Definition:Prime Magic Square

A '''prime magic square''' is a magic square whose elements are all prime numbers, and may or may not also include the number $1$.


Definition:Extremal Embedding in Field of Functional

Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let $\gamma$ be an extremal of $J$. {{explain|Extremum or extremal?}} Let $R$ be a simply connected open region which contains $\gamma$ as a subset. {{explain|GF puts "open" in brackets; does it matter?}} Let a field of functional $J$ be defined at every point of $R$. Let one of the trajectories of the field be $\gamma$. Then $\gamma$ can be '''embedded''' in a field of functional $J$. {{explain|If $\gamma$ "can be" embedded, is it correct to say that $\gamma$ "is" embedded? If not, then is it more grammatically accurate to define $\gamma$ as being "embeddable"? Otherwise it looks as though "can be" introduces a statement that needs to be proven.}}


Definition:Permutable Prime

A '''permutable prime''' is a prime number $p$ which has the property that all anagrams of $p$ are prime.


Definition:Permutable Prime/Sequence

The sequence of '''permutable primes''' begins: :$2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R_{19}, R_{23}, R_{317}, R_{1091}, \ldots$ where $R_n$ denotes the repunit of $n$ digits.


Definition:Anagram

Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $m$ be an integer expressed in base $b$. An '''anagram base $b$''' of $m$ is an integer formed by the digits of $m$ written in a different order. When the number base of $m$ is not specified, base $10$ is assumed.


Definition:Repunit Prime

Let $b \in \Z_{>1}$ be an integer greater than $1$. A '''repunit prime base $b$''' is a repunit base $b$ which is prime.


Definition:Repunit Prime/Index

The '''index''' of a '''repunit prime''' is the number of digits it has. Thus $R_n$ denotes a '''repunit prime''' with $n$ digits.


Definition:Repunit/Index

The '''index''' of a '''repunit''' is the number of digits it has. Thus a '''repunit''' with $n$ digits can be referred to as $R_n$.


Definition:Fermat Set

The '''Fermat set''' is the set of integers: :$\left\{ {1, 3, 8, 120}\right\}$ It has the property of being a Diophantine quadruple.


Definition:Diophantine m-Tuple/Quadruple

A '''Diophantine quadruple''' is a Diophantine $m$-tuple where $m = 4$. That is, a '''Diophantine quadruple''' is a set of $4$ positive integers such that the product of any $2$ of them plus $1$ is a square number.


Definition:Diophantine m-Tuple

A '''Diophantine $m$-tuple''' set of $m$ positive integers such that the product of any $2$ of them plus $1$ is a square number.


Definition:Diophantine m-Tuple/Quintuple

A '''Diophantine quintuple''' is a Diophantine $m$-tuple where $m = 5$. That is, a '''Diophantine quintuple''' is a set of $5$ positive integers such that the product of any $2$ of them plus $1$ is a square number.


Definition:Triperfect Number

A '''triperfect number''' is a positive integer $n$ such that the sum of its divisors is equal to $3$ times $n$.


Definition:Multiply Perfect Number

A '''multiply perfect number''' is a positive integer $n$ such that the sum of its divisors is equal to an integer multiple of $n$.


Definition:Hilbert's Invariant Integral

Let $\mathbf y$ be an $n$-dimensional vector. Let $H$ be Hamiltonian and $\mathbf p$ momenta. Let $\Gamma$ be a curve connecting points $\tuple {x_0, \map{\mathbf y} {x_0} }$ and $\tuple {x, \mathbf y}$. Then the following line integral is known as '''Hilbert's Invariant Integral''': :$\displaystyle \map g {x, \mathbf y} = \int_\Gamma \paren {-H \rd x + \mathbf p \rd \mathbf y}$ {{NamedforDef|David Hilbert|cat = Hilbert}}


Definition:Weierstrass E-Function

Let $\mathbf y, \mathbf z, \mathbf w : \R \to \R^n$ be $n$-dimensional vector-valued functions. Let $\mathbf y$ be such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$. Let $F: \R^{2 n + 1} \to \R$ be twice differentiable {{WRT|Differentiation}} its (independent) variables. Let $J$ be a functional such that: :$\ds J \sqbrk{\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ where:  :$\mathbf y' := \dfrac {\d \mathbf y} {\d x}$ is the derivative of a vector-valued function. === Definition 1 === {{:Definition:Weierstrass E-Function/Definition 1}} === Definition 2 === {{:Definition:Weierstrass E-Function/Definition 2}}


Definition:Multiply Perfect Number/Order

Let $n \in \Z_{>0}$ be a '''multiply perfect number''' such that the sum of its divisors is equal to $m \times n$. Then $n$ is '''multiply perfect of order $m$'''.


Definition:Wonderful Demlo Number

The '''Wonderful Demlo numbers''' are the positive integers defined as the coefficients of the powers of $x$ in the expansion of: :$\dfrac {1 + 10 x} {\left({1 - x}\right) \left({1 - 10 x}\right) \left({1 - 100 x}\right)}$


Definition:Wonderful Demlo Number/Sequence

The sequence of '''Wonderful Demlo numbers''' begins: :$1, 121, 12 \, 321, 1 \, 234 \, 321, 123 \, 454 \, 321, \ldots$


Definition:Obstinate Number

An '''obstinate number''' is a positive odd integer which cannot be expressed as the sum of a power of $2$ and a prime.


Definition:Ore Number

Let $n \in \Z_{>0}$ be a positive integer. $n$ is an '''Ore number''' {{iff}} the harmonic mean of its divisors is an integer.


Definition:Woodall Number

A '''Woodall number''' is a positive integer of the form: :$n \times 2^n - 1$


Definition:Primitive Prime Factor of Fibonacci Number

Let $F_n$ denote the $n$th Fibonacci number. A '''primitive prime factor of $F_n$''' is a prime number $p$ of $F_n$ such that: :$p \divides F_n$ :$\nexists k \in \Z_{>0}: k < n: p \divides F_k$ where $a \divides b$ denotes that $a$ is a divisor of $b$. That is, a prime factor of $F_n$ but of no smaller Fibonacci numbers.


Definition:Primitive Prime Factor

Let $\left\langle{a_n}\right\rangle$ be an integer sequence. A '''primitive prime factor''' for a term $a_n$ is a prime number $p$ of $a_n$ such that: :$p \divides a_n$ :$\nexists k \in \Z_{>0}: k < n: p \divides a_k$ where $a \divides b$ denotes that $a$ is a divisor of $b$.


Definition:Factorion

A '''factorion (base $b$)''' is a positive integer which, when expressed in base $b$, equals the sum of the factorials of its digits.


Definition:Philosophy

'''Philosophy''' is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. As there are considerable overlaps with mathematics, it is sometimes argued that mathematics can be considered a branch of '''philosophy'''. It is the controversial opinion of some mathematicians that philosophy is for people who are not smart enough for mathematics.


Definition:Polynomial Ring/Sequences

Let $R^{\left({\N}\right)}$ be the ring of sequences of finite support over $R$. Let $\iota : R \to R^{\left({\N}\right)}$ be the mapping defined as: :$\iota \left({r}\right) = \left \langle {r, 0, 0, \ldots}\right \rangle$. Let $X$ be the sequence $\left \langle {0, 1, 0, \ldots}\right \rangle$. The '''polynomial ring over $R$''' is the ordered triple $\left({R^{\left({\N}\right)}, \iota, X}\right)$.


Definition:Highly Abundant Number

Let $n \in \Z_{>0}$ be a positive integer. Then $n$ is '''highly abundant''' {{iff}}: :$\forall m \in \Z_{>0}, m < n: \map \sigma m < \map \sigma n$ where $\map \sigma n$ is the $\sigma$ function of $n$.


Definition:Heptagonal Pyramidal Number

'''Heptagonal pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a heptagonal pyramid.


Definition:Lychrel Number/Candidate

No natural number has been '''proved''' to be a '''Lychrel number''' as of time of writing (June $2017$).  However, plenty of numbers have not shown themselves to terminate in a palindromic number, although in some cases millions of iterations have been tested. Hence a '''candidate Lychrel number''' is a natural number which '''is not known to''' form a palindromic number through repeated iteration of the reverse-and add process.


Definition:Sequence/Minimizing/Functional

Let $y$ be a real mapping defined on a space $\MM$. {{Stub|what is $\MM$?}} Let $J \sqbrk y$ be a functional such that: :$\exists y \in \MM: J \sqbrk y < \infty$ :$\ds \exists \mu > -\infty: \inf_y J \sqbrk y = \mu$ Let $\sequence {y_n}$ be a sequence such that: :$\ds \lim_{n \mathop \to \infty} J \sqbrk {y_n} = \mu$ Then the sequence $\sequence {y_n}$ is called a '''minimizing sequence (of the functional $J \sqbrk y$)'''.


Definition:Sequence/Minimizing/Functional/Limit Minimizing Function of

Let $\sequence {y_n}$ be a minimizing sequence of a functional $J$. Suppose: :$\displaystyle \lim_{n \mathop \to \infty} y_n = \hat y$ and :$\displaystyle \lim_{n \mathop \to \infty} J \sqbrk {y_n} = J \sqbrk {\hat y}$ Then $\hat y$ is the '''limit minimizing function''' of $J \sqbrk {y_n}$ and $J \sqbrk {\hat y} = \mu$.


Definition:Ritz Method

Let $\MM$ be a normed linear space. Let $J\sqbrk y$ be a functional defined on space $\MM$. Let $\sequence{\phi_n}$ be an infinite sequence of mappings in $\MM$. Let $\MM_n$ be an $n$-dimensional linear subspace of $\MM$, spanned by first $n$ mapping of $\sequence{\phi_n}$. Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$, where $\boldsymbol \alpha$ is a real $n$-dimensional vector. Minimise $J \sqbrk {\eta_n}$ {{WRT}} $\boldsymbol\alpha$. Then $J \sqbrk {\eta_n}$ is an approximate minimum of $J \sqbrk y$, and is denoted by $\mu_n$. This method is known as '''Ritz method'''.


Definition:Multiplicative Magic Square

A '''multiplicative magic square''' is an arrangement of $n^2$ distinct numbers into an $n \times n$ square array such that: :the product of the elements of each row :the product of the elements in each column :the product of the elements along each diagonal are the same.


Definition:Multiplicative Magic Square/Order

An $n \times n$ multiplicative magic square is called an '''order $n$ multiplicative magic square'''.


Definition:Complete Ritz Sequence

Let $\MM$ be a normed linear space. Let $\sequence {\phi_n}$ be a Ritz sequence in $\MM$. Let $\MM_n$ be an $n$-dimensional linear subspace of $\MM$, spanned by the first $n$ mappings of $\sequence {\phi_n}$. Let $\eta_n$ be of the form: :$\eta_n = \boldsymbol \alpha \boldsymbol \phi$ where $\boldsymbol \alpha$ is an $n$-dimensional real vector. Suppose: :$\forall y \in \MM: \forall \epsilon > 0: \exists \map n \epsilon \in \N: \exists \eta_n \in \MM_n: \size {\eta_n - y} < \epsilon$ {{explain|which norm is it?}} Then the sequence $\sequence{\phi_n}$ is called '''complete''' in $\MM$. {{NamedforDef|Walther Ritz|cat = Ritz}}


Definition:Ritz Sequence

Let $\sequence {\phi_n}$ be an infinite sequence of mappings in a normed linear space, where: :$\phi_n: \R \to \R$ Let this sequence be constrained by the requirements defined in the definition of the Ritz method. {{explain|Specify what these are. For ultimate reusability, consider extracting those requirements into an appropriately-named and transcluded subpage of the above.}} Then the sequence $\set {\phi_n}$ can be called a '''Ritz Sequence'''.


Definition:Plato's Geometrical Number

The actual value of what is generally known as '''Plato's geometrical number''' is uncertain. The passage in question from {{AuthorRef|Plato}}'s {{BookLink|Republic|Plato}} is obscure and difficult to interpret: :''But the number of a human creature is the first number in which root and square increases, having received three distances and four limits, of elements that make both like and unlike wax and wane, render all things conversable and rational with one another.'' There are two numbers which it is generally believed it could be: :$216$ :$12 \, 960, \, 000$ It is believed that the expression: :''three distances and four limits'' refers to cubing. It is further believed that the reference is to the area of the $3-4-5$ triangle, which is $6$. The passage is also deduced to contain a reference to $2 \times 3$. It is also interpreted by other commentators as being $12 \, 960 \, 000$, which is $60^4$.


Definition:Fedorov Group

A '''Fedorov group''' is the symmetry group of a $3$-dimensional configuration in space.


Definition:Space Group

A '''space group''' is the symmetry group of a configuration in $n$-dimensional space.


Definition:Wallpaper Group

A '''wallpaper group''' is the symmetry group of a $2$-dimensional configuration in the plane.


Definition:Amicable Pair/Definition 1

$m$ and $n$ are an '''amicable pair''' {{iff}}: :the aliquot sum of $m$ is equal to $n$ and: :the aliquot sum of $n$ is equal to $m$.


Definition:Amicable Pair/Definition 2

$m$ and $n$ are an '''amicable pair''' {{iff}}: :$\map \sigma m = \map \sigma n = m + n$ where $\map \sigma m$ denotes the $\sigma$ function.


Definition:Amicable Pair/Definition 3

$m$ and $n$ are an '''amicable pair''' {{iff}} they form a sociable chain of order $2$.


Definition:Thabit Pair

A '''Thabit pair''' is an amicable pair formed by an application of Thabit's Rule: :$\tuple {2^n a b, 2^n c}$ where $n$ be a positive integer such that: {{begin-eqn}} {{eqn | l = a       | r = 3 \times 2^n - 1       | c =  }} {{eqn | l = b       | r = 3 \times 2^{n - 1} - 1       | c =  }} {{eqn | l = c       | r = 9 \times 2^{2 n - 1} - 1       | c =  }} {{end-eqn}} are all prime.


Definition:Thabit Number

A '''Thabit number''' is an integer in the form: :$3 \times 2^n - 1$ where $n \in \Z_{\ge 0}$ is a non-negative integer.


Definition:Thabit Number/Sequence

The sequence of Thabit numbers begins: :$2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, \ldots$


Definition:Thabit Prime

A '''Thabit prime''' is a Thabit number which is prime.


Definition:Thabit Prime/Sequence

The sequence of Thabit primes begins: :$2, 5, 11, 23, 47, 191, 383, 6143, 786 \, 431, 51 \, 539 \, 607 \, 551, \ldots$


Definition:Amicable Triplet

Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers. === Definition 1 === {{:Definition:Amicable Triplet/Definition 1}} === Definition 2 === {{:Definition:Amicable Triplet/Definition 2}}


Definition:Amicable Triplet/Definition 1

$\tuple {m_1, m_2, m_3}$ are an '''amicable triplet''' {{iff}} the aliquot sum of any one of them equals the sum of the other two: :the aliquot sum of $m_1$ is equal to $m_2 + m_3$ and: :the aliquot sum of $m_2$ is equal to $m_1 + m_3$ and: :the aliquot sum of $m_3$ is equal to $m_1 + m_2$


Definition:Amicable Triplet/Definition 2

$\left({m_1, m_2, m_3}\right)$ are an '''amicable triplet''' {{iff}}: :$\sigma \left({m_1}\right) = \sigma \left({m_2}\right) = \sigma \left({m_3}\right) = m_1 + m_2 + m_3$ where $\sigma \left({m}\right)$ denotes the $\sigma$ function.


Definition:Ljunggren Equation

The '''Ljunggren equation''' is the indeterminate Diophantine equation: :$x^2 + 1 = 2 y^4$


Definition:Sierpiński Number

There are two kinds of '''Sierpiński number''': === Sierpiński Numbers of the First Kind === {{:Definition:Sierpiński Number of the First Kind}} === Sierpiński Numbers of the Second Kind === {{:Definition:Sierpiński Number of the Second Kind}}


Definition:Sierpiński Number of the First Kind

The '''Sierpiński numbers of the first kind''' are the integers $S_n$ in the form: :$S_n := n^n + 1$ for all integers $n$.


Definition:Sierpiński Number of the Second Kind/Sequence

The  sequence of known '''Sierpiński numbers of the second kind''' starts: :$78\ 557, \ 271\ 129, \ 271\ 577, \ 322\ 523, \ 327\ 739, \ 482\ 719, \ 575\ 041, \ 603\ 713, \ 903\ 983, \ 934\ 909, \ 965\ 431, \ \ldots$


Definition:Sierpiński Number of the Second Kind/Historical Note

{{AuthorRef|Wacław Franciszek Sierpiński}} proved in $1960$ that there is an infinite number of Sierpiński numbers of the second kind.


Definition:Balanced Prime

=== Definition 1 === {{:Definition:Balanced Prime/Definition 1}} === Definition 2 === {{:Definition:Balanced Prime/Definition 2}} === Definition 3 === {{:Definition:Balanced Prime/Definition 3}}


Definition:Balanced Prime/Definition 2

Let $\paren {p_{n - 1}, p_n, p_{n + 1} }$ be a triplet of consecutive prime numbers. $p_n$ is a '''balanced prime''' {{iff}}: {{begin-eqn}} {{eqn | l = p_{n - 1} + d       | r = p_n       | c =  }} {{eqn | l = p_{n - 1} + 2 d       | r = p_{n + 1}       | c =  }} {{end-eqn}} for some $d \in \Z$.


Definition:Balanced Prime/Definition 1

Let $\left({p_{n - 1}, p_n, p_{n + 1} }\right)$ be a triplet of consecutive prime numbers. $p_n$ is a '''balanced prime''' {{iff}}: :$p_n = \dfrac {p_{n - 1} + p_{n + 1} } 2$


Definition:Aliquot Sequence

Let $m$ be a positive integer. Let $\map s m$ be the aliquot sum of $m$. Define the sequence $\sequence {a_k}$ recursively as: :$a_{k + 1} = \begin{cases} m & : k = 0 \\ \map s {a_k} & : k > 0 \end{cases}$ The sequence $\sequence {a_k}$ is known as an '''aliquot sequence'''.


Definition:Eighth Power

A '''eighth power''' is an integer which can be expressed as the $8$th power of an integer.


Definition:Superfactorial

The '''superfactorial of $n$''' is defined as: :$n\$ = \displaystyle \prod_{k \mathop = 1}^n k! = 1! \times 2! \times \cdots \times \left({n - 1}\right)! \times n!$ where $k!$ denotes the factorial of $n$.


Definition:Completely Irreducible

Let $\struct {S, \preceq}$ be an ordered set. Let $p \in S$. An element $p$ is '''completely irreducible''' {{iff}} :$p^\succeq \setminus \set p$ admits a minimum element where $p^\succeq$ denotes the upper closure of $p$.


Definition:Poulet Number

Let $q$ be a composite number such that $2^q \equiv 2 \pmod q$. Then $q$ is a '''Poulet number'''.


Definition:Time/Unit/Lunar Month

The '''lunar month''' is defined as the length of time between $2$ new moons. :$1$ '''lunar month''' $\approx 29 \cdotp 530588$ days or: :$1$ '''lunar month''' $\approx 29$ days, $12$ hours, $44$ minutes and $2 \cdotp 8$ seconds.


Definition:Calendar

A '''calendar''' is a technique to keep track of the passing of time throughout the course of a year. Its objective is to provide a more-or-less standard way of identifying the time of year by associating a name and number to each day in the year.


Definition:Calendar/Julian/Year

A '''Julian year''' is the length of a year as defined using the '''Julian calendar'''. :$1$ '''Julian year''' $= \begin{cases} 366 \, \text {days} & : 4 \divides y \\ 365 \, \text {days} & : 4 \nmid y \end{cases}$ where: :$y$ denotes the number of the year :$4 \divides y$ denotes that $y$ is divisible by $4$ :$4 \nmid y$ denotes that $y$ is not divisible by $4$.


Definition:Calendar/Julian

The '''Julian calendar''' is the calendar which was introduced by [https://en.wikipedia.org/wiki/Julius_Caesar Julius Caesar] in $45 \, \text{BCE}$. It divides the year into: :$365$ days for every $3$ out of $4$ years and: :$366$ days for every $1$ out of $4$ years, known as a leap year. The years themselves are given an index number, and are known by that number. A standard numbering system was introduced by [https://en.wikipedia.org/wiki/Dionysius_Exiguus Dionysus Exiguus]. He identified a particular year as being $525 \, \text{A.D.}$, where $\text{A.D.}$ is an abbreviation for '''Anno Domini''', Latin for '''in the year of the Lord'''. The year $1$ was conventionally supposed to identify the year of the birth of [https://en.wikipedia.org/wiki/Jesus Jesus of Nazareth], although the accuracy of this has since been questioned. Years before $1 \, \text{A.D.}$ are counted backwards and assigned the label $\text{B.C.}$. However, note that the year immediately prior to $1 \, \text{A.D.}$ is $1 \, \text{B.C.}$, not the intuitive '''year $0$''', a discrepancy that can cause confusion. Using this system of numbering, a leap year is identified by this number being divisible by $4$. The $365$ or $366$ days in the year are divided into $12$ approximately equal sections called months, which are assigned both names and index numbers: {{begin-eqn}} {{eqn | l = 1:       | o =        | c = January       | cc= $31$ days }} {{eqn | l = 2:       | o =        | c = February       | cc= $28$ days, or $29$ days in a leap year }} {{eqn | l = 3:       | o =        | c = March       | cc= $31$ days }} {{eqn | l = 4:       | o =        | c = April       | cc= $30$ days }} {{eqn | l = 5:       | o =        | c = May       | cc= $31$ days }} {{eqn | l = 6:       | o =        | c = June       | cc= $30$ days }} {{eqn | l = 7:       | o =        | c = July       | cc= $31$ days }} {{eqn | l = 8:       | o =        | c = August       | cc= $31$ days }} {{eqn | l = 9:       | o =        | c = September       | cc= $30$ days }} {{eqn | l = 10:       | o =        | c = October       | cc= $31$ days }} {{eqn | l = 11:       | o =        | c = November       | cc= $30$ days }} {{eqn | l = 12:       | o =        | c = December       | cc= $31$ days }} {{end-eqn}} Thus, for example, the day following the $31$st of January is the $1$st of February, and the $30$th of June is followed by the $1$st of July.


Definition:Leap Year

The number of days in a year is not an integer, but lies somewhere between $365$ and $366$. However, it is convenient to assign an integer number of days to any given year. It is also desirable to ensure that a particular day of the year occurs at approximately the same time each year. In the Julian and Gregorian calendars: :some years are assigned $365$ days and: :some years are assigned $366$ days. The years which have $366$ days are called '''leap years'''.


Definition:Calendar/Gregorian

The '''Gregorian calendar''' is the calendar which was introduced by [https://en.wikipedia.org/wiki/Pope_Gregory_XIII Pope Gregory XIII] in $1582 \, \text{CE}$. Over the course of the next few centuries it was gradually adopted by the various subcultures of [https://en.wikipedia.org/wiki/Western_culture Western civilization]. It was derived as a refinement of the Julian calendar to correct for the discrepancy between the Julian calendar and the tropical year. The years themselves are assigned the same numbers as their Julian counterparts. An evolving modern convention is to refer to $\text{A.D.}$ and $\text{B.C.}$ as $\text{CE}$ and $\text{BCE}$, for '''common era''' and '''before common era''' respectively. Like the Julian calendar, it divides the year into either $365$ days or $366$ days, according to the year number. The months, similarly, are kept the same as for the Julian calendar. The only difference is in the determination of which years are classified as leap years. Let $y$ be the year number. Then $y$ is a leap year {{iff}} :$y$ is divisible by $400$ or: :$y$ is divisible by $4$ and $y$ is not divisible by $100$. Thus, for example: :$2016$ was a leap year, because $2016$ is divisible by $4$ and not divisible by $100$. :$1900$ was '''not''' a leap year, because $1900$ is divisible by $100$ but not $400$. :$2000$ was a leap year, as $2000$ is divisible by $400$.


Definition:Calendar/Gregorian/Year

A '''Gregorian year''', also known as the '''civil year''', is the length of a year as defined using the '''Gregorian calendar'''. :$1$ '''Gregorian year''' $= \begin{cases} 366 \, \text {days} & : 400 \divides y \\ 365 \, \text {days} & : 400 \nmid  y \text { and } 100 \divides y \\ 366 \, \text {days} & : 100 \nmid  y \text { and } 4 \divides y \\ 365 \, \text {days} & : 4 \nmid y \end{cases}$ where: :$y$ denotes the number of the year :$4 \divides y$ denotes that $y$ is divisible by $4$ :$4 \nmid y$ denotes that $y$ is not divisible by $4$.


Definition:Calendar/Muslim

The '''Muslim calendar''' is a calendar whose period is based on $12$ full cycles of phases of the moon. Each Muslim year lasts either $354$ or $355$ days, depending on precisely when the relevant new moons happen The years are numbered from the year when [https://en.wikipedia.org/wiki/Muhammad Muhammad] and his followers migrated from [https://en.wikipedia.org/wiki/Mecca Mecca] to [https://en.wikipedia.org/wiki/Medina Medina] in $622 \, \text{A.D}$ by the Julian calendar. Years are denoted with the suffix $\text{A.H.}$, meaning '''Anno Hegirae''', literally '''in the year of the Hejira'''. The start and end of the Muslim year are $10$ to $12$ days earlier every civil year. The $354$ or $355$ days in the Muslim year are divided into $12$ approximately equal sections called months, each either $29$ or $30$ days, depending on the phases of the moon. They are assigned both names and index numbers: {{begin-eqn}} {{eqn | l = 1:       | o =        | c = Muḥarram }} {{eqn | l = 2:       | o =        | c = Ṣafar }} {{eqn | l = 3:       | o =        | c = Rabī‘ al-awwal }} {{eqn | l = 4:       | o =        | c = Rabī‘ ath-thānī }} {{eqn | l = 5:       | o =        | c = Jumādá al-ūlá }} {{eqn | l = 6:       | o =        | c = Jumādá al-ākhirah }} {{eqn | l = 7:       | o =        | c = Rajab }} {{eqn | l = 8:       | o =        | c = Sha‘bān }} {{eqn | l = 9:       | o =        | c = Ramaḍān }} {{eqn | l = 10:       | o =        | c = Shawwāl }} {{eqn | l = 11:       | o =        | c = Dhū al-Qa‘dah }} {{eqn | l = 12:       | o =        | c = Dhū al-Ḥijjah }} {{end-eqn}}


Definition:Calendar/Muslim/Year

A '''Muslim year''', also known as the '''Islamic year''' or '''Hijra year''', is the length of a year as defined using the '''Muslim calendar'''. :$1$ '''Muslim year''' $= \begin{cases} 355 \, \text {days} & : \text {Leap year} \\ 354 \, \text {days} & : \text {Non-leap year} \end{cases}$


Definition:Calendar/Jewish/Year

A '''Jewish year''', also known as the '''Hebrew year''', is the length of a year as defined using the '''Jewish calendar'''. Its length is either $354$ or $355$ days, or longer, if an extra month has been inserted.


Definition:Calendar/Jewish

The '''Jewish calendar''' is a calendar based on both the phases of the moon and on the orbital period of earth around the sun. The whose period is based on $12$ full cycles of phases of the moon. The basic Jewish year is $354$ or $355$ days of $12$ months, each alternately $29$ and $30$ days long. When the error between the Jewish year and tropical year becomes approximately $30$ days, a $13$th month is inerted into the year.


Definition:Lucas-Carmichael Number

Let $n$ have the property that: :$p \divides n \implies \paren {p + 1} \divides \paren {n + 1}$ where: :$p$ is prime :$p \divides n$ denotes that $p$ is a divisor of $n$. Then $n$ is classified as a '''Lucas-Carmichael number'''.


Definition:Complement of Subgroup

Let $G$ be a group with identity $e$. Let $H$ and $K$ be subgroups. Let $HK$ be their subset product and $H \cap K$ their intersection. === Definition 1 === {{Definition:Complement of Subgroup/Definition 1}} === Definition 2 === {{Definition:Complement of Subgroup/Definition 2}}


Definition:Inner Semidirect Product

Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $N$ be a normal subgroup of $G$. Let $H$ and $N$ be complementary. Then $G$ is the '''inner semidirect product''' of $N$ and $H$. This is denoted $G = N \rtimes H$ or $G = H \ltimes N$.


Definition:Semidirect Product

Let $H$ and $N$ be groups. Let $\Aut N$ denote the automorphism group of $N$. Let $\phi: H \to \Aut N$ be a group homomorphism, that is, let $H$ act on $N$. The '''(outer) semidirect product of $N$ and $H$ with respect to $\phi$''' is the cartesian product $N \times H$ with the group operation defined as: :$\tuple {n_1, h_1} \circ \tuple {n_2, h_2} = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 \cdot h_2}$ {{explain|What specifically is "$\cdot$" here? This is an instance where it is important (in my belief) to be completely explicit as to what the operations are within the groups $H$ and $N$. It may also be clearer to refer to $G_1$ and $G_2$ because $N$ and $H$ have connotations of "normal subgroup" and "subgroup" respectively, and it is highly desirable to be consistent with other pages.}} It is denoted $N \rtimes_\phi H$.


Definition:Continued Fraction/Expansion of Real Number

=== Irrational Number === Let $x$ be an irrational number. The '''continued fraction expansion of $x$''' is the simple continued fraction $\paren {\floor {\alpha_n} }_{n \ge 0}$ where $\alpha_n$ is recursively defined as: :$\alpha_n = \displaystyle \begin{cases} x & : n = 0 \\ \dfrac 1 {\fractpart {\alpha_{n - 1} } } & : n \ge 1 \end{cases}$ where: :$\floor {\, \cdot \,}$ is the floor function :$\fractpart {\, \cdot \,}$ is the fractional part function. === Rational Number === Let $x$ be a rational number. The '''continued fraction expansion of $x$''' is found using the Euclidean Algorithm. {{explain|how}}


Definition:Derivation on Ring

Let $R$ be a ring. A '''derivation''' on $R$ is a group homomorphism $D$ of the additive group of $R$ which satsfies the Leibniz law: :$\forall a, b \in R: \map D {a b} = \map D a b + a \map D b$


Definition:Differential Ring

A '''differential ring''' is a pair $\struct {R, D}$ where: :$R$ is a ring :$D$ is a derivation on $R$.


Definition:Differential Field

A '''differential field''' is a differential ring which is a field.


Definition:Galois Extension/Finite/Definition 3

$L / K$ is a '''Galois extension''' {{iff}} the order of the automorphism group $\Aut {L / K}$ equals the degree $\index L K$: :$\order {\Aut {L / K} } = \index L K$


Definition:Field Extension/Degree/Finite

$E / F$ is a '''finite field extension''' {{iff}} its degree $\index E F$ is finite.


Definition:Field Extension/Degree/Infinite

$E / F$ is an '''infinite field extension''' {{iff}} its degree $\index E F$ is not finite.


Definition:Field Norm/Definition 1

By Vector Space on Field Extension is Vector Space, $L$ is naturally a finite dimensional vector space over $K$. Let $\theta_\alpha$ be the linear operator: :$\theta_\alpha: L \to L: \beta \mapsto \alpha \beta$ The '''field norm''' $\map {N_{L/K} }\alpha$ of $\alpha$ is the determinant of $\theta_\alpha$.


Definition:Field Norm/Definition 2

Let $L / K$ be Galois. By Finite Field Extension has Finite Galois Group, the Galois group $\operatorname{Gal} \left({L / K}\right)$ is finite. The '''field norm''' $N_{L / K} \left({\alpha}\right)$ of $\alpha$ is $\displaystyle \prod_{\sigma \mathop \in \operatorname{Gal} \left({L / K}\right)} \sigma \left({\alpha}\right)$.


Definition:Nilradical of Ring/Definition 1

The '''nilradical''' of $A$ is the subset consisting of all nilpotent elements of $A$.


Definition:Nilradical of Ring/Definition 2

Let $\Spec A$ denote the prime spectrum of $A$. The '''nilradical''' of $A$ is: :$\displaystyle \Nil A = \bigcap_{\mathfrak p \mathop \in \Spec A} \mathfrak p$ That is, it is the intersection of all prime ideals of $A$.


Definition:Permutation Representation

Let $G$ be a group. Let $X$ be a set. Let $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$. A '''permutation representation''' of $G$ is a group homomorphism from $G$ to $\struct {\map \Gamma X, \circ}$. === Associated to Group Action === {{:Definition:Permutation Representation/Group Action}}


Definition:Faithful Group Action/Definition 1

$\phi$ is '''faithful''' {{iff}} $e$ is the only element if $G$ which acts trivially: :$\forall g \in G: \paren {\forall x \in X: g * x = x \implies g = e}$


Definition:Faithful Group Action/Definition 2

$\phi$ is '''faithful''' {{iff}} its permutation representation is injective.


Definition:Free Group Action

Let $G$ be a group with identity $e$ acting on a set $X$. The group action is '''free''' {{iff}}: :$\forall g \in G: \forall x \in X : g * x = x \implies g = e$


Definition:Kernel of Group Action

Let $G$ be a group with identity $e$. Let $X$ be a set. Let $* : G\times X\to X$ be a group action. === Definition 1 === {{Definition:Kernel of Group Action/Definition 1}} === Definition 2 === {{Definition:Kernel of Group Action/Definition 2}}


Definition:Kernel of Group Action/Definition 1

The '''kernel of the group action''' is the set: :$G_0 = \set {g \in G: \forall x \in X: g * x = x}$


Definition:Kernel of Group Action/Definition 2

The '''kernel of the group action''' is the kernel of its permutation representation.


Definition:Permutation Representation/Group Action

Let $\phi: G \times X \to X$ be a group action. Define for $g \in G$ the mapping $\phi_g : X \to X$ by: :$\map {\phi_g} x = \map \phi {g, x}$ The '''permutation representation of $G$ associated to the group action''' is the group homomorphism $G \to \struct {\map \Gamma X, \circ}$ which sends $g$ to $\phi_g$.


Definition:Group Action/Permutation Representation

Let $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$. Let $\rho: G \to \struct {\map \Gamma X, \circ}$ be a permutation representation. The '''group action of $G$ associated to the permutation representation $\rho$''' is the group action $\phi: G \times X \to X$ defined by: :$\map \phi {g, x} = \map {\rho_g} x$ where $\rho_g : X \to X$ is the permutation representation associated to $\rho$ for $g \in G$ by $\map {\rho_g} x = \map \phi {g, x}$.


Definition:Field of Quotients/Definition 4

A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ which is its total ring of fractions, that is, the localization of $D$ at the nonzero elements $D_{\ne 0}$.


Definition:Fixed Field

Let $F$ be a field. Let $G \le \Aut F$ be a subgroup of the automorphism group of $F$. The '''fixed field''' of $G$ is the set: :$\Fix G = \set {f \in F : \forall \sigma \in G : \map \sigma f = f}$


Definition:Group Action Induced on Subgroup

Let $G$ be a group. Let $X$ be a set. Let $\phi : G \times X \to X$ be a group action. Let $H \le G$ be a subgroup. The '''group action induced on $H$''' is the restriction of $\phi$ to $H \times X$. Equivalently, the '''group action induced on $H$''' is the group action associated to the permutation representation: :$\rho \circ \iota : H \to \struct {\map \Gamma X, \circ}$ where: :$\iota : H \to G$ is the inclusion homomorphism :$\rho$ is the  permutation representation of $\phi$ :$\struct {\map \Gamma X, \circ}$ is the symmetric group on $X$.


Definition:Stable Under Group Action

Let $G$ be a group. Let $X$ be a set. Let $\phi : G \times X \to X$ be a group action. Let $S \subset X$ be a subset of $X$. Then $S$ is '''stable (under $\phi$)''' {{iff}}: :$\map \phi {G \times S} \subset S$ Category:Definitions/Group Actions 80def9y5meto98n8cex3b88rvy86d4n


Definition:Group Action Induced on Stable Subset

Let $G$ be a group. Let $X$ be a set. Let $\phi : G \times X \to X$ be a group action. Let $S \subset X$ be a stable subset. The '''group action induced on $S$''' is the restriction of $\phi$ to $G \times S$.


Definition:Complete Quotient

Let $x$ be an irrational number. The '''sequence of complete quotients''' is defined recursively by: :$\alpha_0 = x$ :$\alpha_{n+1} = \displaystyle\frac1{\{\alpha_n\}}$ where $\{\cdot\}$ denotes fractional part.


Definition:Parametric Equation

Let $\map \RR {x_1, x_2, \ldots, x_n}$ be a relation on the variables $x_1, x_2, \ldots, x_n$. Let the truth set of $\RR$ be definable as: :$\forall k \in \N: 1 \le k \le n: x_k = \map {\phi_k} t$ where: :$t$ is a variable whose domain is to be defined :each of $\phi_k$ is a mapping whose domain is the domain of $t$ and whose codomain is the domain of $x_k$. Then each of: :$x_k = \map {\phi_k} t$ is a '''parametric equation''' where $t$ is the '''parameter'''. The set: :$\set {\phi_k: 1 \le k \le n}$ is a '''set of parametric equations specifying $\RR$'''. === $2$ Dimensions === {{:Definition:Parametric Equation/2 Dimensions}} Category:Definitions/Analysis Category:Definitions/Mapping Theory mske9rzz1wru8u49yo7w5lkbg7xrft2


Definition:Basis Expansion/Recurrence/Period

The '''period of recurrence''' is the number of digits in the recurring part after which it repeats itself.


Definition:Force/Dimension

The dimension of measurement of '''force''' is $\mathsf {M L T}^{-2}$. This arises from Newton's Second Law of Motion and its definition as a mass (of dimension $\mathsf M$) multiplied by an acceleration (of dimension $\mathsf {L T}^{-2}$).


Definition:Force/Unit

The units of measurement of '''force''' are as follows: === Newton === {{:Definition:Newton (Unit)}} === Dyne === {{:Definition:Dyne}} Thus we see: :$1 \ \mathrm N = 10^5 \ \mathrm {dyn}$


Definition:Electromotive Force/Dimension

The dimension of measurement of '''electromotive force''' is $\mathsf {M L}^2 \mathsf T^{-3} \mathsf I^{-1}$. This derives from its definition as: :$\dfrac {\text{Work} } {\text {Charge} }$


Definition:Electromotive Force/Unit

The unit of measurement of '''electromotive force'''  is the volt $\mathrm V$: :$1 \ \mathrm V = 1 \ \mathrm J \ \mathrm C^{-1}$ that is, $1$ joule per coulomb.


Definition:Electromotive Force

'''Electromotive force''' is a quantity that measures the source of potential energy in an electrical circuit. It is defined as the amount of work per unit electric charge.


Definition:Inductance

'''Inductance''' is the property of an electrical conductor by which a change in electric current through it induces an electromotive force in both the conductor itself and also in any nearby conductors.


Definition:Inductance/Dimension

The dimension of measurement of '''inductance''' is: :$\mathsf {M L}^2 \mathsf T^{-2} \mathsf I^{-2}$ This derives from its definition as: :$v = L \dfrac {\mathrm d I} {\mathrm d t}$


Definition:Dynamical System

A '''dynamical system''' is a non-linear system in which a function describes the time dependence of a point in a geometrical space.


Definition:Resistance

'''Resistance''' is a measure of how difficult it is to pass an electric current through a conductor. It is defined as the electromotive force needed to push a unit current through that conductor.


Definition:Resistance/Dimension

The dimension of measurement of '''resistance''' is: :$\mathsf {M L}^2 \mathsf T^{−3} \mathsf I^{−2}$ This derives from its definition as: :$R = \dfrac {\text {EMF} } {\text {current} }$


Definition:Resistance/Unit

The unit of measurement of '''resistance'''  is the ohm $\Omega$: :$1 \ \Omega = 1 \ \mathrm{V} \ \mathrm{A}^{-1}$ that is, $1$ volt per ampere.


Definition:Electric Current/Dimension

'''Electric current''' is one of the fundamental dimensions of physics. In dimensional analysis it is assigned the symbol $\mathsf I$.


Definition:Volume/Dimension

'''Volume''' is a dimension of measurement of physics. The dimension of measurement of '''volume''' is: :$L^3$ length cubed.


Definition:Volume/Unit

The SI unit of '''volume''' is the cubic metre: : $\mathrm m^3$ The CGS unit of '''volume''' is the cubic centimetre: :$\mathrm {cm}^3$ or, less formally: :$\mathrm {cc}$ Thus: :$1 \ \mathrm m^3 = 10^6 \ \mathrm {cm}^3 = 1 \,000 \,000 \ \mathrm {cc}$


Definition:Lucas Prime

A '''Lucas prime''' is a Lucas number which happens to be prime.


Definition:Series (Electronics)

Two or more components in an electrical circuit are connected in '''series''' {{iff}} the current through them flows along a single path through the circuit. {{finish|Needs to be formalised, with the concept of an electrical circuit being mapped by means of a graph. This will need plenty of work.}} Category:Definitions/Electronics obfg4v861zaqqatagkvp8wsqve6x4g3


Definition:Anticlockwise

'''Anticlockwise''' is this direction: :400px


Definition:Clockwise

'''Clockwise''' is this direction: :400px


Definition:Rocket

A '''rocket''' is a device whose means of propulsion works by throwing material out of one end at high speed in order to take advantage of the Principle of Conservation of Momentum to move in the opposite direction. {{finish}} Category:Definitions/Rocket Science 8into5ob296rmdwo4ylnrumq3nrehv9


Definition:Cuboid

A '''cuboid''' is a parallelepiped whose faces are all rectangular. :400px


Definition:Moon

'''The moon''' is the single natural satellite of Earth.


Definition:Integration With Respect To

Let $f$ be a function which is integrable on a domain $D$. Let $F$ be the primitive of $f$: :$\displaystyle F = \int \map f x \rd x$ The operation of evaluating the primitive of $f$ is referred to as '''integration with respect to $x$.'''


Definition:Spherical Triangle

A '''spherical triangle''' is a geometric figure described on the surface of a sphere which is bounded by the arcs of three great circles which intersect in $3$ vertices: :500px


Definition:Spherical Triangle/Side

The '''sides''' of a '''spherical triangle''' $T$ are the arcs of the $3$ great circles which form the boundaries of $T$ between the points at which they intersect.


Definition:Linear Momentum/Dimension

The dimension of measurement of '''linear momentum''' is $\mathsf {M L T}^{-1}$.


Definition:Frequency (Physics)

Let $\map f t$ be a periodic function of time $t$. The '''frequency''' of $f$ is the number of periods of $f$ that occur during a unit time interval.


Definition:Zero Operator

Let $H$ be a Hilbert space. Let $A$ be a linear operator on $H$. Suppose that: :$\forall h \in H: A \left({h}\right) = 0$ Then $A$ is called a '''zero operator''', and is denoted as $\mathbf 0$. Category:Definitions/Hilbert Spaces qijcp7dtvjflubcfb0t86jre0x7cua6


Definition:Space Diagonal

Consider a parallelepiped $P$. The '''space diagonals''' of $P$ are the line segments that can be drawn from each vertex $X$ of $P$ to the vertex $Y$ of $P$ which is not contained in any of the faces of $P$ which contain $X$. :400px In the above diagram, the '''space diagonals''' of the parallelepiped $ABCDEFGH$ are $AF$, $BE$, $CH$ and $DG$.


Definition:Fermat-Torricelli Point

Let $ABC$ be a triangle. The '''Fermat Point''' is defined to be the point $P$ which minimizes the following quantity: :$AP + BP + CP$ which is the total distance from the three vertices to point $P$.


Definition:Weakly Abnormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. === Definition 1 === {{:Definition:Weakly Abnormal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Weakly Abnormal Subgroup/Definition 2}} === Definition 3 === {{:Definition:Weakly Abnormal Subgroup/Definition 3}}


Definition:Weakly Abnormal Subgroup/Definition 1

$H$ is '''weakly abnormal in $G$''' {{iff}}: :$\forall g \in G: g \in H^{\gen g}$ where $H^{\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$.


Definition:Weakly Pronormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. === Definition 1 === {{:Definition:Weakly Pronormal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Weakly Pronormal Subgroup/Definition 2}}


Definition:Trivial Vector Space

Let $V$ be a vector space. Then $V$ is said to be '''trivial''' {{iff}}: :$V = \left\{ {\mathbf 0}\right\}$ where $\mathbf 0$ denotes the zero vector.


Definition:Weakly Pronormal Subgroup/Definition 1

$H$ is '''weakly pronormal in $G$''' {{iff}}: :$\forall g \in G: \exists x \in H^{\gen g}: H^x = H^g$ where: :$H^{\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$ :$H^x$ denotes the conjugate of $H$ by $x$.


Definition:Left Zero Divisor

A '''left zero divisor (in $R$)''' is an element $x \in R$ such that: : $\exists y \in R^*: x \circ y = 0_R$


Definition:Right Zero Divisor

A '''right zero divisor (in $R$)''' is an element $x \in R$ such that: : $\exists y \in R^*: y \circ x = 0_R$


Definition:Exhaust Velocity

Let $R$ be a rocket. The '''exhaust velocity''' of $R$ is the speed at which the propellant of $R$ is emitted out of the back end, to make $R$ travel forwards. Category:Definitions/Rocket Science b3t2yufyhfr3kiisnasodnx14rg98ov


Definition:Usual Ordering

Let $X$ be one of the sets of numbers: $\N$, $\Z$, $\Q$, $\R$. The '''usual ordering''' on $X$ is the conventional counting and measuring order on $X$ that is learned when one is initially introduced to numbers. === Natural Numbers === {{Definition:Ordering on Natural Numbers}} === Integers === {{Definition:Ordering on Integers}} {{expand|This needs to be formalised. The page is included as it is one of the most wanted redline links.}}


Definition:Bid

Let $A$ be an auction. A '''bid''' is a move by a player in $A$ which consists of the submission of a non-negative number. Such a '''bid''' is associated with a payment which is to be made by the winning player in exchange for the prize. Category:Definitions/Game Theory c6zmgj0zfanq5kdfbry41nbd4qjnkip


Definition:Fractal

A '''fractal''' is an object that has the property of self-similarity on all scales.


Definition:Universal Cover

A '''universal cover''' is a covering space which is simply connected. {{explain|Context}} Category:Definitions/Topology ku749z3mkng85xpdpgoubu0l53k7f7p


Definition:Conjugacy Action

Let $\struct {G, \circ}$ be a group. The '''(left) conjugacy action''' of $G$ is the left group action $* : G \times G \to G$ defined as: :$\forall g, x \in G: g * x = g \circ x \circ g^{-1}$ The '''right conjugacy action''' of $G$ is the right group action $* : G \times G \to G$ defined as: :$\forall x, g \in G: x * g = g^{-1} \circ x \circ g$


Definition:Subinterval

Let $I$ be a real interval. A '''subinterval''' $J$ of $I$ is a real interval such that $J \subseteq I$. {{Proofread|A purely intuitive definition which may or may not be accurate -- for example, is $\left({a \,.\,.\, b}\right)$ strictly speaking a subinterval of $\left[{a \,.\,.\, b}\right]$?}} Category:Definitions/Real Analysis kp8w81att27xxfvloskryzhnohdgq9m


Definition:Subdivision (Real Analysis)/Finite

Let $x_0, x_1, x_2, \ldots, x_{n - 1}, x_n$ be points of $\R$ such that: :$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$ Then $\set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a '''finite subdivision of $\closedint a b$'''.


Definition:Subdivision (Real Analysis)/Infinite

Let $x_0, x_1, x_2, \ldots$ be an infinite number of points of $\R$ such that: :$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < \ldots \le b$ Then $\set {x_0, x_1, x_2, \ldots}$ forms an '''infinite subdivision of $\closedint a b$'''.


Definition:Orthonormal Basis

Let $\mathbb F$ be a field. Let $V$ be a vector space over $\mathbb F$. Let $S \subset V$ be a subset of $V$. Then $S$ is an '''orthonormal basis''' {{iff}}: :$(1): \quad S$ is a basis. :$(2): \quad S$ is orthonormal. Category:Definitions/Linear Algebra rn1yswtz371sj368r4xkcfz5q37cxc5


Definition:Left Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. A left module over $R$ is an $R$-algebraic structure $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the '''(left) ring action''', which satisfies the left module axioms: {{:Definition:Left Module Axioms}}


Definition:Right Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. A right module over $R$ is an $R$-algebraic structure $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the '''(right) ring action''', which satisfies the right module axioms: {{:Definition:Right Module Axioms}}


Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences of real or complex numbers. === Definition 1 === {{:Definition:O Notation/Little-O Notation/Sequence/Definition 1}} === Definition 2 === {{:Definition:O Notation/Little-O Notation/Sequence/Definition 2}} This is denoted :$a_n = o \left({b_n}\right)$


Definition:O Notation/Little-O Notation/Sequence/Definition 1

Let $b_n\neq0$ for all $n$. '''$a_n$ is little-O of $b_n$''' {{iff}} :$\displaystyle \lim_{n\to\infty}\frac{a_n}{b_n} = 0$


Definition:O Notation/Little-O Notation/Sequence/Definition 2

'''$a_n$ is little-O of $b_n$''' {{iff}} :$\forall \epsilon \in \R: \epsilon > 0 : \exists n_0 \in \N : \left({n \ge n_0 \implies \left\vert{a_n}\right\vert \le \epsilon \cdot \left\vert{b_n}\right\vert}\right)$ That is: :For all $\epsilon>0$, $\left\vert{a_n}\right\vert \le \epsilon \cdot \left\vert{b_n}\right\vert$ for all sufficiently large $n$.


Definition:O Notation/Little-O Notation/Real Functions/Definition 1

Let $g(x)\neq0$ for $x$ sufficiently large. '''$f$ is little-o of $g$''' as $x \to \infty$ {{Iff}}: :$\displaystyle \lim_{x \to \infty} \ \frac{f \left({x}\right)} {g \left({x}\right)} = 0$


Definition:O Notation/Little-O Notation/Real Functions/Definition 2

'''$f$ is little-o of $g$''' as $x \to \infty$ {{Iff}}: :$\forall \epsilon \in \R : \epsilon > 0 : \exists x_0 \in \R : \forall x \geq x_0 : |f(x)| \leq \epsilon \cdot |g(x)|$


Definition:Ambivalent Group

Let $G$ be a group. Then $G$ is '''ambivalent''' {{iff}} every element of $G$ is conjugate to its inverse: :$\forall g \in G : \exists h \in G : h g h^{-1} = g^{-1}$


Definition:Real Group Element

Let $G$ be a group. Let $g \in G$. Then $g$ is a '''real element (of $G$)''' {{iff}} it is conjugate to its inverse: :$\exists h \in G : hgh^{-1} = g^{-1}$


Definition:O Notation/Big-O Notation/Uniform

Let $X$ be a set. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$. Let $f, g : X \to V$ be mappings. Then '''$f$ is big $\OO$ of $g$ uniformly''' {{iff}}: :$\exists c > 0 : \forall x \in X : \norm {\map f x} \le c \cdot \norm {\map g x}$ This is denoted: :$f = \map \OO g$


Definition:Group Action/Right Group Action

A '''right group action''' is a mapping $\phi: X \times G \to X$ such that: :$\forall \tuple {x, g} \in X \times G : x * g := \map \phi {x, g} \in X$ in such a way that the right group action axioms are satisfied: {{:Definition:Right Group Action Axioms}}


Definition:Right Group Action Axioms

{{begin-axiom}} {{axiom | n = \text {RGA} 1         | q = \forall g, h \in G, x \in X         | m = \paren {x * g} * h = x * \paren {g \circ h} }} {{axiom | n = \text {RGA} 2         | q = \forall x \in X         | m =  x * e = x }} {{end-axiom}}


Definition:Conjugacy Action/Subgroups

Let $X$ be the set of all subgroups of $G$. The '''(left) conjugacy action on subgroups''' is the group action $* : G \times X \to X$: :$g * H = g \circ H \circ g^{-1}$ The '''right conjugacy action on subgroups''' is the group action $* : X \times G \to X$: :$H * g = g^{-1} \circ H \circ g$


Definition:Direct Sum of Module Homomorphisms

Let $R$ be a ring. Let $M,N,P,Q$ be $R$-modules. Let $M\oplus N$ and $P\oplus Q$ be their direct sum. Let $f : M \to P$ and $g : N \to Q$ be module homomorphisms. The '''direct sum''' of $f$ and $g$ is the module homomorphism $f\oplus g : M\oplus N \to P\oplus Q$ defined as: :$(f\oplus g) (m, n) = ( f(m) , g(n) )$


Definition:Transitive Group Action/n-transitive

Let $n\geq1$ be a natural number. The group action is '''$n$-transitive''' {{iff}} for any two ordered $n$-tuples $(x_1, \ldots, x_n)$ and $(y_1, \ldots, y_n)$ of pairwise distinct elements of $S$, there exists $g\in G$ such that: :$\forall i\in \{1, \ldots, n\} : g * x_i = y_i$


Definition:Generated Submodule

=== Definition 1 === {{:Definition:Generated Submodule/Definition 1}} === Definition 2 === {{:Definition:Generated Submodule/Definition 2}}


Definition:Wilson Prime/Sequence

The sequence of Wilson primes begins: :$5, 13, 563$ The next term in the sequence, if there is one, is greater than $2 \times 10^{13}$.


Definition:Field Norm of Complex Number

Let $z = a + i b$ be a complex number, where $a, b \in \R$. Then the '''field norm of $z$''' is written $\map N z$ and is defined as: :$\map N z := \cmod \alpha^2 = a^2 + b^2$ where $\cmod \alpha$ denotes the complex modulus of $\alpha$.


Definition:Value of Continued Fraction/Finite

Let $n\geq0$ be a natural number. Let $(a_k)_{0 \leq k \leq n}$ be a finite continued fraction in $F$. Let $\overline F = F \cup \{\infty\}$ be extended by infinity. === Definition 1 === The '''value''' $[a_0, a_1, \ldots, a_n] \in F \cup \{\infty\}$ is the right iteration of the binary operation: :$[\cdot,\cdot] :  F \times \overline F \to \overline F$: :$[a, b] = a + \dfrac 1 b$. That is, it is recursively defined as: :$[a_0, \ldots, a_n] = \begin{cases}    a_0 & : n = 0 \\   a_0 + \dfrac 1 {\left[{a_1, \ldots, a_n}\right]} & : n > 0 \\ \end{cases}$ or as: :$[a_0, \ldots, a_n] = \begin{cases}    a_0 & : n = 0 \\   \left[a_0, \ldots, a_{n-2}, a_{n-1} + \dfrac 1 {a_n}\right] & : n > 0 \\ \end{cases}$ === Definition 2 === Let the matrix product: :$\begin{pmatrix}a_0 & 1 \\ 1 & 0\end{pmatrix}\cdots\begin{pmatrix}a_n & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix}x_{11} & x_{12} \\ x_{21} & x_{22}\end{pmatrix}$ The '''value''' of the finite continued fraction is $\dfrac{x_{11}}{x_{21}}$


Definition:Value of Continued Fraction/Infinite

Let $\struct {F, \norm {\,\cdot\,}}$ be a valued field. Let $C = (a_n)_{n\geq 0}$ be a infinite continued fraction in $F$. Then $C$ '''converges''' to its '''value''' $x \in F$ {{iff}} the following hold: #For all natural numbers $n \in \N_{\geq 0}$, the $n$th denominator is nonzero #The sequence of convergents $\sequence{C_n}_{n\geq 0}$ converges to $x$.


Definition:Maximal Ideal of Ring/Left

A left ideal $J$ of $R$ is a '''maximal left ideal''' {{iff}}: :$(1): \quad J \subsetneq R$ :$(2): \quad$ There is no left ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.


Definition:Maximal Ideal of Ring/Right

A right ideal $J$ of $R$ is a '''maximal right ideal''' {{iff}}: :$(1): \quad J \subsetneq R$ :$(2): \quad$ There is no right ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.


Definition:Residue Field of Local Ring

Let $R$ be a commutative local ring. Let $m$ be its maximal ideal. The '''residue field''' is the quotient ring $R/m$.


Definition:Asymptotically Equal/Sequences/Definition 1

Let $b_n \ne 0$ for all $n$. $\sequence {a_n}$ is '''asymptotically equal''' to  $\sequence {b_n}$ {{iff}}: :$\displaystyle \lim_{n \to \infty} \dfrac {a_n} {b_n} \to 1$


Definition:Asymptotically Equal/Sequences/Definition 2

$\sequence {a_n}$ is '''asymptotically equal''' to  $\sequence {b_n}$ {{iff}}: :$a_n - b_n = \map o {b_n}$ where $o$ denotes little-o notation.


Definition:Asymptotically Equal/Sequences/Definition 3

$\sequence {a_n}$ is '''asymptotically equal''' to  $\sequence {b_n}$ {{iff}}: :$a_n - b_n = \map o {a_n}$ where $o$ denotes little-o notation.


Definition:Differential/Functional

Let $J \sqbrk y$ be a differentiable functional. Let $h$ be an increment of the independent variable $y$. Then the term linear {{WRT}} $h$ is called the '''differential''' of the functional $J$, and is denoted by $\delta J \sqbrk {y; h}$.


Definition:Compatible Atlases/Definition 1

$\mathscr F, \mathscr G$ are '''$C^k$-compatible''' {{iff}} their union $\mathscr F \cup \mathscr G$ is an atlas of class $C^k$.


Definition:Compatible Atlases/Definition 2

$\mathscr F$ and $\mathscr G$ are '''$C^k$-compatible''' {{iff}} every pair of charts $\struct {U, \phi} \in \mathscr F$ and $\struct {V, \psi} \in \mathscr G$ are $C^k$-compatible.


Definition:Atlas/Maximal Atlas

Let $A$ be a $d$-dimensional atlas of class $C^k$ of $M$. === Definition 1 === {{:Definition:Atlas/Maximal Atlas/Definition 1}} === Definition 2 === {{:Definition:Atlas/Maximal Atlas/Definition 2}} === Definition 3 === {{:Definition:Atlas/Maximal Atlas/Definition 3}}


Definition:Transition Mapping between Charts

Let $M$ be a topological space. Let $d$ be a natural number. Let $(U,\phi)$ and $(V,\psi)$ be $d$-dimensional charts of $M$. Let $U\cap V \neq\emptyset$. The '''transition map from $\phi$ to $\psi$''' is the mapping: :$\psi \circ \phi^{-1} : \phi(U\cap V) \to \psi(U\cap V)$


Definition:Compatible Charts

Let $M$ be a topological space. Let $d$ be a natural number. Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be $d$-dimensional charts of $M$. Then $\struct {U, \phi}$ and $\struct {V, \psi}$ are '''$C^k$-compatible''' {{iff}} their transition mapping: :$\psi \circ \phi^{-1}: \map \phi {U \cap V} \to \map \psi {U \cap V}$ is of class $C^k$. === Smoothly Compatible Charts === {{:Definition:Compatible Charts/Smooth}}


Definition:Compatible Charts/Smooth

$\struct {U, \phi}$ and $\struct {V, \psi}$ are '''smoothly compatible''' {{iff}} their transition mapping: :$\psi \circ \phi^{-1} : \map \phi {U \cap V} \to \map \psi {U \cap V}$ is of class $C^\infty$.


Definition:Chart Compatible with Atlas

Let $M$ be a topological space. Let $A$ be a $d$-dimensional $C^k$-atlas on $M$. Let $\struct {U, \phi}$ be a $d$-dimensional chart of $M$. Then $\struct {U, \phi}$ is '''$C^k$-compatible with $A$''' {{iff}} $\struct {U, \phi}$ is $C^k$-compatible with every chart of $A$. Category:Definitions/Manifolds 9eyy1syk3175jgb8zs3rnvcf1n4i0mw


Definition:Atlas/Maximal Atlas/Definition 1

$A$ is a '''maximal $C^k$-atlas''' of dimension $d$ {{iff}} $A$ is not strictly contained in another $C^k$-atlas.


Definition:Atlas/Maximal Atlas/Definition 2

$A$ is a '''maximal $C^k$-atlas''' {{iff}} $A$ contains all charts of $M$ that are $C^k$-compatible with $A$.


Definition:Atlas/Maximal Atlas/Definition 3

$A$ is a '''maximal $C^k$-atlas''' {{iff}} $A$ is a maximal element of some differentiable structure, partially ordered by inclusion. That is, a maximal element of some equivalence class of the set of atlases of class $\mathcal C^k$ on $M$ under the equivalence relation of compatibility.


Definition:Uniform Contraction Mapping

Let $M$ and $N$ be metric spaces. Let $f : M \times N \to M$ be a mapping. Then $f$ is a '''uniform contraction''' {{iff}} there exists $K<1$ such that for all $x,y\in M$ and $t\in N$: :$d(f(x,t), f(y,t)) \leq K\cdot d(x,y)$.


Definition:Local Diffeomorphism

Let $n$ and $k$ be natural numbers. Let $U \subset \R^n$ be an open set. {{explain|Context of Open Set}} Let $f: U \to \R^n$ be a mapping. Then $f$ is a '''local $C^k$-diffeomorphism''' {{iff}} every $a \in U$ has a open neighborhhood such that the restriction of $f$ to it is a $C^k$-diffeomorphism on its image. === Smooth Local Diffeomorphism === {{definition wanted}} Category:Definitions/Analysis th6qvtq1u3389gwzzwspm3q7qcdgvso


Definition:Open Set/Real Analysis/Real Numbers

Let $I \subseteq \R$ be a subset of the set of real numbers. Then $I$ is '''open (in $\R$)''' {{iff}}: : $\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \openint {x_0 - \epsilon} {x_0 + \epsilon} \subseteq I$ where $\openint {x_0 - \epsilon} {x_0 + \epsilon}$ is an open interval. Note that $\epsilon$ may depend on $x_0$.


Definition:Open Set/Real Analysis/Real Euclidean Space

Let $n \ge 1$ be a natural number. Let $U \subseteq \R^n$ be a subset. Then $U$ is '''open (in $\R^n$)''' {{iff}}: :$\forall x \in U : \exists R > 0: \map B {x, R} \subset U$ where $\map B {x, R}$ denotes the open ball of radius $R$ centered at $x$.


Definition:Smith Number

A '''Smith number''' is a composite number for which the sum of its digits is equal to the sum of the digits in its prime decomposition.


Definition:Smith Brothers

'''Smith brothers''' are two consecutive integers which are both '''Smith numbers'''. === Sequence of Smith Brothers === {{:Definition:Smith Brothers/Sequence}}


Definition:Differentiable Mapping/Real Function/Point/Definition 1

$f$ is '''differentiable at the point $\xi$''' {{iff}} the limit: :$\displaystyle \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$ exists.


Definition:Differentiable Mapping/Real Function/Point/Definition 2

$f$ is '''differentiable at the point $\xi$''' {{iff}} the limit: :$\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists.


Definition:Open Ball/Real Analysis

Let $n \ge 1$ be a natural number. Let $\R^n$ denote a real Euclidean space Let $\left\Vert{\cdot}\right\Vert$ denote the Euclidean norm. Let $a \in \R^n$. Let $R > 0$ be a strictly positive real number. The '''open ball of center $a$ and radius $R$''' is the subset: :$B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert < R}\right\}$


Definition:Closed Rectangle

Let $n\geq1$ be a natural number. Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. The Cartesian product of closed intervals: :$\displaystyle \prod_{i \mathop = 1}^n \left[{a_i \,.\,.\, b_i}\right] = \left[{a_1 \,.\,.\, b_1}\right] \times \cdots \times \left[{a_n \,.\,.\, b_n}\right] \subseteq \R^n$ is called a '''closed rectangle in $\R^n$''' or '''closed $n$-rectangle'''. === Degenerate Case === In case $a_i > b_i$ for some $i$, the rectangle is taken to be the empty set $\varnothing$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.


Definition:Closed Ball/Real Analysis

Let $n \ge 1$ be a natural number. Let $\R^n$ denote real Euclidean space Let $\left\Vert{\, \cdot \,}\right\Vert$ denote the Euclidean norm. Let $a \in \R^n$. Let $R > 0$ be a strictly positive real number. The '''closed ball of center $a$ and radius $R$''' is the subset: :$B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert \le R}\right\}$


Definition:Closed Set/Real Analysis/Real Numbers

Let $S \subseteq \R$ be a subset of the set of real numbers. Then $S$ is '''closed (in $\R$)''' {{iff}} its complement $\R \setminus S$ is an open set.


Definition:Closed Set/Real Analysis/Real Euclidean Space

Let $n\geq1$ be a natural number. Let $S \subseteq \R^n$ be a subset. Then $S$ is '''closed (in $\R^n$)''' {{iff}} its complement $\R^n \setminus S$ is an open set.


Definition:Topology Generated by Synthetic Basis/Definition 1

The '''topology on $S$ generated by $\BB$''' is defined as: :$\tau = \set{\bigcup \AA: \AA \subseteq \BB}$


Definition:Topology Generated by Synthetic Basis/Definition 2

The '''topology on $S$ generated by $\BB$''' is defined as: :$\tau = \set {U \subseteq S: U = \bigcup \set {B \in \BB: B \subseteq U}}$


Definition:Basis (Topology)/Synthetic Basis/Definition 1

A '''synthetic basis on $S$''' is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that: {{begin-axiom}} {{axiom | n = B1         | t = $\mathcal B$ is a cover for $S$ }} {{axiom | n = B2         | q = \forall U, V \in \mathcal B         | t = $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$ }} {{end-axiom}} That is, the intersection of any pair of elements of $\mathcal B$ is a union of sets of $\mathcal B$.


Definition:Basis (Topology)/Synthetic Basis/Definition 2

A '''synthetic basis on $S$''' is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that: : $\mathcal B$ is a cover for $S$ : $\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$


Definition:Topology Induced by Atlas

Let $X$ be a set. Let $A$ be an atlas on $X$. The '''topology induced by the atlas''' is the topology generated by the synthetic sub-basis: :$\displaystyle\bigcup_{(U,\phi) \in A }\left\{\phi^{-1}(V) : V \text{ is open in } U \right\} \subset X$ Category:Definitions/Manifolds h4if4yplnmv9jzgj98we7uukzysh7oo


Definition:Generated Submodule/Definition 1

The '''submodule generated by $S$''' is the intersection of all submodules of $M$ containing $S$.


Definition:Generated Submodule/Definition 2

The '''submodule generated by $S$''' is the set of all linear combinations of elements of $S$.


Definition:Generator of Module/Definition 1

$S$ is a '''generator of $M$''' {{iff}} every element of $M$ is a linear combination of elements of $S$.


Definition:Generator of Module/Definition 2

$S$ is a '''generator of $M$''' {{iff}} $M$ has no proper submodule containing $S$.


Definition:Generator of Module/Definition 3

$S$ is a '''generator of $M$''' {{iff}} $M$ is the submodule generated by $S$.


Definition:Generator of Vector Space

Let $K$ be a division ring. Let $\mathbf V$ be a vector space over $K$. Let $S \subseteq \mathbf V$ be a subset of $\mathbf V$. $S$ is a '''generator of $\mathbf V$''' {{iff}} every element of $\mathbf V$ is a linear combination of elements of $S$.


Definition:Real Submanifold

Let $n,k\geq1$ be natural numbers. Let $M\subset\R^n$ be a subset. === Definition Using Local Diffeomorphisms === $M$ is a '''real $C^k$-submanifold of dimension $d$''' of $\R^n$ {{iff}} for all $p\in M$ there exists a open neighborhood $U$ of $p$ in $\R^n$ and a differentiable function $\phi : U \to \R^n$ that is a $C^k$-diffeomorphism on its image, such that: :$\phi(M \cap U) = \phi(U) \cap (\R^d\times\{0\})$ === Definition Using Local Submersions === $M$ is a '''real $C^\infty$-submanifold of dimension $d$''' of $\R^n$ {{iff}} for all $p\in M$ there exists a open neighborhood $U$ of $p$ in $\R^n$ and a $C^\infty$-submersion $\phi : U \to \R^{n-d}$ such that: :$M \cap U = \phi^{-1}(0)$ === Definition Using Local Embeddings === $M$ is a '''real $C^\infty$-submanifold of dimension $d$''' of $\R^n$ {{iff}} for all $p\in M$ there exists an open neighborhood $U$ in $\R^d$ and a $C^\infty$-embedding $\phi : U \to \R^{n}$ such that: :$p \in \phi(U) \subset M$


Definition:Embedding (Differential Geometry)

Let $m,n\geq1$ be natural numbers. Let $U\subset\R^n$ be open. Let $f : U \to \R^m$ be a mapping. Then $f$ is a $C^k$-'''embedding''' {{iff}} $f$ is: * injective * a $C^k$-immersion * a homeomorphism on its image === Rank === The '''rank''' of an embedding is the rank of its differential at any point. === Smooth Embedding === {{stub}}


=== Real Numbers === {{:Definition:Open Neighborhood/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Open Neighborhood/Real Analysis/Real Euclidean Space}}


Definition:Open Neighborhood/Real Analysis/Real Numbers

Let $x\in\R$ be a real number. Let $I \subseteq \R$ be a subset. Then $I$ is an '''open neighborhood of $x$''' {{iff}} $I$ is open and $I$ is a neighborhood of $x$.


=== On an Open Interval === {{:Definition:Continuously Differentiable/Real Function/Open Interval}} === On an Open Set === {{:Definition:Continuously Differentiable/Real Function/Open Set}}


Definition:Continuously Differentiable/Real Function/Open Interval

Let $I\subset\R$ be an open interval. Then $f$ is '''continuously differentiable on $I$''' {{iff}} $f$ is differentiable on $I$ and its derivative is continuous on $I$.


Definition:Continuously Differentiable/Real Function/Open Set

Let $I \subset \R$ be an open set. Then $f$ is '''continuously differentiable on $I$''' {{iff}} $f$ is differentiable on $I$ and its derivative is continuous on $I$.


=== In an Open Set === {{:Definition:Continuously Differentiable/Real-Valued Function/Open Set}}


Definition:Continuously Differentiable/Real-Valued Function/Open Set

Let $U$ be an open subset of $\R^n$.  Let $f: U \to \R$ be a real-valued function. Then $f$ is '''continuously differentiable in the open set $U$''' {{iff}}: :$(1): \quad f$ is differentiable in $U$. :$(2): \quad$ the partial derivatives of $f$ are continuous in $U$.


=== On an Open Set === {{:Definition:Continuously Differentiable/Vector-Valued Function/Open Set}}


Definition:Continuously Differentiable/Vector-Valued Function/Open Set

Let $U \subset \R^n$ be an open set. Let $f: U \to \R^m$ be a vector-valued function. Then $f$ is '''continuously differentiable in $U$''' {{iff}} $f$ is differentiable in $U$ and its partial derivatives are continuous in $U$.


Definition:Holomorphic Function/Complex Plane

Let $U\subset\C$ be an open set. Let $f : U \to \C$ be a complex function. Then $f$ is '''holomorphic in $U$''' {{iff}} $f$ is differentiable at each point of $U$.


Definition:Derivative/Complex Function/Point

Let $D\subseteq \C$ be an open set. Let $f : D \to \C$ be a complex function. Let $z_0 \in D$ be a point in $D$. Let $f$ be complex-differentiable at the point $z_0$. That is, suppose the limit $\displaystyle \lim_{h \to 0} \ \frac {f \left({z_0 + h}\right) - f \left({z_0}\right)} h$ exists. Then this limit is called the '''derivative of $f$ at the point $z_0$'''.


Definition:Derivative/Complex Function/Open Set

Let $D\subseteq \C$ be an open set. Let $f : D \to \C$ be a complex function. Let $f$ be complex-differentiable in $D$. Then the '''derivative of $f$''' is the complex function  $f': D \to \C$ whose value at each point $z \in D$ is the derivative $f' \left({z}\right)$: :$\displaystyle \forall z \in D : f' \left({z}\right) := \lim_{h \mathop \to 0} \frac {f \left({z + h}\right) - f \left({z}\right)} h$


Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1

$f$ is '''differentiable''' at $x \in \R^n$ {{iff}} there exists a linear transformation $T: \R^n \to \R^m$ and a mapping $r : U \to \R^m$ such that: :$(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \cdot \norm h$ :$(2): \quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$


Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2

$f$ is '''differentiable''' at $x \in \R^n$ {{iff}} for each real-valued function $f_j: j = 1, 2, \ldots, m$: :$f_j: \mathbb X \to \R$ is differentiable at $x$.


Definition:Differential/Real Function/Point

Let $U \subset \R$ be an open set. Let $f: U \to \R$ be a real function. Let $f$ be differentiable at a point $x \in U$. The '''differential of $f$ at $x$''' is the linear transformation $\rd f \left({x}\right) : \R \to \R$ defined as: :$\rd f \left({x}\right) \left({h}\right) = f' \left({x}\right) \cdot h$ where $f' \left({x}\right)$ is the derivative of $f$ at $x$.


Definition:Differential/Real Function/Open Set

Let $U \subset \R$ be an open set. Let $f : U \to \R$ be a real function. Let $f$ be differentiable in $U$. The '''differential''' $\rd f$ is the mapping $\rd f : U \to \operatorname{Hom} \left({\R, \R}\right)$ defined as: :$\left({\mathrm d f}\right) \left({x}\right) = \rd f \left({x}\right)$ where: :$\rd f \left({x}\right)$ is the differential of $f$ at $x$ :$\operatorname{Hom} \left({\R, \R}\right)$ is the set of all linear transformations from $\R$ to $\R$.


Definition:Zuckerman Number

A '''Zuckerman number''' is a positive integer which is divisible by the product of its digits.


{{:Definition:Partial Derivative/Real Analysis/Point}}


Let $U \subset \R^n$ be an open set. Let $f: U \to \R$ be a real-valued function. Let $a = \tuple {a_1, a_2, \ldots, a_n}^\intercal \in U$. Let $f$ be differentiable at $a$. Let $i \in \set {1, 2, \ldots, n}$. ==== Definition 1 ==== {{:Definition:Partial Derivative/Real Analysis/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Partial Derivative/Real Analysis/Point/Definition 2}}


Definition:Partial Derivative/Real Analysis/Open Set

Let $U\subset\R^n$ be an open set. Let $f : U \to \R$ be a real-valued function. Let $f$ be differentiable in $U$. The '''$i$th partial derivative (function) of $f$ with respect to $x_i$''' is the real-valued function which sends each $x\in U$ to the $i$th partial derivative at $x$.


Definition:Partial Derivative/Real Analysis/Point/Definition 1

The '''partial derivative of $f$ with respect to $x_i$ at $a$''' is denoted and defined as: :$\map {\dfrac {\partial f} {\partial x_i} } a := \map {g_i'} {a_i}$ where: :$g_i$ is the real function defined as $\map g {x_i} = \map f {a_1, \ldots, x_i, \dots, a_n}$ :$\map {g_i'} {a_i}$ is the derivative of $g$ at $a_i$.


Definition:Partial Derivative/Real Analysis/Point/Definition 2

The '''$i$th partial derivative of $f$ at $a$''' is the limit: :$\map {\dfrac {\partial f} {\partial x_i} } a = \displaystyle \lim_{x_i \mathop \to a_i} \frac {\map f {a_1, a_2, \ldots, x_i, \ldots, a_n} - \map f a} {x_i - a}$


Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2

$f$ is '''differentiable at $x$''' {{iff}} there exists a linear transformation $T: \R^n \to \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that: :$(1):\quad \map f {x + h} = \map f x + \map T h + \map r h \cdot h$ :$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$


Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1

$f$ is '''differentiable at $x$''' {{iff}} there exist $\alpha_1, \ldots, \alpha_n \in \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that: :$(1):\quad \map f {x + h} = \map f x + \alpha_1 h_1 + \cdots + \alpha_n h_n + \map r h\cdot h$ :$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$


Definition:Differential/Real-Valued Function/Point

Let $U \subset \R^n$ be an open set. Let $f: U \to \R$ be a real-valued function. Let $f$ be differentiable at a point $x \in U$. :$\displaystyle \d f \left({x; h}\right) := \sum_{i \mathop = 1}^n \frac {\partial f \left({x}\right)} {\partial x_i} h_i = \frac {\partial f \left({x}\right)} {\partial x_1} h_1 + \frac {\partial f \left({x}\right)} {\partial x_2} h_2 + \cdots + \frac {\partial f \left({x}\right)} {\partial x_n} h_n$ where: : $h = \left({h_1, h_2, \ldots, h_n}\right) \in \R^n$ : $\dfrac {\partial f} {\partial x_i}$ is the partial derivative of $f$ {{WRT|Differentiation}} $x_i$.


Definition:Differential/Vector-Valued Function/Point

Let $U \subset \R^n$ be an open set. Let $f: U \to \R^m$ be a vector-valued function. Let $f$ be differentiable at a point $x \in U$. The '''differential of $f$ at $x$''' is the linear transformation $\d f \left({x}\right): \R^n \to \R^m$ defined as: :$\d f \left({x}\right) \left({h}\right) = J_f \left({x}\right) \cdot h$ where: : $J_f \left({x}\right)$ is the Jacobian matrix of $f$ at $x$.


Definition:Derivative/Vector-Valued Function/Point

Let $U \subset \R$ be an open set. Let $\mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k: U \to \R^n$ be a vector-valued function. Let $\mathbf f$ be differentiable at $u \in U$. That is, let each $f_j$ be differentiable at $u \in U$. The '''derivative of $\mathbf f$ {{WRT|Differentiation}} $x$ at $u$''' is defined as :$\dfrac {\d \mathbf f} {\d x} \left({u}\right) = \displaystyle \sum_{k \mathop = 1}^n \dfrac {\d f_k} {\d x} \left({u}\right) \mathbf e_k$ where $\dfrac {\d f_k} {\d x} \left({u}\right)$ is the derivative of $f_k$ with respect to $x$ at $u$.


Definition:Lipschitz Continuity/Point

Let $a\in A$. $f$ is a '''Lipschitz continuous at $a$''' {{iff}} there exists a positive real number $K \in \R_{\ge 0}$ such that: :$\forall x \in A: d\,' \left({f \left({x}\right), f \left({a}\right)}\right) \le K d \left({x, a}\right)$


Let $M$ and $N$ be differentiable manifolds. Let $f : M \to N$ be continuous. === Definition 1 === $f$  is '''differentiable''' {{iff}} for every pair of charts $(U, \phi)$ and $(V,\psi)$ of $M$ and $N$: :$\psi\circ f\circ \phi^{-1} : \phi ( U \cap f^{-1}(V)) \to \psi(V)$ is differentiable. === Definition 2 === $f$  is '''differentiable''' {{iff}} $f$ is  differentiable at every point of $M$. === At a Point === {{:Definition:Differentiable Mapping between Manifolds/Point}}


Let $M$ and $N$ be differentiable manifolds. Let $f: M \to N$ be continuous. Let $p \in M$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping between Manifolds/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping between Manifolds/Point/Definition 2}}


Definition:Differentiable Mapping between Manifolds/Point/Definition 1

$f$ is '''differentiable at $p$''' {{iff}} for every pair of charts $\left({U, \phi}\right)$ and $\left({V, \psi}\right)$ of $M$ and $N$ with $p \in U$ and $f \left({p}\right) \in V$: :$\psi \circ f \circ \phi^{-1}: \phi \left({U \cap f^{-1} \left({V}\right)}\right) \to \psi \left({V}\right)$ is differentiable at $\phi \left({p}\right)$.


Definition:Differentiable Mapping between Manifolds/Point/Definition 2

$f$ is '''differentiable at $p$''' {{iff}} there exists a pair of charts $\left({U, \phi}\right)$ and $\left({V, \psi}\right)$ of $M$ and $N$ with $p \in U$ and $f \left({p}\right) \in V$ such that: :$\psi \circ f \circ \phi^{-1}: \phi \left({U \cap f^{-1} \left({V}\right)}\right) \to \psi \left({V}\right)$ is differentiable at $\phi \left({p}\right)$.


Definition:Injective Space

Let $Z = \left({S, \tau_1}\right)$ be a topological space. Then $Z$ is '''injective (space)''' {{iff}} :for all topological spaces $X = \left({H, \tau_2}\right)$ :and for all continuous mappings $f:H \to S$ :and for all topological spaces $Y = \left({T, \tau_3}\right)$ such that $X$ is topological subspace of $Y$: :there exists a continuous mapping $g:T \to S$: $g \restriction H = f$


Definition:Wieferich Prime

A '''Wieferich prime''' is a prime number $p$ such that: :$p^2 \divides 2^{p − 1} − 1$ where $\divides$ denotes divisibility.


Definition:Wieferich Prime/Sequence

The sequence of Wieferich primes begins: :$1093, 3511 \ldots$ No other Wieferich primes exist below $4 \cdotp 968543 \times 10^{17}$.


Definition:Fermat's Equation

'''Fermat's equation''' is the Diophantine equation: :$x^n + y^n = z^n$


Definition:Retraction (Topology)

Let $T_1 = \left({S_1, \tau_1}\right)$ be a topological space. Let $T_2 = \left({S_2, \tau_2}\right)$ be a topological subspace of $T_1$. It means that :$S_2 \subseteq S_1$ Let $f: S_1 \to S_2$ be a mapping. Then $f$ is '''retraction''' of $T_1$ {{iff}} :$\forall s \in S_2: f\left({s}\right) = s$


Definition:Retract (Topology)

Let $T_1 = \left({S_1, \tau_1}\right)$ be a topological space. Let $T_2 = \left({S_2, \tau_2}\right)$ be a topological subspace of $T_1$. Then $T_2$ is a '''retract''' of $T_1$ {{iff}} :there exists a continuous retraction $f: S_1 \to S_2$ of $T_1$.


Definition:Sphenic Number

A '''sphenic number''' is a (positive) integer which has exactly $3$ distinct prime factors all with multiplicity of $1$.


Definition:Imperial/Volume/Cubic Inch

:$1$ '''cubic inch''' $= 16 \cdotp 38706 \, 4$ cubic centimetres $= 0 \cdotp 01638 \, 7064$ litres. This definition is exact, and is derived from the definition of the imperial unit of length as $2 \cdotp 54$ centimetres.


Definition:Imperial/Volume/Cubic Foot

The '''cubic foot''' is the FPS unit of measurement of volume. :$1$ '''cubic foot''' $= 12^3 = 1728$ cubic inches $= 28 \cdotp 31684 \, 6592$ litres $= 0 \cdotp 02831 \, 68465 \, 92$ cubic metres.


Definition:Taxicab Number

A '''taxicab number''' is a positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways.


Definition:Image of Topological Space

Let $T = \left({S, \tau}\right)$ and $Q = \left({X, \tau'}\right)$ be topological spaces. Let $f:S \to X$ be a mapping. Then '''image (of the topological space $T$) of''' $f$ is equal to :$\operatorname{Im}\left({f}\right) := Q_{f\left[{S}\right]} = \left({f\left[{S}\right], \tau'_{f\left[{S}\right]} }\right)$ where $\tau'_{f\left[{S}\right]}$ denotes the subspace topology on $f\left[{S}\right]$.


Definition:Mile

There are a number of definitions for the '''mile'''. === International Mile === {{:Definition:Imperial/Length/International Mile}} === Nautical Mile === {{:Definition:Nautical Mile}} === Admiralty Mile === {{:Definition:Admiralty Mile}} Category:Definitions/Length tcd8q4x2mvlyelr8n81fgj5zhg8v9wt


Definition:Weak Retract (Topology)

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Then $T_1$ is '''a weak retract of''' $T_2$ {{iff}} :there exists a continuous mapping $f:S_2 \to S_2$: $f \circ f = f$ and $\operatorname{Im}\left({f}\right), T_1$ are homeomorphic.


Definition:Lower Topology

Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology. Then $T$ has '''lower topology''' {{iff}} :$\left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$ is sub-basis of $T$ where $x^\succeq$ denotes the upper closure of $x$.


Definition:Ball

=== Open Ball === {{:Definition:Open Ball}} === Closed Ball === {{:Definition:Closed Ball}} === Unit Ball === {{:Definition:Unit Ball}}


Definition:Homogeneous Cyclotomic Polynomial

Let $n\geq1$ be a positive integer. The $n$th '''homogeneous cyclotomic polynomial''' is the homogenization of the $n$th cyclotomic polynomial. Category:Definitions/Cyclotomic Polynomials iqgnx6lwxx9ncuh3cywe3zn7j8718gw


Definition:Special Highly Composite Number

A '''special highly composite number''' is a highly composite number which is a divisor of all larger highly composite numbers.


Definition:Special Highly Composite Number/Sequence

The sequence of special highly composite numbers consists of: :$1, 2, 6, 12, 60, 2520$ {{OEIS|A106037}}


Definition:Modified Kaprekar Mapping

The '''modified Kaprekar mapping''' is the arithmetic function $K: \Z_{>0} \to \Z_{>0}$ defined on the positive integers  as follows: Let $n \in \Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$). Let $n'$ be the positive integer created by: :arranging the digits of $n$ into descending order of size then: :exchanging the last two digits. Let $n''$ be the positive integer created by: :arranging the digits of $n$ into ascending order of size then: :exchanging the first two digits. Then: :$\map {K'} n = n' - n''$ making sure to retain any leading zeroes to ensure that $\map K n$ has the same number of digits as $n$.


Definition:Modified Kaprekar Process

The '''modified Kaprekar process''' is the repeated application of the '''modified Kaprekar mapping''' to a given positive integer.


Definition:Central Tree

A tree $T$ is '''central''' {{iff}} it has a center.


Definition:Bicentral Tree

A tree $T$ is '''bicentral''' {{iff}} it has a bicenter.


Definition:Perfect Digit-to-Digit Invariant

A '''perfect digit-to-digit invariant''' is a number $n$ which is equal to the sum of the digits of $n$ each raised to the power of itself.


Definition:Perfect Digital Invariant

Let $n \in \Z_{>0}$ be a positive integer. $n$ is a '''perfect digital invariant''' {{iff}} when expressed in decimal notation the $n$th powers of its digits add up to some positive integer $m$.


Definition:Imperial/Area/Acre

One '''acre''' is equal to an oblong measuring $1$ chain by $1$ furlong. That is, measuring $4$ rods, poles or perches by $10$ chains. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''acre''' }} {{eqn | r = 4       | c = roods }} {{eqn | r = 10       | c = square chains }} {{eqn | r = 160 = 4 \times 40       | c = square rods, poles or perches }} {{eqn | r = 4840 = 22 \times 220       | c = square yards }} {{end-eqn}} :810px


Definition:Imperial/Area/Square Yard

One '''square yard''' is equal to a square of side $1$ yard in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square yard''' }} {{eqn | r = 9 = 3^2       | c = square feet }} {{eqn | r = 1296 = 36^2       | c = square inches }} {{end-eqn}}


Definition:Imperial/Area/Square Inch

One '''square inch''' is equal to a square of side $1$ inch in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square inch''' }} {{eqn | r = 6 \cdotp 4516 = 2 \cdotp 54^2       | c = square centimetres }} {{end-eqn}} The definition is exact.


Definition:Imperial/Area/Square Foot

One '''square foot''' is equal to a square of side $1$ foot in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square foot''' }} {{eqn | r = 144 = 12^2       | c = square inches }} {{end-eqn}}


Definition:Kaprekar's Constant

Let $n \in \Z_{>0}$ be a positive integer with $4$ digits, not all of which are equal. Let Kaprekar's process be applied to $n$. The number reached after no more than $8$ iterations is known as '''Kaprekar's constant''', and has the value of $6174$. {{NamedforDef|Dattathreya Ramchandra Kaprekar|cat = Kaprekar}}


Definition:Tri-Automorphic Number

A '''tri-automorphic number''' is a positive integer $n$ such that $3 n^2$ ends in a repetition of $n$.


Definition:Support of Module

Let $A$ be a commutative ring with unity. Let $M$ be a unitary $A$-module. The '''support''' $\operatorname{Supp}(M)$ of $M$ is the set of prime ideals $P$ of $A$ such that the localization of $M$ at $P$ is nonzero: :$\operatorname{Supp}(M) = \{P\in \operatorname{Spec}(A) : M_P\neq0 \}$ where $\operatorname{Spec}(A)$ is the spectrum of $A$.


Definition:Integral Ring Extension

Let $A$ be a commutative ring with unity. Let $R\subset A$ be a subring. The ring extension $R \subseteq A$ is said to be '''integral''' {{iff}} for all $a \in A$, $a$ is integral over $R$.


=== Real-Valued Function on an Abstract Set === {{Definition:Support of Mapping to Algebraic Structure/Real-Valued Function}} === General Real-Valued Function in $\R^n$ === === General Algebraic Structure === Let $(A, *)$ be an algebraic structure with an identity element $e$. Let $S$ be a set. Let $f : S \to A$ be a mapping. The '''support of $f$''' is the set: :$\operatorname{supp}(f) = \{s \in S : f(s) \neq e\}$ {{expand}}


Definition:Weakly Locally Compact Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''weakly locally compact''' {{iff}} every point of $S$ has a compact neighborhood.


Definition:Noetherian Ring/Definition 1

A commutative ring with unity $A$ is '''Noetherian''' {{iff}} every ideal of $A$ is finitely generated.


Definition:Noetherian Ring/Definition 2

A commutative ring with unity $A$ is '''Noetherian''' {{iff}} it satisfies the ascending chain condition on ideals.


Definition:Noetherian Ring/Definition 3

A commutative ring with unity $A$ is '''Noetherian''' {{iff}} it satisfies the maximal condition on ideals.


Definition:Noetherian Ring/Definition 4

A commutative ring with unity $A$ is '''Noetherian''' {{iff}} it is Noetherian as an $A$-module.


Definition:Noetherian Module/Definition 1

$M$ is a '''Noetherian module''' {{iff}} every submodule of $M$ is finitely generated.


Definition:Noetherian Module/Definition 2

$M$ is a '''Noetherian module''' {{iff}} it satisfies the ascending chain condition on submodules.


Definition:Noetherian Module/Definition 3

$M$ is a '''Noetherian module''' {{iff}} it satisfies the maximal condition on submodules.


Definition:Sylow p-Subgroup/Definition 1

Let $\order G = k p^n$ where $p \nmid k$. A '''Sylow $p$-subgroup''' is a $p$-subgroup of $G$ which has $p^n$ elements.


Definition:Sylow p-Subgroup/Definition 2

A '''Sylow $p$-subgroup''' of $G$ is a '''maximal $p$-subgroup''' $P$ of $G$. In this context, maximality means that if $Q$ is a $p$-subgroup of $G$ and $P \le Q$, then $P = Q$.


Definition:Cover of Set/Subset

Let $A \subseteq S$ be a subset. Let $\mathcal C$ be a set of subsets of $S$. Then $\mathcal C$ is a '''cover of $A$''' {{iff}} $A \subseteq \displaystyle \bigcup \mathcal C$, where $\cup$ denotes union.


Definition:Open Cover/Subset

Let $H$ be a subset of $S$. Let $\CC$ be a cover of $H$. Then $\CC$ is an '''open cover for $H$''' {{iff}}: :$\CC \subseteq \tau$ That is, {{iff}} all the elements of $\CC$ are open sets.


Definition:Prime Number Race

The set of prime numbers $\Bbb P$ may be partitioned into subsets according to a particular property. A '''prime number race''' is a comparison of the count of the number of prime numbers in each partition with increasing $p \in \Bbb P$.


Definition:Pandigital Set

A set of integers is '''pandigital''' if it contains exactly $1$ instance of each digit. The digits may or may not contain zero, depending on context.


Definition:Pandigital Set/Integer

The element of a singleton pandigital set is itself referred to as a '''pandigital integer''' or '''pandigital number'''.


Definition:Ackermann-Péter Function

The '''Ackermann-Péter function''' $A: \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{> 0}$ is an integer-valued function defined on the set of ordered pairs of positive integers as: :$A \left({x, y}\right) = \begin{cases} y + 1 & : x = 0 \\ A \left({x - 1, 1}\right) & : x > 0, y = 0 \\ A \left({x - 1, A \left({x, y - 1}\right)}\right) & : \text{otherwise} \end{cases}$


Definition:Fermat Number/Naming Conventions

The '''Fermat number''' $F_0$ is often referred to as the ''' $1$st Fermat number''', so (confusingly) this convention dictates that $F_n$ is the '''$n + 1$th Fermat number'''. However, another convention is that $F_0$ can be referred to as the '''zeroth Fermat number''', thus bringing the appellation in line such that $F_n$ is the '''$n$th Fermat number'''. Both conventions are in place, sometimes in the same work. For example, {{AuthorRef|David Wells}}, in his {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}} of $1997$, refers to $5 = F_1$ in Section $5$ as the '''$2$nd Fermat number'''. However, in Section $257$ he defines $F_3 = 2^{2^3} + 1 = 257$ as the '''$3$rd Fermat number'''. Similarly, in Section $65,537$ he defines $F_4 = 2^{2^4} + 1 = 65 \, 537$ as the '''$4$th Fermat number''', and so on. Both of these naming conventions is more or less clumsy. {{ProofWiki}} takes the position that the cat has to jump one way or the other, and so uses the second of these conventions: :$F_n$ is the '''$n$th Fermat number'''.


Definition:Unitary Divisor

Let $n \in \Z$ be an integer. Let $d \in \Z$ be such that: :$d$ is a divisor of $n$ :$d$ and $\dfrac n d$ are coprime. Then $d$ is a unitary divisor of $n$.


Definition:Unitary Perfect Number

A '''unitary perfect number''' is a positive integer which equals the sum of its positive unitary divisors apart from itself.


Definition:Prime Quadruple

A '''prime quadruple''' is a set of $4$ prime numbers of the form $n$, $n + 2$, $n + 6$, $n + 8$. That is, they are two pairs of twin primes separated by $4$.


Definition:Propositional Tableau/Construction/Finite

The '''finite propositional tableaus''' are precisely those labeled trees singled out by the following bottom-up grammar: :{| style="border-spacing:20px;" | $\boxed{\mathrm{Root}}$ | A labeled tree whose only node is its root node is a '''finite propositional tableau'''. |- | colspan=2 | For the following clauses, let $t$ be a leaf node of a '''finite propositional tableau''' $T$. |- | $\boxed{\neg \neg}$ | If $\neg \neg \mathbf A$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$  is a '''finite propositional tableau'''. |- | $\boxed \land$ | If $\mathbf A \land \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$  :a child $r$ to $s$, with $\Phi \left({r}\right) = \mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed{\neg \land}$ | If $\neg \left({\mathbf A \land \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \neg\mathbf A$ :another child $s'$ to $t$, with $\Phi \left({s'}\right) = \neg\mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed \lor$ | If $\mathbf A \lor \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$ :another child $s'$ to $t$, with $\Phi \left({s'}\right) = \mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed{\neg\lor}$ | If $\neg \left({\mathbf A \lor \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \neg\mathbf A$  :a child $r$ to $s$, with $\Phi \left({r}\right) = \neg \mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed \implies$ | If $\mathbf A \implies \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \neg\mathbf A$ :another child $s'$ to $t$, with $\Phi \left({s'}\right) = \mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed{\neg\implies}$ | If $\neg \left({\mathbf A \implies \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$  :a child $r$ to $s$, with $\Phi \left({r}\right) = \neg \mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed \iff$ | If $\mathbf A \iff \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A \land \mathbf B$  :another child $s'$ to $t$, with $\Phi \left({s'}\right) = \neg \mathbf A \land \neg\mathbf B$ is a '''finite propositional tableau'''. |- | $\boxed{\neg\iff}$ | If $\neg \left({\mathbf A \iff \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: :a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A \land \neg \mathbf B$  :another child $s'$ to $t$, with $\Phi \left({s'}\right) = \neg \mathbf A \land \mathbf B$ is a '''finite propositional tableau'''. |} Note how the boxes give an indication of the ancestor WFF mentioned in the clause. These clauses together are called the '''tableau extension rules'''.


Definition:Propositional Tableau/Construction/Infinite

An infinite labeled tree $\left({T, \mathbf H, \Phi}\right)$ is a '''propositional tableau''' {{iff}}: :There exists an exhausting sequence of sets $\left({T_n}\right)_{n \in \N}$ of $T$ such that for all $n \in \N$: ::$\left({T_n, \mathbf H, \Phi \restriction_{T_n}}\right)$ :is a finite propositional tableau, where $\Phi \restriction_{T_n}$ denotes the restriction of $\Phi$ to $T_n$.


Definition:Cyclic Number

A '''cyclic number''' is an integer $n$ whose cyclic permutations of its digits are integer multiples or integer divisors of $n$.


Definition:Sorted Number

A '''sorted number''' is an integer whose digits form a non-decreasing sequence, for example: :$1 \, 123 \, 334 \, 899$


Definition:Rare Number

A '''rare number''' is a non-palindromic integer $n$ which has the property that $n + r$ and $n - r$ are both square, where $r$ is the reversal of $n$.


Definition:Two-Sided Prime

A '''two-sided prime''' is a prime number which remains prime when: :any number of digits are removed from the left hand end and: :any number of digits are removed from the right hand end but, generally, not from both ends at once.


Definition:Cunningham Chain

There are $2$ types of '''Cunningham chain''': === First Kind === {{:Definition:Cunningham Chain/First Kind}} === Second Kind === {{:Definition:Cunningham Chain/Second Kind}}


Definition:Undecidable Statement

An '''undecidable statement''' is a statement which, in the context of a given deductive apparatus, cannot be assigned a definitive truth value.


Definition:Equivalence of Norms

Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be norms on a vector space $V$. $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} there exist real constants $c$ and $C$ such that: :$\forall \mathbf x \in V: c \norm {\mathbf x}_1 \le \norm {\mathbf x}_2 \le C \norm {\mathbf x}_1$


Definition:Language of Propositional Logic/Basson-O'Connor

There are many formal languages expressing propositional logic. The formal language used on {{ProofWiki}} is defined on Definition:Language of Propositional Logic. This page defines the formal language $\LL_0$ used in: * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are a non-empty set of symbols $\PP_0$. See the {{ProofWiki}} definition. ==== Signs ==== ===== Brackets ===== The '''brackets''' used are '''round brackets''': {{begin-eqn}} {{eqn | ll= \bullet       | l = (       | o = :       | r = \)the '''left bracket''' sign\(       | c =  }} {{eqn | ll= \bullet       | l = )       | o = :       | r = \)the '''right bracket''' sign\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \bullet       | l = .       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \supset       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \equiv       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \sim       | o = :       | r = \)the negation sign\(       | c =  }} {{end-eqn}} See the {{ProofWiki}} definition. === Collation System === The collation system used is that of words and concatenation. See the {{ProofWiki}} definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\PP_0$ be the vocabulary of $\LL_0$. Let $Op = \left\{{., \lor, \supset, \equiv}\right\}$. The rules are: {| |- | $\mathbf W: \PP_0$ || | $:$ | If $p \in \PP_0$, then $p$ is a WFF. |- | $\mathbf W: \neg$ || | $:$ | If $\mathbf A$ is a WFF, then $\sim \mathbf A$ is a WFF. |- | $\mathbf W: Op$ || | $:$ | If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF. |} See the {{ProofWiki}} definition.


Definition:Pea Pattern/Sequential

For the '''sequential pea pattern''', the concatenation into $p_n$ is in sequential order of how the first instance of the distinct digits appear in $p_{n - 1}$.


Definition:Pea Pattern

The first term $p_0$ is an arbitrary integer. $p_n$ is formed from $p_{n - 1}$ as follows. First, the distinct digits in $p_{n - 1}$ are counted, and the number of each is noted. The count of each distinct digit is concatenated with an instance of the digit itself. Then those concatenations are themselves concatenated into $p_n$ according to a predetermined order.


Definition:Pea Pattern/Ascending

For the '''ascending pea pattern''', the concatenation into $p_n$ is in ascending order of the distinct digits of $p_{n - 1}$.


Definition:Pea Pattern/Descending

For the '''descending pea pattern''', the concatenation into $p_n$ is in descending order of the distinct digits of $p_{n - 1}$.


Definition:Time/Unit/Great Year of Plato

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''great year of Plato''' }} {{eqn | r = 12 \, 960 \, 000       | c = days }} {{eqn | o = \approx       | r = 36 \, 000       | c = years (of $360$ days) }} {{end-eqn}}


Definition:Counterexample

Let $X$ be the universal statement: :$\forall x \in S: \map P x$ That is: :''For all the elements $x$ of a given set $S$, the property $P$ holds.'' Such a statement may or may not be true. Let $Y$ be the existential statement: :$\exists y \in S: \neg \map P y$ That is: :''There exists at least one element $y$ of the set $S$ such that the property $P$ does ''not'' hold.'' It follows immediately by De Morgan's laws that if $Y$ is true, then $X$ must be false. Such a statement $Y$ is referred to as a '''counterexample to $X$'''.


=== Substitution for Well-Formed Part === {{:Definition:Substitution (Formal Systems)/Well-Formed Part}} === Substitution for Letter === {{:Definition:Substitution (Formal Systems)/Letter}} === Substitution of Term for Variable === {{:Definition:Substitution (Formal Systems)/Term}} === Substitution for Metasymbol === {{:Definition:Substitution (Formal Systems)/Metasymbol}}


Definition:Substitution (Formal Systems)/Letter

Let $\mathbf B$ be a well-formed formula of $\FF$. Let $p$ be a letter of $\FF$. Let $\mathbf A$ be another well-formed formula. Then the '''substitution of $\mathbf A$ for $p$ in $\mathbf B$''' is the collation resulting from $\mathbf B$ by replacing all occurrences of $p$ in $\mathbf B$ by $\mathbf A$. It is denoted as $\map {\mathbf B} {\mathbf A \mathbin {//} p}$.


{{:Definition:Substitution (Formal Systems)/Term/In WFF}}


Definition:Substitution (Formal Systems)/Metasymbol

Let $S_1$ be a statement form. Let $p$ be a metasymbol which occurs one or more times in $S_1$. Let $T$ be a statement. Let $S_2$ be the string formed by replacing ''every'' occurrence of $p$ in $S_1$ with $T$. Then $S_2$ results from the '''substitution of $p$ by $T$ in $S_1$'''. $S_2$ is called a '''substitution instance''' of $S_1$.