IISER pune Notice for Admission to Ph. D. Program 2012

Notice for Admission to Ph. D. Program

Applications are invited for the Ph. D. program scheduled to commence from August  2012  in the following areas of basic sciences:

1.      Biological/Ecological  Sciences :   Application Form

2.      Chemical Sciences :  Application Form

3.      Mathematical Sciences :  Write to mathphdselection@iiserpune.ac.in

4.      Physical Sciences :  Application Form

ONLINE APPLICATION FORM available UNTIL April 30,  2012

Test/Interview Dates: (New!!)

Biological/Ecological Sciences:

Written Test: June 20, 2012

Interviews: June 21-22, 2012

Chemical Sciences:

Written Test and Interviews: May 24-25, 2012

Physical Sciences:

Written Test and Interviews: May 30-31, 2012

Note: For Biological/Ecological Sciences, candidates should come prepared to stay foratleast 3 days. For Chemical and Physical Sciences, candidates should come prepared to stay for atleast 2 days.

Eligibility

General Criteria:

Nationality: Indian

Age: Not more than 28 years on August 01, 2012.  Age relaxation of 5 years for SC/ST/OBC/PwD/Female applicants

Minimum Educational Qualifications/Selection Procedure

Biological Sciences

       60%  marks (55% for SC/ST)  in:

 ·  M.Sc in any branch of Science – Biological Sciences/Ecology & EnvironmentalSciences/Biodiversity/Biochemical/Biophysical/ Biomathematical or equivalent. Including Physics, Chemistry & Mathematics.

·   B.E/B.Tech/M.Tech or MBBS degree holders may also apply.

·   The candidate MUST qualify National Eligibility Test (NET) with valid CSIR-JRF/UGC-JRFor DBT-JRF-A or ICMR-JRF or ICAR-JRF or CSIR-LS or INSPIRE-PhD or DAE-JEST or NBHM or have a valid GATE qualification.

·   Additional criteria may be used for shortlisting of candidates.

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Chemical Sciences60% marks (55 for SC/ST)  in:

M. Sc. or equivalent degree in Chemistry/Physics/Biochemistry/Material Science/Bioinformatics/Pharmacy

The candidate should have qualified National Eligibility Test (NET) with CSIR-JRF/UGC-JRF, CSIR-LS (only inorganic and physical chemistry), JEST or INSPIRE-Ph.D or GATE.

The following ranks for GATE will be eligible.

1. Organic  chemistry applicants: 1-500 ranks
2. Inorganic and Physical chemistry applicants:  All ranks

Additional criteria may be used for shortlisting of candidates.

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Mathematical Sciences.

60 marks (55%  for SC/ST)  in:

M.Sc/M.Stat/M.Tech/M.A in any field,

or in:

B.Sc/B.Stat/B.Tech/B.A in any field.

The candidates must have appeared for the NBHM Ph.D. screening test for the year 2012. Candidates with Undergraduate degree will be admitted to the Integrated  Ph.D  program. Shortlisted candidates from the NBHM exam will be contacted directly.

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Physical Sciences 60% marks (55 for SC/ST)  in:

(i) M. Sc. or equivalent degree in Physics/Material Science OR

(ii) BE / BTech / MTech

The candidate should have qualified at least one of the following tests/fellowship in Physics valid for the current academic year:

(i)                 CSIR-JRF/UGC-JRF

(ii)              JEST/GATE with ranks less than or equal to 500

(iii)            INSPIRE-Ph.D fellowship.

Additional criteria may be used for shortlisting of candidates.

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Selection Procedure

• Only Eligible candidates that meet the minimum educational qualifications and satisfy the additional criteria set by the respective departments will be called for a written test.  The additional criteria may  include cut-off marks at different levels of examinations, rank in the national examinations etc. No correspondence/inquiries will be entertained in this regard.
• Candidates who are shortlisted by the selection committee will be informed the dates and details of the written test/interview by e-mail.
•  Only candidates who qualify the written test will be interviewed.  The interview may be conducted in 1 or 2 rounds.
• Based on their academic record and the performance of the candidates in the written test and interview, the departmental selection committee will recommend candidates to the Chairman for admission to the Ph. D. program.

E-mail will be the primary means of communication. All applicants should specify their e-mail ID clearly and legibly in their application forms.

How to Apply/ Application Deadline

Application form has to be filled ONLINE before  April 30, 2012. Candidates intending to apply to more than one program/Department should apply separately application forms foreach program/ Department.

• Fill the online application form completely. Incomplete application forms may be rejected.
• Attested copies of all supporting documents (mark sheets, certificates, age proof, caste certificate, etc.) are to be brought in at the time of the interview.

Please carry one photograph. In addition, applicants for Biological/Ecological  Sciences should bring along one recommendation letter at the time of interview The last date for receipt of completed applications ONLINE is April 30, 2012.

Contact e-mail

Please contact Coordinator for each Department at the following address in case you have questions:

Biology (biophdselection@iiserpune.ac.in)

Chemistry (chemphdselection@iiserpune.ac.in)

Mathematics (mathphdselection@iiserpune.ac.in)

Physics (physicsphdselection@iiserpune.ac.in)

Test/Interview Dates (New!!)

Biological/Ecological Sciences:

Written Test: June 20, 2012

Interviews: June 21-22, 2012

Chemical Sciences:

Written Test and Interviews: May 24-25, 2012

Physical Sciences:

Written Test and Interviews: May 30-31, 2012

Note: For Biological/Ecological Sciences, candidates should come prepared to stay for atleast 3 days. For Chemical and Physical Sciences, candidates should come prepared to stay for atleast 2 days.

LAST DATE FOR RECEIPT OF COMPLETED APPLICATION FORMS, FILLED ONLINE is April 30, 2012

Fourier Series

Fourier series is an expansion of a periodic function of period $2\pi$ which is representation of a function in a series of sine or cosine such as

$f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)+\sum_{n=1}^{\infty }b_{n}sin(nx)$
 
   where $a_{0}$ , $a_{n}$ and $b_{n}$ are constants and are known as fourier coefficients.
   In applying fourier theorem for analysis of an complex periodic function , given function must satisfy following condition
  (i) It should be single valued
  (ii) It should be continuous.
 
  Drichlet's Conditions(sufficient but not necessary)

  When a function $f(x)$ is to be expanded in the interval (a,b)
  (a) $f(a)$ is continous in interval (a,b) except for finite number of finite discontinuties.
  (b) $f(x)$ has finite number of maxima and minima in this interval.

Orthogonal property of sine and cosine functions
  $\int_{-\pi}^{\pi}sin(mx)cos(mx)dx=0$
  $\int_{-\pi}^{\pi}sin(mx)sin(nx)dx= \int_{-\pi}^{\pi}sin(mx)sin(nx)dx=\begin{bmatrix}
\pi\delta_{mn} &m\neq 0 \\
 0& m=0
\end{bmatrix} $
 $\int_{-\pi}^{\pi}cos(mx)cos(nx)dx=\begin{bmatrix}
\pi\delta_{mn} &m\neq 0 \\
 2\pi& m=0
\end{bmatrix} $
 

Fourier Constants
 $a_{0}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx$
 $a_{0} is the average value of function $f(x)$ over the interval
 $a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx$
 $b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx$
  

For even functions
  $f(-x)=f(x)$ and fourier series becomes
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)$
 

 For odd functions
  $f(-x)=-f(x)$ and fourier series becomes
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}sin(nx)$
  

Complex form of fourier series
  putting $c_{0}=c_{0}$
  $c_{n}=\frac{a_{n}-ib_{n}}{2}$
  and
  $c_{-n}=\frac{a_{n}+ib_{n}}{2}$
  $f(x)=\sum_{-\infty }^{\infty }C_{n}e^{inx}$
  coefficent
  $C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$
  

Fourier series in interval (0,T)
  General fourier series of a periodic piecewise continous function $f(T)$ having period $T=\frac{2\pi}{\omega}$ is
  $f(t)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(nx)+\sum_{n=1}^{\infty }b_{n}sin(nx)$
  where
  $a_{0}=\frac{1}{T}\int_{0}^{T}f(t)dt$
$a_{n}=\frac{2}{T}\int_{0}^{T}f(t)cos(n\omega T)dt$
$b_{n}=\frac{2}{T}\int_{0}^{T}f(t)sin(n\omega T)dt$
 

 Complex Form of Fourier Series
  $f(x)=\sum_{n=-\infty }^{\infty }C_{n}e^{-i\omega t}$
  where
  $c_{n}=\frac{1}{T}\int_{0}^{T}f(t)e^{-i\omega t}dx$
 

 Advantages of Fourier series
  1. It can also represent discontinous functions
  2.  Even and odd functions are conveniently represented as cosine and sine series.
  3.  Fourier expansion gives no assurance of its validity outside the interval.

  Change of interval from $(-\pi,\pi)$ to $(-l,l)$
  Series will be
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(\frac{nx\pi}{l})+\sum_{n=1}^{\infty }b_{n}sin(\frac{nx\pi}{l})$
  with
  $a_{0}=\frac{1}{2l}\int_{-l}^{l}f(x)dx$
  $a_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)cos(\frac{n\pi x}{l})dx$
  $b_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)sin(\frac{n\pi x}{l})dx$
  

Fourier Series in interval $(0,l)$
  Cosine series when function $f(x)$ is even
  $f(x)=a_{0}+\sum_{n=1}^{\infty }a_{n}cos(\frac{n\pi x}{l})$
  $a_{0}=\frac{1}{l}\int_{0}^{l}f(x)dx$
  $a_{n}=\frac{2}{l}\int_{0}^{l}f(x)cos(\frac{n\pi x}{l})dx$
  Sine series when function $f(x)$ is odd
  $f(x)=\sum_{n=1}^{\infty }a_{n}sin(\frac{n\pi x}{l})$
  $b_{n}=\frac{2}{l}\int_{0}^{l}f(x)sin(\frac{n\pi x}{l})dx$

Complex Analysis Part 2

Liouville's Theorem 
If a function $f(z)$ is analytic for all finite values of z, and is bounded then it is a constant.
Note:- $e^{z+2\pi i} = e^z$

Taylor's Theorem
If a function $f(z)$ is analytic at all points inside a circle C, with its centre at point a and radius R then at each point z inside C
$f(z)=f(a)+(z-a)f'(a)+\frac{1}{2!}(z-a)^2f''(a)+.......+\frac{1}{n!}(z-a)^nf^n(a)$
Taylor's theorem is applicable when function is analytic at all points inside a circle.

 Laurent Series
If $f(z)$ is analytic on $C_{1}$ and $C_{2}$ and in the annular region R bounded by the two concentric circles $C_{1}$ and $C_{2}$ of radii $r_{1}$and $r_{2}$ ($r_{1} > r_{2}$) with their centre at a then for all z inside R
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.........$
where,
$a_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(n+1)}}$
$b_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(-n+1)}}$

Singular points
If a function $f(z)$ is not analytic at point z=a then z=a is known as a singular point or there is a singularity of $f(z)$ at z=a for example
$f(z)=\frac{1}{z-2}$
z=2 is a singularity of $f(z)$

Pole of order m
If $f(z)$ has singularity at z=a then from laurent series expansion
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.....+\frac{b_{m}}{(z-a)^m}+\frac{b_{m+1}}{(z-a)^{m+1}}$
if
$b_{m+1}=b_{m+2}=0$
then
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+..........+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+.....+\frac{b_{m}}{(z-a)^m}$
and we say that function $f(z)$ is having a pole of order m at z=a. If m=1 then point z=a is a simple pole.

Residue
The constant $b_{1}$ , the coefficent of $(z-z_{0})^{-1}$ , in the Laurent series expansion is called the residue of $f(z)$ at singularity $z=z_{0}$
$b_{1}=Res_{z=z_{0}}f(z) = \frac{1}{2\pi i}\int_{C_{1}}f(z)dz$

Methods of finding residues
1.  Residue at a simple pole
if $f(z)$ has a simple pole at z=a then
$Res f(a) =\lim_{z\rightarrow a}(z-a)f(z) $
2. If $f(z)=\frac{\Phi(z)}{\Psi (z)}$
and $\Psi(a)=0$ then
$Res f(a)=\frac{\Phi(z)}{\Psi^{'} (z)}$
3. Residue at pole of order m
If $f(z)$ is a pole of order m at z=a then
$Res f(a)= \frac{1}{(m-1)!}\left \{ \frac{d^{m-1}}{dz^{m-1}}(z-a)^{m}f(z) \right \}_{z=a}$

 Residue Theorem

If $f(z)$ is analytic in closed contour C excapt at finite number of points (poles) within C, then
$\int_{C}f(z)dz = 2\pi i \textsl { [sum of the residues at poles within C]}$

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Complex analysis part 2