$a,b,c$ are positive real number. Prove that $(a^2+2)(b^2+2)(c^2+2)&gt;9(ab+bc+ca)$  I tried to it by AM-GM inequality but it is not coming
I have a  beautiful  inequality, but I can only prove part of cases. Given any positive real numbers $a,b,c$, we have   $$(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$$  How can we prove this statement?
I've just started learning Lisp and I can't figure out how to compile and link lisp code to an executable.  I'm using clisp and clisp -c produces two files:   .fas .lib   What do I do next to get an executable? Laplace equation of composite function Suppose we have $f(x,y)$, which satisfies Laplaces equation: $\frac{\delta^2f}{\delta x^2}+\frac{\delta^2f}{\delta y^2}=0$  Does $f(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})$ also satisfy Laplace equation? automatically execute an Excel macro on a cell change Principled way of collapsing categorical variables with many levels? What techniques are available for collapsing (or pooling) many categories to a few, for the purpose of using them as an input (predictor) in a statistical model?    Consider a variable like  (discipline chosen by an undergraduate student). It is unordered and categorical, but it can potentially have dozens of distinct levels. Let's say I want to use major as a predictor in a regression model.  Using these levels as-is for modeling leads to all sorts of issues because there are just so many. A lot of statistical precision would be thrown away to use them, and the results are hard to interpret. We're rarely interested in specific majors -- we're much more likely to be interested in broad categories (subgroups) of majors. But it isn't always clear how to divide up the levels into such higher-level categories, or even how many higher-level categories to use.  For typical data I would be happy to use factor analysis, matrix factorization, or a discrete latent modeling technique. But majors are mutually exclusive categories, so I'm hesitant to exploit their covariance for anything.  Furthermore I don't care about the major categories on their own. I care about producing higher-level categories that are coherent with respect to my regression outcome. In the binary outcome case, that suggests to me something like linear discriminant analysis (LDA) to generate higher-level categories that maximize discriminative performance. But LDA is a limited technique and that feels like dirty data dredging to me. Moreover any continuous solution will be hard to interpret.  Meanwhile something based on covariances, like multiple correspondence analysis (MCA), seems suspect to me in this case because of the inherent dependence among mutually exclusive dummy variables -- they're better suited for studying multiple categorical variables, rather than multiple categories of the same variable.  edit: to be clear, this is about collapsing categories (not selecting them), and the categories are predictors or independent variables. In hindsight, this problem seems like an appropriate time to "regularize 'em all and let God sort 'em out". Glad to see this question is interesting to so many people! Why do we need autoencoders? Recently, I have been studying autoencoders. If I understood correctly, an autoencoder is a neural network where the input layer is identical to the output layer. So, the neural network tries to predict the output using the input as golden standard.  What is the usefulness of this model? What are the benefits of trying to reconstruct some output elements, making them as equal as possible to the input elements? Why should one use all this machinery to get to the same starting point? Horned Mohawk looking insect. What is it? Any ideas? I didn't know any other way to explain it.  I live in Indiana, USA if that helps.   Someone said it might be a "stink bug" but I'm not sure and not willing to find out! There are Recurrent Neural Networks and Recursive Neural Networks. Both are usually denoted by the same acronym: RNN. According to , Recurrent NN are in fact Recursive NN, but I don't really understand the explanation.  Moreover, I don't seem to find which is better (with examples or so) for Natural Language Processing. The fact is that, although Socher uses Recursive NN for NLP in his , I can't find a good implementation of recursive neural networks, and when I search in Google, most of the answers are about Recurrent NN.  Besides that, is there another DNN which applies better for NLP, or it depends on the NLP task? Deep Belief Nets or Stacked Autoencoders? (I don't seem to find any particular util for ConvNets in NLP, and most of the implementations are with machine vision in mind).  Finally, I would really prefer DNN implementations for C++ (better yet if it has GPU support) or Scala (better if it has Spark support) rather than Python or Matlab/Octave.  I've tried Deeplearning4j, but it's under constant development and the documentation is a little outdated and I can't seem to make it work. Too bad because it has the "black box" like way of doing things, very much like scikit-learn or Weka, which is what I really want. Why is the B field of a solenoid equal to $\mu_0 i n$ while that of a loop is $\frac{\mu_0 i R^2}{2(R^2+z^2)^{3/2}}$? EntityFieldQuery order by title Why won't OS X trust GitHub's SSL certificate? Analogue of closed graph theorem This is the analogue of closed graph theorem for compact space          Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset of $X \times K$. Prove that $f$ is continuous.      Because metric space is first countable, all we need is to prove that if $x = \lim_{n \rightarrow \infty} x_{n}$, then $f(x) = \lim_{n \rightarrow \infty} f(x_{n})$. Suppose $x = \lim_{n \rightarrow \infty} x_{n}$, then because $K$ is compact, sequence $\{f(x_{n})\}$ has a subsequence $\{f(x_{n_{i}})\}$ converge. Because the graph of $f$ is closed, we must have $\lim_{n \rightarrow \infty}\{f(x_{n_{i}})\} = f(x)$. All we need to prove now is $\{f(x_{n})\}$ converge to $f(x)$, but I can't prove that. Can any one help me. I really appreciate. Thanks $4$ letter words taken from the letters of CONCENTRATIONS How many words with or without meaning can be made from the letters of the word CONCENTRATIONS by taking $4$ letters at a time?  There are  $14$ letters total present in the word. $4$ letters can be picked in $14\choose 4$ ways. $4$ letters can be arranged in $4!$ ways. So ideally answer has to be ${14\choose 4} * 4!$. But I know this answer is wrong because some letters are repeating. How do I solve this then? I appreciate any help. When you send a group text message in iOS, it appears as if you automatically opt in to receive any and all replies to that message. Is there a way to avoid being included in group replies? Is there a way to remove oneself from a group chat? Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions Proper capitalization of commonly used acronyms and initialisms What are tag badges? How do they work? Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice? How many Shutter Clicks on a New Camera considered normal? How many Shutter Clicks on a New Camera considered normal?    I purchased a quotes new Nikon D750 and checked the shutter count and it has 264 Shutter Clik count on it.    Is that suspect of a returned purchase or used camera? Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$ I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks          Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-n) + 1$ is irreducible over $\mathbb Z$ for all $n\ge1$ and $ n\ne4$.      Suppose $h(x) = f(x) g(x)$, then we must have $f(i)g(i) = 1$ for all $i = 1,2,...n$. So both $f(i)$ and $g(i)$ are $1$ or $-1$. In either case, $m(x) = f(x) - g(x)$ has degree smaller than $n$ and have $n$ different roots ($1,2,...,n$). So we must have $m(x) = 0$. Then $h(x) = f(x)^{2}$. So $n$ must be even. Let $n = 2k$. Because $f(x)$ has degree $k$, there are $k$ values from $\{1,2,...,2k\}$ at which $f(x)$ is $1$ and $k$ values at which $f(x)$ is $-1$.  This is where I got stuck. Hope some one can help me solve this. Thanks. Pre & post increment operator behavior in C, C++, Java, & C#