I was wondering if there is a property that is known to be satisfied for certain "things" but for which we do not know any explicit example.  More explicitly (also more restrictive but possibly easier):     Is there a set $S \subseteq \mathbb{N}$ for which one can prove that $S \neq \varnothing$ and yet there is no known (to us, humans, at this point in time) element of $S$?
There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting. There are situations where such unknown numbers are necessarily natural numbers, for example in Ramsey theory. For example, we know that there is a smallest integer $n$ such that any graph with $n$ vertices contains a complete or an independent subgraph of 10 vertices, but we don't know the exact value of $n$.  What kinds of unknown small (less than 100, say) integers are there? What are the smallest unknown constants which are known to be integers? Or, more rigorously, what is the smallest upper bound for an unknown but definable number that is known to be an integer?  I know that asking for the smallest unknown integer is ill-defined since we do not know the exact values. The more rigorous version of the question is well-posed, but I do not want to keep anyone from offering interesting examples even if they are clearly not going to win the race for the lowest upper bound.  An answer should contain a definition of an integer quantity (or a family of them) and known lower and upper bounds (both of which should be integers, not infinite). Conjectures about the actual value are also welcome. I have given one example below to give an idea of what I'm looking for.
There is a kind of situation that seems to occur somewhat commonly, that goes something like:   Figure out some way to handle a particular situation do it regularly, and make a habit of it, as it seems to work OK … … and only much later actually realize that it very much isn’t OK, that it’s been doing harm all the time, and that it’s too late to fix anything.   Does there exist any good words or phrases for this kind of situation? "Arnold raced out of the door": grammatical or not? How to use HttpWebRequest (.NET) asynchronously? How can I use HttpWebRequest (.NET, C#) asynchronously? Suppose that $X_1, \ldots, X_n \sim N(0,1)$ are independent random variables. I am interested in finding a constant C that satisfies:  $$ E\left[\max_{1\leq i\leq n}|X_i|\right] \leq C \sqrt{log\  n} $$  I know one method is to employ the moment generating function trick, then take logs of both sides. However, I was wondering if there exists a more direct method. thanks! How to find and replace using sed on records starting with Specific number Because of past conflicts, I don't believe my advisor will write me a good LOR for postdoc positions. What should I do? Is it possible to find power armor in the wasteland? How to pretty print XML from Java? I have a Java String that contains XML, with no line feeds or indentations. I would like to turn it into a String with nicely formatted XML. How do I do this?  String unformattedXml = "&lt;tag&gt;&lt;nested&gt;hello&lt;/nested&gt;&lt;/tag&gt;"; String formattedXml = new [UnknownClass]().format(unformattedXml);   Note: My input is a String. My output is a String.   (Basic) mock result:    &lt;?xml version="1.0" encoding="UTF-8"?&gt; &lt;root&gt;   &lt;tag&gt;     &lt;nested&gt;hello&lt;/nested&gt;   &lt;/tag&gt; &lt;/root&gt; Linux network troubleshooting and debugging From time to time Linux and Unix users faced with various network problems. Many of these problems are presented here and at some others troubleshooting forums, but they are very concrete and contains a lot of additional technical information, and sometimes it's rather difficult to understand the main point and the real reason of buggy system behavior.   By asking this question, my intention is to start a community wiki page which allows generalizing our network troubleshooting and debugging experience. I hope the Linux and Unix users could easier recognize and solve("divide and conquer") their network problems using this page.   The parent of this page should be  . But here we should focus on troubleshooting the network problems from user- and kernel-space.  I suppose, if you:    Share the information about using some great network diagnostic tool with concrete usage examples and examples of network bugs, which they help to catch. Share the link to the great network tutorial connected with this subject Tell about a general method or recipe which allows to tackle some class of network problems Share information about your tool-set for network debugging and troubleshooting   it would perfectly fits for this topic.    I'll begin from sharing the link to varios  and . Also  seem to have actual information about our subject. And for diving into linux networking we definetely need to visit . How to unprotect GeneratingFunction This is a recurring question (see ,  and ), but I have a different spin.  Suppose I have a bunch of samples from a generic MCMC sampler. For each sample $\theta$, I know the value of the log likelihood $\log f(\textbf{x} | \theta)$ and of the log prior $\log f(\theta)$. If it helps, I also know the value of the log likelihood per data point, $\log f(x_i | \theta)$ (this information helps with certain methods, such as WAIC and PSIS-LOO).  I want to obtain a (crude) estimate of the marginal likelihood, just with the samples that I have, and possibly a few other function evaluations (but without rerunning an ad hoc MCMC).  First of all, let's clear the table. We all know that the harmonic estimator is the . Let's move on. If you are doing Gibbs sampling with priors and posteriors in closed form, you can use ; but I am not sure how to generalize outside of those cases. There are also methods that require you to modify the sampling procedure (such as via ), but I am not interested in that here.  The approach I am thinking of consists of approximating the underlying distribution with a parametric (or nonparametric) shape $g(\theta)$, and then figuring out the normalization constant $Z$ as a 1-D optimization problem (i.e., the $Z$ that minimizes some error between $Z g(\theta)$ and $f(\textbf{x}|\theta) f(\theta)$, evaluated on the samples). In the simplest case, suppose that the posterior is roughly multivariate normal, I can fit $g(\theta)$ as a multivariate normal and get something similar to a Laplace approximation (I might want to use a few additional function evaluations to refine the position of the mode). However, I could use as $g(\theta)$ a more flexible family such as a variational mixture of multivariate $t$ distributions.  I appreciate that this method works only if $Z g(\theta)$ is a reasonable approximation to $f(\textbf{x}|\theta) f(\theta)$, but any reason or cautionary tale of why it would be very unwise to do it? Any reading that you would recommend?  The fully nonparametric approach uses some nonparametric family, such as a Gaussian process (GP), to approximate $\log f(\textbf{x}|\theta) + \log f(\theta)$ (or some other nonlinear transformation thereof, such as the square root), and  to implicitly integrate over the underlying target (see  and ). This seems to be an interesting alternative approach, but analogous in spirit (also, note that GPs would be unwieldy in my case). Find count of all combination of numbers whose sum is x If Newton's third law is true, why can we sink in sand? Newton's third law of motion states that every action has an equal and opposite reaction. That is the reason we do not sink into the earth, because when our weight exerts a force on the earth it also exerts an equal and opposite force on us.  But when we stand on quicksand or on fluids we can sink in. How is this possible? Does it not exert an equal and opposite force on us? Or are Newton's laws different in the case of fluids and substances of low densities? table side by side with minipage I can't manage to get two tables side by side in a minipage environment. Any kind suggestion would be highly appreciated!   \documentclass{article} \usepackage{array}  \begin{document} \begin{table}[htbp] \centering \begin{minipage}{.2\textwidth}     \begin{tabular}{| m{3cm} | m{4.3cm} | c |}   \hline  number of entity filter (size 1) &amp; number of context filters (size 2, size 3,  size 4) &amp; Accuracy\\ [2ex]   \hline  32  &amp;  32  ,  32  ,  32  &amp;  0.835 \\[0.5ex] \hline 32  &amp;  64  ,  64  ,  64  &amp;  0.821 \\[0.5ex] \hline 32  &amp;  128  ,  128  ,  128  &amp;  0.819 \\[0.5ex] \hline 32  &amp;  256  ,  256  ,  256  &amp;  0.749 \\[0.5ex] \hline 32  &amp;  512  ,  512  ,  512  &amp;  0.683 \\[0.5ex] \hline 32  &amp;  1024  ,  1024  ,  1024  &amp;  0.528 \\[0.5ex] \hline 32  &amp;  2048  ,  2048  ,  2048  &amp;  0.235 \\[0.5ex] \hline 64  &amp;  32  ,  32  ,  32  &amp;  0.850 \\[0.5ex] \hline 64  &amp;  64  ,  64  ,  64  &amp;  0.816 \\[0.5ex] \hline 64  &amp;  128  ,  128  ,  128  &amp;  0.786 \\[0.5ex] \hline 64  &amp;  256  ,  256  ,  256  &amp;  0.795 \\[0.5ex] \hline 64  &amp;  512  ,  512  ,  512  &amp;  0.765 \\[0.5ex] \hline 64  &amp;  1024  ,  1024  ,  1024  &amp;  0.669 \\[0.5ex] \hline 64  &amp;  2048  ,  2048  ,  2048  &amp;  0.331 \\[0.5ex] \hline 128  &amp;  32  ,  32  ,  32  &amp;  0.920 \\[0.5ex] \hline 128  &amp;  64  ,  64  ,  64  &amp;  0.890 \\[0.5ex] \hline 128  &amp;  128  ,  128  ,  128  &amp;  0.886 \\[0.5ex] \hline 128  &amp;  256  ,  256  ,  256  &amp;  0.846 \\[0.5ex] \hline 128  &amp;  512  ,  512  ,  512  &amp;  0.789 \\[0.5ex] \hline 128  &amp;  1024  ,  1024  ,  1024  &amp;  0.755 \\[0.5ex] \hline 128  &amp;  2048  ,  2048  ,  2048  &amp;  0.488 \\[0.5ex] \hline 256  &amp;  32  ,  32  ,  32  &amp;  0.920 \\[0.5ex] \hline 256  &amp;  64  ,  64  ,  64  &amp;  0.922 \\[0.5ex] \hline 256  &amp;  128  ,  128  ,  128  &amp;  0.912 \\[0.5ex] \hline 256  &amp;  256  ,  256  ,  256  &amp;  0.877 \\[0.5ex] \hline 256  &amp;  512  ,  512  ,  512  &amp;  0.872 \\[0.5ex] \hline 256  &amp;  1024  ,  1024  ,  1024  &amp;  0.771 \\[0.5ex] \hline 256  &amp;  2048  ,  2048  ,  2048  &amp;  0.683 \\[0.5ex]  \hline  \end{tabular} \end{minipage}% \hfill \begin{minipage}{.2\textwidth}     \begin{tabular}{| m{3cm} | m{4.3cm} | c |}   \hline  number of entity filter (size 1) &amp; number of context filters (size 2, size 3,  size4) &amp; Accuracy\\ [2ex]   \hline 512  &amp;  32  ,  32  ,  32  &amp;  0.910 \\[0.5ex] \hline 512  &amp;  64  ,  64  ,  64  &amp;  0.922 \\[0.5ex] \hline 512  &amp;  128  ,  128  ,  128  &amp;  0.909 \\[0.5ex] \hline 512  &amp;  256  ,  256  ,  256  &amp;  0.910 \\[0.5ex] \hline 512  &amp;  512  ,  512  ,  512  &amp;  0.883 \\[0.5ex] \hline 512  &amp;  1024  ,  1024  ,  1024  &amp;  0.730 \\[0.5ex] \hline 512  &amp;  2048  ,  2048  ,  2048  &amp;  0.682 \\[0.5ex] \hline 1024  &amp;  32  ,  32  ,  32  &amp;  0.925 \\[0.5ex] \hline 1024  &amp;  64  ,  64  ,  64  &amp;  0.931 \\[0.5ex] \hline 1024  &amp;  128  ,  128  ,  128  &amp;  0.920 \\[0.5ex] \hline 1024  &amp;  256  ,  256  ,  256  &amp;  0.917 \\[0.5ex] \hline 1024  &amp;  512  ,  512  ,  512  &amp;  0.899 \\[0.5ex] \hline 1024  &amp;  1024  ,  1024  ,  1024  &amp;  0.858 \\[0.5ex] \hline 1024  &amp;  2048  ,  2048  ,  2048  &amp;  0.776 \\[0.5ex] \hline 2048  &amp;  32  ,  32  ,  32  &amp;  0.933 \\[0.5ex] \hline 2048  &amp;  64  ,  64  ,  64  &amp;  0.934 \\[0.5ex] \hline 2048  &amp;  128  ,  128  ,  128  &amp;  0.925 \\[0.5ex] \hline 2048  &amp;  256  ,  256  ,  256  &amp;  0.928 \\[0.5ex] \hline 2048  &amp;  512  ,  512  ,  512  &amp;  0.907 \\[0.5ex] \hline 2048  &amp;  1024  ,  1024  ,  1024  &amp;  0.878 \\[0.5ex] \hline 2048  &amp;  2048  ,  2048  ,  2048  &amp;  0.798 \\[0.5ex]  \hline  \hline \end{tabular} \end{minipage} \caption{filter window size} \label{tab:config_filternum} \end{table}   \end{document}   What I have so far  EDIT: I followed the second solution from , and it somehow didn't work. I am not sure why. Gmail App in my Android device shows correctly all the labels that I created using Gmail through web interface.  Can I create new labels in Gmail on Android device using Gmail App?   Also, ideally, once I create these new labels in Gmail App, I obviously should be seeing the same in Web Interface of Gmail too (Sync!!).  Any ideas? I may not be coming in tomorrow...    I might not be coming in tomorrow...       When should I use "may" and "might"? Why did Obi-Wan allow Leia to get adopted by such a high profile figure? I know that Leia is a princess because she was adopted by the Queen of Alderaan, but why would Obi-Wan allow this? Wasn't the original plan of Obi-Wan and Yoda to hide the Skywalkers from Palpatine? Luke was placed into a very low-profile position, being the child of two farmers on a backwater planet. However, Leia occupies a very important political position.  Is there any reason why Obi-Wan would do this if he was trying to hide her? Subgroup of $\mathbb{R}$ either dense or has a least positive element? How do you search for questions filtered by a particular user? I want to do a search on StackOverflow for a few terms, but also by user, since I know which user referenced the item I'm looking for. Differentiation operator and eigenvalues Let $V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\}$. Let $T : V \to V$ be given by differentiation, in essence $$T(p(x)) = p'(x)$$ It seems to me that the only eigenvalue that can exist is $\lambda =0$, since the given operator lowers the degree of any given polynomial and hence makes it impossible for a constant multiple of a polynomial to be equal to the polynomial's derivative. Is my claim correct?