I was trying to find if the the following limit exists: $$ \lim_{(x,y)\to (0,0)} \frac{x^3y^2}{x^6+y^4} $$ From  I tried to switch to polar coordinates: $$ \frac{x^3y^2}{x^6+y^4}=\frac{r^3\cos^3\theta\cdot r^2\sin^2\theta}{r^6\cos^6\theta+r^4\sin^4\theta}=\frac{r\cos^3\theta\sin^2\theta}{r^2\cos^6\theta+\sin^4\theta}\to 0 $$ But apparently this limit does not exists (proved it in another way). Why this method worked for the limit in that topic and here it didn't work?
Given $\frac{a}{c} + \frac{b}{d} = 1$ Prove that $$\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$$ does not exist.  So I have done the proof for strict inequalities. And the limit only exists when the fractions add up to an integer greater than $1$. I'm not sure how to approach this other than counter examples.
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$. Exercise 11, page 45 from Hungerford's book Algebra.     If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every   subgroup of $H$ is normal in $G$.   I am trying to show that $a^{-1}Ka\subset K$, but I got stuck. What I am supposed to do now?  Thanks for your kindly help. I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear.  One such concept is the notion of contra- and co-variant vectors.  describes a contra-variant vector as varying with the co-ordinates, i.e. the inverse of the reference axes. It gives the example of a change of basis matrix $\mathbf{M}$, and a contra-variant vector $\vec{v}$ and states that in the new set of co-ordinates, we have:  $$\vec{v}'=\mathbf{M}\vec{v}$$  It then states that a covariant vector varies with the co-ordinate axes.     Does this mean that for a covariant vector $\vec{u}$ and change of basis matrix $\mathbf{M}$ we have: $$\vec{u}'=\mathbf{M}^{-1}\vec{u}?$$   I am also having problems understanding the concept of raising and lowering indices. For instance, Wikipedia states that for two four-vectors $\vec{A},\vec{B}$ and using Einstein summation we have:  $$\left\langle \vec{A} \middle|\vec{B} \right\rangle=A^{\mu}B_{\mu}=A_{\nu}B^{\nu}$$  And I am having difficulty understanding how this relates to the other definition of the inner product:  $$\left\langle \vec{A} \middle|\vec{B} \right\rangle =\mathbf{A}^{T}\mathbf{\eta}\mathbf{B}$$  If anyone could help give me a more intuitive understanding of these concepts I would be grateful. How to use pagenumbering in the document? I'm having some difficulty understanding how to interpret variable importance output from the Random Forest package. Mean decrease in accuracy is usually described as "the decrease in model accuracy from permuting the values in each feature".   Is this a statement about the feature as a whole or about specific values within the feature? In either case, is the Mean Decrease in Accuracy the number or proportion of observations that are incorrectly classified by removing the feature (or values from the feature) in question from the model?   Say we have the following model:  require(randomForest) data(iris) set.seed(1) dat &lt;- iris dat$Species &lt;- factor(ifelse(dat$Species=='virginica','virginica','other')) model.rf &lt;- randomForest(Species~., dat, ntree=25, importance=TRUE, nodesize=5) model.rf varImpPlot(model.rf)     Call:  randomForest(formula = Species ~ ., data = dat, ntree = 25,  proximity = TRUE, importance = TRUE, nodesize = 5)  Type of random forest: classification Number of trees: 25 No. of variables tried at each split: 2          OOB estimate of  error rate: 3.33% Confusion matrix:           other virginica class.error other        97         3        0.03 virginica     2        48        0.04     In this model, the OOB rate is rather low (around 5%). Yet, the Mean Decrease in Accuracy for the predictor (Petal.Length) with the highest value in this measure is only around 8.   Does this mean that removing Petal.Length from the model would only result in an additional misclassification of 8 or so observations on average?   How could the Mean Decrease in Accuracy for Petal.Length be so low, given that it's the highest in this measure, and thus the other variables have even lower values on this measure? How to import a jar in Eclipse I am flying from the US to Edinburgh, UK. My layover is at Heathrow. Will I need to go through customs and check bags again at Heathrow before I take the next flight? Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$      I need help solving this question. Every reply is appreciated.  Thanks! Testing for linear dependence among the columns of a matrix I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be positive definite.)  My hypothesis is that at least one security is linearly dependent on other securities. Is there a function in R that sequentially tests each column a matrix for linear dependence?  For example, one approach would be to build up a correlation matrix one security at a time and calculate the determinant at each step. When the determinant = 0 then stop as you have identified the security who is a linear combination of other securities.  Any other techniques to identify linear dependence in such a matrix are appreciated. How exactly does a "random effects model" in econometrics relate to mixed models outside of econometrics? I used to think that "random effects model" in econometrics corresponds to a "mixed model with random intercept" outside of econometrics, but now I am not sure. Does it?  Econometrics uses terms like "fixed effects" and "random effects" somewhat differently from the literature on mixed models, and this causes a notorious confusion. Let us consider a simple situation where $y$ linearly depends on $x$ but with a different intercept in different groups of measurements:  $$y_{it} = \beta x_{it} + u_i + \epsilon_{it}.$$  Here each unit/group $i$ is observed at different timepoints $t$. Econometricians call it "panel data".    In mixed models terminology, we can treat $u_i$ as  a fixed effect or as a random effect (in this case, it's random intercept). Treating it as fixed means fitting $\hat \beta$ and $\hat u_i$ to minimize squared error (i.e. running OLS regression with dummy group variables). Treating it as random means that we additionally assume that $u_i\sim\mathcal N(u_0,\sigma^2_u)$ and use maximum likelihood to fit $u_0$ and $\sigma^2_u$ instead of fitting each $u_i$ on its own. This leads to the "partial pooling" effect, where the estimates $\hat u_i$ get shrunk toward their mean $\hat u_0$.  R formula when treating group as fixed:    y ~ x + group R formula when treating group as random:   y ~ x + (1|group)  In econometrics terminology, we can treat this whole model as a fixed effects model or as a random effects model. The first option is equivalent to the fixed effect above (but econometrics has its own way of estimating $\beta$ in this case, called "within" estimator). I used to think that the second option is equivalent to the random effect above; e.g. @JiebiaoWang in his highly upvoted answer to  says that      In econometrics, the random-effects model may only refer to random intercept model as in biostatistics    Okay --- let us test if this understanding is correct. Here is some random data generated by @ChristophHanck in his answer to  (I put  for those who do not use R):    @Christoph does two fits using econometrics approaches:    fe &lt;- plm(stackY~stackX, data = paneldata, model = "within") re &lt;- plm(stackY~stackX, data = paneldata, model = "random")   The first one yields the estimate of beta equal to -1.0451, the second one 0.77031 (yes, positive!). I tried to reproduce it with lm and lmer:  l1 = lm(stackY ~ stackX + as.factor(unit), data = paneldata) l2 = lmer(stackY ~ stackX + (1|as.factor(unit)), data = paneldata)   The first one yields -1.045 in perfect agreement with the within estimator above. Cool. But the second yields -1.026, which is miles away from the random effects estimator. Heh? What is going on? In fact, what is  plm even doing, when called with model = "random"?  Whatever it is doing, can one somehow understand it via the mixed models perspective?  And what is the intuition behind whatever it is doing? I read in a couple of econometrics places that random effects estimator is a weighted average between the fixed effects estimator and the "between" estimator which is more or less regression slope if we do not include group identity in the model at all (this estimate is strongly positive in this case, around 4.) E.g. @Andy :     The random effects estimator then uses a matrix weighted average of the within and between variation of your data. [...] This makes random effects more efficient[.]   Why? Why would we want this weighted average? And in particular, why would we want it instead of running a mixed model? Changing pemalink structure to /%post_id%/%postname%/ I'm currently using permalink structure: /%year%/%monthnum%/%day%/%postname%/  I want to change that to: /%post_id%/%postname%/  I expect it to be:   faster (database is sorted by post_id?) shorter URLs (and clearly visible short verstions, just leaving the %postname%/ out) more SEO friendly - since my content is "evergreen" so publishing date is irrelevant (and written at each post, along with last update date) - so this would give shorter URLs, with no downsides compared to using year/month/day structure.   THE QUESTION: Does WordPress handle all the 301 redirects automatically? I can fix internal links to the new structure (though I'd expect that to be automatic as well), but worry about outsite links, pointing to my posts/articles.  If not, what's the best way to implement this change? Any links or full answers here are welcome.  I have about 120+ posts already, will this change affect the speed since it's not been done from the start? Formula for prime counting function I saw this formula on  page 2  $$\pi (n)=\sum_{j=2}^{n}\frac{\sin^{2}\left(\pi \frac{(j-1)!^{2}}{j}\right)}{\sin^{2}(\frac{\pi }{j})}$$  Where $\pi(n)$ is the prime counting function. Is this true? How to prove it? I am traveling from the US to India via London Heathrow on November 6. I will be at LHR on 7th at 6:20 am and departing at 9:25 pm. I'm an Indian national and have a valid US visa.  Do I need a transit visa to stay airside?  I have been scared that British Airways have stopped a few people from boarding flights (no details just rumors).  Do I need a transit visa to stay landside? I want to do some sightseeing. What are the chances of getting past officials using transit without visa concession?  Real stories will help a lot to calm the butterflies in my stomach. Finding minimum-area-rectangle for given points? I am using pdflatex on a fresh (and current) MikTex 2.9 install with the following MWE:  \documentclass{article} \usepackage{animate} \begin{document}  Test \end{document}   Pdflatex throws the error:  Package: pdfbase 2017/03/23 v0.14 driver independent access to low-level PDF fe atures \g_pbs_page_int=\count175  ! Undefined control sequence. &lt;argument&gt; \luatexversion                            &lt;95 l.924 }   I have read the other posts containing the offending control sequence, especially  but the solutions offered there didn't help in my case.  This problem might have arisen with a recent MikTex update. I have now completely uninstalled, redownloaded, reinstalled and updates MikTex -- with no success. Are immutable arrays possible in .NET? Is it possible to somehow mark a System.Array as immutable.  When put behind a public-get/private-set they can't be added to, since it requires re-allocation and re-assignment, but a consumer can still set any subscript they wish:  public class Immy {     public string[] { get; private set; } }   I thought the readonly keyword might do the trick, but no such luck. Kernel Panic - not syncing: VFS: Unable to mount root fs on unknown-block(0,0) Upon trying to upgrade from 10.10 to 11.04 all seemed to go well until the restart. This error message is what comes up:  Kernel Panic - not syncing: VFS: Unable to mount root fs on unknown-block(0,0)   How do we fix that? Accessing the iPhone's Call log with the iPhone SDK How to find an ellipse , given 2 passing points and the tangents at them? Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ellipse which satisfies it?  Thank you in advance.  Edit : May be , I can say the first answer (by Mr. André Nicolas) is for the general case. Is there no special case where only a finite set of ellipses satisfies the condition?  Edit : According to the answers and comments , I can compute an arbitrary chosen ellipse for my condition (using the method by Mr.Patrick Da Silva). But there are possibly many others which satisfy my conditions. Am I right? best way to pick a random subset from a collection? I have a set of objects in a Vector from which I'd like to select a random subset (e.g. 100 items coming back; pick 5 randomly). In my first (very hasty) pass I did an extremely simple and perhaps overly clever solution:  Vector itemsVector = getItems();  Collections.shuffle(itemsVector); itemsVector.setSize(5);   While this has the advantage of being nice and simple, I suspect it's not going to scale very well, i.e. Collections.shuffle() must be O(n) at least. My less clever alternative is  Vector itemsVector = getItems();  Random rand = new Random(System.currentTimeMillis()); // would make this static to the class      List subsetList = new ArrayList(5); for (int i = 0; i &lt; 5; i++) {      // be sure to use Vector.remove() or you may get the same item twice      subsetList.add(itemsVector.remove(rand.nextInt(itemsVector.size()))); }   Any suggestions on better ways to draw out a random subset from a Collection? Best practices for storing UI settings?