Let $x,y,a,b~\text{and}~ c\in \mathbb{R}$ such that $a,b&gt;0$. Then show that $ax^2+by^2+2cxy \geq 0$ and $(ab-c^2)&gt;0$. Please help to prove this. I tried AM GM inequality but did not  succeeded.
I know that for any nonzero $x,y\in\mathbf{R}$,  $$ax^2+by^2+cxy &gt; 0,$$  where $a,b,c\in\mathbf{R}$. What can I deduce about $a$, $b$, and $c$?    For example, letting $x=1$ and $y=0$, I know that $\boxed{a&gt;0}$. Letting $x=0$ and $y=1$, I know that $\boxed{b&gt;0}$. Letting $x=y=1$, I know that $\boxed{c&gt;-(a+b)}$.  What else can I deduce about $a$, $b$, and $c$? How will I know when it's time to stop looking? (Is that last question even answerable?)
Qt: Best practice for a single instance app protection Django: manage.py does not print stack trace for errors Why could the pictures and paintings in Hogwarts talk and the photos of Harry's parents couldn't? How can I make my IP address appear to be from another country? How do I make it appear that my IP address is coming from one country while I'm located in another? I live in Germany and some websites (like Hulu or Youtube) don't work because my IP isn't in the US. How do I get around this? Do I have to use a proxy or something?  Moderator note Super User does not endorse nor defend any activity which may be used to circumvent local/state/national laws. From a toggle to an impulsion I'm trying to make a circuit (with no programmable circuits, only logic gates, flip flops and latches) that outputs a short impulsion each time it's input toggles from 0 to 1. Do you have any idea of how to do so? OpenGL GLX context is not using direct rendering, which may cause performance problems. (steam) AMD DRIVER When I start steam in a terminal, I get the following error messages   charlie@SteamMPL:~$ steam rm: no se puede borrar «/home/charlie/.steam/steam»: Es un directorio rm: no se puede borrar «/home/charlie/.steam/bin»: Es un directorio Running Steam on ubuntu 14.04 64-bit STEAM_RUNTIME is enabled automatically [2014-08-03 11:50:13] Startup - updater built Jul 15 2014 18:14:26 Installing breakpad exception handler for appid(steam)/version(1405474565_client) libGL error: failed to load driver: swrast Looks like steam didn't shutdown cleanly, scheduling immediate update check Installing breakpad exception handler for appid(steam)/version(1405474565_client) [2014-08-03 11:50:14] Checking for update on startup [2014-08-03 11:50:14] Buscando actualizaciones disponibles... Installing breakpad exception handler for appid(steam)/version(1405474565_client) [2014-08-03 11:50:14] Download skipped: /client/steam_client_ubuntu12 version 1405474565, installed version 1405474565 [2014-08-03 11:50:14] Nothing to do [2014-08-03 11:50:14] Verificando instalación... [2014-08-03 11:50:14] Performing checksum verification of executable files [2014-08-03 11:50:15] Verification complete Gtk-Message: Failed to load module "overlay-scrollbar" Gtk-Message: Failed to load module "unity-gtk-module" Installing breakpad exception handler for appid(steam)/version(1405474565_client) Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element Fontconfig warning: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 78: saw unknown, expected number [0803/115015:WARNING:proxy_service.cc(958)] PAC support disabled because there is no system implementation libGL error: failed to load driver: swrast Error: OpenGL GLX context is not using direct rendering, which may cause performance problems.   Hardware information:   CPU: AMD Athlon x4 760k  GPU: AMD Radeon R7 250 1 GB GDDR5 (Driver: Catalyst 14.4) RAM: 8 GB DDR3 1600   What can I do to circumvent this issue? Let $X$ be a beta distributed variable with parameters $a$, $b$. Let $Y$ be a beta distributed variable with parameters $c$, $d$. Let $Z = \max(X, Y)$.  Does anybody know of a fast way to compute the mean and variance of $Z$ given $(a, b, c, d)$?  $X$, $Y$ don't necessarily need to have beta distributions, but they should be similar -- in $[0, 1]$.  What I really want to find is a way to quickly approximate the distribution of the sample max of a bunch of random variables. Let $Z = \max(\{X_1, X_2,\dots , X_n\})$. If I know parameters (not necessarily beta-distribution) for $X_1, X_2,\dots, X_n$, can I quickly compute a set of parameters (same distribution as $X_i$) to form a distribution that closely approximates $Z$?    EDIT: To clarify, the purpose of this has to do with programming AI for board games. If we are evaluating a position in a chess game and we determine one move leads to a 60% chance of winning while all others have a 20% chance of winning, then we value the position as a 60% win. This is the minimax algorithm... However, what I'm curious about is whether this can be improved upon. In the simple case, we approximate: $$ \mu_c = \max(\{\mu_1, \mu_2,\dots, \mu_n\}) $$ where $\mu_c$ is the expectation from the current position and $\mu_i$ is the expectation from each of the sub-trees.  It seems like we should be able to do better than this. Can we find a function f that replaces max and closely approximates parameters for the distribution of the sample maximum? $$ (\mu_c, \sigma_c) = f\left(\{(\mu_1,\sigma_1), (\mu_2,\sigma_2), \dots, (\mu_n, \sigma_n)\}\right) $$ In this case, we still want $\mu_i$ to be the expectation from each of the nodes. However, $\sigma_i$ does not necessarily need to be variance/standard deviation; just a parameter that represents uncertainty.  Just as: $$ \max(\{a, b, c\}) = \max\left(\{\max(\{a, b\}), c\}\right) $$ We should expect: $$ f\left(\{a, b, c\}\right) \approx f\left(\{f(\{a, b\}), c\}\right) $$ So $f$ does not necessarily need to take parameters for more than two nodes.  One thing, however, is that $f$ should be relatively fast to compute. If the integrals must be calculated, I think doing double exponential integration might work well, but I'm not sure -- Square roots of integers and cyclotomic fields For every $ N \in \mathbb Z$ there exists an integer $n$ such that $ \sqrt N \in \mathbb Q(\zeta_n)$.   I am struggling where to start this question, please suggest me few hints. Interpreting coefficient in a linear regression model with categorical variables I will give my examples with R calls. First a simple example of a linear regression with a dependent variable 'lifespan', and two continuous explanatory variables.  data.frame(height=runif(4000,160,200))-&gt;human.life human.life$weight=runif(4000,50,120) human.life$lifespan=sample(45:90,4000,replace=TRUE) summary(lm(lifespan~1+height+weight,data=human.life))  Call: lm(formula = lifespan ~ 1 + height + weight, data = human.life)  Residuals: Min       1Q   Median       3Q      Max  -23.0257 -11.9124  -0.0565  11.3755  23.8591   Coefficients:              Estimate Std. Error t value Pr(&gt;|t|)     (Intercept) 63.635709   3.486426  18.252   &lt;2e-16 *** height       0.007485   0.018665   0.401   0.6884     weight       0.024544   0.010428   2.354   0.0186 *   --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1   Residual standard error: 13.41 on 3997 degrees of freedom Multiple R-squared: 0.001425,   Adjusted R-squared: 0.0009257  F-statistic: 2.853 on 2 and 3997 DF,  p-value: 0.05781   In order to find the estimate of 'lifespan' when the value of 'weight' is 1, I add (Intercept)+height=63.64319  Now what if I have a similar data frame, but one where one of the explanatory variables is categorical?  data.frame(animal=rep(c("dog","fox","pig","wolf"),1000))-&gt;animal.life animal.life$weight=runif(4000,8,50) animal.life$lifespan=sample(1:10,replace=TRUE) summary(lm(lifespan~1+animal+weight,data=animal.life))  Call: lm(formula = lifespan ~ 1 + animal + weight, data = animal.life)  Residuals: Min      1Q  Median      3Q     Max  -4.7677 -2.7796 -0.1025  3.1972  4.3691   Coefficients:             Estimate Std. Error t value Pr(&gt;|t|)     (Intercept) 5.565556   0.145851  38.159  &lt; 2e-16 *** animalfox   0.806634   0.131198   6.148  8.6e-10 *** animalpig   0.010635   0.131259   0.081   0.9354     animalwolf  0.806650   0.131198   6.148  8.6e-10 *** weight      0.007946   0.003815   2.083   0.0373 *   --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1   Residual standard error: 2.933 on 3995 degrees of freedom Multiple R-squared: 0.01933,    Adjusted R-squared: 0.01835  F-statistic: 19.69 on 4 and 3995 DF,  p-value: 4.625e-16   In this case, to find the estimate of 'lifespan' when the value of 'weight' is 1, should I add each of the coefficients for 'animal' to the intercept: (Intercept)+animalfox+animalpig+animalwolf? Or what is the proper way to do this?  Thanks Sverre How to deal a specific amount of damage using commands? I want to make a datapack where if you touch lime glass, you get damaged 2.5 hearts. I don't want the commands to work for just this specific value. Instead, I want to be able to specify the amount of damage. I can't use the instant damage effect because it's not controllable. I am trying to follow the  and . I tried to push some of my Lightning components which depend on a managed package in App Exchange. I would like to know how to install a managed package using Salsforce DX CLI so that i can use that command in my travis.YML file for my CI integration.  Here is the script for my Travis file  sudo: true dist: trusty cache: false env: - URL=https://developer.salesforce.com/media/salesforce-cli/sfdx-linux- amd64.tar.xz before_install:  - export SFDX_AUTOUPDATE_DISABLE=true - export SFDX_USE_GENERIC_UNIX_KEYCHAIN=true - export SFDX_DOMAIN_RETRY=300 - wget -qO- $URL | tar xJf - - "./sfdx/install" - export PATH=./sfdx/$(pwd):$PATH - sfdx update - sfdx force:auth:jwt:grant --clientid $CONSUMERKEY --jwtkeyfile  assets/server.key --username $USERNAME --setdefaultdevhubusername -a DevHub script: - sfdx force:org:create -f config/project-scratch-def.json -s -a  PackageInstallation - sfdx force:org:list - sfdx force:org:display - sfdx force:user:password:generate -u PackageInstallation - sfdx force:org:display - sfdx force:org:open -u PackageInstallation - sfdx force:source:push -u PackageInstallation - sfdx force:apex:test:run -u PackageInstallation -c -r human   I get the following error as my components depend on the app exchange components:     No COMPONENT named markup://ldt:datatableDev found : Is product of two closed sets closed? If $A$ is closed in $X$ and $B$ is closed in $Y$,   does it follow that $A \times B$ is closed in $X \times Y$ ? What does "All things X" mean? I've seen the expression in some company names; for example , .  Is the expression "All things X" a parody of some other popular phrase, or does it have its own meaning? Or is it just a misuse of grammar? Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t &lt; 0 \, \, \text{for all} \, \, 1 &lt; t &lt; 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed? Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ? How was Dumbledore able to defeat Grindelwald, since Grindelwald had the Elder Wand? How to render a 30K image with cycles? I have to render an image of 27000 x 7000 for a 150dpi print. The scene is quite big, but cycles preview and a render in 2700x7000 works as expected.  If I hit render my memory raises up to 20gb, but my cpu usage is at 5%. Why cycles does not using the full cpu power during the render process?  Here are my rendersettings:  Pathtracing Samples:3000 Tilesize: 64x64 Bounces: Max 12 | Min 3 Diffuse: 4 Glossy: 4 Transmission: 12 Volume: 1   Is there a way to optimize things? CDN: How is it possible that my DNS delivers a different IP depending on the visitors location? I found  how a CDN works. But there is one thing I don't really understand. Let's assume I setup multiple DNS servers at my location and they use the nameserver domains dns1.example.com, dns2.example.com and dns3.example.com. This DNS servers are able to deliver a server IP depending on the visitors location (ping, geo database, browser language or whatever). Now I update this nameserver settings for my domain www.example.org at the registry.  Now, the very first request on www.example.org with an expired TTL tries to resolve the domain. It asks:   the local .hosts/DNS, if TTL expired: the internet providers DNS, if TTL expired: the root DNS, if TTL expired: my local dns1.example.com   But if I understand it correct, the new IP is then added to all these nameserver caches until the TTL expires again. So how is it possible to send other IPs to the visitor depending on his location?  In  theandym said the request is "forwarded", but I don't think this is how a CDN works, because "forwarding" means lengthen the transmission way resulting a longer loading time. Or does a CDN require zero TTL for the domain?  Update1 Through  I found  describing how they optimized CDN performance. It did not explain how the CDN works in general, but there were interesting explanations like the following:     Thereafter, whenever a client attempts to fetch content hosted on the   CDN, the client is redirected to the node determined to have the least   latency to its prefix. This redirection however is based on the prefix   corresponding to the IP address of the DNS nameserver that resolves   the URL of the content on the client’s behalf, which is typically   co-located with the client.   This means Google checks at first the latency of all IP prefixes and defines a DNS resolution table (?) for all available prefixes. And if a visitor has the IP 198.51.100.231 the Google server IP is used, that is set for the prefix 198.51.100.0. But again: How does Google's DNS know which IP the visitor is using? Most visitors resolve Google's domain through their internet provider and by that the resolving is done through those external DNS servers or not?  As an additional example: If I start a DNS resolution for the domain facebook.com with different online tools (hosted in different countries) it is resolved to different IPs with different domains like:   31.13.92.36 Reverse: edge-star-mini-shv-01-frt3.facebook.com 31.13.76.68 Reverse: edge-star-mini-shv-01-sea1.facebook.com 31.13.69.228 Reverse: edge-star-mini-shv-01-iad3.facebook.com 157.240.2.35 Reverse: edge-star-mini-shv-01-ort2.facebook.com   After that I thought it could depend on the DNS server location used by the visitor, but I tried my own (Deutsche Telekom, Germany), Google's (8.8.8.8) and a major one from France (Orange) and they all returned for facebook.com the IP 31.13.92.36. “May I know your good name?” "May I know your good name” is a typically Indian way of honouring another person by asking their name using an adjective like sweet, good, beautiful, et cetera. Of course there won’t ever be any bad or sour name (unless we feel it such) when asking.   If this manner of asking someone’s name is incorrect or inappropriate, then how could I make it sound more polite and amicable without negotiating the basic principles of English? How do I replace the *first instance* of a string in .NET?