I need a code for thermodynamics research that generates a matrix of 32768 by 32768 elements. The code is the followng: s0[MicroNumber_, k_] := If[RealDigits[MicroNumber, 2, 15][[1]][[k]] == 0, -1, 1] s0[MicroNumber_, 0] = 1; s0[MicroNumber_, 16] = 1; s = Table[s0[i - 1, j], {i, 1, 32768}, {j, 0,...

We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no symmetries, no matter how fundamental, would hold? Any special kind of metric, geometry or shape?

I remember a story about a man who bets "his head" with the devil and loses. I do not remember what the wager actually was. The devil is ready to chop the man's head off with an axe, but the man says he bet his head and not his neck, so the devil better take all of the head but none of the neck, ...

hi im new to blender and i want to know if i can paint in different color both side of this faces without using any shader, i tried uv unwraping but always when i paint one side of the face the other is painted too this is the object im trying to paint.

I have a Curve object which I would like to use as a base curve for a GeoNode group; essentially using an existing curve with the curve nodes. This feels like it should be a pretty common situation: wanting to use an existing curve as opposed to the basic primitives, but it seems impossible? From...

I'm trying to understand how to properly structure a C++ project by using CMake, googletest, and gcov for test coverage. I would like to build a general CMakeLists.txt that would work for any platform/compiler. This is my first attempt. However, if I try to build the project and then run lcov (to...

What is known about the classification of knots in a solid torus $S^1 \times D^2$? Is enumerating them a reasonable problem? Do we get a similar classification as for knots in $S^3$? Ideally there would be a simple description of the Seifert fibered knots in $S^1 \times D^2$ like for prime non-s...

Does a complete type contain a complete theory i.e. are sentences considered formulas with a free variable in the definition of a complete type?

The employees of corporate "Clocks" have a fixed rule for working hours. All the employees start working at 16:00 and finish their work by 21:00. They take exactly one break during that time. All the employees stop working when their work is half done and at the same time both the minute and the ...

Any help getting the answer shown here? I have tried adding the complex impedance of L and R in series and then adding that to C in parallel. I'm not super sure about that last step, and then attempting to simplify that. I get annoyingly close, having the two terms shown but they are the denomin...

I'm going through a stats intro and got puzzled by the concept of interval and ratio data. Celsius temps is an example of interval data that can not be multiplied and divided, becasue there is no true 0. Yet, a simple linear transformation changes Celsius into Kelvin, where 0 is defined. Kelvin t...

As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting). When it comes to computing $\log P(x)$, one may use the formula $$ (\log P)' = \frac{P'}{P} $$ to compute the expansion of the logarithm of $P(x)$ with $P(0)=1$ as $$ \log P \eq...

The new book I translate is composed by two volumes A and B. Each volume must have its own index and table of contents and page numbering starting from 1 but of course the cross references between the two volumes must be valid. Lets say that the whole book has (as an example) 8 chapters and each ...

I'm following the examples for ESP32: AsyncWebServer _server(80); _server.on("/", HTTP_GET, [](AsyncWebServerRequest *request) { request->send(SPIFFS, "/index.html", String(), false); }); _server.on("/css/index.css", HTTP_GET, [](AsyncWebServerRequest *request){ request->send(...

Given $x_1=2$ and $x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2}, n\geq 2$ Prove that $y_n=\sum_{k=1}^{n}\frac{1}{x_k^2-1}, n\geq 1$ converges and find its limit. To prove a convergence we can just estimate $x_n > n$, therefore $y_n<z_n$, where $z_n=\sum_{k=1}^{n}\frac{1}{k^2-1}$ and $z_n$ conver...

In The Two Towers, chapter five, soon after Gandalf has returned as Gandalf the White, he says something very peculiar to Aragorn, Gimli and Legolas in Fangorn: ‘You have not said all that you know or guess, Aragorn my friend,’ he said quietly. ‘Poor Boromir! I could not see what happened to him...

Can someone that doesn't speak cantonese nor mandarin get away with watching the movie without subs? I ask because I am in a foreign country where the movie is shown in English but the subs are in the native language.

I understand that an extra Feat is a big deal, but it seems to me that given the attributes changes put everything in line with V. Humans now, that feat isn't a big deal any more since most of the new races have things that are at least that good, but also tend to get another benefit or two as we...

I need to make object with multiple materials, but some parts need to be "irregular" shape. I attached screenshot to explain better what I need to do. What is the best way to select part of object surface (similar to one on screenshot) so I can apply secondary material (mostly that would be some ...

I was looking again at the periodic trends like Ionization Energy, Electron Affinity, Radius and so on. I understood all the general trends and the exeptions for these ones, but I can't fully understand the trends for Atomization Enthalpy/Energy. Quoting my book ( Inorganic Chemistry - Atkins ) : "...

This is most certainly a dup of something, I just can't find it. Mods: merge away! Over the past months my rear spockets have accumulated a fair amount of gunk == chain oil + dirt + sand + hair + grass + wood chips. They look bad, furthermore I fear the chain might be picking some of that up. I k...

I have the following plot: Plot[x^2,{x,0,100}] Is there a function that determines the plot's current height and width? And how can I change these dimensions?

Whenever I find out a company and job I’m interested in has unlimited PTO I’m no longer interested. As a software engineer I find there is always work that can be done. This makes me afraid of “You can take time off as long as the work is done.” Am I wrong…is unlimited PTO generally beneficial to...

The beginning of the James Bond film From Russia with love (1963) portrays the end of a game between the fictional characters Kronsteen (Czechoslovakia) and Macadams (Canada) in the last and decisive round of the Venice International Grandmasters Championship. The moves go like this: [White "Kron...

From this page: So I'm trying to get that result in Mathematica. From the fourier formula above, we can see that {a, b} = {1,-1}. However, the code below returns a different result. FourierTransform[Cos[2 \[Pi] A t], t, w, FourierParameters -> {1, -1}] (*\[Pi] DiracDelta[2 A \[Pi] - w] + \[Pi]...

I was using my raspberry pi pico and testing motors, and it worked just fine(pins was in GND + 3v3 pins). Then I wanted to use On/Off function in C, And i used the pin GP28_A2 to use it. But it wont turn on the motor. I used ampermeter and voltmeter, and everything was same(3.2volts 0.15-0.3amp) ...

Firstly, apologies for this question but I couldn't really wrap my head on how to do this (ultimate aim is to put it in app scripts) Let's say I have a table that is updated daily showing the fruit Tony eats on that day For some reason, I need to find the number of days between eating Pear follow...

I have seen this, but biblatex is used there, I don't use that in my case. Is there a way for me to do: \url{https://example.tld/example/something/} Or, if it's possible, maybe even: \url{https://www.example.tld/example/something/} And, instead of getting https://www.example.tld/example/somethi...

I am a dual EU/UK citizen (recently naturalized in the UK). Since the end of the Brexit transition period, I have tried to collect stamps in my UK passport. I travel frequently between the EU and the UK, and sometimes ask the European immigration officers to stamp my British passport. They ask if...

According to Wikipedia: Functional forms and parameter sets have been defined by the developers of interatomic potentials and feature variable degrees of self-consistency and transferability. When functional forms of the potential terms vary, the parameters from one interatomic potential functio...

I wasn't sure if this proof was correct or not. Proposition. If $(a_n) \to a$ and $(a_n) \to b$, then $a = b$. Proof. Suppose $(a_n) \to a$ and $(a_n) \to b$. Then, $\lim \limits_{n \to \infty}(a_n)=a$ and $\lim \limits_{n \to \infty}(a_n)=b$. So for every $\epsilon > 0$ there exists a $N \in N$ ...

I'm in need of creating a set up that has instances aligned along a spiral but are reflecting a central character's position based on the angle they are positioned relative to it. This is what the set up looks like currently, all characters are revolved around the character in a radiating fashio...

Cross-posted in Wolfram community, an online Mathematica notebook is available there. The problem under investigation is solving the Laplacian equation to model seepage. See the image below for the region and the model. The function u[x,y] represents the total head at the coordinate [x,y]. The ...

I am using Duolingo to practice Japanese, and I'd be able to practice a lot more if I could actually do this while listening to music. Right now, when you get to the next question, it turns down the volume for all other apps, reads the question, and restores the volume again. This is annoying eno...

Title says it all, but for specifics.. Just found out my thesis advisor published a paper last year using a figure from my thesis. The thesis is in embargo until later this year. I do have be honest and say that the figure in question is a redraft (with some slight alteration) from an older study...

The fish-people are hybrids of fish and humans: They have bodies akin to standard teleost fish, such as carp or salmon, with a face, brain, and throat like a human. While it would be quite easy to tape their mouths shut, the world doesn't have any tape: The technology level is around Medieval Eur...

Is there a specific term for an abelian group (or ring) $G$ that satisfies the following property? For every element $g \in G$ and natural number $n$, there exists a $q \in G$ such that $\underbrace{q + q + \dots + q}_{n\text{ times}} = g$. Loosely speaking, this property means that you can "di...

In this paper it is mentioned that ... octenidine can release iodine radicals from the PVP complex, resulting in a tissue irritation as well as strong brown to violet discolouration in the border areas. but elsewhere I read that ... it turns out that ΔG > 0 for radical iodination. Is it reall...

In Futurama, How Hermes Requisitioned His Groove Back (season 2, episode 15), Bender loses his brain (which looks like a floppy disk), causing him to repeatedly say: Bender: I am Bender. Please insert girder. I'm a little confused, since the definition of "girder" is: girder: A beam, as of ste...

We have a cake of length $n$ and we have two arrays $A$ and $B$ of size $n$. The two arrays have order like below. $A[0]<A[1]<\dots <A[n]$, $B[0]>B[1]>\dots >B[n]$. We define a envyless location as a cut in the cake where it satisfies $A[i]=B[i]$. For example, $A=[1,2,3,4,5,6,7]$ and $B=[7,6,5,4,...

Say I have a nested loop, is there a way to break out of the outer loop from the inner one? For example this code: for i in range(3) for j in range(5) echo i j if i == 1 && j == 2 break endif endfor endfor only breaks out of the inner loop, but how cou...

The Ukrainian course on Duolingo just gave me this sentence to translate: "Я ходив у банк вчора". But my understanding so far has been that you use іти for one-time specific instances of going somewhere (on foot), and ходити for repeatedly going somewhere or for walking around in general -- so u...

It looks like a standard, but seems to have fins extending out and is the first thing hit by the 2nd Death Star. It appeared to be holding a small freighter/cruiser similar to a GR-75 transport, when the fly-under was done by the camera.

Let $\mathcal{F}$ denote the set of conservative vector fields on $\mathbb{R}^3$ that are continuous. That is $$ \mathcal{F}=\{F:\mathbb{R}^3 \rightarrow \mathbb{R}^3: F \text{ is continuous and } F=\nabla \phi \text{ for some } \phi \in C^1(\mathbb{R}^3) \}.$$ Is $\mathcal{F} $ closed under com...

This is the fourth exercitationes from chapter nine of Wheelock's 7th edition latin textbook. Im translating some sentences as practice for a final exam. I translated the sentence as "Neither of the other will give full blame to wealth at that time." Can someone check this? I may be missing somet...

When Ingenuity was released on the Martian surface in April 2021, Perseverance had to drive away to let the helicopter recharge its batteries with its solar panel. Was there a concern that the rover could have experienced a fault, preventing it from driving away? What were the contingency measure...

For hydrogen, the 3-21G basis set (in CFOUR format) is H:3-21G 3-21G Split-valence basis set 1 0 2 3 0.5447178000D+01 0.8245472400D+00 0.1831915800D+00 0.1562849787D+00 0.00000000 0.9046908767D+00 0.00000000 0.00000000 1.0000000 My interpretation is that this defines two orb...

In the debates over drug regulation, there is typically one side which argues the government should ban and restrict use of property that is harmful or has high risk of being harmful. They may cite the US Supreme Court case United States v. Miller or bans on more obviously dangerous narcotics lik...

Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture? Things like Green's formula, Stokes formula etc. do not work - if in the beginning we choose to use "left handed" $x, y, z$-coordinate ax...

Why aren't these two integrals equal? $$\int_0^\pi \int_{-1}^{1} e^r \,dr\,d\theta$$ $$\neq$$ $$\int_0^{2\pi} \int_{0}^{1} e^r \,dr\,d\theta$$ Let me explain why I'm asking. The change of variables theorem for double integrals is here: Change of Variables Theorem But now suppose we have the unit ...

GPS approaches being so trivially easy to establish and maintain, are there any/many US airports with instrument approaches which do not have at least one RNAV approach?

Recently, Nanci Pelosi was barred from receiving Eucharist by the Catholic Church, due to her public stance on abortion which contradicts the Catholic faith, considered a mortal sin by the Church. What is the biblical basis for denying someone in mortal sin the Eucharist?

Tedd owns a riding horse. One day, Tedd notices that his horse has somehow gotten itself drunk. Tedd has someplace where he needs to be, so he saddles up his drunken horse and rides to his destination. Has a crime been committed? Answers are welcome from any jurisdiction where alcohol is legal,...

I cannot place Kes' accent. What is it? Is it based on any non-fiction accent? Is it the accent of Jennifer Lien from her daily life? Or was it a choice to make this the accent of the character?

I want to remove the message that is output in the message area when a new node is created. A typical message may look like this. "Article My article has been created." I examine the node in hook_node_insert, and when certain criteria is met, I want to delete the message or prevent the message f...

Appreciate any info on where Gates A8 or A12 in Schiphol airport are located. Are they real gates or airlines like KLM due to crowded terminals instead of keeping passengers waiting inside arriving planes, de-embark them on tarmac and transport them by bus to some terminal?

There is water dripping out of the part of the t-valve adapter connected to the toilet tank. The angle required to tighten the bolt with a tool makes it (seemingly) impossible. Wrench? Not enough space to turn it. Pliers? Too big to reach it. It’s been tightened by hand as much as possible, but ...

INTRO: I am trying to achieve a spaceship that flies through space and the pilot can get out of their seat and walk around the ship and place physics objects in it while the ship is moving, rotating, accelerating, and decelerating EXAMPLES: Subnautica has achieved this to some degree, in their c...

Let's assume the Tarski's axiom: For all set $u$, there is a Grothendieck universe $\mathbb U$ such that $u\in\mathbb U$. From now on I will drop "Grothendieck" and just write "universe." I think what one usually does when doing category theory is that one fixes a universe $\mathbb U$ and conside...

I'm running macOS 12.3.1 I added a couple of lines to my .zshrc, viz. export GREP_OPTIONS='--color=always' export GREP_COLOR='1;35;40' After this, when I pipe grep output to tr, it returns the same number of lines but all of the lines are blank for example: grep ^.....$ /usr/share/dict/words | tr "

I recently got a hankering to play the game again. I've forgotten a fair amount of the game mechanics, and wanted to run through the first bit of the game again. However, my last game is about 66% of the way through, and I don't want to lose that progress. How do I preserve my prior data such tha...

I would like to define the character @ as follows: \def @#1#2{\catcode`#1=13\def #1{#2}} so that when I type @xy x, TeX typesets y. However, when I run this, I get an error. I have seen other answers here and here that ask about changing catcodes in commands. However, none of the answers given so...

I have a handful of cloths that I've used to clean and grease my bike chain over the years. Each is covered in chain grease and other gunk that was once on my chain. I'm starting to wonder what I should do when my latest cloth reaches peak grossness. It seems like there are two obvious choices: ...

First time poster - my apologies if I don't get the etiquette correct. I have a ~200TB RAID6 array with 30 disks and I'm unable to mount it - I just get the message: mount /dev/md125 /export/models mount:/dev/md125: can't read superblock If I run mdadm --detail on it, it shows as clean: /dev/md1...

I have ten copies of each of the twelve pentominoes. Can I use all of them to completely tile twelve 5 x 10 rectangles?

Per the manual, I am trying to define a variable that inserts "p dolce" dynamics. pDolce = \tweak DynamicText.self-alignment-X #LEFT #(make-dynamic-script (markup "p" #:normal-text #:italic "dolce")) \relative c' { c \pDolce c c c } But what if I need to tweak two proper...

I am considering an ANOVA model for such data: Student Class Points on exam Day of studying 1 P1 12 Day 10 2 P3 23 Day 5 3 P3 8 Day 1 4 P2 36 Day 10 5 P1 10 ...

On doing what things a Brahmin lose his caste ? On doing what a Kshatriya lose his caste ?

Recently, I've noticed that the NY-Times Wordle game accepts the word "muist" as a five-letter word attempt. But what does it mean and where does it come from? I've searched online and I stumbled upon a few scrabble-oriented resources that strongly indicate the word is a valid scrabble word (I as...

i made an armature by skin modifier -> create armature. Can i somehow see and change the vertex weights of the created vertex groups?

I am planning on installing Ubuntu on my computer. I studied the installation instructions, and I discovered that one of the steps is to select whether the Ubuntu system will be encrypted. I understand that encryption enhances security. If a computer with no encryption is stolen, the thief cou...

I am looking for some help constructing a function Part[M_,N_] which outputs the list of all integer partitions of positive integers up to $M$ such that in each partition, no number is repeated more than $N+1$ times. For example, when $N=3$, the partition {1,4,5,5,5,5,5,7} of $37$ should not cou...

This is the $2$-dimensional wave equation $$ u_{tt} = u_{xx} + u_{yy} $$ with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$. The inverse Fourier transform used is $$ u(x,y,t) = \iint \hat{u}(\omega_{x}, \omega_{y}, t)e^{(2\pi) \omega_{x} i x} e^{(2\pi) \omega_{y} i y}d\omega_{x} d\om...

I am planning on building a deck with composite boards. The plan is to have a 3x2m platform, however the decking boards I like come in lengths of 3.6m only. I am wondering how I can best use boards to keep costs as low as possible, whilst also having a design that looks nice. A standard horiontal...

I want to create hands for my model. But it must be in a cylinder shape. There are so many tutorials on YT but can't find one I want. Can anyone show me how to model like this?

I want to create tags for equations as sequential asterisks a+b=c (∗) x+y=z (⁑) i+j=k (⁂) How would you typeset them?

I just learned from a YouTube video that in Japan there was an add-on for the original Nintendo Famicom console that included a keyboard (HVC-007) and compact cassette tape player & recorder (HVC-008). Since I've dabbled in disk and tape formats in the past I immediately started Googling for the...

I have a circuit that uses MOSFET to power a resistive plate to heat it up. The problem is that this MOSFET always burns after a few seconds. Let me explain my circuit: First, the outlet AC voltage (230V AC) is turned into DC voltage via a full-bridge rectifier (D1) and with a capacitor (C2.) At ...

This should have been asked here, but apparently it hasn't. How to delete everything from a string except alphanumeric characters and spaces? For example, str = "This is, my string number 5!" I want it to be "This is my string number 5"

Num calcn for Tria Center Can someone please help in finding an analytic expression for the center of a circle through 3 given points by modifying ( Solve in place of NSolve ) a conventional numerical approach above? The program hangs and some more coding is needed. Or, from the following we may ...

Let $K ⊂ R^n$ be convex and compact with $0$ in the interior of $K$. Let $f ∈ C(K, R^n)$ with $f(∂K) ⊂ K$. If this is the case, do we in fact have $f(K) \subset K$. It is probably not the case that the image of a convex set is convex as those are hard to prove, but we at least know it is compact ...

Let $(E, |\cdot|)$ be a n.v.s. We fix $r>0$ and $x,y \in B(0, r)$ such that $x\neq y$. Here $B(0, r)$ is the open ball centered at the origin and having radius $r$. The set of all points in the line though $x$ and $y$ is $$ \{tx+(1-t)y \mid t \in \mathbb R\}. $$ Consider the map $f: \mathbb R \to...

I would like to mount some external flood-lights on a motorhome with a 12V DC system. I may need to install new positive wires to make sure they are sufficiently thick. I wonder if I also need to install thicker negative wires, or if I can simply connect to the existing negative wires, without wo...

As a sequel to this question Formula for matrix (dates in ascending order) how would the formula (=IF($B2="Apple",A2-SORT(FILTER(A:A,B:B="Pear", A:A<$A2),1,FALSE),"") be modified if the dates are in descending order? That is, Tony ate an apple on some day, how many days before did he eat a pear?

I used to use Google DNS servers, i.e. 8.8.8.8 and 8.8.4.4, but I couldn't access certain websites. Then I decided to use Quad9 DNS as my alternate DNS server: Is it OK to use mixed DNS servers?

Definition of XOR-Universal hash functions by Abidin[1]: A class $H$ of hash functions from $M$ to $T$ is XOR-Universal$_2$ ($XU_2$) if there exist at most $|H|/|T|$ hash functions $h$ $\in$ $H$ such that $(h(m_1) = h(m_2$) $\oplus t)$ for any two distinct $m_1$, $m_2$ $\in$ $M$ and any $t \in T...

Savitch's Theorem states that $NSPACE\left( f \left( n \right)\right) \subseteq DSPACE\left( \left( f \left(n \right) \right)^2 \right)$ for any $f\left(n \right) \in \Omega \left( \log{n} \right)$. The proof I am familiar with states the following: we are given a non-deterministic turing machine...

Let us now set aside the question of the submicroscopic structure of space-time and concentrate, instead, on its large-scale properties. In this case, we may imagine that the smooth manifold picture will be adequate and that its structure in the large can be obtained by piecing together smaller "...

I have an Icelandair flight from Paris CDG to Newark with a stopover in Keflavik. I checked in online for the first leg (to Keflavik) and have a boarding pass. As often for flights to the US, I can't check in online for the second. Do I need to get the boarding pass for the second leg in Paris, o...

They were born at the same time. One lived in the USA and the other in Germany. Each would have "episodes" where one would pass out and then would see through the awake one's eyes. This is how they grew up, until one day the boy was kidnapped and she had to go save him. He was born into a royal o...

I know there are questions which might sound similar, but this is a complete diferent one. Firs I am using the word extending and not expanding and second the question is limited to a time period long befor 2008 and even befor Putin came to power. So at the begin of the 90s the Cold War was ofici...

I want someone to correct my code in python. My goal is to create a code that will add a new column that is based on the conditional function of two columns in my dataframe. I want to add a fees column which is a numeric column that is different and based on whether the success is True or False a...

This is my solution for problem 1 of the projectEuler with python: import time start_time = time.time() #Time at the start of program execution def multiple_3_or_5(n): if n % 3 ==0 or n % 5 ==0: return True else: return False sum = 0 for i in range (1,1000): # print (

Eze22:19 Therefore thus saith the Lord God; Because ye are all become dross, behold, therefore I will gather you into the midst of Jerusalem. 20 As they gather silver, and brass, and iron, and lead, and tin, into the midst of the furnace, to blow the fire upon it, to melt it; so will I gather you...

When creating the plant life on alien planets, photosynthetic pigments other than chlorophyll can be the predominant molecule to create foliage of colours other than green, as we see on earth but for the minority of plants. But most trees and woody plants have brown bark, other than the few examp...

For a dice pool of 6d6(4+), where any dice showing 4 or 5 or 6 is a success, what is the anydice formula to have the probability of at least a double among the 4+ dice? The goal is to know the chance for successfull rolls to show doubles and beyond only among the success faces.

How to make a toggle shortcut? The Right Click over "Camera to View" "Change shortcut" used is: Shift+Ctrl+Alt+(numpad) 0 Want to press the same combo to able/disable. currently it only enables it, Have to go back to the the checkbox to disable. Also with the Camera selected the View> "Camera to ...

What are farmers (in Israel) supposed to do with all their free time during the shemita year? Are we, today, supposed to do anything special during the shemita year (in addition to actions related to produce of the land). I am asking about actions we might take that are reflective of what farme...

what is the simplest way to explain how to determine whether the resolution of a triangle (finding all its sides and angles, given 3 of them) using the sine theorem gives one or two solutions and whether the solution found is correct? I have explained congruent triangles to my pupils (ASA, SAS, S...

I have an online virtual debit card which was provided from a company as a refund. To all intents and purposes, this is indistinguishable from a physical card but I've yet to find anywhere online where I can use for part payment so how to you spend the exact amount left on the card? I had thought...

Can I use two relays to increase current rating? I have requirement of 12 A, 240 VAC load, but I have 7 A, 250 VAC relays. Can I add two relays to make 7 A + 7 A, relay which is driven by single input signal? TLDR; Can we add relay to increase current for inductive load?

Given two positive integer fractions \$x\$ and \$y\$ such that \$x < y\$, give the fraction \$z\$ with the smallest positive integer denominator such that it is between \$x\$ and \$y\$. For example \$x=2/5\$, \$y=4/5\$, the answer is \$1/2\$. Other fractions such as \$3/5\$ are also in between t...

I'm looking for a lightweight PDF viewer on PC, that can let me highlight texts and save the highlights. Other than highlighting, I have no needs for other bloated features. Thanks!

I am having some difficulty understanding the concept of measure preserving, invariance, and ergodic. Here is a proof from Theorem 6.2.6 in Durrett's Probability: Theory and Examples, 5e (p.338) (available at https://services.math.duke.edu/~rtd/PTE/pte.html), which states that If A = [a, b), then...

Patrick and Rachel go to a tennis tournament with 7 other couples. Each round is a single's match (1 vs 1). Nobody plays against his/her partner and nobody plays twice against the same player. At the end of the tournament Patrick asked everyone how many people they played against, and found that ...

I created a new wallet in Bitcoin Core v23.x, as expected it created an empty wallet version 169900 descriptors true format sqlite. It seems it created 4 streams of 1000 keys each. Anyway I told it create some P2PKH, P2SH-nested P2WPKH, P2WPKH, P2TR addresses. It created a bech32m address, but it...

It's from the Wall Street Journal article. Twitter Employees Face ‘Chaos Tax’ I understand that the article describes Tweeter in chaotic situation. But I can't guess what chaos tax implies exactly. I can't find any example of the expression.

Assuming an elemental for of Magic that allows the mages to control fire, water, earth, and the wind itself (with some limited power to change states, like turning ice to steam and vice versa), as well as the ability to impose your will upon the weather. Not the climate, just the weather. Climate...

I live in a coastal New England town and while reading a local history I noticed frequent mention of the "partridge" as being a commonplace bird in the area. Now, however, this bird which is more specifically the ruffled grouse, Bonasa umbellus, is completely absent in our community and not to be...

The following image shows a 6-transistor XOR circuit. But I don't know why the last 2 gates are needed at all, the circuit can be simplified to: Is there anything that prevents us from using the last circuit? I tested all 4 scenarios of (A,B) = (0,0) (0,1) (1,0) (1,1), all of them gives a corre...

My knowledge of automorphic forms basically comes from Lang's $SL_2$ and Borel's automorphic forms on $SL_2$. I try to understand them from representation theory. For example, an integral weight $k$ automorphic form corresponding to the $k$-th character eigenspace for the right $SO_2$ action on $...