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
• Notes:Mean value theorem
  ** {{M|\exists c\in(a,b)}} such that {{MM|1=f'(c)=\frac{f(a)-f(b)}{b-a} }} ====For {{M|f:[x,x+h]\subseteq U\rightarrow\mathbb{R} }} for {{M|U}} open in a [[Banach space]
  3 KB (529 words) - 08:07, 4 June 2016
• Notes:Diff algorithm
  # {{C|>{{M|s}}}} - '''Insert''' - append {{M|s}} to the output string, do not mov # {{C|<}} - '''Delete''' - deletes from the input string {{M|A}} by incrementing
  12 KB (2,041 words) - 00:50, 27 June 2016
• Notes:The foundations of Mathematics - Kunen
  | {{M|1=\exists x(x=x)}} | {{M|1=\forall z(x\in x\leftrightarrow z\in y)\rightarrow x=y}}
  2 KB (342 words) - 02:38, 31 July 2016
• Task:Continuity types
  # {{M|f:\mathbb{R}\rightarrow\mathbb{R} }} is continuous at {{M|x\in\mathbb{R} }} if: ...\delta>0\forall a\in\mathbb{R}[\vert a-x\vert < \delta\implies\vert f(a)-f(x)\vert < \epsilon]}} - and a discussion of why this is intuitive, and how to
  3 KB (668 words) - 22:38, 4 August 2016
• Notes:Tangent space
  * {{M|1=M:=\{p\in\mathbb{R}^3\ \vert\ p=(x,y,x^2+y^2)\}\subset\mathbb{R}^3}} (with the subspace topology) and the two char *# {{M|1=i:\mathbb{R}^2\rightarrow M}} by {{M|1=i:(x,y)\mapsto(x,y,x^2+y^2)}} (think of it as like the "identity chart" it is what the manifold
  5 KB (1,002 words) - 19:42, 15 August 2016
• Topological vector space
  ...}} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.|X|J}} is a [[topological space]]. We call the [[tuple]]: ...ogical vector space". The topology is "more implicit" when we speak of {{M|X}} than the field of a vector space is, so often we will just write:
  2 KB (383 words) - 14:03, 16 February 2017
• Semi-ring of sets
  ...easure]] and later a [[measure]]) a {{C|pre-measure on {{M|X}}}} where {{M|X}} is say a semi-ring or something. All we need to do is show the pre-measure on {{M|X}} extends uniquely to a pre-measure to allow the theorems ([[extending pre-
  3 KB (508 words) - 17:25, 18 August 2016
• A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
  {{Stub page|grade=A*|msg=Demote to grade A once fleshed out and grade C once (most of) a proof has been added}} ...s]]'' [[ring of sets generated by|generated by]] a collection of sets, {{M|X}}.</ref>; furthermore this extension is unique{{rMIAMRLS}}. The details fol
  2 KB (390 words) - 22:16, 19 August 2016
• Types of set algebras/Measure theory summary table
  ! {{M|C}} | X
  559 B (79 words) - 20:05, 19 August 2016
• First order language
  ** {{M|C}} - The set of [[logical connective]] symbols. {{Caution|Not all of these a ...neg(\exists x(\neg(A)))}} or define {{M|\exists x(A)}} as {{M|\neg(\forall x(\neg(A)))}} }}
  3 KB (455 words) - 10:45, 8 September 2016
• Term (FOL)
  * {{C|1=Term ::= c {{!}} x {{!}} ft₁...t_n}}
  1,006 B (165 words) - 05:15, 8 September 2016
• Formula (FOL)
  # If {{M|A}} is a formula and {{M|x}} is a {{link|variable symbol|FOL}} then both: # {{M|\exists x(x\doteq y)}} is easier to read than {{M|\exists xx\doteq y}}
  3 KB (500 words) - 05:37, 8 September 2016
• Semantics of terms (FOL)
  ...a {{link|variable symbol|FOL}} then: {{M|1=x_{\mathbf{M}[\sigma]}:=\sigma(x)}} ...hbf{M}[\sigma]}:=c_\mathbf{M} }} (recall {{M|c_\mathbf{M} }} denotes {{M|I(c)}} where {{M|I}} is an {{link|interpretation|FOL}})
  1 KB (216 words) - 07:49, 10 September 2016
• Semantics of logical connectives (FOL)
  Let {{M|\mathscr{L} }} be a given [[first order language]] and let {{M|C}} denote the collection of all {{link|logical connective symbols|FOL}}<ref Suppose {{M|X}} and {{M|Y}} are variables which may only take on truth values (eg, are th
  2 KB (369 words) - 07:46, 11 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
  ...l spaces]]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. Then: ...[continuous map]], {{M|f:X\rightarrow Y}} to another continuous map, {{M|g:X\rightarrow Y}}, if {{M|f}} and {{M|g}} are [[homotopic]] is an equivalence
  2 KB (272 words) - 23:37, 14 October 2016
• Notes:Homotopy terminology/Terminology
  ...{{M|(:X\rightarrow Y)}}. Let {{M|1=I:=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}\subset\mathbb{R} }} # '''Homotopy - ''' any continuous map of the form {{M|H:X\times I\rightarrow Y}} such that: {{M|1=\forall a\in A\forall s,t\in I[H(a,
  900 B (184 words) - 14:40, 16 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces/proof
  # For all {{M|f\in C^0(X,Y)}} that {{M|f\simeq f\ (\text{rel }A)}}, symbolically: #* [[Reflexive]]: {{M|1=\forall f\in C^0(X,Y)[\homo{f}{f}]}}
  3 KB (533 words) - 07:33, 18 September 2016
• Characteristic property of the disjoint union topology/Statement
  {{Stub page|msg=A rewrite, while not urgent, would be nice|grade=C}} ...htarrow\coprod_{\alpha\in I}X_\alpha}} given by {{M|i_\beta:x\mapsto(\beta,x)}} are the [[canonical injections]]<noinclude>
  1 KB (238 words) - 20:05, 25 September 2016
• Disjoint union (set)
  {{Stub page|grade=C|msg=I've created this page to provide some content, it needs references and * {{M|1=(\beta,x)\in\coprod_{\alpha\in I}X_\alpha\iff(\beta\in I\wedge x\in X_\beta)}}
  1 KB (210 words) - 20:21, 25 September 2016
• Doctrine:Homotopy terminology
  * Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] * Let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset]] of {{M|X}}
  3 KB (535 words) - 09:01, 31 October 2016
• Closed map
  ...Y}} be a [[map]] (not necessarily [[continuous]] - just a map between {{M|X}} and {{M|Y}} considered as sets), then {{nowrap|we call {{M|f}} a ''closed ...ink|image|map|s}} (under {{M|f}}) of all {{plural|closed set|s}} of {{Top.|X|J}} are [[closed set|closed]] in {{Top.|Y|K}}
  1 KB (246 words) - 19:59, 26 September 2016
• Every surjective map gives rise to an equivalence relation
  ...surjective property is never used! It is true though that every map, {{M|f:X\rightarrow Y}} gives rise to an equivalence relation, where {{M|x_1\sim x_2 ...case we can {{link|factor|function}} {{M|f}} through {{M|\pi:X\rightarrow X/\sim}} always to yield {{M|\bar{f} }}, and "distil" the information of {{M|
  2 KB (315 words) - 13:54, 8 October 2016
• Dense
  ...{{M|X}}, say {{M|A\in\mathcal{P}(X)}}, we say {{M|A}} is ''dense'' in {{M|X}} if: ...M|\text{Closure}(A)}}]]{{M|\eq X}} (sometimes written: {{M|\overline{A}\eq X}})
  6 KB (1,097 words) - 04:15, 1 January 2017
• Equivalent statements to a set being dense
  {{Stub page|grade=A*|msg=Demote to grade C or D once more theorems have been sought out and the page resembles a page ...rary [[subset of]] {{M|X}}. Then "{{M|E}} is [[dense set|dense]] in {{Top.|X|J}}" is equivalent to any of the following:
  3 KB (490 words) - 20:18, 28 October 2016
• Closure of a set in a topological space
  ...and [[boundary of a set in a topological space]], boundary is the reason {{C|closure (topology)}} couldn't be used as even in [[Topology (subject)|topol :* {{C|[[closure (set, topology)]]}} redirects here, for use with [[template:link]
  2 KB (256 words) - 10:16, 28 September 2016
• Disconnected (topology)
  {{Stub page|grade=C|msg=There's not much more to be said, but it does meet the defining criteri ...op.|X|J}} as given above), then we say {{M|A}} is ''disconnected in {{Top.|X|J}}'' if{{rITTBM}}:
  1 KB (153 words) - 23:59, 30 September 2016
• A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
  Let {{Top.|X|J}} be a [[topological space]], then{{rITTMJML}}{{rITTBM}}: ...e both [[open set|open]] and [[closed set|closed]] in {{Top.|X|J}} are {{M|X}} itself and {{M|\emptyset}}
  461 B (69 words) - 22:52, 30 September 2016
• Every continuous map from a non-empty connected space to a discrete space is constant
  Let {{Top.|X|J}} be a non-empty<ref group="Note">meaning {{M|1=X\neq\emptyset}}</ref> ''{{link|connected|topology}}'' [[topological space]] ...It should go without saying, but a map is constant if {{M|1=\forall p,q\in X[f(p)=f(q)]}}</ref>
  851 B (138 words) - 23:02, 30 September 2016
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
  Let {{Top.|X|J}} be a [[topological space]], and let {{Top.|Y|K}} be a [[discrete topolo * {{Top.|X|J}} is {{link|disconnected|topology}} (ie, not {{link|connected|topology}})
  1 KB (172 words) - 23:12, 30 September 2016
• A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
  Let {{Top.|X|J}} be a [[topological space]], then{{rITTMJML}}: * {{Top.|X|J}} is {{link|disconnected|topology}} (ie: not {{link|connected|topology}})
  833 B (126 words) - 23:15, 30 September 2016
• Notes:Homology
  & & x \ar@/_.25pc/[dll]_a \\ & & z \ar@/_.5pc/[uu]^c \ar@/_2pc/[uu]_d
  6 KB (897 words) - 07:30, 15 October 2016
• The image of a compact set is compact
  ...M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}} and let {{M|f:X\rightarrow Y}} be a ''[[continuous]]'' [[map]]. Then: ...}} being an [[open cover]] of {{M|A}} by sets [[open set|open]] in {{Top.|X|J}}
  2 KB (332 words) - 17:20, 18 December 2016
• Homeomorphic
  Two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}}, are said to be ''homeomorphic'' if there exists a [[h ...nce relation]] on the morphisms (which are continuous maps) between {{Top.|X|J}} and {{Top.|Y|K}}.
  883 B (132 words) - 11:52, 8 October 2016
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
  ...ow Y}} be any [[function]] between them, and let {{M|\sim\subseteq X\times X}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by * {{M|1=\forall x,x'\in X[x\sim x'\iff f(x)=f(x')]}}
  6 KB (1,097 words) - 20:24, 9 October 2016
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
  ...hat would require its inverse was also continuous</ref>, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}}<ref>[[File:MondTop2016ex1.pdf]]</ref>. ...by that map yields an injective continuous map]]''" that {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} is [[injective]] and [[continuous]].
  2 KB (264 words) - 22:32, 9 October 2016
• A subspace of a Hausdorff space is Hausdorff
  {{Stub page|grade=C|msg=Flesh out and check the proof before removing this}} ...nsidered as a [[topological subspace]], {{M|(A,\mathcal{J}_A)}}, of {{Top.|X|J}} is also Hausdorff{{rITTMJML}}.
  2 KB (387 words) - 12:51, 10 October 2016
• Properties of the pre-image of a function
  Let {{M|X}} and {{M|Y}} be [[sets]] and let {{M|f:X\rightarrow Y}} be a [[function]] between them. Then: # For {{M|1=A\in\mathcal{P}(Y)[f^{-1}(Y-A)=X-f^{-1}(A)]}} - [[corollary]] to 3
  2 KB (417 words) - 00:51, 14 October 2016
• A map is continuous if and only if the pre-image of every closed set is closed
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[mapping]], then{{rITTMJML}}{{rITTBM}}{{rTJRM}}: ...{M|1=\forall E\in C(\mathcal{K})[f^{-1}(E)\in C(\mathcal{J})]}} (where {{M|C(\mathcal{H})}} denotes the ''[[set of all closed sets|set of all]]'' [[clos
  2 KB (378 words) - 01:39, 14 October 2016
• Exercises:Mond - Topology - 2/Section B/Question 6
  ...simply the union of a [[Möbius strip]] together with a "solid 8" shape, {{C|oo}} but joined. Take the left "disk" in this formation and lift it, then r [[File:N sliding along pole.JPG|thumb|Showing {{M|N}} is the joined {{C|oo}} I speak of.]]
  7 KB (1,331 words) - 12:27, 19 October 2016
• Notes:Chain complex of modules
  * <div style="overflow:hidden;">{{M|\mathcal{C}: \xymatrix { \cdots \ar@{<-}[r] & C_{n-1} \ar@{<-}[r]^{\partial_n} & C_n \ A ''positive complex'' {{M|\mathcal{C} }} has {{M|1=C_n=0}} (the [[trivial module]]) for all {{M|n<0}} and is usu
  3 KB (440 words) - 17:15, 20 October 2016
• Exercises:Measure Theory - 2016 - 1/Section B/Problem 1
  #***** {{M|\forall C,D\in\mathcal{A}_k[C\cup D\in\mathcal{A}_k]}} ...algebras {{M|1=\mathcal{A}_n:=\sigma(\mathcal{C}_n)}} where {{M|1=\mathcal{C}_n:=\mathcal{P}(\{1,2,\ldots,n\})}} where {{M|1=\{1,2,\ldots,n\}\subset\mat
  10 KB (1,844 words) - 14:09, 23 October 2016
• Subset of
  ...M|\forall a\in A[a\in B]}} which comes from {{M|1=\forall x[x\in A\implies x\in B]}}. {{Requires references|grade=C|msg=Could use something, obvious to layperson}}
  1 KB (208 words) - 19:21, 28 October 2016
• The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set
  {{Stub page|grade=C|msg=Obvious and easy "theorem", created to make the proof of claims in [[De * {{M|1=(A\cap B\ne\emptyset)\iff(\exists x\in A[x\in B])}}. As [[intersection is commutative]], it follows that {{M|1=\iff(\e
  2 KB (287 words) - 19:31, 28 October 2016
• Equivalent conditions to a set being bounded/Statement
  ...space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then the following are all logical equivalent to each other<ref group="N # {{M|1=\exists C<\infty\ \forall a,b\in A[d(a,b)<C]}} - {{M|A}} is [[bounded]] (the definition)
  720 B (124 words) - 23:41, 29 October 2016
• 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
• Omega(X,b)
  {{DISPLAYTITLE:{{M|\Omega(X,b)}}}}{{Stub page|grade=B|msg=Check, maybe find another reference, remove a ...{{M|b\in X}} be given. Then {{M|\Omega(X,b)\subseteq}}[[C(I,X)|{{M|C([0,1],X)}}]] is the set containing all [[loops]] based at {{M|b}}{{rITTMJML}}. That
  2 KB (364 words) - 04:47, 3 November 2016
• The set of continuous functions between topological spaces
  ...,Y)}} denotes the [[set]] of all ''[[continuous]]'' [[functions]] from {{M|X}} to {{M|Y}}, with respect to the {{plural|topolog|y|ies}}: {{M|\mathcal{J} * {{M|\big(f\in C(X,Y)\big)\iff\big(f:X\rightarrow Y\text{ is a continuous function}\big)}}
  1 KB (235 words) - 05:02, 3 November 2016
• Index of spaces, sets and classes
  * [[The set of continuous functions between topological spaces]] - {{M|C(X,Y)}} ** [[C(I,X)|{{M|C([0,1],X)}}]]
  335 B (61 words) - 05:06, 3 November 2016
• Proof that the fundamental group is actually a group
  ...(I,X)}}]]</ref>. We claim that from this we can make a [[group]], {{M|(\pi(X,b),*)}} called [[the fundamental group]] where{{rITTMJML}}: * {{MM|1=\pi(X,b):=\frac{\Omega(X,b)}{\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\}\big)} }}
  2 KB (281 words) - 11:20, 8 November 2016
• Proof that the fundamental group is actually a group/Outline
  ...{M|X}} - but considered only on the subset of {{M|C([0,1],X)}}, {{M|\Omega(X,b)}}. Then we define: {{MM|1=\pi_1(X,b):=\frac{\Omega(X,b)}{\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})\big)} }
  3 KB (454 words) - 18:31, 4 November 2016
• Homotopy concatenation
  ...- the [[closed unit interval]]</ref> be {{plural|homotop|y|ies}} from {{M|X}} to {{M|Y}}. Suppose: * {{M|1=\forall x\in X[H_1(x,1)=H_2(x,0)]}} - that the final stage of {{M|H_1}} is the same as the initial stage
  2 KB (260 words) - 05:09, 6 November 2016
• Homotopy invariance of loop concatenation
  ...is a path and {{M|1=\ell(0)=\ell(1)=b}}.<br/><br/>Furthermore, {{M|\Omega(X,b)}} is not just a [[set]], it does have a [[group]] structure we can imbue * {{M|1=\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)[([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2'])\implies[\ell_1*\ell_2]=[\el
  3 KB (462 words) - 09:21, 6 November 2016
• Homotopy invariance of path concatenation
  ...preserving homotop|y|ies}} (where {{M|H_1,H_2:[0,1]\times [0,1]\rightarrow X}} are the specific {{plural|homotop|y|ies}} of the {{link|path|topology|s}} *** {{M|1=p_1*p_2:[0,1]\rightarrow X}} by {{M|p_1*p_2:t\mapsto\left\{\begin{array}{lr}p_1(2t)&\text{for }t\in[0,
  2 KB (273 words) - 19:11, 9 November 2016
• Notes:Polynomial ring
  ...that he means a ring with identity or what?</ref> in the indeterminate {{M|X}} is a [[mapping]]: * {{M|A:M\rightarrow R}} by {{M|A:X^n\mapsto a_n}} such that {{M|a_n\eq 0}} for "almost all" {{M|n\ge 0}}
  1 KB (220 words) - 19:42, 19 November 2016
• Absolute value (object)
  {{Stub page|grade=C|msg=Needed for work with series weirdly enough!}} ..._v:F\rightarrow\mathbb{R} }} given by {{M|\vert\cdot\vert_v:x\mapsto \vert x\vert_v}}
  2 KB (350 words) - 05:25, 21 November 2016
• UTLOC:1
  ...defined by: {{M|1=b_n:\eq\left\{\begin{array}{lr}a_{n-1}&\text{if }n\ge 2\\c\in\mathbb{R}&\text{if }n\eq 1\end{array}\right.}} also {{link|converges|seq ** Here {{M|c\in\mathbb{R} }} is any value. I used {{M|0\in\mathbb{R} }} on paper as I wa
  4 KB (795 words) - 10:03, 22 November 2016
• Operations on convergent sequences of real numbers
  ...s considered convergent with respect to the [[metric]] {{M|d(x,y):\eq\vert x-y\vert}}). Suppose that [[limit (sequence)|{{M|\lim_{n\rightarrow\infty}(a_ Let {{M|c\in\mathbb{R} }} be any [[real number]]. Immediately we have the following r
  5 KB (900 words) - 05:45, 23 November 2016
• The vector space of all linear maps between two spaces
  ...\} }}, here {{M|\mathcal{F}(U,V)}} denotes the [[set of all functions from X to Y|set of all functions from {{M|U}} to {{M|V}}]]<ref group="Note">You ma ...vector space, however the remainder of the proof is easy and routine|grade=C}}
  2 KB (400 words) - 21:16, 17 November 2016
• Notes:Richard Sharp question
  ...t f\Vert_\infty \frac{1}{n^2\delta^2}\sum_{k\eq 0}^n(k-nx)^2\ {}^nC_kx^k(1-x)^{n-k} }}</span> ...rt f\Vert_\infty \sum_{k\eq 0}^n\frac{(k-nx)^2}{n^2\delta^2}\ {}^nC_kx^k(1-x)^{n-k} }}
  2 KB (424 words) - 21:03, 25 November 2016
• Weierstrass approximation theorem
  ** {{M|\forall f\in C([0,1],\mathbb{R})\forall\epsilon>0\exists n\in\mathbb{N}[\Vert f-B_N(f)\Ver ...]] from the [[closed interval]] {{M|[a,b]:\eq\{x\in\mathbb{R}\ \vert\ a\le x\le b\}\subset\mathbb{R} }} to the [[real line]], {{M|\mathbb{R} }}. We cons
  8 KB (1,610 words) - 08:17, 28 December 2016
• Notes:An introduction to manifolds - Loring W. Tu
  ====1.1: {{M|C^\infty}} vs Analytic functions==== ...eq\lim_{t\rightarrow 0}\left(\frac{f(c(t))-f(p)}{t}\right)\eq\frac{d}{dt}f(c(t))\Big\vert_{t\eq 0} }}
  5 KB (910 words) - 14:07, 1 December 2016
• Characteristic property of the tensor product/Statement
  : '''Notice: ''' this page is supposed to be transcluded, use {{C|1=full=true}} to show claims and extra things ...ue [[linear map]], {{M|\overline{A}:V_1\otimes\cdots\otimes V_k\rightarrow X}} such that:
  2 KB (268 words) - 22:07, 20 December 2016
• Pointed topological space
  {{Stub page|grade=C|msg=Proper stub!}} ...}} be a [[topological space]] and let {{M|p\in X}} be a fixed point in {{M|X}}. Then:
  442 B (72 words) - 21:26, 12 December 2016
• Nth homotopy group
  : '''Note: ''' the [[fundamental group]] is {{M|\pi_1(X,p)}} ...ub page|grade=A*|msg=Not super urgent, but would be useful - demote to {{C|C}} once the content is less note-like}}
  2 KB (409 words) - 22:17, 12 December 2016
• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
  ...]] and let {{M|f_1,f_2:I\rightarrow X}} be {{link|path|topology|s}} in {{M|X}} such that: ...|H:f_1\simeq f_2\ (\text{rel }\{0,1\})}}, where {{M|H:I\times I\rightarrow X}} is the [[homotopy]] between the paths {{M|f_1}} and {{M|f_2}}.
  2 KB (282 words) - 22:38, 12 December 2016
• Finite complement topology
  ...e can give {{M|X}} called "the finite complement topology", such that {{M|(X,\mathcal{J})}} is a [[topological space]]. It is defined as follows{{rFAVID ...p="Note">Many authors give the {{M|U\eq\emptyset}} condition as {{M|X-U\eq X}}. It is easy to see however that:
  1 KB (233 words) - 09:21, 30 December 2016
• List of topological properties
  Here {{M|(X,\mathcal{J})}} is a [[topological space]] or {{M|(X,d)}} is a [[metric space]] in the definitions. | Let {{M|A\in\mathcal{P}(X)}} be given. The ''closure'' of {{M|A}}, denoted {{M|\overline{A} }} is def
  4 KB (630 words) - 19:33, 16 February 2017
• Index of notation for sets of continuous maps/Index
  ...X,Y)}}]] - for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}, {{M|C(X,Y)}} is the [[set]] of all [[continuous maps]] between them. ...}}, set of all {{link|path|topology|s}} on a [[topological space]] {{Top.|X|J}}
  2 KB (463 words) - 06:20, 1 January 2017
• Index of notation/C
  ! [[C(X,Y)|{{M|C(X,Y)}}]] ...example: [[C(I,X)|{{M|C(I,X)}}]] - all {{link|path|topology|s}} in {{Top.|X|J}}. These sets often have additional structure (eg, [[vector space]], [[al
  958 B (151 words) - 06:13, 1 January 2017
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.5
  ...mathbb{Z}^3\ \vert\ \exists x\in\mathbb{Z}^3[A(x)\eq y]\}\eq\{A(x)\ \vert\ x\in\mathbb{Z}^3\} }} ** {{M|\text{Ker}(A):\eq\{x\in\mathbb{Z}^3 \ \vert\ A(x)\eq 0\} }}
  3 KB (473 words) - 19:34, 14 January 2017
• If the intersection of two open balls is non-empty then for every point in the intersection there is an open ball containing it in the intersection
  * Demote to grade {{C|B}} once this is linked to by a few pages. }}{{M|\newcommand{\ball}[1]{B_{r ...that {{M|B_{r_1}(x_1)}} and {{M|B_{r_2}(x_2)}} are [[open balls]] of {{M|(X,d)}}). Let {{M|B_i:\eq B_{r_i}(x_i)}} (for brevity<ref group="Note">The len
  6 KB (1,007 words) - 20:16, 16 January 2017
• Given two open balls sharing the same centre but with differing radius then the one defined to have a strictly smaller radius is contained in the other
  * Demote to {{C|B}} once links have been added}} Let {{M|(X,d)}} be a metric space. Let {{M|x\in X}} be given and let {{M|r_1,r_2\in\mathbb{R}_{>0} }} be given such that {{M|
  2 KB (391 words) - 20:27, 16 January 2017
• Topology induced by a metric
  {{Stub page|grade=A*|msg=Demote to grade {{C|B}} once there are links in place to this page}} ...c topology'' that can be defined in terms of the [[metric]], {{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }}.
  2 KB (388 words) - 12:05, 17 January 2017
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.2
  ...PQRSTUVWXYZ} }}. I notice that {{M|\sf{G} }} here is homeomorphic to a {{M|C}}, so I have included {{M|\underline{\text{G} } }}, this represents {{M|G}} * I also include {{M|\mathcal{Z} }} representing a {{M|\sf{Z} }} with a {{C|-}} through the middle, again due to how common this form is
  17 KB (3,132 words) - 12:03, 18 January 2017
• Closed interval
  {{Stub page|grade=C|msg=Traditional stub page}} * {{M|[a,b]:\eq\left\{x\in\mathbb{R}\ \vert\ a\le x\le b\right\} }}
  1 KB (233 words) - 14:19, 19 January 2017
• Notes:CW-Complex
  ...]]'' [[open cell|open cells]], {{M|\{e_\alpha\}_{\alpha\in I} }}, with {{M|X\eq\bigcup_{\alpha\in I}e_\alpha}}, such that: # {{Top.|X|J}} is a [[Hausdorff space]]
  10 KB (1,736 words) - 01:00, 23 January 2017
• Boundary (topology)
  ...space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then the ''boundary'' of {{M|A}}, denoted {{M|\partial A}} is defined as * {{M|\partial A:\eq X-(\text{Int}(A)\cup\text{Ext}(A))}} - where {{M|\text{Int}(A)}} denotes the
  2 KB (307 words) - 22:57, 23 January 2017
• The ell p spaces
  Let {{M|p\in[1,+\infty]:\eq\{x\in\overline{\mathbb{R} }\ \vert 1\le x\} }} be given. We define the {{M|\ell^p}} [[normed space]] as follows: ...l p|{{MM|\ell^p}}]]{{MM|:\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \sum_{n\eq 1}^\infty \vert x_n\vert^p<+\infty\right\}\right.
  2 KB (296 words) - 14:30, 26 January 2017
• Notes:Stone-Weierstrass theorem
  ...is strictly [[corollary]] to {{M|A}}, that is {{C|A{{M|\implies}}B}} but {{C|B ''(does not imply)'' A}} ...{M|)}} be given^Important:<ref group="Note">There are a lot of {{C|K}}s in play here. As per ''[[Doctrine:Notation for sets of continuous maps
  4 KB (791 words) - 13:29, 27 January 2017
• Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
  ...directly from the definitions (Hatcher, of course...) that {{M|H^\Delta_0(X)\cong\mathbb{Z} }} * {{MM|H^\Delta_n(X):\eq\frac{\text{Ker}(\partial_0)}{\text{Im}(\partial_1)} }}
  13 KB (2,312 words) - 06:33, 1 February 2017
• Successor of a set
  ...{{M|X}} be a [[set]]. The ''successor'' of the set {{M|X}}, written {{M|S(x)}}, is defined as follows{{rITSTHJ}}: * {{M|S(x):\eq x\cup \{x\} }}
  2 KB (305 words) - 15:14, 3 February 2017
• Notes:Delta complex/Formal attempt
  ...complex]] in the back of your mind, and a simplex as being like {{M|\{a,b,c\} }} for a triangle and such. * Let {{M|\#(n):\eq\{1,\ldots,n\}\subset\mathbb{N} }} - I did want to use {{M|C(n)}} for "count" or "consecutive" but given the context that'd be a poor ch
  5 KB (966 words) - 14:36, 6 February 2017
• Convex set
  {{Stub page|grade=C|msg=Needs a reference and tidying up! ...{M\mathbb{C} }} and let {{M|C\in\mathcal{P}(X)}} be given. Then we say {{M|C}} is ''convex'' if:
  799 B (143 words) - 11:27, 9 February 2017
• Convex
  ...ce]] {{M|(X,\mathbb{K})}} a [[subset of]] {{M|X}}, say {{M|C\in\mathcal{P}(X)}} is convex if ** {{M|\forall x,y\in C\forall t\in[0,1]\subset\mathbb{R}[x+t(y-x)\in C]}} - the line between any two points in the set is also in the set
  532 B (92 words) - 14:50, 9 February 2017
• Notes:Functional Analysis II
  ...als]], {{M|\mathbb{R} }}, or the field of [[complex numbers]], {{M|\mathbb{C} }} ** He uses {{M|B_X(0,1)}} - ball notation for {{M|(X,\Vert\cdot\Vert)}} centred at {{M|0}} of radius {{M|1}} - {{Caveat|easily m
  4 KB (818 words) - 12:00, 9 February 2017
• Convex function
  #* In symbols: {{M|\forall x,y\in S\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in S]}}, and ...e line {{M|[x,y]}} is {{M|\le}} the point {{M|t}}-far along the line {{M|f(x)}} to {{M|f(y)}}
  1 KB (224 words) - 10:54, 10 February 2017
• Intermediate value theorem
  ...a ''{{link|connected|topology}}'' [[topological space]] and suppose {{M|f:X\rightarrow}}[[the real line|{{M|\mathbb{R} }}]] is a [[continuous function] ...ally means {{M|(a\le b)\wedge(b\le c)}} - so {{M|a\le b}} ''and'' {{M|b\le c}}.</ref>
  5 KB (979 words) - 17:35, 17 February 2017
• Real projective space
  * Demote to grade C once charts and definition 1 is in place [[User:Alec|Alec]] ([[User talk:Al ...ize:1.6em;"><mm>\frac{\mathbb{S}^n\subset\mathbb{R}^{n+1} }{\langle x\sim -x\rangle}</mm></span>
  2 KB (289 words) - 09:08, 18 February 2017
• Notes:Ell^p(C) is complete for p between one and positive infinity inclusive
  ...\subseteq\overline{\mathbb{R} } }} the space [[ell^p(C)|{{M|\ell^p(\mathbb{C})}}]] is [[complete metric space|complete]]. * Let {{M|(\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})}} be given
  4 KB (664 words) - 18:56, 22 February 2017
• Example:A bijective and continuous map that is not a homeomorphism
  {{Stub page|grade=C|msg=Maybe add others? Tidy the page up a bit!}} ...rbrace{\mathbb{S}^1}_{\subseteq\mathbb{C} } }} by {{M|f:r\mapsto e^{2\pi j x} }}, then:
  856 B (142 words) - 23:04, 22 February 2017
• Path-connected topological space
  Let {{Top.|X|J}} be a [[topological space]], we say that {{M|X}} is ''path connected'' or ''is a path connected (topological) space'' if t * {{M|\forall x_1,x_2\in X\exists p\in }}[[C(I,X)|{{M|C([0,1],X)}}]]{{M|[p(0)\eq x_1\wedge p(1)\eq x_2]}}
  2 KB (249 words) - 12:52, 23 February 2017
• Evenly covered by a continuous map
  ...t {{M|U\in\mathcal{J} }} be given, so {{M|U}} is an [[open set]] of {{Top.|X|J}}, we say that:
  2 KB (394 words) - 21:54, 24 February 2017
• Notes:Covering spaces
  ...t {{M|p:E\rightarrow X}} be [[continuous map]], we say {{M|U}} open in {{M|X}} is ''evenly covered'' by {{M|p}} if: ** {{M|p:E\rightarrow X}} is a covering map if:
  3 KB (658 words) - 19:20, 25 February 2017
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6
  ...homology groups of {{M|T^2:\eq\mathbb{S}^1\times\mathbb{S}^1}} and of {{M|X:\eq\mathbb{S}^1\vee\mathbb{S}^1\vee\mathbb{S}^2}} # Prove that {{M|T^2}} and {{M|X}} are not [[homotopy equivalent spaces]]
  10 KB (1,664 words) - 12:43, 1 March 2017
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6/X Complex
  ! colspan="3" | {{M|\Delta}}-complex for {{M|X}} ...{\mathbf{A} } }|(.7){\LARGE{\mathbf{ B} } } & & \bullet_{v_4} \ar@{<-}[ur]^x \\
  806 B (126 words) - 16:58, 28 February 2017
• Unique lifting property
  ...e {{Top.|Y|K}} is a [[connected topological space]] and {{M|f:Y\rightarrow X}} is a [[continuous map]], then{{rITTGG}}^Partial:{{rITTMJML}}<su {{Requires work|grade=C|msg={{Warning|What follows is VERY messy.}} I was distracted when writing i
  13 KB (2,510 words) - 16:23, 2 March 2017
• A pair of identical elements is a singleton
  * {{M|\exists x[x\in\{t,t\}\wedge\forall y(y\in\{t,t\}\rightarrow y\eq x)]}} (as per definition of {{link|singleton|set theory}} ...is the pairing axiom, in this case {{M|A}} and {{M|B}} are {{M|t}} and {{M|C}} is the (it turns out unique) set {{M|\{t,t\} }}
  2 KB (315 words) - 23:35, 8 March 2017
• Exercises:Saul - Algebraic Topology - 9/Exercise 9.7
  ...at the {{link|cone|topology}} on the [[real projective plane]], {{ie}} {{M|C(\mathbb{RP}^2)}}, is not a [[topological n-manifold with boundary|topologic {{M|\newcommand{\crp}{C(\mathbb{RP}^2)} }}
  8 KB (1,299 words) - 13:33, 15 March 2017
• A sequence consisting of the nth terms of the sequences in a Cauchy sequence of elements in any little-L space is itself a Cauchy sequence of complex numbers
  ...o elements in {{M|\ell^p}} are {{M|(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C} }} such that certain properties hold. * {{ie}} {{M|\big((x^1_n)_{n\in\mathbb{N} },(x^2_n)_{n\in\mathbb{N} },\ldots,(x^k_n)_{n\in\mathbb{N} },\ldots\big)\subseteq\ell^p}} is a Cauchy sequence
  1 KB (238 words) - 17:52, 18 March 2017
• Metrically bounded set
  ...space]] and let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset of]] {{M|X}}. We say {{M|A}} is ''(metrically)<ref group="Note">Alec's terminology to ...C\in}}[[Positive reals|{{M|\mathbb{R}_{>0} }}]]{{M|\forall a,b\in A[d(a,b)<C]}}
  3 KB (673 words) - 23:10, 18 March 2017
• A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
  ...space]] and let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset of]] {{M|X}}. Then we claim{{rFAVIDMH}}: ...\underbrace{\forall x\in X\exists C\in\mathbb{R}_{>0}\forall a\in A[d(a,x)<C]}_{\text{Claim} }\big)}}
  6 KB (1,092 words) - 00:41, 19 March 2017
• If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
  ...ooth atlas]] on some ''[[locally euclidean]]'' [[topological space]], {{M|(X,\mathcal{J})}}<ref group="Note">This page's statement is used to build a [[ Recall that to be considered smooth or {{M|C^\infty}} they must be smooth at each point in the domain, we will show {{M|
  6 KB (1,182 words) - 13:38, 1 April 2017
• Example:Smooth map between the real line and the circle (considered as smooth manifolds)
  ...}_\mathbb{R} }} is the [[identity map]], {{M|\text{Id}_\mathbb{R}:x\mapsto x}}</ref>) and let [[Circle|{{M|\mathbb{S}^1}}]] be considered as a smooth ma ...>0\} }} and {{M|\varphi_+:U^+\rightarrow I}} by {{M|\varphi_+:(x,y)\mapsto x}}
  4 KB (757 words) - 13:25, 2 April 2017
• If an inner product is non-zero then both arguments are non-zero
  Let {{M|((X,}}[[K (field)|{{M|\mathbb{K} }}]]{{M|),\langle\cdot,\cdot\rangle)}} be an [ * {{M|\forall x,y\in X[\langle x,y\rangle\neq 0\implies(x\neq 0\wedge y\neq 0)]}}
  1 KB (181 words) - 23:40, 7 April 2017
• Given a Hilbert space and a non-empty, closed and convex subset then for each point in the space there is a closest point in the subset
  Let {{M|(X,\langle\cdot,\cdot\rangle)}} be a [[Hilbert space]]<ref group="Note">Recall ...nduced by the inner-product]], {{M|\Vert\cdot\Vert:x\mapsto\sqrt{\langle x,x\rangle} }}
  3 KB (592 words) - 00:52, 7 April 2017
• K (field)
  ...e what {{M|\mathbb{K} }} means when encountered as a [[field]] (eg if {{M|(X,\mathbb{K})}} is a [[vector space]] - [[User:Alec|Alec]] ([[User talk:Alec| ...ers]], {{M|\mathbb{R} }}, or the [[field of complex numbers]], {{M|\mathbb{C} }}.
  2 KB (323 words) - 03:54, 8 April 2017
• Distance from a point to a set
  ...be given. For any point {{M|x\in X}} we define the ''distance between {{M|x}} and {{M|A}}''{{rFAVIDMH}} to be: * {{MM|d(x,A):\eq\mathop{\text{Inf} }_{a\in A}\Big(d(x,a)\Big)}}
  1,017 B (196 words) - 20:28, 9 April 2017
• The norm of a space is a uniformly continuous map with respect to the topology it induces
  {{Stub page|grade=C|msg=Check over before removing this, ensure it's linked to, maybe add "unif ...a [[normed space]], we claim that the [[norm]] itself, {{M|\Vert\cdot\Vert:X\rightarrow\mathbb{R} }}, is a [[uniformly continuous]] map, with respect to
  4 KB (687 words) - 20:59, 9 April 2017
• Square root
  ...R}_{\ge 0} }} where {{M|f(x)}} is typically written {{M|\sqrt{x} }} or {{M|x^\frac{1}{2} }} ...qrt{-1}(ad+bc)}}, obviously {{M|(\sqrt{-1})^2\eq-1}} so {{M|(a+b\sqrt{-1})(c+d\sqrt{-1})\eq ac-bd+\sqrt{-1}(ad+bc)}}
  858 B (151 words) - 05:22, 10 April 2017
• Square root (real function)
  {{Stub page|grade=C|msg=Made this page to prove smoothness really [[User:Alec|Alec]] ([[User ta ...}}, note that if {{M|y^2\eq x}} then {{M|-y}} is also a square root of {{M|x}}:
  2 KB (408 words) - 05:46, 10 April 2017
• Monotonicity of the integral of non-negative extended-real-valued measurable functions with respect to a measure
  Let {{M|(X,\mathcal{A},\mu)}} be a [[measure space]] and let {{M|f,g\in}}[[Set of non- * if {{M|f\le g}} - {{ie}}: {{M|\forall x\in X[f(x)\le g(x)]}} - then:
  3 KB (623 words) - 19:41, 14 April 2017
• C( )0,1( ,R) is not complete when considered with the L^1 norm
  {{DISPLAYTITLE:{{M|C([0,1],\mathbb{R})}} is not complete when considered with {{M|L^1}} norm}} Define {{M|(f_n)_{n\in\mathbb{N} }\subseteq C([0,1],\mathbb{R})}} as follows:
  2 KB (353 words) - 18:15, 23 April 2017
• Simply connected topological space
  Let {{Top.|X|J}} be a [[topological space]], we say {{M|X}} is ''simply connected'' if{{rITTMJML}}: * {{M|X}} is a [[path-connected topological space]], and
  4 KB (601 words) - 16:10, 24 April 2017
• Contractible topological space
  Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is a ''contractible topological space'' if{{rITTMJML}}: * {{M|\exists c\in X\big[(:x\mapsto c)\simeq \text{Id}_X\big]}}
  3 KB (544 words) - 20:00, 24 April 2017
• Homotopy equivalent topological spaces
  ...M|Y}}, or {{M|X}} and {{M|Y}} have the same ''homotopy type'', written {{M|X\simeq Y}}, if{{rITTMJML}}: * {{M|\exists f\in}}[[C(X,Y)|{{M|C(X,Y)}}]]{{M|\exists g\in C(Y,X)\big[(g\circ f\simeq }}[[Identity map|{{M|\text{Id}_X}}]]{{M|)\wedge(g\circ
  3 KB (596 words) - 21:13, 24 April 2017
• Square lemma (of homotopic paths)
  Let {{M|F:[0,1]\times[0,1]\rightarrow X}} be a [[continuous map]] (note this is sufficient to make it a [[homotopy] * {{M|f:[0,1]\rightarrow X}} by {{M|f(t):\eq F(t,0)}}
  2 KB (435 words) - 14:51, 25 April 2017
• Experiment:The wx wrapping function
  ...wxGraph'' project folder, and {{C|wxStaticWrappedText}} (my class, not a {{C|wx}} class) ...{{C|wxString}} to the label of a {{C|wxStaticText}}, we then tell it to {{C|Wrap}} passing the usable width as a parameter, the wrap function takes the
  2 KB (371 words) - 16:38, 23 April 2017
• Cauchy-Schwarz inequality for inner product spaces
  ...t\rangle:X\times X\rightarrow\mathbb{K} }} be an [[inner product]] so {{M|(X,\langle\cdot,\cdot\rangle)}} is an [[inner product space]], then{{rW2014LNF * {{M|\forall x,y\in X\left[\vert\ip{x,y}\vert\le\sqrt{\ip{x,x} }\sqrt{\ip{y,y} }\right]}}
  6 KB (1,279 words) - 13:00, 4 April 2017
• Poisson distribution
  |above=<span style="font-size:1.5em;">{{M|X\sim\text{Poi}(\lambda)}}</span> |data2={{MM|\mathbb{P}[X\eq k]:\eq e^{-\lambda}\frac{\lambda^k}{k!} }}
  8 KB (1,401 words) - 00:52, 20 July 2018
• Example:Joint probability distribution
  ...& \mathbf{S} & \mathbf{Y} }</mm><hr/><mm>\xymatrix{ A & 0 \ar[l] \ar[r] & C \\ & 1 \ar[ul] \ar[dr] & \\ B & 2 \ar[l] \ar[r] & D\\ & 3 \ar[ur] \ar[ul] } * {{M|X:S\rightarrow\{A,B\} }}<ref group="Note" name="rvmeasurability"/>
  7 KB (1,100 words) - 19:36, 13 September 2017
• Uniform probability distribution
  * for {{M|a,b\in\mathbb{N}_{\ge 0} }} we have: {{M|X\sim\text{Uni}(a,b)}} to mean: *for {{M|c\in\mathbb{R} }} we define: {{MM|\mathbb{P}[X\eq c]:\eq\left\{\begin{array}{lr}<!--
  1 KB (192 words) - 05:41, 15 January 2018
• Binomial distribution
  |above=<span style="font-size:1.5em;">{{M|X\sim\text{Bin}(n,p)}}</span> |data1={{M|\mathbb{P}[X\eq k]:\eq{}^n\text{C}_k\ p^k(1-p)^{n-k} }}
  4 KB (653 words) - 13:11, 22 September 2017
• Geometric distribution
  {{Stub page|grade=C|msg=Removed previous stub message and demoted [[User:Alec|Alec]] ([[User ta * {{M|\P{X\eq k} :\eq (1-p)^{k-1}p}} - {{link|pmf|statistics}} / {{link|pdf|statistics
  3 KB (557 words) - 15:14, 16 January 2018
• Exponential distribution/Definition
  Set {{c|home}} to something when using this page to change the "the proof of this i </noinclude>Let {{M|\lambda\in\mathbb{R}_{\ge 0} }} be given, and let {{M|X\sim\text{Exp}(\lambda)}} be an ''exponentially distributed'' [[random varia
  1 KB (192 words) - 01:27, 16 March 2018
• The sum of two random variables with Poisson distributions is a Poisson distribution itself
  {{Requires work|grade=C|msg=There's still some work to do on this page, but the gist is very much p # {{M|X\sim\text{Poi}(\lambda)}} and
  3 KB (536 words) - 22:46, 4 November 2017
• Mdm of the Poisson distribution
  ...xt{Poi} }}]]{{M|(\lambda)}} for some {{M|\lambda\in\mathbb{R}_{>0} }}. {{M|X}} may take any value in {{M|\mathbb{N}_0}} * {{MM|\text{Mdm}(X)\eq 2\lambda e^{-\lambda}\frac{\lambda^u}{u!} }}<ref>Alec's own work, I act
  7 KB (1,308 words) - 00:27, 8 November 2017
• Common integrals table
  {{Stub page|grade=C|msg=This page will become more orderly when things are added to it}} {{MM|\newcommand{\plusc}[1][]{ {\ +C#1} } }}
  805 B (135 words) - 08:11, 11 November 2017
• Workings of the Enigma machines
  ..." it is transferred to enter the wheels as the {{M|C}} signal, and the {{M|C}} key corresponds to an {{M|A}} signal entering the wheels :* consider any other item, say {{M|x}}, then {{M|T(T(x))\eq T(x)\eq x}} - as required
  5 KB (834 words) - 14:25, 15 December 2017
• Common data type ranges
  ...e moment, please only use these IF YOU KNOW THEY APPLY - there are various C conventions... }} * In general, given {{N}} [[bit|bits]], the range of {{M|x}} (an {{M|n}}-bit signed two's complement integer) is:
  2 KB (261 words) - 12:11, 24 November 2017
• Projection matrix
  ...}} is to be [[transpose (matrix)|transposed]] to {{M|\left(\begin{array}{c}x\\y\end{array}\right)}} * let {{M|v'\in\mathbb{R}^3}} be given, so {{M|v'\eq(x,y,z)^T}} for some {{M|x,y,z\in\mathbb{R} }}, now
  6 KB (1,051 words) - 07:44, 13 December 2017
• Logical workings of the Enigma machines
  * {{M|A,B,C,D,E\in S_I}} represent the permutations of the rotors, there may only be 3 ** So {{M|F(x)\eq EDCBA(x)\eq A(B(C(D(E(x)))))}} so {{M|A}} represents the first rotor (the one immediately after the
  4 KB (696 words) - 15:24, 15 December 2017
• Reversing the order of coefficients of a polynomial
  * {{M|f(x):\eq a+bx+cx^2+dx^3+ex^4+\cdots+\alpha x^n}} ...{a}{x^n}+\frac{b}{x^{n-1} }+\frac{c}{x^{n-2} }+\frac{d}{x^{n-3} }+\frac{e}{x^{n-4} }+\cdots + \alpha }}
  675 B (131 words) - 13:45, 18 December 2017
• Notes:Reflection of ray given normal
  * {{M|\ell_1}} by {{M|y:\eq mx+c}} and * {{M|\ell_2}} by {{M|y:\eq m'x+c'}}
  2 KB (444 words) - 21:02, 2 January 2018
• Combination
  ** {{M|\{X\in\mathcal{P}(A)\ \vert \#(X)\eq n\} }} ...them. The code might be {{M|1234}}, indicating A{{M|\eq 1}}, B{{M|\eq 2}}, C{{M|\eq 3}} and D{{M|\eq 4}} are the positions required to unlock the lock.
  3 KB (506 words) - 16:32, 14 April 2018
• Poisson mixture
  * {{XXX|Link here}} {{C|If we have a Poisson distribution and each of its events being noticed i.i. ** That is, let {{M|X\sim\text{Poi}(\lambda)}} and let {{M|(X_i)_{i\in\mathbb{N} } }} be the even
  3 KB (509 words) - 00:41, 20 July 2018
• Notes:Collision
  ...v_1'+\frac{1}{m_2}(m_1v_1+m_2v_2)}} (written in the form {{M|y\eq m\cdot x+c}}) As this is the equation for a line (where {{M|x\eq v_1'}} and {{M|y\eq v_2'}}) we can see for huge {{|v_1'}} we get huge {{
  3 KB (666 words) - 11:19, 25 September 2018
• Binary multiplexor
  ...xor a simple table; where A and B are the input lines, S the selector, and C the output: !C
  806 B (143 words) - 15:29, 29 October 2018
• Hexadecimal numbers
  ! {{M|X+Y}} ! {{M|X}}
  3 KB (389 words) - 18:38, 23 February 2019
• Quadratic equation
  |above=<span style="font-size:2em;">{{M|ax^2+bx+c\eq 0}}</span> * {{MM|\frac{d}{dx}\big[ax^2+bx+c\big]\Big\vert_{x}\eq 2ax+b}}
  755 B (122 words) - 11:51, 9 March 2019