Disclaimer: This is one of the many ways to prove the equation. For a more detailed proof, please refer the original document by the man himself or the translation provided here.

The underlying assumptions are:
1) the two postulates of special relativity
2) Newtonian formulation of energy due to motion and momentum .

The proof:

Considering a body at rest emits radiations in all directions with equal intensity, thus the body’s velocity remains unchanged. It just looses energy (E, w.r.t a stationary observer) in the form of radiations. An observer witnesses the even in different scenarios:
1) The body emits energy E and then the observer acquires velocity v (instantaneously)
Thus from the observer’s reference frame the energy of the body should be

2) The observer is already moving with velocity v and then the body emits energy E.
From the observer’s reference frame the energy of the body now should be

Both the scenarios should lead to the same calculation of energy for the system implying:

— (1)

Since and are the Energy of radiation seen from different frames, they can be related as per the transverse Doppler effect:

Since the observer is moving with velocity v, time between the crests measured from the frame of observer is different from the one measured from the frame of the stationary body. Thus:


Or

Substituting in (1) gives:

Since the difference in Kinetic energies for the two scenarios is non zero, although the velocity is same, suggests the mass to be different.
Thus,
Suggesting the conversion of mass of the body into energy.

So here my answer to:

Assume we have 1, 2, 3, 4, 5, 6 and “+” and “*” available – what numbers can we create?

First estimating the number of numbers(;)) possible:

1) Numbers consisting of all six digits :

3⁵ * 6!.. as we can either add or multiply or just combine(3 possibilities) .

ex:- 632514, 12+63*45, 1+2*3*4+5+6

2) Numbers consisting of five digits :

⁶C₅ * 3⁴ * 5!

3) Numbers consisting of four digits :

⁶C₄ * 3³ * 4!

4) Numbers consisting of three digits :

⁶C₃ * 3² * 3!

5)

5) Numbers consisting of two digits :

⁶C₂ * 3 * 2!

6) Numbers consisting of one digit :

⁶C₁

adding these gives 244176.

To find all these numbers can be computationally demanding unless we use some optimization. Below is a code in python that finds all possible combination of the six numbers and also numbers with just one addition and multiplication operation.

(The code uses some functions which I have not included here, to keep it legible)
X = [1, 2, 3, 4, 5, 6]

Y = []
Z = []

# 1 digit numbers (pure)
for i in range (0, len(X)):
Y.append(X[i])

# 2 digit (pure)
for i in range (0, len(X)):
for j in range (0, len(X)):
if (i != j):
temp = twodigit(X[i], X[j])
Y.append(temp)

#3 digit (pure)
for i in range (0, len(X)):
for j in range (0, len(X)):
for k in range (0, len(X)):
if ((i != j) and (j != k) and (i!= k)):
temp = threedigit(X[i], X[j], X[k])
Y.append(temp)

#4 digit (pure)
for i in range (0, len(X)):
for j in range (0, len(X)):
for k in range (0, len(X)):
for l in range (0, len(X)):
if ((i != j) and (j != k) and (k!= i)and (l!= i)and (l!= j)and (l!= k)):
temp = fourdigit(X[i], X[j], X[k], X[l])
Y.append(temp)

#5 digit (pure)
for i in range (0, len(X)):
for j in range (0, len(X)):
for k in range (0, len(X)):
for l in range (0, len(X)):
for m in range (0, len(X)):
if ((i != j) and (j != k) and (k!= i)and (l!= i)and (l!= j)and (l!= k)and (m!= i)and (m!= j)and (m!= k)and (m!= l)):
temp = fivedigit(X[i], X[j], X[k], X[l], X[m])
Y.append(temp)
#6 digit (pure)
for i in range (0, len(X)):
for j in range (0, len(X)):
for k in range (0, len(X)):
for l in range (0, len(X)):
for m in range (0, len(X)):
for n in range (0, len(X)):
if (i != j) and (j != k) and (k!= i)and (l!= i)and (l!= j)and (l!= k)and (m!= i)and (m!= j)and (m!= k)and (m!= l)and (n!= i)and (n!= j)and (n!= k)and (n!= l)and (n!= m)):
temp = sixdigit(X[i], X[j], X[k], X[l], X[m], X[n])
Y.append(temp)

# addition and multiplication on two numbers
for i in range(0, len(Y)):
Z.append(Y[i])
for i in range(0, len(Y)):
for j in range(0, len(Y)):
if(check(Y[i], Y[j])):    #the two numbers should not have any common digits
temp1=Y[i]+Y[j]
temp2=Y[i]*Y[j]
if(notinZ(temp1)):
Z.append(temp1)
if(notinZ(temp2)):
Z.append(temp2)
print len(Y)
print len(Z)

This code generates 6338 including the 1956 numbers which are obtained without any operation (just by combination)

I hope you remember the, very old, Young’s experiment . If you do, you may jump to the next para. Quick recall : When a wave passes through two slits, an interference pattern is observed at the detector. When this double slit experiment is performed with a beam of “macro”particles, say bullets, most bullets are found in the middle of the two slits with less being found the further you get from the center. OK, that’s all usual.

It gets interesting when performed with electrons. The double slit experiment is most  phenomenal in explaining the dual nature of electrons. So let’s see what happens: 

Place an electron gun at the originating point. When the electrons pass through the slits and fall on the detector – the result is just like the waves !!

There must be some explanation. After all, an electron couldn’t go through both slits as waves do. Instead of a continuous stream of electrons, let’s turn the electron gun down so that at any one time only one electron is in the experiment. Now the electrons won’t be able to cause trouble since there is no one else to interfere with. The result should now look like that for bullets. But it doesn’t! It would seem that the electrons do go through both slits.

This is indeed a strange occurrence; we should watch them ourselves to make sure that this is indeed what is happening. So, we put a light behind the wall so that we can see a flash from the slit that the electron went through, or a flash from both slits if it went through both. Try the experiment again. As each electron passes through, there is a flash in only one of the two slits. So they do only go through one slit! But something else has happened too: the result now looks like that for bullets.

Obviously the light is causing problems. Perhaps if we turned down the intensity of the light, we would be able to see them without disturbing them. When we try this, we notice first that the flashes we see are the same size. Also, some electrons now get by without being detected. This is because light is not continuous but made up of particles called photons. Turning down the intensity only lowers the number of photons given out by the light source.The particles that flash in one slit or the other behave like the bullets, while those that go undetected behave like waves.

Well, we are not about to be outsmarted by an electron, so instead of lowering the intensity of the light, why don’t we lower the frequency. The lower the frequency the less the electron will be disturbed, so we can finally see what is actually going on. Lower the frequency slightly and try the experiment again. We see the bullet curve. After lowering it for a while, we finally see a curve that looks somewhat like that of the waves! There is one problem, though. Lowering the frequency of light is the same as increasing it’s wavelength, and by the time the frequency of the light is low enough to detect the wave pattern the wavelength is longer then the distance between the slits so we can no longer see which slit the electron went through.

So have the electrons outsmarted us? Perhaps, but they have also taught us one of the most fundamental lessons – an observation is only valid in the context of the experiment in which it was performed. If you want to say that something behaves a certain way or even exists, you must give the context of this behavior or existence since in another context it may behave differently or not exist at all.

Here are the steps to configure the signal analyzer.

1)

Pretty obvious –  start Quartus

2)

Create New Project – Open the new project wizard.

3)

Add the file containing VHDL code to you project (the new project wizard will lead you here)

4)

Setup the Family of FPGA and device you are using (the new project wizard will lead you here)

5)

After the project has been created, compile it and fix the errors, if any.

6)

Open the signal analyzer tool. Tools > Signal analyzer

7)

Double click on add nodes and choose the nodes except the clock.

8)

Add clock under the Signal Configuration> clock and setup the hardware under JTAG chain configuration.

9)

choose the sof file and Start Compilation

10)

If all is fine you would see the signals.

I hope it won’t be difficult. The tool is very user friendly and self descriptive at most stages.

Happy Tapping !!

Yes i am done with my courses, so I produce this list of the courses I liked most during my Masters at (ISY) Electrical dept, LiU.

#5  Digital Integrated Circuits

Good introductory course for Electronics students, specially for transistor lovers like me. I had been in love with transistors for about 4 years before i took this course, but still it had a lot to offer. Biasing, currents through transistors, optimizations,  in short a lot about good circuit designing at transistor level.

#4  Design of Digital Systems

A project course that takes you from an introduction to VHDL and FPGAs to advance level.  I worked in a group of 4 to  implement a music system on Altera’s DE2_70 FPGA board.

#3  Mixed Signal Processing System

Another project course that gives a good overview of Mixed Signal Processing but the best part is the insight it gives into the use of scripting for Simulations and Co-simulations and makes you think of behind the GUIs & of course a healthy relationship with Prof. Wikner’s mixedsignal blog.

#2  VLSI chip design project

And yet another project course, this one is a complete circuit designing course from High level design to full custom layout in 0.35 µm CMOS process. The best part is that the chip is sent for fabrication and then you get to test your first chip .. I worked in a group of 5 to produce a 6-bit High speed DAC (current steering) & know what .. the chip worked

#1  Analog & Discrete Time Integrated Circuits

Arguable, but my love for analog circuit designing drives it to top this list.  It’s a deeper insight into  circuit designing and a good platform for #3. Amplifiers, mixers,filters, data converters, quality matrices, pros and cons of different architectures and/in different situations, etc.

The number of ones in the code adds up to the number counted.  Its like reading of a thermometer. If  you turned the code 90 degrees with the LSB at the bottom, then, as the number increased or decreased, the place where the 1s stop would rise and fall just like the mercury in an old thermometer.(imagine )

Thermometer coding is used in some high speed ADC, DACs. I have used it in my VLSI chip design project – 6 Bit High Speed DAC. In most current-steering DACs, switches are used to route current into resistors or some sort of current sensor. The switches are driven by the digital value which is to be converted to analog. However, if, when changing digital values, some switches open and others close, you will naturally have a mismatch between the “make” time of the closing switches and the “break” time of the opening switches. That means  you might have a combination of switches that are transitional. This ends up creating a glitch at the output as the values resolve and stabilize.

Thermometer code avoids this problem. When the value changes, the changing switches are either all opening or all closing. There’s no mix of opening and closing switches, and so there’s no glitching. But there are no free lunches!! The tradeoff is in the number of bits required. It’s completely linear   there’s no compression whatsoever..

Example :-         3:7 Thermometer coding

Decimal                         Binary                                                 Themometer

0                                      000                                                      0000000

1                                       001                                                      0000001

2                                       010                                                      0000011

3                                       011                                                       0000111

4                                       100                                                      0001111

5                                       101                                                       0011111

6                                       110                                                       0111111

7                                      111                                                       1111111

1. Boring part

2. Interesting Part

1. The Boring but important Part…..

GOLDEN RATIO !!!!

Two numbers are said to be in golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

OR        a & b are in golden ratio if  :-

The Golden Ratio is also sometimes called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion.

A Golden rectangle is a rectangle made of sides a & b where a, b are in golden ratio.

Golden ratio appears in several forms which we (may) have not observed yet.

2. The Interesting Part

As i have not mentioned that the number of Fibonacci series are in golden ratio, well not in the beginning but, as we go further in the series.

series                                             :   1     1     2     3       5       8        13         21           34

ratio between adjacent terms :      1     2   1.5  1.66  1.6   1.625   1.615     1.619

so you see we approach golden ratio as we go further in the series.

What i want to say is that for any series made of

S_n+1 = S_n-1 + S_n  Ratio of 2 adjacent terms reaches golden as we go further in the series.

this also gives you the value of golden ratio.

divide both sides by S_n

Then you get

Φ² = Φ + 1

solve the quadratic for Φ and you get

the golden ratio.

What is this SMART DUSTTTTT???

OK so here’s the idea. Very small sized (comparable to dust particles) motes, having sensing, computation and communication and power generation capabilities, would be spread all over. These motes, without disturbing any usual activities, would monitor the world.

How??

The sensor mounted on the motes would sense the desired activities which will be processed by motes and sent to the central monitoring station. The information sent by motes can be helpful in preventing events like :
fires, in forest or buildings, any kind of leakage in critical areas of a nuclear power plant or other chemical industries, etc. the idea has not yet been fully realized, due to several constraints and complexities involved in reducing the size and improving the performance. The transistors have to be shrunk, i have heard of intel using 2nm process, not sure though. Power is also a constraint which also limits the range of the motes.
Work is going on. Let’s hope for the best!!

Thanks to prof. Jacob Wikner for letting me in, in his Smart Dust research.

01*01 = 001 ; 100 = 10*10

02*02 = 004 ; 400 = 20*20

03*03 = 009 ; 900 = 30*30

—————————–

10*10 = 100 ; 001 = 01*01

11*11 = 121 ; 121 = 11*11

12*12 = 144 ; 441 = 21*21

13*13 = 169 ; 961 = 31*31

—————————-

100 * 100 = 10000 ; 00001 = 001*001

101*101 = 10201 ; 10201 = 101*101

102*102 = 10404 ; 40401 = 201*201

103*103 = 10609 ; 90601 = 301*301

——————————

110*110 = 12100 ; 011*011 = 00121

111*111 = 12321 ; 12321 = 111*111

112*112 = 12544 ; 44521 = 211*211

113*113 = 12769 ; 96721 = 311*311

———————————-

so there’s a pattern

similarly for 1000 -1003, 1010-1013, 1100-1103,—————–

goes on and on and on…..

this was one : there s another kinda pattern too

1*X = X ; X = X*1                                for all X

10*X = X0 ; 0X = X*01                     for all X

11*AB.. = XYZ.. ; ..ZYX = ..BA*11

for AB.. being element of 0-18 or 100-118 or 1000-1018 and so on

12*AB.. = XYZ.. ; ..ZYX = ..BA*21

for AB.. being element of 0-4 or 10-14 0r100-104 or 110-114 and so on..

Acknowledgement : The idea popped from Prof J. Jacob Wikner’s post http://mixedsignal.wordpress.com/?s=palindrome