be a

polynomial

in x (coeffficients belong to

𝒵

), say

ϕ

⁢

(

x

)

.
Let our definition of a failure be a

composite number

. Then,

x

=

ψ

⁢

(

x

0

)

=

x

0

+

k

⁢

(

ϕ

⁢

(

x

0

)

)

is a failure function because the values of x generated by

ϕ

⁢

(

x

0

)

, when substituted in

ϕ

⁢

(

x

)

, generate only failures.

•

(ii) Let the mother function be an

exponential function

, say

ϕ

⁢

(

x

)

=

a

x

+

c

.
Then

x

=

ψ

⁢

(

x

0

)

=

x

0

+

k

.

E

⁢

u

⁢

l

⁢

e

⁢

r

⁢

p

⁢

h

⁢

i

⁢

(

ϕ

⁢

(

x

0

)

)

is a failure function since the values of x generated by

ψ

⁢

(

x

0

)

, when substituted in the mother function, generate only failures.

•

(iii) Let our definition of a failure be a non-Carmichael number.
Let the mother function

ϕ

⁢

(

x

)

be

2

n

+

49

. Then,

n

=

5

+

6

⁢

k

is its failure function

ψ

⁢

(

x

)

.

1.2

Note

Here too our definition of a failure is a composite number and k belongs to N.

Title

failure function tests

Canonical

name

FailureFunctionTests

Date of creation

2013-03-22 19:33:30

Last modified on

2013-03-22 19:33:30

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Classification

msc 00-02

Classification

msc 00-01

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

LaTeXML
PM Planetary Update on Developmental web Servers and Sites–including PlanetPhysics.org web servers

PM Planetary Update on Developmental web Servers and Sites–including PlanetPhysics.org web servers

1

Web servers Updates:

(In reverse chronological order, with the first website/server listed being the latest one)
Currently all servers are running at full speed…
Books can be edited and saved in PDF format at the Media Wiki version 1.17

http://wiki.planetphysics.info:8888

PlanetPhysics.org website:

(Yes, you need to add :8888 after the URL for protection against spamming!)

“

Planetary.PlanetPhysics.info

” is a new PM and PlanetPhysics.org developmental server/website for PlanetMath.org, running Drupal under Ubuntu 12.04.01 OS with

Planetary.PlanetMath.Beta3B

; it is currently dedicated as a Developmental PM new server for high storage backup and website

development

; it is also planned in the future to add to it a Planetary.PlanetPhysics website, running Drupal under Ubuntu 12.04.1 OS. This is currently the only PM developmental website that allows direct upload of Full (not filtered) HTML source code;

Beta2

”PlanetPhysics.org developmental website”

is actually an earlier PM developmental planetary website running on a different server

Beta.PlanetMath.org

is running Beta3 Planetary for the PM developmental website with Drupal under Ubuntu 12.04.1 OS

PlanetPhysics.org R- wiki is an Organizational Blog and Discussion R-wiki for backing up contents

PlanetPhysics.org R-wiki on Noosphere 1.0 main server running under Ubuntu 10.04

PlanetPhysics.org’s MediaWiki v.1.17 server runs under Scientific Linux version 6.1 OS

PlanetPhysics.us- is the Noosphere 1.0 website of PlanetPhysics.org

PlanetPhysics.org ’s

http://pp-dev.org:8080/

Developmental-redux website runs on internet port 8080 Noosphere v. 1.5.

Title

PM Planetary Update on Developmental web Servers and Sites–including PlanetPhysics.org web servers

Canonical name

PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers

Date of creation

2013-03-22 19:36:31

Last modified on

2013-03-22 19:36:31

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

31

Author

bci1 (20947)

Classification

msc 00-02

Classification

msc 00-01

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

LaTeXML
an example of mathematical induction

an example of mathematical induction

Below is a step-by-step demonstration of how mathematical induction (the

Principle of Finite Induction

) works.

Proposition

.

For any positive integer

n

,

2

n

-

1

≤

n

!

.

Proof.

Setup.

Let

S

be the set of positive integers

n

satisfying the rule:

2

n

-

1

≤

n

!

. We want to show that

S

is the set of

all

positive integers, which would prove our proposition.

Initial Step.

For

n

=

1

,

2

n

-

1

=

2

1

-

1

=

2

0

=

1

, while

n

!

=

1

!

=

1

. So

2

n

-

1

=

n

!

for

n

=

1

and thus

2

n

-

1

≤

n

!

all the more so. This shows that

1

∈

S

.

Induction

Step 1.

Assume that for

n

=

k

,

k

a positive integer,

2

n

-

1

≤

n

!

. In other words, we assume that

k

∈

S

, or that

2

k

-

1

≤

k

!

.

Induction Step 2.

Next, we want to show that

k

+

1

∈

S

. If we let

n

=

k

+

1

, then by the

assumption

of the proposition, showing

n

=

k

+

1

∈

S

is the same as showing

2

n

-

1

≤

n

!

, or

2

k

≤

(

k

+

1

)

!

. This can be done by the following calculation:

2

k

=

2

k

-

1

⁢

2

(1)

≤

k

!

⁢

2

(2)

≤

k

!

⁢

(

k

+

1

)

(3)

=

(

k

+

1

)

!

,

(4)

where Equations (1) and (4) are just

definitions

of the power and the

factorial

of a number, respectively. Step (3) is the fact that

2

≤

k

+

1

for any positive integer

k

(which, incidentally, can be proved by mathematical induction as well). Step (2) follows from the

induction step

, the assumption that we made in the

Induction Step 1.

in the previous paragraph. Because

2

k

≤

(

k

+

1

)

!

, we have thus shown that

n

=

k

+

1

∈

S

, proving the proposition.
∎

Remark.

All these steps are essential in any proof by (mathematical) induction. However, in more advanced math articles and books, some or all of these steps are omitted with the assumption that the reader is already familiar with the

concepts

and the steps involved.

Title

an example of mathematical induction

Canonical

name

AnExampleOfMathematicalInduction

Date of creation

2013-03-22 15:39:46

Last modified on

2013-03-22 15:39:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Example

Classification

msc 00A05

Classification

msc 00A35

Classification

msc 03E25

Related topic

PrincipleOfFiniteInduction

Generated on Tue Feb 6 22:17:24 2018 by

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arg min and arg max

arg min and arg max

For a real-valued function

f

with domain

S

,

arg

⁡

min

x

∈

S

⁡

f

⁢

(

x

)

is the set of elements in

S

that achieve the

global minimum

in

S

,

arg

⁡

min

x

∈

S

⁡

f

⁢

(

x

)

=

{

x

∈

S

:

f

⁢

(

x

)

=

min

y

∈

S

⁡

f

⁢

(

y

)

}

.

arg

⁡

max

x

∈

S

⁡

f

⁢

(

x

)

is the set of elements in

S

that achieve the

global maximum

in

S

,

arg

⁡

max

x

∈

S

⁡

f

⁢

(

x

)

=

{

x

∈

S

:

f

⁢

(

x

)

=

max

y

∈

S

⁡

f

⁢

(

y

)

}

.

Title

arg min and arg max

Canonical

name

ArgMinAndArgMax

Date of creation

2013-03-22 14:27:55

Last modified on

2013-03-22 14:27:55

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

11

Author

kshum (5987)

Entry type

Definition

Classification

msc 00A05

Defines

argmin argmax

Generated on Tue Feb 6 22:17:26 2018 by

LaTeXML
arithmetic

arithmetic

Arithmetic

is ” the science of numbers and

operations

on sets of numbers. Arithmetic is understood to include problems on the origin and development of the

concept

of a number, methods and means of calculation, the study of operations on numbers of different kinds, as well as

analysis

of the axiomatic

structure

of number sets and the

properties

of numbers.” The four basic operations are

addition

,

subtraction

,

multiplication

and

division

; in the absence of parentheses these are performed according to the rules of

operator precedence

. From multiplication follows

exponentiation

.

Most civilizations usually develop arithmetic before any other branches of mathematics, and in modern times

children

are usually taught arithmetic first (though in some curricula they might be taught

set theory

first).

Numeral systems

of ancient civilizations often took letter symbols for numbers. Boyer called these letter symbols

one-to-one

ciphered numerals. The ancient Greeks, for example, used letters in Ionian and Dorian

alphabets

to represent numbers, as did the Romans in one alphabet. These systems were

additive

in one sense (that is, the value of the overall “word” was merely the sum of the values of the

individual

symbols). To convert

rational numbers

into a ciphered numeration system,

Fibonacci

implemented seven rules, one being a LCM

multiple

. For example, four of Fibonacci’s rules were used by earlier cultures. One was an Egyptian scribe. In 1650 BCE Ahmes converted

n

p

by using large multiples. About 200 years earlier, a student converted

n

p

⁢

q

by using small LCM multiples.

It was not just the invention of zero as a symbol for that

integers

between 1 and -1 allowed Hindu-Arabic numerals to significant expand man’s ability to perform arithmetic operations. Significantly larger numbers came into use after zero became a placeholder in the new additive-exponential system. An algorithm written in Hindu-Arabic numbers before 1600 AD was also added. Modern decimals apply few lessons learned during the 3,200 year life of

Egyptian fractions

. One exception is

aliquot parts

, used in the

fundamental theorem of arithmetic

, and

higher arithmetic

.

The next great expansion of arithmetic power occurred with the invention of

calculators

and

computers

in the 20th Century, with both kinds of devices performing arithmetic in a binary algorithm. The algorithm, stated in a cursive form, had been used prior to 2,000 BCE.

As

number theory

generalizes the principles of arithmetic to all numbers or all numbers of a given kind, it is sometimes called the “higher arithmetic.”

0.1

External links

•

http://liberabaci.blogspot.com

Fibonacci

•

http://rmprectotable.blogspot.com

n/p

•

http://emlr.blogspot.com

n/pq

Title

arithmetic

Canonical

name

Arithmetic

Date of creation

2013-03-22 18:16:18

Last modified on

2013-03-22 18:16:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

29

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Synonym

arithmetics

Generated on Tue Feb 6 22:17:28 2018 by

LaTeXML
arithmetic progression

arithmetic progression

Arithmetic progression

of length

n

, initial term

a

1

and

common difference

d

is the

sequence

a

1

,

a

1

+

d

,

a

1

+

2

⁢

d

,

…

,

a

1

+

(

n

-

1

)

⁢

d

.

The sum of terms of an arithmetic progression can be computed using Gauss’s trick:

S

=

(

a

1

+

0

)

+

(

a

1

+

d

)

+

⋯

+

(

a

1

+

(

n

-

2

)

⁢

d

)

+

(

a

1

+

(

n

-

1

)

⁢

d

)

+

S

¯

=

(

a

1

+

(

n

-

1

)

⁢

d

)

+

(

a

1

+

(

n

-

2

)

⁢

d

)

+

⋯

+

(

a

1

+

d

)

+

(

a

1

+

0

)

¯

2

⁢

S

=

(

2

⁢

a

1

+

(

n

-

1

)

⁢

d

)

+

(

2

⁢

a

1

+

(

n

-

1

)

⁢

d

)

+

⋯

+

(

2

⁢

a

1

+

(

n

-

1

)

⁢

d

)

+

(

2

⁢

a

1

+

(

n

-

1

)

⁢

d

)

.

We just add the sum with itself written backwards, and the sum of each of the columns equals to

(

2

⁢

a

1

+

(

n

-

1

)

⁢

d

)

. The sum is then

S

=

(

2

⁢

a

1

+

(

n

-

1

)

⁢

d

)

⁢

n

2

.

Title

arithmetic progression

Canonical

name

ArithmeticProgression

Date of creation

2013-03-22 13:39:00

Last modified on

2013-03-22 13:39:00

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

10

Author

bbukh (348)

Entry type

Definition

Classification

msc 00A05

Classification

msc 11B25

Related topic

MulidimensionalArithmeticProgression

Related topic

SumOfKthPowersOfTheFirstNPositiveIntegers

Generated on Tue Feb 6 22:17:30 2018 by

LaTeXML