MATHEMATICS

Kamis, 09 Desember 2010

Reminder 2011 course registration

Ref: 47911183 Reminder - changes to registration dates for undergraduate courses starting in February 2011 - 22 Dec 2010.

I still haven't decided yet... I really don't know at this moment. Tired, tomorrow...

[Exercise] - 2a: The coconut congruence equation

The coconut problem is what is called an algebra word problem. In a course in elementary number theory we could skip the words and ask to solve the following equation:

$$\frac{4}{5}(\frac{4}{5} (\frac{4}{5} (\frac{4}{5} (x-1) - 1) -1) -1) \equiv 1 \mod{5}$$

Rabu, 08 Desember 2010

[Sign of the times] - Wikileaks: Yes, USA: No

Often heard: "What can I do about it, anyway?"

Well, you can close your MasterCard account for example. Or stop buying books from Amazon. Don't buy American if you have the option. You can also support Wikileaks directly of course. I think they need it and deserve it. - They belong to the good guys on this planet. Wikileaks is more than just Assange, although he is important.

Just my 0,02c.

[Exercise] - 1a : Hint

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

Hint: MST121 / MS221 math suffices to solve this one, we need to find a function $A(n)$ such that $${A(n)}^2 = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$
and
$$f(n) = A(n) \in \mathbb{N}$$
( To be continued. )

Selasa, 07 Desember 2010

[Exercise] - 2

This is a famous problem. I found it on the Internet by searching for "mathematics, monkey, coconut, problem". - My version is in a Dutch book called Algebra by M. Riemersma.

Five men and a monkey were shipwrecked on a desert island, and they spent the first day gathering coconuts for food. Piled them all up together and then went to sleep for the night. But when they were all asleep one man woke up, and he thought there might be a row about dividing the coconuts in the morning, so he decided to take his share. So he divided the coconuts into five piles. He had one coconut left over, and he gave that to the monkey, and he hid his pile and put the rest all back together.

By and by the next man woke up and did the same thing. And he had one left over, and he gave it to the monkey. And all five of the men did the same thing, one after the other; each one taking a fifth of the coconuts in the pile when he woke up, and each one having one left over for the monkey. And in the morning they divided what coconuts were left, and they came out in five equal shares. Of course each one must have known there were coconuts missing ; but each one was guilty as the others, so they did not say anything.

How many coconuts were there in the beginning?”

To be continued ( i.e. answer and comment )

Senin, 06 Desember 2010

[Exercise] - 1

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

For $n=1$ to $5$ we have ($n, \ \sqrt{f(n)}, \ f(n)$):
$\begin{array}{lll}
1. & 3. & 9. \\
2. & 11. & 121. \\
3. & 41. & 1681. \\
4. & 153. & 23409. \\
5. & 571. & 326041.
\end{array}$

Belajar Matematika Online (Wolframalpha)

Belajar matematika bukan saja lewat buku, CD, belajar kelompok atau kursus. Dengan menggunakan sebuah search engine kita dapat mengetikkan formula matematika di kotak pencarian kemudian akan ditampilkan solusi dari persamaan matematika tersebut.

Salah satu keunggulan wolfram alpha dibanding search engine lain adalah dilengkapi dengan database engine yang memudahkan perhitungan matematis.
Misalnya belajar tentang integral, differensial, deret, fungsi kuadrat dan lain-lain,

Jika ingin membuktikan klik disini wolfram alpha


[News] - Math and industry

The European Science Foundation published the ESF Forward Outlook "Mathematics and Industry". Download a PDF copy from this page.

Minggu, 05 Desember 2010

[Sign of the times] - New York /12-2010

From an article in the New York Times: LAPTOPISTAN

credit: Piotr Redlinski for The New York Times 

( A new series of posts [Sign of the times] with mainly pictures. Common theme is of course the theme of this blog: "mathematics and / or study". ) Let me know if you think this series doesn't belong here.

Comment:
Take your laptop with you and you can go out on your own. Have coffee, work, study and enjoy the company of people around you. In the Netherlands public libraries are ( or already have ) transformed themselves into embassies of Laptopistan with free power-points and WiFi.

Jumat, 03 Desember 2010

[Sign of the times] - Abacus still in use

The Corporation is optimistic about the potential of the abacus system of mental arithmetic, which it introduced in two of its primary schools on Friday. The system is aimed at improving the comfort of young students with numbers and the mathematical functions.

Source: express buzz

Link: Abacus ( see the comment )

Link: An introduction to the abacus

Link: Chinese Abacus + Manual

Video Lectures on Number Theory

Among lectures on Calculus I,II and III, ( Introduction to ) Linear Algebra and ( Introduction to ) Differential Equations from the UCCS ( University of Colorado and Colorado Springs ) Department of Mathematics you will find video lectures on Math 311 Number Theory by Professor Dr. Seung Son here. I have watched most of them earlier this year. This week I watched some of them again.

While watching a video on mathematical induction something amazing happened, not sure if I would call it a cognition, but it's close. Since I was able to do proofs by mathematical induction and thus understood it, I thought I was done studying mathematical induction. ( I mean both the MS221 and M208 exams included questions on induction). Well, I close-to-cognited that I didn't understand proofs by mathematical induction -at all-.

Do you? If so:
- state the first principle ( of mathematical induction ) using symbols only,
- state the second principle using symbols only,
- re-formulate: "Show that: ... $$\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$$ ..." using Set Terminology,
- can you explain the difference between the first and second principle?
- give an example of a proof using the first principle,
- give an example of a statement which can only be proved with the second principle.

I failed ( note: past tense ) all answers to the questions above. Post is To Be Continued ...

Triangle Puzzle

Have you ever had a nice problem that you just thought about at odd moments?  Boring meeting, stuck waiting somewhere, few surprise extra minutes in a day?

For a while now, my favorite problem like that has been finding a nice way to divide up a square into the seven triangle types.  I love tangrams, and I like Pierre Van Hiele's mosaic puzzle even better.  If you do too, stop reading right now and try this problem.  It's fun and worth a surprising amount of thought.  (For me, anyway.)  Then suddenly this week, one of my little thumbnail sketches worked out.  I don't know whether to be happy or sad.  Being a geogebra nerd, I wanted to make a sketch of it, and that led to making a puzzle out of it.

You can print this picture of the pieces to try in real life, or try it with the Geogebra file or as a webpage.    (A solution is an option on the file or webpage.)



But... now I'm left wondering what to think about in those rare extra moments.  Then on Twitter, Justin Lanier (@j_lanier) tweets:
Had an insight in the shower this morning. Example: .717171... = .717171.../1 = .717171.../.999999... = 71/99 (!)
 Hmmm.  Really?  Maybe it's a coincidence, because 100 times .717171... minus the original leaves you 99... hmm.  Would it work for .717171.../.6666... ?  It does.  Tweet back:
@ cool. So is .a_1 a_2...a_n repeating / .xxx... =a_1...a_n/xx...x (n times) for any x? Or divided by .b_1 b_2... b_m repeating ...
Which connects to another problem (from Dave Coffey) I like thinking about: how many digits does it take 1/17 to repeat and how can you tell?  In general?

OK.  Deep breath.  There's always more problems.

Soal Ulangan Umum Semester I

Tidak terasa satu semester sudah perjalanan waktu di Tahun Pelajaran 2010 / 2011 ini. Segenap Pengurus MGMP Matematika SMP DKI Jakarta mengucapkan selamat menempuh Ulangan Umum Semester I yang akan dilaksanakan mulai Senin, 6 Desember mendatang.
Berikut soal ulangan umum semester 1 Tahun Pelajaran 2009/2010 sebagai bahan latihan :

  1. Matematika 7 Semester 1 ( Download )
  2. Matematika 8 Semester 1 ( Download )
  3. Matematika 9 Semestar 1 ( Download )

Rabu, 01 Desember 2010

മാത്യു സാറിന് ആസ്ട്രേലിയയിലും നേട്ടം


ആസ്ട്രേലിയായിലേക്ക് ചിലന്തിഗവേഷണത്തിനുപോയ എം.ജെ. മാത്യുസാറിനെ ഓര്‍മ്മയില്ലേ? എറണാകുളത്തെ ഐ.ടി@ സ്കൂള്‍ മാസ്റ്റര്‍ ട്രൈനറായിരുന്ന ആ പ്രതിഭയെ നാം ഈ പോസ്റ്റിലൂടെ പരിചയപ്പെട്ടിരുന്നു. പോകുന്നതിനു മുമ്പ്, തന്റെ അനുഭവങ്ങള്‍ ബ്ലോഗിലൂടെ പങ്കുവെയ്ക്കാമെന്ന് അദ്ദേഹം ഏറ്റിരുന്നതാണ്. പലവട്ടം ചാറ്റിലും മറ്റും കണ്ടിരുന്നുവെങ്കിലും, തിരയ്ക്കിട്ട ഗവേഷണങ്ങള്‍ക്കിടയില്‍ പോസ്റ്റും ചോദിച്ച് ശല്യപ്പെടുത്തേണ്ടെന്ന് കരുതിയിരുന്നതാണ്. എന്നാല്‍ രണ്ടാഴ്ച മുമ്പ്, വൈകിയതിനുള്ള ക്ഷമാപണത്തോടെ,അനുഭവങ്ങള്‍ മെയിലായി പറന്നുവന്നു. നിങ്ങളുടെ ഓരോ കമന്റും, നമ്മുടെ നാടിന്റെ അഭിമാനമുയര്‍ത്താനുള്ള അദ്ദേഹത്തിന്റെ കഠിനശ്രമങ്ങള്‍ക്ക് കരുത്തേകും.

മാത്​സ് ബ്ളോഗിന്റെ എല്ലാ വായനക്കാര്‍ക്കും ഓസ്ട്രേലിയയില്‍ നിന്നും അഭിവാദ്യങ്ങള്‍. യാത്ര തിരിക്കുന്നതിനു മുന്‍പ് മാത്​സ് ബ്ളോഗുമായി ഇന്‍റ്ററാക്റ്റ് ചെയ്യാമെന്ന് ടീമംഗങ്ങള്‍ക്ക് വാക്കു കൊടുത്തിരുന്നെങ്കിലും ഇപ്പോഴാണതിനു സാധിച്ചത്, സദയം ക്ഷമിക്കുമല്ലോ?

ഇവിടെ ഞാന്‍ പെര്‍ത്ത് എന്ന സിറ്റിയിലാണ് താമസിക്കുന്നത്. ഓസ്ട്രേലിയയിലെ ആറ് സംസ്ഥാനങ്ങളിലൊന്നായ വെസ്റ്റേണ്‍ ഓസ്ട്രേലിയയുടെ തലസ്ഥാനമാണ് പെര്‍ത്ത്. ക്രിക്കറ്റ് ആരാധകര്‍ക്ക് ഒരുപക്ഷേ പരിചിതമായ, ബാറ്റ്സ്മാന്‍മാരുടെ പേടിസ്വപ്നമായ, തലയ്ക്കുമുകളില്‍ പന്ത് കുത്തി ഉയരുന്ന, ലോകത്തിലെ ഏറ്റവും വേഗവും ബൗണ്‍സുമുള്ള "വാക്കാ" (WACA – Western Australian Cricket Association) ഗ്രൗണ്ട് ഇവിടെ സ്ഥിതി ചെയ്യുന്നു. ഇവിടെയുള്ള വെസ്റ്റേണ്‍ ഓസ്ട്രേലിയന്‍ മ്യൂസിയം (http://museum.wa.gov.au/), യൂണിവേഴ്സിറ്റി ഓഫ് വെസ്റ്റേണ്‍ ഓസ്ട്രേലിയ (http://www.uwa.edu.au/) എന്നിവിടങ്ങളിലായാണ് ഞാന്‍ ഗവേഷണം ചെയ്യുന്നത്.

തികച്ചും പ്രോത്സാഹജനകമായ കണ്ടുപിടുത്തങ്ങളാണ് ഗവേഷണത്തിലൂടെ ഉരുത്തിരിഞ്ഞിരിക്കുന്നത്. ഓസ്ട്രേലിയ-പസഫിക്ക് മേഘലയില്‍ നിന്നും 7 പുതിയ ഇനം (species) ചിലന്തികളെയും അവയുള്‍പ്പെടുന്ന ഒരു പുതിയ ജീനസ്സി (genus) നേയും ഈ ചുരുങ്ങിയ കാലയളവിനുള്ളില്‍ കണ്ടുപിടിക്കാന്‍ സാധിച്ചു. അന്താരാഷ്ട്ര ജൈവവൈവിധ്യ വര്‍ഷത്തില്‍ തന്നെ ഈ കണ്ടുപിടുത്തം നടത്താന്‍ കഴിഞ്ഞതില്‍ വളരെയധികം സന്തോഷമുണ്ട്. പുതിയ ജീനസ്സിന് പ്ളെബ്സ് (Plebs) എന്നും സ്പീഷീസ്സുകള്‍ക്ക് പ്ളെബ്സ് ഡീവിയസ് (Plebs devius), പ്ളെബ്സ് ആര്‍ലീനിയേ (Plebs arleneyae), പ്ളെബ്സ് ആര്‍ലെറ്റിയേ (Plebs arletteyae), പ്ളെബ്സ് നിയോഹെബ്രൈഡിക്കസ് (Pleb neohebridecus), പ്ളെബ്സ് പട്രീ‍ഷ്യസ് (Plebs patricius), പ്ളെബ്സ് റോസ്മേരിയെ (Plebs rosemaryae), പ്ളെബ്സ് സെബാസ്റ്റിനി (Plebs sebastiani) എന്നുമാണ് നാമകരണം ചെയ്തിരാക്കുന്നത്. ഈ കണ്ടുപിടുത്തം പ്രസിദ്ധീകരണത്തിനായി അന്താരാഷ്ട്ര ശാസ്ത്ര ജേര്‍ണ്ണലായ Invertebrate Systematics -ന് അയയ്കാനായി തയ്യാറാക്കിക്കൊണ്ടിരിക്കുകയാണിപ്പോള്‍.

ഗവേഷണത്തിന്റെ അടുത്ത ഘട്ടം ഈ ചിലന്തികളുടെ പരിണാമ വളര്‍ച്ച (Phylogeny), ജൈവ-ഭൂമിശാസ്ത്രം (Biogeography) എന്നിവയുടെ പഠനമാണ്. സമയപരിധിക്കുള്ളില്‍ ഇവയും പൂര്‍ത്തീകരിക്കാന്‍ സാധിക്കുമെന്നാണ് പ്രതീക്ഷ.

ഇവിടുത്തെ ജോലി സംസ്കാര (work culture) ത്തെക്കുറിച്ച് സൂചിപ്പിച്ച് ഈ കുറിപ്പ് ഉപസംഹരിക്കാമെന്ന് കരുതുന്നു. മ്യൂസിയത്തിലേയും യൂണിവേഴ്സിറ്റിയിലേയും സ്റ്റാഫ് വളരെ കഠിനമായി അധ്വാനിക്കുന്നവരാണെന്നാണ് ഞാന്‍ മനസ്സിലാക്കിയത്. നമ്മുടെ ജോലിസ്ഥങ്ങളില്‍ വളരെ സാധാരണമായ ഗ്രുപ്പിസമോ അലസതയോ പരദൂഷണമോ ഇവിടെ കണ്ടില്ല. ഓരോരുത്തരും തങ്ങളുടെ ജോലിയോട് 100 ശതമാനം ആത്മാര്‍ത്ഥത പുലര്‍ത്താന്‍ ശ്രമിക്കുന്നതായിക്കണ്ടു. സഹപ്രവര്‍ത്തകരോട് വളരെ സ്നേഹത്തോടും ബഹുമാനത്തോടുമാണ് ഇവര്‍ പെരുമാറുന്നത്. തികച്ചും സ്നേഹപൂര്‍ണ്ണവും ആദരപൂര്‍ണ്ണവുമായ ഒരു അനുഭവമാണ് ഏനിക്കും ഇവിടെ ലഭിച്ചുകൊണ്ടിരിക്കുന്നത്.

Mathew M.J., Ph.D.
Research Scientist - Arachnology
Department of Terrestrial Zoology
Western Australian Museum
Locked Bag 49, Welshpool DC, Perth, WA 6986
AUSTRALIA

email: matthew.mundackatharappel@museum.wa.gov.au
mathewmj@asianetindia.com
mathewmj@gmail.com

tel: +61 8 9212 3790
mob: +61 4 2174 2248
fax: +61 8 9212 3882

http://www.museum.wa.gov.au/

Haskell Researchers Announce Discovery of Industry Programmer Who Gives a Shit

The worldwide Haskell community met up over beers today to celebrate their unprecedented discovery of an industry programmer who gives a shit about Haskell.

On Wednesday, researchers issued a press release revealing that 27-year-old Seth Briars of North Carolina, a Java programmer at Blackwater accounting firm Ross and Fordham, actually gives a shit about Haskell.

"Mr. Briars has followed every single one of our press releases for years," the press release stated. "Probably even this one."

Haskell researcher Dutch Van Der Linde explained how they had stumbled on the theoretical possibility of Briars and his persistent interest in Haskell. "We knew that there are precisely 38 people who give a shit about Haskell," said Van Der Linde, "because every Haskell-related reddit post gets exactly 38 upvotes. It's a pure, deterministic function of no arguments -- that is, the result is independent of what we actually announce. But there are only 37 of us on our mailing list, so we figured there was a lurker somewhere."

"That, or it was an off-by-1 error not detectable by our type system," Van Der Linde added. "But we don't, uh, like to dwell on, I mean with good unit testing practices we can, um... sorry, I need to get some water."

As Van Der Linde stumbled off in a coughing fit, his fellow researcher Bonnie MacFarlane outlined their basic dilemma: "Finding a person who gives a shit about Haskell is an inherently NP-complete computer science problem. It's similar in scope and complexity to the problem of trying to find a tenured academic who didn't have the bulk of his or her work done by uncredited graduate students. So even though we suspected Briars existed, we needed a strategy to smoke him out."

She explained the trap they set for Briars: "We crafted a fake satirical post lampooning Haskell as an unusable, overly complex turd -- a writing task that was emotionally difficult but conceptually trivial. Then we laced the post with deeper social subtext decrying the endemic superficiality and laziness of global industry programming culture, to make ourselves feel better. Finally, each of us upvoted the post, which was unexpectedly contentious because nobody could agree on what the reddit voting arrows actually mean."

"And then we waited to see who, if anyone, would give a shit," she said.

MacFarlane concluded, "Our elegant approach didn't work, so we hired a Perl hacker to go dig up the personal details on all 38 accounts that had ever upvoted a Haskell post, and the only one we didn't know was Seth Briars. So we reached out to him, and thankfully so far he hasn't threatened to sue us."

Briars says he is pleased to have been recognized for his apparently unique shit-giving about Haskell. "I've been giving a shit about Haskell for a long as I can remember. I follow all their announcements and developments closely, just in case I ever get the urge to use the language for something someday."

"It's a beautiful, elegant language," Briars observed as he busied himself cleaning a fingernail. "You'd be hard-pressed to find a more expressive and composable core. And they've made astounding advances over the years in performance, interoperability, extensibility, tooling and documentation."

"I'm kind of surprised I'm the only person on earth who gives a shit about it," Briars continued. "I'd have thought there would be more people following the press releases closely and then not using Haskell. But they all just skip the press releases and go straight to the not using it part."

"People see words like monads and category theory," Briars continued, swatting invisible flies around his head for emphasis, "and their Giving a Shit gene shuts down faster than a teabagger with a grade-school arithmetic book. I'm really disappointed that more programmers don't get actively involved in reading endless threads about how to subvert Haskell's type system to accomplish basic shit you can do in other languages. But I guess that's the lazy, ignorant, careless world we live in: the so-called 'real' world."

Haskell researcher Javier Escuella remains hopeful that one day they may be able to double or even triple the number of industry programmers who give a shit about Haskell. "I believe the root cause of the popularity problem is Haskell's lack of reasonable support for mutually recursive generic container types. If we can create a monadic composition-functor wrapper that is perceived as sufficiently sexy by hardened industry veterans, then I think we will see an uptick in giving a shit, possibly as much as a full extra person."

Haskell aficionado Harold MacDougal is not quite as sanguine as his colleague Escuella. "I doubt Haskell will ever be appreciated by the uneducated natives of this industry. As exciting as it is, the discovery of Briars should be considered an anomaly, and not as a sign that more people will ever give a shit. Programmers only seem to pay attention to things when there is humor involved."

"We do have an experimental humor monad," added MacDougal. "But it doesn't seem to be getting much adoption. Haskell fans just don't see the need for it."




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