aarjaneiro/planetmath huggingface dataset:
open question

open question

The adjective

open

is used by mathematicians to a statement which has neither been proven to be true or to be false. (In the light of

incompleteness

thorems, we should perhaps also add “not been proven to be

independent

of the axioms”)

Examples:

“It is an

open question

whether there are an

infinite

number of

prime numbers

all of whose digits are 1.” means that it neither has been proven that there exist an

infinity

of such primes nor been proven that there are only a finite number of such primes.

“It is an

open problem

to determine the smallest number of lines which contain all points of the set

S

.” means that the number in question has not yet been determined.

The term

conjecture

refers to a statement which the speaker has reason to believe the statement is correct even though the speaker cannot prove the statement. (Also, one should be warned about a common abuse of this term. Even after a statement has been proven, sometimes it is still referred to as a conjecture by force of habit.)

The

difference

between the terms

conjecture

and

open question

is that the term

open

is neutral — saying that a statement is open does not connote that the speaker is voicing an opinion regarding the truth or falsity of the statement.

The attachment below gives some examples of famous open problems in mathematics.

http://garden.irmacs.sfu.ca/

Open Problem Garden is an external that contains more examples of open problems in mathematics.

Title

open question

Canonical

name

OpenQuestion

Date of creation

2013-03-22 14:42:41

Last modified on

2013-03-22 14:42:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 00A20

Defines

open problem

Defines

open

Defines

conjecture

Generated on Tue Feb 6 22:20:06 2018 by

LaTeXML
overview of the content of PlanetMath

overview of the content of PlanetMath

http://planetmath.org/TopicEntryOnFoundationsOfMathematics

Foundations of Mathematics

1.

Index for the Foundations of Mathematics

2.

http://planetmath.org/BibliographyForAxiomaticsAndMathematicsFoundationsInCategories

Bibliography for The Foundations of Mathematics

http://planetmath.org/TopicEntryOnAlgebra

Algebra

http://planetmath.org/Geometry

Geometry

http://planetmath.org/TopicEntryOnTopology

Topology

http://planetmath.org/AlgebraicTopology

Algebraic Topology

1.

Index for algebraic topology

2.

http://planetmath.org/CategoricalOntologyABibliographyOfCategoryTheory

Bibliography for algebraic topology

http://planetmath.org/CategoryTheory

Category Theory

1.

Index for category theory

2.

http://planetmath.org/IndexOfCategories

Index of categories

3.

http://planetmath.org/BibliographyForMathematicalPhysicsFoundationsAxiomaticsAndCategories

Bibliography for axiomatics and category theory

http://planetmath.org/HigherDimensionalAlgebraHDA

Higher Dimensional Algebra

1.

Index for higher dimensional algebra

2.

http://planetmath.org/HigherDimensionalAlgebraHDA

Bibliography for higher dimensional algebra

http://planetmath.org/AlgebraicGeometry

Algebraic Geometry

http://planetmath.org/NumberTheory

Number Theory

http://planetmath.org/TopicEntryOnDiscreteMathematics

Discrete Mathematics

http://planetmath.org/TopicEntryOnAnalysis

Analysis

http://planetmath.org/DifferentialEquation

Differential Equations

1.

Index of differential equations

2.

Bibliography for ordinary differential equations

http://planetmath.org/TopicEntryOnAppliedMathematics

Applied Mathematics

1.

Index for Applied Mathematics

2.

http://planetmath.org/FoundationsOfQuantumFieldTheories

Mathematical Foundations

of Quantum Physics

3.

http://planetmath.org/AxiomaticAndCategoricalFoundationsOfMathematicsII

Axiomatic and

Categorical

Foundations

of Mathematical Physics

4.

http://planetmath.org/BibliographyForMathematicalPhysicsFoundationsAxiomaticsAndCategories

Bibliography for Mathematical Physics and Physical Mathematics

http://planetmath.org/TopicEntryOnOrderTheory

Order Theory and

Logic Algebra

Mathematics Education and Educators

1.

Elementary

2.

High school mathematics

http://planetmath.org/TopicEntryOnMiscellaneousMathematics

Miscellaneous

Title

overview of the content of PlanetMath

Canonical

name

OverviewOfTheContentOfPlanetMath

Date of creation

2013-03-22 14:39:38

Last modified on

2013-03-22 14:39:38

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

106

Author

matte (1858)

Entry type

Topic

Classification

msc 00A20

Related topic

AlgebraicGeometry

Related topic

TopicEntryOnAlgebra

Related topic

TopicEntryOnTopology

Related topic

AlgebraicTopology

Related topic

CategoryTheory

Related topic

AxiomsOfMetacategoriesAndSupercategories

Related topic

TopicEntryOnAppliedMathematics

Related topic

BibliographyForMathematicalBiophysics

Related topic

FoundationsOfQuantumFieldTheories

Generated on Tue Feb 6 22:20:07 2018 by

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pathological

pathological

In mathematics, a

pathological

object

is mathematical
object that has a highly unexpected .

Pathological objects are typically percieved to, in some sense, be
badly behaving. On the other hand, they are perfectly properly
defined mathematical objects. Therefore this “bad behaviour” can
simply be seen as a

contradiction

with our intuitive
picture of how a certain object should behave.

Examples

•

A very famous pathological

function

is the

Weierstrass function

, which is a

continuous function

that is nowhere

differentiable

.

•

The

Peano space

filling curve. This pathological curve
maps the unit interval

[

0

,

1

]

continuously onto

[

0

,

1

]

×

[

0

,

1

]

.

•

The Cantor set. This is subset of the

interval

[

0

,

1

]

has the pathological property that it is

uncountable

yet its measure is zero.

•

The Dirichlet’s function from

ℝ

to

ℝ

is continuous at every
irrational point and

discontinuous

at every rational point.

•

Ackermann Function

.

See also

[

1

]

.

References

1

Wikipedia

http://en.wikipedia.org/wiki/Pathological (mathematics)

entry on pathological, mathematics.

Title

pathological

Canonical name

Pathological

Date of creation

2013-03-22 14:41:56

Last modified on

2013-03-22 14:41:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 00A20

Generated on Tue Feb 6 22:20:09 2018 by

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quotations about mathematics

quotations about mathematics

There are many memorable quotations pertaining to mathematics, from mathematicians, scientists, artists, etc. The following small sampling is organized by what kind of person said the given quote, with first priority given to mathematicians.

1

Mathematicians

“God made the integers, and all the rest is the work of man.”

—

Leopold Kronecker

“A mathematician is a device for turning coffee into

theorems

.”

— Pal Erdős

2

Philosophers

“Mathematics possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture.”

— Bertrand Russell

“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

— Bertrand Russell

3

Physicists

“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”

—

Albert Einstein

4

Politicians

“I had a feeling once about Mathematics, that I saw it all — Depth beyond depth was revealed to me — the Byss and Abyss. I saw, as one might see the transit of Venus or even the Lord Mayor’s Show, a quantity

passing through

infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!”

— Winston Churchill

5

Fictional characters

“Math, my dear boy, is nothing more than the lesbian sister of biology.”

— Peter Griffin,

Family Guy

, “When You Wish Upon A Weinstein”

“How about we fire up the old Segway and find a nice quiet field to do

long division

in? I mean, a nice quiet field

in which to

do long division. Sorry, sorry, everybody.”

— Neil Goldman,

Family Guy

, “8 Simple Rules for Buying My Teenage Daughter”

References

1

J. M. Cohen & J. C. Cohen,

Dictionary of Twentieth-Century Quotations

. London: Penguin Books (1995): 115, 328

2

Paul J. Nahin,

Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills

. Princeton: Princeton University Press (2006): xiv

3

Tom Stafford & Matt Webb,

Mind Hacks: Tips and Tools for Using Your Brain

. New York: O’Reilly (2004): 312

Title

quotations about mathematics

Canonical

name

QuotationsAboutMathematics

Date of creation

2013-03-22 18:11:25

Last modified on

2013-03-22 18:11:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Feature

Classification

msc 00A20

Generated on Tue Feb 6 22:20:11 2018 by

LaTeXML
topic entry on algebra

topic entry on algebra

The subject of

algebra

may be defined as the study of

algebraic
systems

, where an algebraic system consists of a set together
with a certain number of

operations

, which are

functions

(or

partial functions

) on this set. A prototypical example of an
algebraic system is the

ring of integers

, which consists of the
set of integers,

{

…

,

-

2

,

-

1

,

0

,

1

,

2

,

…

}

together
with the operations

+

and

×

.

In

addition

to studying

individual

systems, algebraists consider
classes of systems defined by common properties. For

instance

,
the example cited above is an example of a ring, which is an algebraic
system with two operations which

satisfy

certain axioms, such as

distributivity

of one operation over the other.

The reason for considering classes of systems
is in order to save work by stating and proving

theorems

at the
appropriate level of generality. For instance, while the statement
that every integer equals the sum of four squares is specific to the
ring of integers (there are many rings in which this is not the case)
and its proof makes use of specific facts about integers, the proof
of the fact that the

product

of two sums of integers equals the sum
of all products of numbers appearing in the first sum by numbers
appearing in the second sum only involves the distributive law, so
an analogous theorem will hold for any ring. Clearly, it is
wasteful to restate the same theorem and its proof for every ring
so we state and prove it once as a theorem about rings, then apply
it to specific instances of rings.

1.

http://planetmath.org/node/ConceptsInAbstractAlgebra

Concepts in abstract algebra

2.

topics on

group theory

3.

topics on ring theory

4.

topics on ideal

theory

5.

topics on field theory

6.

topics on

homological algebra

7.

topics on

category theory

8.

algebraic k-theory

9.

Special notations in algebra

10.

Topics on polynomials

11.

Topics on field extensions and Galois theory

12.

Entries on finitely generated ideals

13.

http://planetmath.org/node/2530

Topic entry on

linear algebra

14.

http://planetmath.org/node/5663

Concepts in linear algebra

15.

Matrices of special form

16.

http://planetmath.org/MatrixFactorization

Matrix decompositions

17.

Bibliography for group theory

18.

topics on

universal algebra

Title

topic entry on algebra

Canonical name

TopicEntryOnAlgebra

Date of creation

2013-03-22 18:00:02

Last modified on

2013-03-22 18:00:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 00A20

Related topic

TopicEntryOnTheAlgebraicFoundationsOfMathematics

Related topic

OverviewOfTheContentOfPlanetMath

Generated on Tue Feb 6 22:20:25 2018 by

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vacuous

vacuous

Suppose

X

is a set and

P

is a

property

defined as follows:

X

has property

P

if and only if

∀

Y

[

Y

satisfies

condition

1

]

⇒

Y

satisfies condition

2

where condition

1

and condition

2

define the property.
If condition

1

is never satisfied then

X

satisfies property

P

vacuously

.

Examples

1.

If

X

is the set

{

1

,

2

,

3

}

and

P

is the property defined as above with condition

1

=

Y

is a

infinite subset

of

X

, and condition

2

=

Y

contains

7

. Then

X

has property

P

vacously; every infinite subset of

{

1

,

2

,

3

}

contains the number

7

[

1

]

.

2.

The

empty set

is a

Hausdorff space

(vacuously).

3.

Suppose property

P

is defined by the statement :

The present King of France does not exist.

Then either of the following

propositions

is satisfied vacuously.

The present king of France is bald.

The present King of France is not bald.

References

1

Wikipedia

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

entry on Vacuous truth.

Title

vacuous

Canonical

name

Vacuous

Date of creation

2013-03-22 14:42:27

Last modified on

2013-03-22 14:42:27

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Definition

Classification

msc 00A20

Synonym

vacuously

Synonym

vacuously true

Synonym

vacuous truth

Generated on Tue Feb 6 22:20:26 2018 by

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MatheRealism

MatheRealism

MatheRealism

is the position that the amount of

information

in
the

universe

places a limit on the possible contents of mathematics.
Its supporters claim it as a philosophical

foundation of mathematics

.

The

argument

proceeds as follows. The number of atoms is about
10^80 and will remain so forever, limited by the horizon of

observation

and notwithstanding the expansion of the universe; the
remaining part of the universe is causally disconnected from us. The
number of elementary particles is less than 10^100. Although every
atom has infinitely many eigenstates, only a finite number of them can
be distinguished (due to the uncertainty

relation

of quantum
physics). Further the eigenstates, with exception of the ground state,
have a limited life time. All this leads to the result that only a finite number of bits
can be stored in the universe. A

conservative

estimate

is 10^100
bits, but in any case there is an

upper limit

of information

L

.

According to MatheRealism only such numbers exist which are

computable

or can be identified uniquely and addressed individually by any other means. In particular, the supporters of
this view claim that any irrational number which cannot be
represented by less information than is

contained

by its

infinite

string of bits, cannot get addressed at all and that even most

natural
numbers

cannot get addressed because their most economical

representation

requires more information than

L

. According to
MatheRealism, a number which cannot be represented, addressed, or used
otherwise does not exist. This implies that

infinite sets

do not exist.

Note: Although all available numbers have a finite contents of information, there is not a

greatest number

, because, by useful abbreviations, numbers as large as desired can be represented by means of little information.

The expression MatheRealism is touching on materialism. It may not be mixed up with the notion ”realism” of current philosophy of mathematics which in fact is an idealism.

Literature

W. Mückenheim: Die Mathematik des Unendlichen, Shaker, Aachen 2006.

Title

MatheRealism

Canonical name

MatheRealism

Date of creation

2014-10-29 17:12:48

Last modified on

2014-10-29 17:12:48

Owner

WM (16977)

Last modified by

WM (16977)

Numerical id

16

Author

WM (16977)

Entry type

Topic

Classification

msc 00A30

Related topic

DecimalExpansion

Generated on Tue Feb 6 22:20:30 2018 by

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ultra-complex systems

ultra-complex systems

\xyoption

curve

0.1

Ultra-complex systems

An

ultra-complex system

represents

the human mind from the standpoint
of a generalized

Categorical Ontology

Theory of Levels as the highest level of complexity that
emerged through biological and social coevolution over the last

2.2

million years on Earth.

0.1.1

Preliminary Data:

One can represent in square categorical

diagrams

the emergence of ultra-complex
dynamics from the super-complex dynamics of human organisms coupled

via

social interactions
in

characteristic

patterns represented by

http://planetmath.org/RosettaGroupoids

Rosetta biogroupoids, together with the complex–albeit inanimate–systems with ‘chaos’. With the emergence of the ultra-complex system of the human mind– based on the super-complex human organism– there is always an associated
progression towards

higher dimensional algebras

from the lower
dimensions of human neural network dynamics and the simple algebra
of physical dynamics, as shown in the following, essentially

non-commutative

categorical diagram.

Definition 0.1

.

An

ultra-complex system,

U

C

⁢

S

is defined as an object

representation

in the following non-commutative
diagram of systems and dynamic system

morphisms

or ‘dynamic

transformations

’:

\xymatrix

@

C

=

5

p

c

[

S

U

P

E

R

-

C

O

M

P

L

E

X

]

\ar

[

r

]

(

Higher
Dim

)

\ar

[

d

]

Λ

&

(

U

C

⁢

S

=

U

L

T

R

A

-

C

O

M

P

L

E

X

)

\ar

[

d

]

o

⁢

n

⁢

t

⁢

o

C

O

M

P

L

E

X

&

\ar

[

l

]

(

Generic Map

)

[

S

I

M

P

L

E

]

Note that the above diagram is indeed not ‘natural’ (i.e. it is not

commutative

) for reasons
related to the emergence of the higher dimensions of the super–complex
(biological/organismic) and/or ultra–complex (psychological/neural network dynamic) levels in comparison with
the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems.

Title

ultra-complex systems

Canonical name

UltracomplexSystems

Date of creation

2013-03-22 18:11:40

Last modified on

2013-03-22 18:11:40

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

19

Author

bci1 (20947)

Entry type

Topic

Classification

msc 00A30

Classification

msc 18A40

Classification

msc 37B25

Classification

msc 93D15

Classification

msc 18A05

Classification

msc 93D21

Classification

msc 37B10

Synonym

extremely complex systems

Synonym

the mind

Related topic

CategoricalOntology

Related topic

RosettaGroupoids

Related topic

StrongAIThesis

Generated on Tue Feb 6 22:20:32 2018 by

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errors can cancel each other out

errors can cancel each other out

If one uses the

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

change of variable

tan

⁡

x

:=

t

,

d

⁢

x

=

d

⁢

t

1

+

t

2

,

cos

2

⁡

x

=

1

1

+

t

2

(1)

for finding the value of the

definite integral

I

:=

∫

π

4

3

⁢

π

4

d

⁢

x

2

⁢

cos

2

⁡

x

+

1

,

the following calculation looks appropriate and faultless:

I

=

∫

1

-

1

d

⁢

t

3

+

t

2

=

1

3

⁢

/

1

-

1

⁡

arctan

⁡

t

3

=

1

3

⁢

(

5

⁢

π

6

-

π

6

)

=

2

⁢

π

3

⁢

3

(2)

The result is quite .  Unfortunately, the calculation two errors, the effects of which cancel each other out.

The crucial error in (2) is using the substitution (1) when

tan

⁡

x

is

discontinuous

in the point

x

=

π

2

on the

interval

[

π

4

,

3

⁢

π

4

]

of integration.  The error is however canceled out by the second error using the value

5

⁢

π

6

for

arctan

⁡

-

1

3

, when the right value were

-

π

6

(the values of arctan lie only between

-

π

2

and

π

2

; see

cyclometric functions

).  The value

5

⁢

π

6

belongs to a different branch of the

inverse tangent

function

than

π

6

; parts of two distinct branches cannot together form the

antiderivative

which must be

continuous

.

What were a right way to calculate

I

?  The

universal trigonometric substitution

produces an awkward

integrand

2

+

2

⁢

t

2

3

-

2

⁢

t

2

+

3

⁢

t

4

and

2

-

1

and

1

-

2

,  therefore it is unusable.  It is now better to change the interval of integration, using the

properties

of

trigonometric functions

.

Since the (graph of) cosine

squared

is

symmetric about

the line

x

=

π

2

,  we could integrate only over

[

π

4

,

π

2

]

and multiply the

integral

by 2 (cf.

integral of even and odd functions

):

I

=

2

⁢

∫

π

4

π

2

d

⁢

x

2

⁢

cos

2

⁡

x

+

1

.

We can also get rid of the inconvenient

upper limit

π

2

by changing over to the sine in virtue of the

complement formula

cos

⁡

(

π

2

-

x

)

=

sin

⁡

x

,

getting

I

=

2

⁢

∫

0

π

4

d

⁢

x

2

⁢

sin

2

⁡

x

+

1

.

Then (1) is usable, and because

sin

2

⁡

x

=

t

2

1

+

t

2

,  we obtain

I

=

2

⁢

∫

0

1

d

⁢

t

(

2

⁢

t

2

1

+

t

2

+

1

)

⁢

(

1

+

t

2

)

=

2

⁢

∫

0

1

d

⁢

t

3

⁢

t

2

+

1

=

2

3

⁢

/

0

1

⁡

arctan

⁡

t

⁢

3

=

2

⁢

π

3

⁢

3

.

Title

errors can cancel each other out

Canonical

name

ErrorsCanCancelEachOtherOut

Date of creation

2013-03-22 18:59:39

Last modified on

2013-03-22 18:59:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Example

Classification

msc 00A35

Classification

msc 26A06

Classification

msc 97D70

Related topic

UniversalTrigonometricSubstitution

Related topic

SubstitutionNotation

Related topic

IntegrationOfRationalFunctionOfSineAndCosine

Generated on Tue Feb 6 22:20:34 2018 by

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preservation and reflection

preservation and reflection

In mathematics, the word “preserve” usually means the “preservation of properties”. Loosely speaking, whenever a mathematical

construct

A

has some property

P

, after

A

is somehow “transformed” into

A

′

, the transformed object

A

′

also has property

P

. The constructs usually refer to sets and the

transformations

typically are

functions

or something

similar

.

Here is a simple example, let

f

be a function from a set

A

to

B

. Let

A

be a

finite set

. Let

P

be the property of a set being finite. Then

f

preserves

P

, since

f

⁢

(

A

)

is finite. Note that we are not saying that

B

is finite. We are merely saying that the portion of

B

that is the

image

of

A

(the transformed portion) is finite.

Here is another example. The property of being

connected

in a

topological space

is preserved under a

continuous function

. Here, the constructs are topological spaces, and the transformation is a continuous function. In other words, if

f

:

X

→

Y

is a continuous function from

X

to

Y

. If

X

is connected, so is

f

⁢

(

X

)

⊆

Y

.

Many more examples can be found in

abstract algebra

.

Group homomorphisms

, for example, preserve

commutativity

, as well as the property of being

finitely generated

.

The word “reflect” is the dual notion of “preserve”. It means that if the transformed object has property

P

, then the original object also has property

P

. This usage is rarely found outside of

category theory

, and is almost exclusively reserved for

functors

. For example, a

faithful functor

reflects

isomorphism

: if

F

is a faithful functor from

𝒞

to

𝒟

, and the object

F

⁢

(

A

)

is isomorphic to the object

F

⁢

(

B

)

in

𝒟

, then

A

is isomorphic to

B

in

𝒞

.

Title

preservation and reflection

Canonical name

PreservationAndReflection

Date of creation

2013-03-22 17:12:18

Last modified on

2013-03-22 17:12:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 00A35

Defines

preserve

Defines

reflect

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