Probability – Common PMF based on the PMF of another variable

So there is this PMF from $ M $ (a positive integer r.v) as follows:

$$ p_M (m) = 1/3 * m * 3 ^ (- m) $$

With $ m = {1,2,3,4,5, …} $

Another random variable N has a uniform PMF based on $ {1,2,3,4,5, …, 3n} $

What is $ P_M, _N (m, n) $ and $ P_N (n) $?

I'm not sure how to deal with this problem. I have calculated the values ​​of $ P_M (m) $: {0.11,0.074, 0.037, …}, but then I got stuck. I am not sure how to proceed to calculate N. I thought I should share each one $ P_M (m) $ With $ 3N $But then I realized that the value was getting so low that I doubt it's the right way to go.

Any advice? Thanks a lot!

Why is the automatic reopening of previous windows / tabs such a common pattern?

I love being able to continue working where I left off, but not when I specifically tell an application / operating system to stop or when it crashes.

Sometimes I choose to restore what I had previously opened, but most of the time I'm not asked.

It is difficult and sometimes impossible to change this attitude, and it seems that more and more things are adopting this pattern.

Wouldn't an actual choice be the best solution?

Why is the automatic reopening of previous windows / tabs such a common pattern?

I love being able to continue working where I left off, but not when I specifically tell an application / operating system to stop or when it crashes.

Sometimes I choose to restore what I had previously opened.
but I'm not asked most of the time.

It is difficult and sometimes impossible to change this attitude, and it seems that more and more things are adopting this pattern.

Wouldn't an actual choice be the best solution?

Synchronization – After a sleeping client goes online again, what technique does he use to find the last common ancestor in a blockchain?

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Find common elements in arrays

My goal is to get a list of relatives setnteceIds repeated in the order of most.

In my database I have strains (words):

+---------+-----------------+
|  Stem   |   SentenceId    |
+---------+-----------------+
| Hello   | (2,46,62)       |
| Good    | (1,123,45)      |
| Morning | (23,2436,12312) |
+---------+-----------------+

My steps:

  1. Take all SatzIds

    (6, 86),(6),(4),(3, 86),(...)
  2. Create a blank object and increase its value each time I see a number (or set it to 1 if it hasn't been seen before).

     {2: 1, 3: 4, 4: 1, 6: 11, 85: 1, 86: 5, 87: 1, 89: 1}

  3. Sort the object by the attribute and assign it to a relevant array.

     ("6", "86", "3", "89", "87", "85", "4", "2")

Full code:

getAllStatementIdFromStem = async (stems: string()) => {
    const ids: { (key: number): number } = {};
    await this.getBulkStem(stems).toArray((stems: Stem()) =>
      stems.map(({ statements }) =>
        statements.forEach((statementId: number) => {
          typeof ids(statementId) === "undefined"
            ? (ids(statementId) = 1)
            : ids(statementId)++;
        })
      )
    );
    return Object.entries(ids)
      .sort((a, b) => a(1) - b(1))
      .reverse()
      .map(x => x(0));
  };

Could I do something more effective? I feel like I'm taking too many extra steps.

nt.number theory – Largest common divisor as a nonlinear combination $ ax + of + alpha xy $ and question about the related function

There is this well-known elementary theorem:

Every pair of integers $ a $ and $ b $ has a common divisor $ d $ the form $ d = ax + by $, In addition, every common divisor of $ a $ and $ b $ share this $ d $,

This is how the largest common divisor can be represented linear as a function of $ a $ and $ b $,

But it also seems to me to be an interesting question when the greatest common divisor can be represented as follows $ ax + by + alpha xy $, Where $ alpha in mathbb Z $ is a constant.

It seems obvious to some $ alpha $ the greatest common divisor $ d $ will not be expressible in the form $ ax + by + alpha xy $ but for some others $ alpha $ it will be expressible.

At least, $ d $ can be expressed as $ ax + by + alpha xy $ if $ alpha = 0 $ but sure for some $ a, b in mathbb Z $ the vote $ alpha = 0 $ is not the only one.

So let's assume that for $ (a, b) in mathbb Z times mathbb Z $ the sentence $ beta ((a, b)) $ is the crowd of all $ alpha $ so that $ d = ax + by + alpha xy $, for example if for some $ (a, b) $ we have that $ d = ax + by + alpha_r xy $ to the $ r = 1, …, $ then $ alpha_r in beta ((a, b)) $,

If $ text {noe} ( beta ((a, b))) $ designated the number of elements of the set $ beta ((a, b)) $ then I would like to know at least some facts about the function $ (a, b) to text {noe} ( beta ((a, b))) $ and what is generally known about this function, for example some properties of it or some rules that govern its behavior?

price – Magento 2.3.x: How can you set a different tax rule for several shops in a common country?

I am creating two different stores US and US1 Enter image description here

and put control rule US tax rule and US1-control rule
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further Tax zones and tax rates to adjust Country US and Tax rate 5% and 10%,
Enter image description here

Select admin view US1 and US under admin -stores> configuration> General and set the default country USA.
Now as you select in the tax rule in the frontend, the calculation viewer adds both
Control rule%.
Example product price is $ 22.,

when choosing US1 or US stores.

Results display 10% + 5% = $ 25.30
Enter image description here

How can the tax be separated when choosing the branch?

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Web application – what is the most common idea for web application development?

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