How to find this limit and prove it rigorously: $\sqrt{2 + \sqrt{2+\sqrt{2+\sqrt2...)}}}$?
$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
LaTeX for Microsoft Surface I have my new Microsoft Surface and I was wondering whether it's possible to install pieces of software like MikTeX and TexMaker on this tablet. Does anyone know? still frame camera solve I've found a number of how-tos for solving video but I'm just looking to put an element in a photo and can't seem to find a how-to for that.  Is it possible in blender to do a camera solve with a still image?  Thanks. How to install and use Ubuntu from a USB flash drive on Windows 8? Which of these sentences is correct?     Rupees are interchangeable with dollars   or      Rupees is interchangeable with dollars. I'm trying to make a basic Windows application that builds a string out of user input and then adds it to the clipboard. How do I copy a string to the clipboard using Python? Finding Sylow 2-subgroups of the dihedral group $D_n$ I am trying to describe the Sylow $2$-subgroups of an arbitrary dihedral group $D_n$ of order $2n$.   In the case that $n$ is odd, $2$ is the highest power dividing $2n$, so that all Sylow $2$-subgroups have order $2$, and it is fairly easy to describe them.  However, if $n$ isn't odd, we may factor a power of $2$ out and write $|D_n|=2^{k}m$ for some odd integer $m$.   There is  that there exist precisely $m$ Sylow $2$-subgroups, but it does not provide an explicit description of such subgroups in the case $n$ is odd.   Additionally, someone has asked a similar question in the past, and they claim to give a description of the Sylow $2$-subroups in the case $n$ is odd, but I can not find a source or an explanation. The question is .     Can anyone provide a description of how one may determine precisely the Sylow $2$-subgroups for the case $n$ is odd?    Thank you. And now I am unable to chmod it back.. or use any of my other system programs. Luckily this is on a VM I've been toying with, but is there any way to resolve this? The system is Ubuntu Server 12.10.   I have attempted to restart into recovery mode, unfortunately now I am unable to boot into the system at all due to permissions not granting some programs after init-bottom availability to run- the system just hangs. This is what I see:  Begin: Running /scripts/init-bottom ... done [   37.062059] init: Failed to spawn friendly-recovery pre-start process: unable to execute: Permission denied [   37.084744]  init: Failed to spawn friendly-recovery post-stop process: unable to execute: Permission denied [   37.101333] init: plymouth main process (220) killed by ABRT signal   After this the computer hangs. Is it wasteful to use a heating element, instead of doing useful work? Consider a computer CPU consuming electrical energy to perform calculations and consequently emitting heat.  Assumption: That a CPU consuming x Watts of power, emits the same amount of heat as an incandescent heating element which consumes the same amount of power.  The rest of the question assumes that this is true: the energy has to go somewhere, and I can't see that the outcome of the calculations counts as "somewhere".  I've long repeated the claim that "wasteful" appliances such as incandescent lightbulbs, over-full kettles, computers left running, etc. are not wasting power if we make the assumptions:   the appliance is in a room which we aim to maintain at above a certain temperature, warmer than the external temperature (cf a home in winter) the room leaks heat to the outside faster than the "wasteful" appliances emit heat. the room contains an electric heater the electric heater is regulated by a thermostat   In these circumstances, I think (?), if a light bulb produces excess heat, its only effect is to cause the thermostat to switch off sooner, so what energy you've "wasted" in the light bulb, is "saved" at the heater.  I don't think that's controversial.  But my question is:   If a CPU does useful work, and emits all the input energy as heat as a side effect But a heating element emits the same heat, without doing useful work Is it therefore wasteful to use heating elements? Is it more efficient to (for example) run  on a CPU, to get "free" computation as a side-effect of heating?   One reason to heat a room with a heating element, rather than with CPUs, is that the heating element is much cheaper to produce. But ignoring the initial cost of the hardware, would a "heater" made out of 20 CPUs, each running at full capacity and consuming 50W each, output less thermal heat than a 1KW element that does nothing else useful?  To be useful, of course, the CPUs would need to output the result of their computations, which would account for a tiny amount of energy. Let's assume that's negligible.  I'm sure there's a flaw in there somewhere; probably that the assumption above is false. What have I missed? Why are there non-visible objects in my final render? I find myself needing to draw lots of elegantly curved paths in TikZ. Ideally, I'd just specify a series of points, and TikZ would calculate the extra data itself to draw a nice series of curves passing smoothly through these points, perhaps with an optional "looseness" parameter that I could specify. But the only way I can find to draw nice curves is by explicitly giving control points, or by manually specifying in and out angles.  I can think up a simple algorithm to do this, which would certainly be within TikZ's power to perform: just choose the in and out angles in a simple fashion based on the relative angles between each adjacent pair of line segments.  Is something like this already built-in? Or can someone cook something up that does the job?  Edit: Jake has provided an answer using the plot [smooth] functionality. This is almost perfect! But it can't do what I need, because it doesn't let me specify tangent angles manually where needed, which is especially important at the beginning and end of the curve. I would have thought this would be a natural and straightforward addition to the existing plot [smooth] algorithm: for every coordinate, an optional angle should be able to be specified as an argument, which if supplied is treated as the tangent angle for the curve at that point. And while we're at it, it wouldn't hurt to also allow the tension to be modified along the path.  A minimal extension to the algorithm would just accept two optional parameters, for the curve tangent at the beginning and end. What is the definition of a "feature map" (aka "activation map") in a convolutional neural network? Intro Background Within a convolutional neural network, we usually have a general structure / flow that looks like this:  input image (i.e. a 2D vector x)   (1st Convolutional layer (Conv1) starts here...)  convolve a set of filters (w1) along the 2D image (i.e. do the z1 = w1*x + b1 dot product multiplications), where z1 is 3D, and b1 is biases. apply an activation function (e.g. ReLu) to make z1 non-linear (e.g. a1 = ReLu(z1)), where a1 is 3D.   (2nd Convolutional layer  (Conv2) starts here...)  convolve a set of filters along the newly computed activations (i.e. do the z2 = w2*a1 + b2 dot product multiplications), where z2 is 3D, and and b2 is biases. apply an activation function (e.g. ReLu) to make z2 non-linear (e.g. a2 = ReLu(z2)), where a2 is 3D.    The Question The definition of the term &quot;feature map&quot; seems to vary from literature to literature. Concretely:  For the 1st convolutional layer, does &quot;feature map&quot; corresponds to the input vector x, or the output dot product z1, or the output activations a1, or the &quot;process&quot; converting x to a1, or something else? Similarly, for the 2nd convolutional layer, does &quot;feature map&quot; corresponds to the input activations a1, or the output dot product z2, or the output activation a2, or the &quot;process&quot; converting a1 to a2, or something else?  In addition, is it true that the term &quot;feature map&quot; is exactly the same as &quot;activation map&quot;? (or do they actually mean two different thing?)  Additional references: Snippets from :  *The nomenclature is being used loosely here. In particular, I'm using &quot;feature map&quot; to mean not the function computed by the convolutional layer, but rather the activation of the hidden neurons output from the layer. This kind of mild abuse of nomenclature is pretty common in the research literature.   Snippets from :  In this paper we introduce a visualization technique that reveals the input stimuli that excite individual feature maps at any layer in the model. [...] Our approach, by contrast, provides a non-parametric view of invariance, showing which patterns from the training set activate the feature map. [...] a local contrast operation that normalizes the responses across feature maps. [...] To examine a given convnet activation, we set all other activations in the layer to zero and pass the feature maps as input to the attached deconvnet layer. [...] The convnet uses relu non-linearities, which rectify the feature maps thus ensuring the feature maps are always positive. [...] The convnet uses learned filters to convolve the feature maps from the previous layer. [...] Fig. 6, these visualizations are accurate representations of the input pattern that stimulates the given feature map in the model [...] when the parts of the original input image corresponding to the pattern are occluded, we see a distinct drop in activity within the feature map. [...]  Remarks: also introduces the term &quot;feature map&quot; and &quot;rectified feature map&quot; in Fig 1.  Snippets from :  [...] One dangerous pitfall that can be easily noticed with this visualization is that some activation maps may be all zero for many different inputs, which can indicate dead filters, and can be a symptom of high learning rates [...] Typical-looking activations on the first CONV layer (left), and the 5th CONV layer (right) of a trained AlexNet looking at a picture of a cat. Every box shows an activation map corresponding to some filter. Notice that the activations are sparse (most values are zero, in this visualization shown in black) and mostly local.   Snippets from   [...] Every unique location on the input volume produces a number. After sliding the filter over all the locations, you will find out that what you’re left with is a 28 x 28 x 1 array of numbers, which we call an activation map or feature map. All I remember is the guy was sent to a planet to test something or he did something illegal, and time moved really fast on the planet but slow to the outside world. He had to fix something to leave the planet I think. His dad was also dying and he checked how many days he had left every day, but in the end his dad didn't actually die as time was fast. It took place on a sandy planet. How do I install software using the Ubuntu Software Center? The title says it all. I'd like to know how I can install software - an example is the Chromium web browser. Why do we notate the greatest common divisor of $a$ and $b$ as $(a,b)$? In my textbook on elementary number theory from a class last year, as well as elsewhere through my academic experience and even posts here, I often see the greatest common divisor notated as $(a,b)$ (where this represents the greatest common divisor of $a,b$).  My question is, why is such an unusual notation used? I say it's unusual (if not a bit absurd), for a couple of reasons:   It is unintuitive. It does not grant the reader an immediate knowledge of what it means. In particular, I can easily see it being misinterpreted as an ordered pair of points, and not as a function (which it basically is and I touch on in a second) or representing a single value. Good notation would be immediately understood, no? One should not struggle with parsing through the context or meaning. A clearer notation is often used, and just three letters away. You could certainly think of $(a,b)$ as an ordered pair, if we consider it a function. I often see the alternate $gcd(a,b)$: a function, which takes in two inputs ($a$ and $b$), and outputs their greatest common divisor (or factor, if you prefer). Merely adding the three letters makes the notation clearer by clarifying it is  a function, and using a relatively well-accepted acronym. (Or, at least, I often hear it abbreviated "g.c.d." in school.)   I've mulled this over for a while and I can't think of much without borrowing inspiration from other "unusual" notations. For example:   Common understanding in context: consider the $\log(x)$ function. Depending on context, without a base stated as seen here, it can be presumed to mean whatever is most "appropriate." It could be base $e$ (common in mathematics), base $10$ (common in engineering), or base $2$ (common in computer science). The ambiguity is essentially resolved by context in other words - personally, not a fan of that (I like the explicit stating of bases for full clarity), but if such a motivation holds elsewhere, it might stand to reason that $(a,b)$ could just be taken as a shorthand and "understood" in the context of number theory to simply be the greatest common divisor. That it is in fact used despite $gcd(a,b)$ being clearer and somewhat common also does seem to suggest this playing a role. Perhaps it just started that way and it kept being used. For example,  (based on a ) regarding how logarithms, roots, and exponentiation can be unintuitive from a notation perspective, and proposes an alternative, easier to understand notation. I haven't been in middle school in nearly a decade, but guessing from discussions online, that obviously hasn't really picked up too much. In that sense, perhaps it's like that for $(a,b)$: like we still notate exponentiation, etc., the same way today despite how counterintuitive it is (at least for students), perhaps it just "stuck" and proliferated?   I wasn't able to easily find any reason why online, and it bugs me a bit, so I ask:     Why do we so unintuitively notate the greatest common divisor of two numbers $a,b$ as $(a,b)$?   I of course recognize that this is somewhat moot in light of how easily it's made clear - as established above, all I have to do is append the front of the parenthetical by $gcd$ or $gcf$ to make it way better. And of course, people do that. But I'm more curious in where this ambiguous former notation arose and, in light of a better notation, why it is still used today. Some Windows 10 UWP icons not displaying on search, or in settings Alternating binomial sum with intervals of two Fix integer $n\geq 1$. Consider the number $$1-\binom{n}{2}+\binom{n}{4}-\binom{n}{6}+\cdots$$where the sum continues as long as the lower number in the binomial is $\leq n$. Is there a way to simplify this sum?   The first few values are $1, 0, -2, -4, -4$. Help removing subsection and subsubsection numbers from Table of Contents TOC Hello I am trying to further modify my TOC.  I got an answer  as to how I could remove the \section numbers on the left side of the TOC and now I would like to expand that.  I would like to remove the left-hand numbering for \subsection and \subsubsection as well but I do not know how.  (I also want to keep these numbers out of the document and I am using \def\sectionformat{} to do that)  Also, my TOC is slightly over one page and so I would like to adjust the spacing from 2 to 1.5,  is there any way to do this?    \documentclass[12pt, a4paper, twoside]{article}     \usepackage{baskervald}     \usepackage[margin=1in,bindingoffset=15.5mm,heightrounded]{geometry}     \usepackage[T1]{fontenc}     \usepackage[notes]{biblatex-chicago}     \usepackage{filecontents}     \usepackage{microtype}     \usepackage{url}     \usepackage{makeidx}     \usepackage{indentfirst}     \usepackage{setspace}     \usepackage{chngcntr}\counterwithin*{footnote}{section}     \newcommand*{\textsubscript}[1]{$#1{}$}     \DefineBibliographyStrings{english}{references = {Works Cited}}     \DeclareBibliographyCategory{primary}     \makeindex     \addbibresource{bibliography.bib}     \makeatletter     \let\latexl@section\l@section     \def\l@section#1#2{\begingroup\let\numberline\@gobble\latexl@section{#1}{#2}\endgroup}     \def\@seccntformat#1{\ifcsname #1format\endcsname\else\csname the#1\endcsname\quad\fi}     \def\sectionformat{}     \def\subsectionformat{}     \def\subsubsectionformat{}     \makeatother     \author{}     \title{\vspace*{36 pt}\textbf{\emph{\Huge{Big Important Title}}}\vfill}     \date{}     \begin{document}     \tableofcontents     \section{Part I}     Hello World     \end{document} It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. Generate dynamic XY coordinate labels for dataframe corners in ArcGIS layout I need to label the four corners of a dataframe as shown in the image. Can this be done with dynamic text scripts? I want to do something like String.Format("[{0}, {1}, {2}]", 1, 2, 3) which returns:  [1, 2, 3]   How do I do this in Python?