Showing that a particular harmonic function is constant
Two question on harmonic function
Say I have a module $M$ over a ring $R$ that is   $1.$ finitely-generated, i.e. $\exists$ finite $X \subset M$ such that $M = \sum_{i=1}^{n} Rx_{i}, x_{i}\in X$ (but $X$ is not necessarily free)  $2.$ free, i.e. $\exists$ a basis $Y \subset M$ a.k.a. $Y$ is free and generates M, (but $Y$ is not necessarily finite).  Now I want to show, that there exists a finite basis for $M$. For vector spaces, this would obviously be trivial, but for some  $R$-Modul, I'm not sure how to proceed.   Furthermore, is every minimal generating subset of $M$ a basis? The Node Editor won't work for the object selected Is there any physical or mental cost to casting spells? How to delete a registry value in C# Does $\sum_{k=1}^{\infty}\ln(\frac{k}{k+1})$ converge/diverges?? Does this series converge or diverge? $$\sum_{k=1}^{\infty}\ln(\frac{k}{k+1})$$   my thought is that, I can break it down to $$\sum_{k=1}^{\infty}\ln(k) - \sum_{k=1}^{\infty}\ln(k+1)$$ then maybe using comparison test or something? But I don't know exactly how to prove whether this series converges or diverges.  Any help would be appreciated! Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same? Given $n_1 $ number of  $1 $'s,  $n_2 $ number of  $2 $'s,  $n_3 $ number of  $3 $'s,  $n_4 $ number of  $4 $'s.  form a sequence using all these numbers such that two adjacent numbers should not be same.  I have tries lot of things but nothing worked.  can somebody tell me how to solve this problem? How about a "Vote not to close" option to counter the "Vote to close"? My request is simple: Many times, there will be a question where some people feel like it should be closed. At the same time, I'll feel like it's a perfectly legit question, and it should therefore NOT be closed. Right now, I have to wait till it gets enough close votes (which it always does for some reason, as soon as 1 or 2 people vote to close, e/o else decides to vote as well) and then vote to re-open. How about, as soon as there's one vote to close, there should be an option to counter the close vote, something like "vote not to close" or whatever. This would then bring the "Vote to close" number down by one. Only when the vote to close outnumbers the vote not to close by 5, that's when a question is indeed closed. Any thoughts? Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? How to solve 'I found no database files--while read file something.aux' Hi I am trying to include references into my paper but it keeps showing the following when I compile. I am using texstudio. Message: I'm skipping whatever remains of this command The style file: plain.bst I found no database files---while reading file term paper.aux Warning--I didn't find a database entry for "Astrov"  My code is like this:   \documentclass[letter,scriptaddress,twocolumn, prl,showkeys]{revtex4}  \usepackage{amssymb, amsmath, amstext, amscd, verbatim, mathtools, graphicx, float, caption, subcaption} \usepackage[utf8]{inputenc} \usepackage{natbib} \begin{document} This is my reference\cite[1]{Astrov}. \bibliography{term paper} \bibliographystyle{plain} \end{document}   They are all in the same directory. I am really panicing about this at the moment and apologize if similar questions have been asked before. Thanks in advance for your help! Correlations with unordered categorical variables Single service account running all SQL Servers Is there any negative performance impact for using the same service account and agent account to run SQL Server and SQL Agent respectively for all SQL server instances running at a small company of say 35 servers? The biggest database is 0.5 GB. any suggestions are welcome. Thank you. How can a network with only ReLU nodes output negative values? I'm trying to use an api with a feedforward neural network for time series forecasting.  For dense aggregate data it works fine, but for sparse data it sometimes forecasts negative values, even though my historical data has only positive values.   The source code is very dense, and I might be missing a line or two, but as far as I can tell, the input layers and the hidden layers are all ReLU nodes.   Assuming I am correct, how can a network with only ReLU layers lead to negative values, especially if none of the training data has negative values? Maximum number of edges in a bipartite graph Prove that for a bipartite graph $G$ on $n$ vertices the number of edges in $G$ is at most $\frac{n^2}{4}$.     I used induction on $n$.    Induction hypothesis: Suppose for a bipartite graph with less than $n$ vertices the result holds true.    Now take a bipartite graph on $n$ vertices.Let $x,y$ be two vertices in $G$ where an edge exist between $x$ and $y$. Now remove these two vertices from $G$ and consider this graph $G'$. $G'$ has at most ${(n-2)^2}\over4$. Add these two vertices back.  Then the number of edges $G$ can have is at most    $$|E(G')|+d(x)+d(y)-1$$   My question is in my proof I took $d(x) + d(y) \le n$, where $d(x)$ denotes the degree of vertex $x$.  Can I consider $d(x)+d(y) \leq n$? I thought the maximum number of edges is obtained at the situation $K_{\frac n 2,\frac n 2}$ I can't retag or otherwise edit a question while waiting for an edit I reviewed from queue to be approved or rejected give space above and below a DIV Draw Vertices of Regular Polygon I am trying to draw just the nodes (no sides) at the vertices of a regular hexagon.  should be easily modified to accomplish what I want, but I am such a novice that I have difficulty understanding the example. Ideally, each vertex would be a separate named node so that I can easily draw edges between them (I will be making several different graphs using these same nodes). How to secure recommendations and apply for PhD after having worked outside of academia for two years? Will “Erase Ubuntu16.10 and reinstall” keep my Win7 installation? Infinitely many primes of the type 5 mod 6. Problem: Prove that there are infinitely many primes of the type 5 mod 6.  My professor did the problem and the proof was horribly long. Can someone show me a shorter version of the proof of this problem. After Google Play Service update to version 13 I got an error