reference request – On the Hochschild cohomology of the minimal model of an $A_infty$ algebra

Suppose $(A, (mu_k))$ is a (curved) $A_infty$ algebra, and let $(tilde A, (tildemu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(tilde A)$. (It may be also related to $HH^*(A,tilde A)$ or $HH^*(tilde A,A)$ for the diagonal $A_infty$ bimodules.)

Question: Are there any natural relations between $HH^*(A)$ and $HH^*(tilde A)$ ? For instance, does there exist a ring homomorphism between them? Any reference would be greatly appreciated.

reference request – Number of homeomorphisms between spaces

I need help with the following.

We know that those spaces that have a trivial group of homeomorphisms are those called rigid, there are even some references in the group about them here.

My first question was about the existence of spaces that are not homeomorphic to more than one space and I think that with the above we have an answer. My question is, will you have any reference where it talks about spaces that admit only 2 homeomorphic spaces to it, or in general, spaces that only admit $n$ homeomorphic spaces to it?

reference request – Applications of ZFA-Set Theory

The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements.

I have thusfar found two applications of ZFA-Set Theory. Firstly, ZFA can be used to prove the independence of the Axiom of Choice.

Secondly, one can show a correspondence between certain ZFA-models and a certain class of groups.

Both of these applications of ZFA make use of so-called permutation models.

I’d like to know what other applications ZFA might have. I would particularly be interested in applications that do not make use of permutation models.

testnet – Blockcypher new transaction request returning more than input and output

I am new to Bitcoin and Blockchain, and Blockcypher..

I am trying to use Blockcypher APIs to create (and sign) and send a transaction (on testnet). The request i provide has 1 input and 1 output, but the response i receive has 2 inputs/outputs/tosign 50% of the time.. Why would this be? Thanks

Sample Request:

Posting to

    "double_spend": false,
    "inputs": ({
        "addresses": ("mqtDjKw8wCyPBvk61NfkgmNFZDwHvCJTUD")
    "outputs": ({
        "value": 2,
        "addresses": ("tb1qfefhgu4npfqjk36v66eqk6ee0gska720j09nvr")


    "tx": {
        "hash": "b520ccb4b9ab96075c3cf1a91b0c7710a22332f0bb4d25409bcf9693a35477dd",
        "block_height": -1,
        "addresses": ("mqtDjKw8wCyPBvk61NfkgmNFZDwHvCJTUD", "tb1qfefhgu4npfqjk36v66eqk6ee0gska720j09nvr"),
        "total": 55000,
        "fees": 30000,
        "relayed_by": "",
        "received": "2021-07-22T10:19:47.853381661Z",
        "ver": 1,
        "vin_sz": 2,
        "vout_sz": 2,
        "confirmations": 0,
        "preference": "high",
        "double_spend": false,
        "inputs": ({
            "prev_hash": "34e749b34cc75b996e7b28638cc4648958942b046bf9961837af7ab0046e53d1",
            "output_index": 0,
            "output_value": 10000,
            "addresses": ("mqtDjKw8wCyPBvk61NfkgmNFZDwHvCJTUD"),
            "script_type": "pay-to-pubkey-hash"
        }, {
            "prev_hash": "106ad632e2474e683b2180181c1b911bc4d5727db5ab341ee8aded3b78fb2da7",
            "output_index": 1,
            "output_value": 75000,
            "addresses": ("mqtDjKw8wCyPBvk61NfkgmNFZDwHvCJTUD"),
            "script_type": "pay-to-pubkey-hash"
        "outputs": ({
            "value": 2,
            "script": "00144e537472b30a412b474cd6b20b6b397a216ef94f",
            "addresses": ("tb1qfefhgu4npfqjk36v66eqk6ee0gska720j09nvr"),
            "script_type": "pay-to-witness-pubkey-hash"
        }, {
            "value": 54998,
            "script": "76a91471b6e0ce39475cbe5cd43966f5c7eb80a886952088ac",
            "addresses": ("mqtDjKw8wCyPBvk61NfkgmNFZDwHvCJTUD"),
            "script_type": "pay-to-pubkey-hash"
    "tosign": ("c0859bbf0f964af14c01d9ff2da9e5254279d02c5c75c902e3a8180db6bb67e5", "91a6c5b3003047c0af81348f12867918ebc4efd70f3ff1a5ee9e4566bf361de8"),
    "signatures": (),
    "pubkeys": ()


reference request – connectivity function within a neighborhood in lattice percolation

Consider independent bond percolation on $mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster. Let $N_d$ be the set of all points within distance $d$ of the origin in $mathbb{Z}^2$. What is known about the joint distribution of $(Y_x)_{xin N_d}$, rigorously or heuristically?

hybrid solution – HTTP Error 400. The request hostname is invalid using SharePoint Application Proxy

Has anyone else experienced this error after completing the SharePoint Application Proxy deployment?

I have the Proxy connector set up and I have added all the required information in my Azure Active Directory(Internal URL and SPN for SSO). When I run the Test Application Proxy Configuration report, I have all green checkmarks(picture below). So I am confused as to why I am getting this error.

enter image description here

internal dns – Windows 10 / Chrome: HTTP browser request to IP address differs from same request to hostname

Agaoin, this is on Windows 10. I’m bringing in a new Ubuntu test web server ( to our internal network, which is all in the domain. There’s a DNS server on another machine that handles all DNS lookups.

PC configuration for DNS is:

DNS configuration

When I ping the web server by name, I get

DNS resolution

I have rebooted within the last five minutes.

Now, at a Chrome or Firefox or even Edge (which I usually avoid like the plague) web browser, when I access the server by name, I get a failure:

Failed when using hostname

But when I access via IP address, I have success:

Success via IP address

So it looks to me like the DNS lookup either isn’t happening at ANY of the web browsers, or they’re getting results different from the commandline ping. How can I diagnose this, and – more to the point – how can it be fixed?

hyperbolic geometry – Reference request: Discrete subgroup of $mathrm{PO}(n,1)$ preserving proper subspace has infinite covolume

I’m looking for a reference for the following claim:

Let $PP$ denote the group of isometries of $V = mathbb{R}^{n,1}$ preserving the upper sheet of the hyperboloid (i.e. $PP$ is the group of isometries of hyperbolic $n$-space), and $G$ a discrete subgroup of $PP$.

Claim. If $G$ preserves a proper (linear) subspace of $V$, then $G$ is of infinite covolume in $PP$.

The way I think a proof goes is:

  • If $G$ preserves a subspace $U$, it preserves its orthogonal complement $U^perp$ too.
  • Either $U$ or $U^perp$ either intersects the hyperboloid or contains an isotropic vector.
  • Up to swapping $U$ and $U^perp$, if $U$ intersects the hyperboloid, it preserves a proper subspace of hyperbolic space, and is therefore of infinite covolume.
  • Similarly, if $U$ only contains an isotropic vector, it fixes an ideal point of hyperbolic space, and must also be of infinite volume.

I’m not 100% convinced by this argument, and I looked in Ratcliffe’s and Vinberg’s book for a better one and/or a reference, to no avail.


node.js – API request returning status code 404 not found

Im making a fantasy stock app. when making a valid request to the API, the data gets returned as expected. If a stock does not exist, the server returns “Request failed with status code 404”. I am trying to handle the error by logging the status to the console but the app does not catch the 404 error and the request stays pending. I cant find a way to handle the error so that it exits the promise to show the user that the stock is invalid.

my backend is as follows

app.get('/api/stocks', (req, res) => {
    const stockName = req.query.stockName
        method: 'get',
        url: `${stockName}/quote?token=${testSecretKey}`
        .then(response => {
        }).catch(err => {

my front end

    handleSubmit = async e => {
        if (this.state.searchTerm !== '') {
            await axios.get("api/stocks", {
                params: {
                    stockName: this.state.searchTerm
                .then((res, req) => {
                    if (res.status === 200) {
                        this.setState({ stock:, searchTerm: '', isStockValid: true });
                    } else {
                // .catch(() => { console.log('error 404') });
            } else {
                alert('Enter a stock symbol. Example: AAPL for Apple inc.');

javascript – Why is the fetch request to AWS Express server failing?

I have upload an express app to an AWS Elastikbeanstalk instance. When I enter the production url into chrome it returns a JSON with the data I want. Here is that code:

const express = require("express");
const axios = require('axios');
const app = express();
const cors = require('cors');
const port = process.env.PORT || 5000;


app.listen(port, () => {
    console.log("listening on " + port);

app.get("", (req, res) => {
    let login = process.env.USER
    let pass = process.env.PASS
        method: 'get',
        url: process.env.URL,
        auth: {
            username: login,
            password: pass
    }).then(function (response) {
      .catch(function (error) {

So just one express route that gets a JSON from an API using axios. Again when I enter the production URL into chrome or test this locally it works.

Then I make a fetch request from my wordpress front end to the production URL all I get is

TypeError: Failed to fetch

after a few second delay. Here is that fetch request:

//Load cards onlick with API
function wpb_hook_javascript() {
         function loadCards(){
             fetch('<The-URL-of-AWS-SERVER', {
                 method: 'GET'
                .then(response => console.log(response))
                //.then(data => console.log(data))
                .catch((error) => {
                    console.error('Error:', error);
add_action('wp_head', 'wpb_hook_javascript');

What am I doing wrong?