aarjaneiro/planetmath huggingface dataset:
Rydia’s Mathemagic Minute

Rydia’s Mathemagic Minute

Rydia’s Mathemagic Minute

is a mini-game from

Final Fantasy XIII

in which Rydia presents the player a riddle in the form of four numbers, each in the range

-

1

<

n

<

10

which the player must then employ in

conjunction

with a selection of basic

arithmetic operations

(

addition

,

subtraction

,

multiplication

,

division

) in order to obtain 10 as a result. For example, given {8, 8, 6, 4}, one possible

solution

is

8

×

8

-

4

÷

6

. If the player can solve the riddle, he is given another riddle of the same form, until exhausting the time limit of 90 seconds. A player may skip a riddle and move on to another one, but in so doing incurs a penalty in

negative

points. Correctly answering five riddles in a row results in the player receives a time limit

extension

.

Concatenation

is not allowed. For example, given {1, 7, 2, 9},

19

-

(

2

+

7

)

is not a valid answer; the program in fact will not allow the player to even try concatenation. Answers involving

negative numbers

at intermediate steps are not valid either, though when they occur they’re displayed in blue (in lieu of a minus sign).

Fractions

are not at all allowed at intermediate steps, and when the player presents an

operation

that would result in such a result (e.g., dividing a number by a smaller

coprime

number), the

computer

simply rejects the operation.

The mini-game can be played either as a side quest on the full game, or by itself online at quare Enix Members.

Title

Rydia’s Mathemagic Minute

Canonical

name

RydiasMathemagicMinute

Date of creation

2013-03-22 18:57:16

Last modified on

2013-03-22 18:57:16

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Generated on Tue Feb 6 22:18:37 2018 by

LaTeXML
saddle point approximation

saddle point approximation

The

saddle point approximation

(SPA), a.k.a. stationary phase approximation, is a widely used method in quantum field theory (QFT) and related fields.
Suppose we want to evaluate the following

integral

in the limit

ζ

→

∞

:

ℐ

=

lim

ζ

→

∞

⁡

∫

-

∞

∞

d

⁢

x

⁢

e

-

ζ

⁢

f

⁢

(

x

)

.

(1)

The saddle point approximation can be applied if the

function

f

⁢

(

x

)

satisfies certain conditions. Assume that

f

⁢

(

x

)

has a

global minimum

f

⁢

(

x

0

)

=

y

m

⁢

i

⁢

n

at

x

=

x

0

, which is sufficiently separated from other local minima and whose value is sufficiently smaller than the value of those.
Consider the

Taylor expansion

of

f

⁢

(

x

)

about the point

x

0

:

f

⁢

(

x

)

=

f

⁢

(

x

0

)

+

∂

x

⁡

f

⁢

(

x

)

|

x

=

x

0

⁢

(

x

-

x

0

)

+

1

2

⁢

∂

x

2

⁡

f

⁢

(

x

)

|

x

=

x

0

⁢

(

x

-

x

0

)

2

+

O

⁢

(

x

3

)

.

(2)

Since

f

⁢

(

x

0

)

is a (global)

minimum

, it is clear that

f

′

⁢

(

x

0

)

=

0

. Therefore

f

⁢

(

x

)

may be approximated to quadratic order as

f

⁢

(

x

)

≈

f

⁢

(

x

0

)

+

1

2

⁢

f

′′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

2

.

(3)

The above assumptions on the minima of

f

⁢

(

x

)

ensure that the dominant contribution to (

1

) in the limit

ζ

→

∞

will
come from the region of integration around

x

0

:

ℐ

≈

lim

ζ

→

∞

⁡

e

-

ζ

⁢

f

⁢

(

x

0

)

⁢

∫

-

∞

∞

d

⁢

x

⁢

e

-

ζ

2

⁢

f

′′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

2

(4)

≈

lim

ζ

→

∞

⁡

e

-

ζ

⁢

f

⁢

(

x

0

)

⁢

(

2

⁢

π

ζ

⁢

f

′′

⁢

(

x

0

)

)

1

/

2

.

In the last step we have performed the Gaußian integral.
The next nonvanishing higher order correction to (

4

) stems from the quartic term of the expansion (

2

). This correction may be incorporated into (

4

) to yield (after expanding part of the

exponential

):

ℐ

≈

lim

ζ

→

∞

⁡

e

-

ζ

⁢

f

⁢

(

x

0

)

⁢

∫

-

∞

∞

d

⁢

x

⁢

e

-

ζ

2

⁢

f

′′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

2

⁢

(

1

-

ζ

4

!

⁢

(

∂

x

4

⁡

f

⁢

(

x

)

)

|

x

=

x

0

⁢

(

x

-

x

0

)

4

)

.

(5)

…to be continued with applications to physics…

Title

saddle point approximation

Canonical

name

SaddlePointApproximation

Date of creation

2013-03-22 13:38:07

Last modified on

2013-03-22 13:38:07

Owner

msihl (2134)

Last modified by

msihl (2134)

Numerical id

5

Author

msihl (2134)

Entry type

Topic

Classification

msc 00A05

Classification

msc 00A79

Synonym

stationary phase method

Generated on Tue Feb 6 22:18:39 2018 by

LaTeXML
singleton

singleton

A

singleton

is a set containing a single element.

Title

singleton

Canonical

name

Singleton

Date of creation

2013-03-22 13:13:41

Last modified on

2013-03-22 13:13:41

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 00A05

Related topic

ATopologicalSpaceIsT_1IfAndOnlyIfEverySingletonIsClosed

Generated on Tue Feb 6 22:18:41 2018 by

LaTeXML
stable subspace

stable subspace

A subset

S

of a larger set

T

is said to a

stable subset

for a function

f

:

T

→

T

iff

f

⁢

(

S

)

⊂

S

.

Alternative phrasings with the same meaning are:

•

f

is an invariant subset for

f

•

f

stabilizes

S

•

S

is stable under (the action of)

f

•

S

is

invariant

under (the action of)

f

•

S

is left stable by/under

f

•

S

is left invariant by/under

f

Title

stable subspace

Canonical

name

StableSubspace

Date of creation

2013-03-22 17:56:52

Last modified on

2013-03-22 17:56:52

Owner

lalberti (18937)

Last modified by

lalberti (18937)

Numerical id

6

Author

lalberti (18937)

Entry type

Definition

Classification

msc 00A05

Synonym

invariant subspace

Synonym

stable subset

Synonym

invariant subset

Generated on Tue Feb 6 22:18:42 2018 by

LaTeXML
subsequence

subsequence

Given a

sequence

{

x

n

}

n

∈

ℕ

, any

infinite subset

of the sequence forms a

subsequence

. We formalize this as follows:

Definition

.

If

X

is a set and

{

a

n

}

n

∈

N

is a sequence in

X

, then a

subsequence

of

{

a

n

}

is a sequence of the form

{

a

n

r

}

r

∈

N

where

{

n

r

}

r

∈

N

is a

strictly increasing

sequence of

natural numbers

.

Equivalently,

{

y

n

}

n

∈

ℕ

is a subsequence of

{

x

n

}

n

∈

ℕ

if

1.

{

y

n

}

n

∈

ℕ

is a sequence of

elements of

X

, and

2.

there is a

strictly increasing function

a

:

ℕ

→

ℕ

such that

y

n

=

x

a

⁢

(

n

)

for all

⁢

n

∈

ℕ

.

Example

.

Let

X

=

ℝ

and let

{

x

n

}

be the sequence

{

1

n

}

n

∈

ℕ

=

{

1

,

1

2

,

1

3

,

1

4

,

…

}

.

Then, the sequence

{

y

n

}

n

∈

ℕ

=

{

1

n

2

}

n

∈

ℕ

=

{

1

,

1

4

,

1

9

,

1

16

,

…

}

is a subsequence of

{

x

n

}

. The subsequence of natural numbers mentioned in the

definition

is

{

n

2

}

n

∈

ℕ

and the

function

a

:

ℕ

→

ℕ

mentioned above is

a

⁢

(

n

)

=

n

2

.

Title

subsequence

Canonical

name

Subsequence

Date of creation

2013-03-22 12:56:34

Last modified on

2013-03-22 12:56:34

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 00A05

Generated on Tue Feb 6 22:18:44 2018 by

LaTeXML
sum of odd numbers

sum of odd numbers

The sum of the first

n

positive

odd integers

can be calculated by using the well-known of the

arithmetic progression

, that the sum of its is equal to the

arithmetic mean

of the first and the last , multiplied by the number of the :

1

+

3

+

5

+

7

+

9

+

⋯

+

(

2

⁢

n

-

1

)

⏟

n

=

n

⋅

1

+

(

2

⁢

n

-

1

)

2

=

n

2

Thus, the sum of the first

n

odd numbers

is

n

2

(this result has been proved first time in 1575 by Francesco Maurolico).

Below, the odd numbers have been set to form a

triangle

, each

n

th

row containing the next

n

consecutive odd numbers.  The arithmetic mean on the row is

n

2

and the sum of its numbers is

n

⋅

n

2

=

n

3

.

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

⋮

⋮

⋮

Title

sum of odd numbers

Canonical

name

SumOfOddNumbers

Date of creation

2013-03-22 14:38:35

Last modified on

2013-03-22 14:38:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Example

Classification

msc 00A05

Classification

msc 11B25

Related topic

NumberOdd

Generated on Tue Feb 6 22:18:46 2018 by

LaTeXML
time invariant

time invariant

A

dynamical system

is

time-invariant

if its generating formula is dependent on state only, and independent of time. A synonym for time-invariant is

autonomous

. The complement of time-invariant is time-varying (or nonautonomous).

For example, the continuous-time system

x

˙

=

f

⁢

(

x

,

t

)

is time-invariant if and only if

f

⁢

(

x

,

t

1

)

≡

f

⁢

(

x

,

t

2

)

for all valid states

x

and times

t

1

and

t

2

. Thus

x

˙

=

sin

⁡

x

is time-invariant, while

x

˙

=

sin

⁡

x

1

+

t

is time-varying.

Likewise, the discrete-time system

x

⁢

[

n

]

=

f

⁢

[

x

,

n

]

is time-invariant (also called shift-invariant) if and only if

f

⁢

[

x

,

n

1

]

≡

f

⁢

[

x

,

n

2

]

for all valid states

x

and time indices

n

1

and

n

2

. Thus

x

⁢

[

n

]

=

2

⁢

x

⁢

[

n

-

1

]

is time-invariant, while

x

⁢

[

n

]

=

2

⁢

n

⁢

x

⁢

[

n

-

1

]

is time-varying.

Title

time invariant

Canonical name

TimeInvariant

Date of creation

2013-03-22 15:02:14

Last modified on

2013-03-22 15:02:14

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 00A05

Related topic

AutonomousSystem

Defines

time-invariant

Defines

shift-invariant

Generated on Tue Feb 6 22:18:53 2018 by

LaTeXML
Venn diagram

Venn diagram

Note: Currently, overlapping do not seem to be showing up properly in html mode. Therefore, this entry is best viewed using page mode.

A

Venn diagram

is a visual tool used in describing how two or more sets are logically related to one another. The simplest example is a Venn diagram for two sets. Each set is represented by a planar region bounded by a circle, so that the two regions overlap. In the

diagram

below, set

A

is represented by the reddish circular disc and set

B

is represented by the bluish circular disc:

The overlapping region

represents

the

intersection

of the sets

A

and

B

, denoted by

A

∩

B

.

Typically, the two sets are subsets of some bigger set

U

, called a

universe

. Therefore, the corresponding Venn diagram above is shown to be sitting inside a larger rectangular region representing the universe:

Notice that the region representing the universe

U

is partitioned into

4

mutually exclusive regions:

Title

Venn diagram

Canonical

name

VennDiagram

Date of creation

2013-03-22 17:45:43

Last modified on

2013-03-22 17:45:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 00A05

Classification

msc 00A06

Classification

msc 03E99

\@unrecurse

Generated on Tue Feb 6 22:18:58 2018 by

LaTeXML
well-defined

well-defined

A mathematical

concept

is

well-defined

(German

wohldefiniert

, French

bien défini

), if its contents is
on the form or the alternative representative which is used for defining it.

For example, in defining the

http://

planetmath

.org/FractionPower

power

x

r

with

x

a

positive

real and

r

a

rational number

,
we can freely choose the fraction form

m

n

(

m

∈

ℤ

,

n

∈

ℤ

+

) of

r

and take

x

r

:=

x

m

n

and be sure that the value of

x

r

does not depend on that choice (this is justified in the entry fraction power). So,
the

x

r

is well-defined.

In many

instances

well-defined is a synonym for the formal

definition

of a

function

between sets. For example,
the function

f

⁢

(

x

)

:=

x

2

is a well-defined function from the real numbers to the real numbers because
every input,

x

, is assigned to precisely one output,

x

2

. However,

f

⁢

(

x

)

:=

±

x

is not well-defined
in that one input

x

can be assigned any one of two possible outputs,

x

or

-

x

.

More subtle examples include

expressions

such as

f

⁢

(

a

b

)

:=

a

+

b

,

a

b

∈

ℚ

.

Certainly every input has an output, for instance,

f

⁢

(

1

/

2

)

=

3

. However, the expression is

not

well-defined since

1

/

2

=

2

/

4

yet

f

⁢

(

1

/

2

)

=

3

while

f

⁢

(

2

/

4

)

=

6

and

3

≠

6

.

One must question whether a function is well-defined whenever it is defined on a domain of

equivalence classes

in such a manner that each output is determined for a representative of each equivalence class. For example, the
function

f

⁢

(

a

/

b

)

:=

a

+

b

was defined using the representative

a

/

b

of the equivalence class of fractions

equivalent

to

a

/

b

.

Title

well-defined

Canonical

name

Welldefined

Date of creation

2013-03-22 17:31:32

Last modified on

2013-03-22 17:31:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A05

Synonym

well defined

Related topic

function

Related topic

WellDefinednessOfProductOfFinitelyGeneratedIdeals

Generated on Tue Feb 6 22:19:00 2018 by

LaTeXML
Windows Calculator

Windows Calculator

The

Windows Calculator

is a software calculator that comes bundled with the Windows operating system. The basic mode is called “Standard” and is the default, Scientific mode has most of the

operations

available on a typical

scientific calculator

. Note that switching between modes causes the loss of the current value displayed (unless of course that value is 0). For some reason, Standard mode has a

square root

key but Scientific mode does not. As a workaround in scientific mode, one can enter, say,

[2] [x^y] [0] [.] [5]

.

Division by zero

causes an error condition that must be cleared with the C key on the displayed keyboard (or the Escape key on the

computer

’s keyboard). Integer values smaller than

10

32

can be displayed in all their digits. According to the Help, the Windows Calculator truncates

π

to 32 digits, but

rational numbers

are stored internally “as

fractions

”.

Like most scientific calculators, the Windows Calculator can display results in binary, octal and

hexadecimal

, but is limited to

integers

in those bases. Additionally,

negative numbers

are shown in two’s

complement

(and the sign change key performs two’s complement on the displayed value). In those bases, the user can choose the data size: quadruple word (the default), double word, word or byte. Overflows don’t trigger any kind of exception or error notification, the

calculator

quietly discards the more significant digits and displays the least significant digits that will fit in the currently selected data size.

Like the

Mac OS Calculator

, for the Windows Calculator

0

0

=

1

.

References

1

David A. Karp, Tim O’Reilly & Troy Mott,

Windows XP in a Nutshell

Cambridge: O’Reilly (2002): 114 - 117

Title

Windows Calculator

Canonical

name

WindowsCalculator

Date of creation

2013-03-22 16:39:22

Last modified on

2013-03-22 16:39:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Classification

msc 01A07

Generated on Tue Feb 6 22:19:02 2018 by

LaTeXML
doubling and halving algorithm for integer multiplication

doubling and halving algorithm for integer multiplication

Because multiplying and dividing by 2 is often easier for humans than multiplying and dividing by other numbers there is an algorithm for

multiplication

of any two

integers

that takes advantage of multiplication and

division

by 2.

Call the algorithm with two integers.

1.

Use one of the integers to start a column on the left and the other to start a column on the right. (Either number can be put in either column, there are very minor optimizations that are unlikely to make a

difference

in performance, such as not choosing for the left column numbers that end long

Cunningham chains

).

2.

Divide the previous integer on the left column by 2 and write the yield below it, ignoring any

fractional part

there may be. Multiply the previous integer on the right column by 2 and write the

product

below.

3.

Repeat Step 2 until the yield on the left column is 1.

4.

For every

even number

on the left column, cross out the right column’s number of the same row.

5.

Add up the the numbers on the right column that haven’t been crossed out.

For example, to multiply 108 by 255:

108

255

54

510

27

1020

13

2040

6

4080

3

8160

1

16320

27540

This works in any base (as long as one doesn’t get confused about parity in odd bases). For example, 18 times 24 in base 5:

33

44

14

143

4

341

2

1232

1

3014

3212

Naturally one might wonder if this can be applied to binary and used by

computers

. After all, halving and ignoring the fractional part is even easier: it’s just a matter of shifting the bits to the right, and it doesn’t matter what the computer does with the discarded bit (as long as it doesn’t put it back into the original byte or word in the most significant bit or the sign bit). Doubling is also easy, just a shift left, with the only concern being overflow.

1010

111

101

1110

10

11100

1

111000

1000110

Of course this algorithm is not suitable for large integer multiplication as is required in the search for large prime numbers.

References

1

Paul Erdős & János Surányi

Topics in the theory of numbers

New York: Springer (2003): 5

2

Ogilvy & Anderson,

Excursions in

Number Theory

. Oxford: Oxford University Press (1966). Reprinted New York: Dover (1988): 6 - 8

Title

doubling and halving algorithm for integer multiplication

Canonical

name

DoublingAndHalvingAlgorithmForIntegerMultiplication

Date of creation

2013-03-22 17:01:20

Last modified on

2013-03-22 17:01:20

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Algorithm

Classification

msc 00A06

Classification

msc 00A05

Classification

msc 11B25

Generated on Tue Feb 6 22:19:04 2018 by

LaTeXML
multiplication

multiplication

Multiplication

is a mathematical

operation

in which two or more numbers are added up to themselves by a factor of other numbers. For example,

2

×

3

=

2

+

2

+

2

=

3

+

3

=

6

. The numbers may be real,

imaginary

or

complex

, they may be

integers

or

fractions

. Among

real numbers

, if an

odd number

of multiplicands are

negative

, the overall result is negative; if an

even number

of multiplicands are negative, the overall result is

positive

. Two examples:

(

-

3

)

×

(

-

5

)

=

15

;

(

-

2

)

×

(

-

3

)

×

(

-

5

)

=

(

-

30

)

.

The usual

operator

is the cross with its four arms of equal length pointing northeast, northwest, southeast and southwest:

×

. Other options are the central dot

⋅

and the

tacit multiplication operator

. In many computer programming languages the asterisk is often used as it is almost always available on the keyboard (Shift-8 in most American layouts, as well as dedicated key if the keyboard has a numeric keypad), and this is the operator likely to be used in a computer implementation of a

reverse Polish notation

calculator. In

Mathematica

, the space can sometimes

function

as a

multiplication operator

, but more experienced users warn novices not to rely on this feature.

Just as with

addition

, multiplication is

commutative

:

x

⁢

y

⁢

z

=

x

⁢

z

⁢

y

=

y

⁢

x

⁢

z

, etc.

The iterative operator is the Greek capital letter pi:

∏

i

=

1

n

a

i

,

which is a

compact

way of writing

a

1

×

a

2

×

…

×

a

n

.

Multiplication of

complex numbers

is helped by the following

identity

:

(

a

+

b

⁢

i

)

×

(

x

+

y

⁢

i

)

=

(

a

⁢

x

-

b

⁢

y

)

+

(

a

⁢

y

+

b

⁢

x

)

⁢

i

. To give three examples:

(

17

+

29

⁢

i

)

⁢

(

11

+

38

⁢

i

)

=

-

915

+

965

⁢

i

(the result has both real and

imaginary parts

),

(

1

+

2

⁢

i

)

⁢

(

1

-

2

⁢

i

)

=

5

(the result is a

real prime

) and

(

4

+

7

⁢

i

)

⁢

(

7

+

4

⁢

i

)

=

65

⁢

i

(the result has only an imaginary part).

Title

multiplication

Canonical name

Multiplication

Date of creation

2013-03-22 16:35:37

Last modified on

2013-03-22 16:35:37

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

10

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A06

Classification

msc 11B25

Classification

msc 00A05

Related topic

Product

Related topic

ProductOfNegativeNumbers

Related topic

FactorsWithMinusSign

Generated on Tue Feb 6 22:19:06 2018 by

LaTeXML
plus sign

plus sign

There are two main uses of the

plus sign

“

+

” (which is a simplified form of “&”) in the mathematics and the applying sciences:

•

The original use is as the sign for the

binary operation

addition

of numbers and other elements of rings and algebras, vectors, etc.:

a

+

b

:=

the sum of

⁢

a

⁢

and

⁢

b

•

There is also a special use for the unary operation

identity mapping

concerning numbers and other ring

elements

:

+

a

:=

a

(for all

a

)

Title

plus sign

Canonical

name

PlusSign

Date of creation

2013-03-22 17:35:43

Last modified on

2013-03-22 17:35:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 00A06

Classification

msc 00A05

Synonym

plus

Related topic

Sum

Related topic

SumOfSeries

Related topic

SignumFunction

Related topic

OppositeNumber

Related topic

ProductOfNegativeNumbers

Generated on Tue Feb 6 22:19:07 2018 by

LaTeXML
subtraction

subtraction

Subtraction

is a mathematical

operation

in which the value of a number is decreased by the values of one or more other numbers. Subtraction can be seen as a kind of

addition

with

negative numbers

. For example,

7

-

4

=

7

+

(

-

4

)

=

3

.

The usual

operator

looks like a dash:

-

. This operator is used in standard

infix notation

as well as in

Polish notation

and

reverse Polish notation

.

Besides the possibility of overflow or underflow, subtraction presents no problems for

fixed point arithmetic

provided the operands are representable in

fixed point

to begin with.

Title

subtraction

Canonical

name

Subtraction

Date of creation

2013-03-22 16:35:31

Last modified on

2013-03-22 16:35:31

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A06

Classification

msc 00A05

Classification

msc 11B25

Generated on Tue Feb 6 22:19:20 2018 by

LaTeXML