real analysis – Is this version of Clairaut-Schwarz's theorem valid when mixed partial derivatives are of order greater than $ 2 $?

I've asked this question on MSE here. A person gave an answer but then deleted it because my version of Clairaut-Schwarz's theorem is stronger than his. I meant that my version only requires the continuity of a mixed partial derivative while its may require continuity of all partial mixed derivatives.

It seems that this question will not receive any answer in MSE, so I can only post on

I usually meet the theorem of Clairaut-Schwarz where the mixed partial derivatives are of order $ 2 $, that is to say.

$ textbf {Clairaut-Schwarz theorem:} $Let $ X $ to be open in $ mathbb R ^ n $, $ f: X to F $, and $ i, j in {1, ldots, n } $. Assume that $ partial_j partial_i f $ is continuous to $ a $ and that $ partial_j f $ exists in a neighborhood of $ a $. then $ partial_i partial_j f (a) $ exists and $$ partial_i partial_j f (a) = partial_j partial_i f (a) $$

I would like to ask if the Clairaut-Schwarz theorem is valid in the case where the mixed partial derivatives are of arbitrary order $ m $, that is to say.

Let $ X $ to be open in $ mathbb R ^ n $, $ f: X to F $, and $ m in mathbb N $. assume $ j_1, j_2, ldots, j_m in {1, ldots, n } $ and $ sigma $ is a permutation of $ {1, ldots, m } $. Yes $ partial_ {j_1} partial_ {j_2} cdots partial_ {j_m} f $ is continuous to $ a $ and $ partial_ {j _ { sigma (2)}} cdots partial_ {j _ { sigma (m)}} f $ exists in a neighborhood of $ a $then $$ partial_ {j_1} partial_ {j_2} cdots partial_ {j_m} f (a) = partial_ {j _ { sigma (1)}} partial_ {j _ { sigma (2)}} cdots partial_ {j _ { sigma (m)}} f (a) $$

Thank you very much for your help!

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dnd 5th – Why does Night Hag have greater skills in skills than it should be?

Maybe a special bonus, but probably an error.

I looked through the Manuel Monster, and found nothing to explain why Night Hag has +1 to each of these skills, besides this text

A skill bonus is the sum of the monster's relevant ability modifier and
his skill bonus, which is determined by the monster's challenge
ranking (as indicated in the skill bonus table by challenge level).
Other modifiers may apply. For example, a monster might have a bigger bonus than expected (usually double his skill bonus)
to account for his increased expertise.

It should be noted that when an entry in the manual has the "double of his skill bonus", the only indication is the number itself, so it is possible that an unspecified bonus is present on the Night Hag.

In addition, 3 Hags present in the Manuel Monster (Sea, Green, Night), the night Hag is the only one who has objects (Heartstone, Soul Bag). While Heartstone enumerates a beneficial effect for Hag, the bag does not say it, it simply states that it captures the soul of an evil humanoid who dies because of the haunting action of the Night Hag's Nightmare.

So, since all listed Night Hag skills have a +1, it is unlikely that this is due to a bonus that is not listed, most likely to one or both of the objects. that he can have.

However, since nothing in Night Hag's description or statistics mentions anything above or there to produce a definite bonus for a skill, it is probably a mistake.

Regarding the reporting of an error, there are 2 options: 1- post a message on the bug reporting forum on DNDBeyond; 2- try to contact the assistants directly.

In terms of contacting the assistants directly, there are 2 things I found, the first is a customer service number – 1 (800) 324-6496; the second is this webpage:

Be aware, however, that the DNDBeyond bug reporting forum probably only applies to the site itself. If you need to contact the assistants directly if you think this is important.

special functions – Question about ⌈x⌉ greater than or less than x

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dnd 5th – Does the "Fade Away" feature grant greater invisibility?

It does not literally grant greater invisibility but it does not end when you make an attack

Things only do what they say, and the Fade Away option does not say that it ends when an attack is launched. However, you have forgotten an important part of the feature:

You can use a bonus action to magically become invisible, as well as any equipment you wear or carry, until the beginning of your next turn.

Once you use this feature, you will not be able to use it until you have finished resting.

Compare that to the invisibility to spell:

A creature you touch becomes invisible until the end of the spell. Everything that the target wears or wears is invisible as long as it is on the person of the target. The spell ends for a target that attacks or casts a spell […]

You only use Fade Away once per rest which makes it far of breaking the game. Moreover, it only lasts one lap, unlike greater invisibility. Also do not forget that it 's a level 10 feature, so that it only comes into play later in the year. advancement of the character.

php – How to make the IF condition recognize values ​​greater than 100 and / or how to browse a table obtained from a csv file and get the desired value?

Good night!

Gentlemen, I'm new to PHP and have the following problem.
I have two options below to solve my problems. However, I can not make them work completely.


This condition of the IF mine does not calculate the freight value if the distance exceeds 1000 km.

The distance is calculated correctly using the Google Distance Matrix API.

Follows the SI section:

if ($distance > 0 && $distance <= 100)
        $frete = (100 * 2.19);
    }elseif ($distance > 100 && $distance <= 200)
        $frete = (100 * 1.35);
    }elseif ($distance > 200 && $distance <= 300)
        $frete = (100 * 1.18);
    }elseif ($distance > 300 && $distance <= 400)
        $frete = (100 * 1.11);
    }elseif ($distance > 400 && $distance <= 500)
        $frete = (100 * 1.07);
    }elseif ($distance > 500 && $distance <= 600)
        $frete = (100 * 1.04);
    }elseif ($distance > 600 && $distance <= 700)
        $frete = (100 * 1.02);
    }elseif ($distance > 700 && $distance <= 800)
        $frete = (100 * 1.01);
    }elseif ($distance > 800 && $distance <= 900)
        $frete = (100 * 1.00);
    }elseif ($distance > 900 && $distance <= 1000)
        $frete = (100 * 0.99);
    }elseif ($distance > 1000 && $distance <= 1100)
        $frete = (100 * 0.99);
    }elseif ($distance > 1100 && $distance <= 1200)
        $frete = (100 * 0.98);
    elseif ($distance > 1200 && $distance <= 1300)
        $frete = (100 * 0.98);
    }elseif ($distance > 1300 && $distance <= 1400)
        $frete = (100 * 0.97);
    }elseif ($distance > 1400 && $distance <= 1500)
        $frete = (100 * 0.97);
    }elseif ($distance > 1500 && $distance <= 1600)
        $frete = (100 * 0.97);
    elseif ($distance > 1600 && $distance <= 1700)
        $frete = (100 * 0.97);
    }elseif ($distance > 1700 && $distance <= 1800)
        $frete = (100 * 0.97);
    }elseif ($distance > 1800 && $distance <= 1900)
        $frete = (100 * 0.97);
    }elseif ($distance > 1900 && $distance <= 2000)
        $frete = (100 * 0.97);
    elseif ($distance > 3000){
        $distance = 3000;
        $frete = (100 * 0.64);

    echo $distance;
    echo "
"; echo $frete;

Can someone help me with what I am going wrong?


I've tried doing this via an external CSV file. However, I can not scan this file to see if this distance is in an array.

Here is the code below my first try. I would even prefer this option to have a cleaner code and easier maintenance when prices change.

 foreach( $csv->ler() as $linha ) //AQUI TRAGO AS INFORMAÇÕES DO ARQUIVO CSV.{
    if ($distance >= $linha(0) && $distance <= $linha(1)){
            for ($i = 0; $i < count($linha); $i++){
                if ($distance != $linha($i)(0) && $distance != $linha($i)(1)) {
                    echo floatval($distance);
                    echo $linha;                
        var_dump( $linha );//TRÁS TODOS OS DADOS ARRAYS CORRETAMENTE.

Example VARDUMP:

array (size=3) 
  0 => string '1' (length=1)
  1 => string '100' (length=3)
  2 => string '2.19' (length=4)

If anyone can help me with one or the other option, I will be very grateful to you.

Thank you in advance for your attention.

Recognize when an item is greater than or equal to 0.05 in a specific form

I have tried
Count (0.75 (0.043), x ____, x> = 0.05)
But I do not get 1 in return. What's wrong? I have studied Mathematica's help files, but I have not got any clue.
Thank you in advance.

dnd 5th – Can you ever use an ASI to increase a capacity score greater than 20?

From a strict reading of the UPS, you should not be able to, because it does not say "You can not increase a capacity score above your current limit" or something like that, it simply says "You can not raise (…) above 20".

Now, is it planned? Probably not.

In the PHB, the only way to exceed your limit is to use Barbarian 's Primal Champion function, which is at the 20th level and you do not get any more ASI thereafter.

So, a possible read on this is that it is worded as it is for simplicity, since they did not envision articles on DMG.

The problem is that, in my experience, the DMG is written (more) vaguely (than the PHB), which could even be intentional because it is supposed to be read by the DM and it can decide what 'he wants. It means that it is probably implicit Increasing the maximum score from 2 (to 22) allows you to increase your statistical score to 22 by any means, including ASI.

An additional indication that you should be able to do is that each way to increase your stats says something on the lines of "up to 20" (AFAIK), instead of "up to 20". your maximum ". This means that any feature that increases your ceiling does not do anything, since anyway, everything is hard in its own text (Ioun Stone is another example other than ASI).

For the sake of clarity, what I mean by that is that there would be no reason to say it: "your maximum for this score increases by 2" instead of "up to any value" or "no maximum value "or something like that if it was not supposed to influence another element or feature.

About your last question

In fact, 20 is even the maximum for your ability score when you use objects; the quote seems to apply only to the characteristics of the class.

The quote is a general rule. This should not be important, however, because, as far as I know, the objects explicitly indicate their heading (heading from the star-board of Deck of Many Things to 24, cape of Ioun Stones to 20, etc.).

loop – Comparing meta queries for IDs greater than specific IDs

I just want to retrieve all items that have a higher value than a item ID provided

'meta_query' => array(
    'relation' => 'AND',
        'key' => 'rid',
        'value' => $rid,
        'compare' => '=',
        'type' => 'numeric',
        'key'     => ID,
        'value'   => $last_id,
        'compare' => '>',
        'type'    => 'numeric'

So, if $ last_id is "350", I want to retrieve all messages with an ID greater than 350

Aggelematic geometry – Endomorphism ring classification of single abelian patterns greater than $ W (k) / p ^ n $

Yes $ A $ is a simple abelian variety on a closed field algebraically k $ with then $ End (A_k) otimes { mathbb Q} $ is a positive involution divisional algebra. Albert's classification tells us that these types of divisional algebras fall into four types.

Now consider an algebraically closed field k $ of characteristic $ p> 0 $ and a positive integer $ n $is there an (approximate) classification of $ End ( mathcal {A}) $ for simple abelian schemas $ mathcal {A} $ more than $ W (k) / p ^ n $ (here simplicity means $ mathcal {A} _k $ Is simple) ?

If we have an abelian schema $ A $ more than $ W (k) $, can we determine the growth of $ End (A text {mod} p ^ n) $ ?

Motivation: Gross's theory of canonical lift and quasi-canonical emergence for elliptic curves, Grothendieck-Messing theory.