APPENDIX

PROOF OF CHEBYSHEV’S IDENTITY

A.1 First Proof

The following equation for a 2 x 2 matrix is called Chebyshev’s identity

sin sin( 1) sinsin sin ,sin sin sin( 1)sin sin

mA m m B m

A BC D C m D m m

θ θ θθ θθ θ

θ θ

− −⎛ ⎞⎜ ⎟⎛ ⎞

= ⎜ ⎟⎜ ⎟ − −⎜ ⎟⎝ ⎠ ⎜ ⎟⎝ ⎠

θ (A.1)

where θ is defined as

1 1cos ( )2

A Dθ − ⎡ ⎤≡ +⎢ ⎥⎣ ⎦ (A.2)

We also assume that the determinant of the 2 x 2 matrix equals to one

det 1.A B

AD BCC D

= − = (A.3)

To prove the identity, let us denote the ABCD matrix by Ω, and note

(A.4) 2

22

( ).

( )A B A B A BC B A DC D C D C A D D BC

⎛ ⎞+ +⎛ ⎞⎛ ⎞Ω = = ⎜⎜ ⎟⎜ ⎟ + +⎝ ⎠⎝ ⎠ ⎝ ⎠

⎟

Since

1,BC AD= − (A.5)

we obtain

145

2 ( ) 1 ( ),

( ) ( ) 12 cos ,

sin 2 sin .sin

A A D B A DC A D D D A

θθ θθ

+ − +⎛ ⎞Ω = ⎜ ⎟+ +⎝ ⎠

= Ω −Ω −

=

11

−

(A.6)

If we multiply equation A.6 by Ω, we have

23 sin 2 sin ,

sin(2 cos )sin 2 sin ,

sin(2cos sin 2 sin ) sin ,

sin

θ θθ

θ θ θθ

θ θ θ θθ

Ω −ΩΩ =

Ω − −Ω=

Ω − −=

1

1 2

(A.7)

From the trigonometry identity

2cos sin sin sin(2 1) sin(2 1) sin sin 3 ,θ θ θ θ θ θ2 − = + + − − = θ (A.8)

we can modify equation A.7 to

3 sin 3 sin 2 .sinθ θ

θΩ −

Ω =1 (A.9)

From equation A.6 and A.9, we can guess that

sin sin( 1) .sin

m m mθ θθ

Ω − −Ω =

1 (A.10)

This can be easily proved by induction.

We know that equation A.10 works for m = 1, 2 and 3. Suppose that it

works for arbitrary m. We want to know if we multiply equation A.10 by Ω,

whether we will recover the formula in equation A.10. If it does, it proves that

equation A.10 is right. Let us do that in the following equation

146

21 sin sin( 1) ,

sin(2 cos )sin sin( 1) ,

sin(2cos sin sin( 1) ) sin ,

sin(sin( 1) sin( 1) sin( 1) ) sin ,

sinsin( 1) sin .

sin

m m m

m m

m m m

m m m

m m

θ θθ

θ θ θθ

θ θ θ θθ

mθ θ θθ

θ θθ

+ Ω −Ω −Ω =

Ω − −Ω −=

Ω − − −=

Ω + + − − − −=

Ω + −=

1

1

1

1

θ

(A.11)

This just proves the identity.

A.2 Second Proof

Here we use the same notation as that of the above proof. This proof is the

same with Yeh et al [44]. Let V± be the normalized eigenvectors of the ABCD

matrix with eigenvalues ,ie θ± respectively

.iA BV e V

C Dθ±

± ±

⎛ ⎞=⎜ ⎟

⎝ ⎠ (A.12)

It is evident that the two eigenvalues are inverse of each other because the matrix

in A.12 is unimodular. They are given by

1

221 1( ) ( ) 1

2 2ie A D A Dθ± ,⎧ ⎫= + ± + −⎨ ⎬

⎩ ⎭ (A.13)

with the corresponding eigenvectors given by

Vαβ±

±±

⎛ ⎞= ⎜⎝ ⎠

⎟ (A.14)

where

147

122 2

,( )i

B

B e Aθα±

±=⎡ ⎤+ −⎣ ⎦

(A.15)

122 2

,( )

i

i

e A

B e A

θ

θβ

±

±±

−=⎡ ⎤+ −⎣ ⎦

(A.16)

The Chebyshev’s identity (A.1) can be derived by employing the following matrix

equation

1 ,m mA B A B

M M M MC D C D

1− −⎧ ⎫⎛ ⎞ ⎛ ⎞=⎨ ⎬⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎩ ⎭ (A.17)

which says that the mth power of a transformed matrix is equal to the transform of

the mth power of the original matrix. If a matrix M can be found such that

(A.18) 1 0,

0

i

i

A B eM M

C D e

θ

θ−

−

⎛ ⎞⎛ ⎞= ⎜⎜ ⎟

⎝ ⎠ ⎝ ⎠⎟

Then the mth power of the ABCD matrix is immediately given by

1 0.

0

m im

im

A B eM M

C D e

θ

θ−

−

⎛ ⎞⎛ ⎞= ⎜⎜ ⎟

⎝ ⎠ ⎝ ⎠⎟ (A.19)

The matrix M which transforms the ABCD matrix into a diagonal matrix can be

constructed from the eigenvectors of the ABCD matrix. M and its inverse M-1 are

given by

12

1 1 ,( )

Mα αβ βα β α β+ −−

+ −+ − − +

⎛ ⎞= ⎜− ⎝ ⎠

⎟ (A.20)

12

1 .( )

Mβ αβ αα β α β− −

+ ++ − − +

−⎛ ⎞= ⎜−− ⎝ ⎠

⎟ (A.21)

The two columns in equation A.20 are simply the eigenvectors of the ABCD

matrix.

148

The Chebyshev’s identity follows directly from equation A.19 by carrying

out the matrix multiplication

12

01 .( ) 0

m im

im

A B eC D e

θ

θ

α α β αβ β β αα β α β+ − − −

−+ − + ++ − − +

−⎛ ⎞⎛ ⎞ ⎛⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ −−⎝ ⎠ ⎝ ⎠ ⎝⎝ ⎠

⎞⎟⎠

(A.22)

which finally leads to

sin sin( 1) sinsin sin .sin sin sin( 1)sin sin

mA m m B m

A BC D C m D m m

θ θ θθ θθ θ θ

θ θ

− −⎛ ⎞⎜ ⎟⎛ ⎞

= ⎜ ⎟⎜ ⎟ − −⎜ ⎟⎝ ⎠ ⎜ ⎟⎝ ⎠

(A.23)

149

I don't care what is written about me so long as it isn't true.

Dorothy Parker

If an elderly but distinguished scientist says that something is possible, he is almost certainly right;

but if he says that it is impossible, he is very probably wrong.

Arthur C. Clarke

In physics, your solution should convince a reasonable person. In math, you have to convince a person who's trying to make trouble.

Ultimately, in physics, you're hoping to convince Nature. And I've found Nature to be pretty reasonable.

Frank Wilczek

We must not forget that when radium was discovered, no one knew that it would prove useful in hospitals.

The work was one of pure science. And this is a proof that scientific work must not be considered from the point of view of the direct usefulness of it.

It must be done for itself, for the beauty of science, and then there is always the chance

that a scientific discovery may become like the radium a benefit for humanity.

Marie Curie, Lecture at Vassar College

I can live with doubt, and uncertainty, and not knowing. I think it’s much more interesting to live not knowing

than to have answers which might be wrong. I have approximate answers, and possible beliefs,

and different degrees of certainty about different things, but I’m not absolutely sure of anything.

Richard Feynman, No Ordinary Genius

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155

BIOGRAPHICAL NOTE

Work while you have the light. You are responsible for the talent that has been entrusted to you.

Henri-Frédéric Amiel

Yudistira Virgus was born on August 28, 1985, in Palembang city, the

capital of South Sumatra, Indonesia. When he was a child, he used to think that life

was boring. He did not have strong ambitions towards anything. However, his

thought was suddenly changing when he was in senior high school. It began from

his friends’ physics homework. While working on his friends’ physics homework,

he started to love physics more than he had ever done before. By studying physics,

he thought that nature was beautiful and mysterious. He joined Indonesian Physics

Olympiad team and won gold medal in the International Physics Olympiad.

Thereafter, his goal is to become a physicist. He decided to go to Institut Teknologi

Bandung (ITB) to pursue his goal.

Assuming that yahoo is still free email service provider, Yudistira’s email

address will be:

yudistira_virgus@yahoo.com

The following pages contain a formal resume of Yudistira’s professional activities.

156

CURRICULUM VITAE

History of Education

• Institut Teknologi Bandung (ITB), Bandung, West Java, Indonesia B.S.,

Physics, (2004-2008)

• SMU Xaverius 1, Palembang, South Sumatera, Indonesia (2000-2003)

• SMP Xaverius 1, Palembang, South Sumatera, Indonesia (1997-2000)

• SD Xaverius 2, Palembang, South Sumatera, Indonesia (1991-1997)

• TK SION, Palembang, South Sumatera, Indonesia (1989-1991)

Awards

International competition

• Gold medal, The 35th

International Physics Olympiad, Pohang, Korea 2004

• Bronze medal, The 34th

International Physics Olympiad, Taipei, Taiwan

2003

• Gold medal, The 4th

Asian Physics Olympiad, Bangkok, Thailand 2003

National competition

• Silver medal (SMU Xaverius 1 team), the 6th

mathematics and physics

competition, held by Parahyangan University, Bandung, Indonesia 2002

157

• 2nd

place (SMU Xaverius 1 team), national mathematics and physics

competition, held by Sriwijaya University, Palembang, Indonesia 2002

• 1st

place, national Physics Olympiad, held by Ministry of Education in

Denpasar, Bali, Indonesia 2002

• 1st

place, national Physics Olympiad for undergraduate student, held by

Jogjakarta department of education, Jogjakarta, Indonesia 2007

Honors

• Satya Lencana Wirakary

The highest honor in education from the office of President of Indonesia

(Bambang Susilo Yudhoyono) 2006

• Physics undergraduate student of the year

Department of Physics, Institut Teknologi Bandung (2006 & 2007)

• The Dean’s List for academic performance

Faculty of Mathematics and Science, Institut Teknologi Bandung (2004 –

2007)

• An honor from the Indonesia ministry of education for students with

international achievements (2003 & 2004)

• An honor from the President of Indonesia (Megawati Soekarno Putri) for

young Indonesian students with international achievements (2003)

158

Research Experience

• California Institute of Technology, Pasadena, California, USA

Single-molecule Studies of DNA Catalysts and DNA Walkers (2007)

SURF (Summer Undergraduate Research Fellowship) program

Mentor: Prof. Erik Winfree

Co-mentor: Rizal Hariadi and Nadine Dabby

• Independent Undergraduate Research

Magnetic braking on an inclined plane (2005 – 2006)

under supervision of two physics graduate students,

Oki Gunawan (Princeton University) and

Hendra Kwee (College of William and Marry)

• Institut Teknologi Bandung, Bandung, West Java, Indonesia

Photonics crystal (2006 – 2008)

Research advisor: Alexander Iskandar and Tjia May On

Building a small vibration sensor (2006)

Research advisor: Suprijadi Ph.D. and Janto S.Si

• Parahyangan Catholic University, Bandung, West Java, Indonesia

Construction of small Tesla coil for high voltage device (2006)

Research advisor: Janto S.Si

Technical Courses

• Physical Biological Bootcamp, California Institute of Technology, USA,

July 2007

159

• Caltech Workshop on Self-Replicating Chemical Systems, California

Institute of Technology, USA, August 2007

Work and Teaching Experience

• Institut Teknologi Bandung, Bandung, West Java, Indonesia

Teaching assistant for Freshman Physics course (2005)

Teaching assistant for Freshman Physics course (2006)

Teaching assistant for Mechanics course (2007)

Assistant for Modern Physics Lab course (2008)

• The 6th Asian Physics Olympiad, Pekanbaru, Riau, Indonesia

Editor of the theory problem (2005)

• Indonesian Physics Olympiad Team

Leader team for Indonesian Physics Olympiad Team in the 9th

Asian Physics Olympiad in Mongolia (2008)

Official team for Indonesian Physics Olympiad Team in the 37th

International Physics Olympiad in Singapore (2006)

Part-time theory and experimental trainer (2005 – present)

Official team for Indonesian Physics Olympiad Team in the 36th

International Physics Olympiad in Salamanca, Spain (2005)

160