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Proof of the isomorphism $\textrm{Hom}_{A-mod}(\textrm{Hom}_{A-mod}(A,A),A)\cong A$ for any ring $A$

Let $A$ be a commutative ring with identity. Then we have an isomorphism $$A \rightarrow \textrm{Hom}_{A-mod}(A,A), a \mapsto (x \mapsto ax)$$, whose inverse is $$\textrm{Hom}_{A-mod}(A,A) \rightarrow A, \varphi \mapsto \varphi(1).$$Then applying this isomorphism, we get $$\textrm{Hom}_{A-mod}( \textrm{Hom}_{A-mod}(A,A) ,A) \cong \textrm{Hom}_{A-mod}(A,A) \cong A.$$

Differences between localStorage, sessionStorage and cookies

We can store tokens and user information in localStorage, sessionStorage, or cookies. They are all ways of browser that store data locally. In browser, open the developer mode, and click Application. Then you can view the messages stored in the localStorage, sessionStorage, and cookies on the website.Here are some differences between LocalStorage, SessionStorage, and Cookies:Data in LocalStorage have no lifecycle which means it will never expire, and need to be manually deleted. It is merely sto ...

Difference between time and datetime in python

Manipulating time is common during development. Python provides two packages from time, which are datetime and time. We will discuss the difference between the basic functions of these two packages.1. time :To get the current local time, first you need to run time.time() to get the current timestamp.import time #Get the current timestamp time.time() #output: 1715254313.7382145Then you need to format the resulting timestamp to get your desirable time format, e.g. %Y-%m-%d %H:%M:%S.time.localtime ...

moment.js/day.js UTC time to local time

import moment from 'moment'; const time = xxx; moment.utc(time).local().format("YYYY-MM-DD HH:mm:ss") In Nuxt.js, you need to change moment.js to day.js.const dayjs = useDayjs(); const time = xxx; dayjs.utc(time).local().format("YYYY-MM-DD HH:mm:ss")

Vue add whitespaces in a string

In Vue.js, if you simply add multiple whitespaces in a string, it eventually displays a single whitespace.//This will display a single whitespace. 'ab c'If you want to display multiple whitespaces, you could use unicode to represent whitespace.let str1 = 'ab\u00A0\u00A0\u00A0 c'; let str2 = 'ab\u0020\u0020\u0020 c'; let str3 = 'ab\u3000\u3000\u3000 c';These will output multiple whitespaces in strings.

Vue - TypeError: Cannot read properties of null (reading 'insertBefore')

In Vue 3/Nuxt 3, one may occur the error TypeError: Cannot read properties of null (reading 'insertBefore') which has little information and is hard to solve. In the following, we summerize some tips that help you solve this error :1. Change v-if to v-show , or change all v-if , v-else to v-show. Sometimes, the error appears because there are v-if or v-if, v-else pairs.2. Change katex auto-render. The error may appear because you use katex auto-render. To solve this, you could either change kate ...

Keep user logged in using Django and Vue 3

The request.session object will automatically generate a cookie with the default name sessionid, which stores the session's session key value. When the session expires, the cookie will automatically removed. Only when the cookie is cleared can the logged in user log in again.Set the expiration time of the session through the backend, and when the time is up, you can see through the browser that the cookie will automatically disappear. Therefore, it is only necessary to set that the token value i ...

Invalid prop: type check failed for prop "title". Expected String with value "500", got Number with value 500.

Problem.I'm using Nuxt 3.11.1 and my Node version is 20.12.2. And one of my pages unreasonably throw out the misleading error :Invalid prop: type check failed for prop "title". Expected String with value "500", got Number with value 500.I had tried to remove all related "title", but the error was still there. Solution.The error is due to useFetch , just change useFetch to useAsyncData .

Vertically aligning CSS :before and :after content

I am trying to centre the link with the image, but can't seem to move the content vertically in any way.<h4>More Information</h4> <a href="#" class="pdf">File Name</a>The icon is 22 x 22px.pdf { font-size: 12px; } .pdf:before { padding:0 5px 0 0; content: url(../img/icon/pdf_small.png); } .pdf:after { content: " ( .pdf )"; font-size: 10px; } .pdf:hover:after { color: #000; }

Some chunks are larger than 500 KiB after minification

When your vue/nuxt project is large, running npm run build  may output the error : WARN 15:32:15 (!) Some chunks are larger than 500 KiB after minification. Consider: - Using dynamic import() to code-split the application - Use build.rollupOptions.output.manualChunks to improve chunking: https://rollupjs.org/guide/en/#outputmanualchunks ...

Note on perfectoid spaces

In this section, we focus on Section 2 in [Sch], following [Hu], [Hu1], and [Hu2]. Moreover, we need to compare Huber's adic spaces with Berkovich's analytic spaces and Tate's rigid analytic spaces. Hence, we will briefly introduce the notion of Berkovich's analytic spaces in §1.3 and the notion of rigid analytic varieties in §1.4.§1. Adic SpacesDefinition 1.1.  A morphism $f:X\rightarrow Y$ of adic spaces is adic if, for every $x\in X$, there exist open affinoid subspaces $U,V$ of $X,Y$ wi ...

npm run build error: Heap out of memory in Nuxt3

When running npm run build , an error may occur which looked like :<--- Last few GCs ---> [2656:000001DF58927940] 511573 ms: Mark-sweep 2013.9 (2091.6) -> 2012.7 (2091.4) MB, 1022.4 / 0.0 ms (average mu = 0.211, current mu = 0.009) allocation failure; scavenge might not succeed [2656:000001DF58927940] 513611 ms: Mark-sweep 2028.4 (2091.4) -> 2027.2 (2117.4) MB, 2021.5 / 0.0 ms (average mu = 0.083, current mu = 0.008) allocation failure; scavenge might not succeed <--- JS s ...

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