We are asked to show that the function $f(x,s,t) = -\log (st - \|x\|^2)$ is convex on the set $\{x \in \mathbb R^n, s &gt; 0, t&gt;0: \frac{\|x\|^2}{s}&lt;t\}$  I tried proving this with derivatives but it very quickly gets out of hand, and the teacher said the solution is short and simple and not to look for verbose answers.  Another possible way is using composition of convex functions, but sadly that demands that the outer function is non decreasing, which $-\log$ is not.  Any ideas?
Show that the function $f: S \to \mathbb R$ given by $$f(x,s,t):=-\ln(st - ||x||^2)$$ is convex on $$S := \left\{(x,s,t) \in \mathbb R^n \times \mathbb  R \times \mathbb R: \frac{\|x\|^2}{s}&lt;t, s&gt;0, t&gt;0 \right\}$$   $-\ln(x)$ is a convex monotonic decreasing function, hence we can use composition by showing the inner part is concave. Also we know $-||x||^2$ is concave and $st$ is neither convex nor concave on $S$. Also in $S$ we have $st&gt;||x||^2$ but I was unable to show the convexity of the function as a whole
Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$. So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity element, other than 0. Because if 0 is the identity element, then this group won't have inverses.  The set explicitly excludes -1, which I found to be its identity element, which makes going about proving that this is a group mighty difficult. Any way to list similar commands? When you write the command name wrong, bash often does this:  septi@norbert:~$ good No command 'good' found, did you mean:  Command 'gold' from package 'binutils' (main)  Command 'gmod' from package 'gmod' (universe)  Command 'goo' from package 'goo' (universe)  Command 'god' from package 'god' (universe)  Command 'geod' from package 'proj-bin' (universe)  Command 'gord' from package 'scotch' (universe) good: command not found   Or sometimes it does this:  septi@norbert:~$ nftp No command 'nftp' found, but there are 23 similar ones nftp: command not found   Is there any way to ask bash to show these 23 similar commands for me? And, is there a way to show similar commands, including those that aren't yet installed, instead of running the application, ftp for example? Does anyone know how prove the following?   Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. Show that $f$ and $g$ are constant on $\Omega$. Volume of an n-simplex It's rather tedious to show using Fubini's Theorem and induction on $n$ that the volume of the region $x_1+x_2+...+x_n \leq 1$ with $x_1,...,x_n$ nonnegative is $\frac{1}{n!}$.  Is there an easier way to see this? Using Euler's Totient Function, how do I find all values n such that $\phi(n)=12$? How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works? I am totally aware a question of similar nature is asked before. But the answer there is not doing any good for me.  I have installed python 2.7.6 in addition to system default 2.7.3. And now my system is screwed up completely. Now I want to remove the 2.7.6 and get back to my system default. update-alternatives command on python is not working!  I have installed the 2.7.6 by downloading source and triggering "config, make, sudo make install" commands.  Please help. How do I get powdered mammoth tusk? Am I missing something? I can find plenty of tusks, but I just don't know if I can make them into powder myself or do I have to find them for sale? Why isn't .profile sourced when opening a terminal? Just installed Ubuntu 14.04.1 LTS.  According to .profile,  # ~/.profile: executed by the command interpreter for login shells.  # This file is not read by bash, if ~/.bash_profile or ~/.bash_login exists.   There is no ~/.bash_profile or ~/.bash_login.  It does not seem to source .profile when opening a terminal.   Why doesn't it source .profile when opening a terminal? Why is .profile there if it is not sourced? How do I know/control what is sourced when a new terminal is opened? When a new terminal is opened, does it do a "login"? Where is a good place to add $HOME/bin to my path? .bashrc? Will that cause a problem because .profile has the same code? If $f \circ g$ is surjective, then f is surjective Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$  If $f \circ g$ is surjective, then f is surjective, too.  I think that is true.   Question: How can I proove that? I have so far:   $\forall y \in Y: \exists x \in X : f(x)=y$  $\forall x \in Y : \exists y \in Y : f \circ g(y)= x$ How to prove that a simple graph having 11 or more vertices or its complement is not planar? Summation of Central Binomial Coefficients divided by even powers of $2$ Whilst working out  the following summation emerged: $$\sum_{m=0}^n\frac 1{2^{2m}}\binom {2m}m$$ The is equivalent to  $$\begin{align} \sum_{m=0}^n \frac {(2m-1)!!}{2m!!}&amp;=\frac 12+\frac {1\cdot3}{2\cdot 4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}+\cdots +\frac{1\cdot 3\cdot 5\cdot \cdots \cdot(2n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot 2n}\\ &amp;=\frac 12\left(1+\frac 34\left(1+\frac 56\left(1+\cdots \left(1+\frac {2n-1}{2n}\right)\right)\right)\right) \end{align}$$  and terms are the same as coefficients in the expansion of $(1-x)^{-1/2}$.  Once the solution  $$ \frac {n+1}{2^{2n+1}}\binom {2n+2}{n+1}$$ is known, the telescoping sum can be easily derived, i.e.    $$\frac 1{2^{2m}}\binom {2m}m=\frac {m+1}{2^{2(m+1)-1}}\binom {2(m+1)}{m+1}-\frac m{2^{2m-1}}\binom {2m}m$$  However, without knowing this a priori, how would we have approached this problem? how to create a histogram in java Where did the brush size lock go for sculpt mode in 2.8??? What's the difference between an inverter with a bubble at the input and one with a bubble at the output? See , page 2: the internal logic diagram for the MM74HC138.    The diagram shows logic inverters with bubbles at the input or at the output. Is there an actual difference between them? Is there a single word for something that is simultaneously not interesting and not boring? I am looking for a word that would describe a concept or activity, such as playing a game X, that is1 neither boring nor interesting. In other words, it should mean ‘neutral’ wrt intrestingness (~ boringness) but still comparable on the scale of intrestingness to both ‘boring’ and ‘interesting’.  An example sentence would be  As far as I'm concerned, long-distance running is [insert word]; it is not boring but it's also not interesting.  Criteria2   Has to be unambiguous3, and may consist of two words at most. Has to be a single word. Should be formal in the sense that it is included in Cambridge Dictionary, or Merriam Webster, or both. Should not contain negating prefixes, such as ‘un-’.   Does such a word exist in the English language?  For readers' convenience, please state which criteria you believe your word suggestion has met. You may do so, e.g., by adding a disclaimer at the top or bottom of your answer. Simply writing (1., 2.) is sufficient.        1 Assume a ternary system of intrestingness.   2 If not all criteria can be met, you may relax  the requirements from bottom to top, in the following order: 4. -> 3.  -> 2. -> 1. Answers with the smallest number of relaxed criteria in this order will be preferred in accepting. For example, fulfilling  (1., 2., 3.) is better than achieving (1., 3., 4.). Situations where a  higher priority ordering fulfills fewer criteria than a lower priority  vector, such as (1.) vs (3.,  4.), will be assessed on a case-by-case  basis.   3 Words such as ‘unappealing’ thus even fail  criterium 1 because in a three-valued system ‘unappealing’ ~ ‘not  appealing’ ~ ‘not interesting’ can be taken to mean either ‘boring’ or  ‘neutral wrt intrestingness’. What is $dx$ in integration? How do I download a YouTube video? Weighting features prior to SVM I'm building an object detector using HOG features and linear SVM. Some of the regions of the object are more "distinctive" so I would like to give more weight to the features extracted from those regions. (e.g. imagine we want to detect between mugs and glasses, then the most "distinctive" part is the mug's handle)  How could I do this? For sure there should be a more intelligent way than just replicating the features of those "distinctive" regions. Requesting an active user page In my opinion, providing us with an active user page that shows the list of recent active users is useful as we can monitor the active users without visiting each user profile.  The list can be arranged in a grid of m rows by n columns and accompanied by labels representing the their reputations (and other info that you think they are important) and activities  such as "seen 2 minutes ago".  The following figure illustrates my idea. I am sorry. There is a typo in the following figure, Active must be changed to active to make it parallel to the other existing labels.     I think it is not too difficult to implement this. By adding a new action method and editing the view, done!  Of course it is not urgently requested feature. If it is useful, it can be put in the development to-do list so it can be implemented together with other useful requests at once.  Any responses are welcome! Regardless of how difficult it is, could you explain whether or not there are negative effects of this request? Down voting without informative comments looks barbaric. :-)  Edit:  If it is about user privacy, why is there a "seen 2 minutes ago" in the user profile page? I don't think there is a difference if we implement this feature. Is there any way to accept only numeric values in a JTextField?