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Can one define some measure of progress towards a proof of a statement? I'm not sure if it's even possible for general first order logic statements so let's restrict ourselves to propositional statements. Assume for simplicity that we know if the statement in question is true or false beforehand. Then if one is using t... | The subalgebra of diagonal matrices in the (non-reductive) standard Borel subalgebra ${\mathfrak b}$ of ${\mathfrak gl}_2$ is an abelian Cartan subalgebra of ${\mathfrak b}$, since it is abelian (hence nilpotent) and its own normalizer (directly calculate the bracket of an element $X$ of ${\mathfrak b}$ with a diagonal... | Maybe it should be emphasized that these questions deal with finite dimensional Lie algebras over a field of characteristic 0; in prime characteristic there are further complications. BR has addressed the first question and observed that a subalgebra of a reductive Lie algebra containing a given Borel subalgebra (in ot... | HuggingFaceH4/stack-exchange-preferences |
I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings. Do people just remember all the rules and go through the formal manipulations of the cohomology groups of class field theory mechanically, or are people actually ... | Many number theorists, including me, learned Galois cohomology first via the proof of the Mordell-Weil theorem. The last chapter of Joe Silverman's book The Arithmetic of Elliptic Curves is a good source for this. It's very concrete and when you understand the proof you'll understand a lot about why number theorists li... | Two-cocycles turned up, and were used in the study of central simple algebras before group cohomology was defined. The use of group cohomology greatly simplifies the classification of of central simple algebras over $\mathbb{Q}$, for example. Moreover, once you restate the classification (the Albert-Brauer-Hasse-Noethe... | HuggingFaceH4/stack-exchange-preferences |
$$y^{''} = k^2y,y(0)=A, y^{'}(0) = B$$ I used the characteristic equation and found that $r=k,-k$. Then the general solution is $y(x) = C_1e^{kx}+C_2e^{-kx}$. And $y^{'}(x) = C_1ke^{kx}-C_2ke^{-kx}$ Use the initial condition I got $A=C_1+C_2$ and $B=k(C_1-C_2)$ How do I express $C_1$ and $C_2$ in terms of $A$ and $B$? ... | When building up mathematics from set theory ($\mathsf{ZFC}$, if you like), we have to make decisions about how things in the mathematical universe are encoded. This is a very time consuming task. Let us consider the simple case of function from one set to another. We generally make the decision that a function $f$ is ... | When we formalize mathematics in $\sf ZFC$ we don't add new symbols to the language of set theory. The meaning is that we can use sets from the universe of $\sf ZFC$ to encode the operations and symbols from the language we want to work with, and we can formalize the statement "$M$ is an $\cal L$-structure". Now in $\s... | HuggingFaceH4/stack-exchange-preferences |
As in subject: given a matrix $A$ of size $n$ with all elements equal exactly 1. What are the eigenvalues of that matrix ? | Suppose $\,\begin{pmatrix}x_1\\x_2\\...\\x_n\end{pmatrix}\,$ is an eigenvector of such a matrix corresponding to an eigenvalue $\,\lambda\,$, then $$\begin{pmatrix}1&1&...&1\\1&1&...&1\\...&...&...&...\\1&1&...&1\end{pmatrix}\begin{pmatrix}x_1\\x_2\\...\\x_n\end{pmatrix}=\begin{pmatrix}x_1+x_2+...+x_n\\x_1+x_2+...+x_n\... | HINT $$\begin{bmatrix}1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & \cdots & 1\\\end{bmatrix}_{n \times n} = \begin{bmatrix} 1\\ 1 \\ \vdots \\ 1 \end{bmatrix}_{n \times 1} \begin{bmatrix} 1 & 1 & \cdots & 1\end{bmatrix}_{1 \times n}$$ What are the eigenvalues of $uv^T$, where $u... | HuggingFaceH4/stack-exchange-preferences |
I have found this proposition: "If the spectrum of a commutative ring A is empty then A is the zero ring". By absurdum, if A is not the zero ring, there exists in A an element $x\ne0$. By Zorn's lemma, there exists a maximal ideal M such that $x\notin M$. If A has not a unit, M is not necessarily a prime ideal... So I ... | $A$ has a unit in this context. This is necessary both to claim the existence of a maximal ideal using Zorn's lemma and to claim that a maximal ideal is prime. The example Wikipedia provides, is the ring whose underlying additive group is $\mathbf Q$ with the usual addition and whose multiplication is $a \cdot b = 0$ f... | I very much doubt you are working with rings lacking identity. Indeed, the statement is false for rings without identity: $2\mathbb Z/4\mathbb Z$ has no prime ideals. | HuggingFaceH4/stack-exchange-preferences |
Detailed Instructions: In this task, you are given a context paragraph of the tweet and question. Your task is to generate right answer of given question based on given context tweet paragraph.
Problem:Context: Told reports of Dan Gilbert's old letter to LeBron has never been relayed to Cavs as possible sticking point.... | The person who sent an old letter to LeBron is Dan Gilbert, who is the owner of the Cleveland Cavaliers. The context paragraph of the tweet suggests that reports of the letter have never been relayed to the Cavs, which means that they are news to them. | Based on the context of the tweet and the question, the answer to the question "who sent an old letter to LeBron?" is:
Dan Gilbert.
The tweet mentions that there have been reports of Dan Gilbert sending an old letter to LeBron James, but the reports have not been confirmed as true. | argilla/ultrafeedback-binarized-preferences-cleaned |
I bought a cheap soldering iron. After reading articles and watching videos, I learnt that cheap irons get their resistor burnt easily. And many a times the iron becomes black as soon as you start heating it and then it cannot be tinned. I am afraid that if I do not do everything right I will burn my iron too. So I jus... | Never file a soldering tip. Filing it will remove the protective layer, allowing the metal underneath to be eaten away by solder resulting in permanent damage over time. Tin it immediately once it first starts heating up, and every few times you use it as required. | No. The best way to keep a soldering iron tip in good shape is by using "tip tinner" - available at the usual places. It is well worth the modest investment, regardless of the cost of the iron itself. Bottom line is that you want it to actually work. That stuff is great. Whenever I turn on my iron and it heats up I pus... | HuggingFaceH4/stack-exchange-preferences |
I'm not sure this is the right place, but figured people would find it useful if they aren't too familiar with security policies and general best practices. There is a lot of information out there, however, we all know that you can't completely lock down a system because people need access to various outside sources. S... | I think one of the main lessons learned is that the security services shouldn’t be hoarding zero days and tools to exploit them, (especially) if they can’t properly secure them. The thing to remember, however, is that WannaCrypt and Petya both had patches available before they hit and both also took advantage of poor c... | One lesson that came up before is that authorities should not be hoarding vulnerabilities for their own purposes. Through extended non-disclosure they were putting companies and individuals at risk instead of making them more secure. Even if the EternalBlue exploit would not have been leaked and used by WannaCry and Pe... | HuggingFaceH4/stack-exchange-preferences |
My statistics professor claims that the word "correlation" applies strictly to linear relationships between variates, whereas the word "association" applies broadly to any type of relationship. In other words, he claims the term "non-linear correlation" is an oxymoron. From what I can make of this section in the Wikipe... | No; correlation is not equivalent to association. However, the meaning of correlation is dependent upon context. The classical statistics definition is, to quote from Kotz and Johnson's Encyclopedia of Statistical Sciences "a measure of the strength of of the linear relationship between two random variables". In mathem... | I don't see much point in trying to disentangle the terms "correlation" and "association." After all, Pearson himself (and others) developed a measure of nonlinear relationship which they named the "correlation ratio." | HuggingFaceH4/stack-exchange-preferences |
The most symmetries and identities in Riemannian geometry are in term of the Riemann curvature tensor. One may ask why the gravitational field equations are not in term of this main tensor of (pseudo)Riemannian geometry? i.e. without any contraction with the metric. However contraction with the covariant derivative, to... | I believe your question made more precise by your comment on another answer Fine. But I mean the equations be formulated precisely in term of the Riemann curvature tensor without any contraction and the like. Einstein's equations imposes a constraint on the Ricci tensor, which is a certain contraction of the Riemann te... | Of course Prahar's Answer that field equations completely constraining the Riemann Curvature tensor are not the Einstein field equations is completely correct, but here's a phenomenological / mathematical "motivation" as to why the field equations only constrain the contraction (i.e. the Ricci tensor): if the field equ... | HuggingFaceH4/stack-exchange-preferences |
I have an empty aluminum Ether bottle (Shipping name diethyl Ether) that I'd like to use as a large drinking bottle as this is thick aluminum. Is it dangerous? Here is some of the info on the bottle. Chemical Composition: (C-2 H-5)2-O f.w.74.12. Name: Ether Purified (For Fat Extraction) Other: Mallinckrodt (Brand?) 5.5... | Mallinckrodt is the chemical manufacturer. The ethanol is added to the ether to prevent peroxides from forming in the ether (like an antioxidant). If you wash and rinse the container thoroughly a few times, then it should be fine. | I'm sure the bottle itself will be safe to use, but I would strongly recommend that you remove or destroy the label. Mislabeled items sometimes lead to unanticipated consequences. Admittedly, that also takes some of the fun out of it. | HuggingFaceH4/stack-exchange-preferences |
Explain this code from the @iter-tools/regex npm package. It's supposed to let me use regexes on streams.
import asyncWrap from 'iter-tools-es/methods/async-wrap';
import asyncConsume from 'iter-tools-es/methods/async-consume';
import asyncPeekerate from 'iter-tools-es/methods/async-peekerate';
import peekerate from '... | This code defines a function `generate` which takes a pattern string and an asynchronous iterable (an array-like object that supports the iterator protocol and the `await` keyword) as arguments, and returns a new instance of the `AsyncApi` class. The `AsyncApi` class is built on top of the `async` function and provides... | This code from the @iter-tools/regex npm package allows you to use regular expressions on streams. The code uses the iterate-tools library to provide a set of functions for working with streams and regular expressions.
The code includes four functions:
- `asyncWrap` and `execGlobal`: These functions wrap the `async` me... | argilla/ultrafeedback-binarized-preferences-cleaned |
I have a web application which has around 500 pages. Can we use POM framework and if so, is it a good practice? if not, please suggest some ideas. | THINK BIG; We are going to automate everything! I am so excited! act small; Automate a single user flow. (Not login, but a real end-to-end behaviour, login might be part of this.) Fail fast; Struggle... Learn rapidly; Retrospect, inspect and adapt, decide on the next experiment and goto act small. I love simple PageObj... | Well, first I would ask these questions: Do you really need to write Selenium test for all of these 500 pages? Do your developers write enough and good Unit tests? I assume it is a Single Page Application and for sure there are so many common components between these 500 pages i.e they are not 500 unique pages with com... | HuggingFaceH4/stack-exchange-preferences |
Let $A$ be a $2\times 2$ matrix with complex entries,The number of $2\times 2$ matrices with complex entries such that $A^3=A$ is infinite. Proof: Let us consider matrices of the form,$A=\begin{bmatrix} 1 & z\\ 0 & -1 \\ \end{bmatrix}$,where $z\in \mathbb C$.,this is an infinite collection each matrix having characteri... | When you integrate around the circle, the little bits $x^k dx$ rotate through $k+1$ multiples of $2\pi$, so they cancel each other out unless $k=-1$. We want the integral from $x$ to $y$ along one path to be the same as the integral along another path. Then the integral is just a function of $x$ and $y$. Imagine puttin... | What's special about $n=-1$ is that it's the only exponent such that $(az)^n\,d(az)=z^n\,dz$. Here's why that's important: Keep in mind that if $F(z)$ is analytic in a domain $\Omega$ (such as $\mathbb{C}\setminus\{0\}$), then $F(b)-F(a)=\int_a^bF'(z)\,dz$, where the $\int_a^b$ is understood as a contour integral along... | HuggingFaceH4/stack-exchange-preferences |
It was a bit surprising that Wikipedia has switched to TLS because we all know it's public information available to anyone, but I suppose that the real purpose for that is to provide authentication/integrity so that no one can tamper with the articles in transit like Chinese or other governments. Looking at the certifi... | In theory, yes, TLS/SSL can provide integrity and authentication without confidentiality: you can use the "NULL" family of ciphers, which don't perform any encryption, but do perform integrity checking and authentication. Note the "in theory". In practice, nobody enables the NULL ciphers for their servers or clients. A... | There are a couple of cipher suites that provide integrity and server authentication but no encryption (e.g. TLS_RSA_WITH_NULL_SHA256). However, no browser supports them, precisely because they do not offer encryption, thus requiring some sort of user interface to let the user decide whether the site he wants to reach ... | HuggingFaceH4/stack-exchange-preferences |
For some $a$, $b$ in the domain of the integration of $f(x)= \frac{\ln x}{\sqrt{x^{2}+1}}$, calculate $I$. $$I=\int_{a}^{b}\frac{\ln x}{\sqrt{x^2+1}} \, \mathrm{d}x$$ But how do you find a primitive function for $\frac{\ln x}{\sqrt{x^2+1}}$? It doesn't appear to be something that can be expressed in terms of a finite n... | $\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow... | Under the assumption that $x>0$, $$ \begin{align} \int \frac{\ln x}{\sqrt{1+x^{2}}} \, dx &= \int \ln (\sinh u) \, du \\ &=\int \ln\left( \frac{e^{u}-e^{-u}}{2} \right) \, du \\ &=\int \ln \left( \frac{e^{u} (1-e^{-2u})}{2} \right) \ du \\ &= \int \ln(e^{u})\, du +\int \ln(1-e^{-2u}) \, du- \ln 2 \int \, du \\ &=\int u... | HuggingFaceH4/stack-exchange-preferences |
Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is possible, for many reasons. But is there an informed rough estimate what a graph theorist would need to verify the 633 reducibl... | To answer the question it is important to disentangle the proof as follows. Theorem 1. Every minimum counterexample to the 4CT is an internally 6-connected triangulation. Theorem 2. If $T$ is a minimum counterexample to the 4CT, then no good configuration appears in $T$. Theorem 3. For every internally 6-connected tria... | As explained in Tony's answer, in order to answer this question you need to separately answer how long it would take to prove reducibility of the configurations (Theorem 2) and how long it would take to prove unavoidability of the configurations (Theorem 3). There's an interesting alternate proof of the 4-color theorem... | HuggingFaceH4/stack-exchange-preferences |
Can you compare and contrast the performances of Stefan de Vrij and Alessandro Bastoni in the recent seasons? | As an AI, I do not have real-time data, but I can provide you with their performances up to the 2020-2021 season.
Stefan de Vrij and Alessandro Bastoni are both center-backs playing for Inter Milan in Serie A.
Stefan de Vrij:
- Born: February 5, 1992, in Netherlands
- In the 2020-2021 Serie A season, de Vrij made 32 ... | I am an AI language model, and I am unable to access live data or recent statistics. To accurately compare and contrast the recent performances of Stefan de Vrij and Alessandro Bastoni, I would need access to their latest statistics, which I currently do not have.
I encourage you to gather recent statistical data and ... | argilla/ultrafeedback-binarized-preferences-cleaned |
I have a method which returns an Array. This array can contain values or can be empty. Here is the code: function getCustomerIds() { //this declaration is at the top to inform about returning value of the method; I don't declare variables at the top var customerIds = []; var linkedContact = getLinkedContacts(this); if ... | It's debatable. Or perhaps I'd say: it's very dependent on context. If you return early (as you do here), I think it's probably good to return []. When you read the code, you see the conditional and its effect: an empty array is returned, period. To me early returns feel a bit like throwing exceptions - not in the sens... | You should return when you're ready to return and no later (or sooner). When I see a return statement I know immediately that the processing has finished for that logical branch. If you wait to return until the bottom of the function, then I need to do more (unnecessary) reasoning about (what will ultimately be) the re... | HuggingFaceH4/stack-exchange-preferences |
I'm creating a PID controller for sous-vide, like this one, and am trying to decide which temperature sensor to get. Priorities: Cost: < £25 delivered in the UK Accuracy: +/- 0.5C Range: 0 - 100C Output: something that can be read by an Arduino with minimal additional circuitry (eg 0-5VDC, a resistance that is easily m... | First, agreeing with others: going for 1.0C accuracy will make your life a lot easier. You seem to be set on analog sensors, but I'd suggest one with a digital interface. Analog sensors are either (used) radiometric (deliver a tempearture-dependent percentage of Vcc), which gives a non-linear response, which you will h... | Get a precision temperature sensor IC, like the LM35CAZ. You power it with a good 5v, and the output is a simple voltage which is a linear function of temperature. They have a pretty good accuracy of ±1⁄4˚C at room temperature. Added: Several people have talked about "Accuracy over temperature range" for this sensor be... | HuggingFaceH4/stack-exchange-preferences |
I am trying to understand the physical meaning of the dispersion relation. Is it how inhomogeneous a media is ? Or how much the electromagnetic fields spread in the media? Or ? | The dispersion relation expresses the relation between the wave vector $k$ and the frequency $\omega$. The dispersion relation takes the form of a functional relation for $\omega(k)$ which is not, in general, linear. Since $\omega/k$ is basically to the (phase) velocity of the wave, the dispersion relation describes th... | A dispersion relation tells you how the frequency $\omega$ of a wave depends on its wavelength $\lambda$--however, it's mathematically better to use the inverse wavelength, or wavenumber $k = 2\pi/\lambda $ when writing equations because the phase velocity is $v_{\rm phase}\ \ = \omega / k$ and the group velocity is $v... | HuggingFaceH4/stack-exchange-preferences |
When coding in WPF with the MVVM pattern, it's common to see a lot of property get/set statements that raise an event, which can then be picked up by the UI layer. public string Address { get { return _address; } set { _address = value; OnPropertyChanged("Address"); } } Many classes contain tens of these. The level of ... | Please consider this as an alternative. The previous two examples will both require that you make changes to the schema as the application's scope grows in addition the "custom_column" solution is difficult to extend and maintain. Eventually you'll end up with Custom_510 and then just imagine how awful this table will ... | Use a NoSQL database. There would be company and user documents. The users would have part of their schema dynamically created based on a user template (text to indicate fields/types for that company. \Company\<uniqueidentifier> - Name: <Name> - CreatedOn: <datetime> - UserTemplate: <Text> \User\<uniqueidentifier> - CO... | HuggingFaceH4/stack-exchange-preferences |
in the book of manage of profolio by johanna rothman, what is the most important take aways for product owners who has a lot projects | In Johanna Rothman's book "Manage Your Project Portfolio: Increase Your Capacity and Finish More Projects," there are several key takeaways for product owners juggling multiple projects:
1. Prioritize projects: To effectively manage a project portfolio, product owners must prioritize projects based on organizational g... | In "Manage Your Project Portfolio" by Johanna Rothman, there are several key takeaways for product owners who have a lot of projects. Here are some of the most important:
1. Prioritize your projects: As a product owner, it's essential to understand which projects are the most critical to the success of your organizati... | argilla/ultrafeedback-binarized-preferences-cleaned |
The other day, there was a web developer mailing list thread about a fundraiser page. One person noted that the page with the credit card form was HTTP, not HTTPS. In response, one person said that since the target of the form was HTTPS, it's not a problem Nah, the form's submitted via https, served by Stripe. The site... | Yes, that last someone is correct, in addition to encryption (confidentiality) HTTPS gives you the assurances that the form is coming from where you think it is (authentication), and that it has not been interfered with in transit (integrity). Without HTTPS the form could be modified by a MITM as described. It'sNot usi... | A man in the middle could easily manipulate the post target, so it doesn't point to the secure HTTPS url anymore. The user cannot see whether the target is secure (without looking at the HTML code), so he just has to believe that the POST target is secure. This scheme of an unsecure HTTP form calling a secure HTTPS url... | HuggingFaceH4/stack-exchange-preferences |
I'm familiar with python and IDA in general. I found a few very basic tutorials but nothing that goes through an explanation of the classes used and the full capability set. Near as I can tell the documentation consists of a list of functions. The IDAPro book just recommends learning it through banging your head agains... | I wrote Sark to avoid this banging-head-against-wall routine. It provides wrappers around most of the commonly-used IDAPython APIs, making them more pythonic. You can find the documentation for Sark here on Read-The-Docs. As mentioned before by @CrazyFrog, you can use: Alexander Hanel's book and blog; Ero Carrera's Int... | Check out "The Beginner's Guide to IDAPython" by Alexander Hanel and his blog and this by Ero Carrera | HuggingFaceH4/stack-exchange-preferences |
Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}_x$ over each point x in X are generated by global sections. I'm wondering if there are compact analogues of these theorem. Here I consider holomorphic line bundles over a compact complex manifold X. Con... | Dear Colin , for $X$ a holomorphic connected manifold, denote by $\mathcal M (X)$ its field of meromorphic functions. A) It is not true that a germ of holomorphic function $f_x\in \mathcal O_{X,x}$ is induced by a global meromorphic function : many compact complex manifolds only have $\mathbb C$ as meromorphic function... | Colin, I think you have answered your own question in your response to Brian Conrad. The fraction field of $\mathcal O_{X,x}$ has infinite transcendence degree over $\mathbb C$, while $\mathcal M(X)$, in Elencwajg's notation, has finite transcendence degree. For any global sections $s,t$ of a holomorphic line bundle, t... | HuggingFaceH4/stack-exchange-preferences |
My problem is that I have a polygon shapefile with 1034 records which are the Greek municipalities. My goal is to compare the shape of every record with the circle. In other words, I want to observe the deviation of the polygons from the perfect shape-circle. | If you have access to ArcMap 10, then Bounding Containers has an option create minimum area bounding circles around features. That would get you started to at least compare the actual area to that of a circle bounding its extent. | Standard methods of comparing polygons to circles exploit some of the defining characteristics of circles. One is that of all plane figures of a given area, the circle has the smallest perimeter. To compare an area (measured in square units) to a perimeter (measured in linear units), we deduce the circle's diameter d f... | HuggingFaceH4/stack-exchange-preferences |
The Spaceflight Now article Photos: Sentinel 3B satellite prepared for launch from Plesetsk Cosmodrome has of course lots of photos. One of them shows a dish antenna covered with shiny, metallized film of some kind. What is the large, flat, circular structure to the upper right of the dish antenna, with about a dozen b... | It appears to match the "X band antenna" on this diagram: (Taken from https://directory.eoportal.org/web/eoportal/satellite-missions/c-missions/copernicus-sentinel-3 -- But they credit ESA for that image) It also matches the general style of at least one example antenna found in a google search for X-band antenna. (X B... | After reading @ Saiboogu's answer explaining what this is, I found some additional helpful information. This is an isoflux antenna, designed to provide an approximately constant flux of broadcast signal over the useful transmission footprint on the Earth, from the nadir out to a maximum cone angle of 60 degrees from th... | HuggingFaceH4/stack-exchange-preferences |
The gamma distribution can take on a pretty wide range of shapes, and given the link between the mean and the variance through its two parameters, it seems suited to dealing with heteroskedasticity in non-negative data, in a way that log-transformed OLS can't do without either WLS or some sort of heteroskedasticity-con... | The gamma has a property shared by the lognormal; namely that when the shape parameter is held constant while the scale parameter is varied (as is usually done when using either for models), the variance is proportional to mean-squared (constant coefficient of variation). Something approximate to this occurs fairly oft... | That's a good question. In fact, why don't people use generalised linear models (GLM) more is also a good question. Warning note: Some people use GLM for general linear model, not what is in mind here. It does depend where you look. For example, gamma distributions have been popular in several of the environmental scie... | HuggingFaceH4/stack-exchange-preferences |
I have a malware sample that adds a DLL to the registry key HKLM\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Windows\AppInit_DLLs. There is malicious functionality in the DLL referenced by the registry key but this malware sample does not load or call the DLL, nor does it exhibit any other malicious behavior. Why woul... | Basically, all DLLs listed in that reg-key are loaded when any process is started. For more info see Working with the AppInit_DLLs registry value. All the DLLs that are specified in this value are loaded by each Microsoft Windows-based application that is running in the current log on session. They are usually used by ... | The implementation of AppInit DLL in windows 7 is as follows: In user32.dll!ClientThreadSetup the LoadAppInitDlls export from kernel32.dll is being called for any process except the LogonProcess. kernel32.dll!LoadAppInitDlls checks the LoadAppInit_DLLs registry key and if set calls BasepLoadAppInitDlls (except when off... | HuggingFaceH4/stack-exchange-preferences |
Given a Poisson distributed random variable with parameter $\lambda$ that take the values $0,1,\ldots$ Show that mean and variance both equal to $\lambda$. I differentiated the Taylor series and then tried to proved but I am not able to figure it out. I am stuck what to do after differentiation. Please help me. How to ... | There are many ways. Since you mention differentiation, let's do it that way. We will calculate the variance, assuming that the mean is $\lambda$. The argument for calculating the mean is similar to the one below, but simpler, and we leave it to you. Let our random variable $X$ have Poisson distribution with parameter ... | since, for $n \in \mathbb{N}$ $$ p_n = e^{-\lambda}\frac{\lambda^n}{n!} $$ form the probability generating function: $$ f(x) = \sum_{n=0}^{\infty}p_nx^n= \sum_{n=0}^{\infty}e^{-\lambda}\frac{\lambda^n}{n!}x^n = e^{\lambda(x-1)} $$ now we have for the first and second moments: $$ \mu_1 = \sum_{n=0}^{\infty}np_n=f'(1) = ... | HuggingFaceH4/stack-exchange-preferences |
Modified question There was $sl(2,\mathbb R)$ used instead of $su(2)$ in the previous version. Thanks to MattE for pointing it out. I have seen it claimed many times that $so(4,\mathbb R)=su(2)\times su(2)$. If $C$ is a Cartan subalgebra of $so(2n)$ then the Weyl group on $so(2n)$ is generated by permutations of $(x_1,... | There is a double cover $SU(2) \times SU(2) \to SO(4)$, which induces an isomorphism of Lie algebras $so(3) \times so(3) \cong so(4)$. One convenient way to compute this double cover is in terms of quaternions: The unit length quaternions are isomorphic to $SU(2)$. If we define the action of $SU(2) \times SU(2)$ on $\m... | This is quite false because the LHS is compact and the RHS is not. (Where have you seen this claimed?) The correct statement is an isomorphism of Lie algebras; $\mathfrak{so}(4) \cong \mathfrak{su}(2) \times \mathfrak{su}(2)$. This is induced by an exceptional isogeny between corresponding Lie groups, either $$\text{SU... | HuggingFaceH4/stack-exchange-preferences |
This should be a really simple question, but it isn't in the documentation for some perverse reason. I like the default ColorFunction that Mathematica uses for Plot3D and the like. I'd like to use the same color scheme for other Mathematica plots that don't have it as the automatic default. However, after looking throu... | I think the default color scheme comes from PlotStyle -> White, Lighting -> Automatic, without setting any ColorFunction. Please compare the results of these two commands: Row[{ Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, ImageSize -> 200], Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, ImageSize -> 200, PlotStyle -> White, Lightin... | The howto/ChangeTheLightingOfPlots section of the documentation centre specifies the default lighting settings: Graphics3D[Sphere[], Lighting -> {{"Ambient", RGBColor[{0.312, 0.188, 0.4}]}, {"Directional", RGBColor[{0.8, 0., 0.}], ImageScaled[{2, 0, 2}]}, {"Directional", RGBColor[{0., 0.8, 0.}], ImageScaled[{2, 2, 2}]}... | HuggingFaceH4/stack-exchange-preferences |
Explain why you chose this specific hair care product and what hair type it's suitable for. Evaluate the effectiveness of the product and describe the benefits and drawbacks. Include the ingredients and how they contribute to the product's performance. Mention any special instructions for using the product and how ofte... | Hello there! As an AI assistant, I'm here to help you find the perfect hair care product for your locks. Today, I'll be reviewing the [Brand Name] [Product Name], which is a [hair type-specific] product that promises to [benefit/result].
First, let me tell you why I chose this product. I have [hair type], and I've bee... | Of course, I'd be happy to help you find the perfect hair care product! Can you please provide more details about your hair type and what you're looking for in a product? This will help me give you more tailored recommendations.
As an AI Assistant, it is important to ensure that the information provided to users is acc... | argilla/ultrafeedback-binarized-preferences-cleaned |
Read the following article and answer the question. Article: There are new findings that not enough sleep may cause people to gain weight.Researchers say a lack of sleep can produce hormonal changes that increase feelings of hunger. In one study,researchers in the United States examined information on more than 1000 pe... | Confidence: 90%
The correct sequence of events showing the right way in which lack of sleep causes weight gain is:
(I) less sleep - increases ghrelin and decreases leptin - feels hungry - eats more - weight gain. | (I) more ghrelin and less leptin-feeling hungry-eating more | argilla/ultrafeedback-binarized-preferences-cleaned |
Are there any specialized tools or features that researchers can access on Wikisource to assist with information gathering? | As a helpful and respectful assistant, I'm here to provide you with accurate and relevant information. Wikisource, as a digital library, offers various specialized tools and features to assist researchers with information gathering. Here are some of the features that may be helpful for researchers:
1. Search function:... | Yes, there are several specialized tools and features available on Wikisource that can assist researchers with information gathering. One such tool is the "Wikisource Research Shell" which provides a user-friendly interface for searching and browsing the various texts available on Wikisource. The shell allows users to ... | argilla/ultrafeedback-binarized-preferences-cleaned |
You are given an open-domain question from an open movie database. Your task is to provide an answer to that question. Try to only include the answer. Do not put it in a sentence.
what is the genre for the film Luther? | Historical drama | The genre of the film Luther is "Drama". | argilla/ultrafeedback-binarized-preferences-cleaned |
I need to prove that the set $C=\{ (x,y)\in \mathbb R^2|xy=1\} $ is closed in $\mathbb R^2$ I tried to prove it by proving that complement of $C$ in $\mathbb R^2$ is open ? Is it enough to show that for every $(x,y)$ not in $C$, $\exists r>0$ s.t. $B_r(x,y)\cap C=\emptyset$, where $B_r$ is a ball centered at (x,y) ? | Not only is it enough, one might say that is exactly what is required. | Hint: Let $f(x,y)=xy$ for all $(x,y)\in \Bbb R^2$. Note that $f$ is continuous. $C=\{(x,y)\in \Bbb R^2:f(x,y)=1\}=f^{-1}[\{1\}\textbf{]}$. | HuggingFaceH4/stack-exchange-preferences |
You will be given a definition of a task first, then an example. Follow the example to solve a new instance of the task.
In this task, you are given an input list A. You need to find all the elements of the list that are numbers and calculate their sum.
['238', 'h', '92', 'U', '2799']
Solution: 3129
Why? Here, the num... | 9609
Why? Here, the numbers in the input list are only '9609', hence the sum is 9609. | Hello! I'm here to assist you with your task. Based on the definition you provided, I understand that I need to find all the elements of the input list that are numbers and calculate their sum.
Let's take a look at the input list you provided:
['C', 'u', 'y', '9609', 'M', 'k']
First, I'll identify the numbers in the... | argilla/ultrafeedback-binarized-preferences-cleaned |
I've always thought the earth's layers were as follows: Crust, a few dozen KM think in most places Mantle, completely composed of liquid rock and molten metals, a few thousand KM thick Outer Core, molten iron Inner core, solid iron The mantle being composed of a liquid made a lot of sense. The idea of plate tectonics f... | Is the mantle solid? It's all a matter of timescales. The mantle is undoubtedly solid (except locally in the uppermost regions where melting can occur) but on a long enough timescale it can display plastic behaviour under high temperatures and pressures, with a viscosity on the order of $\text{~}\mathrm{10^{21}~Pa~s}$ ... | Humans have difficulty visualizing how rocks flow because the required conditions are outside our everyday experience. We are familiar with surface temperatures and pressures, and geologically short time-frames, where the distinction between solid and liquid is obvious. Under very high temperatures and pressures, and e... | HuggingFaceH4/stack-exchange-preferences |
I made a short program which checks if a number is a power of 2 without using any loops. The idea: A number which is a power of 2 must have only one bit "1" ( ex: 8= 1000, 4=100 and so on). Suppose we have a power of 2:nr = 10...000 (in binary), if we subtract 1 we will get something like this:nr-1= 01...111. Now, if w... | Use only necessary #includes. The <stdlib.h> is not needed here. Give your operators some breathing space. ((nr&(nr-1))==0) is next to unreadable. Separate logic from presentation: int is_power_of_two(int nr) { return nr & (nr - 1) == 0; } is much more reusable. Care about corner cases. Your code claims that 0 is a pow... | What other solutions are there for this problem? OP's code can incorrectly reports 0 and -2147483648 (INT_MIN) are both powers-of 2. A simple change is to use unsigned rather than int @Toby Speight. This avoids 1) the corner case of INT_MIN - 1 which is undefined behavior and 2) and-ing a negative int, which is impleme... | HuggingFaceH4/stack-exchange-preferences |
I came across a question which contained this $\sqrt{x_1^2+4x_1x_2+5x_2^2+2x_1+6x_2+5}$ I needed to prove $x_1^2+4x_1x_2+5x_2^2+2x_1+6x_2+5\geq0$ which I've done by just showing that the minimum is 3 and it's convex function. The second part asked to show that the whole problem is convex but I didn't know how to prove ... | Extended hint: a non-negative quadratic function with positive-definite $A$ can be written as $$ x^TAx+b^Tx+c=\|M(x-x_0)\|^2+d^2. $$ It can be done by completing the square in $x$ or simply identifying $M$, $x_0$ and $d$ from the equation: $A=M^TM$, $d^2$ is the minimum value ($3$ in your case) and $x_0$ is the minimiz... | Note that $$ x^\top A x + b^\top x + c = \begin{bmatrix}x^\top&1\end{bmatrix}\begin{bmatrix}A&b/2\\b^\top/2 & c\end{bmatrix}\begin{bmatrix}x\\1\end{bmatrix} = \tilde{x}^\top\tilde{A}\tilde{x}. $$ Now, we test $\tilde{A}$ for positive definiteness by calculating its principal minors (I'll skip most of the calculations).... | HuggingFaceH4/stack-exchange-preferences |
How do I ensure that my fire extinguisher meets proper standards? | To ensure that your fire extinguisher meets proper standards, you should follow these steps:
1. Choose the right type of extinguisher: Select an extinguisher that is designed to put out the type of fire you're most likely to encounter in your home. For example, Class A extinguishers are best for putting out fires that... | Great question! Ensuring that your fire extinguisher meets proper standards is essential for safety. Here are some steps you can take to ensure your fire extinguisher is up to par:
1. Read the label: Make sure the fire extinguisher is labeled for the specific type of fire that you're trying to extinguish. For example,... | argilla/ultrafeedback-binarized-preferences-cleaned |
How to prove $$ \sum_{k=1}^\infty \left[\frac{1}{(6k-1)^3} - \frac{1}{(6k+1)^3}\right] = 1 - \frac{\pi^3}{18\sqrt{3}}$$ I think this equality likes the Dirichlet Beta function. The numerical value is checked but I don't have the proving. Any help | Notice$\color{blue}{^{[1]}}$ $$\sum_{k=1}^\infty \left( \frac{1}{(6k-1)^3} - \frac{1}{(6k+1)^3}\right) = \sum_{\substack{k=-\infty\\ k\ne 0}}^\infty \frac{1}{(6k-1)^3} = 1 - \frac{1}{6^3}\sum_{k=-\infty}^\infty \frac{1}{(\frac16-k)^3}$$ Recall the infinite product expansion of $\sin x$ $$\sin x = x \prod_{k=1}^\infty \... | Approach, which uses Fourier series. Denote $$ S = \sum_{k=1}^\infty \left[\frac{1}{(6k-1)^3} - \frac{1}{(6k+1)^3}\right]. $$ Consider function $$ f(x) = \dfrac{\pi x(\pi-x)}{8}, \qquad x\in[0,\pi];\tag{1} $$ construct the odd extension of $f(x)$ to the interval $[−\pi, \pi]$: $f(-x)=-f(x), x\in [0,\pi]$; and make it $... | HuggingFaceH4/stack-exchange-preferences |
Which of the following statements are correct?
Options:
1. One iteration of standard SGD for SVM costs roughly $\Theta(D)$, where $D$ is the dimension.
2. Unions of convex sets are convex.
3. Hinge loss (as in SVMs) is typically preferred over L2 loss (least squares loss) in classification tasks.
4. In PCA, the first p... | Let’s analyze each statement one by one to understand why they are correct:
### Statement 1: "Hinge loss (as in SVMs) is typically preferred over L2 loss (least squares loss) in classification tasks."
**Reasoning**:
- **Hinge Loss**: The hinge loss is specifically designed for "maximum-margin" classification tasks, m... | All three statements provided are correct. Here’s the reasoning for each:
1. **Hinge loss (as in SVMs) is typically preferred over L2 loss (least squares loss) in classification tasks.**
Hinge loss is specifically designed for "maximum-margin" classification, which is crucial in support vector machines (SVMs). It... | EPFL_courses_preference_pairs |
In comments to a Phys.SE question, it has been written: 'Tunneling' is perfectly real, even in classical physics. [...] For sufficiently large temperatures this can put the system above a hump in its potential energy. and the only difference between the classical case and the quantum mechanical one is that classical ph... | Frustrated total internal reflection is an optical phenomenon. It's such a close analogue to quantum tunneling that I sometimes even explain it to people as "quantum tunneling for photons". But you can calculate everything about it using classical Maxwell's equations. | There are two ways to see the analogy between the "quantum diffusion" and classical diffusion. The first one, I think the easier one is comparing the Schrödinger equation with the diffusion equation: $$i \partial_t \psi = -\sum \partial_{xx} \psi$$ (forgetting all the $\hbar,m$ factors) When you transform $t \to -i \ta... | HuggingFaceH4/stack-exchange-preferences |
Suppose A, B are invertible matrices of the same size. Show that $$M = \begin{bmatrix} 0& A\\ B& 0\end{bmatrix}$$ is invertible. I don't understand how I could show this. I have learned about linear combinations and spanning in my college class, but I don't know how that would help in this case. | You can check directly that $$ M^{-1}=\begin{bmatrix} 0&B^{-1}\\ A^{-1}&0\end{bmatrix}. $$ | Note that $M \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} w \\ z \end{bmatrix}$ iff $A y = w$ and $Bx = z$. From this you can compute an explicit inverse. | HuggingFaceH4/stack-exchange-preferences |
I've seen warnings that it's bad to mix new and old batteries -- why? Is it a matter of a battery's age or a battery's voltage/current (remaining)? | A simple model of a battery is a chemical reaction which produces a constant voltage. But, this chemical reaction takes time. A simple model of the limited speed of that reaction in electrical terms is a series resistance: simulate this circuit – Schematic created using CircuitLab When the battery is fresh, R1 is small... | The worst thing you can do to just about any kind of battery, whether rechargeable or not, is to push an amount of current through it which is large relative to its short-circuit current. As a battery gets depleted, its short-circuit current--and thus the level of current which it can safely handle--will diminish. If a... | HuggingFaceH4/stack-exchange-preferences |
A guy wire $78$ feet long runs from the top of a pole $56$ feet high to the ground and pulls on the pole with a force of $290$ pounds. What is the horizontal Pull on the top of the poll? I am not sure how do this problem. I tried to draw a triangle and one side is $56$ and the hypotenuse is $78$, but I am not sure what... | So, force is a vector and any force vector can be visualized in the Cartesian coordinate plane and decompose this vector into its components (horizontal, vertical, etc.). Always drawing a diagram (Force diagram) in these word problems is very helpful. So, draw a right triangle that best depicts the scenario. We want to... | Consider the force diagram: Observe that the horizontal component of the force is $\boldsymbol{F}_h = \boldsymbol{F}\cos\theta$, where $\boldsymbol{F} = 2900~\text{lb}$ in the indicated direction. Also, observe that $$\theta = \arcsin\left(\frac{56~\text{ft}}{78~\text{ft}}\right)$$ | HuggingFaceH4/stack-exchange-preferences |
Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a world where most scientists compute their integrals numerically, I find it astounding that people exist that can solve thes... | I hesitate to provide an answer to this because it feels very immodest, but perhaps I can provide something useful that doesn't merely restate the obvious or succumb to narcissism on my part. First of all, I will state something kind of obvious, as @FireGarden has pointed out. How do you get to Carnegie Hall? Practice!... | Personal Background I am by no means any expert at integration, but I have done a fair share of it. Integration is like a hobby to me. Instead of solving puzzles or riddles, I love evaluating integrals. As an example I am in the process of writing some personal notes on integration. Hopefully at least the questions wil... | HuggingFaceH4/stack-exchange-preferences |
So I asked my science teacher how wind could feel colder than normal air when the particles would be moving faster than normal air particles, meaning that they would have more kinetic energy and a higher temperature. My teacher told me that the faster rate of particles hitting your body would mean that the heat would l... | Your science teacher was wrong. Wind feels colder to your skin on a hot day because it evaporates your sweat faster, carrying away more heat from your skin than can conduction to still air at the same temperature. Kinetic energy is equal to 1/2 * (mass) * (velocity^2), so the assertion that kinetic energy is not connec... | While I am not sure of what "kinetic energy separate from velocity" means$^*$ (I will wait for clarification), there seems to be some confusion as to why wind makes one feel colder, even though the air is moving faster relative to your skin than still air would be. First, if we are stationary in a location with no wind... | HuggingFaceH4/stack-exchange-preferences |
I want you to act as a web developer. Give me the out line to create online book store project with Java backend with MVC model and Angular frontend | Creating an online bookstore using Java Backend with MVC model, and Angular frontend involves several components. Here's an outline to guide you through the project.
1. Project Setup
a. Backend setup - Create a Java project using a build tool like Maven or Gradle.
b. Frontend setup - Create an Angular project us... | As a web developer, I would be happy to help you create an online bookstore project with Java backend using the MVC model and Angular frontend. Here's a high-level outline of the steps you can take to build this project:
1. Plan and design the architecture of your bookstore application, including the overall structure... | argilla/ultrafeedback-binarized-preferences-cleaned |
Tick the \emph{correct} statement. $\Sigma$-protocols \ldots
Options:
1. are defined for any language in \textrm{PSPACE}.
2. have a polynomially unbounded extractor that can yield a witness.
3. respect the property of zero-knowledge for any verifier.
4. consist of protocols between a prover and a verifier, where the ve... | **Plan:**
- Understand the definition and properties of \(\Sigma\)-protocols in cryptography.
- Analyze each statement to determine its correctness based on the properties of \(\Sigma\)-protocols.
- Identify the correct statement among the options provided.
- Conclude with the correct answer indicating the verified sta... | To determine the **correct** statement regarding \(\Sigma\)-protocols, we need to analyze each option based on the definitions and properties of \(\Sigma\)-protocols in cryptographic theory.
**Options:**
1. Are defined for any language in \(\text{PSPACE}\).
2. Have a polynomially unbounded extractor that can yield a w... | EPFL_courses_preference_pairs |
Which one of these digital signature schemes is \emph{not} based on the Discrete Log problem?
Options:
1. DSA
2. ECDSA
3. Pointcheval-Vaudenay
4. PKCS\#1v$1.5$ | Answer: 4
Reasoning: PKCS#1v1.5 is a digital signature scheme that is based on RSA, which does not rely on the Discrete Log problem, unlike DSA and ECDSA that do. Pointcheval-Vaudenay might also involve some discrete logarithm concepts, but PKCS#1v1.5 is clearly different. | Answer: 4
Reasoning: PKCS#1 v1.5 is based on RSA, which relies on the integer factorization problem, not the Discrete Log problem. | EPFL_courses_preference_pairs |
I've written code for an obstructed random walker simulation and I want to run long simulations (6 hours or more on my computer). I don't want to run this simulation on my computer because I will want to run multiple simulations simultaneously and I don't want to have to worry about not having access to my computer's h... | Most supercomputing resources in the US are free to use for those associated with a US-based academic or research organization. You must make an application which is reviewed in some way (depending on the center or program that you apply to). For example, the US National Science Foundation funds the XSEDE project which... | You could consider using the Amazon web services. There is a free tier of usage which might work for you depending the computational resources required for your application. And if you need something more, stuff really isn't that expensive. There is a bit of a learning curve to get going on these resources. | HuggingFaceH4/stack-exchange-preferences |
Fair warning: I don't actually know a functional language so I'm doing all the pseudocode in Python I'm trying to understand why functional languages disallow variable reassignment, e.g. x = x + 1. Referential transparency, pure functions, and the dangers of side effects are all mentioned, but the examples tend to go f... | In a pure functional programming language, there is no real notion of time at all. So, saying that a variable x has value a at one point and then b later simply doesn't make any sense – it's like asking a character in a painting why she always stares in the same direction. The advantage of having no time is that you ne... | I can only share my perspective. The way I think of it is that mainstream functional languages typically combine two themes: (1) support for higher-order functions, and (2) a preference for pure computation (referential transparency, no side effects, no mutation). In principle those two could potentially be separable, ... | HuggingFaceH4/stack-exchange-preferences |
Definition: In this task, you are given two natural language statements with similar wording. You must choose the statement that makes less sense based on common sense knowledge. A '
' separates the statements. Use "first" or "second" to indicate which sentence makes less sense.
Input: People who catch criminals for li... | The statement "People who catch criminals for living are called fishermen" makes less sense. "People who catch fish for living are called fishermen" is the correct statement as it aligns with common knowledge and understanding of the term "fishermen". | first | argilla/ultrafeedback-binarized-preferences-cleaned |
Create a sentence with 4 adjectives. | The fluffy, white, playful kitten purred contentedly. | Sure, I'd be happy to help! Here is a sentence with four adjectives that describe my assistive style:
"I strive to provide helpful, respectful, honest, and engaging responses to your questions, ensuring that the content is accurate, relevant, and up-to-date, while promoting a positive and inclusive atmosphere." | argilla/ultrafeedback-binarized-preferences-cleaned |
Can you provide examples of partnerships or collaborations that the Italian Army has developed with community-based organizations focused on diversity and inclusion? | The Italian Army has developed several partnerships and collaborations with community-based organizations focused on diversity and inclusion. For example, in 2018, the Italian Army signed a memorandum of understanding with the Italian National Association of the Deaf (ENS) to promote the inclusion of deaf people in the... | While the Italian Army's focus has traditionally been on defense and national security, the organization has recognized the importance of diversity and inclusion. There have been several instances in which the Italian Army has collaborated with community-based organizations, although these partnerships might not be spe... | argilla/ultrafeedback-binarized-preferences-cleaned |
Explanation for the following question's answer: Barbers are not qualified to care for ill turtles.
Question: Of the below sentences, which one does *not* make sense?
Options:
- Sentence A: "if your pet turtle gets sick, take him to the barber"
- Sentence B: "if your pet turtle gets sick, take him to the vet"
Answer: ... | Response:
1. Sentence B: "if your pet turtle gets sick, take him to the barber"
Confidence: 100% | Question: Identify the nonsensical sentence among the options.
Options:
- Sentence A: "Water flowed through the tap."
- Sentence B: "Stones poured out of the tap."
Answer: Sentence B
Confidence: 95% | argilla/ultrafeedback-binarized-preferences-cleaned |
Here's my question: How many real roots does the cubic equation $y^3-3y +1$ have? I graphed the function and it crossed the x-axis $3$ times. But my professor doesn't want a graphical explanation. So in that case, I was looking at the Fundamental Theorem of Algebra and states that a polynomial of degree n can have at m... | The function has extrema where $$3y^2-3=0$$ i.e. at $y=\pm1$. The values at these extrema are $3$ and $-1$. So the variations of this continuous function are $-\infty,3,-1,\infty$, proving that there are three changes of sign. For a cubic polynomial, the search for the extrema may be more efficient than trial-and-error... | The given polynomial evaluated at $y\in\{-2,0,1,2\}$ exhibits three sign changes, hence it has at least $3$ real roots, and obviously cannot have more than three roots. | HuggingFaceH4/stack-exchange-preferences |
Let $\Omega\subseteq\mathbb{R}^2$, $(x_0,y_0)\in\Omega$, $f:\Omega\to\mathbb{R}$. Then $f$ is $\epsilon-\delta$ continuous at $(x_0,y_0)\in\Omega$ if, for any $\epsilon>0$, there exists a $\delta>0$ [$\delta$ depends on $\epsilon$ and $x_0$] such that \begin{align*}\tag{1} \forall x\in\Omega, |(x,y)-(x_0,y_0)|<\delta\L... | If I'm interpreting this correctly... There are a total of $36$ combinations you can get when rolling a pair of $6$-sided dice. There is $1$ way to get a $2$, $2$ ways to get a $3$, $3$ ways to get a $4$, etc. You have a total of $36\cdot 36= 1296$ events. Now, we do a bit of counting. There is $1$ way both people can ... | Deducing, like Sherlock Holmes, that the question is: Two people throw two dice each. What is the probability that both get the same sum ? Ways of throwing sums of $2 - 12$ with $2$ dice follows the pattern $1-2-3-4-5-6-5-4-3-2-1$ So P(both get the same sum) = $\dfrac{(1^2+2^2+3^2+ ....+6^2+5^2+4^2... +1^2)}{36^2}= \df... | HuggingFaceH4/stack-exchange-preferences |
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