Find isomorphism from $F_5[x]/(x^2+x+2) \rightarrow F_5[x]/(x^2+4x+2) $ I realise both polynomials are irreducible therefore form fields, not sure how to form a isomorphism from one to the other, help please
Find an explicit isomorphism of the map $h: \Bbb F_5[x]/(x^2+x+2) \to \Bbb F_5[x]/(x^2+4x+2)$  I take the question to mean showing the 2 rings are isomorphic by finding an explicit mapping of each element.   I think $x^2+x+2$ and $x^2+4x+2$ are irreducible over $\Bbb F_5$, so these are both fields.  Also $x^2=4x+3$ in the first case and $x^2=x+3$ in the second.  But beyond these basic facts I'm not sure how I should proceed.
How to properly unwrap my mesh? Is there a way to automatically unwrap my character? If not, is there a tutorial or something? Suppose we have a sample of size n= 9, where the sum equals 27, and the sum of the squaredvalues equals 113. Sample mean is 27/9=3 and mean of squared values is 113/9 = 12.56 Can we then say that the sample variance $S^2 = E(\overline X^2) - E(\overline X)^2 = 12.55 - 9 = 3.56$ ? Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$.   Relevant definitions:  $G$ is a simple graph that consists of a vertex set $V(G) = \{v_1, v_2, ..., v_n\}$ and an edge set $E(G) = \{e_1, e_2, ..., e_m\}$ where each edge is an ordered pair of vertices. The edge $\{u,v\}$ is denoted $uv$.  A walk of length $k$ is a sequence $v_0,e_1,v_1,e_2,...,e_k,v_k$ of vertices and edges such that $e_i=v_{i-1}v_i$ for all $i$. A path is a walk with no repeated vertex.   My attempt:  Induction, for $k=1$ it's obvious.  Suppose for $k-1$ and we'll prove for $k$.  Let $G$ be a simple graph such that $\forall v \in V : d(v)\ge k$.  We can assume that there are at least $k+1$ vertices since there are $k$ neighbours to every vertex.  Let $v_0$ be some vertex, it has $k$ neighbours, we'll move to one of its neighbours say $v_1$ and remove $v_0$. So now there are $k$ vertices with a degree of at least $k-1$ and from the induction hypothesis we'll have a path of length $k$.  Is using only $k$ vertices from $n$ right? Does it keep the generality when using only all the neighbours of $v_0$? Yes, I'm really sure I want to delete that post What is the cause for the validity of Newton's third law? What, specifically, causes newton's third law?  For instance, if I push on a wall, why is it that I experience a force in the opposite direction?    I seem to vaguely understand that is has something to do with electronic repulsion or molecular compression (maybe that's completely wrong, I don't know).  As a related question, what would happen if two objects that were /infinitely/ immovable (at the molecular level, it cannot be broken or compressed) collided with each-other? Variable wavelength IR Diode? I was thinking of building a DIY IR Spectroscope but for this, I need to find an IR diode that can do variable wavelengths. So far I have looked at Farnell and Mouser but all the diodes I have looked at seem to be fixed on a wavelength.   Is this possible? or would I need some sort of filters and optical equipment, if so which parts would I need?  Thanks. Before we launched our website, we had set up a "coming soon" page and google picked up the title and description from its contents. So the description in the search results said:     Coming soon! Visit example.org for updates.   It's been a few weeks since we launched our website. We've even created a sitemap and submitted it to Google. In Google Search Console, the pages have been crawled and all the pages are appearing as expected on Google, EXCEPT the homepage which is still not updated! The title and description of the homepage in google search results still says "coming soon.".***  I checked on bing too, but it appears to be fine there.  All other pages seem to be indexed fine on Google. I don't even have any crawl errors in my reports. So what seems to be the problem? I've already waited for 2 weeks. A few months ago I was introduced to the new  type and was glad 's flaws with regard to time zones were finally taken care of.  However, I was left wondering if there were any overhead or problems that could occur from using this new type.  I work on a multi-locale web application. Does anyone know of anything that could sway me from just using it for all my date/time work? Is there a window for abuse here?  Reference: Displaying verbatim content with \tl_to_str:* \documentclass{article} \usepackage{expl3} \ExplSyntaxOn \begin{document} \tl_to_str:n { testing _ things }  \par  \tl_set:Nn \l_tmpa_tl { testing _ things } \tl_to_str:N \l_tmpa_tl \end{document}   gives a strange result (testing ̇things).  How can I get testing_things?  I basically want verbatim content (with line breaking). Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the smallest positive integer $d$ such that $n\mid q^d-1$.  My approach: It's easy to show that if $k$ is a finite field of order $q$ with $\text{char} \ k=p$ then $k=\mathbb{F}_{q}$ with $q=p^m$ with $m\geq 1$. If $K$ is the splitting field of $X^n-1$ over $k$ then $K=\mathbb{F}_{q^r}$ for some suitable $r\geq 1$.  Let $\alpha \in K=\mathbb{F}_{q^r}$ be a root of $X^n-1$ then $\alpha^n-1=0$ and $\alpha\neq 0$.   Thus, $\alpha^n=1$ and $\alpha \in \mathbb{F}^{\times}_{q^r}$ where $\mathbb{F}^{\times}_{q^r}$ is the cyclic group of order $q^r-1$.  But I have some difficulties:  1) From $\alpha^n=1$ it follows that $o(\alpha)\mid n$. How to show that $o(\alpha)=n$? I was trying to show it using that $(n,p)=1$ but I failed.  Remark: Because if we can show that $o(\alpha)=n$ then the desired result follows immediately from cyclicity of multiplicative group of field $\mathbb{F}_{q^r}$.  I have spent one day in order to come up with this approach. So please do not duplicate this question and can anyone show how to answer my question, please.  Detailed explanation would be great! Symmetric powers of modules Understanding the parameters inside the Negative Binomial Distribution I was trying to fit my data into various models and figured out that the fitdistr function from library MASS of R gives me Negative Binomial as the best-fit. Now from the  page, the definition is given as:     NegBin(r,p) distribution describes the probability of k failures and r   successes in k+r Bernoulli(p) trials   with success on the last trial.   Using R to perform model fitting gives me two parameters mean and dispersion parameter. I am not understanding how to interpret these because I cannot see these parameters on the wiki page. All I can see is the following formula:    where k is the number of observations and r=0...n. Now how do I relate these with the parameters given by R? The help file does not provide much information either.   Also, just to say a few words about my experiment: In a social experiment that I was conducting, I was trying to count the number of people each user contacted in a period of 10 days. The population size was 100 for the experiment.   Now, if the model fits the Negative Binomial, I can blindly say that it follows that distribution but I really want to understand the intuitive meaning behind this. What does it mean to say that the number of people contacted by my test subjects follows a negative binomial distribution? Can someone please help clarify this? wanted to get end date for Current Fiscal Quarter in apex any ideas, how to get this information. I am a little bit confused by early stopping and in particular by how it can be inserted inside a CV framework. As far as I understand, I can fix the optimal number of epochs (for NN, or number of trees for XGB) by early stopping, that is:   pick a validation set, train with increasing number of epochs until a predefined metric evaluated on the validation set starts worsening that is the optimal number of epochs to prevent overfitting   and that's fine. But then I would like to insert this early stopping framework inside a CV framework: Suppose I have a model with 10 hyperparameters I want to fix via CV. And suppose that there is an eleventh hyperparameter, the number of epochs. My feeling is that one can do like this:   create the K resampled folds, for each of which you have a training and validation set choose a suitable grid for your 10 hyperparameters for each point on the grid train your model in each fold with early stopping, that is use the validation set of the fold to keep track of the preferred metric and stop when it gets worse take the mean of the K validation metric choose the point of the grid (i.e. the set of hyperparameters) that gives the best metric   Questions:   What number of epochs should I choose as optimal? in each of the K folds I have, in general, a different number of stopping epochs.  seems to suggest they take the mean of the K stopping epochs. Is this right? Is it actually a "fair" practice to use the validation metric coming from the early stopping as a proxy of the out-of-sample metric? As  in section 3.4.5, maybe the best thing to do would be:     ...if you want to do early stopping, then in each fold you should take your training set and split it again, holding out a small early-stopping-set to guide the early stopping, and then evaluate the model on the validation set of that fold.   But this seems to me a really intricate process...   Unfortunately, I wasn't able to find references where this issue is presented in a clear and transparent form. Are any of the USB ports on a Macbook Pro "always on" Always on USB ports will provide power to a device even when the laptop is shut down. Obviously this can be problematic for battery issues, but sometimes a user wants this capability all things considered.  Do the Macbook Pro offer this on any of the USB ports and if not is there any chance its wired and just has to be enabled? What does 's=.*/==' do in sed? I saw sed 's=.*/==' in context of sh script and I'm puzzled. I could not find in sed manual or web search (for sed s=) how s is used, not s///. Apart from s I see only one potential command here = (Print the current input line number), but in such case what the rest is doing...   Running the command in shell produces same output as input for e.g echo 'jkfdsa=335r34',  whereas echo 'jkfdsa=335r34' | sed 's/=.*/==/' does replacement as per manual. Also slightly modifying command to e.g. echo 'jkfdsa=3' | sed 's798=.*/==/' gives sed: -e expression #1, char 11: unterminated 's' command, so original should have some correct meaning. What is it? What price is a limit buy and a limit sell order settled at? A market has an existing limit buy order at $1.00.  The market then receives a limit sell order at $0.95, what price is the transaction settled at? Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices. In C#/VB.NET/.NET, which loop runs faster, for or foreach?  Ever since I read that a for loop works faster than a foreach loop a  I assumed it stood true for all collections, generic collections, all arrays, etc.  I scoured Google and found a few articles, but most of them are inconclusive (read comments on the articles) and open ended.  What would be ideal is to have each scenario listed and the best solution for the same.  For example (just an example of how it should be):   for iterating an array of 1000+ strings - for is better than foreach for iterating over IList (non generic) strings - foreach is better than for   A few references found on the web for the same:               [Edit]  Apart from the readability aspect of it, I am really interested in facts and figures. There are applications where the last mile of performance optimization squeezed do matter. What is the norm of this bounded linear functional? Let $a$, $b$ be two arbitrary but fixed real numbers such that $a &lt; b$, let $C[a,b]$ denote the normed space of all continuous real (or complex) valued functions defined on $[a,b]$ with the maximum norm, and let $f$ be the functional defined on $C[a,b]$ as $$f(x):= \int_{a}^{b} x(t) x_0(t) dt $$ for all $x$ in $C[a,b]$, where $x_0$ is a fixed element of $C[a,b]$.   Now $f$ is evidently linear and bounded. How to compute the norm of $f$? What is this norm?   And what is the norm of the bounded linear functional $g$ defined as follows: $$g(x):= \int_{a}^{(a+b)/2} x(t) \cdot dt - \int_{(a+b)/2}^{b} x(t) \cdot dt$$ for all $x$ in $C[a,b]$?