I have been recently studying binomial theorem and in that we very frequently encounter factorials. But one queer thing which I found is $0!$. Even more queer is its value which is      $0! = 1$.   I was told this fact and I accepted it(without any proof). But I am still confused about it.  How can we prove that $0!= 1 $?
How can I prove  from first principles that $0!$ is equal to $1$?
How to shift EXIF date/time created by time in days, hours, minutes? I have two camera bodies, I was shooting while on vacation, when I downloaded the photos to lightroom, one body was set to the correct date/time, the other was off by 722 days, 4 hours, 32 minutes.  Is there a tool available that can help me fix the date/time so that they appear in sequence while browsing my lightroom library?  I'd like for them to be as close to the right time.     I was able to adjust created year, month, date, but it doesn't appear lightroom will allow me to adjust increments (only hour increments).  Many thanks if you can help me in this task! A lot of people write makefiles that say something like  paper.pdf: paper.tex         pdflatex paper         bibtex paper         pdflatex paper         pdflatex paper   To handle re-running TeX to get new/changed references and so forth. Is there a better way to do this? How do you prove a number is prime? I am a software engineer but try to keep up with maths as I really enjoyed the subject at school. I just saw a great TED talk:   The talk states that the current largest known prime is $2^{57885161}-1$  This got me thinking - how does one actually prove whether a number this large is prime or not?  Is it just done by brute force via a computer algorithm or are there any well known mathematical techniques for doing so?  I would be really grateful for any pointers as I am eager to learn about these techniques (if they exist). Current situation: I'm required to cite "normative" documents (read: standards and manuals) separately from articles and books etc. Using biblatex and biber with the numeric style I'm doing that by issuing multiple printbibliography commands like so:  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} %... lots of stuff \printbibliography[type=standard,title=Standards \&amp; Manuals] \vskip\baselineskip \printbibliography[type=manual,heading=none] \printbibliography[nottype=standard,nottype=manual]   which gives a heading Standards &amp; Manuals, prints a list of manuals, leaves a bit of space then the list of standards and finally, under a new heading Bibliography gives books and articles etc. Thus far everything is fine, enter:  The Problem: I'm using the shorthand field in my database for the standards and manuals which works perfect as far as output within the text is concerned and sets the shorthand as the label in the bibliography and prints the details to the right of it - problem is: my shorthands are considerably longer than the labels generated by the numeric style, but even though they're in three different \printbibliography commands, everything gets indented to the maximum width of the label, making the part for books and articles look quite ridiculous.  The following shows the problem with only two separate \printbibliography (and an obviously exaggeratedly long shorthand):   Now  what I would obviously want is for each of the \printbibliography commands to only use the indentation actually necessary to accommodate the largest label appearing within, i.e. rendering the Manuals section as above, but the Bibliography section like follows:   Can anyone help me to achieve this? Or give me some pointer where to look?  The above was produced with the following MWE on an up-to-date TeXLive 2014:  \documentclass[10pt,british]{scrbook}  \usepackage{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern}  \usepackage{filecontents} \begin{filecontents*}{\jobname.bib} @manual{HowTo,     title = {How to Make the World Go Round},     shorthand = {HowToMakeTheWorlGoRound},     publisher = {The Publisher},     year = {2000} }  @article{Article,     author = {Author, A. and Buthor, B.},     journal = {Journal Title},     pages = {113--126},     title = {The Article},     volume = {65},     year = {1900} } \end{filecontents*}  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} \addbibresource{\jobname.bib}  \usepackage[hidelinks]{hyperref}  \begin{document} \let\cleardoublepage\relax \chapter{The Document} Nothing to see here, but we refer to the \parencite{HowTo} based on a principle discovered by \textcite{Article}.  \printbibliography[type=manual,title=Manuals] \printbibliography[nottype=manual] \end{document} Prove that for all sets A and B, A ∩ B = ∅ implies ( A ∪ B ) - B = A In the next proof we avail ourselves of the next lemma: For all sets A and B, ( A ∪ B ) - B = A. Proof: Let A and B be arbitrary sets and let x ∈ ( A ∪ B ) - B. x ∈ ( A ∪ B ) - B ⇔ x ∈ ( A ∪ B ) ∧ x ∉ B ⇔ ( x ∈ A ∨ x ∈ B ) ∧ x ∉ B ⇔ ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B) For p ∧ ~ p ≡ F and p ∧ q ⇒ p where p and q are prepositions, ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B ) ⇔ x ∈ A ∨ F By modus tollendo ponens, x ∈ A ∨ F ⇔ x ∈ A So then, x ∈ ( A ∪ B ) - B ⇔ x ∈ A And ( A ∪ B ) - B = A Therefore, the statement for all sets A and B, A ∩ B = ∅  implies ( A ∪ B ) - B = A is true trivially. Is this proof right? I'm using PostGIS 1.5.2, with geometries in SRID:900913. I need create a circles using a list of points as centre, with a radius of 600 kilometers. I'm using this query:  INSERT INTO circles (geom) (    SELECT ST_Buffer(point, 600000, 'quad_segs=8')     FROM points );   But circles created have not 600 kilometers of radius (radius is near this length, but not exactly).  Are there other methods to create circles in PostGIS?  NOTE: Information represented is from Spain. Correct projection is 4326, but client use Google rasters, so I am storing data in 900913 to avoid reprojections and increase performance. Let's say I write a mine.service file. Then I use systemctl enable mine.service.  If I later decide to edit mine.service, do I have to tell systemd that mine.service was changed? If so, how do I do that? Cannot pickup Gauldur amulet fragment in Geirmund's Hall I am having trouble retrieving the Gauldur amulet fragment from Geirmund's Hall after clearing it before I was aware of this quest. Now when I return the quest indicator points on the floor in boss' room, specifically on a patch of submerged dirt. The water is quite shallow and I am quite sure that there is nothing to pick up there.  To me it seems the quest has bugged as I killed the boss before embarking on this quest, or killed him while unaware of the quest and not picking the fragment up. How can I get around this? Is there a console command which can help me? I had to create an 800mm x 2000mm print and so I started by actually creating a 800mm x 2000mm file in Illustrator.  The file contained only vectors so my question is:  Can I create my file size dimensions to be smaller as long as it keeps the correct ratio for the final print?  Like I said, vector ONLY, no photos or raster images. The differences between TeX engines I am confused about what TeX and its offspring (LaTeX, pdfTeX/pdfLaTeX, e-TeX, ConTeXt, LuaTeX, etc.) are.  Some, like LaTeX, seem like add ons to TeX (maybe a change to the eyes and mouth, but not stomach). Others, like pdfTeX, seem like changes to the underlying system (i.e., a change to the stomach).  Is there a good reference that explains the differences and the evolution? How to make a series of rectangles follow a pen path I am trying to get a series of shapes (rectangles) to follow a path (sine wave) that I made with the pen tool, but I have no clue how to do that or if it is possible.   I have tried warping a rectangle into the sine wave path, but it takes many different warped rectangles and it isn't smooth.  I thought  might work for rectangles, but it only works for text. How to properly render smoke on a transparent background? Is it possible to use `find -exec sh -c` safely? I'm trying to use find to echo 0 into some files, but apparently this only works with sh -c:  find /proc/sys/net/ipv6 -name accept_ra -exec sh -c 'echo 0 &gt; {}' \;   But using sh -c with find -exec makes me feel very uneasy because I suspect quoting problems. I fiddled a bit with it and apparently my suspicions were justified:   My test setup:  martin@dogmeat ~ % cd findtest  martin@dogmeat ~/findtest % echo one &gt; file\ with\ spaces martin@dogmeat ~/findtest % echo two &gt; file\ with\ \'single\ quotes\' martin@dogmeat ~/findtest % echo three &gt; file\ with\ \"double\ quotes\" martin@dogmeat ~/findtest % ll insgesamt 12K -rw-rw-r-- 1 martin martin 6 Sep 17 12:01 file with "double quotes" -rw-rw-r-- 1 martin martin 4 Sep 17 12:01 file with 'single quotes' -rw-rw-r-- 1 martin martin 4 Sep 17 12:01 file with spaces  Using find -exec without sh -c seems to work without problems - no quoting necessary here:  martin@dogmeat ~ % find findtest -type f -exec cat {} \; one two three  But when I'm using sh -c {} seems to require some kind of quoting:  martin@dogmeat ~ % LANG=C find findtest -type f -exec sh -c 'cat {}' \; cat: findtest/file: No such file or directory cat: with: No such file or directory cat: spaces: No such file or directory cat: findtest/file: No such file or directory cat: with: No such file or directory cat: single quotes: No such file or directory cat: findtest/file: No such file or directory cat: with: No such file or directory cat: double quotes: No such file or directory  Double quotes work as long as no file name contains double quotes:  martin@dogmeat ~ % LANG=C find findtest -type f -exec sh -c 'cat "{}"' \; one two cat: findtest/file with double: No such file or directory cat: quotes: No such file or directory  Single quotes work as long as no file name contains single quotes:  martin@dogmeat ~ % LANG=C find findtest -type f -exec sh -c "cat '{}'" \; one cat: findtest/file with single: No such file or directory cat: quotes: No such file or directory three    I haven't found a solution that works in all cases. Is there something I'm overlooking, or is using sh -c in find -exec inherently unsafe? What does each AMS package do? In my view, any document with a substantial amount of mathematics should load the packages from AMS-TeX. Hence, most of my document's preambles contain  \usepackage{amsmath, amsthm, amssymb, amsfonts}   (I could add to that the non-AMS mathtools, etc.). What do each of those packages do? Do I really need to list all of them, or do some of them load others? In particular, does any package load all of them, so that I can simply type  \usepackage{ams-everything}   shorter than the current list? What is the word for the emotion I feel when I see someone being humiliated? When I see someone else being embarrassed / humiliated, for example a singer forgetting their words in a concert, I would say something like "I feel embarrassed for them". But is there a better word to describe this feeling?  I would say it's a mix of:   Empathetic discomfort for someone else's humiliation Tension; hope that spectators will be forgiving Frustration that I cannot help them   Other scenarios I would feel the same emotion:   Seeing someone make a presentation, the slides are all out of order, and their face gets red with embarrassment while there's a tense silence in the room. Seeing someone make an embarrassing social blunder in front of unforgiving people. Seeing a speaker who keeps making a distracting and annoying gesture, and they don't realize that the whole audience is annoyed. Seeing someone be taken advantage of, like seeing someone naively and happily accept a terrible offer on a car. Seeing a person of strong image (for example the CEO of your company) do something like accidentally leave their webcam on after a conference call, and employees see him/her let out a big burp and pick their nose or something.   In these cases it's important that the audience is not forgiving, even if they don't confront the person about it. The person may or may not know they are being humiliated.  "Pity" as proposed in an answer seems very close, but I think it lacks a few components. I think "pity" has a chronic connotation - that you are referring to the general state of things rather than a specific event.  I think "pity" also depends on the subject suffering, but in my example they are often oblivious.  It could very well be that there is no word for this at all.  NOTE: Some people do not feel the same way I do in this scenario, so you may not think this question makes any sense. How is money held in escrow for a home purchase from a IRA treated by the IRS? Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help. My machine was updated today to Ubuntu 16.04.2. However, the kernel is still "4.4.0-62-generic". What happened (it seemed that the principal advantage of the 16.04.2 version was the update of the kernel to 4.8) ? Do I have to make a fresh install ? what is wrong with this??? please,can anyone help?  \begin{equation} \label{probabilityy} P^{t}({E_{0})=P_{0}+(1-P_{0})P_{1}+ (1-P_{0})(1-P_{1})P_{2}+.......+P_{n}\Pi_{k_{0}}^{n-1}(1-P_{k}) \end{equation} Why is super.super.method(); not allowed in Java? I read  and thought that would easily be solved (not that it isn't solvable without) if one could write:   @Override public String toString() {     return super.super.toString(); }   I'm not sure if it is useful in many cases, but I wonder why it isn't and if something like this exists in other languages.  What do you guys think?  EDIT: To clarify: yes I know, that's impossible in Java and I don't really miss it. This is nothing I expected to work and was surprised getting a compiler error. I just had the idea and like to discuss it.