archiver 8

test

2006-03-08T00:15:00.000-08:00

t test. t log. t this.
log of this blog.

template change
1. from minima to thisaway blue.
2. navbar blue to black.
3. added zoomclouds default as first item of sidebar.
republish entire blog : error
republish index : no error (then?)

next time: technorati tags. republish entire blog again.

status of my blog
1. Opera: beautiful --> no problem
2. Firefox: something inbetween page rendering. long word problem. --> minor problem
3. IE: sidebar items at far below. --> big problem.

problem suspect: zoomclouds as first item, republishing error.

expected size of the largest piece

2006-02-23T05:45:00.000-08:00

loop
expected size of the largest piece of the loop

what to see.
what to not see.

t math. t probability.

my math study

2006-02-20T11:21:00.000-08:00

math study
t study. t math. t my.

undergraduate abstract algebra:
over.

abstract linear algebra:
need study.

undergraduate real analysis:
over. (over because i find i can solve exercises of lebesgue measure theory, which is the next step of college real analysis)

complex analysis:
need study.

point set topology:
<-- I think there's no point (unless for practising rigorousness in mathematics) in studying point set theory on its own. (just like there's no point in studying category theory in its own.) so? study only the needed parts when needed. point set topology and category theory is too general and too abstract.

detail's not important.

bijections from NxN onto N

2006-02-20T11:20:00.000-08:00

bijections from NxN onto N
tag: t math. t function. t bijection. t polynomial. t zakon.

bijections from N x N onto N
f(n,m) = 2^n (2m-1)
g(x,y) = g(s,y) = 1/2 (x+y-1)(x+y) + (1-y)

For f, observe that the expression 2m-1 usually denotes odd numbers. f comes from and gives rise to a multiplication-based decomposition of N into N pieces.

For g, consider s = x+y as fixed and y varing. g comes from and gives rise to a decomposition of N into N intervals.
In fact, g is the famous zig-zag proof of equinumeracy of N x N and N.
Note that g is a polynomial.

Note that, to prove that f and g are bijections, we find meanings in the expressions of f and g. Find meanings and motivations in expressions, formula, formulation, proof, theorem, conjecture, equalities, inequalities

ref:
f from sci.math
g from http://www.trillia.com/distr3/zakon-basic-a4-one.pdf

test for tags

2006-02-19T10:14:00.000-08:00

t find
t:find
t-find

t find this
t:find this
t-find this

condition for Riemann integrability

2006-02-19T09:25:00.000-08:00

criteria for being Riemann integrable
tag: math. proof. divide. sci.math. integrable. criteria. using compact. using refinement. using covering. oscillation. envelope. using closure. using point set topology. proving measure zero. history.

f : [0,1] or [a,b] --> [-M,M] or [-1,1]
D = the set of points where f is discontinuous
D_e = the set of points where f is discontinuous by size larger than e somehow.

"f (bounded) is Riemann integrable on [0,1] iff m(D) = 0"

It says f is Riemann integrable iff the set of badly-behaving points D is small.

See Robert Israel's nice summary of a proof (the 'if' part) in which he directly successfully uses D rather than using D_e in constructing the required partition of [0,1]. Rather than to use directly D and K = [0,1]\D, he covers (and replace) D with a slightly bigger open set U and K with V. (compactness of K is used to construct V and finiteness of F1 is used to construct F2). Uses F1 and F2 to find a refinement partition pi. pi is not directly constructed from D and K, but by first covering and replacing D and K both by F1 and F2 (finite collection of open intervals).
learn: flexibility. using refinments. using covering.
concept: covering. refinement. compact.

See Herman Rubin's proof (of both 'if' and 'only if' part at once) in which he uses D_e (in his notation, A_e) rather than using directly D. He uses that 'the closure of A_e is contained in A_(e/2)'. this is for using compactness to reduce to 'finitely many' intervals.
learn: using closure containment.
concept: closure containement. compact.

off topic.

See Robert Israel's 'only if' proof. (a usual and straightforward proof). The proof can be considered 'divide and conquer' twice. He reduces
m(D)=0
to
m(D_e) < (arbitrary small).
First 'divide and conquere', in that D --> D_e.
Second 'divide and conquere', in that 0 --> arbitrary small.
Observe that this is usual in proving something is measure zero.
learn: "take epsilon -> 0". usual way of proving m(something)=0.

See R3769's where the concept of lower envelope and upper envelope is used.

See Dave L. Renfro's post, for the history. an excerpt:
"Riemann's nontrivial contributions on this topic were (a) giving a
necessary and sufficient condition for integrability based on the
behavior of a function, (b) using this condition to prove the
integrability of a function having a dense set of discontinuities,
and (c) putting the focus on the collection of functions that are
integrable according to some notion of integrability, rather than
defining a notion of integrability only in order to rigorously
prove certain desired integrability properties."

square matrix and it's transpose are similar

2006-02-14T07:58:00.000-08:00

"Are a matrix and it's transpose similar?"
three proofs in the sci.math (google groups) thread.

proof 1 : Divide and Conquer Method: rational canonical form and with that minimal polynomial determines companion matrix.

proof 2 : Divide and Conquer Method: Jordan normal form decomposition J = D + N and showing that N is similar to transpose of N by permutation.

Above two are examples of Divide And Conquere that is not Divide Into Cases. (ways: direct sum decomposition. linear sum)

proof 3 : symmetry in computing g_1,...

tag: math. linear. algebra. proof. divide

definition

2006-02-13T16:59:00.000-08:00

tag: math. definition.

justification of a definition
1. fits mathematical purposes
2. makes the notion precise

monotonic subsequence : example of proof by cases

2006-02-13T16:57:00.000-08:00

Proof By Cases: Example from monotonic subsequence theorem
tag: math. proof. case.

"every infinite sequence of real numbers have a monotonic infinite subsequence."

the conclusion has two CASES: having nonincreasing infinite subsequence and having nondecreasing infinite subsequence.

one proof (in Zakon series) starts by dividing two CASES:
case 1: SOME subsequence of the sequence don't have maximum value.
case 2: ALL subsequence of the sequence have max.
SOME or ALL

another proof (in everything2) starts by dividing cases:
case 1: infinitely many maximal index.
case 2: only finitely many maximal index. so that no index is a maximal index from on some N.
, where maximal index is defined as http://everything2.com/index.pl?node_id=983077
Infinite occurence or only finitely many occurence.

another proof starts by dividing:
case 1: unbounded sequence
case 2: bounded sequence.
, then case 2 is more divided into cases,
, as in http://www.puc.edu/Faculty/George_Hilton/id94.htm
unbounded or bounded.

all the three are proofs by cases.

reposting 3 posts.

2006-02-07T11:29:00.000-08:00

tag: recovery math

My unlisted three posts on February 4:
http://archiver8.blogspot.com/2006/02/ewd975-dijkstras-pythagorean-theorem.html
http://archiver8.blogspot.com/2006/02/thats-not-only-way-largest-example.html
http://archiver8.blogspot.com/2006/02/inductive-proof-example.html

- - - - - - -

link: EWD975 Dijkstra's Pythagorean Theorem

tag: math. auxiliary. construct. formulate.

good example of constructing an auxiliary figure.
good example of symmetric formulation of a given theorem.

pnt: He takes into account that the formulation be made symmetric.

pnt: In constructing the auxiliary figure, Dijkstra considers that the difference alpha+beta-gamma should be constructed and that it should be done in a way that alpha and beta are symmetric. hence that germ of the figure. Lesson is that the auxiliary figure is not pulled out of a magicians hat, but follows naturally from the theorem to be proved.

- - - - - -

an inductive proof: example
link: Wallace-Bolyai-Gerwien Theorem at cut-the-knot

tag: inductive. math. proof. example. divide. step. intermediate. auxiliary.

my notation
a + b : a and b put together.
a = b : a and b are equidecomposable.

the proof of the theorem combines various simple equidecomposability results in these steps:
1. polygon = triangle + triangle + ...
2. triangle = rectangle = square
3. square + square + ... = big square
4. (from 1&2&3) therefore polygon = square
5. (finally) polygon = square = another polygon.

1&3 are examples of "divide and conquer", in particular, decomposing the mathematical object into simpler pieces.

2 are examples of "step by step" or "intermediate object": where rectangle is the intermediat object.
4to5 is an example of introducing "intermediate object", where square is the intermediate object for the proof.

the following three simple equidecomposability results are used in the proof:
* square = rectangle
* rectangle = triangle
* square + square = square

- - - - - - -

That's not the only way: the largest example
link: Pythagorean Theorem and its many proofs at cut-the-knot

54 proofs of the pythagorean theorem.

lesson: there are always other ways to prove a given theorem.

tag: math. alternative. proof. many.

test 1234

2006-02-04T12:37:00.000-08:00

to republish.

third disappearance test

2006-02-04T12:03:00.000-08:00

link test
@D @Firefox.
same.
blummy's BlogThis! button.
a javascript.
longer post.
longer post.
longer post.
longer post.
longer post.
longer post.
resizing.
link test
(link button press test)
test: wait 5 minutes
going to other page. minimizing the window. but i think that was more than 5 minutes, i did go outside for a while maybe.(?)
two post lost? one is: many proofs of pythagorean theorem (from cut the knot) the largest example of "there is always another proof". where did my post go...

second disappearance test

2006-02-04T11:49:00.000-08:00

Google
@D @Firefox.
same.
blummy's BlogThis! button.

first disappearance test

2006-02-04T11:48:00.000-08:00

@D @Firefox
by blummy's BlogThis! button.
whofeij

firefox setting and web browser setting

2006-01-26T03:14:00.000-08:00

tag: firefox. setting. web. browser. bookmarklet.

goto about:config

Let keyword.URL be
http://www.google.com/search?ie=UTF-8&sourceid=navclient&gfns=1&q=
as said in http://kb.mozillazine.org/Firefox_:_Tips_:_Location_Bar_Search
.

Let config.trim_on_minimize be
true
as said in 'one only one memory usage thread site:mozillazine.org'
.

add yubnub from here
as said in 'installing yubnub'

- - - - - my favorite bookmarklets - - - -

Go to del.icio.us/help , or there buttons are.
del.icio.us direc.tor there 'link:'.

Go to blummy and drag bookmarklets into toolbar.

.

The Geometry Junkyard: My Contributions

2006-01-24T09:58:00.000-08:00

The Geometry Junkyard: My Contributions

the first paragraph.

english. recall.

"some of the pages here are ... which is which"

"network is my computer"

2006-01-21T14:17:00.000-08:00

http://66.102.7.104/search?q=cache:CsVZzsGHDqYJ:max.limpag.com/2005/11/04/the-network-is-my-computer/+%22the+network+is+my+computer%22&hl=en&lr=&client=opera&strip=1
"network is my computer"

seems network is down. i want to read that blog.

experience.
Reading Other peoples experience is important cuz it saves my time of experience.

GYUHANG.NET: 신에 가까운

2006-01-21T12:22:00.000-08:00

GYUHANG.NET: 신에 가까운

한줄요약: 예수는 인류 역사상 가장 신에 가까운 인물이었다.

동의.

Browser Buttons For the Future | Search Results Usability (Video) | What Makes a Blog a Blog? | Gmail Information Architecture | Google S...

2006-01-21T11:09:00.000-08:00

Browser Buttons For the Future | Search Results Usability (Video) | What Makes a Blog a Blog? | Gmail Information Architecture | Google S...

My comment on this article.

"Show Links" - Fagan Finder's URLinfo has section for backlinks.

"Translate" - there is a boomarklet for "google translate to English". autotr is a yubnub command for this.

"Search Visited" - Google Desktop Search does this.

- - -
google blogoscoped.

Add Notes to Any Page with MyStickies

2006-01-21T10:47:00.000-08:00

Add Notes to Any Page with MyStickies

no support for Opera. (it's Opera again!)
this is something i wanted.

googling word: annotation system web page. comment on.
tag:

블로그 메타 사이트의 짤막한 역사

2006-01-21T07:57:00.000-08:00

레이딘 트리스탄의 서 : 제3권 Story Maker

한줄요약: AllBlog가 제일 좋다.
(modenblog를 지우자)

Blogger Help : Blogger Browser Matrix

2006-01-21T07:48:00.000-08:00

Blogger Help : Blogger Browser Matrix

blummy BlogThis button with Opera test.

delete. test.

Mathematics as a language

2006-01-21T05:42:00.000-08:00

http://www.cut-the-knot.org/language/hersh.shtml

math lingo vs plain english.
1. I have to learn both.

Distance between strings

2006-01-21T05:22:00.000-08:00

http://www.cut-the-knot.org/do_you_know/Strings.shtml

induction - recursive - finite - minimal number - algorithm.

Sam Loyd's Geometric Puzzle

2006-01-21T04:47:00.000-08:00

tag: math

http://www.cut-the-knot.org/pythagoras/Loyds.shtml

Don't forget to scroll down.

key conceps: rectangular viewpoint(as opposed to trigonometric viewpoint). calculating area within rectangular setting.

geometric viewpoints: trigonometric eye. rectangular eye. Euclid theorems. complex number and vector.

� Firefox accelerating development cycle | Between the Lines | ZDNet.com

2005-12-23T09:07:00.000-08:00

� Firefox accelerating development cycle | Between the Lines | ZDNet.com:

"Annotating, tagging and sharing bookmarks and other content on the Web is one area of development for 2.0. "

This will be cool. I will keep using firefox as my primary web browser.

엠파스 블로그

2005-12-15T08:16:00.000-08:00

엠파스 블로그 머피의 법칙

머피의 법칙 : 잘못될 가능성이 있는 것은 반드시 잘못된다.
세상을 움직이는 59가지 법칙

text editors : my favorite text editor

2005-12-06T04:48:00.000-08:00

notepad:
mpad (metapad), cedit (crimson editor)

folding editors:
jedit, freemind

online editors:
gmail-draft & blogger
(public posting - in blogger and not in gmail,
private postin? - in gmail and not in blogger,
)

math editors:
MathType, Winedt

online math editors:
AoPS texer box , wikipedia sandbox

quick web browsing: how to surf web quickly

2005-12-06T04:34:00.000-08:00

Ie = Internet Explorer. Fx = Firefox. Op = Opera

Image-block, Ad-block : Fx - ImgLikeOpera, Adblock Plus, Flashblock

Offline-browsing : 'Work Offline' button in Fx, software like 'SurfOffline', maf extension for Fx.

Esc Key : Esc key stops animated gifs and image loading.

Background loading : is a benefit of tabbed browsing

google cache :

google browse by name :

.

- not draft. but pub. - from blogthis button
tabbed. next line is empty.

new line.

test with rich formatting

2005-12-06T03:55:00.000-08:00

gmail rich formatting.

bold

italic

my favorite

2005-12-04T23:15:00.000-08:00

my favorite
subject : stuff <- why

web-browser : Firefox aka Fx <- flexible,
blog : blogger <- popular and supported by google, mail-to-blogger, quick?,
e-mail : gmail <- many geek-like ways to use gmail, autosave, quick,

my very first blog message

2005-12-01T05:28:00.000-08:00

my very first message.
\enddocument