Number Theory – Largest prime factor in the gap between two consecutive primes

definition: In the gap between two consecutive odd prime numbers, we have one or more compound numbers. We define the largest of all prime factors of this composition as the maximum prime factor of the gap.

claimEvery prime number is a maximum prime factor for a prime number gap.

I am looking for a proof or a proof.

Update, December 7, 2019: Guess confirmed for $ p le 2 times 10 ^ 9. $

Note: This question was posted in MSE and received many positive votes, but no answer, so it was posted in MO.

What are the main causes of the gender pay gap?

Women get pregnant, men do not.

This, of course, is an excessive simplification, but it is the basis of many differences between men and women, including the pay gap. Women who become pregnant are therefore considered providers. That's why men work more dangerously to earn more. That's why men work overtime. That's why men work more hours than women.

There are many who show many studies that the primary causes of the gender pay gap are men and women who make various labor decisions The Equal Pay Act of 1963 makes it illegal to pay women for equal work.

The real question is why the myth of gender discrimination still exists, given the equal pay law and the numerous studies that show how women make different work decisions than men. The answer can be summarized in one word: feminism.

linear algebra – limiting the spectral gap of a simple symmetric matrix

I have a seemingly innocent linear algebra problem that I can not solve, and I hope you kindly give me an insight. Here is the description: Let $ mathbf {a} = (a_1, a_2, dots, a_d) ^ {T} $ be a positive probability vector, $ d. H. $ $ Vert mathbf {a} vert_1 = 1 $ and $ a_i> 0 $ for all $ i $, Leave matrix $ A $ be defined as follows: $$ A = textrm {diag} ( mathbf {a}) – mathbf {a} mathbf {a} ^ {T} $$ from where $ textrm {diag} ( mathbf {a}) $ means the diagonal matrix with the $ i $the diagonal entry is $ a_i $, It's easy to show that $ mathbf {1} _d $, the all-one vector of dimension $ d $is an eigenvector of $ A $ of eigenvalue $ 0 $, And the Gershgorin circle set shows that too $ A $The eigenvalues ​​of are greater or equal $ 0 $, My question is:

What is the smallest eigenvalue of $ A $ that is not null

I did the calculation when $ d = 3 $ and realized that there may not be a simple analytical formula for it and therefore a nice lower bound is much appreciated.

Thank you very much!

Question about the gender pay gap in the US?

As early as 1987, around 25% of women paid more than their husbands. Today it is more like 40%.

Great efforts have been made in the US to equate women at all levels with men. This equality is reflected in many laws that require women to be paid as well as men, and laws that give women more opportunities than men. Yet women earn only about 80% of a man's earnings.

Why do women earn less than men? A Harvard University study showed that women give higher priority to family time. Women rarely work overtime or work weekends. Women are less likely to work late or on vacation, and women have more free time and receive unpaid leave under the Family Medical Leave Act.

The gender pay gap in the US is about the same as in Germany, Britain, France, Denmark, New Zealand, etc. and much better than in the Ukraine.
https://blogs.elenasmodels.com/en/gender …

Ease of use – How do I make my gap text exercise more obvious?

I've developed a Chinese language learning app that requires the user to fill in the blanks. A word in the sentence is empty, and the user must choose which options are correct. It looks like this:

Enter image description here

Users seem to understand what they should do there. However, some users are confused by exercises like these:

Enter image description here

They do not notice that the sign for "two" already exists and that the space should be filled with "twenty". Instead, they are looking for a button with "twenty-two" and are confused if they can not find it.

Is there a better way to present the exercise to make it clear that it needs to fill the void instead of finding the translation of "twenty-two"?

linux machines are with a time gap despite ongoing chronyd service

We have 3 machines in the cluster (Red Hat 7.2)

  • linux_machine1
  • linux_machine2
  • linux_machine3

That's the machine time

Sun Sep  1 13:54:20 UTC 2019 - linux_machine1
Sun Sep  1 13:34:15 UTC 2019 - linux_machine1
Sun Sep  1 13:53:35 UTC 2019 - linux_machine1

We can detect time differences between machines (this is not good, as all machines should be synced with the name of the NTP server "moon.ntp.org").

the configuration file:

more /etc/chrony.conf
# Use public servers from the pool.ntp.org project.
# Please consider joining the pool (http://www.pool.ntp.org/join.html).
server name moon.ntp.org iburst

We also check the chronyd service and the service runs on all machines

systemctl status chronyd

We also have those

chronyc tracking | grep "Leap status"
Leap status     : Normal

And we get normal condition on all machines

So, if the configuration is ok ( moon.ntp.org is the ntp Server)
And
The jump status is Normal

And all services are running

Why do we get the time gap on all 3 Linux servers?

TO UPDATE:

after we have restarted all chrony service as

systemctl restart chronyd

Now the time is the same on all machines

the question is – why need it here chronyd Restart?

Optimization – Hardness of QUBO problems with small spectral gap

I Solve Quadratic Unconstrained Binary Problems: $ min limits_ {x in { { pm 1 } ^ n}} x ^ top A x $

The quantum adiabatic proposition states that if the spectral gap of $ A $ (the module of least non-zero eigenvalue) is large enough, then these problems can be solved efficiently by quantum adiabatic evolution.

Conversely, if the spectral gap is small, the adiabatic evolution needs exponential time to approach a ground state.

Are there similar results for classic algorithms?
Are problems with the small spectral gap also inherently more difficult for classical algorithms?

8 – Passing a Node Argument to the Form Controller to Generate a Complex Gap Text Query

This is a two-part question, please.

  1. Is it possible to pass a node argument from a custom route to a form controller? I tried to add NodeInterface to an implementation of the buildForm function, which resulted in an error. I was inspired by this tutorial. My buildForm method from the controller code is below:

    public function buildForm(array $form, FormStateInterface $form_state, NodeInterface $node){
        dsm($node, 'Node Argument');
    }
    

I understand that trying to do this is wrong. The code is just here to illustrate what I'm trying to achieve.

And that's my routing code:

quiz.nodetab:
 path: '/node/{node}/quiz'
 defaults:
  _title: 'Quiz'
  _form: 'DrupalquizFormQuiz'
 requirements:
  _permission: 'access content'

But I get the following error message in which the use of the build form Interface is wrong:

Fatal Error: The declaration of Drupal quiz Form Quiz :: buildForm (Array $ form, Drupal Core Form FormStateInterface $ form_state, Drupal node NodeInterface $ node) must be compatible with Drupal Core Form FormInterface :: buildForm ($ form_array, Drupal Core Form FormStateInterface $ form_state) in /var/www/composerDrupalTest/web/modules/custom/quiz/src/Form/Quiz.php on line 9

  1. The reason why I am trying to tie a node into a form controller is that I need to create a blank text question that is a form with many picklist options like this one. The node contains the gap data.

How can I create such a form with multiple picklist options?