aarjaneiro/planetmath huggingface dataset:
probabilistic proof

probabilistic proof

Most exciting results in mathematics make use of standard proofs; however, some emerging mathematics is making use of proofs nearly as exciting as the results. One of these methods in the

Probabilistic method

. The following is an outline of how such a proof might proceed.

Theorem

structure

:

Let

X

be a finite family of objects. We wish to show every element of

X

has some

property

T

.

Proof scheme:

(i)

Show that the probability that an object

A

in

X

has the property

T

is

1

-

ε

for some

0

≤

ε

≤

1

.

(ii)

Suppose

A

∈

X

is some object without property

T

. Use

A

to construct
more than

ε

⁢

|

X

|

other objects in

X

also without property

T

.

(iii)

Conclude that if an object

A

∈

X

does not have property

T

then the probablity that a random element of

X

has property

T

is less than

1

-

ε

thus creating a

contradiction

. So every

element

in

X

has property

T

.

(See

Examples of probabilistic proofs

.)

The surprising discovery is that in some problems it is possible to show both (i) and (ii) thus establishing the contradiction in (iii). One would typically expect that to demonstrate the probablity is

1

-

ϵ

would necesitate knowing that there are no more than

ϵ

counterexamples

, but this is not always the case.

Proofs of this flavor appear in

graph theory

because of a wealth of knowledge about

random graphs

and random walks etc. (Example, the work of Alon on

H

-universal graphs.) Establishing the necessary probabilities is therefore possible without carefully enumerating the elements in

X

. Then beginning with a possible counterexample, combinatorial changes are made to the candidate to create too many counterexamples.

A second technique applies to the mathematics of logic, more explicitly

model theory

. Recent work by Marcus du Sautoy treats problems on counting the number of

p

-groups through model theory.

Both proof techniques are highly non-constructive and can leave a reader wondering if they trust the result, but each is as foundational as a proof by induction.

Title

probabilistic proof

Canonical

name

ProbabilisticProof

Date of creation

2013-03-22 15:53:44

Last modified on

2013-03-22 15:53:44

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Example

Classification

msc 00A35

Generated on Tue Feb 6 22:20:49 2018 by

LaTeXML
mathematics of finance

mathematics of finance

From ancient times, a socially significant application of mathematics
has been in accounting. This entry points to places on
PlanetMath which deal with this topic.

1.

interest

2.

simple interest

3.

compound interest

, continuously compounded

4.

interest rate

5.

effective interest rate

6.

instantaneous effective interest rate

7.

present value

, net present value

8.

coupon rate

9.

rate of return

10.

yield rate, yield curve

11.

force of interest

12.

annuity

13.

perpetuity

14.

call option, put option,

15.

American option, European option

16.

arbitrage, the arbitrage

theorem

17.

risk neutral

18.

Black-Scholes formula

19.

capital asset pricing model (CAPM)

Title

mathematics of finance

Canonical

name

MathematicsOfFinance

Date of creation

2013-03-22 16:40:17

Last modified on

2013-03-22 16:40:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 00A69

Classification

msc 91B28

Classification

msc 00A06

Generated on Tue Feb 6 22:21:13 2018 by

LaTeXML
arrow

arrow

An

arrow

is a line, usually straight, capped at one end, or both ends, with an arrowhead which points in a specific direction. In a flowchart, arrows show the order in which instructions are to be performed (including where branching may occur) or they show the flow of cause and effect. In certain

arrangements

of numbers, such as the Collatz tree, arrows show which numbers are the input values, with the output value being pointed at by the arrowhead.

In writing about vectors, arrows can be written over the symbols for the origin and the

endpoint

, e.g.,

A

⁢

B

→

and

C

⁢

D

←

. TeX provides commands for these.

An arrow, regarded as a

morphism

,

functor

or

natural transformation

also plays

central

roles
in

Category Theory

and

Categorical Dynamics

.

Title

arrow

Canonical name

Arrow

Date of creation

2013-03-22 12:55:46

Last modified on

2013-03-22 12:55:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

14

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Related topic

Morphism

Related topic

Functor

Related topic

NaturalTransformation

Related topic

Mapping

Related topic

Vector

Related topic

Category

Generated on Tue Feb 6 22:21:15 2018 by

LaTeXML
blackboard bold

blackboard bold

Certain capital letters are sometimes written in

blackboard bold

and considered to be a distinct symbol with a distinct meaning from their regular counterparts. This is to help alleviate overloading of the letters of the Latin-1 alphabet in mathematics and physics. For example, the blackboard letter

ℂ

signifies the

complex numbers

, while in the same set of formulas one could also use

C

as some

variable

or constant,

complex

or not, confident that it would not be confused with the complex numbers in general.

Mathematicians are not unanimous in their approval of blackboard bold, however, and

Donald Knuth

did not include support for it in TeX. The

American Mathematical Society

created the TeX package

amsfonts

which

PlanetMath

document preambles contain by default. The entire Latin-1 alphabet is available thus, while Unicode in Letterlike Symbols only makes available a subset of the most commonly used (such as

ℚ

and

ℤ

).

In typewritten dissertations, and even some published books, simple bold of non-italics letters is used.

Title

blackboard bold

Canonical

name

BlackboardBold

Date of creation

2013-03-22 16:45:35

Last modified on

2013-03-22 16:45:35

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:16 2018 by

LaTeXML
empty product

empty product

The

empty product

of numbers is the borderline case of

product

, where the number of is empty.  The most usual examples are the following.

•

The

http://planetmath.org/Introducing0thPower

zeroth power of a non-zero number:

a

0

•

The factorial of 0:  0!

•

The

http://planetmath.org/FundamentalTheoremOfArithmetics

prime factor

presentation

of unity, which has no prime factors

The value of the empty sum of numbers is equal to the

additive identity

number, 0.  Similarly, the empty product of numbers is equal to the

http://planetmath.org/Unity

multiplicative identity

number, 1.

Note.

When considering the

complex numbers

as pairs of real numbers one often identifies the pairs

(

x

,

0

)

and the reals

x

.  In this sense one can think that the

Cartesian product

ℝ

×

{

0

}

is equal to

ℝ

.  This seems to the

equation

ℝ

×

ℝ

0

=

ℝ

1

+

0

=

ℝ

1

=

ℝ

,

although the

http://planetmath.org/GeneralAssociativity

associativity

of Cartesian product is nowhere stated.  Nevertheless, it is sometimes natural to define that the Cartesian product of an empty collection of sets equals to a set with one element; so it may that e.g.

ℝ

0

=

{

0

}

.

One can also consider empty products in

categories

.
It follows directly from the definition that an object in a category
is a

http://planetmath.org/CategoricalDirectProduct

product
of an empty family of objects in the category
if and only if it is a

terminal object

of the category.
Sets are a special case of this:
in the

category of sets

the singletons are the terminal objects,
so the empty product exists and is a singleton.

Title

empty product

Canonical name

EmptyProduct

Date of creation

2013-03-22 14:48:13

Last modified on

2013-03-22 14:48:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A99

Related topic

EmptySet

Related topic

IndeterminateForm

Related topic

LocallyEuclidean

Related topic

AnalyticContinuationOfGammaFunction

Related topic

Introducing0thPower

Related topic

EmptySum

Generated on Tue Feb 6 22:21:18 2018 by

LaTeXML
Jordan’s inequality

Jordan’s inequality

Jordan’s

Inequality

states that

2

π

⁢

x

≤

sin

⁡

(

x

)

≤

x

for all

x

∈

[

0

,

π

2

]

.

Title

Jordan’s inequality

Canonical name

JordansInequality

Date of creation

2013-03-22 11:42:29

Last modified on

2013-03-22 11:42:29

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

25

Author

Koro (127)

Entry type

Theorem

Classification

msc 00A99

Classification

msc 26D05

Related topic

ComparisonOfSinThetaAndThetaNearTheta0

Generated on Tue Feb 6 22:21:20 2018 by

LaTeXML
Köchel number

Köchel number

The

Köchel number

K

n

of a by Wolfgang Amadeus Mozart is its index in the ordered set compiled by Ludwig von Köchel in the conjectured chronological of .

K

n

25

+

10

approximates Mozart’s age at time of writing reasonably well for

n

>

100

(thus

K

n

25

+

1766

approximates the year).

References

1

L. von Köchel,

Chronologisch-thematisches Verzeichnis sämtlicher Tonwerke Wolfgang Amade Mozarts; nebst Angabe der verlorengegangenen, angefangenen, übertragenen, zweifelhaften und unterschobenen Kompositionen

, (1947), Ann Arbor: J. W. Edwards.

Title

Köchel number

Canonical

name

KochelNumber

Date of creation

2013-03-22 16:06:58

Last modified on

2013-03-22 16:06:58

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 00A99

Synonym

Kochel number

Related topic

OpusNumber

Generated on Tue Feb 6 22:21:22 2018 by

LaTeXML
LaTeXsymbol for Cauchy principal value

LaTeXsymbol for Cauchy principal value

The usual symbol (dashed

integral

) used to denote Cauchy’s principal value of an integral can be created in

LaTeX

through macros.

1

1

UK

List of

TeX

is a reference.

These one are given by the following instructions, which must be included on the

preamble

.

\def\Xint#1{\mathchoice
{\XXint\displaystyle\textstyle{#1}}%
{\XXint\textstyle\scriptstyle{#1}}%
{\XXint\scriptstyle\scriptscriptstyle{#1}}%
{\XXint\scriptscriptstyle\scriptscriptstyle{#1}}%
\!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$}
\vcenter{\hbox{$#2#3$}}\kern-.5\wd0}}
\def\ddashint{\Xint=}
\def\dashint{\Xint-}

The commands to execute those macros are

`

⁢

`

\

d

⁢

a

⁢

s

⁢

h

⁢

i

⁢

n

⁢

t

⁢

"

and

`

⁢

`

\

d

⁢

d

⁢

a

⁢

s

⁢

h

⁢

i

⁢

n

⁢

t

⁢

"

for single dash and double dash, respectively. Let us expose a few examples.

•

=

∫

Ω

F

⁢

(

ζ

,

η

)

⁢

𝑑

ζ

⁢

𝑑

η

•

-

∫

z

0

z

f

⁢

(

ζ

)

⁢

𝑑

ζ

•

E

i

(

z

)

=

-

-

∫

-

z

∞

e

-

ζ

ζ

d

ζ

=

-

∫

-

∞

z

e

ζ

ζ

d

ζ

,

ℜ

z

>

0

,

(exponential integral)

•

l

⁢

i

⁢

(

z

)

=

-

∫

0

z

d

⁢

ζ

log

⁡

ζ

≡

E

⁢

i

⁢

(

log

⁡

z

)

,

ℜ

⁡

z

>

1

,

(logarithmic integral)

•

Γ

⁢

(

z

)

⁢

Γ

⁢

(

1

-

z

)

=

-

z

⁢

Γ

⁢

(

-

z

)

⁢

Γ

⁢

(

z

)

=

-

∫

0

∞

ζ

z

-

1

ζ

+

1

⁢

𝑑

ζ

=

π

⁢

csc

⁡

π

⁢

z

,

0

<

ℜ

⁡

z

<

1

,

(Gamma function reflection’s formula)

Title

LaTeXsymbol for

Cauchy principal value

Canonical name

LaTeXSymbolForCauchyPrincipalValue

Date of creation

2013-03-22 17:49:51

Last modified on

2013-03-22 17:49:51

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

5

Author

perucho (2192)

Entry type

Feature

Classification

msc 00A99

Related topic

LogarithmicIntegral2

Generated on Tue Feb 6 22:21:24 2018 by

LaTeXML
opus number

opus number

An

opus number

is a number attached to a particular musical

composition

or set of compositions by a publisher to denote the order of publication by the given composer.
Theoretically, opus numbers would give musicologists a simple means to chart the growth of a composer chronologically, to make statements to the effect that there is a direct

correlation

between such and such composer’s handling of a particular musical technique and the opus numbers.

In practice, however, many factors contribute to making the opus numbers for certain composers less than useful, such as their being assigned by the composer rather than the publisher. Antonín Dvořák, for example, tried to fool publishers in an effort to get the best deal, and thus often gave misleadingly low or high opus numbers to his compositions. The confusion with Carl Nielsen’s opus numbers arose not from any wheeling and dealing, but from his not having an exclusive publisher and not keeping track of which opus numbers he hadn’t used (see sequence A113529 in Sloane’s OEIS for a listing of opus numbers skipped over by Carl Nielsen).

Other composers just never had any opus numbers given to their compositions by either publishers or themselves; musicologists then sometimes try to create a catalog with numbers named after themselves. In the case of Mozart, the Köchel numbers by Ludwig von Köchel are almost universally accepted, but in the case of Domenico Scarlatti, three different musicologists came up with three different catalogs, leading to many of Scarlatti’s sonatas to be known by three different numbers. The lack of opus numbers is not a problem for composers whose repertory works consist of less than a dozen of large-scale works (e.g., Bruckner and Mahler).

References

1

Schönzeler, H. H.

Dvořák

London: M. Boyers (1984): 220 - 239

Title

opus number

Canonical

name

OpusNumber

Date of creation

2013-03-22 16:44:58

Last modified on

2013-03-22 16:44:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Related topic

KochelNumber

Generated on Tue Feb 6 22:21:26 2018 by

LaTeXML
QED

QED

The term “QED” is actually an abbreviation and stands for the Latin

quod erat demonstrandum

, meaning “which was to be demonstrated.”

QED typically is used to signify the end of a mathematical proof. The symbol

□

is often used in place of “QED,” and is called the “

tombstone

”, “Halmos symbol” or “Halmos tombstone” after mathematician Paul Halmos (it can vary in width, however, and sometimes it is fully or partially shaded). Halmos borrowed this symbol from magazines, where it was used to denote “end of article”.

Title

QED

Canonical

name

QED

Date of creation

2013-03-22 12:40:14

Last modified on

2013-03-22 12:40:14

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

8

Author

mathwizard (128)

Entry type

Definition

Classification

msc 00A99

Synonym

Q.E.D

Related topic

QEDInTheoreticalAndMathematicalPhysics

Related topic

QCDOrQuantumChromodynamics

Related topic

MathematicalFoundationsOfQuantumFieldTheories

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Related topic

FoundationsOfQuantumFieldTheories

Related topic

QuantumChromod

Defines

Halmos symbol

Defines

tombstone

Defines

Halmos tombstone

Generated on Tue Feb 6 22:21:32 2018 by

LaTeXML
Q holding bay 47

Q holding bay 47

The holding bay is now empty.

Title

Q holding bay 47

Canonical

name

QHoldingBay47

Date of creation

2013-03-22 17:48:46

Last modified on

2013-03-22 17:48:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:33 2018 by

LaTeXML
seven-segment display

seven-segment display

Most

calculators

use a

seven-segment display

to show both input and output numbers. Here is Champernowne’s constant

C

10

to 9

decimal places

in a seven-segment display:

Technically, the space for the

decimal point

counts as a segment, so for each place value there are actually eight segments. Generally, calculator manufacturers prefer the base 10

pandigital number

1234567980 as a demo number for the calculator packaging.

Sometimes wear and tear can cause a segment to not light up or darken properly; this can occasionally lead to confusion between digits. The simplest way to test for this is to input

8

×

10

x

-

1

9

, where

x

is the the number of decimal place values available on the display (that is, a bunch of 8s). At various times in the past some manufacturers have developed variants of the glyphs for the digits to make it more

obvious

when a segment has burnt out, such as a 0 that looks like a lowercase O, or a 6 that looks like a lowercase B. Given that many

scientific calculators

offer the option of doing calculations in

hexadecimal

, the latter option is not acceptable for such calculators. The 7 on Sharp-brand calculators uses the upper left vertical segment.

Title

seven-segment display

Canonical

name

SevensegmentDisplay

Date of creation

2013-03-22 17:06:51

Last modified on

2013-03-22 17:06:51

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:35 2018 by

LaTeXML
Siegel’s axiom

Siegel’s axiom

In a mathematical text two mathematical symbols that do not form a larger of have to be by at least one word or by a punctuation mark.

Example:

Do not write “… take

n

n

×

n

-matrices”, but rather “take

n

matrices of order

n

×

n

”.

Title

Siegel’s axiom

Canonical name

SiegelsAxiom

Date of creation

2013-03-22 14:11:59

Last modified on

2013-03-22 14:11:59

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

8

Author

mathwizard (128)

Entry type

Axiom

Classification

msc 00A99

Generated on Tue Feb 6 22:21:36 2018 by

LaTeXML
superscript

superscript

A

superscript

is a symbol or group of symbols written above, and usually a bit to the right, of another symbol or group of symbols in order to show a

relation

among the two. The most common use of superscripts in mathematics is for notating

exponentiation

. For example, in the expression

4

7

, the 7 is a superscript to the 4. Iterated sum and

product

notation also uses superscripts most of the time, to show the iterator end values, and the same goes for integral notation. Notation of iterated functions also benefit from using superscripts to cut down on nested parentheses, e.g.,

f

3

⁢

(

x

)

=

f

⁢

(

f

⁢

(

f

⁢

(

x

)

)

)

; examples of iterated functions include the

iterated totient function

ϕ

i

⁢

(

n

)

and

iterated sum of divisors function

σ

i

⁢

(

n

)

. The contravariant indices of tensors are also denoted by superscripts.

Nonfiction literature in general uses number superscripts to indicate footnotes with the corresponding numbers. This usage is often avoided in mathematical books and papers to prevent possible confusion with exponentiation.

Title

superscript

Canonical name

Superscript

Date of creation

2013-03-22 17:06:45

Last modified on

2013-03-22 17:06:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:38 2018 by

LaTeXML
TFAE

TFAE

The abbreviation “TFAE” is shorthand for “

t

he

f

ollowing

a

re

e

quivalent”. It is used before a set of

equivalent

conditions (each implies all the others).

In a

definition

, when one of the conditions is somehow “better” (simpler, shorter, …), it makes sense to phrase the definition with that condition, and mention that the others are equivalent. “TFAE” is typically used when none of the conditions can take priority over the others. Actually proving the claimed equivalence must, of course, be done separately.

Title

TFAE

Canonical

name

TFAE

Date of creation

2013-03-22 12:44:40

Last modified on

2013-03-22 12:44:40

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

4

Author

ariels (338)

Entry type

Definition

Classification

msc 00A99

Synonym

the following are equivalent

Generated on Tue Feb 6 22:21:39 2018 by

LaTeXML
text figures

text figures

Some fonts have a

text figures

variant of the digits 0 to 9 where some of them have ascenders and some have descenders so that they blend better with the general text. Here is Champernowne’s constant

C

10

to 9

decimal places

written in Georgia:

Usually, 6 and 8 have ascenders, 3, 4, 5, 7 and 9 have descenders. Since most texts use base 10 numbers, and each digit usually occurs in about the same proportion as the other digits, this means that numbers have varying heights in just about the same propoertions as the words in the text. Using text figures for base 3 numbers would result in numbers always lacking in ascenders and descenders. For mathematical books and papers, however, text figures are generally eschewed.

Title

text figures

Canonical

name

TextFigures

Date of creation

2013-03-22 17:06:48

Last modified on

2013-03-22 17:06:48

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A99

Generated on Tue Feb 6 22:21:46 2018 by

LaTeXML
topic entry on miscellaneous mathematics

topic entry on miscellaneous mathematics

This topic provides links to those areas that are not restricted to Mathematics, but rather are metamathematical, or relevant to mathematics, though more general in nature.

1.

Applied Mathematics;

some physical applications:

catacaustic

, Heron’s principle, uncertainty principle,

motion in central-force field

, heat equation, wave equation

2.

Definition

(s) of

Metamathematics

3.

http://

planetmath

.org/FormalLogicsAndMetaMathematics

Formal Logics

and Metamathematics

4.

Mathematical logics

5.

Theory of Models, AI, ‘Automatic’ proof generation

6.

Problem solving techniques and guides

7.

Mathematics vocabulary

8.

Automatic translation and Meta-languages

9.

Categorical Ontology

10.

http://planetmath.org/GWLeibnizSQuote

Does Mathematics need Philosophy?

11.

Mathematical philosophy and

http://planetmath.org/BeyondFormalism

Philosophical mathematics
(

Philosophy of Mathematics

and Mathematics of Philosophy)

12.

Mathematical (Theoretical) physics and Physical mathematics

13.

Quotations about mathematics

14.

:

http://planetmath.org/TermsFromForeignLanguagesUsedInMathematics

html version,

http://planetmath.org/TermsFromForeignLanguagesUsedInMathematicsPageImagesVersion

page images version

15.

Mathematics education

16.

Characterisation

17.

http://planetmath.org/PlatosMathematics

History of Mathematics and

http://planetmath.org/AnOutlineOfHilbertsProgramme

mathematical

concepts

18.

Biographies on PlanetMath

19.

Resources for Mathematics: bibliographies, books, textbooks, mathematics resources on the web,

computer

softare/programs, web links to other encyclopedic references concerned with, or related to, Mathematics

20.

Mathematical applications,

paradoxes

and puzzles

Title

topic entry on miscellaneous mathematics

Canonical

name

TopicEntryOnMiscellaneousMathematics

Date of creation

2013-03-22 18:00:42

Last modified on

2013-03-22 18:00:42

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

18

Author

rm50 (10146)

Entry type

Topic

Classification

msc 00A99

Classification

msc 00A20

Related topic

Metamathematics

Related topic

AnOutlineOfHilbertsProgramme

Related topic

BibliographyForAxiomaticsAndMathematicsFoundationsInCategories

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