Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function.   I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$  $\psi(t) := f (x_0 + tv)$, is convex over $I (x_0, v )$.  where $L_{v,x_0}:= \{x_0 + tv : t ∈ \mathbb{R}\}$, and $I(x_0,v):=\{t \in \mathbb{R} : x_0 + tv \in C\}$  Any help is very much appreciated
Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set.  $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ where $L_{v,x_0} := \{x_0 + tv : t \in \mathbb{R}\}$.  i.e. f is convex iff when restricted to any line, it be convex on that certain line.  So essentially to handle this proof I need to show that convexity for $f$ holds when its convex over any arbitrary line, and on the converse it does not hold when $f$ is not convex over that arbitrary line. If $f$ is convex then,  $$\forall x,y \in C, a \in [0,1]$$ it holds that $$f(ax+(1-a)y) \leq af(x) + (1-a)f(y) \space \space\space\space\space\space\space\space\space\space\space\space (1)$$  EDIT: $$**Part II**$$  Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function.   I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$  $\psi(t) := f (x_0 + tv)$, is convex over $I (x_0, v )$.  where $L_{v,x_0}:= \{x_0 + tv : t ∈ \mathbb{R}\}$, and $I(x_0,v):=\{t \in \mathbb{R} : x_0 + tv \in C\}$  Any help is very much appreciated
Why are there no other dragons in ASOIAF? Set same icon for all my Forms Is there any way to set the same icon to all my forms without having to change one by one? Something like when you setup GlobalAssemblyInfo for all your projects inside your solution. Why don't planets have Circular orbits? I've been making a map for 1.10 and it's going great, but 1.11 shows great mobs to use for it. However, most things have channged in command syntax. For example, /kill @e[type=ArmorStand,r=6] is now /kill @e[type=Armor_Stand,r=6]  What about /testfor? I can't find a way to testfor a player to stand on a block and give them an effect.   What's is going on in this image is that a redstone clock is running into this command /testfor @a[1117,20,507,r=1] (if I was to stand on them cords the comparator would turn on and power that command block), but it only works in 1.10... from 1.11 snapshots, I've seen some errors of it, like it now needs to have {} at the beginning and end of it. (e.g. /testfor @a{[1117,20,507,r=1]})   It's still not working, though, so does anyone know how to get it to work? Rename a file to parent directory's name in terminal I have to deal with a large number of files nested inside of directories (one file per directory), resembling something like this:  fred/result.txt john/result.txt mary/result.txt ...   I am currently using the following command to process each of those files:  find . -maxdepth 1 -type d \( ! -name . \) -exec bash -c \"cd '{}' &amp;&amp; processResult result.txt\" \;   I am looking for something I can add into the end of this command that will change the filenames to fred.txt, etc. and then later move the file into the parent directory to eliminate the extra layer of directories.  What would be the best way to do this? Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate solutions.  I also looked up these related questions - , ,  - but was not happy with proofs given there. I seem to need some help understanding these. Any such help is appreciated.      Show that every sequence in $\Bbb R$ either has a monotone increasing sub-sequence or a monotone decreasing sub-sequence.   Let $(x_n)$ be a sequence in $\Bbb R$. Suppose $(x_n)$ is not bounded. Without loss of generality we may assume that $(x_n)$ is not bounded above. Therefore given any real number there is a member of the sequence which is greater. Let $x_{n_1}$ be any member of the sequence.   There is $x_{n_2} \gt \sup\{x_1, x_2, ..., x_{n_1} \}$. For $i \gt 1$ let $x_{n_i} = \{x_1, x_2, ..., x_{n_{i - 1}}\}$ then $(x_{n_k})$ forms a monotone subsequence of $(x_n)$.   Now suppose instead that $(x_n)$ is bounded. By the Bolzano-Weierstrass Theorem there is a subsequence $(y_n)$ of $(x_n)$ which converges to a limit $y$. Without loss of generality there are infinitely many distinct values in $(y_n)$ that are unequal to $y$. Let $y_{k1}$ be the first such element. Let $y_{k2}$ be any element in $\{ y' \in (y_n) \ \ | \ \ |y' - y | \lt |y - y_{k1}|  \}$.   For $i \gt 1$ let $y_{ki} \in \{ y' \in (y_n)  \ \ | \ \ |y' - y | \lt |y - y_{k \ i - 1}|  \}$. Such $y_{ki}$ exists for every $i \in  \Bbb N$ since $ \lim (y_n) = y $. Now let $(y_{kn})$ be the sub-sequence of $(y_n)$ thus formed. At least one of the two following sets must contain infinitely many elements.    $\{ y \in (y_{kn}) \ \ | \ \ y \gt x\}$ $\{ y \in (y_{kn}) \ \ | \ \ y \lt x\}$   The one which does forms a monotone subsequence. (a.k.a. "god rays" and many other names) can be created in Blender Internal by checking the Halo checkbox in the lamp properties box.  How can I create a similar effect using Cycles? Disable the guest user completely in El Capitan without having to disable 'Find My Mac' I know it should be straight forward but unfortunately, it  isn't.  I did disable the guest user in the System Preferences by unchecking the checkbox for Allow guests to login in to this computer, but it still shows up on the login screen.  I have heard that disabling 'Find My Mac' would fix it for OS X Lion, but I want to keep it.  Is there a way to do this? Why doesn't Normal[] work on GeometricTransformation? Moving Grub to a new drive to get rid of Windows When is a gerund supposed to be preceded by a possessive adjective/determiner? I assume that the following sentences are grammatically correct:   He resents your being more popular than he is. Most of the members paid their dues without my asking them. They objected to the youngest girl's being given the command position. What do you think about his buying such an expensive car? We were all sorry about Jane's losing her parents like that.   I'm still getting used to this possessive gerund structure. It sounded so weird to me at first. Is the structure used in both formal and informal contexts? Are there any alternative structures that result in the same meaning and are more frequently used? (Examples taken from ) How to run ffmpeg from command line in android Calculating polygon area within polygons in QGIS? I am new to QGIS - I want to estimate the area of the reef under a polygon grid. My shapefiles are already in meters, however I am having a hard time calculating the area under the grid layer. Attached are the photos. I want to calculate the total area of the bold black polygons (reefs) under each grid square (image 2). How much reef is inside each square, for example, grid1: 25 sq.km, grid 1 sq.km, etc. Each reef polygon has its corresponding area in the layer attribute table.    ADDITIONAL PHOTO: RED IS THE REEF  THE ATTRIBUTES OF THE REEF LAYER How can I get a specific line from a file? How to view the output of a running process in another bash session? I have left a script running on a remote machine from when I was locally working at it. I can connect over SSH to the machine as the same user and see the script running in ps.   $ ps aux | grep ipcheck myuser  18386  0.0  0.0  18460  3476 pts/0    S+   Dec14   1:11 /bin/bash ./ipchecker.sh   It is simply outputting to stdout on a local session (I ran ./ipchecker.sh form a local terminal window, no redirection, no use of screen etc).  Is there anyway from an SSH session I can view the output of this running command (without stopping it)?  So far the best I have found is to use strace -p 18386 but I get hordes of text flying up the screen, its far too detailed. I can stop strace and then sift through the output and find the text bring printed to stdout but its very long and confusing, and obviously whilst it's stopped I might miss something. I would like to find a way to see the script output live as if I was working locally.  Can anyone improve on this? The obvious answer is to restart the script with redirection or in a screen session etc, this isn't a mission critical script so I could do that. Rather though, I see this as a fun learning exercise. Prove by induction. How do I prove $\sum\limits_{i=0}^n2^i=2^{n+1}-1$ with induction? In episode 10 of Full Metal Alchemist: Brotherhood, during the events in the Research Lab 5, Edward Elric is severely hurt. Later in the hospital, he writes down everything he had come to know, and he discusses with Alphonse, Hughes, and Armstrong about what to do with the information to continue the investigation.  At this point, King Bradley arrives with lemon into the room. He finds out what they are doing but acts very calm and chilled out and allows them to continue investigation while being careful.  Why did he not get Hughes and Armstrong killed at that point? Or at least tell them to not investigate at all? Existence of minimal $\sigma$-algebra and transfinite induction Basically, I've designed a booklet on Photoshop CC. It's an informational booklet so it's quite vibrant with lots of colours and pictures in it.   I've finished it and exported all the files into PDF format. Upon doing this, the PDF's are much "duller" for use of a better word. Now I'm not a designer and the whole thing was a learning process but now I'm reading about CMYK and RGB colour models and conversions etc. I designed the whole thing RGB (because CMYK looked duller obviously. The PDF's aren't disastrous, red is still red, black is still black but it's nowhere near as visually appealing anymore.   Is this just because of the monitor and the models each program uses or when I send my brochure to be printed, am I doomed for a much duller booklet? There's obviously a way around it otherwise all print would be dull. If $q: E\rightarrow X$ is a covering map that has a section (i.e. $f: X\rightarrow E, q\circ  f=Id_X$) does that imply that $E$ is a $1$-fold cover?