User uday - MathOverflow

Pseudo-differential operators which are independent of lower order perturbations

2012-05-09T20:03:00Z

In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real principal type operator with principal symbol $q(x,\xi),$ then for a lower order perturbation $R$ we can always find a zeroth order elliptic operator $E$ such that $E^{-1}(P+R)E=q(x,D).$

The reason for this independence of lower terms in elliptic type operator is because the principal term can be inverted and in real principal type operator is there exists a non-degenerate Hamiltonian flow.

Are there any other class of pseudo-differential operators which have this property? If not, is there any work in microlocal analysis which tells us that elliptic and real principal type operators form the largest such class?

Note: As stated in the question, my interest is in 'reductions of operators', i.e., transforming to simpler forms. I am not interested in regularity or solvability problems of pseudo-differential operators. I am aware of examples(not stated in the question) of operators for which the solvability issues are independent of lower order terms. For instance, http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf

functions satisfying "one-one iff onto"

2012-02-17T19:00:00Z

Hello Everybody.

I need some more examples for the following really interesting phenomenon:

   A function from the class ... is one-one iff it is onto.

Some examples I know:

1) Finite set case: functions from $\lbrace 1,2,\dots,n\rbrace$ to itself is one-one iff onto.

2) Linear operators $T\colon V\rightarrow V,$ where $V$ is a finite-dimensional vector space is also one-one iff onto.

3) Linear operators of the from (I-K) where K is some compact operator acting on a Banach space satisfies this property. This is the famous result of Fredholm.

It is very easy to find domains where the result fails.

I remember my teacher telling me that 'compactness is the next best thing to finiteness', hence this result which trivially holds in the finite case can happen only in the compact setting. I would like to know, if this is really the case or are there any other examples?

Thank you in advance.

EDIT: Looking at some answers, I thought it is better if the scope of the question is broadened.

Does injection (surjection) imply surjection (injection) and isomorphism/isometry? (i.e. by assuming one-one can I get ontoness and structure preserving properties free)

Optimization version of the Sylvester equation

2012-12-26T15:40:29Z

The Sylvester equation is a matrix equation of the form $AX-XB=C,$ where $A,B,C$ are given matrices of dimension $m\times m,n\times n$ and $m\times n$ and $X$ is an unknown matrix of dimension $m\times n.$ It is a well known fact that the equation has an unique solution if and only if the matrices $A$ and $B$ have disjoint spectrum. If they do not have disjoint spectrum, then the result in general depends on $C.$

While determining perturbation of eigenvalues in certain context I was naturally drawn to the the problem of determining the minimum, $min_{X}||AX-XB-C||,$ where $||.||$ is the Frobenius norm. Clearly, if the spectrum of $A$ and $B$ is disjoint then there is a choice of $X$ for which the norm is zero. Otherwise, we need to resort to certain optimization techniques. One approach could be to vectorize the matrices using Kronecker products and determine the minimum of a linear system.

The problem is: "What is the choice of $X$ for which the norm $||AX-XB-C||$ attains minimum (if it exists) when the spectra of $A$ and $B$ are not disjoint?"

I have not found any literature on discussion about similar problems. I would be very thankful for any references or suggestions in this direction.

Grothendieck on Topological Vector Spaces

2012-03-10T18:28:39Z

In the short biography article on the Alexander Grothedieck

http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf ,

it is mentioned that after Grothendieck submitting his first thesis on Topological Vector Spaces(TVS), apparently, told Bernard Malgrange that 'There is nothing more to do, the subject is dead.'

Also, after nearly two decades, while listing 12 topics of his interest Grothendieck gives least priority to Topological Tensor Products and Nuclear Spaces.

Now, the questions I have are:
What made Grothendieck make this pronouncement on TVS? Could somebody indicate some significant problems or contributions in this area after Grothendieck? My interest is not in the applications or the impact on the subject on other areas of mathematics but I am interested in knowing about the growth of TVS theory itself.

Thank you, in advance, for your answer.

Triangularizing a function matrix with smooth eigenvlaues

2012-12-12T06:45:30Z

Given a matrix with function entries, which are smooth and homogeneous, and having smooth eigenvalues, can we find a conjugating matrix with smooth and homogeneous entries that triangularize the given matrix? For instance, given $A(x)$ is an $N\times N$ matrix with entries $a_{ij}(x)$ that are smooth and homogeneous in $x$ of order $1.$ Also, given that the eigenvalues of $A(x)$ are smooth. Find an invertible(may be in small neighborhood) matrix $E(x)$ with smooth entries such that $E^{-1}(x)A(x)E(x)$ is upper-triangular.

Sometime back I had asked a question on triangularizing a function matrix. Now, it is clear to me that it is possible to find, by Schur decomposition, a triangularizing matrices which are measurable. Also, one of the answers posted for that question was, it is not always possible to uniformly triangularize especially for certain matrices with non-differentiable eigenvalues. The question in this post is directed towards smoothness and homogenity of such matrices under the condition that they have smooth eigenvalues.

I would be grateful for any reference or insight in this direction. Thank you.

a question about first-order hyperbolic equations

2012-11-09T16:37:44Z

Performing certain manipulations on pseudo-differential equations I have come across the following first order equation: $$ D_{t}u-\lambda(t,x,D_{t},D_{x})u=0, \ \ (*) $$ where $\lambda$ is a scalar pseudo-differential operator with the principal symbol being real-valued and independent of $D_{t}.$ But, the lower-order terms of $\lambda$ depend on $D_{t}$ (or $\tau$ at the symbol level).

I was expecting a hyperbolic equation. But, I find that standard text books(like M.Taylor's 'Pseudo-differential operators') treat only equations in which $\lambda$ term is independent of $D_{t}$ (or $\tau$ at the symbol level).

For the equation $(*)$ to be hyperbolic, is it necessary that $\lambda$ to be independent of $D_{t}$? Are there any references which discuss these issues?

second order operator with real coefficients and not locally solvable

2012-05-18T10:39:23Z

This question is a follow up of the following answer I posted recently:

http://mathoverflow.net/questions/68680/counterexamples-in-pde/97250#97250

Is there a second order partial differential operator with real coefficients which are not solvable in Lewy sense? I am also not aware of first and third order operators but I thought second order operator is a natural question as we may be able to apply some conjugations of known first order non-solvable Lewy like operators with complex coefficients. In this direction, Mizohata's example $\frac{\partial }{\partial x}+ix\frac{\partial }{\partial y}$ was not helpful as it is giving me Grushin like operators whose solvability is proved.

Answer by Uday for second order operator with real coefficients and not locally solvable

2012-09-27T18:46:26Z

The following example is an second order differential operator with real coefficients defined on $\mathbb{R}^{3}$ which is not solvable about origin:

$Pu=(x_{2}^{2}-x_{3}^{2})D_{1}^{2}u+(1+x_{1}^{2})(D_{2}^{2}u-D_{3}^{2}u)-$ $$x_{1}x_{2}D_{1}D_{2}u-D_{1}D_{2}(x_{1}x_{2}u)+x_{1}x_{3}D_{1}D_{3}u+D_{1}D_{3}(x_{1}x_{3}u)$$

This beautiful operator was given by H$\ddot{o}$rmander in his Springer 1963 'Linear Partial Differential Operators' book.

What is the 'non-intuitive' part in sphere eversion (turning inside out)?

2012-08-29T12:25:38Z

Hello, Everybody!

The question does not mean sphere eversion is intuitive to me! In fact, it is just the opposite and that is the purpose of this question.

Recently, I was reading about Smale's paradox, the problem of sphere eversion (turning a sphere inside out). The wiki article is quite clear and gave me a good overview of the topic. I happened to see an animation of the eversion process as well.

The problem of sphere eversion is to construct a homotopy between the inside and outside of a sphere in a three dimensional space. During the continuous deformation self-intersections of the sphere are allowed and creating creases is not allowed.

Given that we can self-intersect the sphere while the process of eversion what could be a possible obstruction to the eversion? What exactly do we mean by self-intersection? Moreover, I find it difficult to imagine why a similar process cannot be employed in the circle case? Why can't we self intersect a circle with itself to turn it inside out? Is there an easy explanation for this phenomenon?

This topic is new to me. I hope the question is not too naive. Thank you in advance.

Why take 'complex powers' of pseudo-differential operators?

2012-08-19T05:21:52Z

Given a pseudo-differential operator $P$ of order zero, Seeley showed that the holomorphic family of operators $\lbrace P^{z} : z\in \mathbb{C} \rbrace$ of all complex powers is contained in the class of pseudo-differential operators.

Apart from knowing that we can take powers of these operators, is there any application of this theory? I understand the utility of raising operators to fractional powers. But, irrational and complex powers are not clear.

Answer by Uday for Why is symplectic geometry so important in modern PDE ?

2012-05-27T16:57:41Z

Fourier Integral Operator is an operator which has its Schwartz kernel as a distribution whose singularities are on a Lagrangian submanifold. In fact, we can associate a FIO with an amplitude and a Lagrangian submanifold in a unique manner. Lagrangian submanifold is a topic of symplectic geometry.

Answer by Uday for Motivation for and history of pseudo-differential operators

2012-05-21T18:52:48Z

A slightly different motivation for fourier integral operators and pseudo-differential operators is given in the first chapter of this book. In this chapter, V.Guillemin presents this subject from the conormal bundles point of view and then shows how pseudo-differential operators are a special case of operators coming out of the Fourier distributions.

Section 3 of this chapter also provides a historic outline of this theory.

Interesting, Guillemin also shows here that pseudo-differential operators can be used to fine tune the theory of conormal distributions themselves -- giving a natural but unexpected application of pseudo-differential operators.

Answer by Uday for Counterexamples in PDE

2012-05-17T18:49:45Z

Lewy's example and most other allied examples of non-solvable operators have only complex coefficients. Leaving open the question of unsolvable operators with real coefficient. F.Treves in 1962 has constructed a fourth order operator iterating Lewy operator.

If $P$ is Lewy operator then $PP\bar{P}\bar{P}$ is a real coefficient fourth order operator which is not local solvable.

Triangularizing a matrix with function entries

2012-05-14T14:22:34Z

Hi Everybody!

Given a matrix, with smooth functions as arguments is there any result which say about its triangularization?

I know that, the question is in affirmative for diagonalizing a matrix which has distinct eigenvalues. Infact, we can show by Implicit function theorem that eigenvalues can be chosen to be $\mathcal{C}^{1}$functions. We can also calculate projection operators for the matrix using Cauchy integral formula.

But, I have not found any result for triangularization. An answer or any references would be great help. Thank you.

ADDED LATER: I have found in the Micheal Taylor's book 'Pseudo-differential operators' Page.72 a claim that for a matrix $K(\xi)$ there is always a measurable unitary matrix $U(\xi)$ such that $U^{-1}(\xi)K(\xi)U(\xi)$ is an upper triangular matrix. In this case $K(\xi)$ is smooth matrix in $\xi.$ He does not say exactly the name of this result though.

Answer by Uday for Combinatorial interpretation of the power of a series

2012-05-13T10:30:06Z

Nice question! You are looking for a combinatorial interpretation of nth-fold self-convolution of a sequence. A quick google serach gave me this result on Catalan sequence. In this paper, a similar result is worked out on a series with Catalan coefficients. May be you can use similar idea for generic sequences.

Game on undirected graphs

2012-04-24T16:38:23Z

One of my friends suggested the following 2-player game.

   Given an undirected graph(not necessarily connected), each player takes turns 
   and removes either one vertex or two adjacent vertices. Removing a vertex from 
   the graph consists of deleting the incident edges as well. The player who does
   not have any move loses.

Though the rules of the game look very simple, this turns out to be an interesting counting and connectivity game. Infact, using symmetry argument, we have also found that there is a winning strategy for the second player when the undirected graph is merely an even length cycle.

The following are my questions:

Does any graph theory concept suggest this game? Is there a standard game of this sort? Can we come up with a winning strategy for any player for the general graph? My guess is this game may be a version of game of Nim over graphs, I am not sure though.

Thank you.

Answer by Uday for Excellent uses of induction and recursion

2012-03-30T17:01:48Z

1)Proof of Euler's formula, V-E+F=2, with induction on F (number of faces).

2)Backward induction proof of generalized AM-GM inequality.

3)Proof of Heine-Borel theorem using ~~Transfinite~~ Topological induction.

Self-dual normed spaces which are not Hilbert spaces

2012-03-15T19:33:05Z

Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-dual it has to be Hilbert space.

Since, we want isomorphism in the norm sense, examples like $\mathbb{R}^{n}$ are ruled out. The norms of the space and its dual have to be equal and not just equivalent.

Thank you.

Answer by Uday for What is the Implicit Function Theorem good for?

2012-02-15T22:41:53Z

One of the major applications of Implicit Function Theorem is the lesson it teaches:

                      Locally, Manifold Theory = Linear Algebra.

That is, locally, we can perform our calculus as if it is linear algebra. Solving simultaneous equations, discussing about linear independence of coordinates, basis set and mapping from one manifold to another can be viewed as linear transformations. Discuss the invertability of functions as if they are linear transformations. Infact by Darboux theorem, in Symplectic manifold theory the linear algebra aspects is more prominent.

Easy and Hard problems in Mathematics

2012-02-26T05:42:47Z

Modified question:

I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context turns them into monstrous-unimaginably difficult to solve problems.

By changing the context I mean, by changing one class of objects in the problem to a related class of objects. For example, from directed graph to undirected graph or Zygmund class to Log-lipshitz class. By changing a 'less-than problem' to 'greater-than problem'. From 2-case problem to 3-case problem. There are plenty of such examples in Theoretical Computer Science or Computational Complexity theory. I need some examples in Mathematics. Lot of examples fall in this category but I am looking for only extreme examples like the ones I stated below. Since, this question is asked for pedagogical purpose it would be interesting if there is a story behind the problem.

Examples of problems:

Linear Programming to integer linear programming
2-coloring to 3-coloring
Eulerian graph to Hamiltonian graph
Undirected graph case to directed graph case in Shannon's switching game
2-SAT to 3-SAT

One thought which motivated me to pose this question is: what if Konigsberg problem has been formulated as a vertex problem. Would Leonard Euler get inspried to create graph theory? No doubt, history speaks differently as Konigsberg problem is stated in terms of edges. Not only Euler solved this problem but created a branch of mathematics out of it! And I am not sure what turn of events would have taken place had the problem been posed in terms of vertices.

IMHO, there are look-alike easy problems and hard problems coexisting but it is the easy problems which saved mathematicians day and hard ones which gave them incentive to work harder.

Some pointers for hardness of problem: problems which need sophisticated tools, techniques which diverge from the routine ones, radical thinking or bold ideas to solve the them, like Poincare conjecture. Or, those problems which do not have adequate tools yet to attempt them, like (NP=P?).

I would appreciate any answers in this direction. Thank you in advance.

Example for pairwise triangularizable but not all three.

2012-03-07T08:28:28Z

I am not able give an example for the following problem on simultaneous triangularization. So, I thought I will post it here.

Give an example of three linear transformations $A,B$ and $C,$ such that the pairs $\lbrace A,B\rbrace$, $\lbrace B,C\rbrace$ and $\lbrace A,C\rbrace$ are simultaneously triangularizable, but the triplet $\lbrace A,B,C\rbrace$ is not simultaneously triangularizable.

Thank you.

ADDED LATER: For my need, I am looking for an example of linear transformations acting on vector spaces over $\mathbb{C}.$

Orthogonality in non-inner product spaces

2012-03-03T17:47:13Z

I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\perp y$) if $||x||\leq ||x+\alpha y||,$ for every $\alpha,$ a scalar.

1) What is the rational behind the definition above? I guess, it has got something to do with minimum overlap between $x$ and $y$.

2) Is this unique generalization of the concept of orthogonality from inner product spaces?

Thank you.

Answer by Uday for Readings for an honors liberal art math course

2012-03-02T17:11:35Z

Another good book though starts at an elementary level but covers lot of fundamental topics such as geometry, topology and calculus and could be highly recommended for the honors programs is: What is Mathematics? By Richard Courant and Herbert Robbins.

Answer by Uday for Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level?

2012-02-29T09:19:03Z

I feel both human-ability and technological-assitance should go hand-in-hand. We have to give equal importance to making students use a calculator and also learning how to do it by hand. I also feel we should encourage students to use softwares like Mathematica and Matlab. Otherwise, what advantage does a future mathematician have over old-timers!

With this background, I feel there should be a clear emphasis on the interpretation of the results a student obtains on performing a calculation.

            'The purpose of computing is insight, not numbers.' -Hamming.

For example, we can use the series

$\frac{1}{1+x} = 1-x+x^{2}+\cdots,$ for $|x|<1$ to demonstrate the fact that if $|x|<<1$ (|x| is far far less than one) then $\frac{1}{1+x} \approx 1-x$ and show the results in a calculator.

Say, $(1.001)^{-1}$ can be easily seen without the use of calculator as approximately equal $1-0.001=0.999.$ Division problem can be turned into a simple subtraction problem. After showing algebraic manipulation, we can show the calculator result and ask students to interpret the precision and give a good explanation.

We could also enhance Mathlete competitions and make students learn to calculate mentally faster than a calculator, for which we need calculators!

Answer by Uday for Readings for an honors liberal art math course

2012-02-22T06:54:59Z

The following books by American Mathematical Society are very good.

1) Fixed points, by Yu.A.Shashkin

2) Stories about Maxima and Minima, By V.M.Tikhomirov

3) Intuitive Topology, By V.V.Prasolov

These books translated from Russian are very insightful and are not voluminous.

You can also look at the following combinatorial game theory book:

Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. By Donald Knuth.

Answer by Uday for Readings for an honors liberal art math course

2012-02-19T20:16:32Z

Few books to consider:

1)Godel, Escher, Bach: An Eternal Golden Braid, By Douglas R Hofstadter : This Pulitzer prize winning book will be good for your needs. Has short and to great extent independent chapters and goes into the the meaning of mathematics, art, music and computing. Though the book looks voluminous it gives a possibility for selected reading.

2)What is the name of this book?: The riddle of Dracula and other logical puzzles, By Raymond M Smullyan ("The most original, most profound, and most humorous collection of recreational logic and math problems ever written." — Martin Gardner.)

3)Letter to young mathematician: the art of mentoring, By Ian Stewart : Subjects ranging from the philosophical to the practical--what mathematics is and why it's worth doing, the relationship between logic and proof, the role of beauty in mathematical thinking, the future of mathematics, how to deal with the peculiarities of the mathematical community are all discussed here.

4)An Imaginary Tale: The Story of i [the square root of minus one], By Paul J. Nahin : Discusses imaginary numbers from a historical perspective.

5) (added later)The Creative Process: Reflections on the Invention in the Arts and Sciences : Collection of essays written by some of the best minds like Poincare, Einstein, Mozart on how they got original ideas. Especially Poincare's essay is very insightful.

6) (added later)I Want to Be a Mathematician. Springer-Verlag. By Paul Halmos : It is good to introduce students to Halmos writing. I am surprised nobody mentioned this.

Answer by Uday for 2 short article vs. a long one

2012-02-19T09:43:01Z

It depends! I am not giving any advice but I am only recalling an anti-measure theoretic quote in this context:

              “The whole is greater than the sum of its parts.” - Aristotle

It may help to think in these lines. All the best!

Sobolev embedding proof without Gagliardo–Nirenberg–Sobolev inequality or Morrey's inequality

2012-02-18T07:31:46Z

Hello Everybody,

Is there a proof of Sobolev embedding theorem without using the GNS or Morrey inequalities? If so, can you provide me with some references?

Background: I happened to attened a talk on Computational PDEs and Sobolev spaces. The speaker made a reference to a proof by S. L. Sobolev using polynomials. But, unfortunately he does not remember any references.

Thank you in advance.

Answer by Uday for How to motivate and present epsilon-delta proofs to undergraduates?

2012-02-15T21:34:40Z

Creating a $\epsilon-\delta$ game is really interesting. Thanks Charles Matthews! BTW, a similar strategy has been stated by Prof.Terry Tao in:Thinking and Explaining, http://mathoverflow.net/questions/38882 (version: 2011-10-12)

One other issue that usually undergrads feel elusive about in $\epsilon-\delta$ method is why is it "for every $\epsilon>0$ there exists a corresponding $\delta>0$" and not the other way round. In this context, the following simple analogy may illustrate the point:

Assuming the discrete maths course is offered to CS students, I will consider software development analogy. In software development, there are essentially two parties. One Developer($\delta$ producer) and the other Client/User($\epsilon$ giver).

We can ask the students which of the below models is preferred:

Model 1: Client gives a specification and developer abides by it. That is, client demands for certain feature in her product and developer accordingly makes the product. Analogously, fix $\epsilon>0$ in the range adjust $\delta$ in the domain.

Model 2: Developer gives certain product and client should accept it however pathetic it may be. Analogously, fix $\delta>0$ and expect $\epsilon>0$ to be satisfied.

Model 1 is naturally preferred. And that is our $\epsilon-\delta$ method.

Of course, we can change the setting depending on the target students(engineers/physicists/biologists etc. ).

Answer by Uday for Functions of Pseudodifferential Operators

2012-02-15T19:36:22Z

Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:

As pointed out by Liviu Nicolaescu in the comment above, Taylor's approach seems to have much wider applicability when it comes to functional calculus. In fact in page 295 of Taylor's book it is mentioned that Seeley's results form a special case of the result.

Moreover, these methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.

Comment by Uday

2012-12-28T19:16:08Z

I think G.B.Folland discusses this in his Introductory book on PDE.

Comment by Uday

2012-12-27T16:35:13Z

@Suvrit Thank you for the answer. Your answer settles the fact that there exists a solution. But, my original interest, which I unfortunately did not make it clear in my post, is to get some quantitative information on the bounds of the solution w.r.t the eigenvalues of $A$ and $B$.

Comment by Uday

2012-12-27T15:37:10Z

I have figured out a way to translate the paper. Thank you once again.

Comment by Uday

2012-12-27T05:52:47Z

@Igor Khavkine Yes, my interest is in the case when the matrices have overlapping spectra. I have edited the question to reflect this.

Comment by Uday

2012-12-27T05:41:39Z

Thank you for the reference.

Comment by Uday

2012-12-12T09:33:00Z

@András Bátkai Thank you for the reference. I will try to get this paper. Is there an English translation of this paper?

Comment by Uday

2012-12-12T09:30:22Z

@Denis Serre Yes. Kato's book discusses Jordan form. But, I find that questions about Jordan form and triangular form are a bit different. For example, the matrix $$ \left(\begin{array}{cc} 1&z\\ 0&1 \end{array}\right) $$ is trivially triangulariable with smooth entries but cannot be written in Jordan form at $z=0$.

Comment by Uday

2012-09-03T11:51:44Z

Real Analysis: Modern techniques and their applications by G.B.Folland is a good reference.

Comment by Uday

2012-08-30T07:13:35Z

@Mark Thank you. The inputs are sufficient for further exploration.

Comment by Uday

2012-08-21T05:27:06Z

Thanks, Rafe. So, 'Spectral theory of pseudo-differential operators' is the answer to my question.

Comment by Uday

2012-06-29T15:18:04Z

Are you referring to Holmgren's uniqueness theorem?

Comment by Uday

2012-06-29T15:06:51Z

Does singular support in the title refer to points of discontinuity of the function? And, does $F_{\sigma}$ refer to a set which is countable union of closed sets?

Comment by Uday

2012-05-26T20:51:32Z

I had asked a similar question in: mathoverflow.net/questions/95042/…. Some of the answers therein may help.

Comment by Uday

2012-05-21T03:51:23Z

Nice example. Thanks!

Comment by Uday

2012-05-14T18:05:39Z

I will check this. Thanks a million!