Determine the convergence of the recursion $x_n=\sin(x_{n-1})$
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
Here is my script:  #!/bin/bash  echo "Digite o local em que deseja instalar o IGRAFU(pressione enter para  instalar em  ${HOME}/IGRAFO):" read caminho if test -z $caminho then caminho="${HOME}/IGRAFO" fi echo "O IGRAFU será instalado no diretório: $caminho" mkdir -pv $caminho  mv -v ./* $caminho echo "Pronto!"   At 'read caminho' I may receive from the user a path like ~/somefolder. When the script receives that kind of path both mv and mkdir won't make tilde expansion, so it will try to create a ~/somefolder and not /home/username/somefolder and therefore fail.  How do I ensure that the tilde will be converted into the HOME variable? I am travelling to Budapest and therefore have a Schengen Visa. I hold an Indian passport and a valid US visa. I'm travelling from Chicago-Heathrow-Budapest and returning by Budapest-Dublin-Chicago.  Would I need a transit visa for UK/Ireland?  UPDATE  Hoping this will help someone someday.   I was not allowed to board my return flight (BUD-DUB-ORD) because I do not hold an overnight Irish Transit Visa. They said I could have transited happily if the connecting flight was in the same day, but because I would have to originally land at 10 PM and catch the connecting flight next morning at 10 AM, I needed a transit visa. Well, I thought it was BS, but couldn't help comply with the rules. I booked another last minute ticket and flew via Warsaw via LOT. Show forms in a modal window I have a form and I am using Drupal 7. I need to show this form in a popup when a user clicks a link. The user should then be able to complete the form whilst in the pop up window. I can use anything like  or modals or anything. I just wanted to know which is the better option and what options do I have for Drupal. For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$ I have only recently been exposed to sets. According to , as seen on the first bullet mark of the link, $ \forall A: \emptyset \subseteq A$  Does this mean that $\emptyset$ is an element of all sets? (This is False, thank you those that answered)  Is the empty set also a set itself?  Assuming these statements are true, then the empty set therefore an element of the empty set. This does not sound right, please clarify for me. Thank you. 2.81.16 No Rock Generator Why Session object destruction failed How can I produce exercises in one part of a LaTeX document with selected answers in another? I could, of course, simply typeset the questions and the answers separately, but this is too likely lead to divergence in the labels for the questions and the answers as the document is edited.   What I'd really like is a command that would allow me to typeset exercises at the point at which the command was used and which took an optional argument for a solution to be stored until a final section (chapter, or part) "Selected Exercise Answers". I'd like the questions (and the selected answers) to be labeled in the output by Chapter.Section.Question or even Chapter.Question. Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$ As an example I have a BSB file from NOAA which is a navigational map. The map has a SRS of WGS84 and a Mercator projection.  I initially translate the file from BSB format to GTiff format using the following command:  gdal_translate -of GTiff 18554_1.kap noaa.tif  I can view this GTiff file using Paint Shop Pro quite fine. I then translate the file from GTiff format to VRT format using the following command:  gdal_translate -of vrt –expand rgba noaa.tif noaa.vrt  I then try to generate a tile matrix set using the following command:  gdal2tiles.py noaa.vrt tiles_folder  Where the map tiles are located in tiles_folder. If I try to open the openmaps.html or google.html, the map zoom in and out looks good. However, on tiles generator like MBTiles, I see tiles swap. See the image below. I am pulling my hair off to get the tiles to line up. I am guessing it is projection error. I am not understanding those projection types. It would great someone can explain it while finding the error. I am planning to use on a tiling  like this. By the way, on GDAL how do I specify the BSB (reference) file that is associated with KAP (data) file? I have seen cases where only one BSB file referred by multiple KAP files. There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process? Is there a problem with defrosting meat on the counter? I generally defrost meat on the kitchen counter. A friend of mine suggested that this was dangerous and suggested that I defrost meat in the refrigerator. I am no biologist but it seems to me that as long as the meat doesn't get warm defrosting it on the counter should be safe.  Generally, I remove the frozen items from the freezer and place them on a plate on the counter in the packaging they were frozen in. When they are mostly / completely thawed I place the meat in the refrigerator.  Am I wrong and should I be defrosting in the refrigerator? For a given data matrix $A$ (with variables in columns and data points in rows), it seems like $A^TA$ plays an important role in statistics. For example, it is an important part of the analytical solution of ordinary least squares. Or, for PCA, its eigenvectors are the principal components of the data.  I understand how to calculate $A^TA$, but I was wondering if there's an intuitive interpretation of what this matrix represents, which leads to its important role? In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$). Draw ellipses with different coloring for their intersection in tikz? How do I color intersection of two ellipses in different color. For instances, the ellipses should themselves be colored as yellow and their intersection as red. My code for drawing ellipses is given below  \draw (-4,0) ellipse (8 and 3); \draw (8,0) ellipse (8 and 3);   UPDATE: In the comments to individual answers, I have requested to show it for three ellipses. What's the canonical way to check for type in Python? What is the best way to check whether a given object is of a given type? How about checking whether the object inherits from a given type?  Let's say I have an object o. How do I check whether it's a str? How should I address a potential FERPA violation by a professor? In the United States, federal law prohibits professors from releasing any student's grades to any party other than the student and any party whom the student authorizes to receive the grades.  One of my professors appears to be violating this law by uploading each student's lab grades in one file, which is accessible to anyone who is enrolled in the class.  Here is my proposed plan of action:   Ask the professor - in private - to post the grades using the online management system the professor already uses for other things.  If the professor doesn't fix this privacy violation, inform the professor that the method by which he/she post grades is a violation of federal law. As a last resort, contact the department dean. What is the probability that $x_1+x_2+...+x_n \le n$? Given that $X_1, X_2...$ are mutually independent random variables. For each $i$ with $1\le i \le n$   the variable $X_i$ is equal to either $0$ or $n+1$ $E(X_i)$ = $1$   also.. if $X_i$ is equal to either $0$ or $n+1$, doesn't that mean that all $X_i$ need to be $0$?  Thanks 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? After using `networksetup -setdnsservers Wi-Fi "8.8.8.8" "8.8.4.4"` how do I clear them? I'm looking to script a temporary DNS server change to override the DHCP response I get from the network. I've figured out I can use networksetup but once we're done testing. Before I push to production, I figure I should know how to undo this.  How can I script to clear these changes and restore the settings the network provides now as if I never ran the following script?  networksetup -setdnsservers Wi-Fi "8.8.8.8" "8.8.4.4"   It's not feasible to save any the previous settings and I'm looking to remove the setting so that the actual new values can be retrieved from the network as opposed to saving what the settings were previously so I don't save an old value that is no longer correct.  Basically, I'm looking to ipconfig /renew on macOS. "At" vs. "in" before verb In a document I found the following sentence:     listeners are more accurate at understanding speech spoken in their own accent...   Would it be an error to use "in" instead of "at"?  Actually in this case "at" sounds better than "in", but in general before verb is "at" always used?