Feedback request – WordPress Hosting by Namecheap

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reference request – Building algebraic geometry without prime ideals

$DeclareMathOperatorSpec{Spec}DeclareMathOperatorev{ev}$Teaching algebraic geometry, in particular schemes, I am struggling to provide intuitive proofs. In particular, I find it counter-intuitive that points are prime ideals. I discovered a trick which I suspect is not new. Basically, you build the functor of points into the definition. I want to modify the definition of $Spec(R)$ as follows:

As a set, $Spec(R)$ is simply all pairs $x=(k_x, ev_x)$ where $k_x$ is a field and $ev_x:Rto k_x$ is a homomorphism. Then as usual, elements of $R$ are called functions and the value of a function $fin R$ at a point $x$ is $f(x)mathrel{:=}ev_x(f)in k_x$. Then it continues as usual: closed set is where some collection of functions vanishes. Basic open set is where some function is invertible.

Of course, there are some problems with this approach:

  1. The class of all fields is not a set. Technically, we can limit ourselves to some very large set of “test fields”. So this can be swept under the rug.

  2. $Spec(R)$ with this definition is not $T_0$. But after getting used to spaces being not Hausdorff it should be easy to take it to the next level with spaces being not $T_0$. Of course, to every non-$T_0$ space there is a canonically associated $T_0$ space where you identify topologically indistinguishable points, so you recover the usual construction of $Spec(R)$ this way.

Nevertheless, I find this approach much more intuitive, because it seems like a natural question to solve some system of equations in some unknown field, rather then studying prime ideals (which is of course basically the same thing, language aside).

Is this not new? Are there any lecture notes following this approach? Of course, the full “functor of points” approach sort of contains this one, but notice that to do what I want I do not need Yoneda lemma, I do not ask for functoriality, so I do not need to sweep under the rug all the tedious checks of naturality. So I find it more basic than functor of points.

Here is an example. When we construct the localization of a ring $R$ with respect to a multiplicative set $S$ we prove that prime ideals of $S^{-1}R$ are in bijection with a subset of ideals of $R$. With this approach the corresponding statement is a simple consequence of the universal property of the localization, there is nothing more to prove.

Another example. Prove that the map $mathbb{A}^1to mathbb{A}^3$ given by $tto (t^3, t^4, t^5)$ has image $Z(xz-y^2, x^3-yz, x^2 y -z^2)$. This becomes simply high school algebra.

reference request – Decomposition of $otimes^{m} mathbb{C}^{n}$ under the action of $text{GL}_{n}times text{S}_{m}$

I want to know the proof of the following theorem. It is refereed somewhere that, a proof can be found in: ” Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and
beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ.,
Ramat Gan, 1995, pp. 1–182, DOI 10.1007/BF02771542. MR1321638 (96e:13006) “.

However, I am not able to access this. Could anyone help me out to find this paper, or may be, is there any other place where I can find the proof?

$textbf{Theorem :}$ Let $F^{lambda}_{n}$ denote the irreducible rational representation
of $text{Gl}_{n}$ with highest weight indexed by $lambda$. Let $W^{lambda}_{m}$ denote the irreducible complex representation of $text{S}_{m}$ indexed by $lambda$. Under the joint action of $text{GL}_{n}times text{S}_{m}$ on $otimes^{m} mathbb{C}^{n}$, we have the multiplicity free decomposition $$ otimes^{m} mathbb{C}^{n} cong bigoplus_{lambda} F^{lambda}_{n} otimes W^{lambda}_{m} $$
where the sum is over all partitions $lambda$ of $m$ with at most $n$ parts. Note that all
irreducible representations of $text{S}_{m}$ appear in the decomposition when $n geq m$.

VPSnet Has Monthly Deals for You on VPSes in Lithuania! (From $2.25/mo, Special Game Server Nodes by Request!)

VPSnet has been posting offers on LEB for the last four years.  Their 2020 Cyber Monday offer is here:

  • 1GB RAM w/10GB SSD and unlimited traffic on a 100mbps link for $2.25/month!
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Pretty cool to see these kinds of discounts on a monthly pay.  Usually we only see sales this deep on an annual commitment.  Also, if you request, they can host you on a Ryzen-based game server – see below for details.  All their host nodes run high frequency CPUs.

VPSnet has been around since 2007 – for perspective, that’s about the time the first iPhone debuted.  Here’s what they had to say about themselves:

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I’m Andrew, techno polymath and long-time LowEndTalk community Moderator. My technical interests include all things Unix, perl, python, shell scripting, and relational database systems. I enjoy writing technical articles here on LowEndBox to help people get more out of their VPSes.

Can a Postgresql trigger send a row to an api via a post request

I have an application where it would be really nice to be able to use an insert trigger on a table to format and send a post request to an external API. This would save me having to write the code to do this work in another application.

I have had a look around and can see some information about using Postgres in a RESTFUL manner, but this seems to be mainly aimed at the database being the server for the api, which in this case isn’t the direction I need.

I have thought about using NOTIFICATIONS to trigger an external app and have played with a nodejs app to handle these, which seems to work (as an aside, does anyone know of a way of subscribing to notifications from a golang app?) so that is an option too.

Any pointers appreciated.


reference request – Are gyrogroups useful for anything else other than the Einstein velocity addition rule?

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security – “A password reset request was sent…” System-level phishing?

Over the course of the last two days I have received upwards of ten notifications alerting me that a password reset request was sent from some device. When I go to check my Apple ID email, I see no notifications to this effect. When I click the “Show” button, I am taken to System Preferences and given the opportunity to type in my administrator password to reset my Apple ID password. I have not done so.

The full message of the notification reads: “A password reset request was sent from a device at the location shown below.”

I have a hard time believing this is a legitimate communication from Apple. The few others who have posted similar questions here and elsewhere have received unhelpful answers. Any idea what’s going on here? Has my system been compromised somehow?

10.15.7 (19H2)

"A password reset request..."

woocommerce – Product variations Edit / Add / Duplicate taking 10 min to process complete request

I am facing issue in saving , duplicating & updating the product that have 2000 variations , round around 10 min to complete the process of saving , duplicating or updating the product information.

Server is in AWS EC2 and RDS.

I will be very thankful if some one help me to fix .

Thanks & Regards
Harshit Varshney

reference request – LGV scheme for lattice paths that move in non-unit spatial positive steps

In the Lindström–Gessel–Viennot lemma (LGV) the lattice paths are taken to move in unit spatial-steps in unit time.

However, there are applications (here) where a version of LGV still “applies”(i.e. the LGV is used as an analogy) even though the paths are jumping in varying non-unit positive increments at each unit step time. In other words, a lattice path might jump two positive integers at time t: $P(t+1)-P(t)=2$ and three positive integers at some other time s: $P(s+1)-P(s)=3$.

So it would be interesting to read of work done in LGV/Vicious-walkers and its generalizations that possibly include non-unit step. Of course, once one drops the unit-step requirement, one must also work with a more general definition of “intersection”.

I was thinking maybe with the bijection to Young Tableaux, one can obtain a generalization in the Young Tableaux side even though there is no corresponding object at the Vicious walkers side.

8 – Is there a way to get the controller class from a Request object or any other service class?

Background: My controller class holds some not so easy logic to receive some data from a route parameter but in a block plugin I need this data. As this block will only be rendered on this specific route I would like to simply get the controller of the route from within of this block plugin and the receive this data from the controller.

Is there any way to get the controller class of the current route from any of the services like current_route_match? I can get the route object or the route name from current_route_match but I did not find a way to get the real controller class.