Alif Wahid

Ridiculously long integers, but still nowhere near infinity!

So, isomorphismes posted something interesting earlier in the week to demonstrate the difference between \(2^{2^{2^2}}\) and \(2^{2^{2^{2^2}}}\). That’s hyper-exponentiating 2 to itself 4 times and 5 times respectively. These are actually part of a fascinating mathematical operator called tetration (i.e., the next operator in the series of operators: addition, multiplication, exponentiation). It’s amazing how many programming languages do NOT support calculating such long integers, which is not surprising given their rarity. However, python is pretty sweet for such calculations. Here is the sequence of numbers corresponding to hyper-exponentiation of \(6^6\), starting with 2 and ending with 5 (and the actual tetration will literally take more than a lifetime to compute on my laptop!). Note that the python operator for exponentiating is ** (e.g, 6**6 means 6 raised to the power of 6, and 6**6**5 means 6 raised to the power of 6**5, which in turn means 6 raised to the power of 5).

$ python -c ‘print 6**6**2’ # \(6^{6^2}\)

10314424798490535546171949056

$ python -c ‘print 6**6**3’ # \(6^{6^3}\)

1204120867648235108202090056857283403336732693457453224358121221145020555710636789704085475234591191603986789604949502079328192358826561895781636115334656050057189523456

$ python -c ‘print 6**6**4’ # \(6^{6^4}\)

3048038843157366051156451562587960914683843070009432577720371743970260248988253388734087776516378135275070793705483810168992174078819457229916982909272972117802705538546043924392880249338669607593617565773503681777562894648411193991668219111024140760301610404587920245316338496104651309961792720642767736988127034593657473200942901910918843867439542131034903455333688218414524168236538840952909280093318279899396079306730083204209917234906371378011735099583017148436711446474063192400558782588545007076563578617663198436110485321365453724515867573206330463738004176650289141225099730033358933544597672427393530251774965047211695506312946306762294998433468262930173135236327015973465126381014128785885984902934363536428840380015651386113570744138512750613093972490684498513211851441804268854063729284096776385783954252916394679043883583422253317285565365659687584910629328675433251717593584447460445269780478280767340984528217318932238565096396250980632747766225123954084888838021720715193156305805969620729856

$ python -c ‘print 6**6**5’ # \(6^{6^5}\)

80190511417718642126823324718367187228561124379028767032642984026696527685909099423272280409907130820856664234534252547383919785792220682688124768661305459764363907411429981465891057029933838727501814441806045135620442558743661835589426589946920649349657656706090250821685723480965941188343685690726218140655579217325748445855297737560689439245320090903450689423418447823641842197996266347921612064380092293936942024867447336260960218766156355104115750573964203330671274400021356103878977554933511538319549310099032097779743184906645434985411235166939435035172411964842142967548250148630273650014462188652334799262982699997472433086018965308982853218279479424824047741627463816736228241352680785451432095209668261788939711558466713720132242293745772921448940790740551844434434008906193034676987240057304500131108010023042597053394274584797206497036333055579458255064407007544868240706439176260524117888597747817247043924561435278271887309056381091805867601619602251796096400239298214815262205815810495851883048734986346152273704541907980517682891333798723716799846126881590621405666624030853266332188998637596226214198907834122541927489293463347160133763014502117756168216336158830114627329202977218109579368237166132156567117925020087348139705459145227331715719630342522870498465476785107571053263453494079678567755889095079940187526351199266190216925889027808671629102384349737214723184859355227570333017933339515713795388860158422658813142610052462552561531124468334021552575519317369712398549893299488022466192324266086303869235263681881809144657510051875031162274098866094419279562380208220324102530098886472069111428433617488472272516055190671056469982414848473047070790257893061962649402322109549904795828661722527648687617928767746379721495747519959211141040916111102472432018152460719051167544236405919983233953117838933243887167089427812364370202619892209018498976682851438682521894475191713352835282030493296589384712919392973226219211191288091922284035764198302804401510674264271313400291750479617586815808002065334610106237612814316692500812416262477849331005382194774509783776249392848253693735848749122479363634821386023094809009260807127069703642131601341758921068404932742749189556771687054015933472600318253567596808221091251211711703641198856155255542413502599219243125231124707010703756432040851991341579197236142864356940729178223076963340376298091195126023533546841565469722388179096534865015625515047046570963420216955624280137393078231569773569948982141887926144207971441215537594906005093536952329848039312778015477469720653882057885248129417138963934082124319879328510703466345181658431317850957327034071471765397226881197993545556865982592007997710424004475702357132496494376641281701478783172600043123929627756814940337917468536651352909682412163154933605051724078476404415853009241046889879088290672699116823567675505259508394940589299351448798962932730350799970185840036495181266341124321852431181496056540339690610156603751845458286632667474065265696737473864354691357207202701527065402487087291412527403277767976883461633028962004285545846440493575225314130774394979967937378817702113126306072419455152323267882594983571298483500465825807896703872181789457381955432647872387911051213467617557987023849695828359459524711163550419985869657676704055817908644687127673576453955210839424436840190659827027252321398501932586759740411729952289617418278134765622813326050166959957384064382813113083786831755203742521598218605765840629154362364687711303817838049012975261098818706031083778779921930338153969952829372320637217705971993553150607385902119752440657964303988303972862883646147475106786443197703235867584836077370838721142011678759973762131722424134687500917686363953045267662773093137815945736556948724190193573407163764867877153195367591431100153449614703833275030770886797919827969802690303977026301264215440127629900242728911768560267326235803994874362448037123613763254450430482381895799210777320387010513081228433695682802772932190357949981416457818029991504540768966753037459786011903710783960269984510243360995482400887126305528142426809242291255927388970092499522644826730634353554532290013554216298408936830014398138795251653589037358576904476827007923274508531053478037943367964176441257037590277013740407417782007327008826098874282368889270784570950786912620185328736577519896968757943687578610897754204026914925858221388080673050441824821755725576167340253305804521182043728264128801559756563257488713680680809133701727450964058594763006137824371369361316200344599880051384402035659367496743923603271929776588780455945342609429175333833732087253316702961877934549090835555674032605356077637644879327372936947591318361663596803630395896131225284879988495303929143762967731049100198363156149538755837425424959700972683697853135492946217817764276303379016406744567350241586674650572185257582725886064487676298551839944386144412978961115582326074861396098373880273079980787032483386367357279417962171668621359717512606596304376531440825003611118804365098297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There’s a bit of buzz around Bitcoin this month, simply due to the volatile exchange rate for buying bitcoins with greenbacks (and vice versa). Expectedly, the lamestream media has focused on all the circus sideshow. I’ve yet to come across one report in TV/Radio/Print media that actually explains what a bitcoin is…sigh. There are the oversimplified terms like “online currency” (which makes no sense, because how is an existing currency not an online currency given that you can buy/sell with it almost anything over the Internet?). Then there are claims that bitcoin is not under any state control or beyond the monetary regulation of a sovereign central bank. Distinctly false! Sovereign states and central banks have simply chosen to leave bitcoin alone so far, given that it’s smaller than a small potato. But that will change, you can bet your bottom dollar on that outcome :P

Anyway, the pertinent question of what, exactly, is a bitcoin was answered by the original academic paper that proposed the damn thing back in 2009! Full-text PDF is available here. It was authored by an anonymous pseudonym called Satoshi Nakamoto. Most likely multiple smart cookies were behind that paper. It’s a marvellous invention, only from a technological perspective! Just read the paper and educate yourself, instead of putting any faith on the non-existent brains of journalists. 

Then there’s the economics and ethics perspective, which makes bitcoin an especially not marvellous invention! Again, you won’t find anyone in the lamestream media capable of doing the requisite analysis to arrive at that conclusion. There are plenty of “financial advisors” that are both recommending and not recommending investment in bitcoin for currency traders. But the only one that has actually analysed bitcoin with a deductive dose of logic is Karl Denninger at Market-Ticker.Org. Read that article to once again educate yourself on all the non-technological things that are wrong with bitcoin.

That’s it folks, just 2 articles is all it takes to bust this buzz into a non-mystery. But the irrational circus circulating throughout the lamestream media would fool everyone and anyone that this is the craziest thing to have happened since the invention of money. What a load of dung poop :P Anyway, just to reiterate: IGNORE everything you hear about bitcoin from both proponents and opponents. Then just read this and this, and rest assured that the earth is still a semi-spherical object in orbit about a magnificent monster.

Radiating Rambling Ruminations

I totally agree with isomorphismes about follow friday being a boring idea, and failure friday being a far more interesting idea. Regardless of how far this little meme radiates, it’s quite fun to ruminate and ramble on about what makes me a fundamentally flawed person :P My trivial failures are obviously easy to guess. So let me see if I can regurgitate juicy tidbits to chew on for some self-twisted, end of week entertainment :)

My academic record has a bunch of notable failures (as in flunked courses), but these I am particularly fond of because they built much needed character and resilience. One that comes to mind readily is that I flunked a major maths course in the third year of undergraduate engineering. The university let me off for doing well in other courses by the end of the year so that I didn’t have to repeat it in the following year, even though I never asked to be let off. But the failed grade is still glaringly visible in my transcript, and rightly so! This was a case of total and utter boredom with a course to such an extent that I didn’t bother doing any of the weekly assignments (which cumulatively carried a big proportion of the final grade) and submitted an almost empty exam paper at the end. Motivation (or lack thereof) is a mysterious manifestation in my case :P

Driving must go down as a failure for me. It took three attempts before I passed my full license test in New Zealand. That was a while ago and I’m more experienced now. But still, in Melbourne I use public transport almost exclusively to save me (and others) from my unsafe driver self. Concentrating on the road in front of me is too boring a task ;-) My mind (and the car by extension) ends up wandering toward poetry, philosophy, physics…and all the more interesting things worth pondering. Unfortunately, that means people around me everywhere are in danger when myself is driving :(

I have a particular propensity to miss deadlines, especially conference submission deadlines :P I missed a few notable deadlines while working on my PhD thesis (e.g., the little beauty was 12 months overdue when I eventually submitted). I was fortunate to have had an awesome supervisor who tolerated a certain number of these on the basis that I am a pedantic prick who cannot express precisely what he wants to say in his writing (some might call it writer’s block). The rate of passage of time in my headspace is far removed from the rate of passage of time in ordinary existence (one more reason why my reflexes are slow when driving). I simply fail to feel the sense of urgency and desperation that comes before a deadline, which everybody else seems to feel innately. So I’m a pain for anyone whose job it is to manage the projects that I’m working on, and I do enjoy being like that ;-)

Speaking of conference submissions, I’ve had a number of brutal rejections that were difficult to accept. But the referees were generally correct in pointing out my errors and I just had to take it on the chin :P This is frequent across the dominion of science. In fact, I would say that this is universal. And that’s how it ought to be! But what I fail to understand is the lack of acknowledgement of rejections and the very deliberate attempt to cover up one’s past rejections. I had four rejections in total during the four years that I was a PhD student. I also only had four publications in total (two journal papers and two conference papers). Both of these counts are very low since most students typically end up with double digit publication counts by the time they submit their theses nowadays, although I’m not aware of any reliable statistic on the typical rejection count.

On the topic of rejections, I’ve been rejected by plenty of employers throughout my short career so far. More than anything this is due to my total failure at wisely selecting which jobs to apply for and which ones I have to hide from. As with every other aforementioned failures, this is very much a work in progress (and will be for ever). So I’m not fussed about this at all. Fortunately, I’m happy with my current job situation being a contractor and all. I sense a strange sort of entrepreneurial urge that might manifest in a profound sort of way. Or maybe not :P Remains to be seen ;-)

It’s plausible that A1 is a superposition of uncountable quantum fluctuations, which cannot be tamed into an orderly existence. The elementary sub-atomic personalites that comprise such a chaotic connectome have only ever been statistically determined to exist in an unobservable form (after all, no one has ever seen me, myself or I). But I used to think that the audible quacks in my ears were just me and myself. Well, not necessarily. It appears that anti used to think the same about anti-me and anti-myself. For fear of annihilation, none of us can be squeezed into too small a confinement, although a jelly lump (about the size of an apple) enclosed inside an empty cavity is occasionally permitted. However, that does make it terribly difficult to string together unique utterances due to the need to translate a loopy graph into a spanning tree. Information is lost along the way that cannot be recovered. Furthermore, the impossibility of reaching a consensus amongst such sub-atomic personalities regarding the expression of a sentiment is entirely inconsequential for the charming quarks that underwrite the right to asymptotic freedom of speech. And none of this makes any sense, whatsoever, to me.

Daylight saving ended earlier today. It always screws up my internal clock (and that of Microsoft Windows OS). I’m pretty sure that humans didn’t evolve to have this kind of messing with their innate sense of time (perhaps more importantly, a sense of the relative passing of time). At any rate, it reminded me of a journal paper that I read long ago and keep referring others to read, simply because it’s an absolute gem. I never developed an intuitive sense for Einstein’s relativity until Leslie Lamport’s paper explained how “Time, Clocks and the Ordering of Events in a Distributed System” are all linked together. Seven and a half pages of heavenly enlightenment.

“Time, Clocks and the Ordering of Events in a Distributed System”, Leslie Lamport, Communications of the ACM, volume 21, number 7, pp. 558-565, July 1978, URL.

PS. Leslie Lamport nowadays works for Microsoft Research :P

(Source: research.microsoft.com)

Rarely read, but highly influential, journal papers

Nope, I’m not going to create a list here, only attempt to leave behind a memento for my future self from my current forgetful self.

“An Empirical Approach to Economic Intelligence in World War II”, Richard Ruggles and Henry Brodie, Journal of the American Statistical Association, volume 42, number 237, pp. 72-91, March 1947, URL. 

I’ve yet to come across someone who has actually read this paper. I read it couple of years ago and keep coming back to it for a number of reasons. Chief among them is the fact that this paper demonstrates a level of humility that simply does not exist today in any form of scholarship. Ruggles was with Harvard and Brodie was with the State Department at the time of publication. I’m not going to paraphrase the paper since people are perfectly capable of reading it on their own, especially when a paper on statistics is such an endearing, exhilarating and enlightening read!

Lyrics of the song “Lanterns” by Birds of Tokyo

Lately I’ve found
When I start to think aloud
There’s a longing in the sound
There is more I could be
In darkness I leave
For a place I’ve never seen
It’s been calling out to me
That is where I should be

We never carried days on our own
But now it’s up to us to know
The weight of being so much more
We will find ourselves on the road

On we march
With a midnight song
We will light our way
With our lanterns on
On we march
Till we meet the dawn
We will light our way
With our lanterns on

As we walk out
Without question without doubt
In the light that we have found
It is finally clear
Our day has come
And we’ll stand for who we are
We are ready we are young
We have nothing to fear

We never carried days on our own
But now it’s up to us to know
The weight of being so much more
We will find ourselves on the road

On we march
With a midnight song
We will light our way
With our lanterns on
On we march
Till we meet the dawn
We will light our way
With our lanterns on

We held the light
To our faces
And realised
We were chasing
Shadows behind
Not worth saving
So burn it bright
Forever illuminating

On we march
With a midnight song
We will light our way
With our lanterns on
On we march
Till we meet the dawn
We will light our way
With our lanterns on

…
In darkness I leave
For a place I’ve never seen
It’s been calling out to me
That is where I should be

(Source: youtu.be)