I mean the distance between the first particle and the second particle. Should I say:  distance between the first and second particle  or  distance between the first and second particles  ?
I'm a little confused about the following sentences. Could you please tell me which one(s) is/are correct?        The first and the second chapter are difficult.   The first and the second chapters are difficult.   The first and second chapters are difficult.   The first and second chapter are difficult.      I believe #1 &amp; #3 are correct, while #2 &amp; #4 are incorrect.  But when seeing the answers to the Quora question , I wonder why they say #2 is correct.
I have to deal with a large number of files nested inside of directories (one file per directory), resembling something like this:  fred/result.txt john/result.txt mary/result.txt ...   I am currently using the following command to process each of those files:  find . -maxdepth 1 -type d \( ! -name . \) -exec bash -c \"cd '{}' &amp;&amp; processResult result.txt\" \;   I am looking for something I can add into the end of this command that will change the filenames to fred.txt, etc. and then later move the file into the parent directory to eliminate the extra layer of directories.  What would be the best way to do this? How to round up the result of integer division? I'm thinking in particular of how to display pagination controls, when using a language such as C# or Java.  If I have x items which I want to display in chunks of y per page, how many pages will be needed? What causes the permittivity and permeability of vacuum? When light travels through a material, it gets "slowed down" (at least its net speed decreases).  The atoms in the material "disturb" the light in some way which causes it to make stops on its path.  This is expressed in the material's permittivity and permeability:  Its ability to transmit electromagnetic waves.  But then what causes the limited permittivity and permeability of vacuum?  There are no atoms there to disturb the light, so what is keeping it from travelling at infinite speed?  Is there a special property of space itself which determines the speed of light?  Or is it the virtual particles in vacuum which interact with the light? Please note: I posted this question about an hour ago on my account (I just made it as I am new to the site). I revisited my post and saw that I had not given definitions of what I was asking (I thought they were common notations) and it would not let me edit the post for some reason so I will ask here now and hopefully someone can delete the other thread.  How to prove for a finite group $G$ that $|\langle g \rangle|=o(g)$?  Here are the definitions:  Let $G$ be a finite group and $g \in G$  $o(g)$ is the smallest $m \in \Bbb{N}$ such that $g^m=e$ (the identity element of the group). (If such an $m$ exists)  $\langle g \rangle=\{g^m : m \in \mathbb{Z}\}$  I really don't know how to show that this is true I tried to say let $o(g)=m$ and then show that $|\langle g \rangle|$ has exactly $m$ elements but I wasn't even sure it did because it would contain the elements $g^0,...g^{m-1}$ so that would mean it's true but what about $g^{-1}$ and the other elements with negative coefficients so it seems like it would actually be bigger than $m$?  Thanks. Do Irrationals form a group? It's true that the irrationals do not form a group under addition or multiplication, but I want to find a binary operation "∗" such that ($\mathbb R \setminus \mathbb Q,∗$) is a group. It is possible by transpose of operation from $\mathbb R$, but I want a explicit example. Is there any text editor which supports Markdown (Ask Ubuntu's formatting style)? Will hardware changes affect system? use contradiction to prove that the square root of $p$ is irrational Can we continue to show the "Take a tour" banner after a user has created an account? When you view a Stack Exchange site without being logged in, you see a friendly banner, prompting you to take a tour and understand the features of the site:    This links off to a high quality resource that can help new users understand how the site works. However, the moment a user signs up to the site, this banner disappears from immediate view and hides itself as a link under the help menu.  My proposal is to continue showing this banner until one of the following conditions are met;   The user gets the "Informed" badge The user reaches a set amount of reputation (250 or 500 should be sufficient) The user chooses to hide the banner   This should increase the exposure of the tour to new users, who are the people who need the tour, rather than hiding it the moment a user signs up. How to get the Input and output in a terminal to a text file? I want to create a text file which will add each command I am going to run on the terminal as a new line entry in the text file. i.e., I need to keep a record of what are the things I'm doing in the terminal. Further if I can keep a record of outputs I'm getting in the terminal, along with the inputs in the terminal, as a text file, it will be much better.   Each time I forget a command I have to search again in the Internet for the command.  Can I can make a file like above? Can a multi-core processor run multiple processes at the same time? I understand that a multi-core processor can run multiple threads of the same process at the same time using its different cores. Can it also run multiple processes at the same time(each core running a different process)? How do I rollback a TFS check-in? I'd like to rollback a change I made recently in TFS. In Subversion, this was pretty straightforward.  However, it seems to be an incredible headache in TFS: Option 1: Get Prior Version  Manually get prior version of each file Check out for edit Fail - the checkout (in VS2008) forces me to get the latest version  Option 2: Get TFS Power Tools  Download Team Foundation Power Tools Issue rollback command from cmd line Fail - it won't work if there are any other pending changes  Option 3: Manually Undo Changes  manually undo my changes, then commit a new changeset  Question How do I rollback to a previous changeset in TFS? Outlook autocleaning my line breaks and screwing up my email format I'm sending an email using the dotnet framework. Here is the template that I'm using to create the message:  Date of Hire: %HireDate% Annual Salary: %AnnualIncome% Reason for Request: %ReasonForRequest%  Name of Voluntary Employee: %FirstName% %LastName% Total Coverage Applied For:  %EECoverageAmount% Guaranteed Coverage Portion: %GICoveragePortion% Amount Subject to Medical Evident: %GIOverage%   When the messages is received in outlook, outlook tells me "Extra line breaks in this message were removed". And the message displays like this:  Date of Hire: 9/28/2001 Annual Salary: $100,000 Reason for Request: New Hire  Name of Voluntary Employee: Ronald Weasley Total Coverage Applied For:  $500,000 Guaranteed Coverage Portion: $300,000.00 Amount Subject to Medical Evident: $200,000   Note how Outlook incorrectly removes needed line breaks after the name, EECoverageAmount, etc...  It's important for the email recepients to get a correctly formatted email, and I have to assume that some of them use outlook 2003. I also can't assume they will know enough to shutoff the autoclean feature to get the message to format properly.   I have viewed these messages in other mail clients and they display correctly  some more information:   I am using UTF-8 BodyEncoding (msg.BodyEncoding = System.Text.Encoding.UTF8) The msg.Body is being read from a UTF-8 encoded text file, and each line is terminated with a crlf.   Question: How do I change the format of the message to avoid this problem? I am reading through a directory with some pictures and such using a pretty simple implementation of readdir() like the following:  if ($handle = opendir($path)) {     while (false !== ($szFilename = readdir($handle))) {     if ($szFilename[0] !== '.') {         if (is_file($path.$szFilename)) {                 // do stuff             }         }      }  }   The problem that I am having is that the files are not being read in alphabetical order as the docs for readdir() state:     Returns the filename of the next file   from the directory. The filenames are   returned in the order in which they   are stored by the filesystem.   Another weird thing is that, on the local testing server, the same code works great.  This is running on a server using the LAMP stack in both cases.  I know that I can build an array and just sort it, but I was wondering if I was missing something in what I was doing. I am trying to show that if $n$ is an odd natural number, then $8$ divides $n^{2}-1.$ I was able to prove that because I know that if $n$ is an odd natural number, then $n^{2}$ can be written as $8k+1$ for some $k\in \mathbb{N}.$ I would like to show this question by using the Euclidian division. Then I wrote $n^{2}-1=8k+r$, where $0\leq r  &lt; 8.$ When $r=2$ we get $n^{2}-1=8k+2.$ Since $n$ is odd, then $n^{2}-1$ is even and I got stuck. Is there way to fix that? How to get horizontal black line when coloring cells? How to avoid perceived circularity when defining a formal language? Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found . What makes me uneasy about this definition is that words such as "set", "countable", "function", and "number" are used in somewhat non-trivial manners. For instance, behind the word "countable" rests an immense amount of mathematical knowledge: one needs the notion of a bijection, which requires functions and sets. One also needs the set of natural numbers (or something with equal cardinality), in order to say that countable sets have a bijection with the set of natural numbers.  Also, in set theory one uses the relation of belonging "$\in$". But relation seems to require the notion an ordered pair, which requires sets, whose properties are described using belonging...  I found the following in Kevin Klement's,  (pages 2-3).  "You have to use logic to study logic. There’s no getting away from it. However, I’m not going to bother stating all the logical rules that are valid in the metalanguage, since I’d need to do that in the metametalanguage, and that would just get me started on an infinite regress. The rule of thumb is: if it’s OK in the object language, it’s OK in the metalanguage too."  So it seems that, if one proves a fact about the object language, then one can also use it in the metalanguage. In the case of set theory, one may not start out knowing what sets really are, but after one proves some fact about them (e.g., that there are uncountable sets) then one implicitly "adds" this fact also to the metalanguage.  This seems like cheating: one is using the object language to conduct proofs regarding the metalanguage, when it should strictly be the other way round.  To give an example of avoiding circularity, consider the definition of the integers. We can define a binary relation $R\subseteq(\mathbf{N}\times\mathbf{N})\times(\mathbf{N}\times\mathbf{N})$, where for any $a,b,c,d\in\mathbf{N}$, $((a,b),(c,d))\in R$ iff $a+d=b+c$, and then defining $\mathbf{Z}:= \{[(a,b)]:a,b\in\mathbf{N}\}$, where $[a,b]=\{x\in \mathbf{N}\times\mathbf{N}: xR(a,b)\}$, as in  question or  on Wikipedia. In this definition if set theory and natural numbers are assumed, then there is no circularity because one did not depend on the notion of "subtraction" in defining the integers.  So my question is:     Question Is the definition of first-order logic circular?   If not, please explain why.   If the definitions are circular, is there an alternative definition which avoids the circularity?   Some thoughts:   Perhaps there is the distinction between what sets are (anything that obeys the axioms) and how sets are expressed (using a formal language). In other words, the notion of a set may not be circular, but to talk of sets using a formal language requires the notion of a set in a metalanguage. In foundational mathematics there also seems to be the idea of first defining something, and then coming back with better machinery to analyse that thing. For instance, one can define the natural numbers using the Peano axioms, then later come back to say that all structures satisfying the axioms are isomorphic. (I don't know any algebra, but that seems right.) Maybe sets, functions, etc., are too basic? Is it possible to avoid these terms when defining a formal language? How can I fit my table of contents into a single page? When I use \tableofcontents I get all my contents in a single page except for a line that goes on the other page, how can I make all in the same page? Is it possible to view the scene in the viewport camera view at 1:1? Why is there an "electric" potential drop across the resistor? An electric potential difference is created when two charges are separated. In a capacitor, there is a clear accumulation of opposite charges on the two separated plates, therefore work has to be done in moving an unit charge from one plate to another, against the electric field.   But, , a potential difference is created in a resistor due to the heat produced by the collision and increased lattice vibrations. So there is an energy loss across the resistor, yes, but how does this energy loss due to heat create an electric potential difference. In other words, do electrons accumulate on one or both sides of the resistor? If not, why should we do electric work in moving from one end of the resistor to another. If the difference in energy is not due to electric work, then that's not electric potential, is it?