partial order – Showing that $F$ is a monotone function

I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Chapter 1.3 Data Flow Analysis says the following:

The least solution. The above system of equations defines the twelve sets
$$text{RD}_text{entry}(1), dots, text{RD}_{text{exit}}(6)$$
in terms of each other. Writing $overrightarrow{RD}$ for this twelve-tuple of sets we can regard the equation system as defining a function $F$ and demanding that:
$$overrightarrow{RD} = F(overrightarrow{RD})$$
To be more specific we can write
$$F(overrightarrow{RD}) (F_text{entry}(1)(overrightarrow{RD}), F_text{exit}(1)(overrightarrow{RD}), dots, F_text{entry}(6)(overrightarrow{RD}), F_text{exit}(6)(overrightarrow{RD}))$$
where e.g.:
$$F_text{entry}(3)(dots, overrightarrow{RD}_text{exit}(2), dots, overrightarrow{RD}_text{exit}(5), dots) = overrightarrow{RD}_text{exit}(2) cup overrightarrow{RD}_text{exit}(5)$$
It should be clear that $F$ operates over twelve-tuples of sets of pairs of variables and labels; this can be written as
$F : (mathcal{P}(mathbf{text{Var}_star times mathbf{text{Lab}_star}))}^{12} to (mathcal{P}(mathbf{text{Var}_star times mathbf{text{Lab}_star}))}^{12}$
where it might be natural to take $mathbf{text{Var}_star} = mathbf{text{Var}}$ and $mathbf{text{Lab}_star} = mathbf{text{Lab}}$. However, it will simplify the presentation in this chapter to let $mathbf{text{Var}_star}$ be a finite subset of $mathbf{text{Var}}$ that contains the variables occurring in the program $mathbf{S_star}$ of interest and similarly for $mathbf{text{Lab}_star}$. So for the example program we might have $mathbf{text{Var}_star} = { x, y, z }$ and $mathbf{text{Lab}_star} = { 1, dots, 6, ? }$.

It is immediate that $(mathcal{P}(mathbf{text{Var}_star times mathbf{text{Lab}_star}))}^{12}$ can be partially ordered by setting
$$overrightarrow{text{RD}} sqsubseteq overrightarrow{text{RD}}^prime text{iff} forall i : text{RD}_i subseteq text{RD}_i^prime$$
where $overrightarrow{text{RD}} = (text{RD}_1, dots, text{RD}_{12})$ and similarly $overrightarrow{text{RD}}^prime = (text{RD}_1^prime, dots, text{RD}_{12}^prime)$. This turns $(mathcal{P}(mathbf{text{Var}_star times mathbf{text{Lab}_star}))}^{12}$ into a complete lattice (see Appendix A) with least element
$$overrightarrow{emptyset} = (emptyset, dots, emptyset)$$
and binary least upper bounds given by:
$$overrightarrow{text{RD}} sqcup overrightarrow{text{RD}}^prime = (text{RD}_1 cup text{RD}_1^prime, dots, text{RD}_{12} cup text{RD}_{12}^prime)$$

It is easy to show that $F$ is in fact a monotone function (see Appendix A) meaning that:
$$overrightarrow{text{RD}} sqsubseteq overrightarrow{text{RD}}^prime text{implies} F(overrightarrow{text{RD}}) sqsubseteq F(overrightarrow{text{RD}})^prime$$
This involves calculations like
$$text{RD}_text{exit}(2) subseteq text{RD}_text{exit}^prime(2) text{and} text{RD}_text{exit}(5) subseteq text{RD}_text{exit}^prime(5)$$
$$text{RD}_text{exit}(2) cup text{RD}_text{exit}(5) subseteq text{RD}^prime_text{exit}(2) cup text{RD}_text{exit}^prime(5)$$
and the details are left to the reader.

Appendix A gives the following definition for monotone function:

The function $f$ is monotone (or isotone or order-preserving) if
$$forall l, l^prime in L_1 : l sqsubseteq_1 l^prime Rightarrow f(l) sqsubseteq_2 f(l^prime)$$

I am trying to do as the author said, and show that $F$ is a monotone function. However, I have so far been unable to make progress. It seems to me that such a proof should proceed by showing that, for some arbitrary element of the set of elements $F(overrightarrow{text{RD}})$, if we use the fact that $overrightarrow{text{RD}} sqsubseteq overrightarrow{text{RD}}^prime$, then we can deduce that said arbitrary element is also an element of the set $F(overrightarrow{text{RD}})^prime$, and so $F(overrightarrow{text{RD}}) sqsubseteq F(overrightarrow{text{RD}})^prime$. However, it seems to me that the textbook is very poorly written, and so it is difficult for me to even understand what said arbitrary elements of the set $F(overrightarrow{text{RD}})^prime$ even are (they seem to be some kind of cartesian product, but I get very confused when trying to figure out precisely what they are). So how is it shown that $F$ is a monotone function?

magento2 – Magento 2 change Ship button in Order functionality

I’m trying to change the functionality of Ship button when you choose an order.

The only thing that I have found so far is this file under /vendor/magento/module-sales/Block/Adminhtml/Order/View.php:

I see this part where the button has an onclick method:

                    'label' => __('Ship'),
                    'onclick' => 'setLocation('' . $this->getShipUrl() . '')',
                    'class' => 'ship'

and then this is the URL

public function getShipUrl()
        return $this->getUrl('adminhtml/order_shipment/start');

I don’t know if that url is a controller, I have not found a route with that id nor that exactly path. Or maybe it’s an api? This is the url that is shown on backend when I inspect


Hope someone can give me a hint!


Get Multiple Order details Magento 1

I am using Magento ver.

I am trying to get order detail’s for 6-7 orders, want to log their data into (var/log)

What’s the best way to do it ?

macos – How do I use AppleScript to open the first note in a folder, in a different order than most recently edited?

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magento2 – Magento 2 REST Api For Place Order gives “transaction declined” error after successful Payment For Payfort Payment Gateway

I am using Payfort extension to process credit Card Payments online for Payfort Payment Gateway. For Website it’s working Fine and Placing an order after successful payment.

But when i am trying to place an order via API after successful payment via it’s iOs/Android SDK.
It’s giving me this Error.

Transaction has been declined. Please try again later.

This is How i am passing each information i received from Payfort after Successful Payment in this API.




“billingAddress”: {
“city”: “Dubai”,
“countryId”: “AE”,
“customerAddressId”: “2885”,
“customerId”: “3438”,
“fax”: “+971521231234”,
“firstname”: “Test”,
“lastname”: “User”,
“postcode”: “1111”,
“region”: “Dubai”,
“regionCode”: “DXB”,
“regionId”: “597”,
“street”: [
“Street 44, Oud Maitha”
“telephone”: “+971521231234”
“cartId”: “24361”,
“paymentMethod”: {
“additional_data”: {
“amount”: “29995”,
“authorization_code”: “614835”,
“card_number”: “411111******1111”,
“card_holder_name”: “Test User “,
“customer_email”: “”,
“customer_ip”: “”,
“expiry_date”: “2102”,
“fort_id”: “169996200000452654”,
“is_active_payment_token_enabler”: false,
“merchant_reference”: “XXXXXXXX”,
“payment_option”: “VISA”,
“sdk_token”: “c85bf903408b45a19a194710a941607a”,
“token_name”: “82f62b316c3246908ba29c8f2e683f45”
“method”: “md_payfort”

I am getting “True” in the response of this API call.

Then, I am using Create Order API of Magento to Place an Order.




“paymentMethod”: {
“method”: “md_payfort”
“shippingMethod”: {
“additionalProperties”: {},
“carrier_code”: “freeshipping”,
“method_code”: “freeshipping”

After debug, i found that this error is coming from this file.


And it’s because the “execute” method in this class is sending the Payment request again to Payfort and they recieve the “Signature mismatch” error from Payfort because of the unauthorized request. It should not send the Payment request again to Payfort.
I am not able to figure out how to prevent Magento to make this Payment request and just Place the Order.

If anyone can help me out to solve this problem and place an order successfully would be really appreciated.

abstract algebra – Characterization of ${rm PSU}(4,2)$ by its order $|{rm PSU}(4,2)|$ and the number $n_5({rm PSU}(4,2))$ of its Sylow $5$-subgroups?

As we know that $|{rm PSU}(4,2)|=25920=2^6.3^4.5$, and the number $n_5({rm PSU}(4,2))$ of Sylow $5$-subgroups of ${rm PSU}(4,2)$ is $1296=2^4.3^4$.

In paper “Two new characterizations for sporadic simple groups”, Amir Khosravi and Behrooz Khosravi proved that “every sporadic simple group $S$ is uniquely determined by its order $|S|$ and the number of its Sylow $p$-subgroups $n_p(S)$, where $p$ is the largest element of $pi(S)={p|p~{rm is~a~divisor~of}~|S|}$” in “Theorem 3.2”. I just want to prove the same result for ${rm PSU}(4,2)$. But I can not follow “Theorem 3.2”. In the first paragraph of the proof of “Theorem 3.2” it says that “… and so $|G/K|$ is a divisor of $n_p(G)$“. I can not catch the idea of this sentence. May someone give me some more detailed expanation?

We can download this paper at

magento2 – getting error of 400 on creating shipment for an order magento 2.3.1p

showing error while sending this request

/order/330/ship, METHOD=POST, HEADERS={Authorization=Bearer *******************, Host=
.in, Accept=application/json, Content-Type=application/json},
“,”title”:”self”}]}, PROXY-HOST=null, PROXY-PORT=null]

Http Sender warning url:, status code:400

drush – Drupal root not found. Pass –root or a @siteAlias in order to see Drupal-specific commands

I have multiple sites inside the same htdocs directory. In one, I removed all the folders except vendor under Web. Now drush will not work (it worked before I moved the folders). But now, it cannot find the database, which is correctly defined in web/sites/default/settings.php

/srv/www/htdocs/jar/drupal # ./drush status
 PHP binary    : /usr/bin/php
 PHP config    : /srv/www/php.ini
 PHP OS        : Linux
 Drush script  : /srv/www/htdocs/jar/drupal/vendor/drush/drush/drush
 Drush version : 10.3.6
 Drush temp    : /tmp
 Drush configs : /srv/www/htdocs/jar/drupal/vendor/drush/drush/drush.yml
 Drupal root   : /srv/www/htdocs/jar/drupal

I am running Drupal 9.0.8 and drush 10.3.6 (in /srv/www/htdocs/jar/drupal). I have removed drush using composer (2.0.3) and reinstalled it (recreating vendor in the process).

/srv/www/htdocs/jar/drupal # ./drush -vvv cr
 (preflight) Config paths: /srv/www/htdocs/jar/drupal/vendor/drush/drush/drush.yml
 (preflight) Alias paths: /srv/www/htdocs/jar/drupal/drush/sites,/srv/www/htdocs/jar/drush/sites
 (preflight) Commandfile search paths: /srv/www/htdocs/jar/drupal/vendor/drush/drush/src
 (debug) Starting bootstrap to site (0.04 sec, 8.17 MB)

In BootstrapHook.php line 32:
  Bootstrap failed. Run your command with -vvv for more information.
Exception trace:
  at /srv/www/htdocs/jar/drupal/vendor/drush/drush/src/Boot/BootstrapHook.php:32
 DrushBootBootstrapHook->initialize() at /srv/www/htdocs/jar/drupal/vendor/consolidation/annotated-command/src/Hooks/Dispatchers/InitializeHookDispatcher.php:44
 ConsolidationAnnotatedCommandHooksDispatchersInitializeHookDispatcher->doInitializeHook() at /srv/www/htdocs/jar/drupal/vendor/consolidation/annotated-command/src/Hooks/Dispatchers/InitializeHookDispatcher.php:36
 ConsolidationAnnotatedCommandHooksDispatchersInitializeHookDispatcher->callInitializeHook() at /srv/www/htdocs/jar/drupal/vendor/consolidation/annotated-command/src/Hooks/Dispatchers/InitializeHookDispatcher.php:29
 ConsolidationAnnotatedCommandHooksDispatchersInitializeHookDispatcher->initialize() at /srv/www/htdocs/jar/drupal/vendor/consolidation/annotated-command/src/CommandProcessor.php:145
 ConsolidationAnnotatedCommandCommandProcessor->initializeHook() at /srv/www/htdocs/jar/drupal/vendor/consolidation/annotated-command/src/AnnotatedCommand.php:296
 ConsolidationAnnotatedCommandAnnotatedCommand->initialize() at /srv/www/htdocs/jar/drupal/vendor/symfony/console/Command/Command.php:221
 SymfonyComponentConsoleCommandCommand->run() at /srv/www/htdocs/jar/drupal/vendor/symfony/console/Application.php:1018
 SymfonyComponentConsoleApplication->doRunCommand() at /srv/www/htdocs/jar/drupal/vendor/symfony/console/Application.php:271
 SymfonyComponentConsoleApplication->doRun() at /srv/www/htdocs/jar/drupal/vendor/symfony/console/Application.php:147
 SymfonyComponentConsoleApplication->run() at /srv/www/htdocs/jar/drupal/vendor/drush/drush/src/Runtime/Runtime.php:118
 DrushRuntimeRuntime->doRun() at /srv/www/htdocs/jar/drupal/vendor/drush/drush/src/Runtime/Runtime.php:49
 DrushRuntimeRuntime->run() at /srv/www/htdocs/jar/drupal/vendor/drush/drush/drush.php:72
 require() at /srv/www/htdocs/jar/drupal/vendor/drush/drush/drush:4

I looked at all the above suggestions, but am stuck. Drush works perfectly on my other 5 sites…

How to order posts by meta_value created inside loop?

Inside the WP_Query, I’m trying to order posts by price.

The problem is that the price value is not manually defined inside an ACF or so, but it comes from an api call made while looping trough posts, like this:

<?php while ( have_posts() ): the_post();

$asin = get_field("asin");?>

<h3><?php the_title(); ?></h3>
<p><?php echo aawp_get_field_value($asin, 'price'); ?></p>

<?php endwhile; ?>

That means that the numeric value (the price) isn’t available before actually running $the_query, so I cannot use something like:

'meta_key' => 'price',
'orderby' => 'meta_value_num',
'order' => 'ASC'

because “price” does not exists at that time.

I searched a lot but I didn’t find a solution to this. Maybe I need to order posts after the loop?
How would you do that?

Thanks in advance!

postgresql – Case Insensitive ORDER BY clause using COLLATE

I have spent a long time looking for this, and I am getting mixed messages.

In other DBMSs (tested in SQLite, Oracle, MariaDB, MSSQL) I can override the default sort order using the COLLATE clause:

FROM orderby

--  SQLite:     BINARY | NOCASE
--  MariaDB:    utf8mb4_bin | utf8mb4_general_ci
--  Oracle:     BINARY | BINARY_CI
--  MSSQL:      Latin1_General_BIN | Latin1_General_CI_AS

I have pored over the documentation and searched high and low, but I can’t find anything so straightforward for PostgreSQL.

Is there a COLLATE clause value that would sort Case Insensitive?

I know there are many questions regarding case sensitivity, but (a) most of them are old and (b) none that I have seen relate to the COLLATE clause.

FWIW, I am testing on PostgreSGL 11.8. I have a test fiddle on!17/05cab/1, but it’s only for PostgreSQL 9.6.