linear algebra – At the limit of eigenvalues ​​for any matrix

I read this pre-print: https://arxiv.org/abs/1605.00531. The authors mention the following:

"We know that eigenvalues $ z_k $ of any matrix $ Z $ are in the rectangle $ text {Re} (z_k) in sigma_1 $, $ text {Im} (z_k) in sigma_2 $ or $ sigma_1 $ is the range of $ frac {1} {2} (Z + Z ^ dagger) $ and $ sigma_2 $ is the range of $ frac {1} {2} (Z-Z ^ dagger) $. "

I thought the range was a set of all the linear combinations of the columns of $ Z $? is $ sigma_1 $ the cardinality of this set?

More generally, how can I prove their statement or where can I find proof? The reference used by the author is inaccessible with my university.

Thank you so much!

How to solve this exponential limit?

Is the limit $$ lim_ {x to 0} x ^ {cosx} $$ exists or not and how does this limit differ from $$ lim_ {x to 0} | x | ^ {[cosx]} $$.

Is there an index theorem in fixed points that deals with the fixed points of the limit?

assume $ f $ is a continuous function of a unit cube $[0,1]^ n $ to himself, then $ f $ has at least one fixed point. Assume further $ f $ is smooth, $ 0 is a regular value of $ f (x) -x $, and the fixed points are eventually on the limit. Reading some manuals, I find that the fixed points of the limit can not be involved in the index of fixed points. I wonder if there is an index theorem that treats fixed points on the boundary?

seo – Will having a lot of relevant internal links affect the ranking of pages and is there a limit?

We are revisiting our internal liaison strategy at my workplace.

The current implementation takes a lot of links and places them on a number of pages accessible from the header. Some links appear more often than others.

Consider these two cases:

Case 1

Source: Men's summer shorts

Internal link to: women's summer shorts
Internal link to: hats
Internal link to: jogging pants

Case 2

Source: Men's summer shorts

Internal Link to Men's Swimwear
Internal Link to: Men's Sunscreen
Internal Link to: Men's Jogging Pants

Case # 1 contains internal links that are not directly related to the source page, unlike Case # 2.

Will one or the other case affect the ranking of pages?

Second question: is there a gold standard for the number of times that a link can appear on the site? What happens if a link is linked to 500 different pages?

functional analysis – Limit value of continuous representative of a Sobolev function

Let $ f in H ^ 1 (a, b) $, or $ a <b $.

Then there is a permanent representative $ tilde {f} (x) = int_a ^ x f (x) dx + c $ or $ f $ l ^ 2 (a, b) $ is the weak derivative of $ f $ and c is a constant.

My question mainly concerns this constant integration.

By the above equality, it follows that $ tilde {f} (a) = c $.

Now, I've read the statement that for $ phi in C ^ { infty} _c (a, b) $ with $ int_a ^ b phi dx = $ 1 the following equation is true:

$ int_a ^ b f (x) phi (x) dx = tilde {f} (a) = c $

In that case $ phi $ seems to behave in the same way as the dirac distribution, but I can not say why this equality should be respected.

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Macro – Is this the limit of the picture quality of the camera or the subject is not it sharp?

I'm using the "macro" mode in the Powershot G9X II. Macro mode is simply a shorter focus range, there is no magnification in place, as in macro lenses for SLRs. I've taken many pictures of the subject, but that's the best. I realize that the proper way to proceed is to use focus bracketing because the camera probably focuses on the wall and the spider has a few millimeters of focus distance, but I'm not sure. I realized it at home in the evening. In any case, this is not the purpose of my question.

I am constantly struggling with the noise and blur of smartphones on this camera. Critics say it should not be like that, but that's what I see. I would therefore like to ask if it is another manifestation of the same problem or if I have simply been focusing poorly.

Consider the 50 harvest. It's nice and sharp. No problem, there is even a nice background blur effect (it is likely that you are simply focusing at extremely close distances and the wall is not sharp.) Now consider cropping 100%. This is not what I would call strong. I think it's not fuzzy either, because I saw the other pictures of the same subject that were really blurry and that looked different. Instead, I would call this "smartphone quality" where the sensor simply can not produce an image that you can view at 100% magnification.

What do you think? Is my perception of the problem correct? Am I expecting too much from the camera? Yes, I know it's not a macro lens and that the subject's body occupies a far too low percentage of the photo, but why is it blurry anyway? I'm getting similar results with big topics. Everything is beautiful at 50%, but as soon as you go down to 100%, everything becomes blurred and the details are lost. As if a low-pass filter was applied.

Edit: 1/60 f5.6 ISO 250 @ the widest FOV (fully retracted lens). The subject was as close as possible for autofocus to work in macro mode ("flower" icon).

Harvest 100%
Harvest 50%
original

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Does the expansion of the universe actually lose the contraction and has it a limit to reach the value of the total length?

Your question contains many invalid statements.

You do not apply special relativity to the expansion of the universe. You use general relativity which, under certain circumstances, allows expansion by gravity. The notion of "total length" does not exist in general relativity.

If you want to understand how the extension works, just ask it. Do not put invalid stuff in a question.

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