Let $A, B \in M_n$ be positive definite and let $α\in (0, 1)$.  Why does $αA^{−1} + (1 −α)B^{−1}\succeq (αA + (1 − α)B)^{−1}$?
I wonder a generalization of Jensen's inequality: let $\mathbf{X,Y}$ be two positive definite matrices, can we obtain the following Jensen like inequality $$(1-\lambda)\mathbf{X}^{-1}+\lambda\mathbf{Y}^{-1} \succeq((1-\lambda)\mathbf{X}+\lambda\mathbf{Y})^{-1}$$ for any $\lambda \in (0,1)$, where the notation $\mathbf{A}\succeq \mathbf{B}$ represents matrix $(\mathbf{A-B})$ positive semi-definite.
Can I use Wilcoxon Signed-Rank Test to test pairs of data in a time series? I have a dataset of measurements (non-normally distributed). Each line of this dataset is composed of two measurements of the same variable at a specific time (i.e. XYY [time, meas1, meas2]). This dataset contains 24 lines.  Is it correct to use the Wilcoxon Signed-Rank Test to compare these two measurements? Is it correct to use this test to compare the mean value of these measurements between different experiments?  Thanks C++ iostream Corruption using stringstream Can I use a custom Splash screen? The default Splash screens are cool, but I'd like to add some of my own work to remind me of how awesome I am (no, not really).    Is it possible to customize/change the splash screen? I thought the new Apple font was called San Francisco, but I cannot find it in the font list of the Keynote.app for my Mac. Is it called another way or it is not available for Keynote?  I am currently using OS X El Capitan. Is the Macbook 12 inch's USB-C charger interchangeable with other USB-C chargers or devices? If there is only the Macbook 12 inch USB-C charger nearby, can it be used to charge other USB-C devices, such as any Android phones that also use USB-C? (will it be "too powerful" and damage other devices?)  Vice versa, if there is a Quick Charge 2.0 or 3.0 USB-C charger from other devices nearby, can it be used to charge the Macbook 12 inch?  The main concern is about whether it could damage the device or battery. Quantum information references How to Disable Copy Paste (Browser) Why no file-name in Inode information? I know that symlinks have a separated inode from the original file and that they only point for the inode containing the data. My question is, where is the hardlink name stored if there's no field in the inode structure refering to "name". I know that this could be a a relatively easy question, but as all hardlinks have the same inode number i assume that no inode is created when creating a hard link. How does Windows 7 licensing work for running the OS as Virtual Machines? I plan on buying Windows 7 Ultimate Edition to run on an old Pentium 4. I also want to run Windows 7 virtual machines inside Windows 7. My question is two fold:   Do I need a license for each Virtual Machine? Does the answer change if I use a third party virtualization platform (e.g. , , , etc) as opposed to ? Firefox suspicious subprocess (turned out to be legitimate FF behavior) I've discovered this yesterday, as it was generating system load on otherwise idle Ubuntu 14.04 Desktop machine.  Today, I've confirmed such behavior also on another Ubuntu instance (17.10 in Virtualbox VM)  The sub-process in command line looks like below ( I've put this as picture below intentionally to prevent askubuntu.com from changing/escaping the content )   For me it looks like exploit/malware. Or at least there is something to hide.  this happens even for url How do you correctly use a meat thermometer? The separation between the two plates of a capacitor is increased I had the following question on a quiz recently:     A capacitor consisting of two parallel plates, separated by a distance $d$, is connected to a battery of EMF ε. What happens if the separation is doubled while the battery remains connected?    The correct answer was "The electric charge on the plates is halved." However, we were previously given a question in which a circuit consists of a capacitor of 6 μF and a resistor of 300,000 Ω connected to a battery, and the separation between the plates of the capacitor is quadrupled. However, in this circumstance, the charge on the capacitor immediately after the plates are separated remains the same, and the potential difference across the capacitor increases.   I am confused as to why these questions have different answers to them, despite having an extremely similar setup. My best guess is that the circuit in the second question also has a resistor, but I am not sure, and help would be greatly appreciated. Using multiple wildcard parameters with arcpy.ListFeatureClasses I know there has to be a way to do this but I'm having trouble figuring out how to do so. I would like to create a list of feature classes to perform operations on however, I want to have the list contain feature classes which have 2 separate and distinct file name identifiers. The code would look something like:  FCS = arcpy.ListFeatureClasses("X_*" OR "*_Y")   I know OR is incorrect however, is there some way to do this without having to re write code to work with the second wildcard. Thanks for helping gout, i know there is a selection method in the Select By Attributes which allows for multiple wildcard use using the OR statement. In the Iron Man movie Tony flies the Iron Man armour from Miami to Afghanistan, roughly 7800 miles.    What is the maximum range of the Iron Man armour?   What sort of speed does the armour cruise at?  How long would the flight from Miami to Afghanistan have taken?   Could Iron Man circumnavigate the globe without stopping, and how long might this journey take him?   EDIT: Just found this (funny) image, New York to Washington in under an hour?!? :) Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ , but I don't know for the general case . Please help . Thanks in advance ) I did apt-get upgrade it never finishes at this point:  Searching for GRUB installation directory ... found: /boot/grub Searching for default file ... found: /boot/grub/default Testing for an existing GRUB menu.lst file ... found: /boot/grub/menu.lst Searching for splash image ... none found, skipping ... Found kernel: /boot/vmlinuz-3.2.0-24-virtual Updating /boot/grub/menu.lst ... done  You already have a NoLOADER configuration in /etc/grub.conf Running boot loader as requested Running /usr/sbin/grub  ...    The system does not freeze (Ctrl-Z works, I get my console back) just that that /usr/sbin/grub seems to never complete.  What should I do? Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $\overline{x}\in X$.      The n-th power is defined inductively: $$ T^{n+1}(x):=T(T^{n}(x)), n\in\mathbb{N}. $$ And for a function $T\colon X\to X$, $q$-contractive means  $$ \exists 0\leq q&lt;1~\forall~x,y\in X: d(T(x),T(y))\leq q d(x,y). $$    Now to the proof.  I think I have to show, that $T$ is q-contractive, because then it follows with Banach, that $T$ does have exactly one fixed point. So I have to show, that there exists a $0\leq q&lt;1$ so that for all $x,y\in X$ it is $$ d(T(x),T(y))\leq q\cdot d(x,y). $$ And most likely I have to use the q-contractivity of $T^n$, i.e. that there exists a $0\leq q &lt;1$ so that for all $x,y\in X$ it is $$ d(T^n(x),T^n(y))\leq q\cdot d(x,y). $$  Can you help me? Where can I ask questions for website recommendations? Is there any Stack Exchange site where I can ask others to recommend me good web sites for solving specific problems?  For example I would like to know if there is any URL shortening website that can allow me to define the url.  For example instead of http://bit.ly/1gX2dD0 I would like it to be http://bit.ly/superflow - if "superflow" is not available then I will invent another word.  I can imagine that such services are paid but I don't need the link to last more than one day, and maybe there is such a website. When did 5 foot squares become standard in D&D? Wizard sickness I remember in an interview, or during a Q&amp;A, Rowling stating that Wizards live longer than Muggles. This got me thinking about Wizard deaths in general, and then deaths being attributed to illness.  So, I ask, has there ever been a Wizard, that has died from something like cancer? Is there anything, even briefly mentioning a Wizard becoming ill?  And finally, are there any illnesses that are exclusive to Wizards?