Continuum hypothesis for layman

We have at least 2 different kinds of infinities:

1. Integers and all the other countable numbers - That is things you can pair with integers.

For instance pairs: {1, 1}, {1,2} ...

These can be easily paired integers along diagonals:


                            {1   1}     {1 2}    {1 3}

                            {2   1}     {2  2}
                            
                             {3   1}

We can easily pair like so:
1 -> {1 1}
2 -> {1  2}
3 -> {2  1}
4 -> {1  3}
5 -> {2  2}
...

So pairs  [fractions being one of them with equivalence class notion on top] are no 'bigger' than integers.

2. On the other hand real numbers are really bigger than integers!

Courtesy:Wikipedia

There is a nice way to prove this due to George Cantor.

In fact we need to only consider real numbers in {0, 1}

if it is countable, We can write this as:

1-> 0.0000...0
2-> 0.0000...1
   0.0000..2
   ..
   0.9999..9
Now if we go along the diagonal and change 0 to 1 and everything else [1, 2,...9] to 0 we get a number which differs from all these numbers in atleast one position and it is not mappable to an integer!

So real numbers are indeed bigger, in fact way way bigger! In fact they are not even real despite their name!

Now the continuum hypothesis asks whether there is an infinity strictly between integers and reals or reals are the next big infinity. Of course hypothesis claims it is true.

No one has been able to prove or disprove this in more than a century!

How do we know neutrinos exist

I was wondering today why should we postulate neutrinos and make life more complicated than it already is.

This article explains why neutrinos are required and in addition as a bonus explains how they are detected. Apparently it is must for accounting for conservation of energy in beta decay.

Story of the Neutrino

Income inequality of US or hail capitalism US style!

This is from wikipedia and made me fall out of my chair!

1. Top 1% owns 35% of wealth.
2. Top 5% owns 60% of wealth or 3 out of every 5 dollars.
3. Bottom 40% owns  just 0.2% of wealth.
4. Bottom 60% owns just 4% of wealth or 96% is owned by top 40%

Really hail capitalism US style - 60% is poor because they don't try and Big banks/wall street firms will get saved by government if they fail [of course with fat bonuses for their executives].

Is there an opposite of absolute zero temperature?

That is do we have a highest possible temperature in the universe?

We know we have a lowest possible temperature  [−273.15° Celsius] when all particles of the system have no kinetic energy at all.

I was wondering is there an opposite of absolute zero and looked up in the net.

These two articles discuss this. Apparently it goes to the heart of physics!

Current contender is Planck temperature at
                     32
1.4 x 10 
 
PBS - absolute hot

and

http://en.wikipedia.org/wiki/Absolute_hot

Why a digital computer is just a DFA?

Pedantically speaking of course!

A digital computer is just a Deterministic finite automaton, which of course is equivalent to the complexity [or power] of Regular expressions.

How?

A digital computer just has 2^[memory size] states it can go to. So it is equivalent to a DFA having 2^[memory size] states. So in that sense it is no more powerful than a DFA.

But given that these 2^memory size states are huge, it does approximate a universal Turing Machine!

But to reiterate pedantically speaking it is just a DFA with huge number of states!

Layman definition of P, NP and NP-complete

Here is my layman intuition for these important complexity classes:

1. P

All problems which are solvable in polynomial time.
Sorting for example.

2. NP

All problems which can be verified in polynomial time. That is given an instance of a problem and its 'solution' we can tell in polynomial time whether it is indeed a solution or not.
Integer factorization for example.

3. NP-hard

All problems which are harder than NP or at least as hard as NP. Any thing whose verification is exponential will trivially belong to this. 
For example factorial of n.  Or producing all permutations of n.
[Technical nit - problems have to be decision problems for them to be in P, NP etc. but we will ignore this for now]

4. NP-complete 

 

It is in NP - hard  as in 3) above and this can be reduced to a problem in NP. So in a sense NP-completeness brings down the complexity a notch from NP-hard to NP.
For example, subset sum problem. In general many optimization problems end up here.

 

In layman terms:

P <= NP

P+NP-Complete = NP

 

NP < NP-hard

NP-Complete < NP-hard [this follows from previous inequality]

Largest factored number

I was listening about Quantum computation in Quantum theory - lecture 15.

Suddenly I started wondering what is the biggest number [known publicly] to have been factored. As usual I turned to wikipedia.

This number known as RSA-768 is the hardest known number [having only 2 prime factors also called semiprime].

1 ]=> (* 33478071698956898786044169848212690817704794983713768568912431388982883793878002287614711652531743087737814467999489 36746043666799590428244633799627952632279158164343087642676032283815739666511279233373417143396810270092798736308917)

;Value: 1230186684530117755130494958384962720772853569595334792197322452151726400507263657518745202199786469389956474942774063845925192557326303453731548268507917026122142913461670429214311602221240479274737794080665351419597459856902143413


If we purely go by digit size it is:

 1061

2   - 1 

it is 320 digits!

Great Perimeter institute physics videos

I found very good set of physics course videos in perimeter institute recorded seminar archive.

I am going through these 2 courses and they are very good and highly recommended.

1.  Advanced General Relativity

This course introduces general relativity at advanced level as the name implies. Prof. Sorkin is very smart and good teacher despite his soft voice!

I enjoyed the explanation of Covariant derivative and Lie derivative in particular. It gave me an intuition in to why covariant derivative is inextricably linked to metric and very intuitive explanation of Lie brackets as the "inability of the parallelograms to close".

2. Quantum Theory

It introduces quantum theory again at an advanced level. Prof. Emerson again is very good teacher and has a good voice as a bonus.

I am only half way through the course so far. I enjoyed the foundation discussions in lecture 7 and more importantly the connection between classical and quantum mechanics in lecture 8. Latter gave an intuition about why jumping of state is not so different from classical probabilistic "state update rule" and surprisingly where they differ is after the measurement we DO NOT see the outcome.
1. In Quantum mechanics coherent superposition is destroyed and it definitely is in one of the eigen states [although we didn't take a look at the actual state!]
2. In classical mechanics "coherent superposition" remains in the sense same probability density function continues to apply.

This was very interesting observation and came us a surprise to me! This point alone was the worth effort I spent in learning these videos.

Top 10 innovations of 20th century

10. Digital camera

9. ATM

8. Man made satellites and GPS

7. GUI (Graphical User Interface)

6. Refrigerator

5.  Television

 4. Internet

3. Cell Phone

2. Air planes

1. Computer

Honourable mentions:

X-rays, MRI and Ultra sound scans

Credit cards:

How to know if you are educated

How to know if you are educated - that is well-educated. It simply doesn't mean you have a degree in something.

1. You know earth is round. 

It is so simple and obvious, but I don't think many people feel it in their bones. For example, How would you tell a kid that earth is round?

I was taught the example of approaching ship. When it approaches we would see its  bottom part first instead of mast. As I hadn't seen a ship approaching, I had no way of knowing whether it is true. I  could simply accept either statement as true!

One illustration I can think of is you can fly to USA from India in two different directions say through Tokyo or through London. It will be possible only if earth is around.

Another is when it is day in say India it is night in USA. It can't happen if earth is flat. All the earth will get sunlight at same time.

Another is all the satellites rotate the earth and in particular some as geo-synchronous. This makes sense only if earth is round and rotating in its axis.

2. You know earth orbits sun and not the other way around.

It is very hard to get an intuitive feel of this. Particularly because earth hardly seems to be moving [except when there are earthquakes!]. Here you need a good education to help you.

Many people would think that seasons occur because of earth's orbit. No - it is actually because of earth's tilt of 23 degrees. In fact most of the northern hemisphere is farthest from sun in summer.

One way to see this is [if we assume earth is moving] if we were not rotating we will fly away [or towards] from sun and it should get colder over years. So only option left is to orbit. Of course we could be standing still. Why that is not an option with sun around is our point 3)

3. You know Newton's theory of Gravity.

That every thing attracts every thing else [not just girls attract boys or vice versa]. It is the reason we feel pressure on us when sit or stand. That's the reason it hurts when we fall. In fact it is the reason we fall at all.
If earth were to stand still sun would simply attract it and earth would have collided with sun.
Once we understand gravity it is the easiest of all to grasp as we feel it all the time.

4. You know Darwin's theory of evolution by natural selection.

You understand all species once shared a common ancestor and evolved into different species due to natural selection.
This one is hard to swallow because it hurts our pride. We have to come to the realization that we are only special to the extend every other species are special as far as earth is concerned.
But all that shared DNA code, chimpanzee and kangaroo loudly proclaims its truth. We are not well-educated we don't understand evolution.

5. Common courtesy

If we don't have basic respect for people then all our education hasn't really done anything for us after all. It simply didn't have the intended effect - to make us civilized.

How long does it take to jump from San francisco bridge?

How long does it take to jump from San francisco bridge? That is how long a person would be in free-fall?

Golden gate bridge elevation is 67 meters.

So using high school physics:

 distance = 1/2 g t^2

67 = 1/2 * 9.8 * t^2

This gives in my scheme interpreter for time of 3.7 seconds:

(expt (/ (* 67 2) 9.8) 1/2)

;Value: 3.697765458727081

Speed

What would his speed be at the time of his free-fall experience broken?

 speed = gt

(* 9.8 (expt (/ (* 67 2) 9.8) 1/2))

;Value: 36.2381014955254



That is 36.24 m/s!

or:

(* 3.6 (* 9.8 (expt (/ (* 67 2) 9.8) 1/2)))

;Value: 130.45716538389144

 130 KM per hour or close 81 M/hour - Nasty!

Force

What would be the force?

F = d(MV)/dt

So for a person of 70 KG, assuming the final splash down happens in 1 micro-sec in Newtons:

(/ (* 70 (* 3.6 (* 9.8 (expt (/ (* 67 2) 9.8) 1/2)))) .01)

;Value: 913200.15768724

This is equivalent to 91 TON on top of the person - or close to 45 elephants on Top crushing the bones! Not very nice way to go!

My reasons for evolution being true

My reasons for evolution being true or should I say for evolution being obvious, at least in hind sight?  In particular, why human beings are just another animal species although with some remarkable abilities, in particular in brain.

1.  Chimpanzee

When we see a Chimpanzee It looks remarkably similar to us. If we are special why do these guys look very similar to us? Just to fool us in to thinking that we are their cousins? In fact, to me these guys are the 'missing' link! Only an idiot can miss the link...

2. DNA

Why do we have the same DNA made up of same 4 bases [Adenine, Thymine, Cytosin, Guanine]? Why didn't we get created through some other genetic code. This makes it look very very obvious that we evolved from same animals [more exactly from our common ancestors].

3. Germs and diseases

We get pretty much the same diseases as other animals from very similar viruses and bacteria. For example, we get swine flu - So pigs and humans have to be quite a bit similar for these viruses to survive in both the species. In fact we are very similar - we are both mammals.

4.  Breast feeding

Both woman and other female animals breast [or udder] feed their young through nipples in very similar ways. It is striking in particular for other primates.

5. Kangaroo

Why are no Kangaroos to be found in say India, United States or Germany? If every species are specially created there is no particular reason to place all species of Kangaroos in Australia or near by.  Why not place red kangaroo in Australia and Grey one in say India?

Doesn't the geographical distribution of all animals make it very clear, all closely related animals evolved from their common ancestor which happened to be in one particular geographical location. 

Before the invention of air planes, one particular species had to be in close proximity for reproduction and survival. It is another matter, now we have become planetary wide pests like cockroaches or rats.

create shopping list online

Now you can use http://nabrus.com to create your shopping list for free and access it using your mobile from the market.

Its tagline - "create list and see from anywhere in any phone or computer"