nt.number theory – Discriminant of order in Quaternion algebra is square of positive integer

$newcommand{ZZ}{mathbb{Z}} newcommand{cO}{mathcal{O}} newcommand{QQ}{mathbb{Q}} newcommand{BB}{mathbb{B}} DeclareMathOperator{disc}{disc} DeclareMathOperator{trd}{trd}$
Let $cO_{d_i} := ZZ(frac{d_i+sqrt{d_i}}{2}) subseteq QQ(sqrt{d_i})$ be the unique quadratic order of discriminant $d_i<0$ for $i = 1,2$. We let $tau_i = frac{d_i+sqrt{d_i}}{2}$, which notably satisfies the quadratic polynomial
begin{equation}
tau^2 – d_i tau + frac{d_i^2-d_i}{4}.
end{equation}

Let $BB_{ell,infty}$ be the quaternion algebra ramified at the finite, integral prime $ell$ and the non-archimedean prime $infty$.

Assume that we have embeddings $cO_{d_i}hookrightarrow BB_{ell,infty}$. Then, we can consider the order generated by the images; namely, $R:= left<1,phi_1,phi_2,phi_1phi_2right>subseteq BB_{ell,infty}$ where the $phi_i$ are the images of the $tau_i$ under the respective embeddings.

One can then compute the discriminant
begin{equation}
disc(R):=det(trd(alpha_ialpha_j)) = left(frac{d_1d_2-(d_1d_2-2trd(phi_1phi_2))^2}{4}right)^2.
end{equation}

($trd$ is the reduced trace in the quaternion algebra– notation of Voight.)

The unique maximal order of $BB_{ell,infty}$ has discriminant $ell^2$, and since containment of orders implies division of their sizes, we must have $disc(cO_{max}) = ell^2mid disc(R)$. My desired conclusion is that

begin{equation}
ellmid frac{d_1d_2-(d_1d_2-2trd(phi_1phi_2))^2}{4}in ZZRightarrow ellleq d_1d_2/4;
end{equation}

however, this requires that the integer is actually positive. I have tried a number of different methods to show this is positive, but to no avail.

My familiarity with quaternion algebras is minimal; so, perhaps there’s some well-known method of showing positivity, but I have had trouble.

Any help is appreciated. Thanks in advance!

How do I rotate the order of images when I click in JavaScript?

I have to show 3 animal pictures "cat", "dog" and "frog" in this order.
Whenever the user clicks on the "cat" image, a warning window will appear with the message "Do not click on me,
click on the frog ".
Whenever the user clicks on the "dog" image, a warning window will appear with the message "Do not click on me,
click on the frog ".
When the user clicks on the image "frog", the frog jumps over cat and dog. It means
In the beginning Cat Dog Frog is displayed, when the user clicks on the frog, it becomes frog
Cat dog, the user clicks on the frog again, then he becomes cat dog frog, etc.

Here is my code:

Function swapDiv (event, elem) {
elem.parentNode.insertBefore (elem, elem.parentNode.firstChild);
}

Move Gmail files in a different order

When I used Outlook, I could just grab the file (from the left column with all the listed files) and move it to any position in the list. That way I was able to leave my emails that were not ready for archiving but ready to be removed at the bottom of the list.

I can not figure out how to do that in Gmail.

Thanks – hope you can find out what I'm trying to say.

There was a Magento 2.3.1 exception while loading the order

To correct the problem in Magento 2.3.1. The order grid can not be opened in administrator mode. The following error message is displayed.

report.CRITICAL:
a: 1: {s: 19: "real_transaction_id"; s: 22: "5682952820116120004012";} Not possible
To unserialize value. Error: syntax error {"exception": "(object)
(InvalidArgumentException (Code: 0):
a: 1: {s: 19: "real_transaction_id"; s: 22: "5682952820116120004012";} Not possible
To unserialize value. Error: Syntax error at
/var/www/html/tupperware/vendor/magento/framework/Serialize/Serializer/Json.php:39) "}
,

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Group theory – Number of marked abelian groups of order n

I found https://oeis.org/A034382, but I can not get it (16).

To let $ C_n $ be a cyclic order group $ n $. $ Aut (G) $ be an automorphism set of $ G $,

The answer is
$ displaystyle { sum frac {N!} { # Aut (G)}} $ where G is isomorphic equivalence.

I have

$ displaystyle {
# Aut (C_ {p ^ n} ^ k) = p ^ {(n-1) k ^ 2} prod_ {j = 0} ^ {k-1} (p ^ {k} -p ^ {j })
} $

and

$ displaystyle {
# Aut ( prod_i C_ {p ^ {n_i}} ^ {k_i})
= prod_i left ((p ^ {(n_i-1) k_i ^ 2} prod_ {j = 0} ^ {k_i-1} (p ^ {k_i} -p ^ {j})) ( prod_ { j neq i} p ^ { min (n_i, n_j) k_j}) ^ {k_i} right)
} $
,

According to the principle of finite abelian groups, there are 5 groups for $ N = $ 16: $ C_ {16}, C_2 times C_8, C_4 ^ 2, C_2 ^ 2 times C_4, C_2 ^ 4 $,

So, the number of groups that is isomorphic …

  • $ C_ {16} $$ displaystyle { frac {16!} {8}} $
  • $ C_2 times C_8 $$ displaystyle { frac {16!} {(1 times 2) times (2 times 4)}} $
  • $ C_4 ^ 2 $$ displaystyle { frac {16!} {16 times 3 times 2}} $
  • $ C_2 ^ 2 times C_4 $$ displaystyle { frac {16!} {((3 times 2) times 2 ^ 2) times (2 ^ 2 times 2)}} $
  • $ C_2 ^ 4 $$ displaystyle { frac {16!} {15 times 14 times 12 times 8}} $

Sum of them is $ 4250979532800 $, OEIS says $ 4248755596800 $,

I could count correctly up to one (16) out of 20.

Where is wrong

Set theory – Density linear order without endpoints.

To let $ (L, prec) $ be dense linear order without endpoints. Show that $ (L, prec) $ is only separable if $ (L, prec) $ This order is isomorphic to a subordination of the real line $ ( mathbb {R}, <). $

I know that when $ (L, prec) $ is then separable $ (L, prec) $ is a countable dense linear order without endpoints. Then there is a real Dedekind cut after the Baire category set $ D subseteq L $ so that $ D $ avoids everyone $ X in mathcal F $ from where $ mathcal F $ is a countable family of closed nowhere dense subgroups $ (L, prec) $, But I do not know how to proceed from here.

Plugins – moving credit card fields from the order pay end point to the checkout page

I'd like to move the credit card fields that appear when you click the Register Now button on this page: https://seowzrd.com/?page_id=4088&lbjlp_checkout=true&first_name=test&last_name=test&address=testvej%201&zip=4180&city=roskilde&phone = 56789012 & email =testmail@gmail.com

I tried to change the template "thankyou.php" because some said that the endpoint "order-pay" could be edited from here – no luck.
I have tried several templates, and nothing seems to be right for the order pay-end point. We use PurePay.eu as a gateway.

Thank you in advance for your help.

WordPress single page template, custom post type, page pagination order

For a while I searched for a method to page on a single page template, sorted by title (the customer likes sorting the first names of team members).

I assume that this would include a way of paginating by title or? Equalize.

The following paginates quite well by mail, not by title.

') ?>
') ?>

How do you manually page / ask for a custom post type (orderby => title) offset by the title of the post that fills in the single page template? Thank you for any help.