Showing validity of a formula in first order logic
Is the Formula Logically Valid?
Aware abbreviating those commands need not always be helpful, I would like to know how to abbreviate them? The following is what I tried, which does not work:  \documentclass[11pt,handout]{beamer} \newcommand{\bframe}{\begin{frame}} \newcommand{\eframe}{\end{frame}} \newcommand{\ftitle}{\frametitle}   \author{Author} \title{Title}   \begin{document} \maketitle  \begin{frame} \frametitle{Outline}  \end{frame}   \bframe \ftitle{Introduction}  \eframe    \end{document}   At first I thought it might be the case that commands such as \bframe was already defined. But using \def instead still gives no desired result. Geometry with Trigonometry From the given figure, prove that      $$\cot\ \theta=\cot\ A+\cot\ B+\cot\ C$$     I could not even start to the first point. Please help me to solve this.  Thanks in advance SQL Server - Best way to get identity of inserted row? What is the best way to get IDENTITY of inserted row?  I know about @@IDENTITY and IDENT_CURRENT and SCOPE_IDENTITY but don't understand the pros and cons attached to each.  Can someone please explain the differences and when I should be using each? If in a UFD every maximal ideal is principal then it is a PID Is it ethical to use a recommendation letter from a deceased professor? She was a really nice professor and she wrote me a good recommendation letter that resulted in a lot of good results with applications. Recently, this professor died from brain cancer, which was a fast and shocking death for me.  Tomorrow is the deadline for a grant. What should I do? Is it professional to use letters from the deceased? I am in trouble with polarization and entanglement. Let's consider three cases : Case 1) : Statistical mixture of $|H\rangle$ and $|V\rangle$ polarized photons Case 2) : Photons in a superposition state $1/\sqrt{2}(|H\rangle+|V\rangle)$ Case 3) : Photons which are entangled with twin ones in $1/\sqrt{2}(|H,H\rangle+|V,V\rangle)$ state Which experiments can be conducted to differentiate the case  statistical mixture from the case  superposition state ?   Which experiments can be conducted to differentiate the case  statistical mixture from the case  entangled photons in superposition state ?   Using a $45^\circ$ polarizer I think you can differentiate case 1/case 2 but not case 1/case 3 I don't know how to differentiate case 1/case 3 except maybe using quantic tomography and Wigner function. Is it true ? Is there any other simpler way ? Thanks a lot for your answer and sorry for this maybe dummy question... Is it time to be rid of automatic down votes on certain closed questions? Right now,  carry automatic downvotes.  The  for this was largely to enforce specific quality bans in part because people were not down voting questions. Now that question down votes are free on questions and it seems to be pretty commonplace for question quality bans to trigger, even without the automatic votes from closure, and we have completely revamped the closure system. Is it time to reevaluate whether or not these automatic votes are still a good idea?  What benefits do the automatic votes provide and have they effectively run their course? Solve congruence system $\ x\equiv m_i-1 \pmod{m_i}\,$ for $\,i = 1,\ldots, k$ Find natural number $x$ so that $$\begin{align}x&amp;\equiv 9\pmod{10}\\ x&amp;\equiv8\pmod9\\ &amp;\ \ \vdots\\ x&amp;\equiv 1\pmod2\end{align}$$ Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix Symbol table for Lightning components to find Dependencies/References Like we do have SymbolTable for apex class, to find their references/dependencies; do we have anything for the Lightning Components(to find dependencies/references of lightning components/application/events).  Please let me know, if there are any other(native to Salesforce) mechanisms. What's the best option to display Unicode text (hebrew, etc.) in VB6 I have some customers who want to use our speech therapy software in Hebrew.  The programs are in VB6. The best option I'm aware of are:   use the Forms 2.0 controls from MS Office, but you can't distribute them.    $899   $499   Any other options? How to minimize residual sum of squares of an exponential fit? I have the following data and would like to fit a negative exponential growth model to it:  Days &lt;- c( 1,5,12,16,22,27,36,43) Emissions &lt;- c( 936.76, 1458.68, 1787.23, 1840.04, 1928.97, 1963.63, 1965.37, 1985.71) plot(Days, Emissions) fit &lt;- nls(Emissions ~ a* (1-exp(-b*Days)), start = list(a = 2000, b = 0.55)) curve((y = 1882 * (1 - exp(-0.5108*x))), from = 0, to =45, add = T, col = "green", lwd = 4)   The code is working and a fitting line is plotted. However, the fit is visually not ideal, and the residual sum of squares seems to be quite huge (147073).   How can we improve our fit? Does the data allow a better fit at all?  We could not find a solution to this challenge on the net. Any direct help or linkage to other websites/posts is greatly appreciated. How does the coin limit work? Code formatting - tag/block rather than indentation - code block more obvious I was trying to install Linux on my MacBook for class when I bricked my computer, which now in an unbootable state.   I made my partition too small, so I was going to delete and create a larger one. However, Disk Utility did not allow me to delete the small partition. I followed the instructions from   However, not only was I still unable to delete it, but upon restarting I realized my primary partition (located on the same disk) changed into FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF, and I had no options to boot.  (I am sure I performed the command on the correct disk and partition)  I saw some similar threads, but my main bootable drive is the one that got corrupt.   I don't want to lose any data. I'm not the most terminal savvy, so ANY help is appreciated, and I will kindly provide any additional information!  (Disk 0, partition 2 is my main partition, and that only one I care about)  Below are the results of  $ diskutil list and $ gpt -r show disk0.        UPDATE: :  I have followed @klanomath 's instructions. At the last step I received the following error:  Checking prerequisites Checking the partition list Checking for an EFI system partition Checking the EFI system partition's size Checking the EFI system partition's file system Problems were found with the partition map which might prevent booting Error: -69766: The partition map needs to be repaired because there's a problem with the EFI system partition's file system Underlying error: 8: POSIX reports: Exec format error    FINAL UPDATE: SUCCESS!  I just wanted thank @klanomath for helping me through this entire process. He spent 11 hours TeamViewing with me until not only my files were accessible, but until I could access my computer as could before this whole debacle (and this was after 5 hours of email correspondence).   To any future reader with this problem: PLEASE don't give up, it can be fixed! Which preposition should follow "guide" here, "on," "to," or "for"? The following link includes a guide ___________ how to use it.  How should I fill in the blank, on, for, or about? Zombie process using 100% CPU A zombie process, which should not be using any resources (RAM and CPU), is hogging up an entire core on my 8-core 64-bit Ubuntu 11.10 Desktop installation.  This is the output of ps -el|grep Z:  F S   UID   PID  PPID  C PRI  NI ADDR SZ WCHAN  TTY          TIME CMD 0 Z  1000 10317     1 99  80   0 -     0 exit   ?        19:27:15 java &lt;defunct&gt;   ... and here is an excerpt from top:    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND                                                                                                                                                 10317 deltik    20   0     0    0    0 Z  100  0.0   1170:17 java &lt;defunct&gt;  1424 root      20   0  612m 193m 158m S   14  2.4 287:17.11 Xorg 12580 deltik    20   0  710m 135m  46m S    5  1.7  27:31.07 compiz   This is how the zombie process came to existence:   I opened a Java JAR called . This launcher downloads another JAR file to ~/.minecraft/bin/minecraft.jar That JAR was executing when I locked my computer with xscreensaver. Then, I entered my password to unlock the computer, and I saw that the Minecraft window was frozen. I tried killing it (PID 10317) with gnome-system-monitor. It became a zombie. kill -9 10317 was ineffective. xkill only closed the window titled "Minecraft"; it didn't close the underlying java process.   Relevant information:   The graphics driver is FGLRX, and it has caused notorious issues with  of  and , including sometimes high CPU usage of Xorg and compiz.   I want to know:   What way is there to destroy this process of java without restarting, if any? Why is this issue happening? What can I do to prevent this issue from happening again? How can we define convexity in one dimension? Definition of a convex set: A set $S\subset\mathbb R^n$ is said to be convex if for each $x_1, x_2\in S$ the line segment $\lambda x_1 + (1-\lambda)x_2$ for $\lambda\in (0,1)$ belongs to $S$. This says that all points on the line connecting two points in the set are in the set. My question is: how can we define convexity in one dimension? So in the documentation for the DescribeFieldResult.getDefaultValue , it says:     Returns the default value for this field   There is a related method called getDefaultValueFormula whose description reads:     Returns the default value specified for this field if a formula is not used.   What is the difference between these methods? I have a field with a default value. When I call getDefaultValue, I get null. When I call getDefaultValueFormula, I get back "This is the default value" (with quotes). I don't understand why getDefaultValue returns null. Why Cooper or Dr. Brand can't send message?