Tricky convergences of harmonic series divided into digit-9-contained denominators and digit-9-not-contained denominators?
Sum of reciprocals of numbers with certain terms omitted
Atiyah-Macdonald, Exercise 5.4 I was having some trouble with the following exercise from Atiyah-Macdonald.     Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Let $\mathfrak{n}$ be a maximal ideal of $B$ and let $\mathfrak{m}=\mathfrak{n} \cap A$ be the corresponding maximal ideal of $A$. Is $B_{\mathfrak{n}}$ integral over $A_{\mathfrak{m}}$?   The book gives a hint which serves as a counter-example. Consider the subring $k[x^{2}-1]$ of $k[x]$ where $k$ is a field, and let $\mathfrak{n}=(x-1)$. I am trying to show that $1/(x+1)$ could not be integral over $k[x^{2}-1]_{\mathfrak{n}^{c}}$.  I have understood why this situation serves as a counterexample. But I am essentially stuck at trying to draw a contradiction. A hint or any help would be great. Adding tags to files in OS X: How to use, where are tags stored, and will they work on a network? I've heard it is possible to add tags to a file's meta data in OS X, and have Spotlight search through such tags.   In a nutshell, how would one access this feature in Finder &ndash; not Spotlight (Command+Space) &ndash; but the adding of tags themselves? Where are these tags stored?  Do they go in .DS_Store (I tend to delete those!) or elsewhere in the filesystem? Does this tagging mechanism work on removable drives?  What are the requirements? Does this tagging mechanism work over a network share?  I have some Windows SMB network shares at home that I frequently store files on.  Would it work on those or is another configuration required? Aside from Spotlight, are there any command-line tools for searching through or accessing the tag data for files?   Thank you! Let $V=\{f : S \to \mathbb{R} \}$ be the set of all the functions $f : S \to \mathbb{R}$. Let $S=\{ 1,2,...,n$}, now my train of thought was that the basis of V must be a linearly independent set that Span V so somehow show that there is a set of functions on S that can compose as a linear combination and map to any number in $\mathbb{R}$. Any idea how to get a general answer for basis for dimension of V? AMD GPU Material in "material preview mode" and "rendered mode", is entirely gray as background. Only outline is displayed When I compile with LuaLaTeX and TL2016 this MWE   \documentclass{article}  \usepackage[no-math]{fontspec} \usepackage{luatex85} \usepackage{shellesc}  \setmainfont{Minion Pro}  \begin{document}  \huge{huge} \textbf{\huge{huge}}  \Huge{Huge} \textbf{\Huge{Huge}}   \end{document}   I have this .pdf.     Any help? With TL2015 all is correct. Thanks How do I embed an image from the web into a Trello card comment or description? So far I have been successful embedding Youtube videos into Trello card comments; simply by including the URL in the comment Trello picks it up and renders an embedded Youtube thumbnail.   What I would like to do is the same thing, but with a simple PNG image. I don't want to upload the image as it is a Balsamiq mockup permalink which will get updated periodically, and I want the latest version to be displayed on the Trello card.   However, pasting the image URL into the Trello card has no effect.  Am I missing somthing? MySQL Database User: Which Privileges are needed? The short installation instruction for WordPress () state that:     Create a database for WordPress on your web server, as well as a MySQL user who has all privileges for accessing and modifying it.   While setting up a new blog professionally I was wondering how that maps to what the MySQL database user privileges/permissions configuration offers me:   Data: SELECT, INSERT, UPDATE, DELETE Definition: CREATE, ALTER, DROP Extra: INDEX More:  LOCK TABLES REFERENCES CREATE TEMPORARY TABLES CREATE VIEW SHOW VIEW CREATE ROUTINE EXECUTE ALTER ROUTINE    I'm pretty sure for the first three groups, I named them Data, Definition and Extra here. But what about the others below the More entry? Normally I would say, those are not needed, but I would like to get a second opinion. why do academics write peer reviews? There is a shop in the starting area on the island of Koholint, and if you are fast enough to dance around the shopkeeper, you can steal from him. Once you steal from him, though, your title in the game changes to "THIEF".    Is there any way to change your title back to your name without reverting to a different past save or starting over? This came to my mind after I learned the following from :  where T : struct   We, C# developers, all know the basics of C#. I mean declarations, conditionals, loops, operators, etc.  Some of us even mastered the stuff like , , , , ...  But what are the most hidden features or tricks of C# that even C# fans, addicts, experts barely know?  Here are the revealed features so far:    Keywords    by  var by  using() statement by  readonly by  as by  as / is by  as / is (improved) by  default by  global:: by  using() blocks by  volatile by  extern alias by    Attributes    by   by   by   and  by   by   by   by    Syntax    (coalesce nulls) operator by  Number flaggings by  where T:new by  Implicit generics by  One-parameter lambdas by  Auto properties by  Namespace aliases by  Verbatim string literals with @ by  enum values by  @variablenames by  event operators by  Format string brackets by  Property accessor accessibility modifiers by  Conditional (ternary) operator (?:) by  checked and unchecked operators by  implicit and explicit operators by    Language Features   Nullable types by  Anonymous types by  __makeref __reftype __refvalue by  Object initializers by  Format strings by  Extension Methods by  partial methods by  Preprocessor directives by  DEBUG pre-processor directive by  Operator overloading by  Type inferrence by  Boolean operators  by  Pass value-type variable as interface without boxing by  Programmatically determine declared variable type by  Static Constructors by  Easier-on-the-eyes / condensed ORM-mapping using LINQ by  __arglist by    Visual Studio Features   Select block of text in editor by  Snippets by      Framework   TransactionScope by  DependantTransaction by  Nullable&lt;T&gt; by  Mutex by  System.IO.Path by  WeakReference by    Methods and Properties   String.IsNullOrEmpty() method by  List.ForEach() method by  BeginInvoke(), EndInvoke() methods by  Nullable&lt;T&gt;.HasValue and Nullable&lt;T&gt;.Value properties by  GetValueOrDefault method by    Tips &amp; Tricks   Nice method for event handlers by  Uppercase comparisons by  Access anonymous types without reflection by  A quick way to lazily instantiate collection properties by  JavaScript-like anonymous inline-functions by    Other   netmodules by     by     by Characterization of weak convergence in $\ell_\infty$ Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets?   I was only able to come up with a characterization of sequential weak convergence using limit along ultrafilters, which I will describe below. I wonder whether there is some insight into this characterization. (E.g. whether there is some simple reformulations which does not use ultralimits.) Moreover, I do not know whether at least this characterization works for nets, too.  We have the following result describing weak convergence in $C(K)$, see e.g. Corollary 3.138,  in Banach Space Theory by Fabian, Habala et al. (It is a consequence of Rainwater theorem and characterization of extreme points of unit ball in $C(K)^*$.)     Let $K$ be a compact topological space. Let $\{f_n\}$ be a bounded sequence in $C(K)$ and $f\in C(K)$. Then, if $f_n\to f$ pointwise, we have $f_n\overset{w}\to f$.   Moreover we have isometric isomorphism between $\ell_\infty$ and $C(\beta\mathbb N)$, which is described e.g. in the Wikipedia article on  or in  of Carothers' book A short course on Banach space theory. This isomorphism assigns to each bounded sequence $(x_n)$ the continuous function $\overline x$ on $\beta\mathbb N$ defined by $$\overline x(\mathscr U) = \operatorname{\mathscr U-lim} x_n,$$ where $\operatorname{\mathscr U-lim} x_n$ denotes the  of $x_n$ w.r.t the ultrafilter $\mathscr U$.  Combining the above results we get the following characterization:      Let $f^{(n)},f\in\ell_\infty$. The sequence $f^{(n)}$ converges to $f$ weakly if and only if for every ultrafilter $\mathscr U$    $$\lim_{n\to\infty} \operatorname{\mathscr U-lim} f^{(n)}= \operatorname{\mathscr U-lim} f.$$   (The above claim for principal ultrafilters is just a pointwise convergence. But in the above claim the equality is required for free ultrafilters, too.) Why are login forms centered? Why are login forms always centered on the screen? Most horizontally, some vertically also. Sites like Apple/Amazon/Adobe/Microsoft/Google/PayPal/Instagram/Twitter etc. are all centered! Is this an undocumented best practice everyone unanimously decided to use?         &ndash; Wireframes created with   Obviously it's most aesthetically pleasing, but is there any other reasons why?  Edit: Many of these sites do not support mobile either, so it wasn't done specifically for that. Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. I have a MySQL database that I "inherited" and I was not given the admin credentials.  I do however have access to the box that it runs on.  Is there a way to either recover the admin credentials or create new ones? "Speak to" vs. "Speak with" What are the differences between these two phrasal verbs and what are the best situations to use each? I've recently updated to Ubuntu 20.04. When English is not the chosen language all shortcuts don't work (such as Ctrl + C, Ctrl + A, etc). It was working fine in the previous versions. What happened and What can I do to fix it? Why not use the Lagrangian, instead of the Hamiltonian, in nonrelativistic QM? What does the bracketed capital letter indicate in the revised version below? Is it essential, per the rules of proper punctuation, to do so? Are the ellipsis points (i.e., the spacing of the dots) spot on in that sentence to indicate the omission of one or more sentences?  Original quotation  "It need hardly be said that shortness is a merit in words. There are often reasons why shortness is not possible; much less often there are occasions when length, not shortness, is desirable. But it is a general truth that the short words are not only handier to use, but more possible in effect; extra syllables reduce, not increase, vigor."  Revised quotation  "It need hardly be said that shortness is a merit in words. . . . [S]hort words are not only handier to use, but more possible in effect; extra syllables reduce, not increase, vigor."  Thank you. Show that if $x$, $y$, $z$ are integers such that $x^2 + y^2 = z^2$, then at least one of them is divisible by $2$, at least one is divisible by $3$, and at least one is divisible by $5$.  I know that $x,y,z$ are sides of a right-angled triangles. If I start putting values, then I can show, but have no clue about mathematical approach.  How do I solve it mathematically, using modular arithmetic? Minecraft crashes on launch with EXCEPTION_ACCESS_VIOLATION, Problematic frame: ig4dev32.dll or ig4dev64.dll or ig4icd32.dll or ig4icd64.dll When I try to launch Minecraft 1.7.4 or newer, the game crashes on startup, and the game output screen says this:  #  EXCEPTION_ACCESS_VIOLATION (0xc0000005) at pc=*, pid=*, tid=* # # JRE version: Java™ SE Runtime Environment (7.0_51-b13) (build 1.7.0_51-b13) # Java VM: Java HotSpot™ Client VM (* mixed mode, sharing windows-* ) # Problematic frame: # C  [ig4dev32.dll+0x3e88] OR # C  [ig4dev64.dll+0x55ec] OR # C  [ig4icd32.dll+0x104a77] OR  # C  [ig4icd64.dll+0x84930a]   This happens with both Java 7 and 8.  What causes this error, and what can I do to fix the issue?