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!

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 $0<sigma_1<sigma_2$, 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 $0<a<b$, $ab=1$, 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 <p < infty $.

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