Why is it not true that if $\Phi \vdash \exists x \phi$ then there is a term $t$ with $\Phi \vdash \phi \frac{t}{x}$?
Why is it not true that if $\Phi \models \exists x \phi$ then there is a term $t$ with $\Phi \models \phi \frac{t}{x}$?
Analogous results to Euler's theorem Fermat's little theorem and its generalization, Euler's theorem holds for pair of relatively prime integers. Are there any analogous results for not relatively prime integers ? Thanks in advance. SQL Server Stored Procedure Update not working with ASP.Net C# GridView RowUpdating Is it possible to experimentally show that observable universe is much larger than ~93B ly? Is it possible to put commands in /etc/motd? Can I put shell commands in the /etc/motd login banner file? I have tried:   $(uptime)   and  `uptime`   Is this possible? We have been asked to prove the above statement through the two sub-proofs:   Firstly we must prove that $e^x$ is increasing and then prove that $\log(x)$ is also an increasing function.   After we are asked to prove that using the delta-epsilion definition of a limit that $\lim_{x \to 0} e^x = 1$.   I have done all of this but still don't see where the instructor is going with this. Replace HTML page with contents retrieved via AJAX I have an HTML page with a typical structure:  &lt;html&gt;   &lt;head&gt;    &lt;script src="..." &gt;&lt;/script&gt;    &lt;style&gt;...&lt;/style&gt;   &lt;/head&gt;   &lt;body&gt;    content   &lt;/body&gt;   &lt;script&gt;     var success_callback = function(data) {       // REPLACE PAGE CONTENT &amp; STRUCTURE WITH "data"     }     ajax(url, params, success_callback);   &lt;/script&gt; &lt;/html&gt;   Do you think it is possible ?  I've already tried to give the html tag an id and doing $(id).replace(data); with no success.  Don't ask me why, but that is what I need (I'm working with a special "mashup builder" site... it is a long story).  EDIT : I forgot to say that scripts in the received content have to be executed, even external scripts included using &lt;script src="..."&gt;. Require duplicates to be imported in Shotwell I have some drawings that I have imported and tagged by date. Now I want to take some of them that I have organized on my disk and put give them a new tag. Shotwell thinks they are duplicates and won't import them. Is there a way to require Shotwell to import these duplicates? What is a natural response to &quot;Good to see you. How you doing?&quot;? Will a natural response include &quot;doing&quot; as in &quot;I'm doing okay&quot; or will &quot;I'm okay&quot; be more natural? I have heard of some methods, but none of them have stuck. Personally I try to avoid complex types in C and try to break them into component typedef.  I'm now faced with maintaining some legacy code from a so called 'three star programmer', and I'm having a hard time reading some of the ***code[][].  How do you read complex C declarations? Is it (and if yes how?) possible to define an an binary operation $*$ for an arbitrary set $M$ such that $(M,*)$ is a group?  If $M$ is finite or countable infinite this is trivial, but is it also possible for any other sets? And if yes how can you construct $*$? If no can you provide a counterexample? Is it true that if $A$ is a nonnegative primitive matrix, then $AA^T$ is also primitive? Obviously $A^T$ is primitive but in general product of primitive matrices is not primitive.  Any hint? How can time dilation be symmetric? Is there an implicit default constructor in C++? In the book I'm reading at the moment () it says that if you don't declare a default constructor for a class, the compiler supplies one for you, which "zeroes out each data member". I've experimented with this, and I'm not seeing any zeroing -out behaviour. I also can't find anything that mentions this on Google. Is this just an error or a quirk of a specific compiler? command to display time (duration) in human readable form I spent 3 hours trying to write a command to format a time (duration) expressed as a floating number (in seconds) in a nice form as follows:  if the time is &lt; 60 write as a float with 3 decimals (i.e. milliseconds)  else let h,m,s be the hours, minutes and seconds corresponding to the time      if h=0 then write as m:ss (ss always with 2 digits, m can have only 1)      else write h:mm:ss (mm and ss with 2 digits, h can have only 1)   So 1.25678 gives 1.256, 125.35 gives 2:05, 18249 (which equals to 5*3600+4*60+9) gives 5:04:09.  I tried several packages for the comparison/formatting. No success. Can someone help me ? Thanks ! I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example? YA dystopia novel of battle school with unusual military control scheme Some sort of battle school (no, not Ender's Game), where teens are being trained as military commanders.   The control system for their army hinges around some sort of system that requires extremely fine motor control -- being able to "push" around digital representations of military units using a special interface that interprets minute finger movements.  The boy protagonist eventually meets a girl in the battle school, and is astonished to learn that most people do not live as well as he does -- for instance, he gets a fresh jumpsuit outfit every day and disposes of the old one; this is a luxury.  One scene I recall is them kissing for a moment in some sort of underground passageway; the boy protagonist notes the exact length of the kiss, as he has been trained to have heightened time perception.  I don't recall much about other commander characters; I think they have different special abilities, and one other major character might have physical deformities, possibly because of postwar-radiation-related birth defects.  They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. Moving a magnet close to a conductor induces a current. If it consists of a superconducting material with resistance $R=0$, then my textbook says:     Then the induced current will continue to flow even after the induced emf has disappeared.   This makes sense physically - there is no resistance to stop charge flow. But then the book draws this conclusion:     Thanks to this persistent current, it turns out that the flux through the loop is exactly the same as it was before the magnet started to move, so the flux through a loop of zero resistance never changes.   If the flux $\Phi$ never changes in a superconductor, from Faraday's law this means - from what I have learned - that no electromotive force $\mathcal{E}$ is induced:  $$\mathcal{E}=-\frac{\mathrm{d} \Phi}{\mathrm{d}t}=0 \:\:\:\:\text{ when }\frac{\mathrm{d} \Phi}{\mathrm{d}t}=0$$  My conclusion is therefor: There would never be induced any current at all. Current can never be induced in a superconductor loop. Is this the case or am I misunderstanding my book? How can I remove duplicates in my .bash_history, preserving order? I really enjoying using control+r to recursively search my command history. I've found a few good options I like to use with it: # ignore duplicate commands, ignore commands starting with a space export HISTCONTROL=erasedups:ignorespace  # keep the last 5000 entries export HISTSIZE=5000  # append to the history instead of overwriting (good for multiple connections) shopt -s histappend  The only problem for me is that erasedups only erases sequential duplicates - so that with this string of commands: ls cd ~ ls  The ls command will actually be recorded twice. I've thought about periodically running w/ cron: cat .bash_history | sort | uniq &gt; temp.txt mv temp.txt .bash_history  This would achieve removing the duplicates, but unfortunately the order would not be preserved. If I don't sort the file first I don't believe uniq can work properly. How can I remove duplicates in my .bash_history, preserving order? Extra Credit: Are there any problems with overwriting the .bash_history file via a script? For example, if you remove an apache log file I think you need to send a nohup / reset signal with kill to have it flush it's connection to the file. If that is the case with the .bash_history file, perhaps I could somehow use ps to check and make sure there are no connected sessions before the filtering script is run? $f(x)f(1/x)=f(x)+f(1/x)$ Find a function $f(x)$ such that:  $$f(x)f(1/x)=f(x)+f(1/x)$$  with $f(4)=65$.  I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$  which leaves $f(1/x)$ as:  $$a_0+a_1{1\over x}+a_2{1\over x^2}+\ldots + a_n{1\over x^n}$$  On comparing the coefficients of both sides, we see that:  $$2a_0=(a_0)^2+(a_1)^2+(a_2)^2+\ldots+(a_n)^2$$  And  $$a_1=(a_0a_1)+(a_1a_2)+ \ldots +(a_{n-1}a_n)$$  I don't know how to proceed further. I know I need to compare coefficients and come to a conclusion based on their values, but I don't see what to do next. In Texas Hold'em poker, there are some cases where a particular card, often called kicker, acts as a tiebreaker between players to determine who wins the pot, or if the pot has to be shared.  For some combinations such as three or four of a kind, there is no doubt: the kicker is the 5th card of the chosen combination, and determines the winner.  Example     Alice has 7 and 10   Bob has 7 and K   On the board are 7, 7, 2, 3, 4   In that case, both have a three of a kind. But because Bob has a king and Alice only a 10, he wins.  Most of poker websites explaining the rules clearly mention the role of the kicker applying to three of a kind and double pairs. But much fewer say something about colors, straights and full houses, and I found many contradicting answers. I also asked the question to some used online poker players and they also gave me contradicting answers.  I have tried to search for official tournament rules, but most of them only explain what happens with bad behaviors, bad deals, incorrect or confusing betting, showing cards when you shouldn't, acting when it's not your turn, etc. without mentioning subtleties about combinations at all.  To simplify my question, I will take three examples; I think it's better to start with examples before getting to the general answer if one exists. So, what's happening in the following 3 examples? Could you give a more generalized answer?  Example 1 - Flushes     Alice has 2&clubs; and 3&clubs;   Bob has 4&clubs; and 5&clubs;   On the board are 6&clubs;, 8&clubs;, 10&clubs;, 2&hearts;, 5&diams;   Contradicting answer 1: it's a tie, because the highest card included in the flush is the 10, which everybody chooses to include in their 5 showdown cards. Contradicting answer 2: Bob wins, because he has the greatest private card that is part of the flush  Example 2 - Straight     Alice has 6 and K   Bob has 6 and J   On the board are 4, 5, 7, 8, 10   Contradicting answer 1: it's a tie, because the greatest card in the straight is the 8 for both Alice and Bob Contradicting answer 2: Alice wins, because she owns an extra king, compared to the jack of Bob  Example 3 - Full house     Alice has 3 and 7   Bob has 3 and 6   On the board are 3, 3, 2, 2, 5   Contradicting answer 1: it's a tie, because one is supposed to choose only five cards to make a combination, and a full house is already five cards. There couldn't be any kicker, and thus their showdowns are strictly equals. Contradicting answer 2: Alice wins because of her extra 7, compared to the 6 of Bob  Thank you very much for your answers.  Note: I'm unable to post next to you, so I edit my own post; strange not be able to answer to an answer.  Ok, So if I summarize what you are saying :   In the flush case, Bob wins because at some point, their hand differs. Technically, I can continue comparing the highest private card to decide who wins. I had it correct. In the straight case, if the highest card of the straight is public, then it's technically always a tie, no matter what the players had as second private card (asuming that only one of the two cards was part of the straight). In a full house if both players have the same triplet and the same pair, it's always a tie, no matter what the players had as second private card (assuming again that only one of the two was part of the full house).   Please confirm me that what I summarized is correct.  Thank you for your answer.