Proving that a language defined by a regular expression is equivalent to a right linear grammar

After thinking for a bit, I am not able to prove a double inclusion proof for the following problem. It seems very interesting to me.

Consider the regular expression $r= ((1(00)^∗1 + 0)1)^∗$ and the right-linear grammar $G= ({S,A},{0,1},S,P)$, where $P$ consists of the following rules:

$Srightarrow 1A|01S|lambda$

$Arightarrow 00A|11S$

Prove that $L(G)subseteq L(r)$ and vice versa.

In general, how exactly do I prove that a regular grammar describes the same language as a regular expression?

plotting – Minimizing a function that is not defined everywhere

So I have the function

7.8*10^11/x^3 + 1600./x^(4/3) + (32266.7 .19^2 E^(-200. .19 .19 (Pi) y) .19(Pi)^2 Sqrt(y))/V - (4.4*10^8 .19 E^(-100. .19 .19(Pi) y) .19(Pi) y)/x^2

or $$ frac{7,8 times 10^{11}}{x^3} + frac{1600.}{x^{4/3}} + frac{32266.7^2 e^{-200. pi y} pi^2 sqrt{y}}{x} – frac{4.4 times 10^8 e^{-100. pi y} pi y}{x^2}. $$

I am not sure why there is a dot after some numbers, it came after some substitutions.

Anyway, now I want to minimize this function.

Minimize(7.8*10^11/x^3 + 1600./x^(4/3) + (32266.7 .19^2 E^(-200. .19 .19 (Pi) y) .19(Pi)^2 Sqrt(y))/V - (4.4*10^8 .19 E^(-100. .19 .19(Pi) y) .19(Pi) y)/x^2, {x, y})

But it returns the error: NMinimize::nnum saying that the function is not a number at {x,y} = {0.918621,0.716689}. However, substituting these values back in the function actually yields a number?

Even if the function would not be a number at that value for x and y, how would I minimize it?

dnd 5e – where are aquatic elves “defined”

Mordenkainen’s Tome of Foes pages 62-63.

A description and set of racial traits for Sea Elves is found on pages 62 and 63 of Mordenkainen’s Tome of Foes, as well as a brief description (including racial traits) on pages 163-164 of Explorer’s Guide to Wildemount.

This information can also be found here at DNDBeyond, but requires that you purchase MToF or at least the Sea Elf subrace.

A basic monster stat block for a sea elf can be found on page 70 of the adventure Storm King’s Thunder.

complex analysis – Function defined by equations is entire

Prove that if $$f(z) =
dfrac{cos z}{z^2-(pi/2)^2}, & text{if }zneq pm pi/2 \
-dfrac{1}{pi}, & text{if } z=pm pi/2
then $f$ is an entire function.

I wrote Taylor series for $cos z$ about the point $pi/2$ and I got the following:
$$cos z=-left(z-frac{pi}{2}right)+frac{1}{6}left(z-frac{pi}{2}right)^3+dots;$$
which is valid for any $zin mathbb{C}$. Also we can write $dfrac{1}{z^2-(pi/2)^2}$ as $dfrac{1}{(z-pi/2)(z+pi/2)}$. I can write that $dfrac{1}{z+pi/2}$ as a power about the point $pi/2$ but this is valid for $|z-pi/2|<pi$.

Can anyone please show how to solve this question, please?

SQL Server sp_describe_first_result_set: find user defined types

I need to find the column definitions from a stored procedure and to do so I’m using the sp_describe_first_result_set procedure. Is there a way to find the User Defined Type from that procedure result? At the moment I’m only able to find the system data type.

google sheets – ReferenceError: OFFSET is not defined

When running the function:

function updateCells() {
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var spreadsheetSummary = ss.getSheetByName('Summary');
  const range = spreadsheetSummary.getDataRange()
  const values = range.getDisplayValues()
  const transposeCheck = values(0).map((_, iCol) => => row(iCol)).some(cell => cell))
  var countBoolean = transposeCheck.filter(Boolean).length + 1
  var spreadsheetCommunityGroups = ss.getSheetByName('Community Groups');
  var cellValueCommunityGroups = spreadsheetCommunityGroups.getRange('A12').getValue();
  var summaryCommunityGroups = OFFSET('B'+countBoolean, 0, 1);

I receive the error ReferenceError: OFFSET is not defined.

magento2 – Magento 2 – How does require know where jquery is defined?

You can use jquery in phtml like this:

),function($) {
     $(document).ready(function() {
        /* Your logic */

But how does the require know where to find jquery.
I searched the whole codebase, but the path to the jquery file is nowhere defined in no requirejs-config.js file, so how does it know where the file is located?

color scheme – User defined pallete for web pages

When the web browsers first came into existence, it was possible to override the intent of the web designer. There weren’t too many settings. You could locally control the color and font of text, unfollowed links, followed links, and the background. This was a continuation of a tradition of command line interfaces that allowed users to achieve settings they found comfortable. As web design developed, using these settings became impractical. I am not even sure if they still exist.

This page advocates for allowing uses turn off dark mode. However the added cognitive burden of doing so my be enough for the user to lose interest and move on. I am interested in exploring the idea that users could consistently view content in a palette they find most comfortable.

Are there any frameworks in existence or development that would allow a user to select a palette they are comfortable with and have it applied to all the content they view?

8 – New language defined and set to default, original language on config translation still remains English

I have installed languages and set one of them as default, but still numerous things, for example Translations for User account menu block (admin/structure/block/manage/bartik_account_menu/translate) has English as original language. What can cause this?

This is an outdated version 8.9.11 which I’m not using in production, but I’m desperately trying to understand why this is happening on this installation, unlike on others where “original language” nicely becomes whatever has been set as the default language, as this is ruining my clever plans on exporting and importing some configs.

ag.algebraic geometry – $F$-isocrystals defined via a lift of a scheme

Let $X$ be a smooth affine scheme over a finite field $k$. Then there exists a smooth affine formal scheme $mathfrak{X}$ over $W(k)$ with a lift $sigma$ of the Frobenius. A convergent $F$-isocrystal on $X$ is a vector bundle $mathcal{V}$ on $mathcal{X}_{K}$ with an integrable connection and an isomorphism $sigma^*mathcal{V}cong mathcal{V}$, where we see $mathcal{X}_K$ as a rigid analytic space.

This is probably naive, but if we have a scheme-theoretic lift to $W(k)$ , that is to say a scheme $mathcal{X}to W(k)$ such that $mathcal{X}times_{W(k)} kcong X$, with a lift of Frobenius $F$, is giving a vector bundle $mathcal{E}$ on $mathcal{X}_K$ with an integrable connection and an isomorphism $F^*mathcal{E}cong mathcal{E}$ equivalent to the datum of a convergent $F$-isocrystal?

One reason why I’d like this to be true would be that in that case presumably we would have
$$H^i_{rig}(X,mathcal{V})cong H^i(mathcal{X}_K,text{DR}(mathcal{E}))$$
extending the comparision of crystaline cohomology with the de Rham cohomology of a lift (under favourable conditions). The reason why I hope this to be true is that we have rigid GAGA, but I’m not sure.