Definition:Autoregressive Model/Autoregressive Operator

Let $\map \phi B$ be defined as: :$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ where $B$ denotes the backward shift operator. Then $\map \phi B$ is referred to as the '''autoregressive operator'''. Hence the '''autoregessive model''' can be written in the following compact manner: :$\map \phi B \tilde z_t = a_t$


Definition:Autoregressive Model/Parameter

The '''parameters''' of $M$ consist of: :the constant mean level $\mu$ :the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shock $a_t$ :the coefficients $\phi_1$ to $\phi_p$.


Definition:Vector Space/Definition 1

Let $\struct {K, +_K, \times_K}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module. Then $\struct {G, +_G, \circ}_K$ is a '''vector space over $K$''' or a '''$K$-vector space'''. That is, a '''vector space''' is a unitary module whose scalar ring is a field.


Definition:Vector Space/Division Ring

Let $\struct {K, +_K, \times_K}$ be a division ring. Let $\struct {G, +_G}$ be an abelian group. Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module. Then $\struct {G, +_G, \circ}_K$ is a '''vector space over $K$''' or a '''$K$-vector space'''. That is, a vector space is a unitary module whose scalar ring is a division ring.


Definition:Vector Space/Definition 2

Let $\struct {K, +_K, \times_K}$ be a field whose unity is $1_K$. Let $\struct {G, +_G}$ be an abelian group. Let $\struct {\map {\mathrm {End} } G, +, \circ}$ be the endomorphism ring of $\struct {G, +_G}$ such that $I_G$ is the identity mapping. Let $\cdot: \struct {K, +_K, \times_K} \to \struct {\map {\mathrm {End} } G, +, \circ}$ be a ring homomorphism from $K$ to $\map {\mathrm {End} } G$ which maps $1_K$ to $I_G$. Then $\struct {G, +_G, \cdot, K}$ is a '''vector space over $K$''' or a '''$K$-vector space'''.


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2

{{begin-axiom}} {{axiom | n = \text B 2         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B }} {{end-axiom}}


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1

{{begin-axiom}} {{axiom | n = \text B 1         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y \in \mathscr B }} {{end-axiom}}


Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. === Definition 1 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} === Definition 2 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2}} === Definition 3 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3}} === Definition 4 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4}} === Definition 5 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5}} === Definition 6 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6}} === Definition 7 === $\mathscr B$ is said to satisfy the base axiom {{iff}}: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7}}


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3

{{begin-axiom}} {{axiom | n = \text B 3         | q = \forall B_1, B_2 \in \mathscr B         | mr= \exists \text{ a bijection } \pi : B_1 \setminus B_2 \to B_2 \setminus B_1 : \forall x \in B_1 \setminus B_2 : \paren {B_1 \setminus \set x } \cup \set {\map \pi x} \in \mathscr B }} {{end-axiom}}


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4

{{begin-axiom}} {{axiom | n = \text B 4         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \setminus \set x} \cup \set y, \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B }} {{end-axiom}}


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5

{{begin-axiom}} {{axiom | n = \text B 5         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y}  \cup \set x \in \mathscr B }} {{end-axiom}}


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7

{{begin-axiom}} {{axiom | n = \text B 7         | q = \forall B_1, B_2 \in \mathscr B         | mr= \exists \text{ a bijection } \pi : B_1 \setminus B_2 \to B_2 \setminus B_1 : \forall x \in B_1 \setminus B_2 : \paren {B_2 \setminus \set {\map \pi x} } \cup \set x \in \mathscr B }} {{end-axiom}}


Definition:Image of Subset under Mapping/Definition 2

The '''image of $X$ under $f$''' is the element of the codomain of the direct image mapping $f^\to: \powerset S \to \powerset T$ of $f$: :$\forall X \in \powerset S: \map {f^\to} X := \set {t \in T: \exists s \in X: \map f s = t}$


Definition:Preimage of Subset under Mapping/Definition 1

The '''preimage of $Y$ under $f$''' is defined as: :$f^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \map f s = t}$ That is, the '''preimage of $Y$ under $f$''' is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.


Definition:Preimage of Subset under Mapping/Definition 2

The '''preimage of $Y$ under $f$''' can be seen to be an element of the codomain of the inverse image mapping $f^\gets: \powerset T \to \powerset S$ of $f$: :$\forall Y \in \powerset T: \map {f^\gets} Y := \set {s \in S: \exists t \in Y: \map f s = t}$ Thus: :$\forall Y \subseteq T: f^{-1} \sqbrk Y = \map {f^\gets} Y$


Definition:Moving Average Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t$ be the deviation from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a finite linear aggregate of the shocks: :$\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ $M$ is known as a '''moving average (MA) process of order $q$'''.


Definition:Moving Average Model/Moving Average Operator

Let $\map \theta B$ be defined as: :$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$ where $B$ denotes the backward shift operator. Then $\map \theta B$ is referred to as the '''moving average operator'''. Hence the '''moving average model''' can be written in the following compact manner: :$\tilde z_t = \map \theta B a_t$


Definition:Moving Average Model/Parameter

The '''parameters''' of $M$ consist of: :the constant mean level $\mu$ :the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$ :the coefficients $\theta_1$ to $\theta_q$.


Definition:Box-Jenkins Model/ARMA

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ $M$ is known as a '''mixed autoregressive (order $p$), moving average (order $q$) process''', usually referred as an '''ARMA process'''.


Definition:ARMA Model/ARMA Operator

Using the autoregressive operator: :$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ and the moving average operator: :$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$ the '''ARMA model''' can be written in the following compact manner: :$\map \phi B \tilde z_t = \map \theta B a_t$ where $B$ denotes the backward shift operator. Hence: :$\tilde z_t = \map {\phi^{-1} } B \map \theta B a_t$


Definition:ARMA Model/Parameter

The '''parameters''' of $M$ consist of: :the constant mean level $\mu$ :the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$ :the coefficients $\phi_1$ to $\phi_p$ :the coefficients $\theta_1$ to $\theta_q$.


Definition:Scalar Ring/Module

Let $\struct {G, +_G, \circ}_R$ be a module, where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {G, +_G}$ is an abelian group :$\circ: R \times G \to G$ is a binary operation. Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {G, +_G, \circ}_R$.


Definition:Scalar Ring/Unitary Module

Let $\struct {G, +_G, \circ}_R$ be a module, where: :$\struct {R, +_R, \times_R}$ is a ring with unity :$\struct {G, +_G}$ is an abelian group :$\circ: R \times G \to G$ is a binary operation. Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {G, +_G, \circ}_R$.


Definition:Scalar Ring/Vector Space over Division Ring

Let $\struct {G, +_G, \circ}_K$ be a vector space over a division ring, where: :$\struct {K, +_K, \times_K}$ is a division ring :$\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$ :$\circ: K \times G \to G$ is a binary operation. Then the division ring $\struct {K, +_K, \times_K}$ is called the '''scalar division ring''' of $\struct {G, +_G, \circ}_K$, or just '''scalar ring'''.


Definition:Box-Jenkins Model/ARIMA

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let: :$w_t = \nabla^d z_t$ where $\nabla^d$ denotes the $d$th iteration of the backward difference operator. Let $M$ be a model where the current value of $w_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks: :$w_t = \phi_1 w_{t - 1} + \phi_2 w_{t - 2} + \dotsb + \phi_p w_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ $M$ is known as an '''autoregressive integrated moving average (ARIMA) process of order $p$, $d$, $q$'''.


Definition:ARIMA Model/ARIMA Operator

Using the autoregressive operator: :$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ and the moving average operator: :$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$ the '''ARIMA model''' can be written in the following compact manner: :$\map \phi B w_t = \map \theta B a_t$ where $B$ denotes the backward shift operator. Hence: :$\map \varphi B z_t = \map \phi B \paren {1 - B}^d z_t = \map \theta B a_t$ where: :$\map \varphi B = \map \phi B \paren {1 - B}^d$


Definition:Scalar/Module

The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called '''scalars'''.


Definition:Scalar/Vector Space

The elements of the scalar field $\struct {K, +_K, \times_K}$ are called '''scalars'''.


Definition:Vector Field

If a vector quantity $\mathbf v$ can be associated with every point in a given space, then $\mathbf v$ is said to be a '''vector field'''.


Definition:Density (Physics)/Dimension

The dimension of '''density''' is $\mathsf {M L}^{-3}$: mass per unit volume.


Definition:Density (Physics)/Units

* The SI units of '''density''' are $\mathrm {kg} \, \mathrm m^{-3}$ (kilograms per cubic metre). * The CGS units of '''density''' are $\mathrm g \, \mathrm{cm}^{-3}$ or, less formally: $\mathrm g / \mathrm {cc}$ (grams per cubic centimetre). Thus: :$1 \, \mathrm g \, \mathrm {cm}^{-3} = 1000 \, \mathrm {kg} \, \mathrm m^{-3}$


Definition:Density (Physics)/Linear/Symbol

The usual symbol used to denote '''linear density''' is $\mu$ (Greek letter '''mu'''). Sometimes $\lambda$ (Greek letter '''lambda''') is also used.


Definition:Mistake/Whoosho

A '''whoosho''' is a mistake caused by a complete failure to understand a concept vital to the accurate communication of an idea. It is also used for a failure to understand or recognise a joke. The word comes from the popular meme of how '''whoosh''' is supposedly the sound of whatever it is going over the person's head. :''"That went so far over my head I didn't even hear it go whoosh."''


Definition:Current Time

The '''current time''' in the context of time series analysis is the timestamp of the most recent observation. By the very nature of time itself, the '''current time''' is always going to be some non-zero instant in the past.


Let $T$ be a time series whose observations are $\map z {\tau_1}, \map z {\tau_2}, \dotsb, \map z {\tau_t}, \dotsb$ at an unbroken sequence of timestamps $\tau_1, \tau_2, \dotsb, \tau_t, \dotsb$. When we have $N$ such '''successive''' observations, without a missing value, we write: :$z_1, z_2, \dotsb, z_t, \dotsb, z_N$ and it is understood that the subscripts correspond directly to the timestamps. === Equispaced Time Series === {{:Definition:Successive Values of Time Series/Equispaced}}


Definition:Successive Values of Time Series/Equispaced

Let $T$ be equispaced with time interval $h$ between adjacent observations.  Then $N$ '''successive observations''', written as: :$z_1, z_2, \dotsb, z_t, \dotsb, z_N$ occur at timestamps: :$\tau_0 + h, \tau_0 + 2 h, \dotsb, \tau_0 + t h, \dotsb, \tau_0 + N h$ Hence we can refer to the observations at timestamp $\tau_0 + t h$ as $z_t$.


Definition:Discrete Time Series/Sampling

A '''discrete time series''' can be obtained by taking observations at predetermined instants from a comtinuous time series of measurements of the process in question.


Definition:Discrete Time Series/Accumulation

A '''discrete time series''' can be obtained by collecting a varying quantity over a period of time. The observations are the measurements of the quantities at the end of each period.


Definition:Space of Bounded Sequences

The '''space of bounded sequences''', denoted $\ell^\infty$ is defined as: :$\displaystyle \ell^\infty := \set{\sequence{z_n}_{n \mathop \in \N} \in \C^\N: \sup_{n \mathop \in \N} \size {z_n} < \infty}$ As such, $\ell^\infty$ is a subspace of $\C^\N$, the space of all complex sequences. {{stub}}


Definition:Field Negative

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $x \in F$. The inverse of $x$ with respect to the addition operation $+$ in the additive group $\struct {F, +}$ of $F$ is referred to as the '''(field) negative''' of $x$ and is denoted $-x$. That is, the '''(field) negative''' of $x$ is the element $-x$ of $F$ such that: :$x + \paren {-x} = 0_F$


Definition:Realization of Stochastic Process

Let $S$ be a stochastic process. Let $T$ be a time series of observations of $S$ which has been acquired as $S$ evolves, according to the underlying probability distribution of $S$. Then $T$ is referred to as a '''realization''' of $S$. Thus we can regard the observation $z_t$ at some timestamp $t$, for example $t = 25$, as the '''realization''' of a random variable with probability mass function $\map p {z_t}$. Similarly the observations $z_{t_1}$ and $z_{t_2}$ at times $t_1$ and $t_2$ can be regarded as the '''realizations''' of two random variables with joint probability mass function $\map p {z_{t_1} }$ and $\map p {z_{t_2} }$. Similarly, the observations making an equispaced time series can be described by an $N$-dimensional random variable $\tuple {z_1, z_2, \dotsc, z_N}$ with associated probability mass function $\map p {z_1, z_2, \dotsc, z_N}$.


Definition:Strictly Stationary Stochastic Process

Let $S$ be a stochastic process giving rise to a time series $T$. $S$ is a '''strictly stationary stochastic process''' if its properties are unaffected by a change of origin of $T$.


Definition:Mean of Stochastic Process

Let $S$ be a stochastic process giving rise to a time series $T$. The '''mean''' of $S$ over an interval $Q$ is the arithmetic mean of the observations of $T$ over all the timestamps within $Q$.


Definition:Variance of Stochastic Process

Let $S$ be a stationary stochastic process giving rise to a time series $T$. The '''variance''' of $S$ is calculated as: :$\sigma_z^2 = \expect {\paren {z_t - \mu}^2} = \displaystyle \int_{-\infty}^\infty \paren {z - \mu}^2 \map p z \rd z$ where $\map p z$ is the (constant) probability mass function of $S$. It is a measure of the spread about the constant mean level $\mu$.


Definition:Histogram

A '''histogram''' is a form of bar graph used to represent the frequency distribution grouped by class intervals.


Definition:Sample Mean of Stochastic Process

Let $S$ be a stochastic process giving rise to a time series $T$. The '''sample mean''' of $S$ over a set of $N$ successive values $\set {z_1, z_2, \dotsb, z_N}$ is defined as: :$\overline z := \dfrac 1 N \displaystyle \sum_{t \mathop = 1}^N z_t$


Definition:Sample Variance of Stochastic Process

Let $S$ be a stochastic process giving rise to a time series $T$. The '''sample variance''' of $S$ over a set of $N$ successive values $\set {z_1, z_2, \dotsb, z_N}$ is defined as: :$\hat \sigma_z^2 := \dfrac 1 N \displaystyle \sum_{t \mathop = 1}^N \paren {z_t - \overline z}^2$ where $\overline z$ denotes the sample mean of $S$ over $\set {z_1, z_2, \dotsb, z_N}$.


Definition:Scatter Diagram

A '''scatter diagram''' is a technique of graphical representation of the distribution of ordered pairs of random variables. The ordered pairs are treated as points in a Cartesian plane: the first coordinate along the $x$-axis and the second coordinate along the $y$-axis.


Definition:Lag

Let $T$ be a time series. A '''lag''' is a constant time interval between two timestamps of $T$. Thus, for every observation $z_t$ of $T$, a pair of observations can be created with a given '''lag''' $k$ between them: :$\tuple {z_t, z_{t + k} }$


Definition:Field (Abstract Algebra)/Addition

The distributand $+$ of a field $\struct {F, +, \times}$ is referred to as '''field addition''', or just '''addition'''.


Definition:Field (Abstract Algebra)/Product

The distributive operation $\times$ in $\struct {F, +, \times}$ is known as the '''(field) product'''.


Definition:Additive Group of Field

The group $\struct {F, +}$ is known as the '''additive group of $F$'''.


Definition:Additive Inverse

=== Additive Inverse in Ring === {{:Definition:Additive Inverse/Ring}} === Additive Inverse in Field === {{:Definition:Additive Inverse/Field}} === Additive Inverse of Number === The concept is often encountered in the context of numbers: {{:Definition:Additive Inverse/Number}}


Definition:Additive Inverse/Field

Let $\struct {F, +, \times}$ be a field whose addition operation is $+$. Let $a \in R$ be any arbitrary element of $F$. The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$: :$a + \paren {-a} = 0_F$ where $0_F$ is the zero of $R$.


Definition:Unity (Abstract Algebra)

=== Unity of Ring === {{:Definition:Unity (Abstract Algebra)/Ring}} === Unity of Field === {{:Definition:Unity (Abstract Algebra)/Field}}


Definition:Unity (Abstract Algebra)/Field

Let $\struct {F, +, \times}$ be a field. The identity of the multiplicative group $\struct {F, \times}$ is referred to as '''the unity of the field $\struct {F, +, \times}$'''. It is (usually) denoted $1_F$, where the subscript denotes the particular field to which $1_F$ belongs (or often $1$ if there is no danger of ambiguity).


Definition:Space of Convergent Sequences

The '''space of convergent sequences''', denoted $c$ is defined as: :$\displaystyle c := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \exists L \in \C : \forall \epsilon \in \R_{>0} : \exists N \in \R: n > N \implies \cmod {z_n - L} < \epsilon}$ As such, $c$ is a subspace of $\C^\N$, the space of all complex sequences. {{stub}}


Definition:Scalar Addition

=== Scalar Addition on Module === {{:Definition:Scalar Addition/Module}} === Scalar Addition on Vector Space === {{:Definition:Scalar Addition/Vector Space}}


Definition:Scalar Addition/Vector Space

Let $\struct {K, +_K, \times_K}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $V := \struct {G, +_G, \circ}_K$ be the corresponding '''vector space over $K$'''. The field addition operation $+_K$ on $V$ is known as '''scalar addition''' on $V$.


Definition:Scalar Addition/Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. Let $M := \struct {G, +_G, \circ}_R$ be the corresponding '''module over $R$''' (either a left module or a right module). The ring addition operation $+_R$ on $M$ is known as '''scalar addition''' on $M$.


Definition:Vector Addition/Vector Space

Let $\struct {F, +_F, \times_F}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $V := \struct {G, +_G, \circ}_R$ be the corresponding '''vector space over $F$'''. The group operation $+_G$ on $V$ is known as '''vector addition''' on $V$.


Definition:Scalar Multiplication/Module

Let $\struct {G, +_G, \circ}_R$ be an module (either a left module or a right module or both), where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {G, +_G}$ is an abelian group. The operation $\circ: R \times G \to G$ is called '''scalar multiplication'''.


Definition:Scalar Multiplication/Vector Space

Let $\struct {G, +_G, \circ}_K$ be an vector space, where: :$\struct {K, +_K, \times_K}$ is a field :$\struct {G, +_G}$ is an abelian group. The operation $\circ: K \times G \to G$ is called '''scalar multiplication'''.


Definition:Negative of Element

=== Ring Negative === {{:Definition:Ring Negative}} === Field Negative === As a field is also a ring, the same definition can be used: {{:Definition:Field Negative}} Category:Definitions/Ring Theory Category:Definitions/Field Theory itnu454mo7keuscqageynxn280saaus


Definition:Space of Zero-Limit Sequences

The '''space of zero-limit sequences''', denoted $c$ is defined as: :$\displaystyle c_0 := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \forall \epsilon \in \R_{>0} : \exists N \in \R_{> 0}: n > N \implies \cmod {z_n} < \epsilon}$ As such, $c_0$ is a subspace of $\C^\N$, the space of all complex sequences. {{stub}}


Definition:Space of Almost-Zero Sequences

The '''space of almost-zero sequences''', denoted $c_{00}$ is defined as: :$\displaystyle c_{00} := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \exists N \in \R_{> 0}: n > N \implies z_n =0}$ As such, $c_{00}$ is a subspace of $\C^\N$, the space of all complex sequences. {{stub}}


Definition:C^k Norm

Let $f \in \map {\CC^k} {\R}$ be a function of differentiability class $k$. Then '''$C^k$ norm''' is defined as: :$\displaystyle \norm {f}_{\map {C^k} \R} := \sum_{i \mathop = 0}^k \sup_{x \in \R} \size {\map {f^{\paren i}} x} = \sum_{i \mathop = 0}^k \norm{f^{\paren i} }_\infty$ where $f^{\paren i}$ denotes the $i$-th derivative {{WRT}} $x$, and $\norm {\, \cdot \,}_\infty$ denotes the supremum norm.


Definition:Autocovariance

Let $S$ be a stochastic process giving rise to a time series $T$. The '''autocovariance''' of $S$ at lag $k$ is defined as: :$\gamma_k := \cov {z_t, z_{t + k} } = \expect {\paren {z_t - \mu} \paren {z_{t - k} - \mu} }$ where: :$z_t$ is the observation at time $t$ :$\mu$ is the mean of $S$ :$\expect \cdot$ is the expectation.


Definition:Autocorrelation

Let $S$ be a stochastic process giving rise to a time series $T$. The '''autocorrelation''' of $S$ at lag $k$ is defined as: :$\rho_k := \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$ where: :$z_t$ is the observation at time $t$ :$\mu$ is the mean of $S$ :$\expect \cdot$ is the expectation.


Definition:Autocovariance Matrix

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ The '''autocovariance matrix''' associated with $S$ for $\sequence {s_n}$ is: :$\boldsymbol \Gamma_n = \begin {pmatrix} \gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix}$ where $\gamma_k$ is the autocovariance of $S$ at lag $k$. That is, such that: :$\sqbrk {\Gamma_n}_{i j} = \gamma_{\size {i - j} }$


Definition:Autocorrelation Matrix

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ The '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$ is: :$\mathbf P_n = \begin {pmatrix} 1 & \rho_1 & \rho_2 & \cdots & \rho_{n - 1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{n - 2} \\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \rho_{n - 1} & \rho_{n - 2} & \rho_{n - 3} & \cdots & 1 \end {pmatrix}$ where $\rho_k$ is the autocorrelation of $S$ at lag $k$. That is, such that: :$\sqbrk {P_n}_{i j} = \rho_{\size {i - j} }$


Definition:Positive Definite Matrix

Let $\mathbf A$ be a square matrix of order $n$. $\mathbf A$ is '''positive definite''' {{iff}}: :$(1): \quad \mathbf A$ is symmetric :$(2): \quad$ for all column matrices $\mathbf x$ of order $n$, $\mathbf x^\intercal \mathbf A \mathbf x$ is strictly positive.


Definition:Matrix/Order/Column

Let $\mathbf A$ be an $n \times 1$ column matrix. Then the '''order''' of $\mathbf A$ is defined as being $n$.


Definition:Induced Norm

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space. Let $Y \subseteq X$ be a subspace. Then the '''induced norm (on $Y$)''' is defined as the restriction of $\norm {\, \cdot \,}_X$ to $Y$. :$\norm {\, \cdot \,}_Y := \norm {\, \cdot \,}_X {\restriction_Y}$


Definition:Wholly Real/Abbreviated Notation

Let $z = a + i b$ be a complex number such that $b = 0$. That is, let $z$ be wholly real: $z = a + 0 i$, or $\tuple {a, 0}$ Despite the fact that $z$ is still a complex number, it is commonplace to use the same notation as if it were a real number, and hence say $z = a$. While it is in theory important to distinguish between a real number and its corresponding wholly real complex number, in practice it makes little difference.


Definition:Pure Mathematics

'''Pure mathematics''' is the branch of mathematics which is not concerned with real-world applications.


Definition:Parallel (Geometry)/Line to Plane

Let $L$ be a straight line. Let $P$ be a plane. Then $L$ and $P$ are '''parallel''' {{iff}}, when produced indefinitely, they do not intersect at any point.


Definition:Method of Least Squares

=== Approximation Theory === {{:Definition:Method of Least Squares (Approximation Theory)}} === Statistics === {{:Definition:Method of Least Squares (Statistics)}} Category:Definitions/Approximation Theory Category:Definitions/Statistics 51987t69f5t5rza6b9j271im7jtcjbs


Definition:Cardinality/Natural Numbers

When the natural numbers are defined as von Neumann construction of natural numbers, the cardinality function can be viewed as the identity mapping on $\N$. That is: :$\forall n \in N: \card n := n$


Definition:Spline

=== Draftsman's tool === {{:Definition:Spline (Tool)}} === Spline Function === {{:Definition:Spline Function}}


Definition:Spline (Tool)

:400px A '''spline''' is a drawing tool which consists of a long strip of uniformly flexible material fixed in position at a number of points whose tension creates a smooth curve passing through those points. Its purpose is to transferring that curve to another material.


Definition:Spline Function

Let $\closedint a b$ be a closed real interval. Let $T : = \set {a = t_0, t_1, t_2, \ldots, t_{n - 1}, t_n = b}$ form a subdivision of $\closedint a b$. Let $S: \closedint a b \to \R$ be a continuous function on $\closedint a b$ whose values on $t_0, t_1, \ldots, t_n$ are known. On each of the intervals $\closedint {t_k} {t_{k + 1} }$, let $P_k: \closedint {t_k} {t_{k + 1} }: \R$ be a polynomial function such that: :for $t$ on each of $t_k < t < t_{k + 1}$: $\map S t = \map {P_k} t$ The function $S: \closedint a b \to \R$ is known as a '''spline function''' on $T$.


Definition:Spline Function/Knot

The points $T := \set {t_0, t_1, t_2, \ldots, t_{n - 1}, t_n}$ of $S$ are known as the '''knots'''.


Definition:Spline Function/Knot/Knot Vector

The ordered $n + 1$-tuple $\mathbf t := \tuple {t_0, t_1, t_2, \ldots, t_{n - 1}, t_n}$ of $S$ is known as the '''knot vector'''.


Definition:Spline Function/Uniform

$S$ is a '''uniform spline''' {{iff}} $T$ is a normal subdivision. That is, {{iff}} the knots of $S$ are equally spaced.


Definition:Spline Function/Degree

The '''degree''' of $S$ is the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$.


Definition:Spline Function/Order

Some sources, instead of referring to the '''degree''' of a spline, use the '''order'''. Let the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$ be $n$. The '''order''' of $S$ is then $n + 1$.


Definition:Fundamental Circuit (Matroid)

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $B$ be a base of $M$. Let $x \in S \setminus B$. The '''fundamental circuit of $x$ in the base B''', denoted $\map C {x, B}$, is the unique circuit such that: :$x \in \map C {x, B} \subseteq B \cup \set x$


Definition:Set/Uniqueness of Elements/Multiple Specification

For a given '''set''', an object is either '''in''' the set or '''not in''' the set. So, if an element is in a set, then it is in the set '''only once''', however many times it may appear in the definition of the set. Thus, the set $\set {1, 2, 2, 3, 3, 4}$ is the same set as $\set {1, 2, 3, 4}$. $2$ and $3$ are in the set, and '''listing''' them twice makes no difference to the set's '''contents'''. Like the membership of a club, if you're in, you're in -- however many membership cards you have to prove it.


Definition:Set/Uniqueness of Elements/Order of Listing

It makes no difference in what order the elements of a set are specified. This means that the sets $S = \set {1, 2, 3, 4}$ and $T = \set {3, 4, 2, 1}$ are the ''same set''.


Definition:Set/Uniqueness of Elements/Equality of Sets

By definition of '''set equality''' {{:Definition:Set Equality/Definition 1}} So, to take the club membership analogy, if two clubs had exactly the same members, the clubs would be considered as ''the same club'', although they may be given different '''names'''. This follows from the definition of '''equals''' given above. Note that there '''are''' mathematical constructs which '''do''' take into account both (or either of) the order in which the elements appear, and the number of times they appear, but these are ''not'' '''sets''' as such.


Definition:Parabola/Vertex

The '''vertex''' of $P$ is the point where the axis intersects $P$.


Definition:Parabola/Axis

The '''axis''' of $P$ is the straight line passing through the focus of $P$ perpendicular to the directrix $D$.


Definition:Age (Time)

The '''age''' of a physical objectis defined as the period of time over which it has been in existence.


Definition:Classical Algorithm

The '''classical algorithms''' are the conventional techniques for performing arithmetic on numbers with more than $1$ digit.


Definition:Multi-Digit Integer

Let $b \in \N_{>1}$ be a natural number greater than $1$. An '''$n$-digit integer base $b$''' is an integer which has no more than $n$ digits when expressed in base $b$. That is, it is strictly less than $b^n$.


Definition:Classical Algorithm/Primitive Operation

The '''primitive operations''' of the '''classical algorithms''' are the following operations which are performed on '''two''' $1$-digit integers $x$ and $y$: === Addition === {{:Definition:Classical Algorithm/Primitive Addition}} === Subtraction === {{:Definition:Classical Algorithm/Primitive Subtraction}} === Multiplication === {{:Definition:Classical Algorithm/Primitive Multiplication}} === Division === {{:Definition:Classical Algorithm/Primitive Division}}


Definition:Carry Digit

A '''carry digit''' is a digit that appears as a result of the operation of one of the primitive operations of the classical algorithms which is to be applied to a further primitive operation in some manner appropriate to the classical algorithm in question.


Definition:Classical Algorithm/Primitive Addition

Let $x$ and $y$ be $1$-digit integers. Let $z$ be a '''carry digit''' such that either $z = 0$ or $z = 1$. Addition of $x$, $y$ and $z$ gives a $1$-digit '''sum''' $s$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = s       | r = \paren {x + y + z} \pmod b }} {{eqn | l = c       | r = \floor {\dfrac {x + y + z} b} }} {{end-eqn}}


Definition:Classical Algorithm/Primitive Subtraction

Let $x$ and $y$ be $1$-digit integers. Let $z$ be a '''carry digit''' such that either $z = 0$ or $z = -1$. Subtraction of $y$ from $x$ with $z$ gives a $1$-digit '''difference''' $d$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = d       | r = \paren {x - y + z} \pmod b }} {{eqn | l = c       | r = \floor {\dfrac {x - y + z} b} }} {{end-eqn}}


Definition:Classical Algorithm/Primitive Multiplication

Multiplication of '''two''' $1$-digit integers $x$ and $y$ gives a $1$-digit '''product''' $p$ and a $1$-digit '''carry''' $c$ such that: {{begin-eqn}} {{eqn | l = p       | r = x \times y \pmod b }} {{eqn | o =  }} {{eqn | l = c       | r = \dfrac {x \times y - p} b }} {{end-eqn}}


Definition:Classical Algorithm/Primitive Division

Let $y$ be a $1$-digit integer. Let $x$ be a $2$-digit integer $x_1 b + x_2$, where $x_1$ and $x_2$ are $1$-digit integers. Suppose $x_1 \ge y$. Then division of $x_1$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that: {{begin-eqn}} {{eqn | l = x_1       | r = q \times y + r }} {{end-eqn}} Suppose $x_1 < y$. Then division of $x = x_1 b + x_2$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that: {{begin-eqn}} {{eqn | l = x       | r = q \times y + r }} {{end-eqn}}


Definition:Classical Algorithm/Primitive Addition/Base 10 Addition Table

The primitive addition operation for conventional base $10$ arithmetic of two $1$-digit integers can be presented as a pair of operation tables as follows: :$\begin{array}{c|cccccccccc} s & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 \\ 2 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 \\ 3 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 \\ 4 & 4 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 \\ 5 & 5 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4  \\ 6 & 6 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5  \\ 7 & 7 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6  \\ 8 & 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7  \\ 9 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8  \\ \end{array} \qquad \begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 4 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 5 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 6 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 7 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 8 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 9 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array}$


Definition:Classical Algorithm/Primitive Subtraction/Base 10 Subtraction Table

The primitive subtraction operation for conventional base $10$ arithmetic on two $1$-digit integers can be presented as a pair of operation tables as follows: :$\begin{array}{c|cccccccccc} d & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ 1 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 \\ 2 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 \\ 3 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 \\ 4 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 \\ 5 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6  \\ 6 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8 & 7  \\ 7 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9 & 8  \\ 8 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 9  \\ 9 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0  \\ \end{array} \qquad \begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 4 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$


Definition:Classical Algorithm/Primitive Multiplication/Base 10 Multiplication Table

The primitive multiplication operation for conventional base $10$ arithmetic of two $1$-digit integers can be presented as a pair of operation tables as follows: :$\begin{array}{c|cccccccccc} p & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 0 & 2 & 4 & 6 & 8 & 0 & 2 & 4 & 6 & 8 \\ 3 & 0 & 3 & 6 & 9 & 2 & 5 & 8 & 1 & 4 & 7 \\ 4 & 0 & 4 & 8 & 2 & 6 & 0 & 4 & 8 & 2 & 6 \\ 5 & 0 & 5 & 0 & 5 & 0 & 5 & 0 & 5 & 0 & 5  \\ 6 & 0 & 6 & 2 & 8 & 4 & 0 & 6 & 2 & 8 & 4  \\ 7 & 0 & 7 & 4 & 1 & 8 & 5 & 2 & 9 & 6 & 3  \\ 8 & 0 & 8 & 6 & 4 & 2 & 0 & 8 & 6 & 4 & 2  \\ 9 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1  \\ \end{array} \qquad \begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 \\ 4 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 2 & 3 & 3 \\ 5 & 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 \\ 6 & 0 & 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 \\ 7 & 0 & 0 & 1 & 2 & 2 & 3 & 4 & 4 & 5 & 6 \\ 8 & 0 & 0 & 1 & 2 & 3 & 4 & 4 & 5 & 6 & 7 \\ 9 & 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \end{array}$


Definition:Classical Algorithm/Addition

Let $u = \sqbrk {u_{n - 1} u_{n - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $n$-digit integers. The '''classical addition algorithm''' forms their $n + 1$-digit sum $u + v$: :$w = \sqbrk {w_n w_{n - 1} \dotsm w_1 w_0}_b$ where $w_n$ is either $0$ or $1$. The steps are: :$(\text A 1): \quad$ Set $j = 0$, $k = 0$. ::::$j$ is used to run through all the digit positions ::::$k$ keeps track of the carry digit between each step. :$(\text A 2): \quad$ Calculate digit $j$: ::::Calculate $\begin {cases} s = \paren {u_j + v_j + k} \pmod b \\ c = \floor {\dfrac {u_j + v_j + k} b} \end {cases}$ using the primitive addition. ::::Set $w_j$ to $s$. ::::Set $k$ to $c$. :$(\text A 3): \quad$ Add $1$ to $j$, using conventional integer addition. ::::If $j < n$, return to $(\text A 2)$. ::::Otherwise, set $w_n$ equal to $k$ and exit.


Definition:Classical Algorithm/Subtraction

Let $u = \sqbrk {u_{n - 1} u_{n - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $n$-digit integers. The '''classical subtraction algorithm''' forms their $n$-digit difference $u - v$: :$w = \sqbrk {w_n w_{n - 1} \dotsm w_1 w_0}_b$ where $w_n$ is either $0$ or $1$. The steps are: :$(\text S 1): \quad$ Set $j = 0$, $k = 0$. ::::$j$ is used to run through all the digit positions ::::$k$ keeps track of the carry digit between each step. :$(\text S 2): \quad$ Calculate digit $j$: ::::Calculate $\begin {cases} d = \paren {u_j + v_j - k} \pmod b \\ c = \floor {\dfrac {u_j - v_j + k} b} \end {cases}$ using the primitive subtraction. ::::Set $w_j$ to $d$. ::::Set $k$ to $c$. :$(\text S 3): \quad$ Add $1$ to $j$, using conventional integer addition. ::::If $j < n$, return to $(\text S 2)$. ::::Otherwise exit.


Definition:Classical Algorithm/Multiplication

Let $u = \sqbrk {u_{m - 1} u_{m - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $m$-digit and $n$-digit integers respectively. The '''classical multiplication algorithm''' forms their $m + n$-digit product $u v$: :$w = \sqbrk {w_{m + n - 1} w_{m + n - 2} \dotsm w_1 w_0}_b$ The steps are: :$(\text M 1): \quad$ Initialise: ::::Set $w_{m - 1}, w_{m - 2}, \dotsc, w_1, w_0 := 0$ ::::Set $j := 0$ :$(\text M 2): \quad$ Is $v_j = 0$? ::::If so, go to Step $(\text M 6)$. :$(\text M 3): \quad$ Initialise loop: ::::Set $i := 0$ ::::Set $k := 0$ :$(\text M 4): \quad$ Multiply and Add: ::::Set $t := u_i \times v_j + w_{i + j} + k$ ::::Set $w_{i + j} := t \pmod b$ ::::Set $k := \floor {\dfrac t b}$ :$(\text M 5): \quad$ Loop on $i$: ::::Increase $i$ by $1$. ::::If $i < m$, go back to Step $(\text M 4)$. ::::Otherwise, set $w_{j + m} := k$ :$(\text M 6): \quad$ Loop on $j$: ::::Increase $j$ by $1$. ::::If $j < n$, go back to Step $(\text M 2)$. ::::Otherwise exit.


Definition:Classical Algorithm/Division

Let $u = \sqbrk {u_{m + n - 1} u_{m - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $m + n$-digit and $n$-digit integers respectively, where $v_{n - 1} \ne 0$ and $n > 1$. The '''classical division algorithm''' forms: :their $m + 1$-digit quotient $\floor {\dfrac u v} = q = \sqbrk {q_m q_{m - 1} \dotsm q_1 q_0}_b$ and: :their $n$-digit remainder $u \pmod v = r = \sqbrk {r_{n - 1} \dotsm r_1 r_0}_b$ The steps are: :$(\text D 1): \quad$ Normalize: ::::Set: {{begin-eqn}} {{eqn | l = d       | o = :=       | r = \floor {\dfrac b {v_{n - 1} - 1} } }} {{eqn | l = \sqbrk {u_{m + n} u_{m - 1} \dotsm u_1 u_0}_b       | o = :=       | r = \sqbrk {u_{m + n - 1} u_{m - 2} \dotsm u_1 u_0}_b \times d }} {{eqn | l = \sqbrk {v_{n - 1} \dotsm v_1 v_0}_b       | o = :=       | r = \sqbrk {v_{n - 1} \dotsm v_1 v_0}_b \times d }} {{end-eqn}} ::::Note that if $d = 1$, all that needs to be done is to set $u_{m + n} := 0$. :$(\text D 2): \quad$ Initialise $j$: ::::Set $j := m$ :$(\text D 3): \quad$ Calculate the quotient $\hat q$ and remainder $\hat r$: ::::Set $\hat q := \floor {\dfrac {u_{j + n} b + u_{j + n - 1} } {v_{n - 1} } }$ ::::Set $\hat r := \paren {u_{j + n} b + u_{j + n - 1} } \pmod {v_{n - 1} }$ ::::Test whether $\hat q = b$ or $\hat q v_{n - 1} > b \hat r + u_{j + n - 2}$. ::::If so, decrease $\hat q$ by $1$ and increase $\hat r$ by $v_{n - 1}$. ::::If $\hat r < b$ repeat this test. :$(\text D 4): \quad$ Multiply and Subtract: ::::Replace $\sqbrk {u_{j + n} u_{j + n - 1} \dotsm u_j}_b$ by $\sqbrk {u_{j + n} u_{j + n - 1} \dotsm u_j}_b - \hat q \sqbrk {0 v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ ::::If the result of this step is negative, add $b_{n + 1}$ to it. :$(\text D 5): \quad$ Test remainder: ::::Set $q_j := \hat q$. ::::If the result of '''Step $(\text D 4)$''' was negative, go to '''Step $(\text D 6)$''' ::::Otherwise go to '''Step $(\text D 7)$''' :$(\text D 6): \quad$ Add back: ::::Decrease $q_j$ by $1$ ::::Add $\sqbrk {0 v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ to $\sqbrk {u_{j + n} u_{j + n - 1} \dotsm u_j}_b$ (ignoring the carry) :$(\text D 7): \quad$ Loop on $j$: ::::Decrease $j$ by $1$ ::::If $j \ge 0$ go back to '''Step $(\text D 3)$''' :$(\text D 8): \quad$ Unnormalise: ::::The required quotient is $\sqbrk {q_m q_{m - 1} \dotsm q_1 q_0}_b$ ::::The required remainder is obtained by dividing $\sqbrk {u_{n - 1} u_{m - 2} \dotsm u_1 u_0}_b$ by $d$.


Definition:Unit Disk

A '''unit disk''' is a disk whose radius is equal to $1$. Category:Definitions/Metric Spaces dbjkoi549op8ywxtt93i4k4cig1yzxz


Definition:Disk/Radius

Let $\map B {a, r}$ be a disk. The '''radius''' of $\map B {a, r}$ is the parameter $r$.


Definition:Disk/Center

Let $\map B {a, r}$ be a disk. The '''center''' of $\map B {a, r}$ is the parameter $a$.


Definition:Transcendental Function

A '''transcendental function''' is an analytic function which is not an algebraic function. That is, it cannot be expressed as a polynomial equation. === Defined by Integral === {{:Definition:Transcendental Function/Defined by Integral}} === Defined by Differential Equation === {{:Definition:Transcendental Function/Defined by Differential Equation}}


Definition:Transcendental Function/Also defined as

Some sources define a '''transcendental function''' as a real function or complex function which is not an elementary function. However, the distinction between what is and is not an elementary function is more or less arbitrary, consisting of both algebraic functions and those derived from the exponential function, which itself is not algebraic. The current school of thought appears to be that this definition: "not an elementary function" is actually considered to be erroneous. However, the distinction is not considered particularly important nowadays. As long as it is made clear which definition is being used at the time, that would be adequate.


Definition:Algebraic Function

=== Real Algebraic Function === {{:Definition:Algebraic Function/Real}} === Complex Algebraic Function === {{:Definition:Algebraic Function/Complex}}


Definition:Algebraic Function/Real

Let $y$ be a solution to the polynomial equation: :$\map {p_0} x + \map {p_1} x y + \dotsb + \map {p_{n - 1} } x y^{n - 1} + \map {p_n} x y^n = 0$ where $\map {p_0} x \ne 0, \map {p_1} x, \dotsc, \map {p_n} x$ are real polynomial functions in $x$. Then $y = \map f x$ is a '''(real) algebraic function''':


Definition:Algebraic Function/Complex

Let $w$ be a solution to the polynomial equation: :$\map {p_0} z + \map {p_1} z w + \dotsb + \map {p_{n - 1} } z w^{n - 1} + \map {p_n} z w^n = 0$ where $\map {p_0} z \ne 0, \map {p_1} z, \dotsc, \map {p_n} z$ are complex polynomial functions in $z$. Then $w = \map f z$ is a '''(complex) algebraic function''':


Definition:Perpendicular Projection

Let $\PP$ denote the plane. Let $L$ denote a straight line in $\PP$. For all $p \in \PP$, let $K_p$ denote the straight line through $P$ perpendicular to $L$. Let $p_L$ denote the point on $L$ where $K_p$ intersects $L$. Let $\pi_L: \PP \to L$ denote the mapping defined as: :$\forall p \in \PP: \map {\pi_L} p = p_L$ That is, $\pi_L$ sends every point $p$ in $\PP$ to the foot of the perpendicular from $p$ to $L$. $\pi_L$ is called the '''perpendicular projection of the plane onto $L$'''.


Definition:Ordering on Integers/Definition 1

The integers are ordered on the relation $\le$ as follows: :$\forall x, y \in \Z: x \le y$ {{iff}}: :$\exists c \in P: x + c = y$ where $P$ is the set of positive integers. That is, $x$ is '''less than or equal''' to $y$ {{iff}} $y - x$ is non-negative.


Definition:Strict Ordering on Integers/Definition 2

The integers are strictly ordered on the relation $<$ as follows: Let $x$ and $y$ be defined as from the formal definition of integers: :$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$. Then: :$x < y \iff x_1 + y_2 < x_2 + y_1$ where: :$+$ denotes natural number addition :$a < b$ denotes natural number ordering $a \le b$ such that $a \ne b$.


Definition:Ordering on Integers/Definition 2

The integers are ordered on the relation $\le$ as follows: Let $x$ and $y$ be defined as from the formal definition of integers: :$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$. Then: :$x < y \iff x_1 + y_2 \le x_2 + y_1$ where: :$+$ denotes natural number addition :$\le$ denotes natural number ordering.


Definition:Imperial/Area/Rood

One '''rood''' is equal to an oblong measuring $1$ chain by $2 \frac 1 2$ chains. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''rood''' }} {{eqn | r = 2 \tfrac 1 2       | c = square chains }} {{eqn | r = 40       | c = square rods, poles or perches }} {{eqn | r = 1210       | c = square yards }} {{eqn | r = 10 \, 890       | c = square feet }} {{end-eqn}} :400px


Definition:Parabola/Latus Rectum

The '''latus rectum''' of $P$ is the chord of $P$ passing through the focus of $P$ parallel to the directrix $D$.


Definition:Work

Let $P$ be a particle whose position vector at time $t$ is $\mathbf r$. Let a force applied to $P$ be represented by the vector $\mathbf F$. Suppose that, during the time interval $\delta t$, $P$ moves from $\mathbf r$ to $\mathbf r + \delta \mathbf r$. The '''work done''' by $\mathbf F$ during $\delta t$ is defined to be: :$\delta W = \mathbf F \cdot \delta \mathbf r$ where $\cdot$ denotes the dot product.


Definition:Cartesian Product/Cartesian Space/Three Dimensions

The '''cartesian $3$rd power of $S$''' is: :$S^3 = S \times S \times S = \set {\tuple {x_1, x_2, x_3}: x_1, x_2, x_3 \in S}$ The set $S^3$ called a '''cartesian space of $3$ dimensions'''.


Definition:Cartesian 3-Space

The '''Cartesian $3$-space''' is a Cartesian coordinate system of $3$ dimensions. === Definition by Axes === {{:Definition:Cartesian 3-Space/Definition by Axes}} === Definition by Planes === {{:Definition:Cartesian 3-Space/Definition by Planes}}


Definition:Cartesian Coordinate System/Z Coordinate

Let $z$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $z$-axis. Then $z$ is known as the '''$z$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $z$-axis, then $z$ is positive. If $Q$ is in the negative direction along the real number line that is the $z$-axis, then $z$ is negative.


Definition:Rectangular Coordinate System

A '''rectangular coordinate system''' is a '''Cartesian coordinate system''' in which each of the coordinate axes are perpendicular to each other.


Definition:Oblique Coordinate System

An '''oblique coordinate system''' is a coordinate system in which the position of a point is determined by its relation to a set of straight coordinate axes which are oblique. :400px


Definition:X-Y Plane

The '''$x$-$y$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $x$-axis and the $y$-axis. It consists of all the points in $S$ such that $z = 0$.


Definition:Y-Z Plane

The '''$y$-$z$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $y$-axis and the $z$-axis. It consists of all the points in $S$ such that $x = 0$.


Definition:X-Z Plane

The '''$x$-$z$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $x$-axis and the $z$-axis. It consists of all the points in $S$ such that $x = 0$.


Definition:Cartesian 3-Space/Definition by Axes

600pxrightthumbA general point in Cartesian $3$-Space Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates  $\tuple {x, y, z}$, as follows: Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis. Recall the identification of the point $P$ with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane. Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis. Identify the point $P''$ on the $z$-axis such that $OP'' = OP$. Identify the $z$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P$ The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$. It is conventional to locate $P''$ as follows. Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.  Then $P''$ is then one unit ''above'' the $x$-$y$ plane. Let the $x$-$y$ plane be identified with the plane of the page or screen. The orientation of the $z$-axis is then: :coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive :going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.


Definition:Decimal Expansion/Size Less than 1

A number $x$ such that $\size x < 0$ has a units digit which is zero. Such a number may be expressed either with or without the zero, for example: :$0 \cdotp 568$ or: :$\cdotp 568$ While both are commonplace, the form with the zero is less prone to the mistake where decimal point is missed when reading it.


Definition:Cartesian 3-Space/Definition by Planes

600pxrightthumb Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates  $\tuple {x, y, z}$, as follows: Identify one distinct point in space as the origin $O$. Let $3$ distinct planes be constructed through $O$ such that all are perpendicular. Each pair of these $3$ planes intersect in a straight line that passes through $O$. Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection. Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively. Select a point $P$ on the $x$-axis different from $O$. Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane. Identify the point $P'$ on the $y$-axis such that $OP' = OP$. Identify the point $P''$ on the $z$-axis such that $OP'' = OP$. The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$. It is conventional to locate $P''$ as follows. Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.  Then $P''$ is then one unit ''above'' the $x$-$y$ plane. Let the $x$-$y$ plane be identified with the plane of the page or screen. The orientation of the $z$-axis is then: :coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive :going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.


Definition:Wave Profile

Let $\phi$ be a disturbance which is propagated along the $x$-axis with velocity $c$. At $t = 0$, let $\phi = \map f x$. Then $\map f x$ is the '''wave profile''' of $\phi$.


Definition:Switching Circuit

A '''switching circuit''' is a mechanical, electronic or electromechanical system for controlling the route of information or resources. The study of '''switching circuits''' can be considered a field of applied logic.


Definition:Electric Potential/Dimension

The dimension of measurement of '''electric potential''' is $\mathsf M \mathsf L^2 \mathsf T^{−3} \mathsf I^{−1}$.


Definition:Electric Potential/Units

The SI unit of '''electric potential''' is the volt $\mathrm V$.


Definition:Electric Potential

An '''electric potential''' is the amount of work needed to move a unit of electric charge from a given reference point to a specific point in an electric field without producing an acceleration. The reference point is usually either Earth or a point at infinity, although any point can be used.


Definition:Scalar Algebra

'''Scalar algebra''' is the application of the rules of conventional '''algebra''' in order to manipulate '''scalar quantities'''.


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6

{{begin-axiom}} {{axiom | n = \text B 6         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \cup \set x} \setminus \set y \in \mathscr B }} {{end-axiom}}


Definition:Cartesian 3-Space/Coordinate Planes

Consider the '''Cartesian $3$-space''' defined by $3$ distinct perpendicular planes through the origin $O$. These $3$ planes are known as the '''coordinate planes''' of the Cartesian $3$-space.


Definition:Electromagnetic Induction

'''Electromagnetic induction''' is the phenomenon whereby an electromotive force is generated across an electrical conductor in a changing magnetic field.


Definition:Electric Field

An '''electric field''' is a vector field that associates to each point in space the (electrostatic or Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point.


Definition:Magnetic Flux Density

'''Magnetic flux density''' is  the force exerted on a charged particle perpendicular to its velocity through a magnetic field. Hence it is a measure of the intensity of that magnetic field.


Definition:Magnetic Flux Density/Units

The SI unit for '''magnetic flux density''' is the tesla $\mathrm T$: :$1 \ \mathrm T = 1 \ \mathrm {kg} \, \mathrm s^{-2} \mathrm A^{-1}$ that is: :$1 \ \mathrm T = 1 \ \mathrm N \, \mathrm m^{-1} \mathrm A^{-1}$


Definition:Magnetic Flux Density/Dimension

'''Magnetic flux density''' has the dimension $\mathsf {M T}^{-2} \mathsf I^{-1}$.


Definition:Rate/Dimension

A '''rate''' has dimension $\mathsf T^{-1}$.


Definition:Charged Particle

A '''charged particle''' is an ideal particle which bears an electric charge. Category:Definitions/Electrostatics Category:Definitions/Electromagnetism Category:Definitions/Physics gopofl6hh9cqhltggjsiuv9s01nhivp


Definition:Elementary Charge

The '''elementary charge''' is the value of the electric charge on a proton. It is defined as ''exactly'' $1.60217 \, 6634 \times 10^{−19}$ coulombs. === Symbol === {{:Definition:Elementary Charge/Symbol}}


Definition:Tesla (Unit)/Base Units

The SI base units of the '''tesla''' are: :$\mathrm T := \mathrm {kg} \, \mathrm s^{-2} \mathrm A^{-1}$ where: :$\mathrm {kg}$ denotes kilogram :$\mathrm A$ denotes amperes :$\mathrm s$ denotes seconds (of time).


Definition:Volume Charge Density

The '''volume charge density''' quantity of electric charge per unit volume, at any given point in that volume.


Definition:Volume Charge Density/Dimension

The dimension of '''volume charge density''' is $\mathsf {I T L}^{-3}$: electric charge per unit volume.


Definition:Volume Charge Density/Units

The SI units of '''volume charge density''' are $\mathrm C \, \mathrm m^{-3}$ (coulombs per cubic metre).


Definition:Electric Displacement Field

The '''electric displacement field''' is a vector quantity defined as the electric field strength multiplied by the permittivity of the medium through which it passes: :$\mathbf D = \varepsilon \mathbf E$ where: :$\varepsilon$ is the permittivity of the medium :$\mathbf E$ is the electric field strength


Definition:Magnetic Field

A '''magnetic field''' is a vector field that gives rise to a force on a moving electric charge perpendicular to both the velocity of the charge and the direction of the '''magnetic field'''.


Definition:Magnetic Field Strength

'''Magnetic field strength''' is the measure of the intensity of a magnetic field. It can be expressed as: :$\mathbf H = \dfrac {\mathbf B} \mu$ where: :$\mathbf B$ is the magnetic flux density :$\mu$ is the permeability of the medium.


Definition:Magnetic Field Strength/Units

The SI unit for '''magnetic field strength''' is given as amperes per metre: :$\mathrm A \, \mathrm m^{-1}$


Definition:Electrostatic Force

An '''electrostatic force''' is the force exerted on an electric charge caused by an electric field.


Definition:Schrödinger Wave Equation

=== Time Independent Version === {{:Definition:Schrödinger Wave Equation/Time Independent}} {{NamedforDef|Erwin Rudolf Josef Alexander Schrödinger|cat = Schrödinger}} Category:Definitions/Quantum Mechanics 8vncojomzji3ry5cp1wj6to7krdhtjz


Definition:Schrödinger Wave Equation/Time Independent

:$-\dfrac {\hbar^2} {2 m} \nabla^2 \psi + V \psi = E \psi$ where: :$\psi$ is the wave function whose nature is to be determined :$V$ denotes the potential energy (usually positional) :$E$ denotes the total energy of the system :$m$ denotes the mass of the particle whose motion is described by $\psi$ :$\hbar$ is Planck's constant divided by $2 \pi$:  ::$\hbar = \dfrac h {2 \pi}$


Definition:Angular Momentum

The '''angular momentum''' of a body about a point $P$ is its moment of inertia about $P$ multiplied by its angular velocity about $P$. Angular momentum is a vector quantity.


Definition:Angular Momentum/Dimension

The dimension of measurement of '''angular momentum''' is $\mathsf {M L}^2 \mathsf T^{-1}$.


Definition:Vector Quantity/Arrow Representation

A '''vector quantity''' $\mathbf v$ is often represented diagramatically in the form of an '''arrow''' such that: :its length is proportional to the magnitude of $\mathbf v$ :its direction corresponds to the direction of $\mathbf v$. The head of the arrow then indicates the positive sense of the direction of $\mathbf v$. It can be rendered on the page like so: :300px It is important to note that a vector quantity, when represented in this form, is not in general fixed in space. All that is being indicated using such a notation is its magnitude and direction, and not, in general, a point at which it acts.


Definition:Vector Sum/Triangle Law

Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that: :$\mathbf u$ is represented by $\vec {AB}$ :$\mathbf v$ is represented by $\vec {BC}$ that is, so that the initial point of $\mathbf v$ is identified with the terminal point of $\mathbf u$. :400px Then their '''(vector) sum''' $\mathbf u + \mathbf v$ is represented by the arrow $\vec {AC}$.


Definition:Terminal Point of Vector

Let $\mathbf u$ be a vector quantity represented by an arrow $\vec {AB}$ embedded in the plane: :300px The point $B$ is the '''terminal point''' of $\mathbf u$.


Definition:Initial Point of Vector

Let $\mathbf u$ be a vector quantity represented by an arrow $\vec {AB}$ embedded in the plane: :300px The point $A$ is the '''initial point''' of $\mathbf u$.


Definition:Vector Sum/Component Definition

Let $\mathbf u$ and $\mathbf v$ be represented by their components considered to be embedded in a real $n$-space: {{begin-eqn}} {{eqn | l = \mathbf u       | r = \tuple {u_1, u_2, \ldots, u_n} }} {{eqn | l = \mathbf v       | r = \tuple {v_1, v_2, \ldots, v_n} }} {{end-eqn}} Then the '''(vector) sum''' of $\mathbf u$ and $\mathbf v$ is defined as: :$\mathbf u + \mathbf v := \tuple {u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n}$ Note that the $+$ on the {{RHS}} is conventional addition of numbers, while the $+$ on the {{LHS}} takes on a different meaning. The distinction is implied by which operands are involved.


Definition:Vector Length/Arrow Representation

Let $\mathbf v$ be a vector quantity represented as an arrow in a real vector space $\R^n$. The '''length''' of $\mathbf v$ is the length of the line segment representing $\mathbf v$ in $\R^n$.


Definition:Deceleration

'''Deceleration''' is the term used in natural language to mean '''negative acceleration'''. Thus, while '''acceleration''' is used to mean '''speeding up''', '''deceleration''' is used to mean '''slowing down'''. As in the applied mathematical context, '''acceleration''' is used to mean '''any''' change in velocity, including a change in direction while moving at a constant speed, the term '''deceleration''' is generally not used in mathematics.


Definition:Rectilinear Coordinate System

A coordinate system whose coordinate axes are straight lines is called a system of '''rectilinear coordinate system'''.


Definition:Curvilinear Coordinate System

A coordinate system such that at least one of the coordinate axes is a curved line is called a system of '''curvilinear coordinates'''.


Definition:Curved Line

A '''curved line''' is a curve which is specifically not a straight line.


Definition:Line/Curve/Space

A '''space curve''' is a curve which cannot be embedded in the plane, but can be embedded in $3$-space.


Let $S$ be a finite set. Let $\mathscr C$ be a non-empty set of subsets of $S$. === Definition 1 === {{:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1}} === Definition 2 === {{:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2}}


Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1

{{begin-axiom}} {{axiom | n = C1         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \implies C_1 \not \subseteq C_2 }} {{axiom | n = C2         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z }} {{end-axiom}}


Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2

{{begin-axiom}} {{axiom | n = C1         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \implies C_1 \not \subseteq C_2 }} {{axiom | n = C2'         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : y \in C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z }} {{end-axiom}}


Definition:Curvilinear Coordinate System/Cartesian Representation

The relation between curvilinear coordinates and Cartesian coordinates can be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \map z {q_1, q_2, q_3} }} {{end-eqn}} where: :$\tuple {x, y, z}$ denotes the Cartesian coordinates :$\tuple {q_1, q_2, q_3}$ denotes their curvilinear equivalents.


Definition:Orthogonal Curvilinear Coordinates

Let $\tuple {q_1, q_2, q_3}$ denote a set of curvilinear coordinates. Let the relation between those curvilinear coordinates and Cartesian coordinates be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \map z {q_1, q_2, q_3} }} {{end-eqn}} where $\tuple {x, y, z}$ denotes the Cartesian coordinates. Let these equations have the property that: :$\dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j} = 0$ wherever $i \ne j$. Then $\tuple {q_1, q_2, q_3}$ are '''orthogonal curvilinear coordinates'''.


Definition:Functional

=== Real Functional === {{:Definition:Real Functional}} Category:Definitions/Functionals Category:Definitions/Calculus of Variations 1krmqjtyftdlzscrs2kbxvkcqf0rfq7


Definition:Permutation Theory

'''Permutation theory''' is a branch of abstract algebra which studies the properties of permutations.


Definition:Landau Symbols

The '''Landau symbols''' are the $\OO$ and $\mathcal o$ that are used in the definition of big-$\OO$ notation and little-$\mathcal o$ notation: === Big-$\OO$ Notation === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Little-$\mathcal o$ Notation === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} {{NamedforDef|Edmund Georg Hermann Landau|cat = Landau}}


Definition:Vector Subtraction/Arrow Representation

Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that: :$\mathbf u$ is represented by $\vec {AB}$ :$\mathbf v$ is represented by $\vec {AC}$ that is, so that the initial point of $\mathbf v$ is identified with the initial point of $\mathbf u$. :300px Then their '''(vector) difference''' $\mathbf u - \mathbf v$ is represented by the arrow $\vec {CB}$.


Definition:Orientation of Coordinate Axes

The '''orientation''' of a coordinate system is the disposition of the coordinate axes relative to each other.


Definition:Orientation of Coordinate Axes/Cartesian Plane

There are $2$ different orientations of a Cartesian plane: :Right-hand Cartesian plane320px $\qquad \qquad \qquad$ Left-hand Cartesian plane320px


Definition:Orientation of Coordinate Axes/Cartesian Plane/Right-Handed

A Cartesian plane is defined as being '''right-handed''' if it has the following property: Let a '''right hand''' be placed, with palm uppermost, such that the thumb points along the $x$-axis in the positive direction, such that the thumb and index finger are at right-angles to each other. Then the index finger is pointed along the $y$-axis in the positive direction. :320px


Definition:Orientation of Coordinate Axes/Cartesian Plane/Left-Handed

A Cartesian plane is defined as being '''left-handed''' if it has the following property: Let a '''left hand''' be placed, with palm uppermost, such that the thumb points along the $x$-axis in the positive direction, such that the thumb and index finger are at right-angles to each other. Then the index finger is pointed along the $y$-axis in the positive direction. :320px


Definition:Orientation of Coordinate Axes/Cartesian 3-Space

There are $2$ different orientations of a Cartesian $3$-space: :450px $\qquad \qquad \qquad$ 320px


Definition:Orientation of Coordinate Axes/Cartesian 3-Space/Right-Handed

A Cartesian $3$-Space is defined as being '''right-handed''' if it has the following property: Let a '''right hand''' be placed such that: :the thumb and index finger are at right-angles to each other :the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm :the thumb points along the $x$-axis in the positive direction :the index finger points along the $x$-axis in the positive direction. Then the $3$rd finger is pointed along the $z$-axis in the positive direction. :450px


Definition:Orientation of Coordinate Axes/Cartesian 3-Space/Left-Handed

A Cartesian $3$-Space is defined as being '''left-handed''' if it has the following property: Let a '''left hand''' be placed such that: :the thumb and index finger are at right-angles to each other :the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm :the thumb points along the $x$-axis in the positive direction :the index finger points along the $x$-axis in the positive direction. Then the $3$rd finger is pointed along the $z$-axis in the positive direction. :320px


Definition:Pairwise Perpendicular

Let $S = \set {L_1, L_2, \ldots, L_n}$ be a set of lines in a Euclidean space. Then the elements of $S$ are '''pairwise perpendicular''' {{iff}} :$\forall i, j \in \set {1, 2, \ldots, n}, i \ne j:$ $L_i$ and $L_j$ are perpendicular to each other.


Definition:Orthogonal Coordinate System

An '''orthogonal coordinate system''' is a coordinate system in which the coordinate axes are pairwise perpendicular.


Definition:Smallest Natural Number

Let $S \subseteq \N$ be a subset of the natural numbers $\N$. The '''smallest''' element $m$ of $S$ is defined as: :$\forall n \in S: m \le n$ That is, it is the minimal element of $S$ under the usual ordering.


Definition:Energy/Unit

The unit of measurement of '''energy'''  is the joule $\mathrm J$: :$1 \ \mathrm J = 1 \ \mathrm N \ \mathrm m$ where: :$\mathrm N$ denotes the newton :$\mathrm m$ denotes the metre.


Definition:Joule (Unit)

The '''joule''' is the SI unit of energy. It is defined as being: :the '''energy''' transferred to (or work done on) a body when a force of $1$ newton acts on that body in the direction of the force's motion through a distance of one metre :the '''energy''' dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second.


Definition:Joule (Unit)/Base Units

The SI base units of the '''joule''' are: :$\mathrm J := \mathrm {kg} \, \mathrm m^2 \mathrm s^{-2}$ where: :$\mathrm {kg}$ denotes kilograms :$\mathrm m$ denotes metres :$\mathrm s$ denotes seconds (of time).


Definition:Power (Physics)

'''Power''' the amount of energy transferred or converted per unit time. '''Power''' is a scalar quantity.


Definition:Power (Physics)/Dimension

The dimension of measurement of '''power''' is $\mathsf {M L}^2 \mathsf T^{-3}$. This derives from its definition as: :$\dfrac {\text{Energy} } {\text {Time} }$


Definition:Power (Physics)/Unit

The unit of measurement of '''power'''  is the watt $\mathrm W$: :$1 \ \mathrm W = 1 \ \mathrm J \ \mathrm s^{-1}$ where: :$\mathrm J$ denotes the joule :$\mathrm s$ denotes the second.


Definition:Watt (Unit)

The '''watt''' is the SI unit of power. It is defined as being: :the rate at which work is done when the velocity of a body is held constant at $1$ metre per second against a constant opposing force of $1$ newton :the rate at which work is performed when an electric current of $1$ ampere flows across an electrical potential difference of $1$ volt.


Definition:Volt

The '''volt''' is the SI unit of electromotive force. It is defined as being the difference in '''electric potential between two points of an electrical conductor: :when an electric current of $1$ ampere dissipates $1$ watt of power between those points :that will impart $1$ joule of energy per coulomb of electric charge that passes through it.


Definition:Volt/Base Units

The SI base units of the '''volt''' are: :$\mathrm W := \mathrm {kg} \, \mathrm m^2 \mathrm s^{-3} \mathrm A^{-1}$ where: :$\mathrm {kg}$ denotes kilograms :$\mathrm m$ denotes metres :$\mathrm s$ denotes seconds (of time). :$\mathrm A$ denotes ampere.


Definition:Ohm (Unit)

The '''ohm''' is the SI unit of electrical resistance. It is defined as being the electrical resistance between two points of an electrical conductor when a constant electric potential of $1$ volt, applied to these points, produces in the conductor an electric current of $1$ ampere, the  conductor not being the seat of any electromotive force.


Definition:Ohm (Unit)/Base Units

The SI base units of the '''ohm''' are: :$\mathrm W := \mathrm {kg} \, \mathrm m^2 \mathrm s^{-3} \mathrm A^{-2}$ where: :$\mathrm {kg}$ denotes kilograms :$\mathrm m$ denotes metres :$\mathrm s$ denotes seconds (of time). :$\mathrm A$ denotes ampere.


Definition:Strong Nuclear Force

The '''strong nuclear force''' is a force which acts in the range of the order of $10^{-15} \ \mathrm m$. It is what holds protons and neutrons together in the nucleus of an atom. Beyond the range of approximately the diameter of a large atom its effect is diminished. This may be the reason why stable isotopes of large atoms are rare.


Definition:Elementary Charge/Symbol

The symbol used to denote the '''elementary charge''' is usually $\E$ or $e$. The preferred symbol on {{ProofWiki}} is $\E$.


Definition:Zero Vector/Vector Quantity

A '''vector quantity''' whose magnitude is zero is referred to as a '''zero vector'''.


Definition:Scalar Multiplication/Vector Quantity

Let $\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. The operation of scalar multiplication by $m$ of $\mathbf a$ is denoted $m \mathbf a$ and defined such that: :the magnitude of $m \mathbf a$ is equal to $m$ times the magnitude of $\mathbf a$: ::$\size {m \mathbf a} = m \size {\mathbf a}$ :the direction of $m \mathbf a$ is the same as the direction of $\mathbf a$.


Definition:Vector Quantity/Component/Cartesian Plane

Let $\mathbf a$ be a vector quantity embedded in a Cartesian plane $P$. Let $\mathbf a$ be represented with its initial point at the origin of $P$. Let $\mathbf i$ and $\mathbf j$ be the unit vectors in the positive direction of the $x$-axis and $y$-axis. Then: :$\mathbf a = x \mathbf i + y \mathbf j$ where: :$x \mathbf i$ and $y \mathbf j$ are the '''component vectors''' of $\mathbf a$ in the $\mathbf i$ and $\mathbf j$ directions :$x$ and $y$ are the '''components''' of $\mathbf a$ in the $\mathbf i$ and $\mathbf j$ directions.


Definition:Vector Quantity/Component/Cartesian 3-Space

Let $\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$. Let $\mathbf a$ be represented with its initial point at the origin of $S$. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then: :$\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$ where: :$x \mathbf i$, $y \mathbf j$ and $z \mathbf k$ are the '''component vectors''' of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions :$x$, $y$ and $z$ are the '''components''' of $\mathbf a$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions. It is usual to arrange that the coordinate axes form a right-handed Cartesian $3$-space.


Definition:Variational Principle

A '''variational principle''' is a statement defining a condition that is required to be fulfilled in order to solve a problem in the calculus of variations.


Definition:Vectorial Mechanics

'''Vectorial mechanics''' is an approach to problems in mechanics which handles quantities as vector quantities and uses vector algebra and vector calculus for their solution.


Definition:Kinematics

'''Kinematics''' is the branch of the mechanics describing the motion of bodies without considering the forces that may affect this motion.


Definition:Like Vector Quantities

Let $\mathbf a$ and $\mathbf b$ be vector quantities. Then $\mathbf a$ and $\mathbf b$ are known as '''like vector quantities''' {{iff}} they have the same direction.


Definition:Negative of Vector Quantity

Let $\mathbf a$ be a vector quantities. The '''negative''' $-\mathbf a$ of $\mathbf a$ is the vector quantity which has the same magnitude but the opposite direction.


Definition:Scalar Division/Vector Quantity

Let $\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. The operation of scalar division by $m$ of $\mathbf a$ is defined as: :$\dfrac {\mathbf a} m = \dfrac 1 m \size {\mathbf a}$ that is, scalar multiplication by $\dfrac 1 m$ of $\mathbf a$.


Definition:Scalar Division

=== Vector Quantity === {{:Definition:Scalar Division/Vector Quantity}}


Definition:Scalar Multiplication by Negative Scalar

Let $\mathbf a$ be a vector quantity. Let $-m$ be a negative scalar quantity. The operation of scalar multiplication by $-m$ of $\mathbf a$ is defined as: :$-m \mathbf a := \paren {-m} \mathbf a = m \paren {-\mathbf a}$ that is, the negative of the scalar product of $m$ with $\mathbf a$.


Definition:Direct Product Norm

Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces. Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\tuple {x, y} \in V$, $x \in X$ and $y \in Y$. Then the '''direct product norm''' on $V$ is defined as: :$\norm {\tuple {x, y} } := \map \max {\norm x, \norm y}$ where $\max$ denotes the max operation.


Definition:Coplanar Vectors

Three or more vectors are '''coplanar''' if they can be embedded in the same plane.


Definition:Coplanar Vectors/Non-Coplanar

Three or more vectors are '''non-coplanar''' if they are not '''coplanar'''. That is if they cannot be embedded in the same plane.


Definition:Direction Cosine

Let $\mathbf a$ be a vector quantity embedded in a Cartesian $3$-space. Let the angles which $\mathbf a$ makes with the $x$-axis, $y$-axis and $z$-axis be $\alpha$, $\beta$ and $\gamma$ respectively. Then the '''direction cosines''' of $\mathbf a$ are $\cos \alpha$, $\cos \beta$ and $\cos \gamma$, defined individually such that: :$\cos \alpha$ is the '''direction cosine of $\mathbf a$ with respect to the $x$-axis''' :$\cos \beta$ is the '''direction cosine of $\mathbf a$ with respect to the $y$-axis''' :$\cos \gamma$ is the '''direction cosine of $\mathbf a$ with respect to the $z$-axis'''.


Definition:Birkhoff-James Orthogonality

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed linear space. Let $x, y \in V$. Then $x$ and $y$ are '''Birkhoff-James orthogonal''' {{iff}}  :$\norm {x + \lambda y} \ge \norm y$ for all scalars $\lambda$. This is denoted: :$x \perp_B y$


Definition:Orthogonal Basis of Vector Space

Let $V$ be a vector space. Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $V$. Then $\BB$ is an '''orthogonal basis''' {{iff}} $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ are pairwise perpendicular.


Definition:Orthonormal Basis of Vector Space

Let $V$ be a vector space. Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $V$. Then $\BB$ is an '''orthonormal basis of $V$''' {{iff}}: :$(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an orthogonal basis of $V$ :$(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_1} = 1$


Definition:Vector Quantity/Component/Einstein Summation Convention

Let $\mathbf a$ be a vector quantity. $\mathbf a$ can be expressed in component form using the Einstein summation convention as: :$\mathbf a = a_i \mathbf e_i$


Definition:Dot Product/Einstein Summation Convention

Let $\mathbf a$ and $\mathbf b$ be vector quantities. The dot product of $\mathbf a$ and $\mathbf b$ can be expressed as: {{begin-eqn}} {{eqn | l = \mathbf a \cdot \mathbf b        | o = :=       | r = a_i b_j \delta_{i j}       | c =  }} {{eqn | r = a_i b_i       | c =  }} {{eqn | r = a_j b_j       | c =  }} {{end-eqn}}


Positive Integer Greater than 1 has Prime Divisor 0 130 373934 291994 2018-10-29T23:01:52Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Every positive integer greater than $1$ has at least one divisor which is prime. </onlyinclude> == Proof 1 == {{:Positive Integer Greater than 1 has Prime Divisor/Proof 1}} == Proof 2 (not using FTA) == {{:Positive Integer Greater than 1 has Prime Divisor/Proof 2}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Prime Number/Sequence|next = Euclid's Theorem}}: Chapter $1$: Properties of the Natural Numbers: $\S 22 \alpha$ Category:Prime Numbers Category:Positive Integer Greater than 1 has Prime Divisor 2oc0nobqc3qlfa2d3tcqy9rvhai5fqi


Axiom:Axiom of Extension

Axiom:Axiom of Extension 100 149 444633 444632 2020-01-23T20:49:07Z Prime.mover 59 wikitext text/x-wiki == Axiom == The '''axiom of extension''' is the fundamental definition of the nature of a collection: it is completely determined by its elements. <onlyinclude> === Set Theory === {{:Axiom:Axiom of Extension/Set Theory}} === Class Theory === The '''axiom of extension''' in the context of class theory has the same form: {{:Axiom:Axiom of Extension/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Set Equality * Definition:Equals == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} Extension Extension Extension nyvyz3hodadhs77097a0v1mjkjr01o3


Axiom:Axiom of Empty Set

Axiom:Axiom of Empty Set 100 151 449936 449902 2020-02-18T23:25:43Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> The '''axiom of the empty set''' posits the existence of a set which has no elements. Depending on whether this axiom is declared in the context of set theory or class theory, it exists in different forms. === Set Theory === {{:Axiom:Axiom of Empty Set/Set Theory}} === Class Theory === In class theory, the existence of the empty class is not axiomatic, as it has been derived from previous axioms. Hence the '''axiom of the empty set''' takes this form: {{:Axiom:Axiom of Empty Set/Class Theory}} where $V$ denotes the basic universe. </onlyinclude> == Also known as == {{:Axiom:Axiom of Empty Set/Also known as}} == Also see == * Definition:Zermelo-Fraenkel Axioms * Definition:Empty Set * Empty Set is Unique Empty Set Empty Set Category:Definitions/Basic Universe qrgtgfa41x2x4dt8v416smyu66epkeq


Axiom:Axiom of Pairing

Axiom:Axiom of Pairing 100 152 444595 444583 2020-01-23T20:27:31Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Pairing/Set Theory}} === Class Theory === {{:Axiom:Axiom of Pairing/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Definition:Doubleton * Definition:Ordered Pair {{LinkToCategory|Axiom of Pairing|the axiom of pairing}} Pairing Pairing Pairing cuzikepuwbqucmjwfidhnaql1cq6qna


Axiom:Axiom of Specification

Axiom:Axiom of Specification 100 153 493722 453375 2020-10-10T15:04:29Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. Because we cannot quantify over functions, we need an axiom for every condition we can express. Therefore, this axiom is sometimes called an axiom ''schema'', as we introduce a lot of similar axioms. This axiom schema can be formally stated as follows: === Set Theory === {{:Axiom:Axiom of Specification/Set Theory}} === Class Theory === The '''axiom of specification''' in the context of class theory has a similar form: {{:Axiom:Axiom of Specification/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Specification/Also known as}} == Also see == * Axiom:Comprehension Principle -- do not confuse that with this {{refactor|Include the following in its own theorem page, and present that deduction formally}} The '''axiom of specification''' can be deduced from the Axiom of Replacement. == Historical Note == {{:Axiom:Axiom of Specification/Historical Note}} {{Languages|Axiom of specification}} {{language|German|Aussonderungsaxiom|lit = axiom of segregation}} {{end-languages}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Class (Class Theory)|next = Definition:Finite Set}}: Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets Specification Specification Specification 2mhp95orxrykoudc2fopbmzs46tzk0a


Axiom:Axiom of Unions

Axiom:Axiom of Unions 100 158 445003 444875 2020-01-27T23:08:19Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Unions/Set Theory}} === Class Theory === {{:Axiom:Axiom of Unions/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Unions/Also known as}} Unions Unions Unions 614u0u5kgomeahmakt1wp1tl5beot67


Axiom:Axiom of Powers

Axiom:Axiom of Powers 100 159 445020 444039 2020-01-27T23:25:39Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Powers/Set Theory}} === Class Theory === {{:Axiom:Axiom of Powers/Class Theory}}</onlyinclude> == Also known as == {{:Axiom:Axiom of Powers/Also known as}} == Also see == * Definition:Power Set Powers Powers Powers ehj2nolsk55rwqbwequ79o8o70oaewj


Axiom:Axiom of Infinity

Axiom:Axiom of Infinity 100 160 449889 449832 2020-02-18T22:40:16Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> === Set Theory === {{:Axiom:Axiom of Infinity/Set Theory}} === Class Theory === {{:Axiom:Axiom of Infinity/Class Theory}}</onlyinclude> Infinity Infinity Infinity 9gjyglkd40zx3kol5ox0syoks4l30ng


Axiom:Axiom of Replacement

Axiom:Axiom of Replacement 100 163 444040 443865 2020-01-18T01:52:31Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\map f S$. More formally, let us express this as follows: Let $\map P {y, z}$ be a propositional function, which determines a function. That is, we have: :$\forall y: \exists x: \forall z: \paren {\map P {y, z} \iff x = z}$. Then we state as an axiom: :$\forall w: \exists x: \forall y: \paren {y \in w \implies \paren {\forall z: \paren {\map P {y, z} \implies z \in x} } }$ </onlyinclude> == Also presented as == The two above statements may be combined into a single (somewhat unwieldy) expression: :$\paren {\forall y: \exists x: \forall z: \paren {\map P {y, z} \implies x = z} } \implies \forall w: \exists x: \forall y: \paren {y \in w \implies \forall z: \paren {\map P {y, z} \implies z \in x} }$ == Also known as == The '''axiom of replacement''' is also known as the '''axiom of substitution'''. == Historical Note == {{:Axiom:Axiom of Replacement/Historical Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Zermelo Set Theory/Historical Note|next = Axiom:Axiom of Replacement/Historical Note}}: Chapter $1$: General Background: $\S 9$ Zermelo set theory * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Replacement cb6yglqcwu8vyjbv8t8n9hr3wisjls5


Axiom:Axiom of Foundation

Axiom:Axiom of Foundation 100 164 444036 444035 2020-01-18T01:50:48Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For all non-empty sets, there is an element of the set that shares no element with the set. That is: :$\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$ The antecedent states that $S$ is not empty. </onlyinclude> == Also defined as == It can also be stated as: :For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint. :A set contains no infinitely descending (membership) sequence. :A set contains a (membership) minimal element. :The membership relation is a foundational relation on any non-empty set. == Also known as == The '''axiom of foundation''' is also known as the '''axiom of regularity'''. == Sources == * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Foundation tj1k4ptei585ohnkgatal3y7fl9um99


Axiom:Axiom of Choice

Axiom:Axiom of Choice 100 165 462986 462981 2020-04-19T09:47:14Z Prime.mover 59 wikitext text/x-wiki == Axiom == === Formulation 1 === <onlyinclude>{{:Axiom:Axiom of Choice/Formulation 1}}</onlyinclude> === Formulation 2 === {{:Axiom:Axiom of Choice/Formulation 2}} === Formulation 3 === {{:Axiom:Axiom of Choice/Formulation 3}} == Comment == Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to infinite sets of sets this axiom leads to non-intuitive results, notably the famous Banach-Tarski Paradox. For this reason, the '''Axiom of Choice''' (often abbreviated '''AoC''' or '''AC''') is often treated separately from the rest of the Zermelo-Fraenkel Axioms. Set theory based on the Zermelo-Fraenkel axioms is referred to ZF, while that based on the Z-F axioms including the '''AoC''' is referred to as ZFC. == Additional forms == The following are equivalent, in ZF, to the '''Axiom of Choice''': * Zorn's Lemma * Kuratowski's Lemma * Hausdorff Maximal Principle * Tukey's Lemma * Tychonoff's Theorem * Kelley's Theorem * Vector Space has Basis == Historical Note == {{:Axiom:Axiom of Choice/Historical Note}} == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Definition:Surjection/Definition 2|next = Definition:Commutative Diagram}}: Chapter $\text{I}$: Sets and Functions: Composition of Functions * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Surjection iff Right Inverse/Proof 1|next = Definition:Inverse Mapping/Definition 2}}: Appendix $\text{A}.7$: Inverses * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Continuum Hypothesis is Independent of ZFC/Historical Note|next = Axiom:Axiom of Choice/Historical Note}}: Chapter $1$: General Background: $\S 5$ The continuum problem * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Choice Choice Choice Category:Philosophical Positions ohexnx43nsaw0l6fvvv4e34lo39rmrx


Axiom:Comprehension Principle

Axiom:Comprehension Principle 100 170 490600 454411 2020-09-25T06:24:01Z Prime.mover 59 wikitext text/x-wiki == Definition == The '''comprehension principle''' states: <onlyinclude> :Given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$: ::$\set {x: \map P x}$ </onlyinclude> From the definition of a set: :''A set is any aggregation of objects, called elements, which can be precisely defined in some way or other.'' == Also known as == The '''comprehension principle''' can also be referred to as: :the '''abstraction principle''' :the '''axiom of abstraction''' :the '''unlimited abstraction principle''' == Also see == * Axiom:Axiom of Comprehension -- do not confuse that with this * Definition:Set Definition by Predicate == Historical Note == {{:Axiom:Comprehension Principle/Historical Note}} == Sources == * {{BookReference|Axiomatic Set Theory|1972|Patrick Suppes|ed = 2nd|edpage = Second Edition|prev = Definition:Free Variable|next = Axiom:Comprehension Principle/Historical Note}}: $\S 1.3$ Axiom Schema of Abstraction and Russell's Paradox * {{BookReference|Introduction to Graph Theory|1993|Richard J. Trudeau|prev = Square Root of 2 is Irrational/Classic Proof|next = Russell's Paradox}}: $2$. Graphs: Paradox * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Abstraction|next = Definition:Abstract Space|entry = abstraction|index = 2}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Abstraction|next = Definition:Abstract Space|entry = abstraction|index = 2}} * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Frege Set Theory|next = Definition:Set Definition by Predicate}}: Chapter $1$: General Background: $\S 7$ Frege set theory Category:Axioms/Frege Set Theory 6hwmglfowxv66erjlmffiztpsgsz0zp


Rule of Conjunction

Rule of Conjunction 0 266 244265 244259 2016-01-15T17:27:36Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''rule of conjunction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Conjunction/Proof Rule}} === Sequent Form === {{:Rule of Conjunction/Sequent Form}} == Explanation == {{:Rule of Conjunction/Explanation}} Thus a conjunction is added to a sequent. == Also known as == {{:Rule of Conjunction/Also known as}} == Also see == * Rule of Simplification Category:Conjunction Category:Rule of Conjunction h7ohkmvyd9hjrf7hwtrhsis8p4uaiba


Rule of Simplification

Rule of Simplification 0 267 244269 244247 2016-01-15T17:31:54Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''rule of simplification'''  is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Simplification/Proof Rule}} === Sequent Form === {{:Rule of Simplification/Sequent Form}} == Explanation == {{:Rule of Simplification/Explanation}} == Also known as == {{:Rule of Simplification/Also known as}} == Also see == * Rule of Conjunction Category:Rule of Simplification Category:Conjunction d5y2fu6s06nxhtrbuhm14bqvrmbmcym


Rule of Addition

Rule of Addition 0 268 244636 244253 2016-01-16T14:22:07Z Prime.mover 59 wikitext text/x-wiki == Sequent == The '''rule of addition'''  is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Addition/Proof Rule}} === Sequent Form === {{:Rule of Addition/Sequent Form}} == Explanation == {{:Rule of Addition/Explanation}} == Also known as == {{:Rule of Addition/Also known as}} == Also see == * Proof by Cases Category:Disjunction Category:Rule of Addition k35h9024bd9cfpt88usbkj6vldgzcaj


Proof by Cases

Proof by Cases 0 269 244744 244633 2016-01-17T06:23:15Z Prime.mover 59 wikitext text/x-wiki == Sequent == '''Proof by cases''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Proof by Cases/Proof Rule}} === Sequent Form === {{:Proof by Cases/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Formulation 1 === {{:Proof by Cases/Formulation 1}} === Formulation 2 === {{:Proof by Cases/Formulation 2}} === Formulation 3 === {{:Proof by Cases/Formulation 3}} == Explanation == {{:Proof by Cases/Explanation}} Thus a disjunction is '''eliminated''' from a sequent. == Also known as == {{:Proof by Cases/Also known as}} == Also see == * Rule of Addition * Principle of Dilemma Category:Proof by Cases Category:Disjunction fow9jm911mpvt1vg4uw0ptluhlxxo4f


Modus Ponendo Ponens

Modus Ponendo Ponens 0 272 493613 467111 2020-10-10T12:25:12Z Prime.mover 59 wikitext text/x-wiki == Proof Rule == The '''modus ponendo ponens''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Modus Ponendo Ponens/Proof Rule}} === Sequent Form === {{:Modus Ponendo Ponens/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Modus Ponendo Ponens/Variant 1}} === Variant 2 === {{:Modus Ponendo Ponens/Variant 2}} === Variant 3 === {{:Modus Ponendo Ponens/Variant 3}} == Also known as == {{:Modus Ponendo Ponens/Also known as}} == Linguistic Note == {{:Modus Ponendo Ponens/Linguistic Note}} == Also see == The following are related argument forms: * Modus Ponendo Tollens * Modus Tollendo Ponens * Modus Tollendo Tollens == Sources == * {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell|volume = 1|prev = Law of Identity/Formulation 2|next = Modus Ponendo Ponens/Also known as}}: Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Method of Truth Tables|next = Modus Tollendo Tollens}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Direct Proof|next = Definition:Distinction between Logical Implication and Conditional}}: Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Four Color Theorem for Finite Maps implies Four Color Theorem for Infinite Maps|next = Reductio ad Absurdum/Variant 2}}: $\S 1.12$: Valid Arguments * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Rigidity Modulus|next = Modus Tollendo Ponens|entry = ''modus ponens''}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Modulus Sign|next = Modus Tollendo Ponens|entry = ''modus ponens''}} Category:Implication Category:Modus Ponendo Ponens akrpifwojcsk0lbtfid8j6h49tf4cwq