Summer Holidays Are Here

Summer holidays are here and its time to do something productive....
Luckily I've found myself some challenges to make myself busy.

Scipher

My first shot at entrepreneurship. Me and my friends have decided to conduct a national level competitive exam 'Scipher'. There are several challenges lying ahead of us.

• None of us have any experience whatsoever. It is a real challenge to coordinate so many people to make something in this scale a success.
• We need an official website. Unfortunately none of us know anything about designing a website.I'm the only one with some programming background. So I have just 1 week to learn HTML, CSS and Javascript and cook up a website.
  Update:Check out w3schools.com. Building a website is simple and easy.
  Update: YAY..We have an official website. Scipher.in
• The exam is supposed to contain only multiple choice questions. We need to come up with a solution to evaluate thousands of answer paper. We plan to use OMR sheets. But commercial OMR readers cost a lot and we are on a tight budget (practically broke). All we have is access to a normal scanner and we need to develop a software to do the magic.

Summer Project

We had an introductory course on electronics and microcontrollers in this semester (Sem 2). Frankly the course seemed boring especially the labs. All we could do in the lab is to check out the circuit diagram, assemble the circuit elements and get the 'desired output'. 

What did we actually learn from this? How to blink LEDs. How to maintain the temperature of a heater using a temperature sensor. It would have better if we had to do it without the circuit diagram or the microcontroller codes handed out to us. So to really learn something I thought I should take up a project. 

I've got an Idea. Lets make a car capable of driving itself.. No not a google car, more like a toy.

Quantum Filter Theory (Quantum Consciousness)

Coming soon...
Due to academic pressure (where I'm forced to learn a bunch of things I hate and which my brain is unable to comprehend, the so called interdisciplinary approach to science education) I'm unable to pursue my interests in theoretical physics. So I have to wait till my semester vacation to stop 'learning' and start thinking.

BTW I don't have any  grudge against interdisciplinary science education, the thing is it doesn't suite me.

Quantum Entanglement, Multiverse And Arrow Of Time

I don't have much mathematical knowledge of these topics.So the below discussion will be purely conceptual.


In the following post we will be discussing the implications of combining Quantum Entanglement , Multiverse Theory and the concept of Arrow of Time to create a deeper understanding of the universe.

Lets first begin with Quantum Entanglement .

What is quantum entanglement?
Lets suppose a spin 0 particle decays into two particles of half integer spins.By conservation of spin we know that if one is spin up the other should be spin down. But as per quantum mechanics a particle is in a superposition of various states until we make an observation.So suppose we observe the first particle and find it to be spin up, then the other particle 'will' measure spin down.
The act of determining the spin of the first particle somehow affects the second particle!!
Some information should travel from the first particle to the second instantly to convey the information since the second particle should 'collapse' to spin down state from the superposed state.But 'nothing can travel faster than light'.

We had nothing to worry if we were dealing with classical mechanics.But quantum mechanics state that the spin of the second particle cannot be predetermined because it is in a state of superposition.

Another example:
This one might not be as apparent as the last.
I think everyone knows about the double slit experiment.
In a double slit experiment whenever we make a measurement to determine which hole the electron passed through the interference pattern disappears.
That is when we detect the electron at one slit, the 'other part' of the electron should know not to go through the second hole.(The electron should know where it is!!).

Its better to consider the first example if you don't understand the second.

Now we try to explain this 'anti-quantum mechanical' phenomenon in terms of quantum mechanics.
The explanation is more or less philosophical.


Intuitively I feel that the simplest law that could govern anything is 'just to exist'.This is somewhat apparent from quantum mechanics.

We make an assumption that our 'world' is only governed by 'Just Exist' law, which is in turn controlled by 'potentials'. This will give us the 'multi-verse' i.e our universe exist in all possible states it can.We are part of one such state.



Now think about the universe as a tree. From the main stem it splits up into different branches.Like-wise at each 'decision point' the universe split into several state with different possible 'decisions' which result in different outcomes. That is each time you break an entanglement, you are actually becoming part of one such states. One state of universe in which you measure spin up and another in which you measure spin down.

Initially the universe can be thought of as super-state which is a superposition of several states, which then splits up into different states which is in turn made up of superposition of several sub-states.
Before you make the measurement you are part of one such super-state.When we make the measurement you are collapsing to one of the sub-states.

This splitting of states can be interpreted as the direction of time. That is direction of splitting of states gives the direction of arrow of time. Also this splitting requires that we need more and more information to describe the state of the universe and by statistical interpretation of entropy, entropy increases.

Odd Perfect Number

A perfect number is a positive integer, the sum of whose factors(except the number itself) add upto the number.

Example: 28 = 14 + 7 + 4 + 2 + 1

There are about 48 known perfect numbers at the time of writing this article.But none of them are odd. Computers have checked upto $10^{300}$ but could never find one. So I tried my luck disproving their existence.I had a little experience proving this (pefect squares have odd number of factors). But rather than disproving their existence I ended up finding constraints on their existence.

Let $x$ be a odd perfect number.

Then $x$ can be denoted as $x = p^a.q^b.r^c ..….$ where $p, q, r.....$ are prime factors.

Theorem 1: One and only one of the prime factors have an odd power i.e. only one of $a, b, c....$ is odd and all others are even.

Proof: 

All the combination of factors of $x$ can be denoted by varying the powers of the prime factors.

The sum of all the factors of $x$ can then be written as

$s = (p^0 + p^1 + p^2 + ....p^a)(q^0 + ..... q^b)(r^0 + ...… r^c).....$

The Three Point Conjecture

Don't get too excited about the title. It's simply a puzzle(The three utilities puzzle).

There are three houses and three factories(water,gas,electricity) in a city block.Each house should be supplied with water,gas and electricity.The supply lines should not overlap each other.Draw the connections.

In short there are 2 sets of three points.Each point in Set-1 should be connected to every point in Set-2 without the lines overlapping.

Seems easy enough.But there's a catch, no matter how you try, you won't be able to make one last connection.You don't want to know how much time I spend on it and was convinced of its impossibility.

Now one last challenge remained - the proof.
The difficult part about the proof was that I've got no idea where to begin or which path to follow.Proving something like this was entirely new. It disturbed me day and night. It's amazing how desperation could make you do things that you thought you couldn't. And this fine morning something strange struck me.

Balancing Chemical Equations [LifeHack]

Its not often that we come across chemical equations with coefficients like 48 and 50.But when we do most of us are meant to screw up.Thats exactly why a friend of mine asked me to come up with something.(Its hard enough just to memorise all those equations.)

I'll show you how to balance complicated chemical equations with huge coefficients easily and quickly.All you need to know is how to solve linear equations in variables.

The method might seem long but the equations that we get are simple and the method is much quicker than manually trying to balance the atoms.

Steps: 

1.Assign coefficients to each term in the equation (a,b,c.....)

2.Now balance each element i.e. solve for the coefficients.

(You can skip some steps by assigning coefficients directly as shown below)

Lets start with..

S₈ + HNO₃ = H₂SO₄ + NO₂ + H₂O

Prime Factorial

While doing some random computing I accidently found out something.

(2!+1)/3=1
(3!+1)/4=Fraction
(4!+1)/5=5
(5!+1)/6=Fraction
(6!+1)/7=103

Only when the denominator is a prime the quotient will be an integer.

Let n be prime
Then ((n-1)!+1)/n will be an integer.

I computed it to really large numbers and seemed to hold.

Don't yet know if its generally true.

Update: This theorem is called Wilson's Thoerem.