So there is this question that I have been trying to solve but am struggling to do so. As far as I can tell, it is a differential equation however, I am unsure how to go about solving it. It would be very much appreciated if any help was given on the question as I am unsure of how to proceed with it. Thank you.     A rumor is spread at a rate proportional to the number of people who have heard it. If there are $200$ people in a room and the amount of people who have heard the rumor is unknown, then it must be assumed that $200 - x$ have heard the rumor. Therefore the rate is:   $$\dfrac{dx}{dt} = ax(200 - x)$$   It is assumed that the rumor started with one person and that after $1.5$ hours, $12$ people have heard it. How long will it take for $100$ people to hear the rumor?
From Stewart 7e pg 614 # 9  "One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard th eremor and the fraction who have not hear the rumor. a) Write a differential equation that is satisfied by y. b) Solve the differential equation c) A small town has 1000 inhabitants. At 8 am 80 people have heard a rumor. By noon half the town has heard it. At what will 90 percent of the population have heard the rumor? "  The wording of this is very ambiguous to me and I can't really make sense of it.  They mention a product, so I know that something is being multiplied and that y is a fraction which belongs to the population who have seen it so I think that "have not" heard is a constant, and that y is a fraction taht represents who have. I tried to set this up and it is the wrong answer. I am not sure what they want from that, the English usage is too ambiguous to make sense of it. The complete lack of punctuation is what really does it.
I navigate with Ctrl-n,p,f,b (readline/ keys), so that I don't have to move my hands too much and get  again. Is there a way to disable Ctrl-b being hotkey for bold?  Note that this is not just an Emacs thing. Standard web browsers, like Firefox and Chrome, use things like  Ctrl + K and Ctrl + L to navigate to the Google search bar and address bar respectively. In bibliography, how to print the page in my text where I cited certain paper? How to get, for instance,  [19] S. Roth and M. J. Black. Fields of experts: A framework for learning image priors. In IEEE Conf. on Computer Vision and Pattern Recognition, 2005. 1, 2, 7, 8  where the last numbers (1, 2, 7, 8) are the pages where I cited the paper in MY TEXT?  This would be like a reversed reference, or a index for bibliography. Help resizing posted images I often add pictures to my questions. Most of the time they come out the right size. Well, that used to be OK always, up until I started posting screenshots taken by my mobile. For example, this one:  This way, I'm sure you'll agree it is huge, decidedly too big, it takes up the whole body of the question, I mean. At present, the only way I have to resize it is the following code: &lt;sub&gt;(Click image to enlarge)&lt;/sub&gt;  [&lt;img src=&quot;http://i.stack.imgur.com/hKmRp.jpg&quot; width=&quot;210&quot; height=&quot;350&quot; alt=&quot;&quot;&gt;][1]  which I do not remember, and always have to recover from . Is there a simpler, less HTML-hacky (and, I'd expect, more Markdown-based) way of resizing it? PS The output of the above code is: (Click image to enlarge) INT comparison without WHERE statement Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every Hausdorff space has an infinite discrete subspace, there is a subsequence $R=(\mathcal{V}_n)_n$ of $R=(\mathcal{U}_n)_n$ such that $\{\mathcal{V}_n|n \in \mathbb{N} \}$ is a discrete subspace of $\beta X$. In particular, there is a sequence $(A_n)_n$ of sets with $A_n \in \mathcal{V}_m$ if and only if $m =n$. If we let $B_n=A_n - (A_0 \cup...\cup A_{n−1})$, then the sequence $(B_n)_n$ is pairwise disjoint and $B_n \in \mathcal{V}_m$  iff $m=n$. If we let $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even. In other words, if $B =\{ \mathcal{V} \in βX|B \in \mathcal{V}\}$, then $B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even. Therefore, we conclude that the sequence $(\mathcal{V}_m)_m$ cannot converge to any point, so the sequence $(\mathcal{U}_n)_n$ cannot converge to any point either.     I would like to know that:      (1) Why "$B_n \in \mathcal{V}_m$  iff $m=n$"? And " $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even "?      (2)$B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even Connecting GSM SIM900A module with Ardunio I have bought an SIM900A Mini V4 GSM module. Here is the module,   I want to configure with arduino to send and receive SMS.   Do I have to connect TXD of this module--Tx of Arduino &amp; RXD of this module--Rx of Arduino or other way around?  There is no datasheet or documents on this new module.  When I search about RXD &amp; TXD, I found this in      TxD means "Data to be transmitted over the comms link" - which will clearly be an input to the modem;    Similarly, RxD means "Data that has been received from the comms link" - which will clearly be an output from the modem.    But I couldn't send or receive SMS using this configurations.  As I am new to this, It would be very helpful if anyone help on this.  Thank you in advance. Group theoretic meaning of natural isomorphisms between certain functors So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to  $G$ specified by  $\forall g$ $\in G$ $\exists$   $\phi(1)=g$. Each of these homomorphisms is in 1-1 relation with the elements of $G$.  Consider groups for a moment in the context of category theory, viewing the groups as one object categories and thereof group homomorphism as functors. What would a natural isomorphism mean in group theory terms for G?  My thoughts are that the natural transformations equate to maps between homomorphisms and since the homomorphisms are in 1-1 relation with the elements of G, then a natural transformation is a map between group elements, i.e another element of the group. I'm not entirely convinced by this.   Anyway continuing my reasoning a natural isomorphism is a map between the homomorphisms that has an inverse, the thing is in relation to the elements of G this is every element? Furthermore I'm looking to find an equivalence class defined by this natural isomorphism, is this just a collection of the elements that are each others inverses?  Thanks in advance for any assistance. Normality of dependent variable = normality of residuals? This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!).   The assumptions of general linear models (t-test, ANOVA, regression etc.) include the "assumption of normality", but I have found this is rarely described clearly.   I often come across statistics textbooks / manuals / etc. simply stating that the "assumption of normality" applies to each group (i.e., categorical X variables), and we should we examining departures from normality for each group.   Questions:     does the assumption refer to the values of Y or the residuals of Y?  for a particular group, is it possible to have a strongly non-normal distribution of Y values (e.g., skewed) BUT an approximately (or at least more normal) distribution of residuals of Y?   Other sources describe that the assumption pertains to the residuals of the model (in cases where there are groups, e.g. t-tests / ANOVA), and we should be examining departures of normality of these residuals (i.e., only one Q-Q plot/test to run).  does normality of residuals for the model imply normality of residuals for the groups? In other words, should we just examine the model residuals (contrary to instructions in many texts)?  To put this in a context, consider this hypothetical example:   I want to compare tree height (Y) between two populations (X).   In one population the distribution of Y is strongly right-skewed (i.e., most trees short, very few tall), while the other is virtually normal  Height is higher overall in the normally distributed population (suggesting there may be a 'real' difference). Transformation of the data does not substantially improve the distribution of the first    population.  Firstly, is it valid to compare the groups given the radically different height distributions? How do I approach the "assumption of normality" here? Recall height in one population is not normally distributed. Do I examine residuals for both populations separately OR residuals for the model (t-test)?     Please refer to questions by number in replies, experience has shown me people get lost or sidetracked easily (especially me!). Keep in mind I am not a statistician; though I have a reasonably conceptual (i.e., not technical!) understanding of statistics.  P.S., I have searched the archives and read the following threads which have not cemented my understanding: would this be considered a 'methodology' or a 'method'? Say I developed something called "Methodology for Establishing Remaining Life of Components". Basically, at work, we have components (or equipment). I wanted to develop a way to identify how much longer the components / equipment can last before they ware out and cannot be used.  It takes into account "the degree of damage currently in the component, the rate of future damage accumulation based on realistic operating scenarios, and the degree of damage required to cause failure." Taking these three into account, the remaining life of a component can be assessed. This is what I call "Methodology for Establishing Remaining Life of Components".   My question is, would this be considered a 'methodology for establishing remaining life of components' or just a 'method for establishing remaining life of components'? Or do you need more information to figure out if it is a method or methodology?   I cannot provide all the information for business purposes, but if additional  information is required to figure out if it is a method or methodology, I will provide it. in the street or on the street Why do German umlauts and sharp s (ß) work without a package in LaTeX? I am used to the fact that LaTeX will only know letters like ä, ö, ü, or ß if I use packages like  \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage[T1]{fontenc}   Today, I tried LaTeX on a Mac for the first time, using the MacTeX distribution and texmaker. Without writing down any packages, all umlauts and ß were accepted correctly from scratch. Why is that and will the usage of special letters without using the packages cause any problems? JQuery to load Javascript file dynamically Labeling stacked polygons with different records I work in Oil and Gas and trying to label leases is a huge pain.  In most cases there are multiple polygons of the exact same size and shape stacked on each other with records that represent different lessees.    The number of lessees per section is unlimited.  How can i label these polys so that they look like the annotation in the picture below. Optimally the labels would be sorted on the date field as well. This label is also an expression containing the Net Acres and the Expiration Date. resizing a BUTTON through CSS What does capital letter I mean in this formulas? Hello I don't understand meaning of the symbol represented by a capital letter "I" in the formulas below. There is  to the paper with the formulas.    At the beginning I thought it mean "where c is not equal to the result of classifier T(m)", however in step (d) there is a multiplication symbol before it, so it probably mean that "I" should "return" a value. Find out which process is locking a file or folder in Windows How can I find out which process is locking a file or folder in Windows?  For instance, when trying to delete a folder, Windows reports this:     The action can't be completed because the folder is open in another program   Happens the same with a file, but how do I find out what program or application is currently using it and preventing me from deleting the file or folder? Determining possible minimal polynomials for a rank one linear operator I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method.  I think that the facts needed to solve this problem come from the section on Nilpotent operators from Hoffman and Kunze's "Linear Algebra".     Question: Let $V$ be a vector space of dimension $n$ over the field $F$ and consider a linear operator $T$ on $V$ such that $\mathrm{rank}(T) = 1$.  List all possible minimal polynomials for $T$.   Sketch of Proof:   If $T$ is nilpotent then the minimal polynomial of $T$ is $x^k$ for some $k\leq n$.  So suppose $T$ is not nilpotent, then we can argue that $T$ is diagonalizable based on the fact that $T$ must have one nonzero eigenvalue otherwise it would be nilpotent (I am leaving details of the proof of diagonalization but it is the observation that the characteristic space of the nonzero eigenvalue is the range of T and has dimension $1$).  Thus the minimal polynomial of $T$ is just a linear term $x-c$.    Did I make a mistake assuming that $T$ can have only one nonzero eigenvalue?  Thanks for your help How to add a subclass as a subview so that background of subview is set in front of superview What is this strange function definition syntax in C? Update Windows 8 Enterprise to Windows 8.1 Enterprise KMS activation So it is shown that volume license packages of Windows 8, like Pro and Enterprise, are unavailable for upgrading.  I received a link last night to download the Windows 8.1 Enterprise 90-day Evaluation, but it will replace my current Windows 8 Enterprise version.  Is it known when these versions will become available for upgrade? I've been waiting patiently for weeks to get this upgrade.  Thanks.