Reputation
18,475
Next privilege 20,000 Rep.
Access 'trusted user' tools
Badges
2 5 31
Newest
 Revival
Impact
~82k people reached

2h
answered Inverse of $I +T^*T$
13h
answered Laplace transform of the wave equation
14h
accepted How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?
14h
comment $\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.
Thank you. I almost forgot about this question, and to mark your answer.
14h
accepted $\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.
1d
comment Laplace transforms to solve heat equation
@copper.hat : Could be. I see how you're thinking now.
1d
comment Laplace transforms to solve heat equation
@copper.hat : The Laplace transform would be in the t variable, not in x.
2d
comment Showing that an operator generates a contraction semigroup
@user3482534 : $\Re(Dx,x) \le \|D\|\|x\|^{2}=\|D\|\Re(x,x) = \Re ((\|D\| I)x,x)$
2d
comment Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue
@user3482534 : If you have a continuously differentiable function $F(t) \ge 0$ on $[0,\infty)$ and $F'(t) \le 0$, then $0 \le F(t) \le F(0)$ for all $t \ge 0$.
2d
comment Showing that an operator generates a contraction semigroup
@user3482534 : The real part of the inner product is bounded by the absolute value of the inner product, which, by Cauchy-Schwarz, is bounded by the products of norms of vectors.
2d
comment What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?
The Fourier series can represent the signals over a period of the Fourier series. After that, the signal being represented will not agree, unless the signal being represented has the same periodicity. You can represent a voice snippet over an interval $[0,T]$ and the Fourier series signal will repeat itself periodically after that.
2d
comment What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?
It's an offset, a center point about which everything vibrates.
2d
comment How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory
@GiuseppeNegro : your expression would be a special case where a function has one singularity at $x=0$. Stone's formula deals with the resolvent of a selfadjoint operator, which may have a continuum of singularities through the spectral measure.
May
25
comment What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?
@pandu : an offset.
May
25
comment Proving identity of a $C_0$ semigroup
@Leibniz1337 : An antiderivative is unique only up to a constant. I chose the constant so that one of the evaluation terms would vanish; I wanted the evaluation term at $s=t$ to vanish. It's a standard trick for deriving a truncated Taylor series. $\frac{d}{ds}(s-t) = \frac{d}{ds}s$.
May
25
comment solve integral equation using the theory of compact operator
@user3382078 : The functions $1,\cos x,\cos 2x,\cdots$ are orthogonal. The only ones appearing on the right are $1,\cos 2x$.
May
24
answered Proving identity of a $C_0$ semigroup
May
24
comment Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue
@user3482534 : $\inf\{ \lambda : \lambda\in\sigma(-\Delta)\}$. The inf is always a member of the spectrum because the spectrum is always closed.
May
23
comment Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications
If you have a sequence $x_{n}=\sum_{k=1}^{m_{n}}a_{nk}e_k$ that converges to $x$, then $\|x-P_{m_{n}}x\| \le \|x-x_{n}\|$ by the previous remark, which means that $\|x-P_{l}x\| \le \|x-x_{n}\|$ for all $l \ge m_{n}$. That's enough to see that $\lim_{l}\|x-P_{l}x\|=0$.
May
23
comment Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue
If $A$ is selfadjoint with minimum spectral element $\lambda_1$, then $(Ax,x) \ge \lambda_(x,x)$. In your case, $-\Delta$ has minimum spectral element $\lambda_1$. So $\lambda_1(f,f) \le (-\Delta f,f)$.