Matrices – How to maximally L1 norm problem?

I have encountered a problem these days.
begin {equation}
underset { omega} { max} quad Vert text {diag} ( mathbf {h} ^ H) mathbf {G} ^ H mathbf { omega} Vert_1 \
quad mathbf { omega} ^ H mathbf {G} mathbf {G} ^ H mathbf { omega} = t
quad mathbf { omega} ^ H mathbf { omega} leq 1
end {equation}

from where $ mathbf {h} $ and $ mathbf {G} $ are repaired; Supreme H and $ Vert * Vert_1 $ denote complex conjugate transpose or complex 1-norm.

Given for everyone $ t $how to solve this problem. In other words, how to build a relationship between $ t $ and objective function.

Many thanks.

Differential geometry – local uniqueness of the metric for locally maximally symmetric spaces

I'm currently studying maximally symmetrical spaces, physics style. So I am mainly interested in purely local results.

I define a (locally) maximally symmetric space as a pseudo-Riemannian manifold that has $ n (n + 1) / 2 $ independent killing vector fields.

I have found that this is equivalent to the curvature tensor that is of the shape $$ R _ { kappa lambda mu nu} = K (g _ { mu kappa} g _ { nu lambda} -g _ { mu lambda} g _ { nu kappa} ), $$ from where $ K $ is a constant.

The book Gravity and cosmology von Weinberg has a sentence that if $ bar g _ { mu ^ { prime} nu ^ prime} (x ^ prime) $ and $ g _ { mu nu} (x) $ are two metrics (since this is purely local, I basically work in an open set of $ mathbb R ^ n $) which have the same signature and are both maximally symmetric, so that (assuming Einstein summation convention in this post) $$ bar R _ { kappa ^ prime lambda ^ prime mu ^ prime nu ^ prime} = K ( bar g _ { mu ^ prime kappa ^ prime} bar g _ { nu ^ prime lambda ^ prime} – bar g _ { mu ^ prime lambda ^ prime} bar g _ { nu ^ prime kappa ^ prime}) \ R _ { kappa lambda mu nu} = K (g _ { mu kappa} g _ { nu lambda} -g _ { mu lambda} g _ { nu kappa}) $$for the equal $ K $ constant, then the two metrics $ bar g _ { mu ^ prime nu ^ prime} $ and $ g _ { mu nu} $ differ by a coordinate transformation, z. There are functions $$ x ^ { mu ^ prime} = Phi ^ { mu ^ prime} (x) $$ so that $$ g _ { mu nu} (x) = bar g _ { mu ^ prime nu ^ prime} ( Phi (x)) frac { partial Phi ^ { mu ^ prime}} { partial x ^ mu} (x) frac { partial Phi ^ { nu ^ prime}} { partial x ^ nu} (x). $$

Weinberg proves this by explicitly constructing a coordinate transformation over a power series. It is ugly and long.


I thought there is probably an easier way.

Namely, if $ bar g $ and $ g $ are two metrics of the same signature and $ bar theta ^ {a ^ prime} $ is a $ bar g $while $ theta ^ a $ is a $ g $-orthonormal coframe, then the two metrics are the same if and only if the two coframes differ by a generalized orthogonal transformation (Lorentz transformation for general relativity), z. There is a $ mathrm O (n-s, s) $-valued function $ Lambda $ on the open set so that $$ bar theta ^ {a ^ prime} = Lambda ^ {a ^ prime} _ { a} theta ^ a. $$

But even if that is not true, there is must be on $ mathrm {GL} (n, mathbb R) $-valued function $ L $ so that $$ bar theta ^ {a ^ prime} = L ^ {a ^ prime} _ { a} theta ^ a. $$

So I thought I could probably prove that statement by proving that $ L $ is actually a (generalized) orthogonal transformation.


The curvature shape for (locally) maximally symmetrical spaces has a simple form $$ mathbf R ^ {ab} = K theta ^ a wedge theta ^ b \ mathbf R ^ {a ^ prime b ^ prime} = K bar theta ^ {a ^ prime} wedge bar theta ^ {b ^ prime}. $$

My strategy was to take the "locked" sizes in the "prepared" frame and transform them (possibly through non-orthogonal ones) $ L $) in the "unprimed" framework.

For example, for the metric we have $ bar g_ {a ^ prime b ^ prime} equiv eta_ {a ^ prime b ^ prime} $ (from where $ eta $ is the canonical symbol associated with the metric of a given signature, e.g. the Minkowski symbol for general theory of relativity), but in the unpainted frame it is $ bar g_ {ab} $ This is not necessarily "Minkowskian".

I tried to construct the curvature shape directly out of the frame and compare it with the expression that I listed above in the hope that I come to a relationship that implies one of $$ bar g_ {ab} = eta_ {ab} \ bar gamma ^ {ab} = – bar gamma ^ {ba}, $$ that would mean that immediately $ L $ is actually a generalized orthogonal transformation, but I have come to no useful conclusion.

Question: Can I prove this statement (namely, that two locally maximal symmetric spaces of the same dimension, signature, and the same value of $ K $ will be locally isometric) using this orthonormal framework method?

If yes, how does it work? I'm pretty stuck with it.

Properties of the collection of maximally independent sets of a graph

To let $ G $ be a graph and define

$ mathscr {I} (G) = {S subset V (G) | S $ is a maximum independent group of $ G } $

  1. What is known? $ mathscr {I} (G) $?

  2. What are some of the properties of $ mathscr {I} (G) $?

  3. How does it work $ mathscr {I} (G) $ refers to other properties of $ G $ for example chromatic number?

  4. Is it possible to decide if there is a collection? $ mathscr {A} $ corresponds to $ mathscr {I} (H) $ for a diagram $ H $?

java – How can I maximally work against any group in Android Studio?

I have a question about a program I'm doing right now. In principle, I create a configuration so that the user can choose how many groups he wants to create and how many players should be created per group. When it's created, I can give it the ability to insert players and select which group I want you to go. My question is, how can I form a limiter by groups, ie if I target players in group 1, and that is the maximum, in which I do not get more players in group 1, and that happens the same applies to the number of groups , that I have. I've tried creating an array list to put several array lists into others and try to get them to work, but I can not get them. Thank you