which one is correct ? 1- this kind of toothbrushes (has) sensors to detect... etc 2- this kind of toothbrushes (have) sensors to detect... etc
I have a question about using the phrase "some kind of _".  Is it incorrect to say "some kind of + plural noun"?     For example, is it wrong to say "are these rock-like things some kind of chocolates?"   I didn't use "kinds" and that kind of adds to my question. Is "kind" always plural, like "some kinds of chocolates"? Because although I'm using "some", it feels like "kind" shouldn't be a countable noun.  Anyway, I don't think I've heard of "some kind of + plural", but I think it makes more sense then using singular, since I think chocolates are countable. Please help!
Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? From what I understood, cardinal numbers are defined as:  $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$  The continuum hypothesis states that the cardinality of $\mathbb{R}$ is  $\aleph_1$. I more or less understand this cannot be proven (does not follow from ZFC).  However, can it be proven dat there in fact is a 'least cardinal number' greater than $\aleph_0$?  Is it possible that there are sets with cardinality $\aleph_{1/2}$, $\aleph_{1/3}$, et cetera, so to speak?  Or expressed differently: is it possible that for every set A with cardinality CA > $\aleph_0$, there exists another set B with cardinality CB where CA > CB > $\aleph_0$? Suppose I have a document with multiple include or input statements  \input{fileA} \input{fileB}   etc.   Is there an easy way to generate a single .tex file where \input{fileA} is replaced by the actual content of fileA etc. without copying it manually? Do I have to buy or need a 3D-enabled phone to use Google Cardboard? Google cardboard comes with 3D glasses. Does that mean that we will be able to watch 3D videos using a smartphone like the Moto-G (1st edition) which is not 3D enabled? Prefer composition over inheritance? If $A=\left( \begin{array}{ccc} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \end{array} \right)$, find the rank of $A+A^2+A^3$  We have $\rho(A+A^2+A^3)\leq\rho(A)+\rho(A^2)+\rho(A^3) $ where $\rho(A)$ means rank of A.  Here $\rho(A)=3.$ Also $\rho(A^2)=3$ and $\rho(A^3)=3$  as $\det A\neq 0$ and $\det(A^2)=\det(A)\det(A)\neq 0$ and so on.   But how to get the desired result. It's my understanding that "easily" is the regular adverb for "easy", e.g. "He makes friends easily".  But in some collocations one still uses "easy", e.g. "Take it easy", "Rest easy" or "Slow down easy". Is there any rule when "easy" would be used over "easily", or does one just have to know these collocations? I have many tags marked as "interesting" and others as "ignored". (I use Java at work, I'm a Cocoa guy by night, any anything dealing with Windows/.NET/C# simply doesn't apply to me.) When tags are added to the list in the right navbar, they appear in the order in which they were added. There are several problems with this:   It can be difficult to visually locate a tag quickly, unless you have memorized its location. The tags appear as the same gray boxes like everywhere on SO, but with a small "x" afterward, and they wrap based on size. This tends to leave a ragged right margin and wasted space. Ordering based on when the tag was added is not always the most intuitive default. Various users might prefer to view them alphabetically, by popularity, recent activity, etc.   It would be nice to introduce a tabbed interface to switch between various orderings, and remember the value in the cookie, profile, etc. Coupled with this, perhaps users could specify a custom ordering that works best for them? (Currently the workaround is to copy all the tag names, remove them all, and re-add them in the desired order, but this is obviously annoying.)  Thoughts? How to see if SkyDrive is still syncing in windows 8.1? Windows 8 displayed overlay icons on my SkyDrive files, so I could see which files are current and which are in sync.  After I upgraded to Windows 8.1 the files still sync, but I can't see if the files are current or not. The only way is to load a file, check for myself if the latest changes are there (if I remember them) and refresh until the file is synced. Which is a pain! The sync overlay does only show for files which modified locally and are waiting to upload, it does not show up for files which had been changed on the server (other computer) and are waiting to download!!  So how do I get the file sync Status on Windows 8.1? Unable to upgrade Windows 10 edition from Home to Pro I have a Lenovo laptop which upgraded from Windows 8.1 Home to Windows 10 Home.  I have a (legitimate) Windows 10 Pro key which I ran through the "Change Product Key" wizard.  After a reboot I get an error which states "We couldn't upgrade your edition" and asks me to try again or seek help from my system administrator (me!) or Microsoft.  There is nothing in the event logs except Info entries in the Setup log stating it failed to change the installed state to Pro Edition with a hex error code (I don't have access to the laptop right now, I will update later with the full log details)  Googling the hex error code led me to a Windows 7 KB Article regarding Windows Update failure due to locked files. The article stepped through using MSCONFIG to perform a clean boot which I did, but still got the error.  Any pointers on how to upgrade this laptop? Or will I have to get the sledgehammer out and do a clean install? Changing CSV layer attribute value from text to number in QGIS? I read in a CSV file for linking to a .shp file.   All the variables that are just numbers came in as text.   How do I change the text variables to numeric in QGIS? Confidence Interval for variance given one observation This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory":     Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters unknown, give a confidence interval for $\sigma^2$ with a confidence level of at least 99%.   It seems to me that this should be impossible. I have the solution, but haven't read it yet. Any thoughts?  I'll post the solution in a couple days.  [Follow-up edit: Official solution posted below. Cardinal's solution is longer, but gives a better confidence interval. Thanks also to Max and Glen_b for their input.] Why does old PMOS/NMOS logic needed multiple voltages like  +5, -5, and +12 volts? For example, old Intel 8080 processors, old DRAMs, e.t.c...  I'm interested in the causes on the physical/layout level.  What was the purpose of these additional voltages?  Yes, this question is about stuff which was used 35 years ago. I am able to use the split command successfully to split large file into multiple smaller files. This is being achieved by following command  split -b 1G $temp_path $final_filepath   But only caveat is that these files many times contain last line which is split across 2 files. Is there any way to avoid that using split or any other command ? How can you enumerate an enum in C#?  E.g. the following code does not compile:  public enum Suit {     Spades,     Hearts,     Clubs,     Diamonds }  public void EnumerateAllSuitsDemoMethod() {     foreach (Suit suit in Suit)     {         DoSomething(suit);     } }   And it gives the following compile-time error:     'Suit' is a 'type' but is used like a 'variable'   It fails on the Suit keyword, the second one. How do berry mutations work? Looking for title & author - short story - possibly called "Arms Race" - about rediscovery of manual arithmetic short story - possibly called "Arms Race" - about rediscovery of manual arithmetic.   The hero rediscovers manual calculation - e.g., how to do long division with pencil and paper - in a world where everyone has ben using calculators for aeons. Brought before the president to demonstrate this, the great line in the story is the president, after using his calculator to check the hero's answer, says, "My God -- he's right." A great national project to exploit this advance over the enemy is obviously necessary... Why do eukaryotic organisms have introns in their DNA? We touched on introns and exons in my bio class, but unfortunately we didn't really talk about why Eukaryotes have introns. It would seem they would have to have some purpose since prokaryotes do not have them and they evolved first chronologically, but I could easily be wrong. Did the junk sections of DNA just evolve there by some sort of randomness or necessity as opposed to an actual evolutionary advantage? Why hasn't evolution stopped us from having introns since they seem to be a 'waste' of time and DNA? Why do prokaryotes not have introns? Why is this sentence correct? “She suggested that he go to the cinema.” Why is this sentence correct?     She suggested that he go to the cinema.   I would definitely use goes instead of go. show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant How to break the Y axis in an Excel sheet I want to break the Y-axis in an Excel sheet. How can I do that?  For example, suppose I have the data in a range from 0-7000 and then range from 22000-28000. I want a break in the Y axis from 8000-20000.