Definition:Laws of Conservation

The '''laws of conservation''' are laws of physics that states that certain particular measurable properties of isolated physical systems do not change as the system evolves over time.


Definition:Multiplication/Natural Numbers/Addition

Let $+$ denote addition. The binary operation $\times$ is recursively defined on $\N$ as follows: :$\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \left({n + 1}\right) & = m \times n + m \end{cases}$ This operation is called '''multiplication'''. Equivalently, '''multiplication''' can be defined as: :$\forall m, n \in \N: m \times n := +^n m$  where $+^n m$ denotes the $n$th power of $m$ under $+$.


Definition:Ideal Theory

'''Ideal Theory''' is a branch of abstract algebra which studies ideals of rings.


Definition:Algebraic Topology

'''Algebraic Topology''' is a branch of topology which uses tools from abstract algebra to study topological spaces.


Definition:Fractional Calculus

'''Fractional calculus''' is the branch of calculus which considers the concept of derivatives of fractional order.


Definition:Approximation Theory

'''Approximation theory''' is a branch of numerical analysis and applied mathematics concerned with the approximation of a function over an interval can be approximated by a simpler function (usually polynomial).


Definition:Module Theory

'''Module Theory''' is the branch of abstract algebra which studies modules.


Definition:Polynomial Theory

'''Polynomial Theory''' is a branch of abstract algebra which studies polynomials.


Definition:Vector Algebra

'''Vector Algebra''' is the branch of mathematics which studies the algebra of vector spaces.


Definition:Discrete Mathematics

'''Discrete mathematics''' is the branch of mathematics which studies mathematical structures that are discrete rather than continuous.


Definition:Discrete (Word)

'''Discrete''' is a word used to define an object which is not continuous.


Definition:Quantum Mechanics

'''Quantum mechanics''' is the branch of physics concerned with the behaviour of matter at the smallest scale accessible to observation.


Definition:Ergodic Theory

'''Ergodic Theory''' is the study of dynamical systems with an invariant measure.


Definition:Economics

'''Economics''' is the branch of applied mathematics which studies the movement of money.


Definition:Mathematical Programming

'''Mathematical programming''' is the branch of applied mathematics which applies techniques from computer science to solve problems in economics.


Definition:Game Theory

'''Game theory''' is the branch of discrete mathematics which studies mathematical models of conflict and cooperation between intelligent rational decision-makers. Problems in '''game theory''' are a special case of those of linear programming.


Definition:Numerical Analysis

'''Numerical analysis''' is the branch of applied mathematics concerned with the processes by which certain kinds of numerical solutions to problems in (mainly) physics.


Definition:Perturbation Theory

'''Perturbation Theory''' is a branch of applied mathematics which studies systems by investigating their behaviour when certain parameters are changed slghtly. That is, it is the study of the effects of small disturbances.


Definition:Model Theory

'''Model theory''' is a sub-branch of mathematical logic which studies mathematical structures by means of formal languages.


Definition:Fourier Analysis

'''Fourier analysis''' is a branch of analysis which seeks to express functions (usually periodic) by means of series of sines and cosines.


Definition:Operator Theory

'''Operator theory''' is the study of linear operators on function spaces, beginning with differential operators and integral operators.


Definition:Non-Euclidean Geometry

'''Non-Euclidean geometry''' is branch of geometry in which {{AuthorRef|Euclid}}'s fifth postulate does not hold.


Definition:Ordered Geometry

'''Ordered geometry''' is a sub-branch of geometry which featured the concept of betweenness but, like projective geometry, omitting the basic notion of distance.


Definition:Hyperbolic Geometry

'''Hyperbolic geometry''' is a branch of non-Euclidean geometry in which there more than one straight line can pass through a point parallel to another straight line.


Definition:Algebraic Geometry

'''Algebraic geometry''' is the branch of geometry which studies objects in multi-dimensional space using the techniques of abstract algebra. In particular, techniques from commutative algebra are mainly used.


Definition:Brill-Noether Theory

'''Brill-Noether Theory''' is the branch of algebraic geometry which studies special divisors.


Definition:Tarski-Grothendieck Set Theory

'''Tarski-Grothendieck set theory''' is an implementation of axiomatic set theory which allows for the existence of a universe.


Definition:Kummer Theory

'''Kummer Theory''' is a branch of abstract algebra and number theory which investigates certain types of field extensions.


Definition:Tensor Theory

'''Tensor theory''' is the branch of linear algebra which studies tensors.


Definition:Vector Analysis

'''Vector analysis''' is the branch of linear algebra concerned with differentiation and integration of vector spaces, primarily in Euclidean space of $3$-dimensional space.


Definition:Fluid Mechanics

'''Fluid mechanics''' is the branch of mechanics concerned with the behaviour of fluids.


Definition:Electromagnetism

'''Electromagnetism''' is the branch of physics concerned with the study of the combined phenomena of electricity and magnetism, and their interconnected nature.


Definition:Partition Theory

'''Partition theory''' is the subfield of combinatorics which studies integer partitions.


Definition:Proof Theory

'''Proof theory''' is the subfield of mathematical logic which studies proofs as mathematical objects, thus allowing them to be analysed mathematically.


Definition:Fluid Dynamics

'''Fluid dynamics''' is the branch of fluid mechanics concerned with the behaviour of fluids in motion.


Definition:Class Theory

'''Class theory''' is an extension of set theory which allows the creation of collections that are not sets by classes.


Definition:Descriptive Set Theory

'''Descriptive set theory''' is a branch of set theory which studies certain classes subsets of the real number line and other Polish spaces.


Definition:Recursion Theory

'''Recursion theory''' is a branch of computability theory which concerns itself with recursive structures.


Definition:Inner Model Theory

'''Inner model theory''' is a branch of set theory and model theory which studies certain models of ZFC.


Definition:Divisor (Algebra)/Notation

The conventional notation for '''$x$ is a divisor of $y$''' is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by {{AuthorRef|Donald Ervin Knuth|Knuth}} etc. From {{BookReference|Concrete Mathematics: A Foundation for Computer Science||Ronald L. Graham|author2 = Donald E. Knuth|author3 = Oren Patashnik|ed = 2nd|edpage = Second Edition}}: :''The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.'' An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$. Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \ $ or similar to denote non-divisibility.


Definition:Cantor-Bendixson Derivative

Let $\struct {X, \tau}$ be a topological space. Let $S \subseteq X$. Then for all ordinals $\beta$, the $\beta$th '''Cantor-Bendixson derivative''' of $S$ is defined by Transfinite Recursion thus: :$S^{\paren \beta} = \begin {cases} S & : \beta = 0 \\ \paren {S^{\paren \alpha} }' & : \beta = \alpha^+ \\ \displaystyle \bigcap_{\alpha \mathop < \lambda} S^{\paren \alpha} & : \beta = \lambda \end{cases}$ where: :$\paren {S^{\paren \alpha} }'$ is the derived set of $S^{\paren \alpha}$ :$\lambda$ is a limit ordinal. {{NamedforDef|Georg Cantor|name2 = Ivar Otto Bendixson|cat = Cantor|cat2 = Bendixson}}


Definition:Directed Hamilton Cycle Problem/Function Version

:Given a directed graph $G$ with $n$ vertices, to find a Hamilton cycle in $G$.


There are two versions of the '''Directed Hamilton Cycle Problem'''. === Function Version === {{:Definition:Directed Hamilton Cycle Problem/Function Version}} === Decision Version === {{:Definition:Directed Hamilton Cycle Problem/Decision Version}}


Definition:Finite Subset

Let $S$ be a set. Let $T$ be a subset of $S$ which is a finite set. Then $T$ is a '''finite subset''' of $S$. Category:Definitions/Subsets 18w2vinwyqq86kfnhdobc63kpxnfvps


Definition:Quotient Module

Let $M$ be an $R$-module and $M^\ast$ be the underlying group. Let $N$ be a submodule of $M$ and $N^\ast$ be the underlying subgroup. Let $a + N$ denote the coset of the quotient group $M^\ast / N^\ast$. Define the $R$-action on $M^\ast / N^\ast$ as: :$\forall r \in R, \forall a \in M / N: r \left({a + N^\ast}\right) := r a + N^\ast$ Then $(M^\ast / N^\ast,\circ)$ is a '''quotient $R$-module'''.


Definition:Right Module Axioms

{{begin-axiom}} {{axiom | n = \text {RM} 1         | lc= Scalar Multiplication Right Distributes over Module Addition         | q = \forall \lambda \in R: \forall x, y \in G         | ml= \paren {x +_G y} \circ \lambda         | mo= =         | mr= \paren {x \circ \lambda} +_G \paren {y \circ \lambda} }} {{axiom | n = \text {RM} 2         | lc= Scalar Multiplication Left Distributes over Scalar Addition         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= x \circ \paren {\lambda +_R \mu}         | mo= =         | mr= \paren {x \circ \lambda} +_G \paren {x\circ \mu} }} {{axiom | n = \text {RM} 3         | lc= Associativity of Scalar Multiplication         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= x \circ \paren {\lambda \times_R \mu}         | mo= =         | mr= \paren {x \circ \lambda} \circ \mu }} {{end-axiom}}


Definition:Tensor Algebra

'''Tensor algebra''' is the branch of module theory which studies tensor products.


Definition:Module Direct Product

=== Finite Case === {{:Definition:Module Direct Product/Finite Case}} === General Case === {{:Definition:Module Direct Product/General Case}}


Definition:Ramsey Theory

'''Ramsey theory''' is a branch of combinatorics that studies the conditions under which order must appear.


Definition:Angle Inscribed in Circle

:300px Let $AB$ and $BC$ be two chords of a circle which meet at $B$. The angle $\angle ABC$ is the '''angle inscribed at $B$ (with respect to $A$ and $C$)'''. Category:Definitions/Circles 62ne3b5n8dhn4vvvb4wya45hixd45f4


Definition:Path (Topology)/Endpoint

The initial point and final point of $\gamma$ can be referred to as the '''endpoints of $\gamma$'''


Definition:Infinite Path

An '''infinite path''' is a path of infinite length Category:Definitions/Graph Theory 9zbjdb17ulh1ua4rmn8i5zy8hutbh33


Definition:Supremum/Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the '''least upper bound of $T$''' and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$. Some sources refer to the '''supremum of a set''' as the '''supremum ''on'' a set'''. Some sources refer to the '''supremum of a set''' as the '''join of the set''' and use the notation $\bigvee S$. Some sources introduce the notation $\displaystyle \sup_{y \mathop \in S} y$, which may improve clarity in some circumstances. Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a '''supremum''' as an '''upper limit'''.


Definition:Supremum/Also defined as

Some sources refer to the supremum as being '''''the'' upper bound'''. Using this convention, any element greater than this is not considered to be an upper bound.


Definition:Infimum/Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the '''greatest lower bound of $T$''' and denoted $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$. Some sources refer to the '''infimum of a set''' as the '''infimum ''on'' a set'''. Some sources introduce the notation $\displaystyle \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances. Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to a '''infimum''' as a '''lower limit'''.


Definition:Infimum/Also defined as

Some sources refer to the infimum as being '''''the'' lower bound'''. Using this convention, any element less than this is not considered to be a lower bound.


Let $\Phi$ be a differential equation defined on a domain $D$. Let $\phi$ be a function which satisfies $\Phi$ on the whole of $D$. Then $\phi$ is known as '''a solution''' of $\Phi$. Note that, in general, there may be more than one '''solution''' to a given differential equation. On the other hand, there may be none at all. === General Solution === {{:Definition:Differential Equation/Solution/General Solution}} === Particular Solution === {{:Definition:Differential Equation/Solution/Particular Solution}}


Definition:Differential Equation/Solution/General Solution

The '''general solution''' of $\Phi$ is the set of ''all'' functions $\phi$ that satisfy $\Phi$.


Definition:Linear Representation/Group

Let $\left({\mathbb k, +, \circ}\right)$ be a field. Let $V$ be a vector space over $\mathbb k$ of finite dimension. Let $\operatorname {GL} \left({V}\right)$ be the general linear group of $V$. Let $\left({G, \cdot}\right)$ be a finite group. A '''linear representation of $G$ on $V$''' is a group homomorphism $\rho: G \to \operatorname {GL} \left({V}\right)$.


Definition:Linear Representation/Algebra

Let $K$ be a field. Let $A$ be an associative unitary algebra over $K$. Then a '''(linear) representation of $A$''' is a vector space $V$ over $K$ equipped with a homomorphism of algebras: :$\rho: A \to \map {\operatorname {End} } V$ where $\map {\operatorname {End} } V$ is the endomorphism ring of $V$.


Definition:Scalar Triple Product

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k }} {{eqn | l = \mathbf b       | r = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k }} {{eqn | l = \mathbf c       | r = c_i \mathbf i + c_j \mathbf j + c_k \mathbf k }} {{end-eqn}} where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$. The '''scalar triple product''', denoted as $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$, is defined as: :$\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = \begin {vmatrix} a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \\ \end {vmatrix}$


Definition:Singleton Graph

The '''singleton graph''' $N_1$ is the simple graph with one vertex: :60px


Definition:Complete Distributive Lattice

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Then $S$ is a '''complete distributive lattice''' {{iff}} :$(1): \quad S$ is a complete lattice :$(2): \quad S$ is a distributive lattice. Category:Definitions/Lattice Theory 19ry08xqa1fj2agz8s5bokuc36ohn0t


Definition:Cauchy Sequence/Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$. Then $\sequence {z_n}$ is a '''Cauchy sequence''' {{iff}}: : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {z_n - z_m} < \epsilon$ where $\size {z_n - z_m}$ denotes the complex modulus of $z_n - z_m$.


Definition:Commutative and Unitary Ring/Axioms

A commutative and unitary ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in R         | m = a * b \in R         | lc= Closure under addition }} {{axiom | n = \text A 1         | q = \forall a, b, c \in R         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \text A 2         | q = \forall a, b \in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \text A 3         | q = \exists 0_R \in R: \forall a \in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \text A 4         | q = \forall a \in R: \exists a' \in R         | m = a * a' = 0_R = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = \text M 0         | q = \forall a, b \in R         | m = a \circ b \in R         | lc= Closure under product }} {{axiom | n = \text M 1         | q = \forall a, b, c \in R         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = \text M 2         | q = \forall a, b \in R         | m = a \circ b = b \circ a         | lc= Commutativity of product }} {{axiom | n = \text M 3         | q = \exists 1_R \in R: \forall a \in R         | m = a \circ 1_R = a = 1_R \circ a         | lc= Identity element for product: the unity }} {{axiom | n = \text D         | q = \forall a, b, c \in R         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{end-axiom}} These criteria are called the '''commutative and unitary ring axioms'''.


Definition:Accumulation Point/Set

Let $x \in S$. Then $x$ is an '''accumulation point''' of $A$ {{iff}}: :$x \in \cl {A \setminus \set x}$ where $\operatorname{cl}$ denotes the (topological) closure of a set.


Definition:Set Derivative

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \subseteq S$. The '''derivative''' of $A$ in $T$ is the set of all accumulation points of $A$. It is usually denoted on {{ProofWiki}} as $A'$.


Definition:Pythagoras's Constant

'''Pythagoras's constant''' is sometimes used to mean $\sqrt 2$, the square root of $2$. === Decimal Expansion === {{:Square Root of 2}} {{NamedforDef|Pythagoras of Samos|cat = Pythagoras}} Category:Number Theory m3ug0rekc6r54r0n4b62jnacyopnvvr


Definition:Existential Quantifier/Unique

The symbol $\exists !$ denotes the existence of a unique object fulfilling a particular condition. :$\exists ! x: \map P x$ means: :'''There exists exactly one object $x$ such that $\map P x$ holds''' or: :'''There exists one and only one $x$ such that $\map P x$ holds'''. Formally: :$\exists ! x: \map P x \dashv \vdash \exists x: \map P x \land \forall y: \map P y \implies x = y$ In natural language, this means: ::''There exists exactly one $x$ with the property $P$'' : is logically equivalent to: ::''There exists an $x$ such that $x$ has the property $P$, and for every $y$, $y$ has the property $P$ only if $x$ and $y$ are the same object.''


Definition:Existential Quantifier/Exact

The symbol $\exists_n$ denotes the existence of an exact number of objects fulfilling a particular condition. :$\exists_n x: \map P x$ means: :'''There exist exactly $n$ objects $x$ such that $\map P x$ holds'''.


Definition:Set Union/General Definition

Let $S$ be a collection, which could be either a set or a class. The '''union of $S$''' is: :$\ds \bigcup S := \set {x: \exists X \in S: x \in X}$ That is, the set of all elements of all elements of $S$ which are themselves sets.


Definition:Ordered Pair/Informal Definition

An '''ordered pair''' is a two-element set together with an ordering. In other words, one of the elements ''is'' distinguished above the other - it comes first. Such a structure is written: :$\tuple {a, b}$ and it means: : '''first $a$, then $b$'''.


Definition:Ordered Pair/Kuratowski Formalization

The concept of an ordered pair can be formalized by the definition: :$\tuple {a, b} := \set {\set a, \set {a, b} }$ This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory.


Definition:Inverse Mapping/Also defined as

Let $S$ and $T$ be sets. Let $f: S \to T$ be an injection. Then its '''inverse mapping''' is the mapping $g$ such that: :its domain $\Dom g$ equals the image $\Img f$ of $f$ :$\forall y \in \Img f: \map f {\map g y} = y$ Thus $f$ is seen to be a surjection by tacit use of Restriction of Mapping to Image is Surjection.


Definition:Weight of Topological Space

Let $T$ be a topological space. Let $\mathbb B$ be the set of all bases of $T$. === Definition 1 === {{:Definition:Weight of Topological Space/Definition 1}} === Definition 2 === {{:Definition:Weight of Topological Space/Definition 2}}


Definition:Operation Compatible with Equivalence

Let $F$ be a (unary) operation which can be applied to sets. Then $F$ is '''compatible with set equivalence''' {{iff}}: :$F \left({A}\right) = F \left({B}\right) \iff A \sim B$ where: :$A$ and $B$ are arbitrary sets :$\sim$ denotes set equivalence.


Definition:Weight of Topological Space/Definition 1

The '''weight''' of $T$ is defined as: :$\displaystyle \map w T := \bigcap_{\BB \mathop \in \mathbb B} \card \BB$ where $\card \BB$ denotes the cardinality of $\BB$.


Definition:Weight of Topological Space/Definition 2

The '''weight''' of $T$ is the smallest cardinality of the elements of $\mathbb B$: :$\map w T := \min \set {\card \BB: \BB \in \mathbb B}$


Definition:Character of Point in Topological Space

Let $T$ be a topological space. Let $x$ be a point of $T$. Let $\mathbb B \left({x}\right)$ be the set of all local bases at $x$. The '''character of (the point) $x$ in $T$''' is the smallest cardinality of the elements of $\mathbb B \left({x}\right)$: :$\chi \left({x,T}\right) := \min \left\{ {\left\vert{\mathcal B}\right\vert: \mathcal B \in \mathbb B \left({x}\right)}\right\}$


Definition:Character of Topological Space

Let $T$ be a topological space. The '''character of (the topological space) $T$''' is the supremum of the set of characters of the points of $T$: :$\chi \left({T}\right) := \sup \left\{ {\chi \left({x, T}\right): x \in T}\right\}$


Definition:Bound Variable/Examples/Universal Statement

In the universal statement: :$\forall x: \map P x$ the symbol $x$ is a '''bound variable'''. Thus, the meaning of $\forall x: \map P x$ does not change if $x$ is replaced by another symbol. That is, $\forall x: \map P x$ means the same thing as $\forall y: \map P y$ or $\forall \alpha: \map P \alpha$. And so on.


Definition:Bound Variable/Examples/Existential Statement

In the existential statement: :$\exists x: \map P x$ the symbol $x$ is a '''bound variable'''. Thus, the meaning of $\exists x: \map P x$ does not change if $x$ is replaced by another symbol. That is, $\exists x: \map P x$ means the same thing as $\exists y: \map P y$ or $\exists \alpha: \map P \alpha$. And so on.


Definition:Existential Statement/Conditional

A '''conditionally existential statement''' is an existential statement which states the existence of an object fulfilling a certain propositional function dependent upon the existence of certain other objects.


Definition:Conditional/Formal Implication

'''Formal implication''' is a usage of an implication in which it is necessary for there to be a formal connection between the antecedent and the consequent in order for the implication to have any semantic meaning.


Definition:Stoic School

The '''stoic school''' was a school of Greek philosophy which emphasized the importance of goodness and peace of mind gained from living a life of Virtue in accordance with Nature. The writings of its philosophers reveal the first beginnings of the discipline of propositional logic.


Definition:Truth Table/Matrix Presentation

A two-element truth table can be presented in '''matrix form''': :{{:Definition:Conditional/Truth Table/Matrix Form}}$\qquad${{:Definition:Conjunction/Truth Table/Matrix Form}}$\qquad${{:Definition:Disjunction/Truth Table/Matrix Form}}$\qquad${{:Definition:Biconditional/Truth Table/Matrix Form}} that is, in the form of a Cayley table. This, however, can be used only when the operation being displayed has two elements.


Definition:Fundamental Truth Table

The '''fundamental truth tables''' are the characteristic truth tables for: : Logical Not : Conjunction : Disjunction : Conditional : Biconditional as follows: === Logical Not === {{:Definition:Logical Not/Truth Table}} === Conjunction === {{:Definition:Conjunction/Truth Table}} === Disjunction === {{:Definition:Disjunction/Truth Table}} === Conditional === {{:Definition:Conditional/Truth Table}} === Biconditional === {{:Definition:Biconditional/Truth Table}}


Definition:Derivative Truth Table

A '''derivative truth table''' is a truth table for a statement form which is not mapped by one of the fundamental truth tables: : Logical Not : Conjunction : Disjunction : Conditional : Biconditional


Definition:Truth Table/Row

A '''row''' of a truth table is one of the horizontal lines that consists of instances of the symbols $T$ and $F$. Each '''row''' contains the truth values of each of the boolean interpretations of the statement forms according to the propositional variables that comprise them. There are as many '''rows''' in a truth table as there are combinations of $T$ and $F$ for all the propositional variables that constitute the statement forms.


Definition:Truth Table/Column

A '''row''' of a truth table is one of the vertical lines headed by a statement form presenting all the truth values that the statement form takes. Each entry in the '''column''' corresponds to one specific combination of truth values taken by the propositional variables that the statement form comprises.


Definition:Conjugate Statements

Let $p$ and $q$ be statements. The conditional statement: :$p \implies q$ its inverse: :$\lnot p \implies \lnot q$ its converse: :$q \implies p$ and its contrapositive: :$\lnot q \implies \lnot p$ are all '''conjugate statements'''.


Definition:Rubik's Cube

'''Rubik's cube''' is a toy consisting of a plastic cube divided into $27$ smaller cubes, interlocking at the center cube and allowing each face to rotate independently of the others. Each of the $6$ faces of the main cube is colored with one of six contrasting colors. Rotating the faces of the cube causes these $6$ faces to become mixed up. The object of the exercise is to restore the $6$ faces to their original single colors.


Definition:Huntington Algebra

A '''Huntington algebra''' is an algebraic structure $\struct {S, \circ, *}$ such that: {{begin-axiom}} {{axiom | n = \text {HA} 0         | lc=         | t =$S$ is closed under both $\circ$ and $*$ }} {{axiom | n = \text {HA} 1         | lc=         | t = Both $\circ$ and $*$ are commutative }} {{axiom | n = \text {HA} 2         | lc=         | t = Both $\circ$ and $*$ distribute over the other }} {{axiom | n = \text {HA} 3         | lc=         | t = Both $\circ$ and $*$ have identities $e^\circ$ and $e^*$ respectively, where $e^\circ \ne e^*$ }} {{axiom | n = \text {HA} 4         | lc=         | t = $\forall a \in S: \exists a' \in S: a \circ a' = e^*, a * a' = e^\circ$ }} {{end-axiom}} The element $a'$ in $(\text {HA} 4)$ is often called the '''complement''' of $a$. A '''Huntington algebra''' can also be considered as a mathematical system $\set {S, O, A}$ where $O = \set {\circ, *}$ and $A$ consists of the set of axioms $(\text {HA}  0)$ to $(\text {HA}  4)$ as defined above.


Definition:Cardinality/Finite

Let $S$ be a finite set. The '''cardinality''' $\card S$ of $S$ is the '''number of elements in $S$'''. That is, if: :$S \sim \N_{< n}$ where: :$\sim$ denotes set equivalence :$\N_{<n}$ is the set of all natural numbers less than $n$ then we define: :$\card S = n$


Definition:Rubik's Cube/Basic Moves

The set of '''basic moves''' of the Rubik's cube can be denoted: :$\set {U, D, L, R, F, B}$ where $U$, $D$, $L$, $R$, $F$, $B$ are in Singmaster notation.


Definition:Singmaster Notation

Let the cube be facing the user with the $U$ and $D$ faces horizontal and the $F$ face towards the user. In the following, '''clockwise''' denotes rotation of a layer of the cube in the direction of the clock '''as though that face were towards the user'''. :$U$: turns the up face one quarter turn clockwise. :$D$: turns the down face one quarter turn clockwise. :$L$: turns the left face one quarter turn clockwise. :$R$: turns the right face one quarter turn clockwise. :$F$: turns the front face one quarter turn clockwise. :$B$: turns the back face one quarter turn clockwise. Appending a $'$ symbol to one of the letters denotes a turn of that layer one quarter turn anticlockwise. Appending a $2$ to one of the letters denotes a half turn of that layer.


Definition:Rubik's Cube/Facet

Each of the $9$ small squares into which the faces of Rubik's cube is subdivided is called a '''facet'''.


Definition:Symmetric Difference/Definition 5

:$S * T := \set {x: x \in S \oplus x \in T}$


Definition:Rubik's Cube/Corner Facet

Of each of the $9$ facets of each face of Rubik's cube, the $4$ in the corners of the face are called the '''corner facets'''. The set of '''corner facets''' can be denoted $S_C$.


Definition:Rubik's Cube/Edge Facet

Of each of the $9$ facets of each face of Rubik's cube, the $4$ on the edges of the face are called the '''edge facets'''. The set of '''edge facets''' can be denoted $S_E$.


Definition:Rubik's Cube/Center Facet

Of each of the $9$ facets of each face of Rubik's cube, the ones in the centers of the faces are called the '''center facets'''. The set of '''center facets''' can be denoted $S_Z$.


Definition:Trivial Partition/Singleton

The '''singleton partition''' on $S$ is defined as: :$\mathcal P = \left\{{S}\right\}$ That is, it is a partition with only one component.


Definition:Trivial Partition/Partition of Singletons

The '''partition of singletons''' on $S$ is defined as: :$\mathcal P = \set {\set x: x \in S}$ That is, it is a partition such that every component is a singleton.


Definition:Prime Number/Definition 5

A '''prime number''' $p$ is an integer greater than $1$ that has no (positive) divisors less than itself other than $1$.


Definition:Prime-Counting Function/Examples

The values of the prime-counting ($\pi$) function for the first few integers are as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $\map \pi n$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $0$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $4$ |}


Definition:Prime-Counting Function/Examples/16

The value of the prime-counting ($\pi$) function for $16$ is determined as follows. The prime numbers less than $16$ are: :$2, 3, 5, 7, 11, 13$ Hence: :$\map \pi {16} = 6$


Definition:Mapping/Definition 1

A '''mapping''' from $S$ to $T$ is a binary relation on $S \times T$ which associates each element of $S$ with exactly one element of $T$.


Definition:Mapping/Definition 2

A '''mapping $f$ from $S$ to $T$''', denoted $f: S \to T$, is a relation $f = \struct {S, T, G}$, where $G \subseteq S \times T$, such that: :$\forall x \in S: \forall y_1, y_2 \in T: \tuple {x, y_1} \in G \land \tuple {x, y_2} \in G \implies y_1 = y_2$ and :$\forall x \in S: \exists y \in T: \tuple {x, y} \in G$


Definition:Mapping/Definition 3

A '''mapping $f$ from $S$ to $T$''', denoted $f: S \to T$, is a relation $f = \struct {S, T, R}$, where $R \subseteq S \times T$, such that: :$\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in f: y_1 \ne y_2 \implies x_1 \ne x_2$ and :$\forall x \in S: \exists y \in T: \tuple {x, y} \in f$


Definition:Mapping/Definition 4

A '''mapping from $S$ to $T$''' is a relation on $S \times T$ which is: :$(1): \quad$ Many-to-one :$(2): \quad$ Left-total, that is, defined for all elements in $S$.


Definition:Countable Set/Also known as

When the terms '''denumerable''' and '''enumerable''' are encountered, they generally mean the same as countably infinite. Some modern pedagogues (for example {{AuthorRef|Vi Hart}} and {{AuthorRef|James Grime}}) use the term '''listable''', but this has yet to catch on.


Definition:Countable Set/Countably Infinite/Cardinality

The cardinality of a countably infinite set is denoted by the symbol $\aleph_0$ ('''aleph null''').


Definition:Countable Set/Countably Infinite/Definition 1

$S$ is '''countably infinite''' {{iff}} there exists a bijection: :$f: S \to \N$ where $\N$ is the set of natural numbers.


Definition:Countable Set/Countably Infinite/Definition 2

$S$ is '''countably infinite''' {{iff}} there exists a bijection: :$f: S \to \Z$ where $\Z$ is the set of integers.


Definition:Countable Set/Also defined as

Some sources define a countable set to be what is defined on {{ProofWiki}} as a countably infinite set. That is, they use '''countable''' to describe a set which has ''exactly the same'' cardinality as $\N$. Thus under this criterion $X$ is said to be countable {{iff}} there exists a bijection from $X$ to $\N$, that is, {{iff}} $X$ is equivalent to $\N$. However, as the very concept of the term '''countable''' implies that a set '''can be counted''', which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive. Hence, on {{ProofWiki}}, the term countable set will be taken in the sense as to include the concept of finite set, and countably infinite will mean a countable set which is specifically ''not'' finite.


Definition:Countable Set/Definition 3

$S$ is '''countable''' {{iff}} there exists a bijection between $S$ and a subset of $\N$.


Definition:Polish Notation/Formal Definition

Let $\mathcal A$ be an alphabet. Let each $s \in \mathcal A$ be assigned a natural number called its arity. The formal grammar for Polish notation is given by the single bottom-up rule: ::If $s$ has arity $n$ and $\phi_1, \ldots, \phi_n$ are well-formed formulas, then: :::$s \phi_1 \cdots \phi_n$ ::is also a well-formed formula. Notably, in the case where $s$ has arity $0$, this is a vacuous truth, so any such $s$ constitutes a well-formed formula.


Definition:Matrix Theory

'''Matrix theory''' is the field of mathematics which studies matrices.


Definition:Unit Matrix/Examples/3 x 3

The '''$3 \times 3$ unit matrix''' is as follows: :$\mathbf I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$


Definition:Matrix/Lower Triangular Elements

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix. The elements $a_{i j}: i > j$ are called the '''lower triangular elements'''.


Definition:Matrix/Upper Triangular Elements

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix. The elements $a_{i j}: i < j$ are called the '''upper triangular elements'''.


Definition:Permutation Matrix

A '''permutation matrix (of order $n$)''' is an $n \times n$ square matrix with: : exactly one instance of $1$ in each row and column : $0$ elsewhere.


Definition:Monomial Matrix

A '''monomial matrix (of order $n$)''' is an $n \times n$ square matrix with: :exactly one element in each row and column which is not $0$ :$0$ elsewhere.


Definition:(0, 1)-Matrix

A '''$\tuple {0, 1}$-matrix''' is a matrix whose elements consist only of instances of $0$ and $1$.


Definition:Submatrix/Notation

A submatrix of $\mathbf A$ is denoted as follows. Let: : $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ be the indices of the $r$ selected rows : $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ be the indices of the $s$ selected columns where all of $a_1, \ldots, a_r$ are between $1$ and $m$, and all of $b_1, \ldots, b_s$ are between $1$ and $n$. Then the '''submatrix''' formed from rows $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ and columns $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ is denoted as: :$\mathbf A \left[{a_1, a_2, \ldots, a_r; b_1, b_2, \ldots, b_s}\right]$ It is usual to specify the rows and columns in ascending numerical order.


Definition:Submatrix/Notation/Order (m-1) x (n-1) Submatrix

Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $\left({m - 1}\right) \times \left({n - 1}\right)$. Then it is usual to denote $\mathbf B$ by indicating the (single) row and column of $\mathbf A$ which has been '''removed''', as follows: Let: : $a_j$ be the row of $\mathbf A$ which is not included in $\mathbf B$ : $b_k$ be the column of $\mathbf A$ which is not included in $\mathbf B$. Then the '''submatrix''' $\mathbf B$ formed from the remaining rows and columns of $\mathbf A$ can be denoted as: :$\mathbf A \left({a_j; b_k}\right)$


Definition:Determinant/Matrix/Order

The '''order''' of a determinant is defined as the order of the square matrix on which it is defined.


Definition:Minor of Determinant/Notation/Order n-1

Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $n - 1$. Let: :$j$ be the row of $\mathbf A$ which is not included in $\mathbf B$ :$k$ be the column of $\mathbf A$ which is not included in $\mathbf B$. Thus, let $\mathbf B := \map {\mathbf A} {j; k}$. Then $\map \det {\mathbf B}$ can be denoted: :$D_{i j}$ That is, $D_{i j}$ is the minor of order $n - 1$ obtained from $D$ by deleting all the elements of row $i$ and column $j$.


Definition:Minor of Determinant/Notation

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $D := \map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Let: :$\set {a_1, a_2, \ldots, a_k}$ be the indices of the $k$ selected rows of $\mathbf A$ :$\set {b_1, b_2, \ldots, b_k}$ be the indices of the $k$ selected columns of $\mathbf A$ where all of $a_1, \ldots, a_k$ and all of $b_1, \ldots, b_k$ are between $1$ and $n$. Let: :$\mathbf B := \mathbf A \sqbrk {a_1, a_2, \ldots, a_k; b_1, b_2, \ldots, b_k}$ be the '''submatrix''' formed from rows $\set {a_1, a_2, \ldots, a_k}$ and columns $\set {b_1, b_2, \ldots, b_k}$. The '''order-$k$ minor''' of $D$ formed from rows $r_1, r_2, \ldots, r_k$ and columns $s_1, s_2, \ldots, s_k$ can be denoted: :$\map D {a_1, a_2, \ldots, a_k \mid b_1, b_2, \ldots, b_k}$. Each element of $D$ is an order $1$ minor of $D$, and can be denoted: :$\map D {a_i \mid b_j}$


Definition:Cofactor/Element

Let $a_{r s}$ be an element of $D$. Let $D_{r s}$ be the determinant of order $n-1$ obtained from $D$ by deleting row $r$ and column $s$. Then the '''cofactor''' $A_{r s}$ of the element $a_{r s}$ is defined as: :$A_{r s} := \paren {-1}^{r + s} D_{r s}$


Definition:Cofactor/Minor

Let $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ be a order-$k$ minor of $D$. Then the '''cofactor''' of $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ can be denoted: :$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ and is defined as: :$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right) = \left({-1}\right)^t D \left({r_{k+1}, r_{k+2}, \ldots, r_n \mid s_{k+1}, s_{k+2}, \ldots, s_n}\right)$ where: : $t = r_1 + r_2 + \ldots + r_k + s_1 + s_2 + \ldots s_k$ : $r_{k+1}, r_{k+2}, \ldots, r_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{r_1, r_2, \ldots, r_k}\right\}$ : $s_{k+1}, s_{k+2}, \ldots, s_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{s_1, s_2, \ldots, s_k}\right\}$ That is, the '''cofactor of a minor''' is the determinant formed from the rows and columns not in that minor, multiplied by the appropriate sign. When $k = 1$, this reduces to the cofactor of an element (as above). When $k = n$, the "minor" is in fact the whole determinant. For convenience its '''cofactor''' is defined as being $1$. Note that the '''cofactor''' of the '''cofactor''' of a minor is the minor itself (multiplied by the appropriate sign).


Definition:Non-Invertible Matrix/Definition 1

Let $\mathbf A$ have no inverse. Then $\mathbf A$ is referred to as '''non-invertible'''.


Definition:Non-Invertible Matrix/Definition 2

Let the determinant of $\mathbf A$ be equal to $0$. Then $\mathbf A$ is referred to as '''non-invertible'''.


Definition:Zero Vector/Euclidean Space

Let $\struct {\R^n, +, \times}_\R$ be a real vector space. The '''zero vector''' in $\struct {\R^n, +, \times}_\R$ is: :$\mathbf 0_{n \times 1} := \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$ where $0 \in \R$.


Definition:Congruence (Number Theory)/Integers/Remainder after Division

'''Congruence modulo $m$''' is defined as the relation $\equiv \pmod m$ on the set of all $a, b \in \Z$: :$a \equiv b \pmod m := \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$ That is, such that $a$ and $b$ have the same remainder when divided by $m$.


Definition:Congruence (Number Theory)/Remainder after Division

We define a relation $\mathcal R_z$ on the set of all $x, y \in \R$: :$\mathcal R_z := \left\{{\left({x, y}\right) \in \R \times \R: \exists k \in \Z: x = y + k z}\right\}$ This relation is called '''congruence modulo $z$''', and the real number $z$ is called the '''modulus'''. When $\left({x, y}\right) \in \mathcal R_z$, we write: :$x \equiv y \pmod z$ and say: : '''$x$ is congruent to $y$ modulo $z$'''.


Definition:Congruence (Number Theory)/Historical Note

The concept of '''congruence modulo an integer''' was first explored by {{AuthorRef|Carl Friedrich Gauss}}. He originated the notation $a \equiv b \pmod m$ in his work {{BookLink|Disquisitiones Arithmeticae|Carl Friedrich Gauss}}, published in $1801$.


Definition:Equivalence Class/Notation

The notation used to denote an equivalence class varies throughout the literature, but is often some variant on the square bracket motif $\eqclass x \RR$. Other variants: * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts}} uses $\overline x$ for $\eqclass x \RR$. * {{BookReference|Introduction to Topology|1962|Bert Mendelson}} uses $\map \pi x$ for $\eqclass x \RR$. * {{BookReference|Sets and Groups|1965|J.A. Green}} uses $E_x$ for $\eqclass x \RR$. * {{BookReference|Modern Algebra|1965|Seth Warner}} uses $\bigsqcup_\RR \mkern {-28 mu} {\raise 1pt x} \ \ $ for $\eqclass x \RR$, which is even more challenging to render in our installed version of $\LaTeX$ than $\eqclass x \RR$ itself. * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson}} uses $\map {p_\RR} x$. * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}} uses $x / \RR$ for $\eqclass x \RR$ (compare the notation for quotient set). This source also suggests $x_\RR$ as a variant.


Definition:Equivalence Class/Representative

Let $\eqclass x \RR$ be the equivalence class of $x$ under $\RR$. Let $y \in \eqclass x \RR$. Then $y$ is a '''representative of $\eqclass x \RR$.


Definition:Word (Natural Language)

A '''word''' in natural language is intuitively understood as a sequence of sounds which expresses a concept. When written down, it appears as a sequence of letters, each one of which either is, or contributes to, a [https://en.wikipedia.org/wiki/Phoneme phoneme].


Definition:Scope of Occurrence

Let $\FF$ be a formal language with alphabet $\AA$. Let $\mathbf A$ be a well-formed formula of $\FF$. Let $a$ be a fixed occurrence of some element of $\AA$ in $\mathbf A$. Then the '''scope''' of $a$ is the smallest well-formed part of $\mathbf A$ that contains $a$.


Definition:Universal Closure of Well-Formed Formula

Let $\mathcal L_1$ be the language of predicate logic. Let $\mathbf A$ be a well-formed formula of $\mathcal L_1$. A '''universal closure''' of $\mathbf A$ is a sentence $\mathbf B$ of $\mathcal L_1$ of the form: :$\forall x_1: \cdots \forall x_n: \mathbf A$ By definition of sentence, this means that at least the variables occurring freely in $\mathbf A$ should be quantified over.


Definition:Conjunction/Truth Table/Number

The truth table number of the conjunction operator $p \land q$ is as follows: Ascending order: :$0001$ or $\F \F \F \T$ Descending order: :$1000$ or $\T \F \F \F$


Definition:Truth Table/Number

The '''truth table number''' of the truth table of a statement form is a string consisting of the truth values in numerical form of the main connective. The order in which the '''truth table number''' is presented is usually in ascending or descending numerical order of the truth values of the simple statements contributing to that statement form, either: :Ascending order: $00, 01, 10, 11$, that is: $FF, FT, TF, TT$ or: :Descending order: $11, 10, 01, 00$, that is: $TT, TF, FT, FF$ Which format is being used needs to be specified. Hence the '''truth table number''' can be found by reading either '''up''' or '''down''' the column holding the main connective in the truth table of the statement form.


Definition:Conditional/Truth Table/Number

The truth table number of the conjunction operator $p \land q$ is as follows: Ascending order: :$1101$ or $\T \T \F \T$ Descending order: :$1011$ or $\T \F \T \T$


Definition:Disjunction/Truth Table/Number

The truth table number of the conjunction operator $p \lor q$ is as follows: Ascending order: :$0111$ or $\F \T \T \T$ Descending order: :$1110$ or $\T \T \T \F$


Definition:Biconditional/Truth Table/Number

The truth table number of the biconditional operator $p \iff q$ is as follows: Ascending and descending order: :$1001$ or $\T \F \F \T$


Definition:Exclusive Or/Truth Table/Number

The truth table number of the exclusive or operator $p \oplus q$ is as follows: Ascending and descending order: :$0110$ or $\F \T \T \F$


Definition:Structure for Predicate Logic

Let $\LL_1$ be the language of predicate logic. A '''structure $\AA$ for $\LL_1$''' comprises: :$(1): \quad$ A non-empty set $A$; :$(2): \quad$ For each function symbol $f$ of arity $n$, a mapping $f_\AA: A^n \to A$; :$(3): \quad$ For each predicate symbol $p$ of arity $n$, a mapping $p_\AA: A^n \to \Bbb B$ where $\Bbb B$ denotes the set of truth values. $A$ is called the '''underlying set''' of $\AA$. $f_\AA$ and $p_\AA$ are called the '''interpretations''' of $f$ and $p$ in $\AA$, respectively.


Definition:Assignment for Structure/Formula

Let $\mathbf A$ be a well-formed formula of $\mathcal L_1$. Denote with $V \left({\mathbf A}\right)$ the variables which occur freely in $\mathbf A$. An '''assignment for $\mathbf A$ in $\mathcal A$''' is a mapping $\sigma$ with codomain $A$, whose domain is subject to the following condition: :$V \left({\mathbf A}\right) \subseteq \operatorname{dom} \left({\sigma}\right) \subseteq \mathrm{VAR}$ That is, the domain of $\sigma$ contains only variables, and at least those with a free occurrence in $\mathbf A$.


Definition:Assignment for Structure/Term

Let $\tau$ be a term of $\mathcal L_1$. Denote with $V \left({\tau}\right)$ the variables which occur in $\tau$. An '''assignment for $\tau$ in $\mathcal A$''' is a mapping $\sigma$ with codomain $A$, whose domain is subject to the following condition: :$V \left({\tau}\right) \subseteq \operatorname{dom} \left({\sigma}\right) \subseteq \mathrm{VAR}$ That is, the domain of $\sigma$ contains only variables, and at least those which occur in $\tau$.


Definition:Language of Predicate Logic/Formal Grammar/Term

The '''terms''' of $\LL_1$ are identified by the following bottom-up grammar: {{begin-axiom}} {{axiom | n = \mathbf T \ \textrm {VAR}         | t = Any variable of $\LL_1$ is a '''term'''; }} {{axiom | n = \mathbf T \ \FF_n         | t = Given an $n$-ary function symbol $f \in \FF_n$ and '''terms''' $\tau_1, \ldots, \tau_n$: :$\map f {\tau_1, \ldots, \tau_n}$ is also a '''term'''. }} {{end-axiom}}


Definition:Local Minimum in Set of Reals

Let $X$ be a subset of $\R$, the set of all real numbers. Let $x$ be a real number, $x \in \R$. $x$ is '''local minimum in set''' $X$ {{iff}} :$x \in X \land \exists y \in \R: y < x \land \left({y \,.\,.\, x}\right) \cap X = \varnothing$


Definition:Variable/Restricted

A '''restricted variable''' is a variable whose values are confined to some only of those of which it is capable.


Definition:Variable/Unrestricted

An '''unrestricted variable''' is a variable whose values are not confined in any way to some only of those of which it is capable.


Definition:Prime Number/Definition 6

Let $p \in \N$ be an integer such that $p \ne 0$ and $p \ne \pm 1$. Then $p$ is a '''prime number''' {{iff}} :$\forall a, b \in \Z: p \divides a b \implies p \divides a$ or $p \divides b$ where $\divides$ means '''is a divisor of'''.


Definition:Twin Primes

'''Twin primes''' are pairs of prime numbers whose difference is $2$.


Definition:Quadratic Residue/Non-Residue

If there is no such integer $x$ such that $x^2 \equiv a \pmod p$, then $a$ is a '''quadratic non-residue of $p$'''.


Definition:Quadratic Residue/Character

$a$ is either a quadratic residue or a quadratic non-residue of $p$. Whether it is or not is known as the '''quadratic character of $a$ modulo $p$'''.


Definition:Heptagon

A '''heptagon''' is a polygon with exactly $7$ sides. :300px === Regular Heptagon === {{:Definition:Heptagon/Regular}}


Definition:Octagon

An '''octagon''' is a polygon with exactly $8$ sides. :300px === Regular Octagon === {{:Definition:Octagon/Regular}}


Definition:Nonagon

A '''nonagon''' is a polygon with exactly $9$ sides. :300px === Regular Nonagon === {{:Definition:Nonagon/Regular}}


Definition:Hendecagon

A '''hendecagon''' is a polygon with exactly $11$ sides. :300px === Regular Hendecagon === {{:Definition:Hendecagon/Regular}}


Definition:Dodecagon

A '''dodecagon''' is a polygon with exactly $12$ sides. :300px === Regular Dodecagon === {{:Definition:Dodecagon/Regular}}


Definition:Heptadecagon

A '''heptadecagon''' is a polygon with exactly $17$ sides. :300px === Regular Heptadecagon === {{:Definition:Heptadecagon/Regular}}


Definition:Pentagon/Regular

A '''regular pentagon''' is a pentagon which is both equilateral and equiangular. That is, a regular polygon with $5$ sides. That is, a pentagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Hexagon/Regular

A '''regular hexagon''' is a hexagon which is both equilateral and equiangular. That is, a regular polygon with $6$ sides. That is, a hexagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Heptagon/Regular

A '''regular heptagon''' is a heptagon which is both equilateral and equiangular. That is, a regular polygon with $7$ sides. That is, a heptagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Octagon/Regular

A '''regular octagon''' is an octagon which is both equilateral and equiangular. That is, a regular polygon with $8$ sides. That is, an octagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Nonagon/Regular

A '''regular nonagon''' is a nonagon which is both equilateral and equiangular. That is, a regular polygon with $9$ sides. That is, a nonagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Decagon/Regular

A '''regular decagon''' is a decagon which is both equilateral and equiangular. That is, a regular polygon with $10$ sides. That is, a decagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Hendecagon/Regular

A '''regular hendecagon''' is a hendecagon which is both equilateral and equiangular. That is, a regular polygon with $11$ sides. That is, a hendecagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Heptadecagon/Regular

A '''regular heptadecagon''' is a heptadecagon which is both equilateral and equiangular. That is, a regular polygon with $17$ sides. That is, a heptadecagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Dodecagon/Regular

A '''regular dodecagon''' is a dodecagon which is both equilateral and equiangular. That is, a regular polygon with $12$ sides. That is, a dodecagon in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:RSA Algorithm

The '''RSA algorithm''' is a technique of encoding a message such that the method of encoding can be made public without compromising the security. Let Alice be sending a message to Bob. Alice and Bob both agree on two large primes $p$ and $q$, each having at least $100$ digits. Let $M = p q$. $M$ can be made public if they so wish. Let $K = \left({p - 1}\right) \left({q - 1}\right)$. $K$ is to be kept secret. Let Alice choose some number $a$ such that $a \perp K$. $a$ is made known to Bob, and may even be made public. Alice represents her message in a series of numbers in the range $0$ to $M$. Let $x$ be one such number. Alice calculates $y \equiv x^a \pmod M$. The sequence of numbers $y$ is sent to Bob as the coded message. Bob needs to know a number $b$ such that $a b \equiv 1 \pmod K$. This number needs to be kept secret. To decode $y$, Bob computes $y^b \pmod M$. This works because of Fermat's Little Theorem: :$y^b \equiv \left({x^a}\right)^b \equiv x^1 = x \pmod M$ The method works because: :$(1): \quad$ There are efficient tests to find large primes :$(2): \quad$ There are no known methods for finding the prime factors of large numbers efficiently. So making $p q$ public does not help $p$ and $q$ be found, and without those it is impossible to work out what $b$ is. {{NamedforDef|Ronald Linn Rivest|name2 = Adi Shamir|name3 = Leonard Max Adleman|cat = Rivest|cat2 = Shamir|cat3 = Adleman}}


Definition:Cryptography

'''Cryptography''' is the branch of mathematics which concerns the encoding of information for secrecy and security.


Definition:Real Function/Multivariable

Let $f: S_1 \times S_2 \times \cdots \times S_n \to \R$ be a mapping where $S_1, S_2, \ldots, S_n \subseteq \R$. Then $f$ is defined as a '''(real) function of $n$ (independent) variables'''.  The expression: :$y = \map f {x_1, x_2, \ldots, x_n}$ means: :(The dependent variable) $y$ is a function of (the independent variables) $x_1, x_2, \ldots, x_n$.


Definition:Derivative/Real Function/Derivative at Point/Definition 1

That is, suppose the limit $\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ exists. Then this limit is called the '''derivative of $f$ at the point $\xi$'''.


Definition:Derivative/Real Function/Derivative at Point/Definition 2

That is, suppose the limit $\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists. Then this limit is called the '''derivative of $f$ at the point $\xi$'''.


Definition:Secant Line

Let $f: \R \to \R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px A '''secant''' to $f$ is a straight line which intersects the graph of $f$ in (at least) two points. In the above diagram, the '''secant''' is the line $AB$ in $\color {blue} {\text {blue} }$.


Definition:Tangent/Analytic Geometry

Let $f: \R \to \R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px Let $A = \tuple {x, \map f x}$ be a point on $G$. The '''tangent to $f$ at $A$''' is defined as: :$\displaystyle \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$ Thus '''tangent to $f$ at $x$''' can be considered as the secant $AB$ to $G$ where: :$B = \tuple {x + h, \map f {x + h} }$ as $B$ gets closed and closer to $A$. By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$. Hence the '''tangent''' to $f$ is a straight line which intersects the graph of $f$ locally at a single point. :400px In the above diagram, the '''tangent''' is the straight line passing through $A$.


Definition:Celestial Mechanics

'''Celestial mechanics''' is the branch of astronomy which studies the motion of celestial bodies such as planets, stars and comets.


Definition:Orbit (Physics)

The '''orbit''' of body $A$ around body $B$ is the path taken by $A$ as it travels around $B$ under the influence of the force acting between $A$ and $B$. === Period of Orbit === {{:Definition:Orbit (Physics)/Period}}


Definition:Orbit (Physics)/Period

The '''period''' of an orbit of body $A$ around body $B$ is the length of time it takes for $A$ to travel once around $B$ and return to its original position.


Definition:Astrophysics

'''Astrophysics''' is the branch of physics which studies the internal nature of celestial bodies as opposed to their positions and movements.


Definition:Celestial Body

A '''celestial body''' is a more-or-less internally coherent body of matter that exists outside the Earth's atmosphere.


Definition:Planet

A '''planet''' is a celestial body which orbits a star (or the remains of a star) that: :$(1): \quad$ has enough mass so that its own gravitational field makes it approximately spherical :$(2): \quad$ does not have enough mass to trigger thermonuclear fusion :$(3): \quad$ has cleared its neighbouring region of planetesimals.


Definition:Star (Physics)

A '''star''' is a celestial body that consists of a luminous sphere of plasma held together by its own gravitational field.


Definition:Sun

The '''Sun''' is the star around which Earth (and the other planets of the solar system) is in orbit.


Definition:Earth

'''Earth''' is the planet on which we live. It is the third planet of the solar system from the sun. Its orbit lies between those of Venus and Mars.


Definition:Solar System

The '''solar system''' is the system of celestial bodies which are under the direct influence of the gravitational field of the sun.


Definition:Mercury (Planet)

'''Mercury''' is the innermost planet of the solar system. Its orbit lies within that of Venus. Category:Definitions/Solar System plp5e2aklqfluk6yspee40yd9qjgu7b


Definition:Venus

'''Venus''' is the second planet of the solar system from the sun. Its orbit lies between those of Mercury and Earth. Category:Definitions/Solar System 24j4o8crdk8571medw9pcgt6ngffo04


Definition:Mars

'''Mars''' is the fourth planet of the solar system from the sun. Its orbit lies between those of Earth and Jupiter.


Definition:Jupiter

'''Jupiter''' is the fifth planet of the solar system from the sun. Its orbit lies between those of Mars and Saturn.


Definition:Saturn

'''Saturn''' is the sixth planet of the solar system from the sun. Its orbit lies between those of Jupiter and Uranus. Category:Definitions/Solar System 158qpaeeuovkq9yz648dncuu4nzyb1r


Definition:Uranus

'''Uranus''' is the seventh planet of the solar system from the sun. Its orbit lies between those of Saturn and Neptune. Category:Definitions/Solar System mavyjtjplsijg8xj0d8uvoilw3g6ur3


Definition:Neptune

'''Neptune''' is the eighth planet of the solar system from the sun. Its orbit lies mainly between those of Uranus and Pluto, although the latter comes closer to the sun than '''Neptune''' on part of its orbit.


Definition:Pluto

'''Pluto''' is a dwarf planet of the solar system. Its orbit lies mostly outside that of Neptune. Category:Definitions/Solar System jljizc0dsjw0y2cdo2fyfxxb9rbpteb


Definition:Dwarf Planet

A '''dwarf planet''' is a celestial body which orbits a star (or the remains of a star) that: :$(1): \quad$ has enough mass so that its own gravitational field makes it approximately spherical :$(2): \quad$ does not have enough mass to trigger thermonuclear fusion :$(3): \quad$ has not cleared its neighbouring region of planetesimals.


Definition:Planetesimal

A '''planetesimal''' is a celestial body consisting of solid matter such that: :$(1): \quad$ it is held together by its own gravitational field :$(2): \quad$ it is not large enough for its gravitational field to significantly affect other celestial bodies nearby.


Definition:Time/Dimension

'''Time''' is one of the fundamental dimensions of physics. In dimensional analysis it is assigned the symbol $T$ or $\mathbf T$. It is a scalar quantity which can be mapped directly to the real number line.


Definition:Time/Unit Time

The usual units of '''time''' are seconds, and compound units consisting of multiples of seconds. However, when discussing a general physical process it is convenient to discuss the general situation, in which case the term '''unit time''' is used. Thus the specific '''units''' are not mentioned.


Definition:Time/Length

A quantity of '''time''' is usually referred to in natural language in the same terms as linear measure. Thus it is usual to refer to a '''length of time''' of an elapsed quantity of time.


Definition:Time/Unit

The units of measurement of time are universal in all systems of measurement. === Second === {{:Definition:Time/Unit/Second}} Derived units of time include: === Minute === {{:Definition:Time/Unit/Minute}} === Hour === {{:Definition:Time/Unit/Hour}} === Day === {{:Definition:Time/Unit/Day}} There are longer time periods, all of which are more or less approximate: === Week === {{:Definition:Time/Unit/Week}} === Fortnight === {{:Definition:Time/Unit/Fortnight}} === Month === {{:Definition:Time/Unit/Month}} === Lunar Month === {{:Definition:Time/Unit/Lunar Month}} === Season === {{:Definition:Time/Unit/Season}} === Quarter-Year === {{:Definition:Time/Unit/Quarter-Year}} === Year === {{:Definition:Time/Unit/Year}} === Decade === {{:Definition:Time/Unit/Decade}} === Century === {{:Definition:Time/Unit/Century}} === Millennium === {{:Definition:Time/Unit/Millennium}} === Great Year of Plato === {{:Definition:Time/Unit/Great Year of Plato}}


Definition:Derived Unit

'''Derived units''' are units of measurement created by taking specific multiples or fractions of fundamental units.


Definition:System of Measurement

A '''system of measurement''' is a set of fundamental units of measurement with which one can measure any measurable physical property.


Definition:Time/Unit/Minute

The '''minute''' is a derived unit of time. :$1$ '''minute''' $= 60$ seconds.


Definition:Time/Unit/Second/Symbol

The symbol for the '''second''' is $\mathrm s$.


Definition:Time/Unit/Minute/Symbol

The symbol for the '''minute''' is $\mathrm {min}$.


Definition:Time/Unit/Hour/Symbol

The symbol for the '''hour''' is $\mathrm {hr}$.


Definition:Time/Unit/Hour

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hour''' }} {{eqn | r = 60       | c = minutes }} {{eqn | r = 60 \times 60       | rr= = 3600       | c = seconds }} {{end-eqn}}


Definition:Time/Unit/Day

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''day''' }} {{eqn | r = 24       | c = hours }} {{eqn | r = 60 \times 24       | rr= = 1440       | c = minutes }} {{eqn | r = 60 \times 60 \times 24       | rr= = 86\, 400       | c = seconds }} {{end-eqn}}


Definition:Time/Unit/Day/Symbol

The symbol for the '''day''' is $\mathrm {day}$ or $\mathrm d$.


Definition:Sidereal Day

A '''sidereal day''' is (approximately) the length of time it takes for Earth to rotate once on its axis relative to the stars. The mean duration of a '''sidereal day''' is: : $23$ hours, $56$ minutes and $4.0916$ seconds : $23.9344699$ hours : $0.99726958$ (solar) days. The duration of any given '''sidereal day''' is not constant because of various astronomical effects.


Definition:Algorithm/Input

An '''algorithm''' has zero or more '''inputs'''. These are values supplied either: : before the algorithm starts : as the algorithm runs. These '''inputs''' are taken from specified sets of objects.


Definition:Algorithm/Output

An '''algorithm''' has one or more '''outputs'''. These are values which are specifically determined by the inputs.


Definition:Computational Problem

A '''computational problem''' is a mathematical object which represents a collection of questions that are to be solved by algorithmic techniques.


Definition:Algorithm/Finiteness

An '''algorithm''' must '''terminate''' after a finite number of '''steps'''.


Definition:Algorithm/Definiteness

Each '''step''' of an algorithm must be '''precisely defined'''.


Definition:Algorithm/Effectiveness

An '''algorithm''' is supposed to be '''effective'''. That is, its '''operations''' must be basic enough to be able to be done '''exactly''' and in a finite length of time by, for example, somebody using pencil and paper.


Definition:Algorithm/Analysis

There are two primary methods for analyzing '''algorithms''' formally: === Correctness === {{:Definition:Algorithm/Analysis/Correctness}} === Complexity === {{:Definition:Algorithm/Analysis/Complexity}}


Definition:Algorithm/Analysis/Complexity

An algorithm may be analysed in the context of its '''complexity'''. This is typically via measures of run-times using big-O, or big-omega, or big-theta notation. Complexity can be through 3 main types of analysis: # Average-Case Analysis, using a distribution of cases to find an average case of the algorithm run-time for analysis # Best-Case Analysis, using the best possible problem instance of the algorithm for analysis # Worst-Case Analysis,using the worst possible problem instance of the algorithm for analysis


Definition:Stationary

A body is '''stationary''' if its velocity (in a given frame of reference) is zero.


Definition:Infinitesimal

An '''infinitesimal''' is a mathematical object $\delta$ resembling a (real) number such that: :$(1): \quad \delta > 0$ :$(2): \quad \forall x \in \R_{>0}: \delta < x$ That is, an '''infinitesimal''' is a (strictly) positive real number which is smaller than every other (strictly) positive real number.


Definition:Solid of Revolution

Let $F$ be a plane figure. Let $L$ be a straight line in the same plane as $F$. Let $F$ be rotated one full turn about $L$ in $3$-dimensional space. The '''solid of revolution generated by $F$ around $L$''' is the solid figure whose boundaries are the paths taken by the sides of $F$ as they go round $L$.


Definition:Differential Equation/Order

The '''order''' of a '''differential equation''' is defined as being the order of the highest order derivative that is present in the equation.


Definition:Differential Equation/Ordinary

An '''ordinary differential equation''' (abbreviated '''O.D.E.''' or '''ODE''') is a '''differential equation''' which has exactly one independent variable. All the derivatives occurring in it are therefore ordinary. The general '''ODE''' of order $n$ is: :$\map f {x, y, \dfrac {\d x} {\d y}, \dfrac {\d^2 x} {\d y^2}, \ldots, \dfrac {\d^n x} {\d y^n} } = 0$ or, using the prime notation: :$\map f {x, y, y', y'', \ldots, y^{\paren n} } = 0$


Definition:Differential Equation/Partial

A '''partial differential equation''' (abbreviated '''P.D.E.''' or '''PDE''') is a '''differential equation''' which has: :one dependent variable :more than one independent variable. The derivatives occurring in it are therefore partial.


Definition:Differential Equation/Linear

A '''linear differential equation''' is a '''differential equation''' where all dependent variables and their derivatives appear to the first power. Neither are products of dependent variables allowed.


Definition:Differential Equation/Non-Linear

A '''non-linear differential equation''' is a '''differential equation''' which is not linear.


Definition:Differential Equation/System

A '''system of differential equations''' is a set of simultaneous '''differential equations'''. The solutions for each of the differential equations are in general expected to be consistent.


Definition:Differential Equation/System/Autonomous

A '''system of differential equations''' is '''autonomous''' if all of the '''differential equations''' which it comprises are themselves autonomous.


Definition:Differential Equation/Explicit

A '''differential equation''' is called '''explicit''' if it can be written in the form: :$y^{\left({n}\right)} = f \left({x, y, y', y'', \dots, y^{\left({n-1}\right)}}\right)$


Definition:Differential Equation/Autonomous

A '''differential equation''' is '''autonomous''' if none of the derivatives depend on the independent variable. The $n$th order '''autonomous differential equation''' takes the form: :$y^{\paren n} = \map f {y, y', y'', \dots, y^{\paren {n - 1} } }$


Definition:Legendre's Differential Equation

:$\displaystyle \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + p \paren {p + 1} y = 0$


Definition:Integral Curve

Let $\map f {x, y}$ be a continuous real function within a rectangular region $R$ of the Cartesian plane. Consider the first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \map f {x, y}$ :500px Let $P_0 = \tuple {x_0, y_0}$ be a point in $R$. The number: :$\paren {\dfrac {\d y} {\d x} }_{P_0} = \map f {x_0, y_0}$ determines the slope of a straight line passing through $P_0$. Let $P_1 = \tuple {x_1, y_1}$ be a point in $R$ close to $P_0$. Then: :$\paren {\dfrac {\d y} {\d x} }_{P_1} = \map f {x_1, y_1}$ determines the slope of a straight line passing through $P_1$. Let $P_2 = \tuple {x_2, y_2}$ be a point in $R$ close to $P_1$. Then: :$\paren {\dfrac {\d y} {\d x} }_{P_2} = \map f {x_2, y_2}$ determines the slope of a straight line passing through $P_2$. Continuing this process, we obtain a curve made of a sequence of straight line segments. As successive points $P_0, P_1, P_2, \ldots$ are taken closer and closer to each other, the sequence of straight line segments $P_0 P_1 P_2 \ldots$ approaches a smooth curve $C$ passing through an initial point $P_0$. By construction, for each point $\tuple {x, y}$ on $C$, the slope of $C$ at $\tuple {x, y}$ is given by $\map f {x, y}$. Hence this curve is a solution to $(1)$. Starting from a different point, a different curve is obtained. Thus the general solution to $(1)$ takes the form of a set of curves. This set of curves are referred to collectively as '''integral curves'''.


Definition:Parameter of Differential Equation

Let $f$ be a differential equation with general solution $F$. A '''parameter''' of $F$ is an arbitrary constant arising from the solving of a primitive during the course of obtaining the solution of $f$.


Definition:Substance

A '''substance''' is a particular kind of matter with properties specific to it.


Definition:First-Order Reaction

Suppose a body has a tendency to decompose spontaneously into smaller bodies at a rate independent of the presence of other bodies. Then the number of bodies that decompose in a single unit of time is proportional to the total number present. Such a reaction is called a '''first-order reaction'''.


Definition:Matter

'''Matter''' is the stuff of which the physical universe is composed. At a certain level, can be considered as a form of energy. '''Matter''' is (to a first degree of approximation) composed of atoms.


Definition:Chemical Element

A '''chemical element''' is a substance which is composed of atoms of one type. A '''chemical element''' is characterised uniquely by the number of protons it has in the nucleus of its atoms.


Definition:Atom (Physics)

An '''atom''' (in the context of physics and chemistry) is the smallest piece of matter that can exist of a particular type of substance. '''Atoms''' can be subdivided into smaller particles, but then it ceases to be that substance.


Definition:Radioactive Decay

'''Radioactive decay''' is a first-order reaction in which the atoms of a substance may spontaneously break apart, thereby turning into atoms of a different substance.


Definition:Radioactive Decay/Half-Life

The '''half-life''' of a radioactive isotope is the time it takes for exactly half of an arbitrary quantity of that isotope to undergo radioactive decay. Thus it is the time it takes for exactly half of an arbitrary quantity of that isotope to remain.


Definition:Radioactive Decay/Radioactive Element

A chemical element whose isotopes are ''all'' subject to '''radioactive decay''' is known as a '''radioactive element'''.


Definition:First-Order Reaction/Rate Constant

Let $S$ be a substance which decomposes spontaneously in a '''first-order reaction'''. By First-Order Reaction, this is governed by the equation: :$x = x_0 e^{-k t}$ where: :$x$ is the quantity of $S$ at time $t$ :$x_0$ is the quantity of $S$ at time $t = 0$ :$k$ is a positive number. The number $k$ is called the '''rate constant'''. $k$ is constant for a particular substance, and different substances, in general, have different '''rate constants'''.


Definition:Chemical Element/Isotope

'''Isotopes''' are instances of chemical elements which have the same number of protons in the nuclei of their atoms, but different numbers of neutrons. Thus it is usual to discuss '''an isotope''' of a particular element.


Definition:Atom (Physics)/Nucleus

The '''nucleus''' of an atom is the relatively dense core at its center. It consists (to a first degree of approximation) of a number of protons and a similar number of neutrons.


Definition:Elementary Particle

An '''elementary particle''' is one of the basic building blocks of matter. To a first degree of approximation, those building blocks can be considered to be protons, neutrons and electrons. Modern research in particle physics suggests the existence of smaller constituents of protons and neutrons, but at time of writing, electrons are believe to be truly, fundamentally indivisible. Category:Definitions/Particle Physics 2yejk9j01j4hb0trbc4tynec4cuny6z


Definition:Particle Physics

'''Particle physics''' is the branch of physics which studies the structure and interactions of the elementary particles of matter.


Definition:Radioactive Decay/Radioactive Isotope

An isotope of a chemical element which is subject to '''radioactive decay''' is known as a '''radioactive isotope'''.


Definition:Time/Unit/Year

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''year''' }} {{eqn | o = \approx       | r = 365 \cdotp 24219 \, 878       | c = days }} {{eqn | o = \approx       | r = 365 \cdotp 24219 \, 878 \times 24 \times 60 \times 60       | c = seconds }} {{end-eqn}} That is: {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''year''' }} {{eqn | o = \approx       | r = 365       | c = days, }} {{eqn | o =        | r = 5       | c = hours, }} {{eqn | o =        | r = 48       | c = minutes, and }} {{eqn | o =        | r = 45 \cdotp 9747       | c = seconds }} {{end-eqn}} It is defined to be equal to the orbital period of Earth round the sun.


Definition:Time/Unit/Year/Symbol

The symbol for the '''year''' is $\mathrm {yr}$ or $\mathrm y$. In some scientific contexts $\mathrm a$ (for '''annus''', Latin for '''year''') is used.


Definition:Time/Unit/Year/Calendar

A '''calendar year''' is the number of days between dates with the same designation in adjacent years. As a '''year''' is not an integral number of days, it is necessarily an approximation to the actual year. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''calendar year''' }} {{eqn | r = 12       | c = months usually, depending on the calendar used }} {{eqn | r = 365 \text{ or } 366       | c = days, usually, but it depends on the calendar used }} {{end-eqn}}


Definition:Radioactive Decay/Stable

An isotope of a chemical element which is ''not'' subject to '''radioactive decay''' is described as '''stable'''.


Definition:Density of Topological Space

Let $T$ be a topological space. The '''density''' of $T$ is the smallest cardinality of dense subset of $T$: :$d \left({T}\right) := \min \left\{{\left\vert{A}\right\vert: A \subseteq T \land A}\right.$ is dense$\left.{}\right\}$


Definition:Chemical Compound

A '''(chemical) compound''' is a (chemical) substance that consists of molecules composed of more than one type of chemical element.


Definition:Molecule

A '''molecule''' is an arrangement of atoms, possibly of different elements, which are held together in a configuration which requires the application of external energy to break apart. The '''molecule''' is the basic unit of a chemical compound.


Definition:Chemical Reaction

A '''chemical reaction''' is a physical process in which the atoms  composing the molecules of the chemical substances involved change their configurations into different molecules. Category:Definitions/Chemistry Category:Definitions/Physics qoic8xfj9qwghpvwl74fqa2ik9qbkue


Definition:Second-Order Reaction

Let $\AA$ and $\BB$ be two chemical substances in solution which react together to form a compound $\CC$. Let the reaction occur by means of the molecules of $\AA$ and $\BB$ colliding and interacting as a result. Then the rate of formation of $\CC$ is proportional to the number of collisions in unit time. This in turn is jointly proportional to the quantities of $\AA$ and $\BB$ which have not yet transformed. Such a chemical reaction is called a '''second-order reaction'''.


Definition:Thermodynamics

'''Thermodynamics''' is the branch of physics which studies the behaviour of heat.


Definition:Heat

'''Heat''' is a form of energy which is held within a body in the form of the physical motion of the molecules or atoms of which that body is composed. Category:Definitions/Physics Category:Definitions/Thermodynamics dij5rrbowaoa9k8gdp108dyza63xjcl


Definition:Energy

'''Energy''' is a property of physical bodies which can be defined as the ability to do work. Like time, it is difficult to define exactly what '''energy''' actually ''is''.


Definition:Linear Measure/Thickness

'''Thickness''', like breadth, is used as a term for linear measure in a dimension perpendicular to both length and depth. However, whereas breadth has connotations of '''across''', '''thickness''' is used for distances '''through''' the solid figure.


Definition:Kinetic Energy

Let $B$ be a body. The '''kinetic energy''' of $B$ is defined as the energy appertaining to $B$ due to its motion. It is a scalar quantity.


Definition:Potential Energy

Let $B$ be a body. The '''potential energy''' of $B$ is the energy appertaining to $B$ as a result of its position in a force field or because of the internal configuration of its components.


Definition:Force Field

A '''force field''' is a field in which the physical quantity associated with every point is a force which applies to a body placed at that point. Category:Definitions/Physics lg50nsn2tmwne5pkz9xntjsy0t777br


Definition:Constant (Physics)

A physical quantity is defined as being '''constant''' if it does not change with the passage of time.


Definition:Pendulum/Ideals

The ideals of a pendulum are as follows: :The rod is of zero thickness, infinite stiffness and zero mass. :The pivot is immovable and  has zero friction. :The bob is of zero volume. :The gravitation field is (usually) taken to be constant and uniform. :The medium in which the pendulum moves offers no resistive force, unless otherwise specifically stated.


Definition:Pendulum/Bob

The point mass at the free end of the rod of a pendulum is called the '''(pendulum) bob'''.


Definition:Pendulum

A '''pendulum''' is an ideal body consisting of two parts: :$(1): \quad$ a rod attached at one end to a pivot :$(2): \quad$ a point mass at the other end of the rod.


Definition:Rod

A '''rod''' is a rigid straight body whose width and mass are approximated to zero.


Definition:Escape Velocity

Let a body be propelled vertically upwards from the surface of a planet Let the body keep travelling away from the planet indefinitely. Let resistance from the atmosphere of the planet be considered as negligible. The smallest initial speed at which this is possible is known as the '''escape velocity''' of the planet.


Definition:Terminal Velocity

Let a body travelling under a constant force through a medium which resists its progress. The '''terminal velocity''' of the body is the limiting speed through that medium for the given force.


Definition:Speed/Also known as

Some authors erroneously or carelessly refer to the '''speed''' of a body as its velocity. But this is technically wrong if the author does not specify its direction as well as its magnitude.


Definition:Sine/Real Function/Arch

Each section of the sine function between adjacent zeroes is called an '''arch''' of the sine function


Definition:Lemniscate of Bernoulli/Lobe

Each of the two loops that constitute the '''lemniscate''' can be referred to as a '''lobe''' of the lemniscate.


Definition:Brachistochrone

Let a point $A$ be joined by a wire to a lower point $B$. Let the wire be allowed to be bent into whatever shape is required. Let a bead be released at $A$ to slide down without friction to $B$. The shape of the wire so that the bead takes least time to descend from $A$ to $B$ is called the '''brachistochrone'''.


Definition:Wire

A '''wire''' is a rigid body modelled by a specific curve whose width and mass are approximated to zero.


Definition:Cycloid/Generating Circle

The circle which rolls along the straight line is called the '''generating circle''' of the cycloid.


Definition:Confocal Conics

Let $F$ be a set of conic sections. Then $F$ is a set of '''confocal conics''' {{iff}} they all share the same foci.


Definition:Self-Orthogonal Trajectories

Let $\map f {x, y, c}$ define a one-parameter family of curves $F$. Let the family of orthogonal trajectories of $F$ be $F$ itself. Then $\map f {x, y, c}$ is '''self-orthogonal'''.


Definition:One-Parameter Family of Curves/Parameter

The value $c$ is the '''parameter''' of $F$.


Definition:T1/2 Space

Let $T = \left({S, \tau}\right)$ be a topological space. Then $T$ is a '''$T_{\frac 1 2}$ space''' {{iff}}: :$\forall A \subseteq S: A'$ is closed where $A'$ denotes the derivative of $A$.


Definition:Confocal Ellipses

Let $F$ be a set of ellipses. Then $F$ is a set of '''confocal ellipses''' {{iff}} they all share the same foci. Category:Definitions/Conic Sections l20rp6jtvnamympx7ddmlkq7prci5is


Definition:Confocal Hyperbolas

Let $F$ be a set of hyperbolas. Then $F$ is a set of '''confocal hyperbolas''' {{iff}} they all share the same foci. Category:Definitions/Conic Sections r8883h0y4x4pw2nxv9c8eyajujtws6h


Definition:Asymptote

:500pxrightthumbAsymptotes to $y = x + \dfrac 1 x$ (blue): [[Definition:Y-Axis$y$-axis, and the line $x = y$ (red)]] Let $C$ be a curve in the plane. Let $L$ be a straight (infinite) line such that the distance between $C$ and $L$ approaches zero as they tend to infinity. Then $L$ is an '''asymptote to''' (or '''of''') $C$.


Definition:Conic Section/Center

The '''center''' of a section is the point midway between the foci. === Center of Circle === The circle is usually treated differently. From Circle has Two Coincident Foci, the foci and the center are all the same point. :300px {{:Definition:Circle/Center}} === Center of Ellipse === Let $K$ be an ellipse. :500px {{:Definition:Ellipse/Center}} === Center of Hyperbola === Let $K$ be a hyperbola. :500px {{:Definition:Hyperbola/Center}} === Center of Parabola === {{:Definition:Parabola/Center}}


Definition:Conic Section/Reduced Form

Let $K$ be a conic section. Let $K$ be embedded in a cartesian plane such that: :$(1)$ the center is at the origin :$(2)$ the foci are at $\left({\pm c, 0}\right)$ for some $c \in \R_{\ge 0}$. This can be interpreted in the contexts of the specific classes of conic section as follows: === Reduced Form of Ellipse === {{:Definition:Conic Section/Reduced Form/Ellipse}} === Reduced Form of Hyperbola === {{:Definition:Conic Section/Reduced Form/Hyperbola}} === Reduced Form of Circle === {{:Definition:Conic Section/Reduced Form/Circle}} === Reduced Form of Parabola === {{:Definition:Conic Section/Reduced Form/Parabola}}


Definition:Parabola/Center

The parabola has no center. From Parabola has One Focus, the definition of the center as being the point midway between the foci, it follows that for a parabola the definition is inapplicable.


Definition:Conic Section/Reduced Form/Ellipse

Let $K$ be an ellipse embedded in a cartesian plane. $K$ is in '''reduced form''' {{iff}}: :$(1)$ its major axis is aligned with the $x$-axis :$(2)$ its minor axis is aligned with the $y$-axis. :500px


Definition:Conic Section/Reduced Form/Hyperbola

Let $K$ be a hyperbola embedded in a cartesian plane. $K$ is in '''reduced form''' {{iff}}: :$(1)$ its major axis is aligned with the $x$-axis :$(2)$ its minor axis is aligned with the $y$-axis. :500px


Definition:Conic Section/Reduced Form/Circle

Let $K$ be a circle embedded in a cartesian plane. $K$ is in '''reduced form''' {{iff}} its center is located at the origin. :500px


Definition:Conic Section/Reduced Form/Parabola

Let $K$ be a parabola embedded in a cartesian plane. As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section. Instead, $K$ is in '''reduced form''' {{iff}}: :$(1)$ its focus is at the point $\tuple {c, 0}$ :$(2)$ its directrix is aligned with the line $x = -c$ for some $c \in \R_{> 0}$. :400px


Definition:Hyperbola/Equidistance

:400px Let $F_1$ and $F_2$ be two points in the plane. Let $d$ be a length less than the distance between $F_1$ and $F_2$. Let $K$ be the locus of points $P$ which are subject to the condition: :$\size {d_1 - d_2} = d$ where: :$d_1$ is the distance from $P$ to $F_1$ :$d_2$ is the distance from $P$ to $F_2$ :$\size {d_1 - d_2}$ denotes the absolute value of $d_1 - d_2$. Then $K$ is a '''hyperbola'''.


Definition:Set of Condensation Points

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. The '''set of condensation points''' of $A$ is the set of all condensation points of $A$, :$A^0 = \left\{{x \in S:x}\right.$ is condensation point of $\left.{A}\right\}$


Definition:Homogeneous Function/Real Space/Degree

The integer $n$ is known as the '''degree''' of $f$.


Definition:Homogeneous Function/Real Space

Let $f: \R^2 \to \R$ be a real-valued function of two variables. $f \left({x, y}\right)$ is a '''homogeneous function of degree zero''' {{iff}}: :$\exists n \in \Z: \forall t \in \R: f \left({t x, t y}\right) = t^n f \left({x, y}\right)$


Definition:Homogeneous Function/Real Space/Zero Degree

$f \left({x, y}\right)$ is a '''homogeneous function of degree zero''' or '''of zero degree''' {{iff}}: :$\forall t \in \R: f \left({t x, t y}\right) = t^0 f \left({x, y}\right) = f \left({x, y}\right)$


Definition:Linear Ordinary Differential Equation

A '''linear ordinary differential equation''' is an '''ordinary differential equation''' where all dependent variables and their derivatives appear to the first power. Neither are products of dependent variables allowed.


Definition:Linear Second Order Ordinary Differential Equation

A '''linear second order ordinary differential equation''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$ where, as is indicated by the notation, $\map P x$, $\map Q x$ and $\map R x$ are functions of $x$ alone (or constants).


Definition:Niemytzki Plane

The '''Niemytzki plane''' is the topological space $T = \struct {S, \tau}$ defined as: {{begin-eqn}} {{eqn | l = S       | r = \set {\tuple {x, y} \in \R^2: y \ge 0} }} {{eqn | l = \map \BB {x, y}       | r = \set {\map {B_r} {x, y} \cap S: r > 0}       | c = if $x, y \in \R, y > 0$ }} {{eqn | l = \map \BB {x, 0}       | r = \set {\map {B_r} {x, r} \cup \set {\tuple {x, 0} }: r > 0}       | c = if $x \in \R$ }} {{eqn | l = \tau       | r = \set {\bigcup \GG: \GG \subseteq \bigcup_{\tuple {x, y} \mathop \in S} \map \BB {x, y} } }} {{end-eqn}} where $\map {B_r} {x, y}$ denotes the open $r$-ball of $\tuple {x, y}$ in the $\R^2$ Euclidean space.


Definition:Neighborhood System

Let $T = \struct {S, \tau}$ be a topological space. A '''neighborhood system''' is a family $\family {\NN_x}_{x \mathop \in S}$ indexed by points of $S$, such that $\NN_x$ is a local basis at $x$ for $x \in S$.


Definition:Tractrix

Let $S$ be a cord situated as a (straight) line segment whose endpoints are $P$ and $T$. Let $T$ be dragged in a direction perpendicular to the straight line in which $S$ is aligned. The curve along which $P$ travels is known as a '''tractrix'''. :400px


Definition:Tractrix/Linguistic Note

The word '''tractrix''' derives from the Latin '''traho (trahere, traxi, tractum)''' meaning '''to pull''' or '''to drag'''. The plural is '''tractrices'''.


Definition:Pursuit Curve

A '''pursuit curve''' is a curve constructed to model pursuers and pursuees. The '''pursuit curve''' is the curve traced by the pursuers. The pursuee is always on the pursuer's tangent.


Definition:Rate of Change

The '''rate of change''' of a physical quantity is its (first) derivative with respect to time. Loosely speaking, it means '''how fast something changes''', with a wider scope than change in physical displacement. === Dimension === {{:Definition:Rate of Change/Dimension}}


Definition:Steady-State

=== First Order ODE === {{:Definition:Steady-State/First Order ODE}} === Second Order ODE === {{:Definition:Steady-State/Second Order ODE}}


Definition:Steady-State/Electronics

The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R + \left({I_0 - \dfrac {E_0} R}\right) e^{-R t / L}$ The term $\dfrac {E_0} R$ is known as the '''steady-state''' part of $(1)$.


Definition:Transient/Electronics

The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R + \left({I_0 - \dfrac {E_0} R}\right) e^{-R t / L}$ The term: :$\left({I_0 - \dfrac {E_0} R}\right) e^{-R t / L}$ which dies away to zero over time, is known as the '''transient''' part.


Definition:Transient

Consider the Decay Equation: :$\dfrac {\d y} {\d x} = k \paren {y_a - y}$ where: :$k \in \R: k > 0$ :$y = y_0$ at $x = 0$ which has the particular solution: :$(1): \quad y = y_a + \paren {y_0 - y_a} e^{-k x}$ The term: :$\paren {y_0 - y_a} e^{-k x}$ which tends to zero with increasing $x$, is known as the '''transient''' part. === Electronics === The term is often seen in the context of the current in an $LR$ electric circuit $K$: {{:Definition:Transient/Electronics}}


Definition:Electronics

'''Electronics''' is the branch of electromagnetism which studies how to control electricity.


Definition:Barrier

A complex function $\varphi \in \map \C {\overline \Omega}$ is a '''barrier''' for $\Omega$ at $z \in \partial \Omega$ {{iff}}: :$\varphi$ is subharmonic :$\map \varphi z = 0$ :$\varphi < 0$ on $\partial \Omega \setminus \set z$ {{explain|Define $\map \C {\overline \Omega}$}} {{explain|Define $\Omega$}} {{explain|Define $\partial \Omega$}} {{explain|What is the domain and range of $\varphi$?}} {{explain|If $\varphi$ is a complex function, its range is also complex, and so cannot be ordered, so the meaning of $\varphi < 0$ needs to be explained carefully}}


Definition:Surface of Revolution

Let $F$ be a plane figure. Let $L$ be a straight line in the same plane as $F$. Let $F$ be rotated one full turn about $L$ in $3$-dimensional space. The '''surface of revolution generated by $F$ around $L$''' is the surface of the solid of revolution generated by $F$ around $L$.


Definition:Riccati Equation

The '''Riccati equation''' is the first order ordinary differential equation: :$y' = p \left({x}\right) + q \left({x}\right) y + r \left({x}\right) y^2$


Definition:Special Riccati Equation

The '''special Riccati equation''' is the first order ordinary differential equation: :$y' + b y^2 = c x^m$ {{NamedforDef|Jacopo Francesco Riccati|cat = Riccati}}


Definition:Burnout Velocity

Let $R$ be a rocket which is propelled by backwards expulsion of the exhaust products of the burning of its internal fuel supply. Let $R$ be fired straight up from the surface of a planet. The '''burnout velocity''' of $R$ is the velocity attained by $R$ at the instant when its fuel has become exhausted.


Definition:Dynamics

'''Dynamics''' is the branch of the mechanics describing the motion of bodies under the influence of forces.


Definition:Burnout Height

Let $R$ be a rocket which is propelled by backwards expulsion of the exhaust products of the burning of its internal fuel supply. Let $R$ be fired straight up from the surface of a planet. The '''burnout height''' of $R$ is the height attained by $R$ at the instant when its fuel has become exhausted.


Definition:Homogeneous Linear Second Order ODE

A '''homogeneous linear second order ODE''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ where, as is indicated by the notation, $\map P x$ and $\map Q x$ are functions of $x$ alone (or constants).


Definition:Nonhomogeneous Linear Second Order ODE

A '''nonhomogeneous linear second order ODE''' is a differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$ where: :$\map P x$, $\map Q x$ and $\map R x$ are functions of $x$ alone (or constants) :$\map R x$ is not the zero constant function. That is, a '''nonhomogeneous linear second order ODE''' is a linear second order ODE which is not homogeneous.


Definition:Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE

Let $C_1$ and $C_2$ be real numbers. Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ Then: :$C_1 \, \map {y_1} x + C_2 \, \map {y_2} x$ is known as a '''linear combination of particular solutions to $(1)$.


Definition:Linearly Dependent Real Functions

Let $f \left({x}\right)$ and $g \left({x}\right)$ be real functions defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ and $g$ be constant multiples of each other: :$\exists c \in \R: \forall x \in \left[{a \,.\,.\, b}\right]: f \left({x}\right) = c g \left({x}\right)$ or: :$\exists c \in \R: \forall x \in \left[{a \,.\,.\, b}\right]: g \left({x}\right) = c f \left({x}\right)$ Then $f$ and $g$ are '''linearly dependent'''.


Definition:Linearly Independent Real Functions

Let $f \left({x}\right)$ and $g \left({x}\right)$ be real functions defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ and $g$ be such that they are not linearly dependent. Then $f$ and $g$ are '''linearly independent'''.


Definition:Wronskian/General Definition

Let $\map {f_1} x, \map {f_2} x, \dotsc, \map {f_n} x$ be real functions defined on a closed interval $\closedint a b$. Let $f_1, f_2, \ldots, f_n$ be $n - 1$ times differentiable on $\closedint a b$. The '''Wronskian''' of $f_1, f_2, \ldots, f_n$ on $\closedint a b$ is defined as: :$\map W {f_1, f_2, \dotsc, f_n} = \begin {vmatrix} \map {f_1} x & \map {f_2} x & \cdots & \map {f_n} x \\ \map { {f_1}'} x & \map { {f_2}'} x & \cdots & \map { {f_n}'} x \\ \vdots & \vdots & \ddots & \vdots \\ \map { {f_1}^{\paren {n - 1} } } x & \map { {f_2}^{\paren {n - 1} } } x & \cdots & \map { {f_n}^{\paren {n - 1} } } x \\ \end{vmatrix}$ where: :$\begin{vmatrix} \cdots \end{vmatrix}$ denotes the determinant :$\map { {f_1}^{\paren {n - 1} } } x$ denotes the $n - 1$th derivative of $f_1$.


Definition:Wronskian

Let $\map f x$ and $\map g x$ be real functions defined on a closed interval $\closedint a b$. Let $f$ and $g$ be differentiable on $\closedint a b$. The '''Wronskian''' of $f$ and $g$ is defined as: :$\map W {f, g} = \begin {vmatrix} \map f x & \map g x \\ \map {f'} x & \map {g'} x \\ \end {vmatrix} = \map f x \, \map {g'} x - \map g x \, \map {f'} x$ === General Definition === {{:Definition:Wronskian/General Definition}}


Definition:Homogeneous Linear Second Order ODE with Constant Coefficients

A '''homogeneous linear second order ODE with constant coefficients''' is a second order ODE which can be manipulated into the form: :$y'' + p y' + q y = 0$ where $p$ and $q$ are real constants. Thus it is a homogeneous linear second order ODE: :$y'' + \map P x y' + \map Q x y = 0$ where $\map P x$ and $\map Q x$ are constant functions.


Definition:Auxiliary Equation

Let: :$(1): \quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. The '''auxiliary equation''' of $(1)$ is the quadratic equation: :$m^2 + p m + q = 0$


Definition:Cauchy-Euler Equation

The linear second order ordinary differential equation: :$x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$ is the '''Cauchy-Euler equation'''.


Definition:Cauchy-Euler Equation/General Form

Let $n \in \Z_{>0}$ be a strictly positive integer. The linear ordinary differential equation: :$a_n x^n \, \map {y^{\paren n} } x + \dotsb + a_1 x \, \map {y'} x + a_0 \, \map y x = 0$ is the '''$n$th order Cauchy-Euler equation'''.


Definition:Directed Subset

Let $\struct {S, \precsim}$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is a '''directed subset''' of $S$ {{iff}}: :$\forall x, y \in H: \exists z \in H: x \precsim z$ and $y \precsim z$


Definition:Filtered Subset

Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is a '''filtered subset''' of $S$ {{iff}}: :$\forall x, y \in H: \exists z \in H: z \precsim x$ and $z \precsim y$


Definition:Spring/Ideal

An '''ideal spring''' obeys Hooke's Law: :$\mathbf F = -k \mathbf x$ where: : $\mathbf F$ is the force caused by a displacement $\mathbf x$ : $k$ is the constant of proportion. The negative sign indicates that the force pulls in the opposite direction to that of the displacement imposed.


Definition:Spring/Force Constant

The constant of proportion: :$k \in \R_{>0}$ is known as the '''(spring) force contant'''. Its value depends on the particular spring under investigation. The SI unit of the '''force constant''' is newtons per metre: $\mathrm N \ \mathrm m^{-1}$.


Definition:Wall

A '''wall''' is an immovable vertical plane in space to which other physical objects may be attached.


Definition:Cart

A '''cart''' is a body with a given mass which is free to move with zero friction in a single dimension, that is, back and forth. It is usually imagined as having wheels on which it can roll.


Definition:Immovable

An '''immovable''' body is one whose position cannot be changed by applying a force to it. It can be considered as having an infinite mass and zero momentum.


Definition:Spring/Equilibrium Position

The '''equilibrium position''' of a body $B$ attached to a spring $S$ is the position it occupies when $S$ is exerting no force upon $B$. For an ideal spring obeying Hooke's Law $\mathbf F = -k \mathbf x$, the '''equilibrium position''' is set to be the point $\mathbf x = \bszero$.


Definition:Simple Harmonic Motion/Amplitude

The parameter $A$ is known as the '''amplitude''' of the motion.