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 Curious
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  • 0 posts edited
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  • 46 votes cast
Mar
15
awarded  Curious
Mar
14
revised Asymptotic expansion of integrals of the form $\int_{\mathcal{D}} \exp(\lambda\, \phi(x))\, g(x) \,dx$ for small $\lambda.$
added 172 characters in body
Mar
14
asked Asymptotic expansion of integrals of the form $\int_{\mathcal{D}} \exp(\lambda\, \phi(x))\, g(x) \,dx$ for small $\lambda.$
Feb
5
asked Is it possible to construct a 1-D linear differential operator with given spectrum $0\leq\lambda_0\leq \lambda_1\leq\dots\leq\lambda_n\le\dots$?
Feb
1
comment How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$?
Thanks. But I don't want an answer as a power series in b. Actually I am trying to see the behaviour of $\displaystyle\sum_{k=1}^{\infty}\displaystyle\int_{0}^{\infty}dx\: \frac{\exp(-a(x+1)^2+2\pi i k x)}{x+1}$ for small values of $a$. Expanding in $b=-2a+2\pi i k$ doesn't seem helpful in this problem.
Feb
1
asked How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$?
Dec
8
comment Proof of a theorem about oscillation
You are right. I see it now. Thanks.
Dec
8
comment Proof of a theorem about oscillation
Can you include this calculation in your answer ? I am not sure if the method of variation of parameters applies to nonlinear equations
Dec
8
comment Proof of a theorem about oscillation
Hmm.. but I can't see how the derivative of second equation is same as the first?
Dec
8
comment Proof of a theorem about oscillation
I think the difference of x and y satisfies $\displaystyle \frac {d (x-y)}{dt}=A(x-y)+R$?
Dec
8
answered Proof of a theorem about oscillation
Dec
8
comment Proof of a theorem about oscillation
Since the first derivatives of $x$ and $y$ are given so we can write a Taylor series expansion of $x(t)$ and $y(t)$. It gives a result of the form $x(t)-y(t)= x_0^2 h(t,x_0)$ for some function $h$. If one can show that $h$ is bounded in a rectangular region $[0,T]\times[-a,a]$ with some upper bound $M$, then the theorem will follow. Since then $|x(t)-y(t)|<\delta \epsilon$ if we take $|x_0|<\delta$ and $\delta<\epsilon/M$
Sep
1
awarded  Scholar
Sep
1
accepted Probability for the sum of two random numbers being a prime number?
Aug
31
comment Probability for the sum of two random numbers being a prime number?
If the choice of numbers is in the range $N_1$ to $N_2$ (where both $N_1$ and $N_2$ are 'large') then the probability will be roughly $N_2/((N_2-N_1)log(N_2))−N_1/((N_2-N_1)log(N_1))$. Right ?
Aug
30
awarded  Commentator
Aug
30
asked Probability for the sum of two random numbers being a prime number?
Jun
28
awarded  Yearling
Jun
28
answered why symplectic form should be closed when we work on a manifold
Jun
28
comment why symplectic form should be closed when we work on a manifold
Now when you are asking for an example I think my statement is not correct;) It should be other way around - every symplectic vector space is also a symplect manifold. Doesn't the condition $d\omega=0$ follow from bilinearity? Since matrix elements of $\omega$ will have no dependence on coordinates.