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dg.differential geometry – Can we choose smoothly the singular vectors of a matrix?

$newcommand{GLm}{text{GL}_n^-}$

Let $GLm$ be the space of real $n times n$ matrices with strictly negative determinant.

Define $X$ to be the set of all matrices $A in GLm$ whose smallest singular value of is strictly smaller than all the other singular values. $X$ is an open subset of $GLm$.

Can we choose locally the singular vectors of matrices in $X$ in a smooth manner?

Here is a more precise description of what I want:

Let $A in X$. Do there exist an open neighbourhood $O$ of $A$ in $X$, and smooth maps $U:O to text{SO}_n,V:O to text{O}_n^-$ such that
$$ A=U(A)Sigma(A)V(A)^T,$$ holds for every $A in O$, where
$Sigma(A) = text{diag}left( sigma_1(A),dotssigma_n(A) right)$, and $sigma_1(A)$ is the smallest singular value of $A$?

I don’t care about the ordering of $sigma_2,dots,sigma_n$, but I specifically want the minimal singular value to be in a fixed position.


Comment 1: If we replace the requirement that $sigma_1 $ is strictly smaller than the others by the requirement that all the singular values are distinct, than the answer is positive. Indeed, in that case the map

begin{align*}
mu: text{SO}_ntimes mathcal Dtimes text{O}_n^-to X\
(U,Sigma,V)mapsto USigma V^T
end{align*}

is a local diffeomorphism, so is locally invertible (here $mathcal D$ is the space of $ntimes n$ diagonal matrices with strictly increasing positive entries).

Comment 2: If such maps $U,V$ exist, then the map $A to Sigma(A)$ is also smooth. In general the ordered singular values cannot be chosen smoothly when they “cross”, but here I don’t require to keep the ordering of $sigma_2,dots,sigma_n$ fixed, so I don’t think there is an obstruction, but I may be wrong.

client – What is a good resource to help newcomers choose their first wallet / s?

My recommendation would be to use bitcoin.org. It's the original Bitcoin site and also the best, IMO. They include almost all wallets, from multisig to hardware.

But if you asked me, I would recommend limiting your search to hardware wallets, desktop / full wallets. The latter should be used in conjunction with the cold storage method, where you host a client on an offline computer, sign a transaction, and then send it over a computer connected to the Internet.

Here is a tutorial and explanation for cold storage in the wiki.


My recommendation would be the Trezor wallet, although I own a Ledger HW.1.

Definitely one of the best wallets out there, it can withstand a lot and it has great software. It's a cold storage for the pros! :]

Algorithms – Do you choose a random number in a pool with duplicates in O (1) time and less than O (n) space?

Imagine you want to select a random number from a number pool that contains:

i1 times the number 1
i2 times the number 2
i3 times the number 3
… etc

For example, a random number in a pool of
30 times the number 1
90 times the number 2
would yield 1 with 33% probability and 2 with 67% probability.

let NU be the number of unique numbers
NT is the total number of numbers
In our example, NU = 2, NT = 120.

Is there a way to do this in O (1) time and less than O (NT) space?

sharepoint online – For our HR site you can choose between a website for modern teams and a website for modern communication

We have a new SharePoint Online Tenant. and within the integrated root site collection (which is the classic team site type), we've added some document libraries and news Web Parts. Now we want to create our human resources department. and one of the questions I have Whether we should create this HR site as a Modern Team site or as a Modern Communication site?

I read this link at https://docs.microsoft.com/en-us/sharepoint/planning-hub-sites, which mentions that team sites are used for collaboration (group of users sharing information) Communication websites are used when they are restricted Number of users publishing content for other users. Our human resources department will be better focused on a communications page, as the HR manager and 2 HR staff will publish travel policies, holidays, general HR documents, etc. for all users (who only have reading rights) about this content. and later, we might define this site as a hub site and have more detailed site collections for the HR capabilities.

However, one problem with using the communication site is that the links may not be displayed on the left side of the site (as in classic and modern team sites). And our end users said they needed a consistent navigation experience. Since our root site collection has links to the left, they want the same for all site collections. So, I am not sure if I should use the modern team website instead of the communications website just because the HR website should have the navigation links on the left instead of the top (as in the case of the communications website)?

GUI Design – How To Choose The Best Type Of Toolbar Controls?

I have a 3D view that shows two objects. One is the reference object, the other is the user object. By jointly displaying the two objects, the user can compare his own object with a reference object (these objects are color scales).

Each object has three display modes (smooth, wireframe, and flat) and an opacity selection (a slider between 0 and 100).

There are also controls that are "independent of gamut": a selection between 2D and 3D, a "top view" and a "bottom view" button, and a slider called gamut-slice in the following image.

The picture shows what we have today, but we want to redesign this window.

Enter image description here

(The two check boxes "New profile" and "Reference profile" do not make sense and will be deleted.)

I'd like to integrate all the controls into a toolbar, but I do not know what criteria controls are best for.

In particular, I would like to summarize the display modes Smooth, Wireframe and Flat somehow. I thought of two possible solutions: using three separate buttons (one for each mode) or using a combo box with the three modes. There are probably many other possible solutions.

According to which criterion should I make my choice?

dnd 5e – When does an UA Sea Sorcerer choose the added effect of Curse of the Sea?

The sea wizard curse of the sea says in part:

Once per turn, if you cast a spell, you can trigger the spell if that spell deals cold or lightning damage to the accursed target or forces it to move. If you do so, the target will be subjected to the corresponding additional effect. The curse ends when the spell is not a cantrip (you choose the effect to use when more than one effect is applied):

cold damage, If the affected target suffers cold damage from your spell, the ()

lightning damage, If the affected target suffers lightning damage from your spell, your (…)

forced movement, When the target is moved by your spell, you can (…)

I do not know when The sorcerer actually chooses this ability, because it is different from other features such as: B. the overchannel function of Evokationsassistenten, has unusual formulations.

If you cast a level 5 or lower spell that deals damage, you can use your spell to deal maximum damage.

In the question "When must the assistant decide for an override?" It has been decided that this feature will be used if you speak a magicNot if the spell actually causes damage. However, the Curse of the Sea feature uses a different wording, and I'm not sure how this (if anything) affects the timing of the feature.

Does the sorcerer choose to use this feature while casting the spell or if the spell actually causes movement that causes or forces damage?

Once it matters:
If you have used this feature if you speak a magic and decided to, and then the creature proved immune to cold damage, the extra effect would not occur.

Choose a Broker – Forex Training

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