The Cantor function (which I will define recursively below) is Hölder continuous and the exponent there is ln(2)/ln(3).   The function is defined as the limit function of a sequence of functions, where $f_{n}$ looks as below:   \begin{cases}       f_{n-1}(3x/2)  &amp; 0 \leq x\leq 1/3\\      0.5 &amp; 1/3\leq x\leq 2/3 \\       0.5 + f_{n-1}((3x-2)/2) &amp; 2/3\leq x \leq 1    \end{cases}  I've proved that $f_{n}$ converges uniformly to the Cantor function and that $||f_{n} - f_{n-1}||_{\infty}$ $\leq$ $\frac{1}{2^n}$, but I have no idea where to proceed from there. Any inputs will be appreciated. My final goal, as I mentioned above, is to prove Hölder continuity with Hölder exponent ln(2)/ln(3).
Let $C_k$ be the set obtained in the $k-$th stage of building the Cantor set, where  $$C_1=[0,\frac{1}{3}]\cup[\frac{1}{3},\frac{2}{3}]$$  $$C_2=[0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1]$$  $$\dots$$  So we have   $$C_k=\bigcup_{j=1}^{2^k}I_{k,j}$$  where $I_{k,j}$ are pairwise disjoint closed intervals of length $3^{-k}$ in $[0,1]$, let  $$[0,1]\setminus C_k=\bigcup_{j=1}^{2^k-1}J_{k,j}$$  where $J_{k,j}$ are pairwise disjoint open intervals and each $J_{k,j}$ are located between intervals $I_{k,j},I_{k,j+1}$  define  $$F_k:[0,1]\to[0,1]$$  where  $$F_k(0)=0,F_k(1)=1,F_k(x)=\frac{j}{2^k}\quad x\in J_{k,j}$$  and $F_k$ is linear on each subinterval of $C_k$. We can show that $F_k$ converges to the Cantor function $F:[0,1]\to[0,1]$. Moreover, it can be shown that  $$|F_k(x)-F_k(y)|\leq\frac{3^k}{2^k}|x-y|$$  How can we show that $F$ is Hölder continuous of order $\log_32$ ?
Where can I find out about Stack Overflow history? I'd like to know how Stack Overflow and other sites started - which was first, who initiated it (OK, that's on the link below) and why, how the different sites evolved, how the rules evolved, etc.  Where can I find such information?  Of course, I know I can look at  for a brief overview but I'm wondering if there is info on this site, particularly more detailed.. Kids' fantasy movie from the 80s or 90s? Why didn't Escobar's hippos introduced in a single event die out due to inbreeding Today I read a  about how Pablo Escobar had once imported 4 hippos (1 male, 3 female) into his estate in Colombia for his private zoo. After his downfall, while other species were shipped out, hippos were considered too big to move and expected to not survive. However, to the surprise of all the hippos are thriving and are so numerous that there have been calls to cull them. From the report:  Numbers are projected to only get bigger. [Colombian biologist Nataly] Castelblanco and her peers say the population will reach over 1,400 specimens as early as 2034 without a cull - all of them descended from the original group of a male and three females. In the study, they envisaged an ideal scenario in which 30 animals need to be culled or castrated every year to stop that happening.  My understanding is that since there was only 1 male, the gene pool would be limited and lead to lot of inbreeding in the descendants. This would cause population to not explode because some individuals would be unfit to survive. Why has this not happened in case of hippos? Is it because there are 3 females (probably unrelated to each other) which keeps gene pool large enough? Or can mutations explain this phenomenon? Would results have been different if originally 2 females had been moved and only 1 retained? EDIT - I have just found that  is maybe less than 100 individuals which though big is not massive. EDIT2 - I have edited the question title to keep focus on the hippos although a general answer would be welcome. Programmatically, how does "rooting" a device enable access? In what kind of rings a divisor of a product is a product of divisors? In a , if $a|bc$, then $a$ can be written as $a = a_1a_2$ so that $a_1|b$ and $a_2|c$.  Is this property of a commutative ring strictly weaker than the property of being a UFD?  If so, does it have a name, or is it equivalent to some other well-known property? Infinite Cartesian product of countable sets is uncountable Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets? Redirect www to non-www using Apache htaccess does not work At the end of the movie we are shown      a bowl with berries in it and Seneca Crane looked like he was locked in a room with them and was supposed to eat them. Did he eat them?   Why did that happen? Trying to remember the name of a grade-school book about an alien in a cereal box When I was a kid (Grade school, Elementary, can't remember the exact grade, but it was between 1992 and 1999) I read a book about a kid who found an alien in his cereal.  This alien informed him that he was there to defeat an evil threat to Earth, that took the form of the school's most popular student.  The alien came with a gun that did nothing, and the alien in the popular kid disguise had the power to make the kid's fingernails bleed by damaging his nerve endings.  In the end, it was revealed that the evil alien had a huge craving for, and vulnerability towards, pizza flakes, and was defeated by gorging himself on them and then exploding. biblatex: Cite commands with the same author prints authors name twice I'm using biblatex 1.4b, and Biber 0.9. However, I think the problem is with my settings, not with the versions of those programs.  I'm trying to cite a single author, and produce their name, followed by the dates of two papers, using the authoryear-ibid style, and natbib compatability. Here's my biblatex preamble:  \usepackage[style=authoryear-ibid,natbib=true,backend=biber]{biblatex}   What I want is for the following command: The works by \citet{Blau81, Blau88} ... to produce: The works by Blau (1981; 1988) ... however, what I'm getting is: The works by Blau (1981); Blau (1988)...  I know that biblatex knows they're the same author, since in the bibliography they fall under the same name (i.e. after the first instance of the name, there's just a dash instead of the name)  What am I doing wrong here? I've also tried it with the commands \cite and \cites, and neither produces what I want.   (crossposted with comp.text.tex) High-Swing Cascode Current Mirror I am currently dealing with current mirrors and came across the circuit shown below, a High-Swing Cascode Current Mirror. I read that this implementation, as the name suggests, has the advantage of a high voltage headroom, since the minimum potential at the drain terminal of M4 is: $$ V_{D4\_min}=V_{DS4}+V_{DS2}=(V_{GS4}-V_{TH})+(V_{GS2}-V_{TH}) $$ So the drain potential at M4 only needs to be twice the overdrive voltage, in case of same transistor dimensions, in order to hold M2 and M4 in saturation. But cannot this be said for many other cascode current mirrors, since that behaviour simply occurs by "stacking" together two transistors (M2 and M4)? Additionally, why is there a connection from the gate of M1 to the drain of M3? I have a hard time understanding whats going on in this circuit, help is greatly appreciated.    EDIT: A bias voltage source is setting the potential at the gate of M3/4. Sorry its not shown in the circuit.  EDIT 2: I just read that the mentioned bias voltage is implemented by a MOSFET in diode configuration, . For further discussion lets call it M5. Install Ubuntu from ISO image directly from hard disk of a system running Linux? Iphone 4 won't update past a 7.1.2 Find the first occurrence/starting index of the sub-array in C# i want to write a device driver but not able to find the header file can someone please help me find them?Also if someone can point out some important site links that would be really appreciated The question is, what elements on a DM screen are essential? I'm new at playing and DMing, altough a lot of experience watching games, just got a group to play with now. Started with the starter set in 5E and the Lost Mine of Phandelver adventure, however I've noticed the game can come to a stop when checking the rules or how to do something, while getting the books and stuff like that. So I thought to make a DM screen with some cheat sheets, of the most usual things new players, and new DM's have doubts on, and how to make the game flow well without long rule-checking breaks. What are the essential things to have on that screen? So far this is what I found  but not sure if it has all I need or so much stuff that'll also make the game stop a lot while cheking it out. Is it legal to develop a game using D&D rules? For a while now I've been thinking about trying my hand at creating a game similar in spirit and execution to Baldur's Gate, Icewind Dale and offshoots. I'd rather not face the full bulk of work in implementing my own RPG system - I'd like to use D&amp;D rules.  Now, reading about the subject it seems there is something called "The License" which allows a company to brand a game as D&amp;D. This license seems to be exclusive, and let's just say I don't have the money to buy it :p.  Is it still legal for me to implement and release such a game? Commercially or open-source? I'm not sure exactly which edition would fit the best, but since Baldur's Gate is based of 2nd edition, could I go ahead an implement that?  in short: what are the issues concerning licensing and publishing when it comes to D&amp;D?  Also: Didn't see any similar question... The glorified global variable - becomes a gloried global class. Some say breaking object-oriented design.  Give me scenarios, other than the good old logger where it makes sense to use the singleton. I am quite new to Javascript. I have made a html script that includes services I published to ArcGIS for server. It all does exactly what I want and I made it by copying working scripts together and getting help from more experienced scripters. I want to finetune my service by adding a legend. Unfortunately I am not able to create a working script. I have added the parts of my script I assume that are necessary to create my legend. Does anybody see what I am doing wrong?        &lt;script&gt;         var map;         var wmsLayer;         var URL;         var legend;         require(["dojo/query", "esri/map", "esri/layers/WMSLayer", "esri/dijit/Legend", "esri/config", "dojo/domReady!"],         function (query, Map, WMSLayer, Legend, esriConfig) {          esriConfig.defaults.io.proxyUrl = "/proxy/proxy_gcx.php";            map = new Map("map", {               basemap: "topo",               center: [5.5, 52.485],               minZoom: 8,               maxZoom: 13,               zoom: 8,               slider: true           });            wmsLayer = new WMSLayer("URL", {               format: "png",               visibleLayers: ["31", "30", "29", "28"]           });            map.on("layer-add", function (results) {               //add the legend               legend = new Legend({                   map: map,                   layerInfos: [{                       layer: results,                       title: "test"                   }]               }, "legendDiv");               legend.startup();           });             map.addLayer(wmsLayer);       });     &lt;/script&gt; &lt;/head&gt;  &lt;body&gt;     &lt;div id="map"&gt;     &lt;/div&gt;     &lt;div id="legendDiv"&gt;     &lt;/div&gt; &lt;/body&gt;   EDIT: I have edited the code. I am still not done. It does give me a legend on my browser but it says there is no content in there. On my arcgis server manager there is a legend. What happened to the voting buttons? Starting from yesterday, the up-/downvote buttons seem to have moved from the left of a post towards above (in the middle) of the bodies of all questions and answers. Here is a screenprint of a sample (containing such buttons for both a question and an answer to it):    This happens on an iOS device (running 9.3.5), using various browsers and this in any SE site. No idea so far how it looks like in the most recent iOS version. Same for the (green) checkbox below those voting buttons (to mark an answer as accepted). Is this intended to be like this (IMO it looks ugly ... and a huge waste of available space), or is it rather some sort of CSS related issue that needs to be fixed? PS: Looking at the same page around the same time on a Win PC (using any of the common browsers) seems to work fine.