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Page title matches

• Notes:Exceptions and C++
  c: 41 54 push %r12 c: 55 push %rbp
  31 KB (3,827 words) - 14:42, 28 September 2015
• C-ring
  __TOC__{{DISPLAYTITLE:c-ring}} A ''c-ring'' is a [[ring#Properties|commutative ring]]^[[ASN]]:<ref>[[A
  426 B (52 words) - 04:30, 16 October 2016
• C(I,X)
  {{DISPLAYTITLE:{{M|C([0,1],X)}}}} ...{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[closed unit interval]]. Then {{M|C(I,X)}} denotes the [[set of continuous functions]] between the interval, co
  1 KB (258 words) - 05:08, 3 November 2016
• C(X,Y)
  125 B (14 words) - 05:55, 1 January 2017
• Index of notation/C
  ! [[C(X,Y)|{{M|C(X,Y)}}]] ...cial cases of what {{M|X}} and {{M|Y}} might be, for example: [[C(I,X)|{{M|C(I,X)}}]] - all {{link|path|topology|s}} in {{Top.|X|J}}. These sets often h
  958 B (151 words) - 06:13, 1 January 2017
• A cap (B-C) = (A cap B) - C
  {{DISPLAYTITLE:{{M|A\cap (B-C)\eq(A\cap B)-C}}}} ...\ B}} and {{M|C}} be [[sets]]. Then we claim: {{M|A\cap (B-C)\eq (A\cap B)-C}}<ref name="Alec">Alec's own work</ref>
  424 B (71 words) - 15:07, 31 January 2017
• Ell^p(C) is complete for p between one and positive infinity inclusive
  See: [[Notes:Ell^p(C) is complete for p between one and positive infinity inclusive]]
  153 B (26 words) - 18:12, 22 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
• The little-L(C) spaces are complete
  {{DISPLAYTITLE:The {{M|\ell^p(\mathbb{C})}} spaces are complete}} ...b{R} } }}]] be given and consider [[little-L^p(C) space|{{M|\ell^p(\mathbb{C})}}]] then we claim{{rFAVIDMH}}:
  1 KB (236 words) - 19:03, 18 March 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

Page text matches

• Topology
  ...Should be easy to flesh out, find some more references and demote to grade C once acceptable}}
  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
  2 KB (268 words) - 13:37, 20 April 2016
• Nabla
  ...ok (Vector Analysis and Cartesian Tensors - Third Edition - D E Borune & P C Kendall - which is a good book) distinguishbetween the <math>\nabla</math>s
  1 KB (245 words) - 18:35, 13 February 2015
• Connected (topology)
  ** <math>U\cap V\subseteq C(Y)</math> and ...</math> such that: <math>A\subseteq U\cup V</math>, <math>U\cap V\subseteq C(A)</math>, <math>U\cap A\ne\emptyset</math> and <math>V\cap A\ne\emptyset</
  5 KB (866 words) - 01:52, 1 October 2016
• Subspace topology
  {{Requires proof|grade=C|msg=Really easy, hence low importance|easy=true}} {{Requires proof|grade=C|msg=Really easy, hence low importance|easy=true}}
  6 KB (1,146 words) - 23:04, 25 September 2016
• Covering
  ...{{C|<nowiki>[[covering]]</nowiki>}} can be used, rather than the longer {{C|<nowiki>[[cover|covering]]</nowiki>}}
  190 B (24 words) - 15:15, 2 December 2015
• The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
  {{Requires references|grade=C|msg=Requires them, but is very widely known and borderline implicit!}}
  4 KB (814 words) - 22:16, 16 January 2017
• Closure, interior and boundary
  <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
  ...is a space with some useful property, this always means {{M|f:A\rightarrow C}}, for example: ...ent in {{M|B}} and maps these to a [[tuple]] whose first element is in {{M|C}}, second in {{M|D}} and third in {{M|E}}".
  4 KB (659 words) - 13:01, 19 February 2016
• Set theory axioms
  | For any {{M|A}} and {{M|B}} there is a set {{M|C}} such that <math>x\in C\iff x=A\text{ or }x=B</math> | <math>\forall A\forall B\exists C\forall x(x\in C\leftrightarrow x=A\vee x=B)</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:
  4 KB (762 words) - 20:07, 20 April 2016
• Editing guide
  ! rowspan="2" | Blackboard<br/>{{C|\mathbb{letters} }} ! rowspan="2" | Text<br/>{{C|\text{letters} }}
  5 KB (759 words) - 18:48, 24 September 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
  A relation {{M|R}} is transitive if for all {{M|a,b,c\in A}} we have {{M|aRb\text{ and }bRc\implies aRc}}
  3 KB (522 words) - 15:18, 12 February 2019
• Norm
  ...'obvious' as if the image 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 wou ...langle\cdot,\cdot\rangle:V\times V\rightarrow(\mathbb{R}\text{ or }\mathbb{C})}} induces a ''norm'' given by:
  6 KB (1,026 words) - 20:33, 9 April 2017
• Cauchy-Schwarz inequality
  ...<math>f:\mathbb{R}\rightarrow\mathbb{R}</math> give by <math>f(x)=ax^2+bx+c</math> We conclude from this that if a quadratic <math>ax^2+bx+c</math> is to be <math>\ge0</math> then <math>b^2-4ac\le 0</math>
  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
  Rather than working out the transform from {{M|C}} to {{M|S'}} or whatever we can simply notice: We will use <math>(x,y)''</math>to denote a point in {{M|C}}
  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
  ...an still 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
  2 KB (327 words) - 10:25, 12 March 2015
• Algebra of sets
  ...orall A\in\mathcal{A}[A^C\in\mathcal{A}]}}<ref group="Note">Recall {{M|1=A^C:=X-A}} - the [[complement]] of {{M|A}} in {{M|X}}</ref> * <math>A\in R\implies A^c\in R</math>
  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
  *** {{M|1=[a,b)^c=(-\infty,a)\cup[b,+\infty)}} *** {{M|1=[a,d)-[b,c)}} where {{M|1=a< b< c< d}} is {{M|1=[a,b)\cup[c,d)}}
  5 KB (782 words) - 01:49, 26 July 2015
• Measure
  {{Requires proof|Trivial|easy=Yes|grade=C}}
  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
  * {{M|g:(B,\mathcal{B})\rightarrow(C,\mathcal{C})}} is measurable * {{M|g\circ f:(A,\mathcal{A})\rightarrow(C,\mathcal{C})}} is measurable.
  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
• Tangent space
  <math>\omega\in T_p(\mathbb{R}^n)\iff \omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} </math> is a [[Derivation|derivat ...ghtarrow\mathbb{R}</math> which satisfies the [[Leibniz rule]]. Recall {{M|C^\infty(M)}} is the set of all [[Smooth function|smooth functions]] on our [
  6 KB (1,190 words) - 19:27, 14 April 2015
• Set of all derivations of a germ at a point
  ...ref> denotes the set of all [[Derivation|derivations]] of the form <math>D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}</math>, that is - it is a deriv
  772 B (132 words) - 21:49, 13 April 2015
• Leibniz rule
  Take: <math>D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}</math> - a [[Derivation]] if it
  723 B (123 words) - 00:57, 5 April 2015
• Set of all derivations at a point
  .../math> to denote this set (the set of derivations of the form <math>\omega:C^\infty\rightarrow\mathbb{R}</math>)<ref>John M. Lee - Introduction to smoot ...Derivation at a point|derivations (at a point)]] of [[Smooth|smooth or {{M|C^\infty}}]] functions from {{M|A}} at a point {{M|p}} (assume {{M|1=A=\mathb
  2 KB (291 words) - 21:51, 13 April 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: * <math>f:U\rightarrow\mathbb{R}</math> which is <math>C^\infty</math> ([[Smooth|smooth]])
  2 KB (285 words) - 01:36, 5 April 2015
• The set of all germs of smooth functions at a point
  ...the special case of smooth functions, that will be assumed here (so <math>C^k_p(\mathbb{R}^n)</math> is a notation for this set, but there is no genera ...>" in the absence of a space. That is, assume {{M|C^\infty_p}} denotes {{M|C^\infty_p(\mathbb{R}^n)}}
  794 B (140 words) - 01:50, 5 April 2015
• Derivation
  If {{M|a\in\mathbb{R}^n}}, we say that a map, {{M|\alpha:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} }} is a '''''derivation at {{M|a} * Given {{M|f,g\in C^\infty(\mathbb{R}^n)}} we have:
  2 KB (325 words) - 18:08, 14 October 2015
• Smooth
  ...Second Edition</ref> <math>F:U\rightarrow V</math> is '''smooth''', <math>C^\infty</math> or ''infinitely differentiable'' if:
  870 B (148 words) - 06:38, 7 April 2015
• Smooth manifold
  * {{M|C^\infty}} manifold structure
  3 KB (413 words) - 21:09, 12 April 2015
• Smooth structure
  * {{M|C^\infty}} structure
  2 KB (246 words) - 07:10, 7 April 2015
• Motivation for tangent space
  Take the line {{M|1=y=mx+c}}, where {{M|m}} is the gradient and {{M|c}} is the intercept with the y axis, writing this we see the line can be giv | {{M|1=y=mt+c}}
  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
• Smooth function
  ...f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty]}} ...nowledge of [[Smooth manifold|smooth manifolds]] we may already define {{M|C^\infty(\mathbb{R}^n)}} - the set of all functions with continuous partial d
  3 KB (560 words) - 16:16, 14 April 2015
• Motivation for tangent space definitions
  ...R^n)\rightarrow\mathbb{R}</math> where {{M|\cdot}} is where we'd put a {{M|C^\infty}} function (eg <math>D_vf|_p</math>)
  4 KB (790 words) - 22:25, 12 April 2015
• Differential of a smooth map
  * <math>v\in T_p(M)</math> that is to say <math>v:C^\infty(M)\rightarrow\mathbb{R}</math> * <math>f\in C^\infty(N)</math>
  886 B (158 words) - 20:58, 13 April 2015
• The fundamental group
  ...maps, {{M|\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)}}]] on {{M|C(I,X)}} and restricted to {{M|\text{Loop}(X,b)}}, then:
  3 KB (393 words) - 16:10, 4 November 2016
• Paths and loops in a topological space
  ** Note that {{M|a*(b*c)}} performs {{M|b}} and {{M|c}} at 4 times their normal speed and {{M|a}} at just double whereas: ...{M|(a*b)*c}} performs {{M|a}} and {{M|b}} at 4x their normal speed and {{M|c}} at just double - these are clearly different paths - so we don't even hav
  2 KB (347 words) - 19:36, 16 April 2015
• Hausdorff space
  ...q A}} and {{M|D\subseteq B}} then if {{M|A\cap B\eq\emptyset}} we have {{M|C\cap D\eq\emptyset}} - this could be worth factoring out
  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
  ...Vector space|H}} (where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}) with an [[Inner product]] <math>\langle\cdot,\cdot\rangle</math> such
  573 B (93 words) - 17:34, 21 April 2015
• Addition of vector spaces
  '''Operations:''' (given {{M|u_i,v_i\in V_i}} and {{M|c}} is a scalar in {{M|F}}) * <math>c(v_1,\cdots,v_n)=(cv_1,\cdots,cv_n)</math>
  4 KB (804 words) - 18:02, 18 March 2016
• Limit
  ...>\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n> C]</math> ...>\forall C<0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n< C]</math>
  2 KB (310 words) - 18:23, 8 January 2016
• Xypic
  &0\ar[rr]|(.475)\hole&&A'_2\ar[rr]|\hole&&B'_2\ar[rr]|\hole&&C'_2&&\\ 0\ar[rr]&&A'_1\ar[rr]\ar[ur]&&B'_1\ar[rr]\ar[ur]&&C'_1\ar[ur]&&
  2 KB (325 words) - 17:14, 23 October 2016
• Commutator subgroup
  ...[[Commutator|commutators]] of a [[Group|group]] {{M|(G,\times)}}. Then {{M|C}} is a [[Subgroup|sugroup]] of {{M|G}}, furthermore it is a [[Normal subgro * <math>C=\langle\{[g,h]\in G\ |\ g,h\in G\}\rangle</math>
  394 B (65 words) - 11:21, 12 May 2015
• Exercises:Serge Lang - Undergraduate Algebra - C2S1
  ...{{M|a,\ b,\ c}} be elements of {{M|G}}. If {{M|1=ab=ac}} show that {{M|1=b=c}}
  322 B (54 words) - 00:52, 27 February 2016
• Examples of rings
  The complex numbers {{M|\mathbb{C} }} is a commutative ring with unity. {{M|\mathbb{R} }} is a subring, and s * As {{M|1=x=(a+b\sqrt{2})\in\mathbb{R} }} and {{M|1=y=(c+d\sqrt{2})\in\mathbb{R} }} we know automatically {{M|xy=yx}} as multiplicat
  2 KB (269 words) - 17:11, 19 May 2015
• Division algorithm
  ...}"<ref>The mathematics of ciphers, Number theory and RSA cryptography - S. C. Coutinho</ref>, we call:
  3 KB (594 words) - 16:48, 7 December 2017
• Greatest common divisor
  ...Crypt">The mathematics of ciphers, Number theory and RSA cryptography - S. C. Coutinho</ref> is the greatest positive integer, {{M|d}}, that [[Divisor|d
  1 KB (252 words) - 08:33, 21 May 2015
• Divisor
  ...A cryptography - S. C. Coutinho</ref> if there exists a third integer, {{M|c\in\mathbb{Z} }} such that: ...s a ''divisor'' of, ''factor of'', or ''divides'' {{M|a}} if {{M|1=\exists c\in\mathbb{Z}[a=bc]}}
  783 B (141 words) - 08:20, 21 May 2015
• Euclidean algorithm
  ...ws<ref>The mathematics of ciphers, Number theory and RSA cryptography - S. C. Coutinho</ref>:
  1 KB (187 words) - 08:53, 21 May 2015
• Passing to the quotient (function)
  ...under the [[function]] {{M|f}}, {{M|1=f^{-1}(C):=\{a\in A\ \vert\ f(a)\in C\} }}. Just as for {{M|D\in\mathcal{P}(A)}} (a subset of {{M|A}}) we use {{M
  8 KB (1,644 words) - 20:49, 11 October 2016
• Integral domain
  ...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>) *** We can write this as: <math>\exists c\in R[c\ne 0\wedge(ac=0\vee ca=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
• Index of properties
  ...losed under [[Complement|complement]] then <math>\forall A\in\mathcal{A}[A^c\in\mathcal{A}]</math>
  2 KB (360 words) - 20:43, 15 June 2015
• Class of sets closed under complements properties
  * <math>\forall A\in\mathcal{A}[A^c\in\mathcal{A}]</math> where {{M||A^c}} denotes the [[Complement|complement]] of {{M|A}} - That is to say "{{M|\m
  1 KB (165 words) - 20:50, 15 June 2015
• Properties of classes of sets closed under set-subtraction
  {{Requires proof|easy=1|grade=C|msg=Outlines follow for each claim:
  3 KB (490 words) - 11:38, 21 August 2016
• De Morgan's laws
  * <math>A^c\cup B^c=(A\cap B)^c</math> * <math>A^c\cap B^c=(A\cup B)^c</math>
  364 B (69 words) - 13:14, 16 June 2015
• Site projects:Patrolling measure theory/Task list
  ...S is in A for every S in G|A map from two sigma-algebras, A and B,...]] {{C|Needs minor work}} * [[Composition of measurable maps is measurable]] {{C|Work required}}
  5 KB (645 words) - 11:40, 21 August 2016
• Trinary index
  ! class="unsortable" | {{M|C}}
  964 B (165 words) - 20:55, 22 June 2015
• Bounded set
  ...\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
  ...n an interval|space of continuous functions on an interval]], {{M|\mathcal{C}[a,b]}} and the distance function: ...|1=d(f,g)=\sqrt{\int^b_a\vert f(x)-g(x)\vert dx} }} for {{M|f,g\in\mathcal{C}[a,b]}}
  2 KB (382 words) - 15:36, 24 November 2015
• Complex conjugate
  Given a complex number, {{M|z\in\mathbb{C} }} where {{M|1=z=a+bj}} and {{M|j}} denotes {{M|\sqrt{-1} }} the ''complex
  316 B (55 words) - 18:27, 10 July 2015
• Inner product examples
  Here the space is {{M|\mathcal{C}_\mathbb{C}[a,b]}} - the [[Continuous map|continuous functions]] over the interval {{M : (this is simpler then it sounds as for {{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
  * [[Vector space]] (over the [[Field|field]] {{M|\mathbb{R} }} or {{M|\mathbb{C} }}, which we shall denote {{M|F}}) {{M|(X,F)}}, equipped with an
  949 B (161 words) - 21:08, 11 July 2015
• Normed space
  ** 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} }}) *** If the {{M|x_i\in\mathbb{C} }} then this is still the usual length of a (complex) vector (to see this
  2 KB (398 words) - 14:17, 12 July 2015
• Inequalities for inner products
  | For a {{M|\mathbb{C} }} inner product: {{MM|1=\langle x,y\rangle=\frac{1}{4}\Vert x+y\Vert^2-\f
  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} }}
  893 B (141 words) - 15:47, 12 July 2015
• Dynkin system generated by
  * {{MM|1=\delta(\mathcal{G}):=\bigcap_{\begin{array}{c}\mathcal{D}\text{ is a Dynkin system}\\ \text{and }\mathcal{G}\subseteq\mat Claim 1: {{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
  :* Let {{M|A}} be any non empty subset of {{M|X}}, then define {{M|1=B:=A^c}} which is also a subset of {{M|X}}, thus {{M|B}} is open. Then {{M|1=A\cap
  1 KB (263 words) - 13:04, 27 July 2015
• Dynkin system/Definition 1
  * For any {{M|D\in\mathcal{D} }} we have {{M|D^c\in\mathcal{D} }}
  556 B (92 words) - 01:52, 19 March 2016
• Composition of measurable maps is measurable
  * {{M|g:(B,\mathcal{B})\rightarrow(C,\mathcal{C})}} is measurable * {{M|g\circ f:(A,\mathcal{A})\rightarrow(C,\mathcal{C})}} is measurable.
  830 B (129 words) - 00:04, 3 August 2015
• Borel sigma-algebra generated by/Claim 1
  Claim {{{1|1}}}: {{M|1=\sigma(\mathcal{O})=\sigma(\mathcal{C})}}
  170 B (22 words) - 08:30, 6 August 2015
• Borel sigma-algebra generated by
  ! {{M|1=\mathcal{B}(X)=\sigma(\mathcal{C})}} - the closed sets | {{M|1=\mathcal{B}(X):=\sigma(\mathcal{O})=\sigma(\mathcal{C})}} - see '''claim 1''' below
  2 KB (244 words) - 08:30, 6 August 2015
• Borel sigma-algebra
  ! {{M|1=\mathcal{B}^n=\sigma(\mathcal{C})}} - closed<ref name="MIM"/> ...sigma(\mathcal{O})}} and {{M|1=\sigma(\mathcal{O})\subseteq\sigma(\mathcal{C})}} - see '''Claim 1'''
  5 KB (854 words) - 09:25, 6 August 2015
• Dynkin system/Proof that definitions 1 and 2 are equivalent
  :#* 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 ...closed under complements and disjoint unions, we see that {{M|(A^c\udot B)^c\in\mathcal{D} }} thus
  2 KB (326 words) - 05:09, 22 August 2015
• Set subtraction
  * {{M|1=A-B=(A^c\cup B)^c}} '''Claim:''' {{M|1=(A-B)-C=A-(B\cup C)}}
  1 KB (237 words) - 00:48, 21 March 2016
• Notes:Pick function
  Let {{M|c}} be a {{M|4}}-vector in clip-space, let {{M|\text{Ndc}:\text{clip}\rightar ...c_y \\ c_z \\ c_w\end{array}\right]\mapsto\frac{1}{c_w}\left[\begin{array}{c}c_x\\c_y\\c_z\end{array}\right]}}
  4 KB (686 words) - 01:43, 15 September 2015
• LibSR:Project
  ...roup="Note">With {{C|LIBSR_BAD_LUCK_ENABLE_ODD_ALIGNED_POINTERS}} set to {{C|1}}</ref> will have all allocations that do not request alignment aligned t * {{C|LIBSR_STENCIL_BUFFER_BITS}} - defaults to value '''8''' - OpenGL ES 2.0 req
  6 KB (1,087 words) - 04:11, 15 June 2016
• LibSR:Memory
  ...Allocator}} is used. This allocator is just a front for {{C|malloc}} and {{C|free}} (and the aligned memory allocator) ...by using {{C|malloc_usable_size}} - a GNU extension to C's stdlib (from {{C|malloc.h}}.
  1 KB (167 words) - 10:14, 23 January 2016
• Notes:Spherical coordinates
  The 4 charts we use cover sections {{M|A,\ B,\ C}} and {{M|D}} of the circle. There's a lot of overlap. We must define "cano ...<-\frac{\sqrt{2} }{2} }} then we use the bottom chart ({{M|\psi_3}} on {{M|C}})
  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 -
  2 KB (347 words) - 00:36, 27 September 2015
• Notes:Exceptions and C++
  c: 41 54 push %r12 c: 55 push %rbp
  31 KB (3,827 words) - 14:42, 28 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
• Classes of continuously differentiable functions
  ...{M|C^k}} on {{M|U}}, {{M|C^k(U)}} for {{M|U\subseteq\mathbb{R}^n}} and {{M|C^k(\mathbb{R}^n)}} such. * ''Of class {{M|C^k}}'' or
  3 KB (632 words) - 20:32, 16 October 2015
• Measure/Infobox
  |data1={{M|1=\forall\overbrace{(A_n)_{n=1}^\infty }^{\begin{array}{c}\text{pairwise}\\\text{disjoint}\end{array} }\subseteq\mathcal{R}[\mu\left(
  637 B (95 words) - 12:53, 28 March 2016
• Class of smooth real-valued functions on R-n
  * {{M|C^\infty(\mathbb{R}^n)}} {{Note|The conventions concerning the {{M|C^k}} notation are addressed on the page: ''[[Classes of continuously differe
  2 KB (259 words) - 23:41, 21 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
• Pre-measure/New page
  {{Requires references|grade=C|msg=I've sort of forged my own path with regards to what we call a pre-meas
  3 KB (422 words) - 21:25, 17 August 2016
• Sandbox
  C \ar@(dr,dl)@2{_{(}->}[r] _a = "a" & D \ar@{_{(}-^{)} }[r]_b = "b" & Y B \ar@/^/@{_{<}-^{>} }[r] \ar@{_{(}->}[r] & D_5 \ar@{_{(}->}[r] & C
  695 B (132 words) - 22:15, 26 October 2015
• Quotient vector space
  {{Refactor notice|grade=C|msg=See [[/New page]] for current work}}
  5 KB (879 words) - 23:09, 1 December 2016
• Open balls
  ...elper page, quite often one wants to write "open balls" and link it, but {{C|<nowiki>[[open ball|open balls]]</nowiki>}} is rather cumbersome
  229 B (36 words) - 10:56, 1 December 2015
• Compact
  This is a convenience page, now you can say things are {{C|<nowiki>[[compact]]</nowiki>}} inline.
  162 B (22 words) - 14:24, 1 December 2015
• Books:Analysis - Part 1: Elements - Krzysztof Maurin
  ...rm]] may be a function of the form {{M|\Vert\cdot\Vert:V\rightarrow\mathbb{C} }}, see the norm page (there's a warning note on the first line of the def
  1 KB (215 words) - 22:32, 26 February 2016
• Convergent (sequence)
  I keep forgetting if I made {{C|<nowiki>[[Convergence (sequence)]]</nowiki>}} or this one, so I've made bot
  204 B (28 words) - 16:16, 1 December 2015
• Norm/Heading
  ...is a [[vector space]] over the [[field]] {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
  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} }}. For {{M|V}} a [[vector space]] over {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
  1 KB (182 words) - 13:25, 14 February 2016
• Metric/Infobox
  For {{M|V}} a [[vector space]] over {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
  1 KB (180 words) - 10:39, 11 March 2016
• TestNavbox
  |list1=B, C
  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}}: * {{M|1=T:\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}}: * {{M|1=S:\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
  * {{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]}} * {{M|1=\forall X\in\text{Ob}(\mathscr{C})\forall f,g\in\text{Arw}_\mathscr{C}(B,X)[(f\circ e=g\circ e)\implies f=g]}}
  1,012 B (181 words) - 14:43, 6 February 2016
• Bimorphism
  ...bimorphism'' is a [[morphism]] or [[arrow]] in a [[category]] {{M|\mathscr{C} }}{{rAITCTHS2010}}: ...in {{M|\mathscr{C} }} are isomorphisms however, we say that {{M|\mathscr{C} }} is ''[[balanced (category theory)|balanced]]''}}
  577 B (82 words) - 14:23, 13 March 2016
• Types of category arrows/Diagram
  ...}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->
  738 B (101 words) - 14:50, 6 February 2016
• RealQ16s:Provisional instruction listing/Opcode table/2 byte instructions
  ====Prefixed with {{C|1111 1111}}==== Every instruction in this table has 2 bytes, the first byte is {{C|1111 1111}} and the second byte is given in the table.
  412 B (50 words) - 23:02, 12 February 2016
• RealQ16s:Provisional register listing
  ** There are 4 16 bit general purpose registers, A, B, C and D. Each of these is aliased to 2 8 bit registers, they are: ...\text{CL}\ \ \ \vert\ \ \ \overbrace{\text{8 bits} }^\text{CH}\ \ }^\text{C} }} {{M|\Big\vert}} {{M|1=\overbrace{\ \ \overbrace{\text{8 bits} }^\text{D
  2 KB (276 words) - 14:31, 14 February 2016
• RealQ16s:Provisional register listing/Core infobox
  |label3={{R|C}} (16bit)
  745 B (109 words) - 13:53, 14 February 2016
• RealQ16s:Provisional register listing/Core infobox mark II
  | style="text-align:center;" colspan="16"| C ...="width:auto; background:white; color:black;" rowspan="2"| {{M|\Big\} }}'''C'''ount
  3 KB (438 words) - 14:30, 14 February 2016
• Category Theory (subject)
  * {{M|A}}, {{M|B}}, {{M|C}} which are say, groups or topologies, and given * the maps {{M|f:A\rightarrow B}} and {{M|g:B\rightarrow C}} which are say, group homomorphisms or continuous maps
  2 KB (311 words) - 11:46, 19 February 2016
• Site projects:Navboxes
  ...do in this project is [[:Category:Todo: Navbox project]], please use the {{C|<nowiki>{{ProjectTodo|Navbox project|(optional note)</nowiki>}} to mark tod
  834 B (110 words) - 14:44, 27 March 2016
• Negation of implies
  {{Stub page|grade=C|msg=Formal logic pages to not exist!}} {{Requires cleanup|grade=C|msg=A low grade as the content is and the page isn't important}}
  861 B (153 words) - 21:05, 17 November 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/1 implies 2
  :* So {{MM|1=\exists C>0\ \forall N\in\mathbb{N}\ \exists n\in\mathbb{N}[n>N\wedge\Vert L(x_n-p)\V ...at {{M|d(x_n,x)>\epsilon}}, we shall later call such an {{M|\epsilon}} {{M|C}} and construct a subsequence out of the {{M|n}}s
  5 KB (1,064 words) - 02:24, 28 February 2016
• Lesson:Introduction to computing machines
  ...as required knowledge for the [[Lesson:Introduction to C++|Introduction to C++ lesson]].
  504 B (71 words) - 16:33, 27 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:
  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
• Equivalence of Cauchy sequences/Proof
  * {{M|d(a,c)\le d(a,b)+d(b,c)}} for all {{M|b}} in the space. If we can show that {{M|d(a,b)+d(b,c)<\epsilon}} then we'd be done.
  7 KB (1,404 words) - 21:02, 20 April 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}}: ...MM|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}}: ...MM|1=\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
• Isomorphism (category theory)
  ...B}} and {{M|B\mathop{\longrightarrow}^g A}} in a [[category]] {{M|\mathscr{C} }}, we say that {{M|f}} and {{M|g}} are ''isomorphisms''{{rAITCTHS2010}} i
  753 B (117 words) - 14:19, 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}}
  779 B (135 words) - 21:23, 19 April 2016
• Predicate
  ...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
  ...style="font-size:1.2em;text-align:left;">{{MM|1=\forall A\in\mathcal{A}[A^C\in\mathcal{A}]}}<br/>{{MM|1=\forall\{A_n\}_{n=1}^\infty\subseteq\mathcal{A}
  635 B (92 words) - 01:13, 19 March 2016
• Sigma-algebra/Definition
  * {{M|\forall A\in\mathcal{A}[A^C\in\mathcal{A}]}} - Stable under [[complement|complements]]
  779 B (122 words) - 01:25, 19 March 2016
• Limsup and liminf
  {{Stub page|Flesh out|grade=C}}
  187 B (28 words) - 22:17, 19 April 2016
• Symmetric difference
  '''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
• Notes:Hereditary sigma-ring
  ...{M|\sigma_R(\mathcal{H}(S))}} is hereditary {{M|\forall C\in\mathcal{P}(B)[C\in\sigma_R(\mathcal{H}(S))}}
  2 KB (449 words) - 19:26, 24 May 2016
• Site projects:Measure theory plan
  ! {{M|\mathcal{C} }}
  2 KB (309 words) - 15:31, 27 March 2016
• Algebra of sets/Infobox
  |data1={{M|\forall A\in\mathcal{A}[A^C\in\mathcal{A}]}}
  427 B (68 words) - 18:43, 1 April 2016
• Site:Refactoring
  ! [[:Category:Refactoring grade C pages|C]]
  2 KB (425 words) - 02:21, 12 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
• Notes:Utf8
  | Excluding the string-termination character, {{C|0000 0000}} ...n: ''' there's a lot of overlap here. This suggests that a first byte of {{C|1100 0000}} is invalid, because that'd overlap with the 1 byte characters!
  3 KB (426 words) - 22:12, 29 April 2016
• Topological spaces
  This is a "convenience" redirect. Writing {{C|<nowiki>[[Topological space|topological spaces]]</nowiki>}} often is annoyi
  180 B (21 words) - 21:31, 3 May 2016
• Regular topological space/Definition
  ...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
  ...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
  {{Requires proof|grade=C|msg=Not an easy proof}}
  583 B (101 words) - 00:21, 4 May 2016
• Geometric independence
  {{Stub page|Suggest why this definition is useful|grade=C}}
  1 KB (177 words) - 22:01, 4 May 2016
• Notes:Boundary operator
  A ''{{M|p}}-chain'' on {{M|K}} is a function, {{M|c}}, from the set of oriented {{M|p}}-simplicies of {{M|K}} to {{M|\mathbb{Z} # {{M|1=c(\sigma)=-c(\sigma')}} if {{M|\sigma}} and {{M|-\sigma}} are ''opposite'' orientations
  1 KB (257 words) - 00:29, 8 May 2016
• Lebesgue number
  {{Stub page|grade=C|msg=Remember to replace the diameter reminder with a subpage transclusion i
  1 KB (214 words) - 07:44, 10 May 2016
• Subspace (topology)
  This page exists because {{C|<nowiki>[[Subspace (topology)|subspace]]</nowiki>}} can be more "natural" a
  259 B (35 words) - 10:09, 11 May 2016
• Notes:Types of retractions
  * {{C|"The key is that if one want a continuous function r:X-->I ,then r can not
  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
• Kronecker delta
  {{Requires references|grade=C|msg=This is VERY common and the name isn't disputed, there's no ambiguity b {{Stub page|grade=C|msg=Once fleshed out demote even lower or remove this stub notice entirely}
  716 B (112 words) - 12:24, 14 May 2016
• 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
• Notes:Outer-measures to measures
  * '''Theorem C:''' (two parter) *# '''Theorem C (i): ''' Every set of outer-measure {{M|0}} belongs to {{M|\bar{S} }}
  6 KB (1,067 words) - 22:19, 23 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
• Cantor's construction of the real numbers
  {{Stub page|grade=C|msg=Requires proof of claims and such}}
  331 B (41 words) - 11:48, 2 June 2016
• The real numbers
  {{Stub page|grade=C|msg=Once cleaned up and fleshed out, demote to D
  1 KB (200 words) - 21:31, 26 February 2017
• Axiom of completeness
  {{Stub page|grade=B|msg=Demote to grade C once fleshed out better}}
  467 B (69 words) - 13:06, 30 July 2016
• Mean value theorem
  {{Stub page|grade=A*|msg=A fundamental page of analysis, demote to grade C once the content is fleshed out and referenced properly. Demote to A once s
  931 B (144 words) - 06:48, 4 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} }} ...row\mathbb{R} \in}} [[Classes of continuously differentiable functions|{{M|C^1}} - the class of functions with continuous partial derivatives]]. Let {{M
  3 KB (529 words) - 08:07, 4 June 2016
• LibSR:Prototype
  ...l" and address the (many) shortcomings of the [[LibSR:Proof of concept|P.O.C.]]; Due to the "research" nature of the project (I don't know what will act #* A long term and low priority goal is to have a single pool, say {{C|256mb}} (or whatever) that LibSR uses for everything.
  4 KB (719 words) - 22:54, 18 July 2016
• LibSR:Assertions
  {{Stub page|msg=Take time do document properly|grade=C}} # {{C|'''LIBSR_ASSERT('''expr,"message"''')'''}} - put these everywhere, they can
  1 KB (208 words) - 05:46, 15 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
• LibSR:Bad luck mode
  ...find bugs which may not show themselves due purely to luck. For example, {{C|malloc}} calls return memory that is usually aligned to 4096 bytes, this is * {{C|LIBSR_BAD_LUCK_ODD_ALIGN_UNALIGNED_ALLOCATIONS}} - any memory allocations w
  1 KB (216 words) - 23:06, 18 July 2016
• Klein-4 group
  ...Klein-4 [[group]]'' ({{AKA}}: ''Viergruppe''{{rAAPAG}}), {{M|1=V_4=\{1,a,b,c\} }} is a group on 4 elements with the operation {{M|*:V_4\times V_4\righta \begin{array}{|c|cccc|}
  1 KB (167 words) - 00:19, 22 July 2016
• Two-phase commit
  ...re 3 actions that we wish to commit "atomically", {{M|A}}, {{M|B}} and {{M|C}} ...omicity, as {{M|A}} has committed! What should you do with {{M|B}} and {{M|C}} now? {{M|B}} cannot happen, so this situation is already "broken".
  957 B (158 words) - 18:03, 26 July 2016
• Notes:The foundations of Mathematics - Kunen
  ...ghtarrow x\cap y=\emptyset)\rightarrow \exists C\forall x\in F(\text{Sing}(C\cap x))}}
  2 KB (342 words) - 02:38, 31 July 2016
• Task:Continuity types
  # Metric space version of {{C|1)}} (for {{M|(X,d_1)}} and {{M|(Y,d_2)}} being [[metric space|metric space # '''''(CLAIM: 1)''''' - {{Warning|Unproven claim}} - {{C|1)}} and {{C|3)}} are in some sense equivalent. Obviously definition 1 doesn't work for
  3 KB (668 words) - 22:38, 4 August 2016
• Notes:Dual basis
  ...hcal{K}^n}} is the [[concrete-to-abstract isomorphism]]<ref group="Note">{{C|'s inverse}} - as we take the concrete to abstract function to be of the fo
  5 KB (1,020 words) - 08:43, 12 August 2016
• Notes:Tangent space
  ...w M}} by {{M|1=c:(r,\theta)\mapsto(r\cos(\theta),r\sin(\theta),r^2)}} ({{M|c}} for chart) ...2+y^2)=(\sqrt{x^2+y^2},\arctan(\frac{y}{x}))}} and we'll define {{M|1=p_c:=c^{-1}(p)=(\sqrt{x^2+y^2},\arctan(\frac{y}{x}))}}
  5 KB (1,002 words) - 19:42, 15 August 2016
• R^n is a topological vector space
  ...ough to work out though once the topological basis stuff gets sorted|grade=C}}
  805 B (129 words) - 14:25, 16 August 2016
• Topological vector space
  ...\eq\mathbb{R} }}, or [[the complex numbers]], so {{M|\mathbb{K}:\eq\mathbb{C} }} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.|
  2 KB (383 words) - 14:03, 16 February 2017
• Epsilon form of inequalities
  {{Stub page|grade=C|msg=This is not an important page. It was created so anyone unsure of what
  528 B (92 words) - 17:03, 17 August 2016
• Semi-ring of sets
  ...[pre-measure]] (and thus an [[outer-measure]] and later a [[measure]]) a {{C|pre-measure on {{M|X}}}} where {{M|X}} is say a semi-ring or something. ...intersect with {{M|a<c}} and {{M|b<d}}, then clearly {{M|1=[a,b)\cap[c,d)=[c,b)}}, this is the most difficult case.
  3 KB (508 words) - 17:25, 18 August 2016
• Distributivity of intersections across unions
  # {{M|1=A\cap(B\cup C)=(A\cap B)\cup(A\cap C)}} {{Requires proof|easy=yes|grade=C|msg=First one is routine chapter-1 for first-years, second one is easy usin
  730 B (128 words) - 23:19, 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}}
  2 KB (390 words) - 22:16, 19 August 2016
• Types of set algebras/Type table
  ...e that {{M|1=A-B=A\cap B^c=(A^c\cup B)^c}} - or that {{M|1=A-B=(A^c\cup B)^c}} - so we see that being closed under union and complement means we have cl ...under complements, so {{M|\emptyset^c\in\mathcal{A} }} and {{M|1=\emptyset^c=\Omega\in\mathcal{A} }}</ref>
  4 KB (573 words) - 20:00, 19 August 2016
• Types of set algebras/Measure theory summary table
  ! {{M|C}}
  559 B (79 words) - 20:05, 19 August 2016
• Doctrine:Measure theory terminology
  ...}_{\text{Collectively called "partial measures"} } } & & & { \begin{array}{c} {\bf\text{on hereditary }\sigma\text{-ring} }\\ {\bf\text{(or power-set if
  2 KB (333 words) - 17:49, 27 February 2017
• Hereditary set
  {{Requires references|grade=C|msg=Check that "foundations of mathematics" book from the same series as th
  810 B (131 words) - 16:46, 22 August 2016
• First order language
  ** {{M|C}} - The set of [[logical connective]] symbols. {{Caution|Not all of these 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)
  # {{M|A\vee B\vee C\vee D}} is easier to read than {{M|(((A\vee B)\vee C)\vee D)}} * {{C|1=Formula ::= Term {{M|\doteq}} Term {{!}} PREDICATE (Term)<sub>{{n|times}}
  3 KB (500 words) - 05:37, 8 September 2016
• Herbrand domain
  ...{{M|c}} is a {{link|constant symbol|FOL}} of {{M|\mathscr{L} }}) then {{M|c\in H}}
  1 KB (209 words) - 08:35, 8 September 2016
• Editing guide/Templates/Page categorisation
  | {{C|<nowiki>{{Definition|</nowiki>Subject₁<nowiki>|...|Subject</nowi | {{nowrap|{{C|<nowiki>{{Definition|Topology|Geometry}}</nowiki>}}}}
  653 B (80 words) - 10:11, 8 September 2016
• Domain (FOL)
  # For each {{link|constant symbol|FOL}}, {{M|c}} in {{M|\mathscr{L} }}, {{M|I(c)}} is an element of {{M|M}} ...y}}]]'' {{link|function symbol|FOL}}, {{M|f}} in {{M|\mathscr{L} }}, {{M|I(c)}} is an element of {{M|\mathcal{F} }}
  4 KB (672 words) - 06:42, 10 September 2016
• Interpretation (FOL)
  # For each {{link|constant symbol|FOL}}, {{M|c}} in {{M|\mathscr{L} }}, {{M|I(c)}} is an element of {{M|M}} ...y}}]]'' {{link|function symbol|FOL}}, {{M|f}} in {{M|\mathscr{L} }}, {{M|I(c)}} is an element of {{M|\mathcal{F} }}
  2 KB (290 words) - 06:54, 10 September 2016
• Semantics of terms (FOL)
  ...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
  2 KB (369 words) - 07:46, 11 September 2016
• If A is a logical consequence of Gamma then the formula set of Gamma union the negation of A is not satisfiable
  {{Stub page|grade=C|msg=At least flesh out the statement}} {{Requires proof|grade=C|msg=Could use a proof to be honest..., see page 31 of<ref name="MLFFISWL"/>
  417 B (64 words) - 11:25, 11 September 2016
• Demonstrating why category arrows are best thought of as arrows and not functions
  Consider the [[category]], {{M|\mathscr{C} }}, whose {{link|object|category theory|s}} are ''[[finite]]'' {{plural|se ...efine composition of arrows, given {{M|\xymatrix{A \ar[r]^f & B \ar[r]^g & C } }}, as:
  2 KB (275 words) - 12:29, 15 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
  ** Let {{M|C^0(X,Y)}} denote the [[set]] of all [[continuous maps]] of the form {{M|(:X\ ** we define a [[relation]], {{M|\mathcal{R}\subseteq C^0(X,Y)\times C^0(X,Y)}} given by
  2 KB (272 words) - 23:37, 14 October 2016
• Notes:Homotopy terminology/Terminology
  ...]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. Let {{M|C^0(X,Y)}} denote the set of all [[continuous maps]] of the form {{M|(:X\righ # '''Homotopic - ''' a relation on maps {{M|f,g\in C^0(X,Y)}}. We write {{M|f\simeq g\ (\text{rel }A)}} if there exists a homoto
  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
• Unix philosophy/Eric S. Raymond
  {{Stub page|grade=C|msg=I link to this so much it'd actually be good to have my own version. Th
  3 KB (550 words) - 02:12, 20 September 2016
• Unix philosophy
  {{Stub page|grade=C|msg=I link to this so much it'd actually be good to have my own version. Th
  538 B (96 words) - 02:12, 20 September 2016
• Notes:Ascii table
  ! {{C|0}} | {{C|0x30}}
  536 B (74 words) - 07:32, 23 September 2016
• Notes:Parsing
  * {{M|\text{a},\text{b},\text{c} }} terminal symbols {{Caution|Check}} * {{M|a,b,c}} terminal or non terminal symbols {{Caution|Check}}
  2 KB (346 words) - 19:06, 24 September 2016
• Characteristic property of the disjoint union topology/Statement
  {{Stub page|msg=A rewrite, while not urgent, would be nice|grade=C}}
  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
  1 KB (210 words) - 20:21, 25 September 2016
• Canonical injection of the subspace topology
  {{Stub page|grade=C|msg=I have documented the canonical injection on the subspace topology page {{Requires proof|grade=C|msg=Easy, did on paper, it is easy. Haven't found reference but to be fair
  1 KB (195 words) - 23:22, 25 September 2016
• Doctrine:Homotopy terminology
  ...ans continuous with continuous first and second derivatives, so forth, {{M|C^\infty}} means [[smooth]]. ...s functions; and we have {{M|C^0\supset C^1\supset C^2\supset\cdots\supset C^\infty}}</ref>
  3 KB (535 words) - 09:01, 31 October 2016
• Closed map
  * {{M|1=\forall U\in C(X,\mathcal{J})[f(U)\in C(Y,\mathcal{K})]}} - that is, that the {{link|image|map|s}} (under {{M|f}}) ** {{M|C(X,\mathcal{J})}} denotes the set of all {{plural|closed set|s}} of the [[to
  1 KB (246 words) - 19:59, 26 September 2016
• Every surjective map gives rise to an equivalence relation
  ...page|grade=C|msg=Flesh out, check existing content then demote to grade {{C|F}}}} {{Requires proof|grade=C|easy=True|This is easy. Truly}}
  2 KB (315 words) - 13:54, 8 October 2016
• Dense
  {{Requires proof|easy=true|grade=C|msg=It is obvious that {{M|1=(B_\epsilon(x)\cap E\ne\emptyset)\iff(\exists {{Requires proof|easy=true|grade=C|msg=Easy for someone informed of what a metric space and topology is. The c
  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 ...(U\subseteq X-E)]}}{{rFAVIDMH}} (I had to use [[negation]]/{{M|\neg}} as {{C|\not{\subseteq}<nowiki/>}} doesn't render well ({{M|\not{\subseteq} }}))
  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
  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
  {{Requires proof|grade=C|easy=true|See [[Connected_(topology)#Equivalent_definition]] if stuck, but
  461 B (69 words) - 22:52, 30 September 2016
• Every continuous map from a non-empty connected space to a discrete space is constant
  {{Requires proof|grade=C|msg=Easy proof to do, I've done it on paper and there is nothing nasty abou
  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
  {{Requires proof|grade=C|easy=true|msg=Easy proof to do, exercise on page 87 in Lee's top. manifolds
  1 KB (172 words) - 23:12, 30 September 2016
• Disjoint
  {{Stub page|grade=C|msg=Proper stub page, created just to minimally document something}}
  2 KB (294 words) - 03:19, 1 October 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
  {{Requires proof|grade=C|easy=true|msg=Exercise in Lee's top. manifolds, didn't take me very long to
  833 B (126 words) - 23:15, 30 September 2016
• Notes:Homology
  & & z \ar@/_.5pc/[uu]^c \ar@/_2pc/[uu]_d & {\langle a,b,c,d\rangle\cong\mathbb{Z}^4} & {\langle x,y,z\rangle\cong\mathbb{Z}^3}
  6 KB (897 words) - 07:30, 15 October 2016
• Exercises:Mond - Topology - 1
  ==Section C==
  317 B (29 words) - 18:29, 10 October 2016
• The image of a compact set is compact
  {{Requires proof|grade=C|msg=Proof isn't that important as it is easy and routine.
  2 KB (332 words) - 17:20, 18 December 2016
• Homeomorphic
  {{Requires references|grade=C|msg=Trust me, this is true. But references are none the less important! Jus
  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
  ...e's a commented out note and a requires-proof template at the top of the {{C|1===proof==}} heading, remove that later!}}
  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
  {{Requires proof|grade=C|msg=This could be more formal}}
  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}}
  2 KB (387 words) - 12:51, 10 October 2016
• Properties of the pre-image of a function
  {{Requires proof|grade=C|msg=These are easy and routine|easy=true}}
  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
  ...{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
• U-ring
  * [[c-ring]] - a commutative ring.
  455 B (57 words) - 16:32, 19 October 2016
• Q-ring
  * [[c-ring]] - a commutative ring
  435 B (56 words) - 04:28, 16 October 2016
• U-ring/Motivation
  To get the best of both, I decided to use "u-ring", "c-ring" (for a commutative ring) and "cu-ring" for a "commutative ring with u
  647 B (110 words) - 04:11, 16 October 2016
• C-ring
  __TOC__{{DISPLAYTITLE:c-ring}} A ''c-ring'' is a [[ring#Properties|commutative ring]]^[[ASN]]:<ref>[[A
  426 B (52 words) - 04:30, 16 October 2016
• Ring/New page
  **# {{M|1=\forall a,b,c\in R[(a\oplus b)\oplus c=a\oplus(b\oplus c)]}} - [[associativity]] --><ol><li value="5">{{M|1=\forall a,b,c\in R[(a\odot b)\odot c=a\odot(b\odot c)]}}</li></ol>
  4 KB (728 words) - 16:29, 19 October 2016
• Exercises:Mond - Topology - 2
  ==Section C==
  1,014 B (155 words) - 12:23, 19 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
• Exercises:Rings and Modules - 2016 - 1/Problem 1
  ====Part C====
  957 B (158 words) - 16:05, 19 October 2016
• If and only if
  {{Stub page|grade=C|msg=Not important, as I assume readers know this by default}}
  208 B (38 words) - 16:21, 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
  {{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 {{Requires proof|grade=C|easy=true|msg=Review, tidy up... I did this quickly to provide some content
  2 KB (287 words) - 19:31, 28 October 2016
• Equivalent conditions to a set being bounded/Statement
  # {{M|1=\exists C<\infty\ \forall a,b\in A[d(a,b)<C]}} - {{M|A}} is [[bounded]] (the definition) # {{M|1=\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]}}{{rFAVIDMH}}
  720 B (124 words) - 23:41, 29 October 2016
• Equivalent conditions to a set being bounded/1 implies 2
  {{Requires proof|grade=C|easy=true|msg=Easy and routine proof. If stuck see page 13 in Functional An
  123 B (21 words) - 23:45, 29 October 2016
• Equivalent conditions to a set being bounded/2 implies 1
  {{Requires proof|grade=C|easy=true|msg=Easy and routine proof. If stuck see page 13 in Functional An
  123 B (21 words) - 23:45, 29 October 2016
• Equivalent conditions to a set being bounded
  ...b)<C]\big)\implies\big(\forall x\in X\exists C<\infty\forall a\in A[d(a,x)<C]\big)}}, that boundedness implies condition 2 ...in A[d(a,x)<C]\big)\implies \big(\exists C<\infty\ \forall a,b\in A[d(a,b)<C]\big)}}, that condition 2 implies boundedness
  1 KB (167 words) - 23:12, 18 March 2017
• Exercises:Mond - Topology - 4/Section B/Question 6
  ====Part C====
  433 B (60 words) - 15:49, 31 October 2016
• C(I,X)
  {{DISPLAYTITLE:{{M|C([0,1],X)}}}} ...{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[closed unit interval]]. Then {{M|C(I,X)}} denotes the [[set of continuous functions]] between the interval, co
  1 KB (258 words) - 05:08, 3 November 2016
• Omega(X,b)
  ...and let {{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 * [[C(I,X)|{{M|C(I,X)}}]]
  2 KB (364 words) - 04:47, 3 November 2016
• The set of continuous functions between topological spaces
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Then {{M|C(X,Y)}} denotes the [[set]] of all ''[[continuous]]'' [[functions]] from {{M * {{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