Search results

Create the page "C(X,C)" on this wiki! See also the search results found.

Page title matches

• C(I,X)
  {{DISPLAYTITLE:{{M|C([0,1],X)}}}} ...=I:=[0,1]\subset\mathbb{R} }} - the [[closed unit interval]]. Then {{M|C(I,X)}} denotes the [[set of continuous functions]] between the interval, consid
  1 KB (258 words) - 05:08, 3 November 2016
• C(X,Y)
  125 B (14 words) - 05:55, 1 January 2017

Page text matches

• Topology
  ...Should be easy to flesh out, find some more references and demote to grade C once acceptable}} ...of {{M|X}}. This means that if {{M|U\in\mathcal{J} }} then {{M|U\subseteq X}}</ref> such that{{rITTMJML}}{{rFAVIDMH}}:
  3 KB (543 words) - 09:28, 30 December 2016
• Topological space
  {{Requires references|grade=C|Need references for larger/smaller/stronger/weaker, Check Introduction To T * Given any set {{M|X}} we can always define the following two topologies on it:
  2 KB (268 words) - 13:37, 20 April 2016
• Nabla
  <math>\nabla(\ )=\mathbf{i}\frac{\partial(\ )}{\partial x}+\mathbf{j}\frac{\partial(\ )}{\partial y}+\mathbf{k}\frac{\partial(\ )}{\p <math>\nabla\cdot\nabla(\ )=\nabla^2(\ )=\frac{\partial^2(\ )}{\partial x^2}+\frac{\partial^2(\ )}{\partial y^2}+\frac{\partial^2(\ )}{\partial z^2}<
  1 KB (245 words) - 18:35, 13 February 2015
• Connected (topology)
  Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is ''connected'' if{{rITTMJML}}: ...}[U\ne\emptyset\wedge V\neq\emptyset\wedge U\cap V=\emptyset\wedge U\cup V=X])}}
  5 KB (866 words) - 01:52, 1 October 2016
• Subspace topology
  ...let {{M|S}} be a subset of {{M|X}}, possibly empty, possibly equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\ ...|(S,\mathcal{J}_S)}} are precisely the intersection of open sets of {{Top.|X|J}} with {{M|S}}
  6 KB (1,146 words) - 23:04, 25 September 2016
• The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
  ...X\rightarrow\mathbb{R}_{\ge 0} }} be a [[metric]] on that set and let {{M|(X,d)}} be the resulting [[metric space]]. Then we claim: * {{M|\mathcal{B}:\eq\left\{ B_\epsilon(x)\ \vert\ x\in X\wedge \epsilon\in\mathbb{R}_{>0}\right\} }} satisfies the condition [[topol
  4 KB (814 words) - 22:16, 16 January 2017
• Closure, interior and boundary
  ...verline{A}=\bigcap\{B\subset X|A\subset B\text{ and }B\text{ is closed in }X\}</math> ...text{Int}(A)=\bigcup\{C\subset X|C\subset A\text{ and }C\text{ is open in }X\}</math>
  1 KB (210 words) - 00:20, 9 March 2015
• Function
  * {{M|f\subseteq X\times Y}} ...imes Y}} we have {{M|1=\forall x\in X\forall y,z\in Y[(x\mathcal{R}y\wedge x\mathcal{R}z)\implies y=z]}}
  4 KB (659 words) - 13:01, 19 February 2016
• Set theory axioms
  ...{M|Y}} and every element of {{M|Y}} is also an element of {{M|X}} then {{M|X=Y}}<br/> |<math>\forall X\forall Y(\forall u(u\in X\leftrightarrow u\in Y)\rightarrow X=Y)</math>
  3 KB (619 words) - 10:25, 11 March 2015
• Relation
  ...sets{{rAPIKM}}<ref name="TAPL">Types and Programming Languages - Benjamin C. Peirce</ref>, that is: * {{M|\mathcal{R}\subseteq X\times Y}}
  4 KB (762 words) - 20:07, 20 April 2016
• Ordering
  | <math>\forall a\in A\forall b\in A\forall c\in A([aRb\wedge bRc]\implies aRc)</math> ...=\mathbb{N} }} then <math>a\le b\wedge b\le c\iff a\le b\le c\implies a\le c</math>
  5 KB (1,006 words) - 13:21, 1 January 2016
• Equivalence relation
  ...{{M|X}}<ref group="Note">This terminology means {{M|\sim \subseteq X\times X}}, as described on the [[relation]] page.</ref> is an ''equivalence relatio ...|\forall x\in X[(x,x) \in \sim]}}. Which we write {{M|\forall x\in X[x\sim x]}}.
  3 KB (522 words) - 15:18, 12 February 2019
• Norm
  ...of {{M|\Vert\cdot\Vert}} could be in {{M|\mathbb{C} }} then the {{M|\Vert x\Vert\ge 0}} would make no sense. What ordering would you use? The [[canonic # <math>\forall x\in V\ \|x\|\ge 0</math>
  6 KB (1,026 words) - 20:33, 9 April 2017
• Cauchy-Schwarz inequality
  * {{MM|1=\vert\langle x,y\rangle\vert\le\Vert x\Vert \Vert y\Vert}} - the rare but more general ...a proof of the second form - note that {{M|\Vert x\Vert:\eq\sqrt{\langle x,x\rangle} }} is the [[norm induced by the inner product]] [[User:Alec|Alec]]
  3 KB (609 words) - 13:04, 4 April 2017
• Index of notation
  ...to {{M|\mathbb{R} }} - this is unlikely to be given any other way because "C" is for continuous. | {{M|\mathbb{S}^n}}, {{M|l_2}}, {{M|\mathcal{C}[a,b]}}
  9 KB (1,490 words) - 06:13, 1 January 2017
• Basis and coordinates
  ...e, so the coordinate {{M|(x,y)}} is on our paper, and {{M|(x,y)'}} or {{M|(x',y')}} is on their paper. ...[Linear map|linear transform]]? Well recall to be linear <math>T(ax+by)=aT(x)+bT(y)</math>
  9 KB (1,525 words) - 16:30, 23 August 2015
• Group
  ||<math>\forall a,b,c\in G:[(a*b)*c=a*(b*c)]</math> ...*}} is [[Associative|associative]], because of this we may write <math>a*b*c</math> unambiguously.
  7 KB (1,332 words) - 07:17, 16 October 2016
• Cardinality
  ...be injective, but would not be surjective if <math>\exists x(x\in C\wedge x\notin B)</math>, thus not bijective.<ref>p65 - Introduction to Set Theory,
  2 KB (327 words) - 10:25, 12 March 2015
• Algebra of sets
  ...roup="Note">Recall {{M|1=A^C:=X-A}} - the [[complement]] of {{M|A}} in {{M|X}}</ref> ...} in {{M|\mathcal{A} }} the [[complement]] of {{M|A}} (with respect to {{M|X}}) is also in {{M|\mathcal{A} }}
  3 KB (507 words) - 18:43, 1 April 2016
• Sigma-algebra
  # {{M|X\in\mathcal{A} }} as {{M|\emptyset^C\in\mathcal{A} }} :: As {{M|1=A-B=(A^c\cup B)^c}} and a {{sigma|algebra}} is closed under complements and unions, this show
  8 KB (1,306 words) - 01:49, 19 March 2016
• Pre-measure
  * Here {{M|\mathcal{A} }} is an algebra of sets (a system of subsets of {{M|X}}) and {{M|\mu_0:\mathcal{A}\rightarrow[0,+\infty]}} such that: Here {{M|(X,\mathcal{A},\mu_0)}} is a [[Pre-measure space|pre-measure space]], and {{M|
  5 KB (782 words) - 01:49, 26 July 2015
• Measure
  | if the measure of {{M|X}} is finite * Symbolically, if {{M|\mu(X)<\infty}}
  6 KB (941 words) - 14:39, 16 August 2016
• Complement
  ...{M|X}} the complement of {{M|A}} (often denoted {{M|A^c}}, {{M|A'}} or {{M|C(A)}}) is given by: <math>A^c=\{x\in X|x\notin A\}=X-A</math>
  726 B (145 words) - 13:28, 18 March 2015
• Ring
  | <math>\forall a,b,c\in R[(a+b)+c=a+(b+c)]</math> | Now writing {{M|a+b+c}} isn't ambiguous
  7 KB (1,248 words) - 05:02, 16 October 2016
• Measurable map
  Let {{M|(X,\mathcal{A})}} and {{M|(X',\mathcal{A}')}} be [[Measurable space|measurable spaces]] then a map: * <math>T:X\rightarrow X'</math>
  5 KB (792 words) - 02:31, 3 August 2015
• Curve
  Given a <math>f:\mathbb{R}^n\rightarrow\mathbb{R}</math> and a {{M|c\in\mathbb{R} }} we define the level curve as follows<ref> <math>\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}</math>
  1 KB (224 words) - 21:30, 28 March 2015
• Germ
  ...int {{M|p\in\mathbb{R}^n}}, we define an equivalence relation on the <math>C^\infty</math> functions defined in some neighbourhood of {{M|p}} as: ...p V</math> (where <math>W</math> is open) that <math>x\in W\implies f(x)=g(x)</math> - that is {{M|f}} and {{M|g}} agree when restricted to {{M|W}}
  2 KB (285 words) - 01:36, 5 April 2015
• Smooth structure
  ...}_2=\{(B_1(x),\text{Id}_{B_1(x)})|x\in\mathbb{R}^n\}</math> (where {{M|B_r(x)}} denotes an [[Open ball]]) * {{M|C^\infty}} structure
  2 KB (246 words) - 07:10, 7 April 2015
• Motivation for tangent space
  ...(giving things as an angle and a distance from the origin, rather than {{M|x}} and {{M|y}}) We will have two ways of looking at points, as {{M|(x,y)}} - traditionally, and {{M|(r,\theta)}} where:
  6 KB (975 words) - 00:18, 11 April 2015
• Polar equation of a line
  * {{M|1=y=mx+c}} * <math>r=\sqrt{t^2(m^2+1)+2mtc+c^2}</math>
  1 KB (223 words) - 22:43, 10 April 2015
• Motivation for tangent space definitions
  \frac{\delta r}{\delta x} & \frac{\delta r}{\delta y} \\ \frac{\delta \theta}{\delta x} & \frac{\delta \theta}{\delta y}
  4 KB (790 words) - 22:25, 12 April 2015
• The fundamental group
  ...t{rel }\{0,1\})\big)}}]] on {{M|C(I,X)}} and restricted to {{M|\text{Loop}(X,b)}}, then: * {{M|1=\pi_1(X,b):=\frac{\text{Loop}(X,b)}{\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)} }} has a [[group]
  3 KB (393 words) - 16:10, 4 November 2016
• Paths and loops in a topological space
  ...in {{M|X}} is any [[Continuous map|continuous map]] {{M|p:[0,1]\rightarrow X}}<ref>Introduction to topology - lecture notes nov 2013 - David Mond</ref>. Given two paths {{M|p_0}} and {{M|p_1}} in a topological space {{M|X}} with {{M|1=p_0(1)=p_1(1)}} we can obtain a new path by performing {{m|p_0
  2 KB (347 words) - 19:36, 16 April 2015
• Hausdorff space
  Given a [[topological space]] {{M|(X,\mathcal{J})}} we say it is ''Hausdorff''{{rITTBM}} or ''satisfies the Haus ...e">Note that if {{M|X}} is the empty set, then there are no {{M|x_1,x_2\in X}}, so the statement is [[vacuously true]].</ref>
  4 KB (679 words) - 22:52, 22 February 2017
• Inner product
  ...{{Vector space}} (where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}), an ''inner product''<ref>http://en.wikipedia.org/w/index.php?title=In ...(or sometimes <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}</math>)
  6 KB (1,016 words) - 12:57, 19 February 2016
• Hilbert space
  ...] with respect to the associated [[Norm|norm]] <math>\|x\|=\sqrt{\langle x,x\rangle}</math>
  573 B (93 words) - 17:34, 21 April 2015
• Limit
  ...\mathbb{N}[n> N\implies d(a_n,a)<\epsilon] </math> - [[Metric space]] {{M|(X,d)}} ...[a\in U\wedge(n> N \implies a_n\in U)]</math> - [[Topological space]] {{M|(X,\mathcal{J})}}
  2 KB (310 words) - 18:23, 8 January 2016
• Examples of rings
  The complex numbers {{M|\mathbb{C} }} is a commutative ring with unity. {{M|\mathbb{R} }} is a subring, and s Let <math>S=\{x+y\sqrt{2}\in\mathbb{R}|x,y\in\mathbb{Z}\}</math>, defining multiplication and addition in the usual
  2 KB (269 words) - 17:11, 19 May 2015
• Passing to the quotient (function)
  ...0em;}}Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}}<ref group="Note">I have chosen {{M|W}} to mean "whatever"</r # <math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math>
  8 KB (1,644 words) - 20:49, 11 October 2016
• Integral domain
  * A commutative ring, that is: <math>\forall x,y\in D[xy=yx]</math> ...h> or if (by writing {{M|e_+}} as {{M|0}} we can say: <math>\exists c\in R[c\ne 0\wedge ac=0]</math>)
  2 KB (327 words) - 11:09, 20 February 2016
• Index of norms and absolute values
  | <math>\|\cdot\|_{C^k}</math> | <math>\|f\|_{C^k}</math>
  1 KB (207 words) - 09:16, 9 June 2015
• Trinary index
  ! class="unsortable" | {{M|C}} ! class="unsortable" | {{M|X}}
  964 B (165 words) - 20:55, 22 June 2015
• Bounded set
  ...n arbitrary [[subset of]] {{M|X}}. Then we say "{{M|A}} is bounded in {{M|(X,d)}}" if{{rFAVIDMH}}: ...\forall a,b\in A[d(a,b)<C]}} - where {{M|C}} is real<ref group="Note">{{M|C\in\mathbb{R}_{\ge 0} }} should do as {{M|0}} could be a bound, I suppose on
  2 KB (409 words) - 23:31, 29 October 2016
• Complete metric space
  ...chy sequence]] converges to a [[limit (sequence)]] within {{M|X}} then {{M|X}} is a ''complete metric space''<ref name="FA">Functional Analysis - George ...[[sequence]] {{M|1=(x_n)_{n=1}^\infty}}, it converging to a limit {{M|x\in X}} or being a [[Cauchy sequence]] are equivalent. Or in symbols:
  2 KB (382 words) - 15:36, 24 November 2015
• Inner product examples
  Here the space is {{M|\mathcal{C}_\mathbb{C}[a,b]}} - the [[Continuous map|continuous functions]] over the interval {{M ...{M|f\in\mathcal{C}_\mathbb{C}[a,b]}} we really have {{M|1=f(x)=f_r(x)+jf_i(x)}} where {{M|1=j:=\sqrt{-1} }})
  3 KB (678 words) - 16:16, 11 July 2015
• Inner product space
  ...{M|\mathbb{R} }} or {{M|\mathbb{C} }}, which we shall denote {{M|F}}) {{M|(X,F)}}, equipped with an We denote this {{M|(X,\langle\cdot,\cdot\rangle,F)}} or just {{M|(X,\langle\cdot,\cdot\rangle)}} if the field is implicit.
  949 B (161 words) - 21:08, 11 July 2015
• Normed space
  * [[Vector space|vector space]] over the [[Field|field]] {{M|F}}, {{M|(X,F)}} ** where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
  813 B (129 words) - 22:13, 11 July 2015
• Euclidean n-space
  ...uple|tuples]]}} of the form {{M|(x_1,\cdots,x_n)}} where {{M|x_i\in\mathbb{C} }}) #* Defined by {{M|1=X=\mathbb{R}^n}} (so {{M|X}} consists of all {{n|[[Tuple|tuples]]}} of the form {{M|(x_1,\cdots,x_n)}}
  2 KB (398 words) - 14:17, 12 July 2015
• Inequalities for inner products
  ...\vert\le\Vert x\Vert\Vert y\Vert}} for {{M|1=\Vert x\Vert:=\sqrt{\langle x,x\rangle} }} (equality if ''lin dependent'') ...duct space|i.p.s]] we have {{MM|1=\Vert x+y\Vert^2+\Vert x-y\Vert^2=2\Vert x\Vert^2+2\Vert y\Vert^2}}
  1 KB (214 words) - 14:52, 12 July 2015
• Index of spaces
  ...exed as "num" (notice the lower-case) so a space like {{M|C^k}} is under {{C|C_num}}. We do subscripts first, so {{M|A_i^2}} would be under {{C|A _num ^num:2}}
  3 KB (612 words) - 21:06, 29 February 2016
• Space of square-summable sequences
  * {{MM|1=(x_n)_{n=1}^\infty\subset\mathbb{C} }} * For {{M|x,y\in l_2}} we define {{M|1=\langle x,y\rangle:=\sum^\infty_{n=1}x_i\overline{y_i} }}
  893 B (141 words) - 15:47, 12 July 2015
• Dynkin system generated by
  Given a set {{M|X}} and another set {{M|\mathcal{G}\subseteq\mathcal{P}(X)}} which we shall call the ''generator'' then we can define ''the [[Dynkin * {{MM|1=\delta(\mathcal{G}):=\bigcap_{\begin{array}{c}\mathcal{D}\text{ is a Dynkin system}\\ \text{and }\mathcal{G}\subseteq\mat
  2 KB (245 words) - 15:16, 16 December 2016
• Dynkin system
  ...s closed under complements and {{M|X\in\mathcal{D} }} by definition, {{M|X^c\in\mathcal{D} }} : {{M|1=X^c=\emptyset}} so {{M|\emptyset\in\mathcal{D} }}
  1 KB (184 words) - 01:54, 19 March 2016
• Discrete metric and topology/Summary
  ...te topology|discrete topology]] - which is the topology {{M|(X,\mathcal{P}(X))}} (where {{M|\mathcal{P} }} denotes [[Power set|power set]]) ...in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r\le 1 \\ X & \text{otherwise}\end{array}\right. }}
  1 KB (263 words) - 13:04, 27 July 2015
• Dynkin system/Definition 1
  ...sets of {{M|X}}, which we shall denote {{M|\mathcal{D}\subseteq\mathcal{P}(X)}} is a ''Dynkin system''{{rMIAMRLS}} if: * {{M|X\in\mathcal{D} }}
  556 B (92 words) - 01:52, 19 March 2016
• Borel sigma-algebra generated by
  ...Measures, Integrals and Martingales - Rene L. Schilling</ref>: (where {{M|(X,\mathcal{O})}}<ref group="Note">Note the letter {{M|\mathcal{O} }} for the ...f the topology on {{M|X}} is obvious, we may simply write: {{M|\mathcal{B}(X)}}<ref name="MIM"/>
  2 KB (244 words) - 08:30, 6 August 2015
• Borel sigma-algebra
  ...=\mathcal{B}(X,\mathcal{J}):=\sigma(\mathcal{O})}} or just {{M|\mathcal{B}(X)}} if the topology is implicit. ! {{M|1=\mathcal{B}^n=\sigma(\mathcal{C})}} - closed<ref name="MIM"/>
  5 KB (854 words) - 09:25, 6 August 2015
• Dynkin system/Proof that definitions 1 and 2 are equivalent
  :# {{M|X\in\mathcal{D} }} is satisfied by definition :#* Note that {{M|1=A-B=(A^c\udot B)^c}} (this is not true in general, it requires {{M|B\subseteq A}}{{Note|Includ
  2 KB (326 words) - 05:09, 22 August 2015
• Set subtraction
  * {{M|1=A-B=\{x\in A\vert x\notin B\} }} * {{M|1=A-B=(A^c\cup B)^c}}
  1 KB (237 words) - 00:48, 21 March 2016
• Notes:Pick function
  ...4}} (projection AND view operator) you are using (that is for a point {{M|x}}, {{M|Px}} is the complete transformation to clip coordinates) Given a point in the world, {{M|x}} this matrix applies the camera position/angle transformation, then the pr
  4 KB (686 words) - 01:43, 15 September 2015
• Notes:Spherical coordinates
  ...cap}} denotes intersection of sets, {{M|x\in A\cap B\iff x\in A\text{ and }x\in B}}, {{M|\emptyset}} denotes the empty set, so here we are saying "there Then for an {{M|x\in U\cap V}} we have two sets of "coordinates", we have:
  10 KB (1,899 words) - 18:48, 23 September 2015
• Category
  A '''Category {{M|C}}''' consists of 3 things<ref name="EOAT">Elements of Algebraic Topology - ...}<ref group="Note">Munkres calls the class of objects {{M|X}} and uses {{M|X}} for specific objects. Not sure why, so checked definition with [[https://
  2 KB (347 words) - 00:36, 27 September 2015
• Notes:RealQ Architecture
  ...C|XL}} refers to the first 8 bits (or byte) of whatever is in register {{C|X}}. ...write: {{C|MOV r1, r2}}, or just {{C|SWP r}}) and {{C|r}} may be written {{C|rr}} if it is neater.
  2 KB (302 words) - 19:38, 2 October 2015
• Notes:RealQ Architecture/1-Byte operations
  | {{C|[[Notes:RealQ instruction LOAD|LOAD]] r1,r2}} | colspan="2" | {{C|00}}
  2 KB (210 words) - 20:26, 2 October 2015
• Class of smooth real-valued functions on R-n/Structure
  ==Structure of {{M|C^\infty(U)}} where {{M|U\subseteq\mathbb{R}^n}} is open== * {{M|C^\infty(U)}} is a [[vector space]] where:
  636 B (103 words) - 23:43, 21 October 2015
• Sandbox
  A \ar[r]^-f & B\otimes B\otimes B \ar[r]^g & X \\ A \ar[r]^f & B\otimes B\otimes B \ar[r]^g & X \\
  695 B (132 words) - 22:15, 26 October 2015
• Quotient vector space
  {{Refactor notice|grade=C|msg=See [[/New page]] for current work}} ...his means that {{M|1=[u]+[v]=\pi(\pi^{-1}([u])+\pi^{-1}([v]))=\underbrace{[x\in\pi^{-1}([u])+y\in\pi^{-1}([v])]}_\text{Well-defined-ness}=[u+v]}}<ref gr
  5 KB (879 words) - 23:09, 1 December 2016
• Books:Analysis - Part 1: Elements - Krzysztof Maurin
  ...ne in the middle that reads: "{{M|1=x-x_0:=h}}, {{M|1=f(x+h)-f(x)=f'(x)h+r(x,h)}}" should read: ...rgument is negated so it still sort of works out, either way replacing {{M|x}} with {{M|x_0}} is the easiest and most straightforward solution. This is
  1 KB (215 words) - 22:32, 26 February 2016
• Norm/Heading
  ...is a [[vector space]] over the [[field]] {{M|\mathbb{R} }} or {{M|\mathbb{C} }} * {{M|1=d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert}}
  1 KB (194 words) - 19:28, 25 January 2016
• Inner product/Infobox
  ...athbb{F} }}<br/> {{M|\mathbb{F} }} may be {{M|\mathbb{R} }} or {{M|\mathbb{C} }}. * {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle} }}
  1 KB (182 words) - 13:25, 14 February 2016
• Metric/Infobox
  ...2em;">{{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }}</span><br/>Where {{M|X}} is any [[set]] * {{M|1=d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert}}
  1 KB (180 words) - 10:39, 11 March 2016
• TestNavbox
  |list1=B, C |group2=X
  1 KB (132 words) - 20:11, 25 January 2016
• Covariant functor
  * 3 objects, {{M|X}}, {{M|Y}} and {{M|Z}} in a [[category]] {{M|\mathscr{C} }} * a (covariant) functor from {{M|\mathscr{C} }} to another category, {{M|\mathscr{D} }}
  1 KB (205 words) - 16:27, 2 February 2016
• Covariant functor/Definition
  A ''covariant functor'', {{M|T:C\leadsto D}} (for [[category|categories]] {{M|C}} and {{M|D}}) is a pair of [[mapping|mappings]]{{rAIRMACCF}}: ...\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & TX \end{array}\right. }}
  2 KB (253 words) - 15:47, 2 February 2016
• Contravariant functor/Definition
  A ''covariant functor'', {{M|S:C\leadsto D}} (for [[category|categories]] {{M|C}} and {{M|D}}) is a pair of [[mapping|mappings]]{{rAIRMACCF}}: ...\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & SX \end{array}\right. }}
  2 KB (263 words) - 16:53, 2 February 2016
• Types of category arrows
  | align=center | <span style="font-size:1.7em;">{{M|\xymatrix{X \ar@<-.5ex>[r]_g \ar@<.5ex>[r]^f & B \ar[r]^m & A} }}</span> ...M|1=\forall X\in\text{Ob}(\mathscr{C})\forall f,g\in\text{Arw}_\mathscr{C}(X,B)[(m\circ f=m\circ g)\implies f=g]}}
  1,012 B (181 words) - 14:43, 6 February 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/1 implies 2
  ...\Vert_X)}} and {{M|(Y,\Vert\cdot\Vert_Y)}} and also a [[linear map]] {{M|L:X\rightarrow Y}} then we have: * {{M|L}} is continuous at some {{M|p\in X}}
  5 KB (1,064 words) - 02:24, 28 February 2016
• Wedge (category theory)
  ...r of [[object|objects]] {{M|A}} and {{M|B}} in a [[category]] {{M|\mathscr{C} }} we define{{rAITCTHS2010}}: ...{{M|\mathscr{C} }} together with a pair of arrows (also from {{M|\mathscr{C} }}) as follows:
  992 B (149 words) - 23:00, 28 February 2016
• Cone (category theory)
  ...category theory)|objects]] {{M|A}}, {{M|B}} in a [[category]] {{M|\mathscr{C} }}, a ''cone''{{rAITCTHS2010}} is: ...coupled with two [[arrow (category theory)|arrows]] also from {{M|\mathscr{C} }} as follows:
  1 KB (197 words) - 22:27, 28 February 2016
• Cocone (category theory)
  ...category theory)|objects]] {{M|A}}, {{M|B}} in a [[category]] {{M|\mathscr{C} }}, a ''cocone''{{rAITCTHS2010}} is: ...coupled with two [[arrow (category theory)|arrows]] also from {{M|\mathscr{C} }} as follows:
  1 KB (182 words) - 22:28, 28 February 2016
• Product and coproduct compared
  ...pair {{M|A}}, {{M|B}} of [[object|objects]] in a [[category]] {{M|\mathscr{C} }} a: | align="center" | {{M|1=\xymatrix{ & A\\ X \ar[ur]^{f_A} \ar[dr]_{f_B} & \\ & B } }}
  2 KB (351 words) - 16:59, 1 March 2016
• Coproduct (category theory)
  Given a pair of objects {{M|A}} and {{M|B}} in a [[category]] {{M|\mathscr{C} }} a ''coproduct (of {{M|A}} and {{M|B}})'' is a{{rAITCTHS2010}}: ...{{M|1=\xymatrix{ A \ar[r]^{i_A} & S & B \ar[l]_{i_B} } }} (in {{M|\mathscr{C} }}) such that:
  1 KB (192 words) - 19:46, 1 March 2016
• Product (category theory)
  Given a pair of objects {{M|A}} and {{M|B}} in a [[category]] {{M|\mathscr{C} }} a ''product (of {{M|A}} and {{M|B}})'' is a{{rAITCTHS2010}}: ...{{M|1=\xymatrix{ A & S \ar[l]_{p_A} \ar[r]^{p_B} & B} }} (in {{M|\mathscr{C} }}) such that:
  1 KB (192 words) - 23:32, 29 February 2016
• Monic
  ...arrow]], {{M|B\mathop{\longrightarrow}^mA}} in a [[category]] {{M|\mathscr{C} }} is ''monic'' if{{rAITCTHS2010}}: ...1=\forall X\in\text{Ob}(\mathscr{C})\ \forall f,g\in\text{Hom}_\mathscr{C}(X,B)[(m\circ f=m\circ g)\implies f=g]}}
  986 B (163 words) - 13:52, 13 March 2016
• Epic
  ...arrow]], {{M|A\mathop{\longrightarrow}^eB}} in a [[category]] {{M|\mathscr{C} }} is ''epic'' if{{rAITCTHS2010}}: ...\forall X\in\text{Ob}(\mathscr{C})\ \forall f,g\in\text{Hom}_\mathscr{C}(B,X)[(f\circ e=g\circ e)\implies f=g]}}
  987 B (163 words) - 13:59, 13 March 2016
• Every convergent sequence is Cauchy
  ...to get the ball rolling. Page is of low grade due to ease of proof.|grade=C}} If a [[sequence]] {{M|1=(a_n)_{n=1}^\infty}} in a [[metric space]] {{M|(X,d)}} [[convergence (sequence)|converges]] (to {{M|a}}) then it is also a [[
  779 B (135 words) - 21:23, 19 April 2016
• Predicate
  ...|n}}-place relation]]{{M|\subseteq \underbrace{X\times X\times\ldots\times X}_{n\ \text{times} } }}</ref>. ...if {{M|x\in P}}<ref name="TAPL">Types and Programming Languages - Benjamin C. Peirce</ref>
  916 B (160 words) - 18:44, 18 March 2016
• Sigma-algebra/Infobox
  ...ght]}}</div>For a {{sigma|algebra}} {{M|(X,\mathcal{A}\subseteq\mathcal{P}(X))}} ...he properties like being closed under set-subtraction, containing both {{M|X}} and {{M|\emptyset}}}}
  635 B (92 words) - 01:13, 19 March 2016
• Sigma-algebra/Definition
  ...{{M|\mathcal{A} }}<ref group="Note">So {{M|\mathcal{A}\subseteq\mathcal{P}(X)}}</ref>, such that{{rMIAMRLS}}: * {{M|\forall A\in\mathcal{A}[A^C\in\mathcal{A}]}} - Stable under [[complement|complements]]
  779 B (122 words) - 01:25, 19 March 2016
• Symmetric difference
  Let {{M|A,B\in\mathcal{P}(X)}} be two [[subset|subsets]] of a [[set]] {{M|X}}. We define the ''symmetric difference'' of {{M|A}} and {{M|B}} as{{rMTH}} '''Claim 1: ''' this is equivalent to {{M|1=A\triangle B:=(A\cap B^C)\cup(A^C\cap B)}}<ref name="MTH"/>
  830 B (139 words) - 00:59, 21 March 2016
• Algebra of sets/Infobox
  ...l{P}(X)}}</span><br/>For an ''algebra of sets'', {{M|\mathcal{A} }} on {{M|X}} |data1={{M|\forall A\in\mathcal{A}[A^C\in\mathcal{A}]}}
  427 B (68 words) - 18:43, 1 April 2016
• Notes:IPv6
  ...tes = 8 blocks of 4 hex digits<ref name="Naming"/>, separated by colons ({{C|:}})<ref name="Naming"/>, subject to the following rules: # {{C|::}} may occur 0 or 1 time in an address. It means "the missing blocks (whe
  5 KB (837 words) - 06:12, 24 April 2016
• Regular topological space/Definition
  </noinclude>A [[topological space]], {{Top.|X|J}} is ''regular'' if{{rITTGG}}: ...thcal{J}[U\cap V=\emptyset\implies(E\subset U\wedge x\in V)]}} - (here {{M|C(\mathcal{J})}} denotes the [[closed set|closed sets]] of the [[topology]] {
  574 B (93 words) - 23:53, 3 May 2016
• Normal topological space/Definition
  </noinclude>A [[topological space]], {{Top.|X|J}}, is said to be ''normal'' if{{rITTGG}}: ...es(U\cap V=\emptyset\wedge E\subseteq U\wedge F\subseteq V)]}} - (here {{M|C(\mathcal{J})}} denotes the collection of [[closed set|closed sets]] of the
  468 B (73 words) - 00:00, 4 May 2016
• Urysohn's lemma
  ...nd {{M|F}} be a pair of ''[[disjoint]]'' [[closed set|closed sets]] of {{M|X}}, then{{rITTGG}}: * there exists a [[continuous function]], {{M|f:X\rightarrow [0,1]\subset\mathbb{R} }} such that {{M|f}} is {{M|0}} on {{M|E}
  583 B (101 words) - 00:21, 4 May 2016
• Lebesgue number
  {{Stub page|grade=C|msg=Remember to replace the diameter reminder with a subpage transclusion i ...} be a [[metric space]], and {{M|\mathcal{U} }} be a [[open cover]] of {{M|X}}. We define the ''Lebesgue number''{{rITTMJML}} as follows:
  1 KB (214 words) - 07:44, 10 May 2016
• Notes:Types of retractions
  Here {{Top.|X|J}} is a [[topological space]] ...|a\in A}} is called a ''retraction'' and {{M|A}} is the ''retract'' of {{M|X}}.
  6 KB (1,008 words) - 11:56, 2 June 2016
• LibSR:Proof of concept
  # A natural handling of shader variables, code must read {{C|1=z=x*y}} not {{C|varyings.assign("z",MAT4F::Multiply(varyings.get("y"),varyings.get("z"));}} #* However there was a problem, due to the {{C|{{ckw|virtual}}}} methods in play (among other things) it was slow, also it
  1 KB (215 words) - 08:18, 1 October 2017
• Notes:Differential (manifolds)
  * '''Smoothness of a map ({{AKA}}: {{M|C^\infty}}''' - a map, {{M|f:U\subseteq\mathbb{R}^n\rightarrow V\subseteq\mat * '''[[Derivation]]''' - a map, {{M|\omega:C^\infty(M)\rightarrow\mathbb{R} }} that is [[linear map|linear]] and satisfi
  4 KB (716 words) - 14:24, 16 May 2016
• Cantor's construction of the real numbers/Definition
  ...the [[quotient space (equivalence relation)|quotient space]], {{M|\mathscr{C}/\sim}} where:{{rAPIKM}} * {{M|\mathscr{C} }} - the [[set]] of all [[Cauchy sequence|Cauchy sequences]] in {{M|\mathb
  899 B (134 words) - 11:47, 2 June 2016