Can the relative pronoun "where" be omitted in a relative clause with the preposition "in"? Can the following sentences:  It's the church where the painting burned./It's the church in which the painting burned.  be transformed so that they do not contain the pronoun which/where in the following way:  It's the church the painting burnt in.
Relative pronouns "where" and "when": where can they be omitted? I know the "omitting-rules" regarding the relative pronouns who/which/that and whose. How does it work with where and when? In the first sentence I cannot omit where but I can easily omit when in the second sentence. What is the rule?   I saw John in the shop where you bought your brown T-shirt. I met Laura on the day (when) I missed the train to Barcelona.
Which is correct: "one or more is" or "one or more are"? Should the phrase be "one or more is...", or "one or more are..."? Meaning of the word "axiom" Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises to infer conclusions, which are called "theorems" of this theory.  For example, we can use the  to prove theorems of arithmetic.  This is one meaning of the word "axiom". But I recognized that the word "axiom" is also used in quite different contexts.  For example, a group is defined to be an algebraic structure consisting of a set $G$, an operation $G\times G\to G: (a, b)\mapsto ab$, an element $1\in G$ and a mapping $G\to G: a\mapsto a^{-1}$ such that the following conditions, the so-called group axioms, are satisfied:   $\forall a, b, c\in G.\ (ab)c=a(bc)$, $\forall a\in G.\ 1a=a=a1$ and $\forall a\in G.\ aa^{-1}=1=a^{-1}a$.      Why are these conditions (that an algebraic structure has to satisfy to be called a group) called axioms? What have these conditions to do with the word "axiom" in the sense specified above? I am really asking about this modern use of the word "axiom" in mathematical jargon. It would be very interesting to see how the modern use of the word "axiom" historically developed from the original meaning.   Now, let me give more details why it appears to me that the word is being used in two different meanings:  As peter.petrov did, one can argue that group theory is about the conclusions one can draw from the group axioms just as arithmetic is about the conclusions one can draw from the Peano axioms. But in my opinion there is a big difference: while arithmetic is really about natural numbers, the successor operation, addition, multiplication and the "less than" relation, group theory is not just about group elements, the group operation, the identity element and the inverse function. Group theory is rather about models of the group axioms. Thus: The axioms of group theory are not the group axioms, the axioms of group theory are the axioms of set theory.  Theorems of arithmetic can be formalized as sentences over the signature (a. k. a. language) $\{0, s, +, \cdot\}$, while theorems of group theory cannot always be formalized as sentences over the signature $\{\cdot, 1, ^{-1}\}$. Let me give an example: A typical theorem of arithmetic is the case $n=4$ of Fermat's last theorem. It can be formalized as follows over the signature $\{0, s, +, \cdot\}$: $$\neg\exists x\exists y\exists z(x\not = 0\land y\not = 0\land z\not = 0\quad\land\quad x\cdot x\cdot x\cdot x + y\cdot y\cdot y\cdot y = z\cdot z\cdot z\cdot z).$$ A typical theorem of group theory is  which states that for any finite group G, the order of every subgroup H of G divides the order of G. I think that one cannot formalize this theorem as a sentence over the "group theoretic" signature $\{\cdot, 1, ^{-1}\}$; or can one? Universal Chord Theorem Let $f \in C[0,1]$ and $f(0)=f(1)$.   How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$?  In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$.  For any other non-zero real $r$ (i.e not of the form $\frac{1}{n}$), there is a continuous function $f \in C[0,1]$, such that $f(0) = f(1)$ and $f(a) \neq f(a+r)$ for any $a$.  This is called the Universal Chord Theorem and is due to Paul Levy.  Note: the accepted answer answers only the first question, so please read the other answers too, and also this answer by Arturo to a different question:     This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . How do I install different (upgrade or downgrade) PHP version in still supported Ubuntu release? Runtime error : ERROR 000358: Invalid expression using SelectLayerByAttribute in ArcPy? I cannot for the life of me persuade SelectLayerByAttribute to work (I need to automate this for a complex script).  Working in ArcMap's python console on already loaded shapefile, the format below:  import arcpy from arcpy import env arcpy.env.workspace = r"C:/Users/.../working" arcpy.SelectLayerByAttribute_management("layer", "NEW_SELECTION", " 'ATTRIBUTE' = 'Criterion' ")   results in error:     Runtime error : ERROR 000358:   Invalid expression Failed to execute (SelectLayerByAttribute).   An alternative method:  import arcgisscripting import arcpy gp = arcgisscripting.create(9.3) gp.workspace = "C://...//working" sFile = "C://...//working//shapefile.shp" lyr = "layer" gp.makefeatureLayer_management(sFile, lyr) gp.toolbox = "management" gp.SelectLayerByAttribute("layer", "NEW_SELECTION", " 'ATTRIBUTE' = 'Criterion' ")   .. selects everything, including features that don't meet the criterion!  Does anyone have any idea what's going wrong?  I've read some pages on , but found no solution that works here. The working directory is correct. The attribute field and criterion are correctly specified (case sensitive etc), etc. Its very confusing. Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Then S is:   one-one but not onto onto but not one one invertible non-invertible   (One or more correct options)  My attempt: $S=T^4+3T^3-4I=T(T^3+3T^2)-4I=T(4I)-4I=4T-4I$  How do I go about proving or disproving my options?   My thoughts on options # 1. and 2.: I'm guessing $S$ is one-one if $Ker(S)=\theta$ is one way to go, but how do I obtain the kernel when I don't know what the transformation is? $Ker(S)=\theta$ if $T=I$, which does satisfy $T^3+3T^2=4I$. But so can other $T$s.   My thoughts on options # 3. and 4.: $S$ is invertible if $det(S)\neq0$, which is possible $det(T-I)\neq0$, i.e. $T\neq I$. How do I show that?  Please help! I have little experience with testing and I'm being hired to work on a team where I am the only tester. I'm struggling to find the best approach for the process. Being alone should I write down a test script or not? Wouldn't it take too much time? I have no experience in writing test scripts, could anyone help with a good reference (book, video or whatever), please? How to keep font preferences in TexWorks 4 The default font is too small on my screen at the resolution I have it, so I like to reset the font to size 12, but it appears I have to do this everytime I launch the program.  How do I make it remember my preferences? Is it OK to use a forward slash '/' between words other than the obvious 'and/or'? I don't want to get into the proper use of the phrase 'and/or', but rather to investigate the use of the forward slash between other words. Examples:   I have an interview with him/her this afternoon. He has a large binder/notebook.   And even separating just letters:   I connected the i/o ports. (input/ouput) Mark the field as n/a. (not applicable) I did it w/o her help. (without)   It seems as thought the forward slash can be used to indicate an option, display ambiguity, shorten 2 words and shorten 1 word (and others).  Is it OK to use a slash to separate these words/letters (yes, that was intended), or is it better to explicitly say 'his or her'? One more thing, is it correct to 'chain' these phrases together? I'm horrible with examples but here's one:   I don't open letters/mail that aren't/isn't addressed to me. Send email with an attachment located in Google Drive I want to send a PDF attachment located on Google Drive from a Sheet.   Code I'm using is:  function sendEmails() {   var sheet = SpreadsheetApp.getActiveSheet();   var startRow = 2;  // First row of data to process   var numRows = 8;   // Number of rows to process   // Fetch the range of cells A2:B3   var dataRange = sheet.getRange(startRow, 1, numRows, 2)   // Fetch values for each row in the Range.   var file = DriveApp.getFilesByName('ScriptCarPass.pdf')   var data = dataRange.getValues();   for (i in data) {      var row = data[i];     var emailAddress = row[0];  // First column     var message = row[1];       // Second column     var subject = "Sending emails from a Spreadsheet";     MailApp.sendEmail(emailAddress, subject, message, {attachments: file});   } }   Everything works fine except the attachment - i.e. if I run without the attachment it works fine.  I keep getting error Invalid argument: inlineImages  I've also tried with:   attachments: [file.getAs(MimeType.PDF)]   but that gives error that .getAs isn't recognised. I want to view the fonts that are installed on my system. Specifically, I want the client-side (fontconfig) fonts, not the old-style X11 bitmap fonts on the server (for which xfontsel does an adequate job). I guess what I'm looking for would be called a font viewer or font gallery. I want to see a bit of sample text for each font, ideally configurable sample text.  Being able to filter by font name, available character sets and other characteristics would be a plus.  A search of packages in Debian wheezy found gnome-font-viewer and kfontview, but they can only display a single font. I'd rather avoid depending on software that isn't widely available, including on the current Debian stable and Ubuntu LTS. What should be the intuition when working with compactness? I have a question that may be regarded by many as duplicate since there's a  one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}^n$ the compact sets are those that are closed and bounded, however the guy who answered this question and had his answer accepted says that compactness is some analogue of finiteness. That's the first problem: In my intuitive view of finiteness, only boundedness would suffice to say that a certain subset of $\mathbb{R}^n$ is in some sense &quot;finite&quot;. On the other hand there's the other definition of compactness (in terms of covers) which is the one I really need to work with and I cannot see how that definition implies this intuition on finiteness. Also, I feel it's pretty strange the covers people use when they want to deal with compact sets. To prove a set is compact I know they must show that for every open cover there's a finite subcover; the problem is that I can't see intuitively how one could show this for every cover. Also when trying to disprove compactness the books I've read start presenting strange covers that I would have never thought about. I think my real problem is that I didn't yet get the intuition on compactness. So, what intuition should we have about compact sets in general and how should we really put this definition to use? Can someone provide some reference that shows how to understand the process of proving (and disproving) compactness? How can I take shallow depth of field photos with a point-and-shoot camera? I have a compact digital camera, and in macro mode I can manage to achieve a blurred background, if the background is far enough away. Can this be done when taking non-macro pictures, and if not, is there a way to simulate it?  When I shoot portraits with my brother's DSLR, it is quite easy to fade out the background. But with my point and shoot camera, I haven't been able to take such photos.   I choose a large aperture for my camera but it still doesn't work. I know that standing further and zooming will result in a smaller depth of field, but it doesn't seem to be enough to get the really nice "bokeh" look. I've tried macro mode, and that will work if the subject is close enough and the background far enough away, but how can it be done with non-macro pictures?   Why are DSLR cameras are so much better in this area? What are the general recommendations to do this using a compact or super-zoom camera? How to Mount Guest DOS OS with HOST OS.? I am newbie to VM concept. actually i have installed DOS6.22 under virtualbox and using 32bit windows 7 OS as HOST Operating System, now i want to copy some folders/files from HOST OS to DOS6.22 Guest OS. how can i copy.? Increasing the maximum number of enumerate environments? barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: Are there philosophers who examined the reasons for believing in the existence of other people? While I am familiar with Descartes and the oft-quoted idea, I think therefore I am, I am not familiar with any philosophy that actively sought to rationalize that anybody else exists.  Are there philosophers who analyzed this question? Or is it taken for granted that it is (im)possible for anybody to validate their own existence to somebody else? How to hide the .html extension with Apache mod_rewrite I use babelbib and I have a source with two authors. Contrarily to when using the plain bib style, as soon as I switch to babplain, the author names are formatted like this:  [1] Müller, Jan und Eva Schulze: Name of the book, 2015.   While I would like both author names to be formatted the same way, that is one of these:  [1] Müller, Jan und Schulze, Eva: Name of the book, 2015. [1] Jan Müller und Eva Schulze: Name of the book, 2015.   sources:  % test.tex \usepackage[ngerman]{babel} \usepackage[fixlanguage]{babelbib} \selectbiblanguage{ngerman} \bibliographystyle{babplain} \bibliography{test}  % test.bib @manual{foo,     author = "Müller, Jan and Schulze, Eva",     title = "Name of the book",     year = "2015", }   I can fix it by specifying and {Schulze, Eva}, but that way all is treated as the last name, that cannot be the solution, can it? I'm trying to update my system using apt-get, but it's giving me this error  Err http://bd.archive.ubuntu.com raring Release.gpg          Could not connect to bd.archive.ubuntu.com:80 (116.193.170.18), connection timed out Err http://bd.archive.ubuntu.com raring-updates Release.gpg   Unable to connect to bd.archive.ubuntu.com:http: Err http://bd.archive.ubuntu.com raring-backports Release.gpg   Unable to connect to bd.archive.ubuntu.com:http: Reading package lists... Done W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring/Release.gpg  Could not connect to bd.archive.ubuntu.com:80 (116.193.170.18), connection timed out W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring-updates/Release.gpg  Unable to connect to bd.archive.ubuntu.com:http: W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring-backports/Release.gpg  Unable to connect to bd.archive.ubuntu.com:http: W: Some index files failed to download. They have been ignored, or old ones used instead.   I am not a Ubuntu master but I can understand this is not a computer error. Any clues on how to fix it?