String concat/join performance
String.Join vs. StringBuilder: which is faster?
Excel INDIRECT function and conditional formatting - highlighting a row First, this question is motivated by the imprecise question: Is there a sensible notion of parity (evenness and oddness) for real numbers?   Here are some properties a notion of parity for $\mathbb{R}$ should have:    It should be an equivalence relation on $\mathbb{R}$ with exactly two equivalence classes.  Each equivalence class should be dense in $\mathbb{R}$. There should be some kind of symmetry between the two equivalence classes. (This is intentionally imprecise.)   If we divide $\mathbb{R}$ into rationals and irrationals, this lacks symmetry, since the rationals have measure zero and are countable, while the rationals have infinite measure and are uncountable.   So we could require that each equivalence class be uncountable and/or have positive (or infinite) measure. Then take one equivalence class to be the rationals, union with countably many fat cantor sets with positive measure, and the other class its complement. Then both classes are now dense, uncountable, and have infinite measure. But again, symmetry is lacking. One of these classes is "uncountable everywhere" and has "positive measure everywhere" while the other does not.   I propose some definitions: A subset $A \subset \mathbb{R}$ is uncountable everywhere if for any open interval $(a,b)$, the intersection $A \cap (a,b)$ is uncountable. A subset $A \subset \mathbb{R}$ has positive measure everywhere if for every open interval $(a,b)$, the intersection $A \cap (a,b)$ has positive measure. We can see immediately that having positive measure everywhere implies being uncountable everywhere, since a positive measure set must be uncountable.   Finally, my questions:    Is there a partition of $\mathbb{R}$ into two sets that are both uncountable everywhere?  Is there a partition of $\mathbb{R}$ into two sets that both have positive measure everywhere?  Is there a partition of $\mathbb{R}$ into two sets that split each interval $(a,b)$ into two parts of equal measure?   Potential generalizations for extra credit: What about partitions of $\mathbb{R}$ with $n$ equivalence classes, where $n \in \mathbb{N}$, or even with countably many, or even uncountably many equivalence classes?  Note: If you've seen these definitions of uncountable everywhere or positive measure everywhere somewhere under a different name, please let me know. I've never found anything where other people were thinking about these notions.  EDIT: By "measure," I mean Lebesgue outer measure, so we don't have to worry about anything being measurable. If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$? If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$?  Here's what I've done:  $1^3$ congruent to $1 \pmod p$ thus, 1 is a cubic residue, Also $(p-1)^3=(p-1) \pmod p$ implies  $$(p-1)^3 - (p-1) = (p-1)[(p-1)^2 -1] = (p-1)(p^2 -2p) = p(p-1)(p-2) $$ implies $$p|(p-1)^3 -(p-1)$$ Thus  $p-1$ is a cubic residue $\mod p$. Now I don't know how to show for $1\lt a \lt p-1$ are all cubic residues when $p$ is congruent to $2\pmod 3$. I'm having problems with a WFS GetFeature request. I'm using OpenLayers with an OSM map as base layers which use the Google projection. Overlaying the layer with WMS works perfect. When I do a request to return all features within the bounds of a polygon, no features are found.  My layer uses EPSG:28992. the base layer uses: EPSG:900913  My layer configuration in GeoServer:    When I set the declared SRS to EPSG:900913 and set "SRS handling" to "Reproject native to declared" it works fine, but I need OpenLayers to reproject it, not GeoServer.  Below is an example request I would do. I think the problem is that GeoServer does not convert the spatial filter to the native SRS.  http://mydomain.com/wfs?request=GetFeature&amp;version=1.1.0&amp; srsName=epsg:28992&amp; typeName=Namespace:Layer1&amp; propertyName=(*)&amp; CQL_FILTER=year = 2011 and INTERSECTS(geometry,POLYGON((565585.61613069 6816272.786200799,565585.61613069 6817032.3791692,564362.62367835 6817032.3791692,564362.62367835 6816272.786200799,565585.61613069 6816272.786200799)))&amp; outputFormat=GMl2&amp;   Update  I just tried re-projecting some points from OpenLayers (EPSG:900913) via proj4js to EPSG:28992 (PostgreSQL and GeoServer). I took 3 points and  compared them. They all showed an offset.   Converted: POINT(233492,24349424947 582191,0681508556)  In Database: POINT(233530,852147064 582315,680920578)  Difference x: 38,60865281453 Difference y: 124,6127697224 Converted: POINT(133459,1980658749 455770,7481867363)  In Database: POINT(133486,356985081 455879,67223342)  Difference x: 27,1589192061 Difference y: 108,9240466837 Converted: POINT(176920,43262818802 317930,36182729155)  In Database: POINT(176954,154083767 318023,515324756)  Difference x: 33,72145557898 Difference y: 93,15349746445   If the offset was constant I could correct it. Now I'm not sure if the offset is caused by GeSserver or by proj4js. I'm guessing proj4js since the WMS layer does not show any offset. How do I Insert a carriage return on a card in Trello? Prove that: $\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{n}}}}<3,\,\forall n\in\mathbb N.$ Explicit vs implicit SQL joins Why is my bash prompt getting bugged when I browse the history? Often when I start looking at history of commands some of the characters from a command displayed aren't erased, for example:  What's happening:  prompt$ some_command prompt$ some_commanother_command   What should have happened:  prompt$ some_command prompt$ another_command   I can't erase those characters and bash ignores them when executing the command. They also disappear when a new prompt is being displayed (after pressing Enter for example).  I asked other people who work on Linux at my workplace and they said that they have that problem sometimes too, but they didn't have an idea on how solve that issue.  I have Ubuntu 11.10 and I'm using guake. Here is my PS1:  \e[0;31m\u \A ${PWD##*/}\e[0;32m$(parse_git_branch)\e[0;31m$\e[m   where parse_git_branch is  parse_git_branch () {     git name-rev HEAD 2&gt; /dev/null | sed 's#HEAD\ \(.*\)# (\1)#' }   As far as I know my colleagues have that problem even with less "fancy" PS1. Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ I'm trying to prove using induction that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have:   Base case: $P(1) = (1) + (8) + (27) = 36, 36$ can be divided by $9$ so the base case is valid Inductive step: let $P(n)$ be the statement $9$ divides $n^3 + (n+1)^3 + (n+2)^3$. Assume $P(k)$ is true, so $9$ divides $k^3 + (k+1)^3 + (k+2)^3$.   And this is where I'm stuck. I'm not sure how to demonstrate that $9$ divides $P(n)$ when $n = k+1$. If someone could step me in the right direction that would be awesome. Thanks! I think the answer is yes but I know a lot of people disagree. So, I would like to ask these people when exactly does it stop being me.  Let's say I want to upload my brain into a computer using the following procedure:  I have each of my neurons replaced by electronic neurons. By electronic neurons I mean chips with the necessary hardware and software to perform the exact same function that the neuron being replaced was performing. This means that given the same inputs that the biological neuron was given it would produce the same outputs. Then I connect that electronic neuron to all the neurons that former biological neuron was connected to.   At the end of this process is it still me? If not when did it stop being me? When one neuron was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  (assuming it is still me)  Then, I divide these electronic neurons into groups and for each group I perform the following:   Write software to simulate the group of electrical neurons in my computer such that given the same inputs that go into the group of electronic neurons in my brain, it will produce the same outputs. Replace each group of neurons with another chip that communicates with my computer. Each of the chips will communicate to the corresponding software module in order to determine how to transform its inputs into the necessary outputs. Connect all of the inputs and outputs of the group of electronic neurons to that new chip.   At the end of this process is it still me? If not when did it stop being me? When one group was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  (assuming it is still me) Now, what I have is a bunch of these new chips in my head, connected to each other and for each of these chips I have a small neural network software module in my computer. So, now for each pair of chip/software module I do this:   Transfer all connections going into and out of the chip into the computer version (i.e. connect the software module to the other software modules that represent the other chips that this chip is connected to) Disconnect the chip from the other chips and remove it   At the end of this process is it still me? If not when did it stop being me? When one of the new chips was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  If it is still me then I have successfully moved my brain into a computer without ceasing being myself.  So, again the question is if at the end it is not me anymore when exactly does it stop being me and why does it stop being me?  EDIT: This question assumes that our universe and everything in it (including ourselves) follows the (testable) laws of physics. As it stands it also assumes that neuroscience is mostly right, although the question could also be considered without that requirement if instead of neural simulations we were to implement physics simulation of the brain. Of course this is a lot less practical. However, if your view is that our minds have a supernatural component then the question does not make any sense.  It was also pointed out that this is related to the Ship of Theseus, i.e. is something still the same if all its parts have been changed. With regards to that I think it is useful to consider that most of the cells in our body (not including brain cells) are regularly replaced anyway and some people even get entire organs replaced. Yet, we don't say that they stopped being themselves. Note, however, that if you are assuming a supernatural component of the mind that this explanation is not really adequate or relevant. VMware Linux Server -- how can you tell if you are a vm or real hardware? An interesting question.  I have logged into a Linux (most likely SuSE) host.  Is there some way that I can tell programmatically that I am a VM host or not?    Also assume that the vmtools are not installed. I've noticed this issue on a handful of other communities, but making the browser window narrow (mine is 1080x960) makes the .post-layout and #sidebar elements overlap. Maybe something to do with the grid CSS? The #mainbar element seems to be the correct width and changing the .post-layout's overflow-x to auto seems to "fix" it  From retrocomputing:    From this post, because we are meta after all:    Specs:   Google Chrome Version 78.0.3904.108 (Official Build) (64-bit) Ubuntu 18.04.3 LTS Make gedit embedded terminal colours the same as the default terminal I've just updated to 11.10 and now my embedded terminal in gedit does not match my default terminal. I hate looking at a white and black terminal. Does anyone know how to make the embedded terminal match the default terminal?  This is a screenshot:    I want my embedded terminal in gedit to match my default terminal. I'm using LyX for technical documentation and want to automate the process of creating PDF files.  Within LyX, I can simply click the button "View PDF (XeTeX)", and LyX performs the whole build process for me: it creates a .tex file, does some bibliography magic, and uses XeTeX to create the final PDF file.  What I'm looking for is a way to call LyX from the command line and get the same result. So far, I've only managed to generate a plain .tex file (using --export-to xetex). However, when I run xelatex on the generated file, it keeps showing errors that don't occur when building using LyX. Is there a way to get LyX to do the work?  My system: Windows 7 64bit, LyX 2.1.0 If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$ If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$. Here is my attempt at a proof. It feels very elaborate and I am not sure if it is correct. Can someone please point out any mistakes or places where I may improve. Thanks!  By the definition of continuity at $a$ of $f,g$ we have that $\lim\limits_{x\to a}f(x)=f(a)$ and $\lim\limits_{x\to a}g(x)=g(a)$. Suppose that $f(a)&gt;g(a)$. Then there exists a $\delta&gt;0$ such that $f&gt;g$ for all $x$ satisfying $|x-a|&lt;\delta$. Then for $x$ satisfying $|x-a|&lt;\delta$ we have $h(x)=f(x)$ and $k(x)=g(x)$ and thus $h$ and $k$ are continuous at $a$ because $f$ and $g$ are continuous at $a$. Now if $f(a)&lt;g(a)$ we simply relabel $f=\tilde{g}$ and $g=\tilde{f}$, so that $\tilde{f}(a)&gt;\tilde{g}(a)$ and we apply the previous result to show that again $h$ and $k$ are continuous at $a$.\  Now if $f(a)=g(a)$ then we can distinguish three cases. $f\geq g$ in a small neighborhood about $a$, $f\leq g$  in a small neighborhood about $a$ or $f\geq g$ on one side of $a$ and $f\leq g$ on the other side. In the first case we assume that $f\geq g$ in some small neighborhood about $a$. Then, since $f(a)=g(a)$ we have $h(x) = f(x)$ in this neighborhood and $k(x) = g(x)$ and again we see that $h$ and $k$ are continuous at $a$ due to the continuity of $f$ and $g$ at $a$. Similarly if $f\leq g$ in a neighborhood around $a$ then $h(x)=g(x)$ and $k(x)=f(x)$ in this neighborhood and $h$ and $k$ are continuous at $a$.   Suppose that just to the left of $a$ we have $f\geq g$ and just to the right of $a$ we have $f\leq g$. Then $\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}f(x)=f(a)=h(a)=g(a)=\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^+}h(x)$. Similarly $\lim\limits_{x\to a^-}k(x)=\lim\limits_{x\to a^-}g(x)=g(a)=k(a)=f(a)=\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^+}k(x)$. So $\lim\limits_{x\to a}h(x)=h(a)$ and $\lim\limits_{x\to a}k(x)=k(a)$ which means that $h$ and $k$ are continuous at $a$. Lastly if $f\leq g$ just to the left of $a$ and $f\geq g$ just to the right of $a$ we can relabel $f=\tilde{g}$ and $g=\tilde{f}$ and apply the last result to show $h$ and $k$ are continuous at $a$.   Again, please point out any mistakes I may have made. Thanks!! Is there any research or data on horizontal vs. vertical navigational menus? I've seen many sites using both horizontal (top) navigation, vertical (left) navigation, and both across web sites for a long time. My understanding is that they're both conventions, but I would assume there must be some data out there supporting one route over the other.   My question is: are any studies or data for or against horizontal or vertical navigation (as suggested above) with regards to primary navigation (specifically on a smaller site or web app).  I've read some interesting things that are flipping me back and forth between liking horizontal more than vertical and vice versa. For example if you look at Jacob Nielsen's post on  you get the feeling that more important content should be on the left -- but you could take that as it being more important for the navigation (the user's escape routes) to be in that valuable space. What is "superuser" and how do I get that privilege? Combinatorics identity question I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? How do the tens­es and as­pects in English cor­re­spond tem­po­ral­ly to one an­oth­er? What is the proof that the universal constants ($G$, $\hbar$, $\ldots$) are really constant in time and space?