Linearly independent eigenvectors of $2 \times 2$ matrix with same eigenvalue
Prove that T is diagonalizable if and only if the minimal polynomial of T has no repeated roots.
Does every linearly independent set of n vectors in $R^n$ forms a basis in $R^n$? Basically does a vector set that is linearly independent in $R^n$ automatically span $R^n$?  My initial thought is yes, but is there some counterexample that can disprove this? How do I calculate rotation caused by bounce friction? I got the brand new Custodian badge today for reviewing posts. Inspired by that, I reviewed some more suggested edits. Now I ran into a limit:     You have no more suggested edit votes today; come back in x hours.   How is that limit calculated? I made 61 reviews today which seems to be some calculated limit.  I searched for an answer on meta but found nothing helpful. Show reference label when it is undefined When a reference is undefined, LaTeX shows simply [??] which doesn't give much information. I would like it to show the label instead. This is useful when more collaborators work on the same preprint and the references are yet to be done.  I'm using neither biblatex nor natbib. I do use bibtex but it is irrelevant for this question.  MWE:  \documentclass{article}  \begin{document}  This: \cite{undef} should look like: [{\bfseries ?? undef}]  \end{document} LWC to display salesforce objects as a picklist onclick I have a requirement where I need to display list of salesforce objects as a picklist when a radio button is clicked.  Being new to LWC development, I could not figure out what is the right approach to do so. I have come across  from Salesforce articles where we can display radio buttons but as per my scenario, i need to display list of all salesforce objects as picklist when a radio button option is selected.  Can anyone please suggest the right approach to do so. I'm somewhat new to using logistic regression, and a bit confused by a discrepancy between  my interpretations of the following values which I thought would be the same:    exponentiated beta values  predicted probability of the outcome using beta values.    Here is a simplified version of the model I am using, where undernutrition and insurance are both binary, and wealth is continuous:  Under.Nutrition ~ insurance + wealth   My (actual) model returns an exponentiated beta value of .8 for insurance, which I would interpret as:     "The probability of being undernourished for an insured individual is .8 times the probability of being undernourished for an uninsured individual."   However, when I calculate the difference in probabilities for individuals by putting in values of 0 and 1 into the insurance variable and the mean value for wealth, the difference in undernutrition is only .04. That is calculated as follows:   Probability Undernourished = exp(β0 + β1*Insurance + β2*Wealth) /                              (1+exp(β0 + β1*Insurance + β2*wealth))   I would really appreciate it if someone could explain why these values are different, and what a better interpretation (particularly for the second value) might be.     Further Clarification Edits As I understand it, the probability of being under-nourished for an uninsured person (where B1 corresponds to insurance) is:  Prob(Unins) = exp(β0 + β1*0 + β2*Wealth) /               (1+exp(β0 + β1*0+ β2*wealth))   While the Probability of being under-nourished for an insured person is:  Prob(Ins)= exp(β0 + β1*1 + β2*Wealth) /            (1+exp(β0 + β1*1+ β2*wealth))   The odds of being undernourished for an uninsured person compared to an insured person is:  exp(B1)   Is there a way to translate between these values (mathematically)? I'm still a bit confused by this equation (where I should probably be a different value on the RHS):  Prob(Ins) - Prob(Unins) != exp(B)   In layman's terms, the question is why doesn't insuring an individual change their probability of being under-nourished as much as the odds ratio indicates it does?  In my data, Prob(Ins) - Prob(Unins) = .04, where the exponentiated beta value is .8 (so why is the difference not .2?) Earth Engine convert list with coordinates and values into a feature collection for export I need to create a feature collection from a list containing coordinates and band values.  These are extracted from a landsat image as shown in this linked script.    I'm unable to find a solution in creating a feature collection.  Here is the tail end code.  var valuesList = joinedImage.reduceRegion({   reducer: ee.Reducer.toList(4),   geometry: myGeometry }).values().get(0);  var feature = ee.Feature(null, valuesList);   // Wrap the Feature in a FeatureCollection for export. var myFeatures = ee.FeatureCollection([feature]);   // Export the image, specifying scale and region. Export.table.toDrive(myFeatures, "data", "myData", "B4-B5",  "CSV");   This doesn't work, because it is clearly the wrong way to create the feature collection.  I've tried to create it similar to Rodrigo E. Principe's solution of mapping a Feature Collection but I can't seem to create the initial feature collection to start with. Cancelling upvote on comment? As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click). Is there any reference where Captain America's shield is made from stolen Wakanda vibranium? Normal distribution tail probability inequality I am trying to show that $$P(X&gt;t)\leq \frac{1}{2}e^\frac{-t^2}{2}$$ for $t&gt;0$ where $X$ is a standard normal random variable.  Perhaps this is simple.  I have been starting with  $$ \int_{t}^{\infty} \frac{1}{\sqrt{2\pi}} e^\frac{-x^2}{2}dx \leq \int_{t}^{\infty}\frac{x}{t}\frac{1}{\sqrt{2\pi}}e^\frac{-x^2}{2}dx = \frac{1}{t\sqrt{2\pi}}e^\frac{-t^2}{2}$$  but this is not what I want...I am looking for a much stronger inequality.   Any help in the direction of getting the $1/2$? Distribution of $Z_n=n \log\left(\frac{\max\left(X_{(n)},Y_{(n)}\right)}{\min\left(X_{(n)},Y_{(n)}\right)}\right)$ The random variables $X_1,X_2,...X_n,Y_1,Y_2,...Y_n $ are independent and $U(0,a)$-distributed. Determine the distribution of $$Z_n=n\cdot \log\left(\frac{\max\left(X_{(n)},Y_{(n)}\right)}{\min\left(X_{(n)},Y_{(n)}\right)}\right)$$ where $X_{(n)},Y_{(n)}$ denotes $n$th order variable.  This problem is giving me a headache. The answer is that $Z_n$ is $\text{Exp}(1)$-distributed. But i can't derive that conclusion. I'd like to show you my derivation, but its so long and contains so many integrals and fractions that it would take a while to write it here. But basically my approach was to derive the distribution of $M=\max(X_{(n)},Y_{(n)})$ and $L=\min(X_{(n)},Y_{(n)})$. I get the following cdf's: $F_M=(F_X\cdot F_Y)^n=(F_X)^{2n}$ and $F_L=1-(1-(F_X)^n)\cdot (1-(F_Y)^n)=1-(1-(F_X)^n)^2$ ,  correct? $F_X $ stands for the cdf of $X_i$  Then i define the random variable $T=\frac{M}{L}$ and compute its cdf to finally be able to derive the distribution of $Z_n=n\cdot \log(T)$. But thats where i think everything goes wrong. Any help appreciated, thanks. Let $d=(a^n,b^n), d_1=(a,b)^n$; need show $d\mid d_1$ &amp; $d_1\mid d$ to prove.  For the general case of $(a,b)\ne 1, (a,b)= (d_1)^\frac{1}{n}$, else $(a,b)=d_1$.  Let, $d_2= (a,b)=d_1^\frac{1}{n}\implies a=a'd_2, b = b'd_2\implies a^n= (a')^nd_2^n, b^n= (b')^nd_2^n$  Unable to progress further. A complete residue system from an arithmetic progression I'm solving a problem, and the solution makes the following statement: "The common difference of the arithmetic sequence 106, 116, 126, ..., 996 is relatively prime to 3. Therefore, given any three consecutive terms, exactly one of them is divisible by 3." Why is this statement true? Where does it come from? Is it generalizable to other numbers? Is it possible to assign the same value to multiple keys in a dict object at once? Elementary argument for conservation laws from symmetries *without* using the Lagrangian formalism It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. Determine which app is drawing on top of other apps? Is there a way to determine which application is drawing on top of other applications?  For example, Facebook Messenger app can draw the chat bubbles on top of everything, but how do I know which app is doing this?  My problem:  I have an app which is drawing ads on top of everything and I wanna know which app is doing this. Not just to uninstall it, but to warn everyone that uses this app. (I didn't install any application recently and I already verified the recent updates.)  I tried to use the Dump View Hierarchy for UI Automator from the Android Studio, but it seems that the applications that draw on top can't be selected this way.  (Click image to enlarge) A generic list of anonymous class Find all primes $p$ such that  $ p^3-4p+9 $ is a perfect square.   I tried a few different values for $p$, namely $2,3,5,7,$ and $11$. The prime $p =2,7,11$ all worked but $p =13$ didn't so it makes me wonder. How can I find all primes such that it is a perfect square?  EDIT: it turns out this is problem P25 in post number #63 at How does "Reputation" work? On Stack Exchange, users may gain a certain level of reputation.  What does reputation do? How can a user gain or lose reputation?   See also   in the Help Center  - Maximum amount of votes a user can use in a day and reputation requirements to vote.  - How to get an exact overview of the actions that got you your reputation.  - A full list of all privileges granted at each reputation level on all sites. Can I simplify an expression into form which uses my own definitions?