Linearly Independent

The Purist

2010-03-08T04:58:00.002-05:00

by ogden nash.

I give you now Professor Twist,
A conscientious scientist,
Trustees exclaimed, "He never bungles!"
And sent him off to distant jungles.
Camped on a tropic riverside,
One day he missed his loving bride.
She had, the guide informed him later,
Been eaten by an alligator.
Professor Twist could not but smile.
"You mean," he said, "a crocodile."

The problem with Josephus

2010-01-17T20:37:00.000-05:00

Titus Flavius Josephus or Yosef ben Matityahu

was a jewish historian of the first century. He was a military leader in the first Jewish-Roman war where his troop was defeated by Roman forces. Read up if you want about his very interesting life

Josephus's problem:

A legend says that caught upon a cave with the Jewish army being persecuted by the Romans, they decided to commit mass suicide to avoid the humiliation of being defeated, enslaved or tortured (a la Masada). However, Josephus was not too keen on this self sacrifice and to refuse to participate would have surely meant betrayal to his friends. The story goes on to say that they were to form a circle and that every third person would be killed and taken out of the circle until there was one standing that would commit suicide. J's only way of surviving was to cleverly position himself in the position of the last man standing. This turns out NOT to be a trivial calculation for large groups.

of course this story is not mentioned until the middle ages so its authenticity is very dubious.

PLAY THE JOSEPHUS GAME

It is very complicated to solve this problem for 'N people' and 'every K skips'
so we can try to solve for n number of people for a particular number of skips.

unfortunately an explicit solution can't be found for all n and k but instead must be the solution to the following difference equation:

If Josephus had a buddy and you would like to know where would he stand so that two people survive the problem becomes even more complicated.

Variations of this problem have entertained computer scientists for decades since it poses very good scenarios for optimization.
don't think however that you must be caught in an ethnic strife against the world power to be able to put skills like this to good use, this could be useful in other much more common situations like being stuck in a dessert island without food and trying to decide which survivors to eat first. It pays to know math.

Engineering Supervillains

2010-01-09T16:35:00.003-05:00

The proportion of students that enroll in engineering majors has been decreasing since the 80s.

Depiction of engineers in comic books and some cartoons has been much better than TV-shows for older audiences. after all there are many shows about doctors and lawyers and lame office jobs (the office)

Here are three popular engineer supervillains:

Doctor Octopus (Otto Octavius)

A nuclear Physicist/engineer that also must be involved with some kind of Biomedical work since he invented Brain-computer interfaced robotic arms that characterize him. Otto Octavius is irresponsible about his work, greedy and antisocial. Not much of a role model for kids in engineering, unless they are megalomaniac psycopaths.

Ivo Robotnik (Dr Eggman)

Dr Ivo 'Eggman' Robotnik is a gifted mechanical engineer with an IQ of 300. His research deals with making improved cyborgs of animal creatures whom he does not kill, city planning and general robot making. For some reason a pesky blue hedgehog called Sonic, who was the fastest kid in his PE class but didn't do much else with his life, spends every day sabotaging Robotniks plans. This sends a very bad message of brawns over brains to our kids. Allthough he is portrayed as a supervillain he really just is misunderstood. In fact he bears a deliberate resemblance to Teddy Roosevelt. Card carrying member of the bullmoose party and a feminist activist, we should learn to appreciate him more.

finally the baddest of the bad:

Anakin Skywalker (Darth Vader)

Young Anakin was raised in a third world planet and still managed to educate himself very well in areas of mechanical and aerospace engineering. Later in his career like most engineers he ended up in management where he proved his great leadership skills. And so he started wars and destroyed planets ...boohoohoo. He also built the biggest ship ever! too bad he didn't teach himself engineering ethics.

Happy New YearraeY weN yppaH

2010-01-04T11:57:00.005-05:00

I plan to post more this year due to increased time availability and newyearresolutionwishfulthinking.

My last semester in college is fast approaching and big changes are probably coming to my life soon. Hopefully I will have time to update this after i finish in May.

We are fascinated with symmetry
that's what we see in pretty faces, necessary but not sufficient, maybe not even necessary ask Cindy Crawford.

Symmetry can be any self similarity with respect to any of the dimensions of an object and a frame of reference.

I want to talk about a reflective symmetry.
Our right hand is the mirror image of our left hand. d is the mirror image of b (more or less graphically). Generally in order for mirror images to be constructed there must be a frame of reference or frame of symmetry.

in the case of 1 dimensional objects there is a point of symmetry in the 1D space that denotes the two reflections.
in the case of 2 dimensional objects there has to be an axis (line) of symmetry.
for 3d solids there has to be a plane of symmetry (mirrors).

Couldn't you say that reference symmetries require N-1 dimensional manifold of symmetry for N dimensional objects?

Now let's talk about something not always associated with geometry: words, sentences, language.

Palindromes are sentences of 1d (like all sentences not in hyperfiction or madlibs) that express reflective symmetry.
I prefer pi.
reading it backwards we can see it says the same thing. the 0dimensional symmetry point is the 'f'.
race car
is also a palindrome. right hannah?
and the day before yesterday (do you know there's a single word for this in spanish) 01-02-2010 is a palindrome if you consider the string without any kind of separators. NOT ONLY THAT but it is a palindrome that could also be read as a base 3 number since only the digits 0 1 and 2 appear. In the next 100 years this will only happen in:
01022010, 11022011, 02022020, 12022021, 10122101. (5 times)
for 012 only.
and for general palindrome dates
01022010, 11022011, 02022020, 12022021, 03022030, 04022040, 05022050, 06022060, 07022070, 08022080, 09022090, 10122101. (12)
and almost 01122110 (8 days short).

but I know what you're thinking. "I don't live in America!/I dont write dates mm/dd/yyyy!" well there is an endless number of ways to write dates if you follow ddmmyyyy then there are even more palindromes because you are not restricted by the first two numbers to be a maximum of 12 (months) but a maximum of 31 days when appropiate and 28 worst case. a more annoying problem to compute.
if you write dates like mmddyy or ddmmyy then also the possibilities change.

also there can be symmetry with a non unique symmetry reference in any dimension.
to show this the trivial example of an infinite (yet onedimensional) string ...11111111.... or a self connected *{11111}* string of any length. Where is the point of symmetry? everywhere? anywhere?

Physicists often exploit symmetries to simplify problems. sometimes they find a solution and backtrack to the supposed symmetries that allowed for that solution. it's almost a mystical thing to them. A coincidence.

Engineers don't really believe in perfect symmetry that's why they've devised all sorts of measurements and procedures to quantify 'symmetricness' in real objects.

but to stop myself from a combinatoric rant that will put everyone to sleep i leave you with the challenge of making palindromic new year resolutions.

Platypuses Platypi Platypodes

2009-07-09T03:37:00.009-04:00

The plural of platypus is cause for debate among many many third-graders.
I personally like Platypodes best platypuses second and I don't use platypi.

The platypus is a wonderful animal
It is one of the few mammals that lays eggs. But that's not the cool part
the Platypus has electric field sensors in the beak like nose it has.
It uses these electric field sensors to sense preys from the nerve impulses of their muscles.

Did you know that a moving electric field is felt like a magnetic field. I don't plan to get into all of maxwell's equations in this post but if you know what i'm talking about you are probably already thinking the same thing I am:

Imagine a Platypus getting an MRI while blindfolded. It must feel like a hundred thousand river eels are swimming counter clockwise circles around him. If he was hungry you would probably hear the loudest platypus there ever was.

Oh yes! platypodes have poisonous talons so don't annoy them too much. and make sure they take off all metal jewelery and accessories before stepping into the MRI machine and platypodes with pacemakers or implanted metallic prostheses may not get MRIs

If you're nice to platypodes they may help you in your physics homework.

Oliver Heaviside

Egyptian Fractions

2009-03-19T20:21:00.005-04:00

The Egyptian were a clever lot. They discovered a lot of science/math out of necessity and availability of many strange phenomena. The flooding valley of the Nile left them with really fertile ground and gave them several opportunities. Having a large amount of the valley area flood every year fomented community organization and the establishment of a city authority to delegate zoning. Grain surpluses translated in hyperaccelerated development and trade.

The grain trade led to the creation of a set of measurements and instruments. One popular unit was the Heqat that measured volume. You would go to a bar in Men-nefer (Memphis) sit between the sun priest and the scribe and order 1/5 Heqat (1pint) of fine Egyptian beer which you would drink through a straw.

Out of the grain trade arose the necessity to create a system of fractional numbers. since grain could not be sold by the number a heqat was determined to be an important unit and any fraction of it would be represented as a sum of unit fractions. this was likely due to having to carry a different set of containers that could measure each fractional ammount.

so an egyptian merchant would have his heqat bag, a half heqat bag a third heqat bag a fourth heqat bag a fifth heqat bag....
so they decided to represent numbers as sums of unit fractions.

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

if you were an important egyptian mathematician and wanted to know the value of pi.you would make using a compass you draw a circle in the ground with one cubit of radius.
then the egyptian representation of pi would be
3 + 1/8 + 1/61 + 1/5020 +1/128541455 + 1/162924332716605980
this number system only has one advantage to reduce the number of bits necessary to express common units like 1/2 or 1/4 or 3/4 or 2/3.

Perhaps it was this awefull system that plateaued egyptian mathematical progress and let the summerians and other civs improve much more.

Generalizations of Calculus I

2008-11-27T20:40:00.008-05:00

This is the beginning of a series of posts on this subject. I'll stick to the basic stuff and go from here, a primer if you will.

Calculus is a wonderful topic and I'd like to assume you know enough about it. feel free to email me or comment if you think i said anything wrong or if you don't understand anything.

Everybody knows (everybody who matters) about the operation of differentiation. lets call it D

D^1(f(x))= d/dx(f(x))
and successive iterations of these function are taken we obtain a function of two variables that is the nth derivative of a function 'f' at the value 'x'.

The first derivative of a well-behaved-single-real-variable function can be understood as the slope of the line formed by the ordered pair (x, f(x)), or f'(x). the second derivative is a measure of the concavity of said function. each succesive derivative represents the rate of change of the rate of change of the rate of change of the rate of change.....n of a function x with respect to x.

n does not have to be a positive number. it could be a negative one. the operator can simultaneously include its inverse operator and antiderivatives can just be derivatives of negative order. so the operator D^[Z](f(x)) can represent the fundamental operations of calculus: integration and differentiation as a single process and the successive repetitions of them in the same way.

There is however one problem.

Derivatives lose information and integrals require information.

a function f(x) := g(x) + c where c is a constant independent of x will have derivative f'(x)= g'(x) and extra information will be required to reverse the process to the initial state. more so if this process is repeted several times. an Nth order integral will be indeterminate to (N-1) polynomial order unless this information is provided. this problem is hurtful to the generalization of the differintegral.

but for now I'll give you one of these generalizations
this is Cauchy's formula for repeated integration

this is the Riemann-Liouville differintegral

which generalizes derivatives. please note the difference in variables .
(will fix this later)

Think about that

smooth operator

2008-11-16T15:46:00.001-05:00

what is an operator?
you've seen them before: +*-/^ are common examples.
operatos map sets unto other sets. for example a + b the addition operator is mapping the addition of a and b unto a single quantity. but what is addition? we all know how to do it but that's not the same thing as knowing what it is. we can define a lot of concepts in arithmetic by startng at the beginning: 0.
then we can define another number/ quantity as its succesor. we invent the operator k. so that 0k is 1 and 0kk is 2 then we can define addition as m+n as n succesions to m and multiplication as j*k as k additions of j. the reader can prove commutativity laws for all these by noticing that the fifth succesor to one and the first succesr to 5 are the same quantity. we can define powers as g^h as h multiplications of g. this however is not commutative.
what's the next step? what does y powers of x. what's before the succesor operator. can we generalize all these orderings to any order? (valga la redundancia).

I think about this sometimes thanks to m. shachar.

Gravity Express: 42 minute pizza delivery.

2008-10-16T17:39:00.004-04:00

Recipe:
Make a hole through the earth from any two points in the surface connected by a cylindrical cavity. make sure to reinforce the boundaries of the cavity and then lubricate with superfluid butter or frictionless substance evacuate all the air from tunnel and insulate. By dropping yourself, or your sister, or a pizza box into the cavity it is possible to reach the otherside (wherever that may be) in exactly 42 minutes. It is not even necessary for the tunnel to go directly through the center of the earth. Maximum velocity will be achieved when the cart/train/sister/pizza is at the very middle of the path.
proof

d(Kinetic Energy + Potential Energy)/dt=0
KE + PE= c
1/2*mv^2 + PE= c
v=sqrt((c-2PE)/m) so velocity will be maximum when potential energy is minimum.

since gravitational potential energy has radial dependence inside a sphere it will be smallest at the point closest to the center. getting closer to the center increases an objects velocity while getting away from the center implies getting farther from another massive objects center of gravity and thus experience a deceleration. You better have someone catch you on the other side or you're coming right back.
even though the period of the oscillation is constant the maximum velocity isn't tunnels through the very middle of the earth will experience the most velocity since they cover the most distance and it can get pretty fast. relativistically it might even feel shorter to travel straight through the middle. id have to calculate that and get back to you.

so forget 30 minute free pizza and order yourself 42minute Chinese Food done right.

Colombi days

2008-10-14T16:42:00.004-04:00

my grandparents are originally from Spanish North Africa they had seen Barcelona before they decided to move to the Americas and remembered the famous Christopher Columbus statue:

Once they were moved in to Caracas my grandmother sent a letter to her relatives back in Morocco written in ladino:

Its very nice out here, the weather is great and their customs so different. They have built a statue of columbus that sends a very powerful message. If the Barcelona one says: Go to America it's great. This one says: So what? if you don't like it go back!

The Barcelona statue points toward America and the Caracas one points back to Spain.

Quarter Imaginary Number System

2008-09-22T20:46:00.005-04:00

There are 10 kinds of people in the world. Those who know binary and those who don't.

In case you have never seen this joke before and you don't know what I'm talking about:

The decimal number system, is the most common numbers system used by people today.
I think the Sumerians liked 60gesimal number systems.
we like the decimal system because having 10 fingers it's easy for us to count when we learn arithmetic. If you have good memory or really bad math skills you'll remember the last time you used your fingers to aid yourself in calculation.

The way the number system works is the adoption of ten arbitrary symbols to represent the digits one having greater value than the next by a singular unit which is the first digit

therefore 0 is the first digit 1 is the succesor of 0 (S0), 2 is the succesor of 1 S1 (SS0), 3 = SSS0, 4=SSSS0, 5=SSSSS0,6=SSSSSS0, 7=SSSSSSS0, 8=SSSSSSSS0,9=SSSSSSSSS0.
when you reach the end of the scale or when you need to represent S9 the system places the smallest unit in the following 'space' so 10=S9 and 11=S10 and so on.
to expand from other number systems to decimal the value of the second 'digit' with the minimal input is considered the base and an expansion in the following manner is computed.
XYZ=X*B^2+Y*B+Z*B^0 usually the digits XYZ... may not exceed the base so as to avoid two possible ways to represent the same number (non-uniqueness )

Binary only uses the digits 1,0 actually they are called bits when binary is referred to.
therefore 10 in binary is 2 in decimal (1*2+0*1).

THE COOL THING IS
that you can make your base anything you want even negative numbers or imaginary numbers.

if you make the base to be (2i) every number real or complex can be represented only using the numbers 0-4 without the need for negative signs or i's .
Try it!

2323 in quarter imaginary system would be

2*(2i)^3+3*(2i)^2+2*(2i)+3= -2*8i+3*4+4i+3= 15-12i

or qigits. The drawback is that odd imaginary components require 'decimal' or quarterimaginarinal 'digits'

(0.2)= 2*(2i)^-1=-i so (10.2) = i

proving there is only one exact way to represent each number (except for infinite repeating decimals of highest value number like 0.9999999999999...=1 is much harder)

but Donald Knuth came up with this when he was only in middle school.

so to restate the begining joke but this time in the quarter imaginary system

There are 10.2 kinds of people I admire, those with the power to imagine

Donald Knuth

Pass the ball

2008-08-15T16:42:00.003-04:00

Here's something cool that i stumbled upon

Pass the ball

slightly hypnotizing
and bad physics.
it wouldnt have cost them too much to program some physics and walls and bouncing.

Drilling Square Holes

2008-07-28T21:40:00.011-04:00

If you like handy work and tools and sawdust and grease you might have already considered what the best method for making square holes is.
this depends on many things like the material, the size of the hole and your budget.

With a good budget, a not too large hole and some appreciation for mathematics you will find the method described in this entry to be the coolest.

Reuleaux Triangle
the fat triangle formed from the intersection of three circles with origins at the vertices of an equilateral triangle is the Reuleaux Triangle. Interestingly it has the same maximum width regardless of how it is rotated. this property was thought to be only possessed by circles once and yet here's a simple and apt counter example.

Constant width shapes can make great manhole covers, as no orientation of the cover will let it fall through the manhole. A few years ago Microsoft interviewers asked job applicants why manhole covers were round and this was thought to be one of the best answers.
Another more important property of constant width shapes is that they can rotate inside parallel lines.

rotation of a Reuleaux Triangle is possible inside a square and it can encompase almost all the area of said square with the proper parameters.
This shape created by the rotating reuleaux is about 99% the size of the theoretical square it is trying to reproduce. Using a triangle like this a drill bit can be made to drill nearly square holes. The difficulty lies in reproducing the path that the centroid of the R.triangle travels in a real tool. It can be closely approximated by a superellipse |X|^r+|Y|^r= a^r and thanks to Harry Watt you can go by the special drill.

the actual drillbit looks like this
with concave parts to be able to cut into the material.

you can also do cool things like this to get linear alternating motion from circular continuous.

one of the most interesting applications of Reuleaux triangles is the Wankel engine:

how awesome is that engine. Pistons are kinda NSFW.

the three dimensional R. triangle is the R. Tetrahedron

several claims in the literature claim the Reuleaux tetrahedron to be of constant width but this is not true. a similar tetrahedronesque shape exists called Meissner Tetrahedron or body.
here's an animation

homework: can a Meissner body or a Reuleaux tetrahedron have a cube orbit?

imagine a robot holding three transmitters that emit a wave in 60 degree angles ( and do not overlap) from each other in 2d. the transmitter is spinning around hyperelliptically model a the propagation of this wave.

[click the pictures for sources or search wikipedia]

"Hi my name is Franz Reuleaux. I discovered the triangle one day while shaving. I trimmed my beard like the intersection of three circles and noticed how I could unfog a square in the bathroom mirror by rubbing my beard circularly against the glass"

oh yea don't believe that. Even though it's a million times more amazing than Mendeleev dreaming up the periodic table.

Anger Producing Mathematics

2008-07-06T05:20:00.004-04:00

Two very unintuitive results of mathematics anger me whenever i think about them:
The Monty Hall Problem
and the Banach-Tarski paradox.

The Monty Hall problem concerns the old gameshow where there are three doors: one with a prize and two with goats.
once the contestant picks one the host reveals the contents of one of the other two doors that does not contain the prize. the player is then given the choice to switch to the remaining door. Is switching beneficial?

yes it is.

The Banach-Tarski paradox uses the axiom of choice (one of the most complicated concepts in mathematics) to say that a solid ball in 3-dimensional space can be split into several (a finite number) non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.
except that by "pieces" they mean weird noncountable measureless point scatterings.
and of course... you can't actually do that with real solids (see definition of atoms)
in fact you can make thousands of these balls through Banach-Tarski mathemagical duplication but you can't sell them (because they don't exist)

Total Perspective vortex

2008-06-30T02:00:00.004-04:00

Here's some short Douglas Adams fiction enjoy it much

Total Perspective Vortex
The Total Perspective Vortex derives its picture of the whole Universe on the principle of extrapolated matter analyses. Since every piece of matter in the Universe is in some way affected by every other piece of matter in the Universe, it is in theory possible to extrapolate the whole of creation – every sun, every planet, their orbits, their composition and their economic and social history from, say, one small piece of fairy cake.
The man who invented the Total Perspective Vortex did so basically in order to annoy his wife.
Trin Tragula – for that was his name – was a dreamer, a thinker, a speculative philosopher or, as his wife would have it, an idiot. She would nag him incessantly about the utterly inordinate amount of time he spent staring out into space, or mulling over the mechanics of safety pins, or doing spectrographic analyses of pieces of fairy cake.
“Have some sense of proportion!” she would say, sometimes as often as thirty-eight times in a single day.
And so he built the Total Perspective Vortex, just to show her. Into one end he plugged the whole of reality as extrapolated from a piece of fairy cake, and into the other end he plugged his wife: so that when he turned it on she haw in one instant the whole infinity of creation and herself in relation to it.
To Trin Tragula’s horror, the shock completely annihilated her brain; but to his satisfaction he realized that he had proved conclusively that if life is going to exist in a Universe of this size, then the one thing it cannot have is a sense of proportion.

:
you cannot have an absolute sense of proportion and talk to Einstein if you don't believe me.

Information Entropy

2008-06-28T23:05:00.006-04:00

A while back I brought up the infinite monkey theorem.

we seemed to hit a paradox at some point.

Suppose that you have a message of N characters which we will call M.

how much information does such a message contain? assume spaces or punctuation don't count.

one way to measure the amount of information that a message contains is to measure the amount of information that it would take you to store such message.

you could have N number of boxes with C number of subchambers one for each possible subunit
like a four dial lock with the digits 0-9 in each dial as a possible choice.
this message could be stored in four boxes with ten subcompartments by naming the boxes ABC & D and the sub compartments 0-10. therefore with four pebbles you can mark down the message.
to universalize this have a strip of paper with 40 circles in a row and a hole punched in the first 10 then another in the second ten and so on. it takes 40 bits to store the password of this lock on this mechanism.

however this information is compressible note the fact that 9999 the highest possible password is the binary number 10011100001111 that's 14 bins (digits are for decimals) with 2 possibilities thats only 14 circles on a strip of paper that are either punched or unpunched.
and we just compressed our information from 40 bits to 14 without compromising any part of the information.

If you have a gigantic keyboard with all the alphabets in the multiverse (i.e. hebrew, latin, cyrillic, Babylonian Cuneiform, Vogon, Martian and Alphacentaurian) then you would have a lot of characters to pick from and it would not likely be very efficient to store messages.

One pretty cool danish ( if Kenedy is "ein Berliner" then this guy can be a pastry too.) Piet Hein wrote

Problems worthy of attack prove their worth by hitting back
is 26^50= 5*10^70 bits of information in characters
but there are no Zs Xs Qs Js or Ds.
so 21^50= 1*10^66.
but noting that those letters are missing also takes information.

if we use words instead of characters it would make sense to think that this would reduce the amount of information needed since we can't ever make a combination like stakfl ssd kkjkjjjkjkjjksdjsdsdsdsda fammvodfdfd. because at least all the words would make sense, it would take less storage space.
86000^10=2*10^49
for long enough messages. if synonyms are eliminated imagine the possibilities.
its a tradeoff of universality and for efficiency and a more complex coding mechanism.

http://www.edict.com.hk/textanalyser/wordlists.htm

the link directs you to a place where you can see what the frequency of usage of each word
apply it to this formula.

therefore in a N words message the amount of disorder or entropy contained in a message is equal to:

where p(xi) is the probability of word number i in a message

lets find the entropy of a message whose language uses all words with equal frequency and has M words so the probability of each word is 1/M and n is the number of words then the entropy of this message would be n/M*log2(M)
for Piet Hein's message

Word Probability
Problems-----------0.000243
worthy-------------0.000028
of-------------------0.035839
attack--------------0.000103
prove--------------0.000052
their---------------0.002628
worth--------------0.000093
by------------------0.005224
fighting-------------0.000071
back----------------0.000952

will have to be calculated by Derrida (you) as homework.

Scientists get some style

2008-05-25T12:14:00.004-04:00

Neither Alexander Graham Bell's first words transmitted through telephone nor Marconi's first transatlantic transmission equal the approximate the greatness of the first telegraphed message by Samuel Morse.

Marconi sent a string of SSSSSSSSSS
Bell said “Do you understand what I say?”

come on guys! should have been a little more creative like Morse was.

the first telegraphed message carried the phrase

What hath God wrought?

well not the telegraph.

Summer reading

2008-05-10T20:18:00.003-04:00

Summer reading had me a blast
summer reading happened so fast...lol

So here's the official (but incomplete) summer reading list. I'll divide it for the sake of keeping the peace into two categories:

Technical, Mathematical, Scientific

Biological Physics-Philip Nelson
Difference Equations-Saber
Fundamentals of Fluid Mechanics-Munson
Introduction to Physical Chemistry for Biologists and Medical Students-Kruyt
Nonlinear Dynamics of Chaotic and Stochastic Systems-Anishchenko
Organic Chemistry- Bruice
[MCAT PREP complex]

Literary, Historical, Fictional

The Sun also Rises
Farewell to Arms
A Man Without a Country
Sirens of Titan
Jailbird
Guns, Germs and Steel
War and Peace

I'll try to get through all of it but I might add or subtract a few titles.
I need a good math subject/book (dynamics one is very challenging) and i need something electrical.

I also need to go to the Beach

Pathosynthesis.

2008-04-08T02:44:00.014-04:00

How does disease affect art?
beyond the obvious many psychological problems that famous masters of the arts may have had, more tangible/diagnosable diseases are sometimes the cause of a distinct artistic creation.

Frontotemporal Dementia: this family of neurodegenerative disorder has been linked to sudden behavioral changes before the condition becomes problematic. (a).
Dr. Adams, a mathematician and chemist leaves her career as a scientist to take care of a very sick son that miraculously recovers. Afterwards she decides to take up art and abandon science.

She painted this and she titled it Image of a migraine

This other one she called Pi

and yes the colors mean something she represented certain digits with colors
to show the patterns in the decimal digit representation.
in 94 as her condition worsened she became fascinated with this song

and began trying to capture the song in a painting.

This project was very important to her, so imagine all the minute details with which the painting tries to describe the song that she hid into this representation.

The funny thing is that Ravel composed the song when he was exactly the same age as Dr Adams when she started painting "Bolero Unraveled" and Ravel probably suffered of the exact same condition as he had all the same symptoms Dr Adams had.
Unfortunately, she passed away last year, but she left lots of very interesting artwork and an illustrated Invertebrate Zoology book that's just marvelous.

Aphakia, or the absence of a crystalline lens inside the eye is another interesting condition.

Claude Monet, started painting in redder tones towards the end of WWI as he had developed cataracts :a denaturation of the protein inside the eye's own lens due generally to prolonged exposure to UV light. Once Monet's eyes got cloudy the not-so-ultra-violet light (bluish purplish) had a harder time going through his eye than the redder tones with bigger wavelengths.

Around 6 years before his death Monet had the cloudy crystalline lenses removed so he would be able to see again. His aphakia probably made him able to sense Ultra Violet light, as is reported in many cases of the same condition, which would have made him see the world much bluer than the average person. I sometimes wonder what crazy things Monet would see with an increased spectrum of colors to work with.
I mean, just coming up with "Impression, Sunshine" without extra color sensation...

His paintings made after the operation are somewhat bluer and have a different distribution of colors like in these Water Lilies

Leonardo of Pisa

Hybrids

2008-03-31T19:00:00.005-04:00

lets mix it up a little:

mammalian hybrids are sometimes infertile because their genetic make up
Horses have 64 chromosomes while donkeys have 62
mules (male donkey-fem horse crosses) have 63 chromosomes and are mostly infertile.

mules have been made since ancient times as they are better for farming task than their parents.
Recently the first mule was cloned

No Humanzees, but i'll show some cool pictures i found in the InternetsCama (Camel Lama hybrid)
Grolar Bear
The Toast of Botswana Goat-Sheep hybrid
Wholphin
Zorse
Zonkey (on the right)
Jaglion

And the world famous Liger bred for his magic skills

Maria Agnesi

Price Stability

2008-03-30T01:12:00.010-04:00

Economics presents a great opportunity to learn new math.

A difference equation defines a sequence recursively

X(n+1)=F(X(n)) is a first order difference equation
X(n+j)=F(X(n),X(n+1),X(n+2),X(n+3)...X(n+j-1)) is a jth order difference eq.

In order to solve for X(n) in terms of n initial conditions are necessary.
the number of initial conditions necessary is equal to the order of the equation.

The Fibonacci Sequence is a second order DE defined as
F(n)=F(n-1)+F(n-2) and the 2 initial conditions required are F(0)=0 and F(1)=1

where that symbol is the golden ratio.

On a graph a first order differential equation can be drawn as the function X(n+1) in the vertical axis and X(n) in the horizontal. it is also required to draw the line at X(n+1)=X(n) a straight diagonal 45degree line (you'll see why)

so if you draw that graph the places where X(n+1)=X(n) these are equilibrium points
some of which are stable and some which are not. the stability depends on the slope of the X(n+1).

With this the Cobweb economic model was created.
read about it and look at the pretty pictures.

It all makes sense graphically
a good proof i found uses series expansions to show that successive iterations near equilibrium converge into it if the demand is more price elastic.
derivation is left to the reader as an exercise

John Nash

a little Econ

2008-03-19T20:27:00.004-04:00

Venezuela's state-run oil company begins demanding payment in euros as US dollar weakens

the dollar loses value faster than the euro since the contracts are made until the money is spent.
not a dumb move.
so bigger demand for euros mean.

Price of Euro vs quantity sold

sending the $ further down wanting more oil producing countries to switch to euros
the only problem with this argument is that the oil exporting countries will want to buy more things from the US with a weak dollar...
it seems like a destabilizing move whose only antidote is a much stronger dollar soon.

I'm playing around with difference equations and economics
unlike differential equations some problems in difference eqs can be solved completely by graphical means.

More on that later
and no more politics (I promise)

Series and Parallel Inefficiencies

2008-03-06T21:04:00.000-05:00

I'm very distracted and I can't seem to concentrate on work.
So I started thinking:

How does slacking present itself?

Parallel Slacking (or sources of inefficiency):

Listening to music while you work
an itch
non-ideal temperatures cause sweat and cold (at extreme T it is not possible to continue working at all: 0 kelvin makes it hard to breathe and at 451F paper catches fire)
RL conversations at near distances.
small foods

Series Slacking

Inemuri (my favorite)
IMing
large foods
blogging
surfing
bathroom

It's also interesting to see how break periods affect the productivity of a chain.
I used to work as a bank teller and lunch periods were the worst. the lines would get really long since there were less tellers on the floor and they'd feel the stress in the environment and work faster.
I measured it it one time during break. So the 'man.' basically gets the free labor out of the employees' overdrive mode. If the whole bank went on lunch break not only would the free overdrive be lost but there would be transition costs in opening and closing the whole show.

Ok i g2g back to work...

Hypnerotologia Portalia

2008-02-28T20:55:00.000-05:00

So you're stuck on a dungeon____One guard will lie, always
And there are two exits ________The other will lie never.
And two corresponding guards.___ You are only allowed to
One door leads to freedom __________Ask one question.
The other, a slow painful death._ How will you find freedom?

____________________ ___________________
............................................................................
....................?.................................... ?...............
___________________________________ __

.
.
.
If you've never thought about this before or heard a possible solution to this riddle it's a very interesting problem. I have come across one solution, but i have a feeling that all of them have something in common.

Solution: Ask either guard: ' which door would the other guy tell me leads to freedom'? which ever door he tells you, pick the other one.

First proof (exhaustive)
you either ask the liar or the honest guard.
so if you ask the liar what the honest would say he will lie and give you the wrong answer.
and if you ask the honest what the liar would say he will not lie but he will still tell you the wrong answer (because that is truthfully what the lying guard would tell you.)
in either case you pick the other door and you're home free.

so lets represent with some bools.

Liar=will lie <0>&&<1> about the truth= <0> lie
Honest=will tell the truth<1>&&<0>about a lie=<0>lie

now what you can't do is both be free and know who is the liar.
if you attempt to find out who the liar is, you spend your 1 question.
the truth as it turns out will not set you free.

A good twist on this comes from the Werner Herzog movie Kaspar Hauser: Every Man for Himself and God Against All.
in the movie , Kaspar, a feral child is taken care of and instructed ultimately developing a peculiar way of thinking. In one scene a University professor gives him a riddle similar to the previous one except the only thing he needs to find out is if the "guard" is honest or not.
kaspar does not like the convoluted answer the logician gives him because any question with an obvious answer is a much simpler way to find out.

Like: Are you a tree-frog?

This is one of my favorite pictures in all of literature

I am in love with woodcuts.
Particularly crypto-occult ones like this
a man is made to choose between divine glory, worldly glory or mother of love.
guess what he picks?

one day ill have a place like this in my house and put random closets behind the doors
Gloria Dei: Coconut Jello
Mater Amoris: Baseball caps
Gloria Mundi: a stockpile of metal wires and plates.

[mad bonus for identifying the picture]
Arrivederci

George Boole

With Ice Cream

2008-02-18T22:03:00.000-05:00

On Modes

In statistics the word mode is used to describe the value with highest frequency in a data set.

In discrete distributions, the statistician must pick the width of intervals in which to represent a set of data. A person conducting a cross sectional study on the popularity of television shows would have to divide the population into a finite number of equidistant groups.

If the conductor of this study called 1,000,000 people asked them for their age, wrote it down, asked them if they watch the Tonight Show regularly and organized the number of tonight show viewers into each age, she'd be left with the mess of a million ages and each of their tv watching habits (not very useful).

so the conductor decides she's going to have 6 age groups: 0-14 ;15-29; 30-44; 45-59;60-74; 75-89.
and represents the number of Tonight Show fans in a histogram.
so which age group has the most tonight show fans?
this is the mode of the data set but...
what if the conductor -lets give her a name: Lucretia- had picked other age groups with narrower margins or by the remainder obtained when the age in years is divided by 6!
then the values for the mode would certainly change. sometimes a more inclusive category will retain the status of mode, but sometimes it wont.

To illustrate this Lucretia and I conducted a hypothetical study at a hypothetical bar named "100 kPa" where the amount of drinks each customer had was compiled.

Here we have a discrete distribution of the frequency of quantity of drinks per person.
2 is the most popular # of drinks to have.

with another Cat. width

we now see the data differently: 7-8 is the mode.

new mode! 4-6

and

yet another one.

so modes tend to be ambiguous like that, even for continuous distributions.

If we tried to discretize a distribution like this one

the mode would end up being several different values depending on the width of the discrete categories.
but as a thought experiment we can calculate the mode of this distribution for the infinitesimal category width. It is the global maximum of the distribution. it isn't till the width of the categories becomes bigger than the peak for the maximum that the mode changes place.

An interesting thing to construct is a graph of the mode vs the category width of discretization.
such a graph is a good measure of chaos in a distribution. in such a graph an unchaotic distribution would conserve its mode with all step sizes and a distribution. A problem with this tool is multimodality, the existance of more than one mode.

lets call this the modal transform M{f(x)}(l) where f is the probability density function of x and M is the mode with category width l.

what other properties would this transform have?

if the distribution is continuous, but bounded to a finite interval then the range space cardinality reduces to 2 when the category width is half of the domain interval.
at 0+, or the infinitesimal category width, M(0+)= the value of x at the maxima of f(x).
if f(x) does not obey the first 2 dirichlet conditions
non-linearity (think about it if f(x) has a mode "a" and g (x) has a mode "b" then (f+g)(x)'s mode need not be either b or a unless f=g in which case a =b). so the transform is unaffected by a scalar multiplication but not necessarily by the addition of another function.
for every function f(t) there exist one and only one modal transform. uniqueness.
inverse modal transforms are non unique.

well no more about my transform for now.
going back to the meaning of mode, we said it was the value for which maximum frequency was shown. we can easily use anything else other than a frequency distribution and carry this analysis. modes aren't just for discrete data, they aren't just for frequency data

Mode also means fashion. Very surprising is the speed at which modes change in our society.
the clothing industry banks on this. but clothes are slow compared to the rest.
new music, every day. new fads, new conflicts, new jokes, language evolves at trans-warp speeds on the Internets (sic)

A lot of people wonder how the ancient Egyptians managed to build the pyramids, I always wondered how they managed to dress the same way for thousands of years.
you know what their modal transform would look like?

Farewell Farewell, Till we meet again.
May the forces of evil become confused on the way to your house.

Savasorda

Murder on the Dance Floor

2008-01-04T14:57:00.001-05:00

There is a great song by Sophie Ellis Bextor with the same title as this entry. this entry is not about the song, it's about pirates.

"Yarrr me maties! let's sail the internets pillaging, plundering and infringing copyright"
OK, so maybe Johnny Depp wouldn't have said that but the reality is that millions of internet users infringe copyrights daily by obtaining and distributing movies, music, books and images through questionable sources and P2P networks.

Imagine that a person commits the most heinous crime he is able to commit in a place that is so crowded that no one will notice. that person will be far away before anyone realizes what he has done and questioning everyone in the area of the murder proves very difficult.

this scenario I call Murder on the Dance Floor.

Cybernetic Copyright infringement is a murder on the dance floor. Very few get caught because its so difficult to enforce the law. In turn, since its so easy to pirate and it's very unlikely to get caught more and more people do it. Too many murderers makes it even harder to enforce copyrights.

In ancient times there were bandits that would assault traders on their long journeys from city to city. highways and main roads were unpoliceable. But as cities sprung up, trade became more frequent:

the density of travelers on the road increased
the distance to the nearest city from any point decreased
trade is united to reduce costs in the form of caravans
technology on roads and carriers improves the speed of travelers.

These first pirates saw the decline of their trade because of the past four reasons and... boats.
When a new mode of transportation was implemented it took pirates a little while to get boats themselves and start attacking traders. this new mean of transportation was even more unpoliceable than roads creating the infamous sea pirates. GPS and new boats have nearly eradicated maritime piracy. Air piracy is a completely different story for someone else's blog.

The cyberspace, unlike roads, is a non planar graph of millions of terminals with millions of changing connections to police the entire internet it would take a computer many times more powerful than the sum of all the computers connected to the internet. Even then, Laplace's Demon would have something so say about it.

The enforcing agencies are even trying to prosecute websites that link to illegal content. the most well known online video community hosts so much copyrighted content that by the time the right holders complain the same video has been re-uploaded several times by other users who even have the gonads of putting their screen/stage/hacker-name at the beginning of the video.

"Nothing is more destructive of respect for the government and the law of the land than passing laws which cannot be enforced." Einstein

So is all the original content being made is destined to have little protection from an unenforceable and perhaps obsolete institution?

By the way, did you youtube "Murder on the Dance Floor"? tsk tsk shame on you! (and shame on the other 1.5 million people)

Music and movies are going to have to evolve, just like video games did when the whole MMORPG phenomenon sprung and restored the profits lost to piracy. perhaps they should offer the movies and the music through a streaming channel with few commercials and settle for whatever small profits that brings. In the end, all that you can always sow from original content is the ability to sell it before anyone else can. What else can be done without turning our internet into china's?

You better not kill the groove

Ian Malcolm

Infinite Monkey Theorem

2007-12-28T23:25:00.001-05:00

A monkey with a typewriter and infinite time will almost surely produce all the works of Shakespeare. this is known as the Infinite Monkey Theorem.

The Infinite Monkey theorem was disproved in 2003 with the help of an English Art Council grant that allowed some college students to put a computer inside a monkey cage in a local zoo.
The monkeys stopped typing after about five pages and left nothing but a long string of Ss before they destroyed the keyboard with stones, urinated and defecated on the computer.
Great waste of £2,000.

I think the monkeys did it out of frustration with the zoologists and art majors that were conducting the project as they wouldn't listen to the monkeys' complaints about them not being true random number generators and to show that they were only capable of stochastic processes of the excretory type.

So in an infinitely long random string the probability of not finding any finitely long string is 0.

too bad infinite random strings can't be recorded anywhere. even if they could be it would take a very very long time to find it betwixt all the mess.
But probability is complicated enough. Lets not deal with infinities so we don't go crazy like Cantor.

It would take this monkey, and any other hypothetical monkey, a long time to type a random string long enough so that it is 99.99% likely to contain all of Shakespeare's works.

lets not get greedy

lets say we want 50% probability of containing a specific n-long string.
we make sure that the monkey's keyboard only contains the standard latin alphabet with no special characters, or function keys.
Problem: HOW LONG MUST A RANDOM STRING WITH 50% PROBABILITY OF CONTAINING AN N-LONG STRING BE?

The string Monkey has a 26^(-6) P or 0.0000003% of appearing in a 6-string
in a 7 string there are two possibilities
Monkey_ and _Monkey so the probabilities are added to form (26^-6)*(7-6+1)
Generalizing this to obtain the probability of obtaining an n-string in a L-long uniform random string. where N<=L P(N,F,B)=(B^-N)*(L-N+1). Where B is the number of different characters in the string N is the number of characters in the string that is sought. L is the number of characters in the random string.
When P=1/2 and N and B are known.
0.5 = -(B^-N)*(N-1) +(B^-N)*L
L=0.5B^N +(N-1)

This Means the random string L long will be 50% likely to contain the N-long string that has been defined.

tell me if my math is wrong please.

there are many more things to consider that increase the probability of finding it by redefining what one is looking for:
would you accept a wrap around version of the string? like
-ymonke
-eymonk
-keymon
-nkeymon
-ey_ _ monk ... you get the idea.
The effect of this is related to N
would you accept a backwards version of your string?
yeknom (backwards wrap around is also possible)
this doubles the probability of finding the string
except in the case where the string is a palindrome.
in any case statistical independence ceases to exist if you do accept the word backwards.

Do you only care if the string appears once?
XXXMONKEYXXMONKEYXX.

the answers to these questions are useful depending on what you mean by "finding". if you intend to go through a magazine that prints random strings and cut out the word monkey then you might be happy to find out the word monkey appears a few times, you might decide that a cutout of yeknom is sufficient for whatever reason you are extracting the word monkey. However say you come upon the following piece of string monkeyeknom you can either cut out monkey or cut out yeknom.

I found a pretty cool application of this of which i'll post pictures when i can.

Just something to think about

Émile Borel

Filling Space

2007-11-12T13:15:00.001-05:00

Filling space is a topic I find extremely interesting.
Imagine you had a number of books you had to fit on a bookcase that has three shelves.
You can arrange books in a way that will fill each shelf most (unless you are fond of the Dewey system). This is a 1D problem when the height and depth of the books are not considered.

If there is only one shelf, you have to decide which books can go on it depending on what's important to you. But lets assume for the sake of argument, that your objective is to minimize the horizontal waste of space. Then you will pick a combination of books with thicknesses adding up to the approximate the width of the shelf from a lower bound.

If you have more than one shelf, the problem is still unidimensional but now it has become more interesting.

Using what the Gestalt dudes call Productive Thinking. The brain can figure it out pretty fast.
Computers can too, if the number of books and shelves are kept low. but if the objective is to find the most optimal solution then it becomes much more complex.

Transcending the first dimension, you get tired of watching TV and its time to do some work!
so you take all your books out of your shelves (remembering the arrangement to avoid calculating again until you buy a new book) and you want to place them all on your desk.
The total area of the books' covers is less than the area of the table but you are still having a hard time figuring out how to put all the books on the table.

Not only you have to worry about an extra coordinate, you also have to think about rotation!
Rotating any book 90 degrees will be necessary in the previous case. Of course it may well be that there is no solution to a problem of this nature because the geometry of the objects will not allow good packing. In that case you get rid of any books written by Jane Austen and try try again.

Packing of squares and and hexagons might be familiar to you. These tessellations show up in several Scientific problems and are for the most part either intuitive or easy to grasp.
Tessellations of circles (in Euclidean Space) become a special case of hexagonal tilings but now space must be wasted!

But periodic tiles are boring.
Check these out:
The Hirschhorn Tiles

And the Penrose Tiles

Penrose Tiles are very interesting because their discovery has led to advances Crystallography.
Before crystallography only dealt with periodic arrangements of atoms. Whereas this arrangement is not periodic it is certainly ordered in an interesting way. Material Scientists have found alloys that obey this pattern.

But all of this is baby food.
What about 3 dimensional packing?
What about 3 dimensional tessellations?
What about 2 dimensional tessellations in non Euclidean space?

I'd really like to learn more about this stuff.

Cya Later

Grigori Perelman

La Primera Vez

2007-11-04T14:57:00.000-05:00

The Vikings were pretty cool people. very misunderstood.
If I lived in the middle ages i would've loved to be a viking because:

Lots of Booty
Nice weapons
Beer
Nice ships, cheap fares to anywhere and the only ones to America.
Cool names.
More women's rights.

(does not include victims of skirmishes)

Of course i would've hated:

Long cold winters.
bloodbaths
weird rituals

think about how different the world would be if Scandinavia had not been Christianized.

This is something i learned today

This is a drawing of the Piraeus Lion in a famous Greek port
A fleet of Viking mercenaries defaced the Lion after putting down an revolt in the city.
They gave him a pretty nice tattoo.

Nos vemos en el espejo.

Niels Henrik Abel

Do what you want

2007-11-02T02:22:00.001-04:00

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subjects i am really interested in (right this instant):
Partial Differential Equations
Ordinary
Fluid Flow
difference equations and dynamic systems.
fractional calculus?
Biomaterials
Russian
German History late 19th century.
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P.S. Laplace

Classes

2007-10-30T00:09:00.001-04:00

I met with the department head today for a nice conversation.
Public Classtype
{
Molecular biology
Physics Lab 3
Linear Circuits
World Literature I
Research Design
Biomaterials
Thermodynamics (from the physics department)
}

I wanted to come up with a wittier object oriented joke but its been a long time since i last opened my C++ book. I'm putting programming on the backburner for now.
This is the Lorenz map which proves that all butterflies arise as solutions complex dynamic systems.

Foucault, Léon (not that other crazy bastard!)

Incipit Blog

2007-10-28T20:58:00.000-04:00

An orgy of motion
the pendula swing
springs bounce
and waves sing

This is my first post in a journal that i intend to use to figure some things out. at home i can do that with a whiteboard and some markers, but there is an added benefit in trying to express coherent thoughts in correct grammar and without too many integral signs

See you in the mirror.

Le Comte de Saint Germain