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A graph with n points satisfies some conditions.

(APMO 1990) Let $ G $ is a diagram that meets the following conditions. (1) No point has edges to all other points. (2) There are no triangles. (3) Any two points $ A, B $ s.t. there is no edge AB, there is exactly one point $ C $ that there are edges AC and BC. Prove that every point has the same degree.

I've found proof online, but he uses common additions that make no sense to me. I tried to take that into account $ N_a, N_b $ from two points A, B, where there is no edge $ AB $ and to prove that they have no edges in between $ N_a, N_b $ without and their common point $ C $but then I'm stuck.

apache – Can anyone explain what these .htaccess rules and ".well-known" conditions apply to?

Why are there two types of conditions and rules? Can they be combined into a set?

These are two rules that serve two different purposes – they can not be combined. Taking into account the (unnecessary) "known" conditions we have:

Rewrite Engine On
RewriteCond% {SERVER_PORT} 80
RewriteRule ^ (. *) $ Https://www.example.com/$1 [R,L]

RewriteCond% {REQUEST_METHOD} = POST
RewriteCond% {HTTP_USER_AGENT} ^. * (Opera | mozilla | firefox | msie | safari). * $ [NC]
RewriteCond% {THE_REQUEST} ^[A-Z]{3.9}  /.+/ Trackback /?  HTTP / [NC]
RewriteRule.? - [F,NS,L]

The first is a redirect from HTTP to HTTPS. (This should actually be a permanent 301 redirect, not a temporary 302 redirect as currently written.)

The second rule seems to block trackbacks. This is a "feature" that can be used to validate websites / blogs (especially WordPress) LinkBacks to articles and is often called comment, However, this can be abused by spammers, hence the desire to block them.

I'm confused about the purpose of .very famous Exceptions and so many conditions. Is this just a temporary setup by the host or is it advisable to keep it?

These conditions They do not play any role in the rules themselves, they just create exceptions so that the rules are not executed when these demands are made. This occurs when your host automatically renews security certificates.

They are needed only temporarily (if at all). However, it is likely that they will be needed again after a few (3?) Months at the next renewal of the allowances.

As mentioned by @Stephen in comments, however, the conditions for these special rules are most likely superfluous anyway. CPanel does not differentiate, but adds these conditions blindly each RewriteRule Directive. If you remove them manually, cPanel will probably add them back during certificate renewal (as mentioned).

For more information about these conditions, see my answers to the following ServerFault questions:

ca.classical analysis and odes – Under what conditions is this family normal?

To let $ mathcal {S} = {s in mathbb {C} , mid , | Im (s) | <1 } $ be a strip of the complex plane. To let $ q (s, z) $ a holomorphic function $ mathcal {S} times mathbb {C} $, Rent $ mathcal {K} $ a CD of $ mathbb {C} $We can accept:

$$
sum_ {j = 1} ^ infty || frac {1} {n} q (s- frac {j} {n}, z) || _ { mathcal {K}} < infty
$$

For all $ n in mathbb {N} $With $ n ge 1 $where the border continues as $ n to infty $ converges evenly into $ s $ on compact sentences of $ mathcal {S} $, Note that $ || h || _ mathcal {K} = sup_ {z in mathcal {K}} | h (z) | $, Also note that the boundary to the integral converges

$$ int _ {- infty} ^ s || q (t, z) || _ { mathcal {K}} , dt $$

Suppose we have a family of holomorphic functions $ F_n (s): mathcal {S} to mathbb {C} $ in which:

$$
F_n (s) to z , , text {as} , , Re (s) to – infty
$$

$$
F_n (s + frac {1} {n}) – F_n (s) = frac {1} {n} q (s, F_n (s))
$$

I want to show that this family approximates the differential equation $ F # (s) = q (s, F (s)) $ with any precision. Or that $ F_n to F $, To get there, I have to show that this family is normal. My guess is that I lack a condition $ q $ that provides for normality, or I miss an obvious fact about this family.

Basically, I wonder if there is a limit anyway $ n to infty $ from $ F_n $ could escape to infinity, or if the fact that its satisfying equation approaches the differential equation $ y # = q (s, y) $ Forces the convergence to uniquely solve the differential equation. Where the unique solution is a function $ ell $ in which $ ell (- infty) = z $ and $ ell & # 39; = q (s, ell) $that can be shown to exist in a neighborhood of and to be unique $ – infty $ by a slight modification of the usual Picard-Lindelöf theorem. Where by the neighborhood of $ – infty $ I mean $ Re (s) <R $. $ | Im (s) | < tau <1 $ to the $ | R | $ big enough.

All in all, the family is $ {F_n } _ {n = 1} ^ infty $ normal? If not, what would be a reasonable condition to ensure this?

Any help would be appreciated.

Thanks, Richard.

google sheets – Conditional formatting cells with multiple conditions and relative references across a range

I have a Google spreadsheet with arbitrary data. Three adjacent columns (for example, A, B, and C) have been set up to allow only values from 1 to 10 during data validation.

I am looking for a way to check if all three columns of a single row (for example, A4, B4, and C4) have a value of 10 and all three cells should be formatted differently.

This formatting should apply to a full range from A / B / C1 to A / B / C300, and this is the problem.

I've tried many ways to do that with data validation, built-in functions, and conditional formatting, but apart from using the formatting on a line-by-line basis, I have not found a really effective way to do it. Had to resort to the script editor and write a function called when onEdit () is triggered, but this is slow and the coloring is actually applied when the code is triggered because each line must be checked individually.

Is there a way to do this without having to resort to scripts and without having to do it line by line?

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