Can I omit part of a compound word when it's repeated?
Can a hyphen be used without anything on the right side?
Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct. Gaming with Virtualbox, has it worked for you? Currently I am using Windows 8. I have a virtual box with Ubuntu set up for development. Aside from that I sometimes just want to relax and enjoy a game (Battlefield 3, Skyrim, Saints Row: The Third, etc) after hours of research and development.   I want to have my main machine with linux, and install windows on a virtual box.   What I want to know, from those who have tried or seen it:   How is the performance? I've never been clear on this but, even if I install the correct drivers on the windows box, will it matter? At least on this part, I'm just curious how it offload. Which drivers will matter the most.  Mouse and Keyboard integration, I'll be honest, sometimes the mouse integration in Virtual Box is a bit frustrating, mouse cursor disappears, etc.    I'm not sure if this is the right place to ask. I'm merely trying to come back to linux, yet I'm a PC gamer so I can't throw away Steam. I'm aware of the linux client, but most of those games won't work.   If there's a right place to ask this whether StackExchange sites or any other site let me know.   Cheers and I hope those in the same position find a middle ground. New York is a great place to live. New York is a great place to live in.  I've seen the former usage a lot and I've started wondering what the grammar aspects of it are. The main question I'm asking involves the grammar when the preposition in the end of the sentence is deleted. Is it equally formal or correct to omit it? What are the grammar properties of this? Some more examples of similar usage:  This organisation is a great place to work. This is a great place to stay/sit/study/travel/go.  Just to show that 'live' isn't special in this case. Note: I'd posted the same question  before, but the answers were not satisfying enough (the answers there did not seem to provide the precise grammar aspects of this, those that could be in a serious grammar book), so I'm posting this here to get some more views on the question. I hope it is acceptable. Realtime audio processing using Dynamic Which is correct: I'll be moving next month or I'll be shifting next month? For changing one's home from one place to another, I've heard people in western part of the world using the sentence:      I'll be moving next month.   In India, even in the English news channels, I've heard many people using the phrase:     I'll be shifting next month.   Which one is the correct usage? Or are both of them correct and the difference in usage is regional? QGIS 2.99 "qgis._core import error: DLL load failed" I am trying to run a standalone application. However, it does not run the program because of mentioned error in title. Here is my code:  from qgis.core import * from qgis.gui import * from qgis.utils import * from PyQt5.QtCore import * from PyQt5.QtGui import *  app = QgsApplication([], True) path = "C:/OSGeo4W64/apps/qgis-dev" app.setPrefixPath(path, True) app.initQgis()  canvas = QgsMapCanvas() title = "PyQGIS Standalone Application Example" canvas.setWindowTitle(title) canvas.setCanvasColor(Qt.white)  layer_info = 'LineString?crs=epsg:4326' layer = QgsVectorLayer(layer_info, 'MyLine', "memory") pr = layer.dataProvider() linstr = QgsFeature() wkt = "LINESTRING (1 1, 10 15, 40 35)" geom = QgsGeometry.fromWkt(wkt) linstr.setGeometry(geom) pr.addFeatures([linstr]) layer.updateExtents() QgsProject.instance().addMapLayer(layer)  canvas.setExtent(layer.extent()) canvas.setLayers([layer]) canvas.zoomToFullExtent() exitcode = app.exec_() QgsApplication.exitQgis() sys.exit(exitcode)   When I run it, I get this error:  Traceback (most recent call last):   File "C:/Users/DELL/Desktop/MyMap/myMap.py", line 1, in &lt;module&gt;     from qgis.core import *   File "C:\OSGeo4W64\apps\qgis-dev\python\qgis\__init__.py", line 72, in &lt;module&gt;     from qgis.core import QgsFeature, QgsGeometry   File "C:\OSGeo4W64\apps\qgis-dev\python\qgis\core\__init__.py", line 34, in &lt;module&gt;     from qgis._core import *   ImportError: DLL load failed: The specified module could not be found.  Python version is: 3.6 PyQt   version is: PyQt5.9 Windows 10 - 64bit  I also add many path:  %SystemRoot%\system32 %SystemRoot% %SystemRoot%\System32\Wbem %SYSTEMROOT%\System32\WindowsPowerShell\v1.0\ C:\OSGeo4W64 C:\OSGeo4W64\apps C:\OSGeo4W64\apps\gdal-dev C:\OSGeo4W64\apps\gdal-dev\bin C:\OSGeo4W64\apps\gdal-dev\pymod3\Lib\site-packages C:\OSGeo4W64\apps\grass\grass-7.2.2\bin C:\OSGeo4W64\apps\Python36 C:\OSGeo4W64\apps\Python36\Scripts C:\OSGeo4W64\apps\Python36\DLLs C:\OSGeo4W64\apps\Python36\lib\site-packages C:\OSGeo4W64\apps\qgis-dev C:\OSGeo4W64\apps\qgis-dev\bin C:\OSGeo4W64\apps\qgis-dev\lib C:\OSGeo4W64\apps\qgis-dev\python C:\OSGeo4W64\apps\qgis-dev\python\qgis C:\OSGeo4W64\apps\Qt5 C:\OSGeo4W64\apps\Qt5\lib C:\OSGeo4W64\bin C:\OSGeo4W64\apps\qgis-dev\python C:\OSGeo4W64\apps\Python36\lib C:\OSGeo4W64\include C:\Users\Mustafa Uçar\AppData\Roaming\QGIS\QGIS3\profiles\default C:\Users\Mustafa Uçar\AppData\Roaming\QGIS\QGIS3\profiles\default\QGIS   How can I fix this error? Get mp3 duration in android Applying an ellipsis to multiline text Overstayed in the Philippines for years: How to go back to the US without paying a big fine? I'm a U.S. Citizen and I have been overstaying in the Philippines for 5 years. I want to go back to U.S what's the penalty?   I don't have enough money to pay a big fine, is there a way to avoid that? How can a file be copied? What, precisely, is a confidence interval? I know roughly and informally what a confidence interval is.  However, I can't seem to wrap my head around one rather important detail:  According to Wikipedia:     A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained.     I've also seen similar points made in several places on this site.  A more correct definition, also from Wikipedia, is:     if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level   Again, I've seen similar points made in several places on this site.  I don't get it.  If, under repeated experiments, the fraction of computed confidence intervals that contain the true parameter $\theta$ is $(1 - \alpha)$, then how can the probability that $\theta$ is in the confidence interval computed for the actual experiment be anything other than $(1 - \alpha)$?  I'm looking for the following in an answer:   Clarification of the distinction between the incorrect and correct definitions above. A formal, precise definition of a confidence interval that clearly shows why the first definition is wrong. A concrete example of a case where the first definition is spectacularly wrong, even if the underlying model is correct. How do we see? Where do the photons disappear? I know that the light is reflected from a object to my eyes, but I don't understand exactly how. The photons appear from the light source and disappear in my eye! Can someone explain the phenomenon of where the photons go and do to allow us to see? Maximal Ideals in Ring of Continuous Functions Dummit and Foote, 7.4.33(a): Let $R$ be the ring of all continuous functions $[0,1] \to \mathbb{R}$ and let $M_c$ be the kernel of evaluation at $c \in [0,1]$, i.e. all $f$ such that $f(c) = 0$. Show that if $M$ is a maximal ideal in $R$ then $M = M_c$ for some $c$.    I am aware of proofs that use the compactness of $[0,1]$; however I have been told that a simpler proof exists using only the isomorphism theorems for rings along with basic facts about ideals. I have tried on my own to prove it this way but still have no success; is it possible?  (My attempts go something like: since $M$ is maximal, the quotient ring $R/M$ is a field. Then it would be nice to use the fact that $R/M$ has no zero divisors, but I can't quite make it work. Alternatively, consider $M \cap M_c$.) Schengen visa validity start date My Chinese in-laws intend to visit us in one month from now. Their visa is valid from Nov 23rd. Would it be possible for them to take the airplane on November 22nd in China and arrive in Schengen area on November 23rd?  Or does the visa already have to be valid upon departure?  In our particular case, they still need to transfer in Beijing first. The actual flight to Schengen zone effectively only departs on the 23rd so I guess that should be completely fine then? Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Probability brain teaser with infinite loop I found this problem and I've been stuck on how to solve it.     A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours. If we assume that the miner is at all times equally likely to choose any one of doors, what is the expected length of time until he reaches safety?   The fact that the miner could be stuck in an infinite loop has confused me. Any help is greatly appreciated. As a new user without much rep, what should I do if someone else has already asked a question I have, but it has no answers? Pass values from parent to child component on click lightning I have this below code where i am trying to first search an account name and on hit button it displays the result set.  On the button click i am also loading an child component which is displaying the id of the record.  I need the child component should get displayed once i click on the record.  How can i achieve this -   Parent Component -   &lt;aura:component controller="SimpleAccountController"&gt;     &lt;aura:attribute name="account" type="account[]"/&gt;     &lt;ui:outputText aura:id="outName" value="" class="text"/&gt;     &lt;ui:inputText aura:id="name" label="Enter Name:" placeholder="Your Name" /&gt;     &lt;ui:button aura:id="button" buttonTitle="Click to see what you put into the field" class="button" label="Click me" press="{!c.getInput}"/&gt;     &lt;aura:iteration var="acc" items="{!v.account}"&gt;             &lt;a onclick="{!c.accountselected}" data-recId="{!acc.Id}"&gt;         &lt;p&gt;{!acc.Name} : {!acc.Type} : {!acc.Industry} : {!acc.Rating}&lt;/p&gt;             &lt;/a&gt;         &lt;c:HelloWorkdChildCmp selectedaccount="{!acc.Id}" /&gt;     &lt;/aura:iteration&gt; &lt;/aura:component&gt;   COntroller -   ({     getInput : function(cmp, evt) {         var myName = cmp.find("name").get("v.value");         var myText = cmp.find("outName");         var greet = "Hi, " + myName;         myText.set("v.value", greet);         var action = cmp.get("c.seaAcc");         action.setParams({ "s" : myName });         action.setCallback(this, function(response) {             var state = response.getState();             if (state === "SUCCESS") {                 cmp.set("v.account", response.getReturnValue());                 //alert("From server: " + response.getReturnValue());             }         });         $A.enqueueAction(action);     },      accountselected : function(cmp, evt) {        console.log(evt.currentTarget.getAttribute("data-recId")) ;     }  })   Child component -   &lt;aura:component &gt;     &lt;aura:attribute name="selectedaccount" type="String"/&gt;     &lt;div&gt;Id: {!v.selectedaccount}&lt;/div&gt; &lt;/aura:component&gt; How can I install Visual Studio? How can I install Visual Studio?    I'd prefer to use it in Wine or PlayOnLinux if possible. I would like to get the path to the execution directory of a Windows Forms application. (That is, the directory in which the executable is located.)  Does anyone know of a built-in method in .NET to do this?