Difference between revisions of "User:Alec"

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: Or [[User:Alec/ParametricPages]]
 
: Or [[User:Alec/ParametricPages]]
 
: Or [[/Modules]]
 
: Or [[/Modules]]
 +
: Or [[/Unnamed inequality]]
 
==[[User:Alec/Noticeboard|Noticeboard]]==
 
==[[User:Alec/Noticeboard|Noticeboard]]==
 
{{:User:Alec/Noticeboard}}
 
{{:User:Alec/Noticeboard}}

Revision as of 00:09, 29 October 2016

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

Noticeboard

As of Monday, 14/Apr/2025 at 08:40:

Watchdogs

Things not to forget

Pages being worked on

Notation

Symbol Command
\trianglelefteq \trianglelefteq
\triangleq \triangleq
\ntrianglelefteq \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:

\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}
Diagram

Characteristic property of the subspace topology:

  • \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]

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.
             */
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