## 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.

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## SEO – How many links per page can we maximally have in an HTML sitemap?

There is a limit of 50,000 per Sitemap in the Sitemap. Is there a limit on links that may be present in the HTML-based sitemap?

If I have more than 100,000 pages or posts, should I use them as a pagination method for HTML Sitemaps?

PS: XML sitemap is different from HTML based sitemap.

## What are the best merchant category codes for a maximally successful transaction?

Hello everyone,

I see a much greater decline than successful debit card capture.

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## 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 $$H$$?

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## 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