## Integrate conjugate transpose

I can’t find the

Integrate[conjugate Transpose[u[x, t]] *D[u[x, t], {x, 1}],x]

In mathematica.
Thanks

## fa.functional analysis – Conjugate of Composition in Bochner Spaces

Let $$H$$ be a separable Hilbert space (of non-zero dimension), let $$(Omega,Sigma,mu)$$ be a finite measure space, and let $$L^2(mu;H)$$ be the Bochner-space $$mu$$-integrable $$H$$-valued functions. Consider proper, lower semi-continuous, and convex functions $$phi:Hrightarrow (-infty,infty)$$ and $$psi:L^2(mu;mathbb{R})rightarrow (-infty,infty)$$ and define the map:
$$L^2(mu;H) ni f(omega) mapsto Phi(f)(omega) in L^2(mu;mathbb{R});$$
and suppose that $$Phi$$ is Frechet differentiable and coercive on all of $$H$$.

Can the convex conjugate $$(psicirc Phi)^{star}$$ be expressed in terms of $$psi$$ and $$Phi$$?

## convex optimization – Conjugate gradient and the eigenvectors corresponding to the large eigenvalues

I am working on an optimization problem (for example, conjugate gradient) to solve $$Ax=b$$, where $$A$$ is a symmetric positive definite matrix. I can understand that the CG (conjugate gradient) has better performance when the matrix $$A$$ has a smaller conditioner number. But I am wondering is there a relationship between the eigenvectors corresponding to the largest few eigenvalues and the first few update directions of the CG? Any suggestions would be helpful. Thanks!

## how to take derivative of complex function including complex conjugate of variable?

How can i take the derivative of f(z) = z + z*, where z is complex, i.e. a+ib?
My question is actually for f(z, z*) in general.

And My complex variables book doesn’t seem to include such an example – why not?

## Can a non-inner automorphism map every subgroup to its conjugate?

Let $$G$$ be a finite group. Can a non-inner automorphism map every subgroup to its conjugate? Namely, can there be a non-inner automorphism $$alpha$$ that, for every $$Hle G$$, there exists some $$g$$ in $$G$$ such that $$alpha(H)=H^g$$?

## dg.differential geometry – Are a map with constant singular values and its inverse always conjugate through isometries?

Let $$U subseteq mathbb R^2$$ be an open, connected, bounded subset. Fix $$0, such that $$sigma_1 sigma_2=1$$.

Suppose that there exist a diffeomorphism $$f:U to U$$ such that the singular values of $$df$$ equal $$sigma_1,sigma_2$$ everywhere on $$U$$.

Question: Do there exist smooth isometries $$phi_1, phi_2:U to U$$ such that $$phi_1 circ f circ phi_2=f^{-1}$$?

We must have $$phi_i in operatorname{O}(2)$$ (isometries are affine), but the point here is that I want them to map $$U$$ into $$U$$.

Note that $$df^{-1}=(df)^{-1}$$ has singular values $$sigma_1,sigma_2$$, the same as $$df$$.

Here are two examples where this phenomena happens:

1. Affine maps on ellipses:

Let $$0, $$ab=1$$, and let
$$U=U_{a,b}=biggl{(x,y) ,biggm | , frac{x^2}{a^2} + frac{y^2}{b^2} < 1 biggr}.$$

Take $$f(x,y)=Apmatrix{x\y}$$, where begin{align*} & A=A(theta)= begin{pmatrix} a& 0 \ 0 & b end{pmatrix} begin{pmatrix} costheta & -sintheta \ sintheta & cos theta end{pmatrix}begin{pmatrix} 1/a& 0 \ 0 & 1/b end{pmatrix}= begin{pmatrix} costheta & -frac ab sintheta \ frac ba sintheta & cos theta end{pmatrix} end{align*}.

Then $$A(theta)^{-1}=A(-theta)=JA(theta) J$$, where $$J=begin{pmatrix} 1& 0 \ 0 & -1 end{pmatrix}$$ is the reflection around the $$y$$ axis.

2. Non-affine maps on the disk:

Let $$U=Dsetminus{0}$$ where $$D subseteq mathbb R^2$$ is the unit disk.

$$f_c: (r,theta)to (r,theta+clog r )$$. Then we have $$f_{c}^{-1}=f_{-c}=Jf_{c}J$$.

Note that
$$(df_c)_{{ frac{partial}{partial r},frac{1}{r}frac{partial}{partial theta}}}=begin{pmatrix} 1 & 0 \ c & 1end{pmatrix},$$
so the singular values of $$f_c$$ are constants which depend on $$c$$.

Now, in general there are many local solutions to the PDE $$sigma_i(df)=sigma_i$$, so I don’t expect such a special relation between $$f$$ and its inverse in general. But I don’t have a counter-example yet.

## Product of the complex conjugate module

I came across something strange while doing the arithmetic and hoped someone could point out what's wrong with my reasoning?

$$mid z mid <1$$

$$w = z.z ^ *$$

$$mid w mid = mid z.z ^ * mid$$ $$=$$ $$mid z mid mid z ^ * mid$$

$$mid z mid$$ $$= mid z ^ * mid$$

$$mid w mid$$ = $$2 mid z mid$$

Now follows:$$frac { mid w mid} {2} <1$$

Therefore: $$mid w mid <2$$

Where was I wrong in the logic above? The answer is

$$mid w mid <1$$and I get it when I say

$$mid z mid <1$$ and $$mid z ^ * mid <1$$, that's why $$mid z.z ^ * mid <1$$therefore $$mid w mid <1$$

Thanks a lot

## linear algebra – the weight of the conjugate partition is greater than the weight of the nullity partition

This question comes from Exercise 4.1 of the lectures on geometric constructions, Kamnitzer-arXiv-Link, and is a consequence of a question that I have asked here that deals with Part 1 of the exercise. This question is part 2.

We get that in this exercise $$X: mathbb {C} ^ N to mathbb {C} ^ N$$ is a not potent matrix with $$X ^ n = 0$$. The partition is connected to it $$mu = ( mu_1, dots, mu_n)$$ With $$mu_i = dim ker (X ^ i) – dim ker (X ^ {i-1}).$$

To $$X$$ We can also map the partition $$nu = ( nu_1, dots, nu_m)$$ where everyone $$nu_i$$ is the size of the $$i$$-th Jordan Block from $$X$$Placing an order to make the 1st Jordan block the largest size, and so on. The young diagram of $$nu$$ has a conjugate partition $$lambda$$where everyone $$lambda_i$$ is the number of $$j$$ so that $$nu_j geq i$$ (i.e. it is the number of Jordan blocks one size larger or equal $$i$$).

The first part shows that for everyone $$k$$, $$mu_1 + dots + mu_k leq lambda_1 + dots + lambda_k.$$ Now I have to show that as $$GL_n$$ Weights, $$lambda geq mu$$.

We have $$lambda – mu = (( lambda_1- mu_1), dots, ( lambda_n- mu_n)),$$ what I want to express as a sum $$k_1 a_1 + dots + k_ {n-1} a_ {n-1},$$ Where $$k_i$$ are not negative integers and $$a_1 = (1, -1.0, points, 0), points, a_ {n-1} = (0, points, 0, 1, -1)$$. In other words, the setup will $$(( lambda_1- mu_1), dots, ( lambda_n- mu_n)) = (k_1, k_2-k_1, dots, k_ {n-1} -k_ {n-2}, -k_ { n-1}).$$

Starting from the left, we get an inductive equation $$k_i = ( lambda_1 + dots + lambda_i) – ( mu_1 + dots + mu_i),$$ but I can't show that $$– k_ {n-1} = lambda_n- mu_n tag {*}.$$ I'm pretty sure my calculations are correct, and the only place we use the part 1 inequality is to show that the constants $$k_i$$ are not negative, but that's all right – the only part I'm sticking to is showing (*).

## Intuition – topologically equivalent to conjugate fields

Two fields $$f: U subseteq mathbb {R} ^ n to mathbb {R} ^ n$$, $$g: V subseteq mathbb {R} ^ n to mathbb {R} ^ n$$ are continuously differentiable in their respective areas topologically conjugated if there is a homeomorphism $$h: U to V$$ so for everyone $$t$$ we have
$$varphi_t = h ^ {- 1} , phi_t , h$$
Where $$varphi$$ and $$phi$$ are the respective streams of $$f$$ and $$g$$. The fields are topologically equivalent if there is a homeomorphism $$h: U to V$$ that takes tracks from $$f$$ in lanes of $$g$$ and does not change the direction of the orbits.

My lecture notes state that the conjugation is strong and the equivalence is weaker, but 1) the only difference I see is that the conjugation may not keep the direction of the orbits and 2) the equivalence (?) Implies conjugation , so the conjugation is weaker

Can someone clarify?

Thank you so much!

## real analysis – finding the conjugate of a function

I know that the fennel conjugate is a function
$$f ^ * (x ^ *) = sup_x { langle x, x ^ * rangle – f (x) }.$$
How do I find the fennel conjugate of the function, however?
$$f (x) = frac {1} {p} sum limit_ {i = 1} ^ n | x_i | ^ p$$ Where $$1 .

I tried to differentiate the equation and see it as such $$= 0$$ but I can't find an answer.