Let $\Gamma$ be some set of well formed formulas and let $\alpha \in \mathrm{WFF}$. Prove the following statement:   $\Gamma \models \alpha$ if and only if $\Gamma \cup \{(\lnot\alpha)\}$ is not satisfiable.   So far I have done this:      edit: whatever I did was wrong, so can you tell me what could be done?
Let $\Gamma$ be a set of formulas and $\phi$ be a formula. Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$.   This seemed pretty obvious but I wanted to see if my proof made sense:  Proof: $(\Rightarrow)$  To derive for a contradiction, suppose that: $\Gamma \models \phi$. That means for all truth assignments $v$, for $\gamma \in \Gamma$, if $v(\gamma) = T$, then $v(\phi) = T$.  But this contradicts our assumption that $\Gamma \cup \{\neg \phi \}$ is satisfiable by the fact that $v(\phi) = T$ cannot happen so: $\Gamma \not \models \phi$ .  $(\Leftarrow)$   So by the definition of $\Gamma \not \models \phi$, we have that there is some truth assignment $v$ which satisfies $\Gamma$ but does not satisfy $\phi$. So  that means $v(\phi) = F$ since $v(\phi) \not = T$ which implies that $v(\neg \phi) = T$ which means $v$ satisfies $\Gamma \cup \{\neg \phi\}$.    I feel like I'm missing something in the forward direction, but at the same time... It looks pretty trivial as well. Am I missing anything crucial?  Thank you!
Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$ I've recently changed to the new Minecraft launcher, and I have noticed a change in the .minecraft folder layout. I can no longer install mods just by copying it over into the versions/1.6.1 jar file, as this file refreshes every time it is launched.  For those who don't know what I am talking about, here is the new layout:    I am specifically trying to install Optifine, which does work with 1.6.1. Where do I drag in the mod files now? I can't seem to find a jar file that is suited to it, except in the version folder, but as I stated above, this refreshes every time.  After copying a recent version and renaming it in the version folder, I get this error in the development console:   Unexpected exception refreshing version list java.lang.IllegalArgumentException: Version '1.6.1' is already tracked     at net.minecraft.launcher.updater.VersionList.addVersion(VersionList.java:91)     at net.minecraft.launcher.updater.LocalVersionList.refreshVersions(LocalVersionList.java:44)     at net.minecraft.launcher.updater.VersionManager.refreshVersions(VersionManager.java:47)     at net.minecraft.launcher.Launcher$2.run(Launcher.java:164)     at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source)     at java.util.concurrent.FutureTask$Sync.innerRun(Unknown Source)     at java.util.concurrent.FutureTask.run(Unknown Source)     at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)     at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)     at java.lang.Thread.run(Unknown Source) Reverse (parse the output) of Arrays.toString(int[]) Make HttpURLConnection load web pages with images Solving multiple congruences by CRT = Chinese Remainder Theorem How do I get the solution given by CRT to match another solution, e.g. the least positive? For example say I have X = 1234. I choose mi as 5, 7, 11, 13. This satisfies the simple requirements of . More precisely given in my example k = n = 4, and the product of any k - 1 is smaller then X how come simply computing the remainder of each won't give equations that solve to X = 1234. In the case of the example, x = 4 mod 5 x = 2 mod 7 x = 2 mod 11 x = 12 mod 13  Which resolves to  (won't CRT produce the smallest?) Any hints?  Below is the solution linked to on &quot;Math Celebrity&quot; (cached to avoid link rot). How does water evaporate if it doesn't boil? When the sun is out after a rain, I can see what appears to be steam rising off a wooden bridge nearby. I'm pretty sure this is water turning into a gas.  However, I thought water had to reach 100 degrees C to be able to turn into a gas.  Is there an edge case, for small amounts of water perhaps, that allows it to evaporate? Which attacks are possible against raw/textbook RSA? made me think about the movie versions of the 7th book.  In Deathly Hallows Part II      Voldemort has Nagini attack Snape because Voldy thinks that Snape has the Elder Wand's allegiance. We know that Snape didn't actually have the allegiance, Draco did (and by then Harry did) and his plan had failed.   But if Voldy's plan had succeeded with Snape, would he have gotten allegiance?      Since voldemort didn't kill Snape, Nagini did... AppleScript to choose item on web page Is it possible with AppleScript to choose a clickable element in a web page and click it and if so, can you give me an example? My Custom Skin won't Change How can I automatically save screen shots by the press on the Prt Scr button? I want a screen shot to be saved automatically to a location I specify, as soon as I press the Prt Scr button on my keyboard, or the Alt+Prt Scr combo. I also want the date and time to be used as file name for each new screen shot. And I want the files to be saved as PNG images.  Is there a Windows program that would allow me to do just this? I don't want any other fancy features like editing and making notes, etc. just a plain simple program that will either automatically save the screen shots to a pre-set location or prompt me where I want to save it. I don't want to go through MS Paint to save the shots.  I guess what I'm after is a little bit of Linux behavior of Print Screen, but only for Windows. And I additionally want the files to be named by the date and time they were taken.  Update:  Meanwhile, I have found one called Greenshot. But I don't know how to get the file name pattern right.  This is the default pattern.  ${capturetime:d"yyyy-MM-dd HH_mm_ss"}-${title}   Can someone explain this? Why do they have the title capturetime and the quotation marks? It produces file names like this.  2013-01-26 15_24_28-Greenshot.png   And this is what it says about the file patterns in the program description.  ${YYYY} year, 4 digits ${MM} month, 2 digits ${DD} day, 2 digits ${hh} hour, 2 digits ${mm} minute, 2 digits ${ss} second, 2 digits ${NUM} incremental number, 6 digits ${title} Window title ${user} Windows user ${domain} Windows domain ${hostname} PC name   How do I get files that include year, month and day, separated by a dash, followed by hour, minute and second? I want to include a time for uniqueness of the file name. I have a trained logistic regression model that I am applying to a testing data set. The dependent variable is binary (boolean). For each sample in the testing data set, I apply the logistic regression model to generates a % probability that the dependent variable will be true. Then I record whether the acutal value was true or false. I'm trying to calculate an $R^2$ or Adjusted $R^2$ figure as in a linear regression model.  This gives me a record for each sample in the testing set like:  prob_value_is_true         acutal_value    .34                          0    .45                          1    .11                          0    .84                          0     ....                        ....             I am wondering how to test the accuracy of the model. My first attempt was to use a contingency table and say "if prob_value_is_true > 0.80, guess that the actual value is true" and then measure the ratio of correct to incorrect classifications. But I don't like that, because it feels more like I'm just evaluating the 0.80 as a boundary, not the accuracy of the model as a whole and at all prob_value_is_true values.  Then I tried to just look at each prob_value_is_true discrete value, as an example, looking at all samples where prob_value_is_true=0.34 and measuring the % of those samples where the acutal value is true (in this case, perfect accuracy would be if the % of samples that was true = 34%). I might create a model accuracy score by summing the difference at each discrete value of prob_value_is_true. But sample sizes are a huge concern here, especially for the extremes (nearing 0% or 100%), such that the averages of the acutal values are not accurate, so using them to measure the model accuracy doesn't seem right.  I even tried creating huge ranges to ensure sufficient sample sizes (0-.25, .25-.50, .50-.75, .75-1.0), but how to measure "goodness" of that % of actual value stumps me. Say all samples where prob_value_is_true is between 0.25 and 0.50 have an average acutal_value of 0.45. Is that good since its in the range? Bad since its not near 37.5% (the center of the range)?  So I'm stuck at what seems like should be an easy question, and hoping someone can point me to a resource or method to calculate an accuracy stastic for a logistic regression model. Background Nowadays, I see &quot;them&quot; used to mean &quot;those&quot; a lot. I don't know if it was as common in the past. For example, take &quot;one of &quot;. On researching about it, I found some people say it comes from a dialect of British English. Some others say it is a &quot;non-standard&quot; usage. I see this usage in Canadian English also, and it seems some people use it in a sarcastic way. Moreover, I have seen a song titled &quot;one of them days&quot;. I also read, in the book called &quot;A Broken Promise&quot;, &quot;&quot; Finally, Wikipedia says that it is a usage in  (a common name for the Southern Midland dialect of American English):  Pronouns and demonstratives &quot;Them&quot; is sometimes used in place of &quot;those&quot; as a demonstrative in both nominative and oblique constructions. Examples are &quot;Them are the pants I want&quot; and &quot;Give me some of them crackers.&quot;    Question(s): What would you say about the usage of this word? Is it correct? Could we use it in daily speech? Could this usage go beyond a specific dialect and be used in other dialects, regions, etc.? Does it really originate from Appalachian English? Why did this usage become popular among other English speakers? Note: I have already seen a similar question:  However, it only says, &quot;ungrammatical,&quot; there. This question is specific to this situation only, and there is more to it. Do spice containers need to be hermetic? Spice containers purchase  I'm planning on upgrading my spice storage. Reading some advice online (e.g. , and ) I came to the conclusion that spice containers should:   Not be exposed to direct sunlight and high temperatures Be airtight   Which means that either you store it inside of some closed cupboard/wardrobe/room or you have containers which do block light and are in a relatively shady place. And the containers need to be airtight.  Available solutions  I've been browsing for some containers available online in my country and the 2 most sensible options I've seen are these:       The problem  The first one is hermetic (it has some rubber around the lid) but it's clear glass. The other one does not have the rubber, just a regular twist-off lid, but it's amber glass.  It'll be hard for me to find any space inside of cupboards to store all of my spices (I need at least 30), so I'd rather buy amber glass container, which I believe will effectively block the sunlight, even when kept on the counter. But I can't find any hermetic amber-glass containers, only ones with regular lid.  The question  Is that regular twist-off lid enough to keep spices airtight and fresh? Or should I focus on buying a hermetic container (like the one in clear glass). I could then for example paint them black to block the sunlight. Do these new insights into prime numbers affect encryption security? reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time? Add a custom curve mapping property for an add-on? Question The title says it all. I want to be able to make this in Blender's UI, and sample it at specific points :  Problem There is very insufficient documentation on this, and I can't find any examples for how to create a . I have worked with templates before, and , but it requires a curve mapping prop. Extra Info I found a few threads about this, and one gives me the impression that this cannot be done at all, but it is also very old.   This similar question has been asked here before, but it didn't even get a comment:  Is this even possible to access in python (as in only accessible in C)? After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again. Is that normal?  Also, can I set a newly added user as a sudoer or do I have to change that only after adding it? How can I do that via the shell?  Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? I've heard that exposing database IDs (in URLs, for example) is a security risk, but I'm having trouble understanding why.  Any opinions or links on why it's a risk, or why it isn't?  EDIT: of course the access is scoped, e.g. if you can't see resource foo?id=123 you'll get an error page. Otherwise the URL itself should be secret.  EDIT: if the URL is secret, it will probably contain a generated token that has a limited lifetime, e.g. valid for 1 hour and can only be used once.  EDIT (months later): my current preferred practice for this is to use UUIDS for IDs and expose them. If I'm using sequential numbers (usually for performance on some DBs) as IDs I like generating a UUID token for each entry as an alternate key, and expose that. Since I installed TeXLive 2016 I cannot build documents with the standalone document class any more using LuaLaTeX (PDFLaTeX works fine). For example:  \documentclass{standalone} \begin{document} hi \end{document}   Gives me:  ! Undefined control sequence. \sa@placebox -&gt;\newpage \global \pdfpagewidth                                                =\wd \sa@box \global \pdfpageh...  l.4 \end{document}   How to fix that? CSS to make HTML page footer stay at bottom of the page with a minimum height, but not overlap the page I have the following page (deadlink: http://www.workingstorage.com/Sample.htm ) that has a footer which I can't make sit at the bottom of the page. I want the footer to  stick to the window bottom when the page is short and the screen is not filled, and stay at the document end and move down as normal when there is more than a screenful of content (instead of overlapping the content).  The CSS is inherited and befuddles me. I can't seem to change it properly to put a minimum height on the content or make the footer go to the bottom.