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 Jul 29 accepted Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ Jul 28 comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ First, thanks. I don't see immediately why $f(M_g) = M_{f \circ g}$? Also, I don't know this version of spectral theorem. Where can I find it? (I know variant: for self-adjoint $A$ there exist spectral measure $E$ such that $A=\int_{\sigma(A)} \lambda \, dE(\lambda)$) Jul 26 comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ @PhoemueX Can you show me solution? Because I really don't know how to attack problem. Only I can see that from $\lambda_1(A) \geq \lambda_2(A) \geq \ldots \geq 0$ (if we arrange in that way) we get $f(\lambda_1(A)) \geq f(\lambda_2(A)) \geq \ldots \geq 0$ and I think that is very important. Jul 25 comment Evaluate the Integral: $\int(x^5+5^x)\ dx$ You can split in sum of two integrals. Jul 25 comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ @PhoemueX I am mainly interested in eigenvalues (that is singular values (but that is the same here because $A$ is positive)). I would like (if that is possible) to see also why $\sigma(f(A)) \neq f(\sigma(A))$. Jul 25 revised Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ edited title Jul 25 comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ @DanielFischer Yes, you are right, I didn't thinking enough. And yes, we assume that $A$ is continuous linear operator. Jul 25 asked Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$ Jul 22 comment Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$ Hm, just for note: derivative of $x^2+2x$ is $2(1+x)$ and derivative for $\cos^2 x$ is $-\sin 2x$. Jul 22 comment Serge Lang Never Explains Anything Round II @D_S I don't know this things. But if you changed $K'$ with $M$, then, do you need to change it everywhere? (you have in text $K'/k'$....) Jun 16 comment Spectral Measures: Uniqueness @Freeze_S Thanks, but no, I didn't deserve it. Jun 16 comment Spectral Measures: Uniqueness @Freeze_S I don't need really reference. I was just wondering. For example, Birman and Solomyak have some nice books about it. I know also that in Rudin (Function Analysis) we can find also some stuff, but not too much. Jun 16 comment Spectral Measures: Uniqueness Hi, just wondering, from which book are you learning stuff about spectral measure? Jun 4 comment Estimate for the power of a integral Look at en.wikipedia.org/wiki/… May 27 reviewed Approve What is corresponding Lie group for Lie algebra of vector fields in dynamical systems? May 27 reviewed Approve How to find the area of the triangle formed by the lines $y=ax$ , $x+y-a=0$ and the $y$ axis? May 27 comment $\frac{1}{{1 + {\left\| A \right\|} }} \le {\left\| {{{(I - A)}^{ - 1}}} \right\|}$ Hi, is $||| \cdot |||$ unitarily invariant norm or? May 22 revised How to evaluate this indefinite integration $\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$? added 18 characters in body May 22 comment use parseval's identity to evaluate the integral $\int_{-\pi}^{\pi}\sin^4 xdx$ Call me crazy, but I would wrote that in reverse order, that is $\pi \cdot \frac{1}{4} + ...$, because $a_0=\frac{1}{2}$. I mean, your answer is correct, but... That just me. And oh, yes, answer is $3\pi/4$, or? May 22 comment Feynman technique of integration for $\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$ Just for note: There is similar, but harder problem, when you have $dy$ instead of $dx$. If you want to solve that problem (where $x$ is parameter, say $x>0$ (but you can do all for $x \in \mathbb{R}$)), first find $I'(x)$ and then use substitution $y \mapsto \frac{x}{t}$ for $I(x)$, where $I(x)$ is your integral. You should end with something like $I'(x)+2I(x)=0$, that is $I(x)=Ce^{-2x}$ (you can find that $C=\frac{\sqrt{\pi}}{2}$ from $I(0)=\lim_{x \to 0^+} I(x)$).