User:Alec

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Hi, I'm Alec Teal; the admin of this site and creator of most of the content. Write a note on my talk page for requests!

Don't forget about: Category:Finished notes
Or User:Alec/BookTemplate
Or User:Alec/NewTodoTemplate
Or Characteristic property of the disjoint union topology/Proof
Or User:Alec/ParametricPages
Or /Modules
Or /Unnamed inequality
Or /Exam Qs
Or /Questions to do
Or /Pages to do
Or /Goals
Or /Manifolds pages
Or /Link templates added: Alec (talk) 02:07, 31 December 2017 (UTC)
Or /Template redirect test added Alec (talk) 19:26, 31 December 2017 (UTC)

Noticeboard

As of Wednesday, 21/Feb/2018 at 01:35:

Watchdogs

Things not to forget

Pages being worked on

Notation

Symbol Command
[ilmath]\trianglelefteq[/ilmath] \trianglelefteq
[ilmath]\triangleq[/ilmath] \triangleq
[ilmath]\ntrianglelefteq[/ilmath] \ntrianglelefteq

Current projects

Currently ensuring that all existing measure theory articles are "up to a decent standard" and working on Measure Theory, following Halmos' example (rings).

Notes

Diagram for Characteristic property of the product topology:

[ilmath]\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}X_\alpha \ar[dd] \\ & & \\ Y \ar[uurr]^f \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.55){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{f_c,\ f_b,\ f_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}[/ilmath]
Diagram

Characteristic property of the subspace topology:

  • [math]\forall S\in\mathcal{P}(X)\left[\forall Y\forall\mathcal{K}\in\mathcal{P}(\mathcal{P}(Y))\left(\overbrace{\mathcal{T}(Y,\mathcal{K})\implies}^{\text{if }(Y,\mathcal{K})\text{ is a topology then} }\Big[\underbrace{\forall f\in\mathcal{P}(Y\times X)}_{\text{relations from }X\text{ to }Y}\big(\overbrace{\mathcal{F}(f)\implies}^{\begin{array}{c}\text{if }f\text{ is a}\\\text{function then} \end{array} }[\mathcal{C}(f)\iff\mathcal{C}(\iota_S\circ f)]\big)\Big]\right)\right][/math]

Things I want to use

Linked list documentation diagram:

            /*
             * B is the node to be removed.
             * We do this by binding A forward to C
             * 
             * BEFORE: 
             * ---> +---+---------> +---+---------> +---+-----
             *....  | A |           | B |           | C |    ....
             * -----+---+ <---------+---+ <---------+---+ <---
             * 
             * AFTER:       ,------------------.
             * ---> +---+--'        +---+-------'-> +---+-----
             *....  | A |           | B |           | C |    ....
             * -----+---+ <-.-------+---+        ,--+---+ <---
             *               `------------------'
             * 
             *  Note that B.previous.bindForwards(B.next) is the same
             *  as B.next.bindBackwards(B.previous)
             *  A nice side-effect of this is that provided one has a 
             *  ref counting scheme in place (which you need basically
             *  need for safe multi-threaded schemes anyway) any iterators
             *  "pointing" to B remain valid.
             *  Make sure that this "rail-road" situation where going
             *  forward and back MIGHT not be inverses is well documented
             *  and expected.
             */