For $x+y+z=3$ prove that $4\geqq x^2y+y^2z+z^2x$
Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
Since I have an onboard Intel graphics card, and a ATI Radeon display controller, I was wondering if I could switch between the two as my graphics in use, as I could in 14.04 LTS through ATI Catalyst?  16.04 LTS is using the onboard, and the hardware listing lists ATI as well as display controler, but not in use. Additional drivers don't provide anything.   So is AMDGPU being used, or is ATI disabled?  lspci:  00:02.0 VGA compatible controller: Intel Corporation Broadwell-U Integrated Graphics (rev 09) 04:00.0 Display controller: Advanced Micro Devices, Inc. [AMD/ATI] Venus XTX [Radeon HD 8890M / R9 M275X/M375X] (rev ff)   lspci -nnk | grep VGA -A1  00:02.0 VGA compatible controller [0300]: Intel Corporation Broadwell-U Integrated Graphics [8086:1616] (rev 09)     Subsystem: Lenovo Broadwell-U Integrated Graphics [17aa:3824] Get the raw HTML of selected content using javascript Avoiding if statement inside a for loop? Moving from HTTP to HTTPS in Google Search Console I'm moving from HTTP to HTTPS for my entire site. The site is currently added to Google Search Console (formerly Google Webmaster Tools) as www.example.com and all the pages are indexed without using SSL.  How do I go about moving to the new HTTPS URLs on Google Search Console?   Do I just submit an updated sitemap which has the HTTPS URLs or do I add a new site as https://www.example.com and submit the sitemap with HTTPS URLs? All current URLs are set to redirect to their HTTPS counterparts. Do I add multiple properties to Google Search Console? How do I create a key binding that allows me to mount my google drive? I want to mount/unmount my google drive using with a key binding without having to open a terminal. How can I set this up? Catch Regex too complicated when splitting a string in salesforce I have created a function wherein I split a string from the csv filecsvAsString.split('\n') . It is working when the string is not so big. But when I try to split a very large string, an error will occur      regex too complicated   what I want to do is to catch if an error occur during the split process. How will I do it ? I've tried to use try and catch but the problem still occur. Is there any other possible solution? Is there proof that the DSA construction, also used by ECDSA, is secure assuming that discrete logarithms in the relevant group representation are difficult? Restoring a backup to an older version of SQL Server When trying to restore a backup to a SQL Server Express 2008 database I got the following error:  Restore failed for Server '...\SQLEXPRESS'.  (Microsoft.SqlServer.SmoExtended) System.Data.SqlClient.SqlError: The database was backed up on a server running version 10.50.1600.  That version is incompatible with this server, which is running version 10.00.2531.  Either restore the database on a server that supports the backup, or use a backup  that is compatible with this server. (Microsoft.SqlServer.Smo)   Is there a way to get a backup which is compatible with the older (in my case 10.00.2531) version from the newer (in my case 10.50.1600) version of SQL Server Express? Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. Turning fullscreen on made Minecraft crash, and now it crashes on start every time I switched to full screen on minecraft, the screen turned black as if it were loading it took a while so i closed minecraft and reopened it. When I logged on it gave me a program on my taskbar saying "minecraft minecraft" then when i closed the program it gave me this error message:   java(tm) platform SE binary has stopped working.    Is there a way to fix this? Open Finder window from current Terminal location? If I am in a specific path in a Terminal window, how can I open that same window in a new Finder window?   Note: This is the opposite of . Change in "under review" date Increasing sequence of step functions and diverging Lebesgue Integrals Let $\{s_n\}$ be an increasing sequence of step functions which converges pointwise on an interval $I$ to a limit function $f$.  If $I$ is unbounded and if $f(x) \geq 1$ everywhere on $I$ except on a set of measure zero, prove that the sequence $\{\int_I s_n\}$ diverges.  (Here $\int_I s_n$ refers to the Lebesgue Integral.)  Conceptually, this makes sense.  Clearly, if $f(x) \geq 1$ almost everywhere on an unbounded interval, then the sequence of Lebesgue integrals should diverge.  However, in proving this, I am running into issues with the convergence of $\{s_n\}$ being pointwise.  If it were uniform, I could easily show this.  How do I show that (even for pointwise convergence) I can choose $n$ large enough to make $\int_I s_n$ as big as I want? I took a bunch of photos at my son's martial arts club using a D90 with a 50mm F1.8.  The club is lit with overhead fluorescent lighting, and I was getting some weird results where some shots are "white", others have an "orange cast", and still others are partly white and partly orange in the same shot.  I suspect it has to do with the combination of the frequency of the light flicker and with the camera shutter speed. Here is a sequence of 3 shots that show the problem.  The shots were taken with 1/500s at F2 and ISO-800 and with Auto-WB.  First shot ("white/normal"):    Second shot ("orange cast" at top):   Third shot ("orange cast" at bottom):   These were taken in burst mode within a second of each other.  Can anybody tell me what is going on?  And, how I can avoid this? Why do positive and negative variants of the same question elicit the same answer? Ordered N-Tuples as Cartesian Products of Indexed Families In Introduction to Topology, Mendelson defines the Cartesian Product of an Indexed Family as follows:     DEFINITION  Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product  of the sets $\{X_\alpha\}_{\alpha \in I}$,    written  $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$.    Halmos, in Naive Set Theory, makes the following observation about this sort of product:     Ordered triples, ordered quadruples, etc., may be defined as families whose index sets are unordered triples, quadruples, etc.   However, it is not immediately apparent to me why this is true. It makes sense if we understand the prior assertions in Halmos that $\prod_{i \in I} X_i = X_i$ and that $\prod_{i \in I} X_i = X_a \times X_b$ when $I = \{a, b\}$, but these already pose difficulty for me.  Let $I = \{0, 1\}$, let $X_0 = \{A, B\}$, and let $X_1 = \{C, D\}$. Then by Mendelson's (and Halmos's) definitions, it would seem to me that . . .  $$\prod_{i \in I} X_ i = \{\{(0, A), (1, C)\}, \{(0, A), (1, D)\}, \{(0, B), (1, C)\}, \{(0, B), (1, D)\}\}$$  This would appear to me to be the set of all functions matching the definition given, but it is not equal to $X_0 \times X_1$. Moreover, extending this to a case where the index set $I$ is a triple, we would not end up with ordered triples, but with each subset of the product containing three ordered pairs.  When Halmos posits the equalities he does, the idea seems easy to grasp, but the peculiar wording of Mendelson's definition suggests something different to me. I'm looking for a symbol for skeptical consequence so that I can write that something, B, is a skeptical consequence of something else, A.  In a book I read it's written like \(A \vdash B\) or \(A \sststile{}{} B\) but the horizontal line is wavy like \sim, has the same width as the vertical line and does not vary in width. So, what I'm looking for is a binary relation symbol that is similar to the combination of the first two and the last relation in the following image:    That is, something similar to the following mockup but the wavy line should be such that it has a constant width and the same width as the vertical line:    I've tried  without success. Is there any way I can specify a standard or custom numeric format string to always output the sign, be it +ve or -ve (although what it should do for zero, I'm not sure!) Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ Let IG denote Inverse-Gamma distribution   .   If $X\sim IG(\alpha,1)$ and $Y\sim IG(\beta,1)$. Show that   $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$   I tried with jacobian transformation taking $Z=\frac{X}{X+Y}$ and $W=Y$ then $$X=\frac{WZ}{1-Z};Y=W$$  $$\frac{\partial(z,w)}{\partial(x,y)}=\frac{1}{(1-z)^2}$$  $$f_{z,w}(z,w)=f_x(\frac{wz}{1-z})f_y(w)(1-z)^2$$ $$f_{z,w}(z,w)=\frac{1}{\Gamma{(a)}}(\frac{wz}{1-z})^{-\alpha-1}e^{-\frac{(1-z)}{wz}}\frac{1}{\Gamma{(\beta)}}w^{-\beta-1}e^{-\frac{1}{w}}(1-z)^2$$  but I'm stuck, in some place I read that $\frac{X}{X+Y}$ is a type 3 Beta distribution, but I can't show that. How do I get add-apt-repository to work through a proxy?