I'm currently learning mathematical induction from this site:    What I'm confused about is how it presents mathematical induction. It says that there are 3 steps to induction:   Show it true for $n=1$. Assume it is true for $n=k$ Prove it is true for $n=k+1$ (we can use the $n=k$ case as a fact.)   There are many things I am confused about here, about all $3$ steps.  Starting from the first step, why is it necessary to prove it true for $n=1$? I don't get why this step is needed. Second, why choose $1$ of all numbers; can't a number like $2$ be chosen?   Moving on to the second step, why is it legitimate to assume it true for all $n=k$? Is this assumption proved true by the third step, if so, how?   On the final step, first, how can we prove it true for all $n=k+1$ ? Because this prove fundamentally assumes that it is true for $n=k$, but there is no way to verify this. Second, what happens if the set we're doing induction on has only a limited amount of numbers, let's say 100 numbers. So if we go up to the 100th number, then how can $n=k+1$ still be true? Because there is no 101st term for it to be true on; there are only 100 numbers!  Please explain this as simply as possible; I'm still a beginner. I will not be able to understand complicated proof notation.  DETAILS: This question is different from the question above since the question above uses $\lim_{x\to\infty}$ notation and other pieces of calculus knowledge. However, I have not learnt calculus, and nothing about this question suggest prior calculus knowledge. An answer where induction is explained without calculus would benefit me greatly.
I've never really understood why math induction is supposed to work.  You have these 3 steps:   Prove true for base case (n=0 or 1 or whatever) Assume true for n=k.  Call this the induction hypothesis. Prove true for n=k+1, somewhere using the induction hypothesis in your proof.   In my experience the proof is usually algebraic, and you just manipulate the problem until you get the induction hypothesis to appear.  If you can do that and it works out, then you say the proof holds.  Here's one I just worked out,  Show $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^n}{x} = 0$  So you go:   Use L'Hospital's rule. $\displaystyle\lim_{x\to\infty} \frac{\ln x}{x} = 0$. Since that's $\displaystyle\lim_{x\to\infty} \frac{1}{x} = 0$. Assume true for $n=k$. $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^k}{x} = 0$. Prove true for $n=k+1$.  You get $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^{k+1}}{x} = 0.$   Use L'Hospital again: $\displaystyle\lim_{x\to\infty} \frac{(k+1)(\ln x)^{k}}{x} = 0$.  Then you see the induction hypothesis appear, and you can say this is equal to $0$.  What I'm not comfortable with is this idea that you can just assume something to be true ($n=k$), then based on that assumption, form a proof for $n=k+1$ case.  I don't see how you can use something you've assumed to be true to prove something else to be true.
Find the derived subgroup of $A_4$.   Since it is $A_4$, for a permutation $\sigma$ to be in $A_4$, $\sigma$ must have a cycle structure of $2$ cycles. Therefore, $\sigma=(ab)(cd)$.  The commutator of such elements, would be obviously another permutation of $2$-cycle (of length $2$).  Does it mean that $A'_4=A_4$? How can I develop my Android game for different phone resolutions? Inheritance of static members in PHP In PHP, if a static attribute is defined in the parent class, it cannot be overridden in a child class. But I'm wondering if there's any way around this.   I'm trying to write a wrapper for someone else's (somewhat clunky) function. The function in question can be applied to lots of different data types but requires different flags and options for each. But 99% of the time, a default for each type would suffice.  It would be nice if this could be done with inheritance, without having to write new functions each time. For example:  class Foo {     public static $default = 'DEFAULT';      public static function doSomething ($param = FALSE ) {         $param = ($param === FALSE) ? self::$default : $param;         return $param;     } }  class Bar extends Foo {     public static $default = 'NEW DEFAULT FOR CHILD CLASS'; }  echo Foo::doSomething() . "\n";  // echoes 'DEFAULT'  echo Bar::doSomething() . "\n";  // echoes 'DEFAULT' not 'NEW DEFAULT FOR CHILD CLASS'  // because it references $default in the parent class :( List of CSS features not supported by IE6 Which of these sentences is correct?     Rupees are interchangeable with dollars   or      Rupees is interchangeable with dollars. I want to output two different views (one as a string that will be sent as an email), and the other the page displayed to a user.  Is this possible in ASP.NET MVC beta?  I've tried multiple examples:  1.      If I use this example, I receive the "Cannot redirect after HTTP   headers have been sent.".   2.      If I use this, I seem to be unable to do a redirectToAction, as it   tries to render a view that may not exist. If I do return the view, it   is completely messed up and doesn't look right at all.   Does anyone have any ideas/solutions to these issues i have, or have any suggestions for better ones?  Many thanks!  Below is an example. What I'm trying to do is create the GetViewForEmail method:  public ActionResult OrderResult(string ref) {     //Get the order     Order order = OrderService.GetOrder(ref);      //The email helper would do the meat and veg by getting the view as a string     //Pass the control name (OrderResultEmail) and the model (order)     string emailView = GetViewForEmail("OrderResultEmail", order);      //Email the order out     EmailHelper(order, emailView);     return View("OrderResult", order); }   Accepted answer from Tim Scott (changed and formatted a little by me):  public virtual string RenderViewToString(     ControllerContext controllerContext,     string viewPath,     string masterPath,     ViewDataDictionary viewData,     TempDataDictionary tempData) {     Stream filter = null;     ViewPage viewPage = new ViewPage();      //Right, create our view     viewPage.ViewContext = new ViewContext(controllerContext, new WebFormView(viewPath, masterPath), viewData, tempData);      //Get the response context, flush it and get the response filter.     var response = viewPage.ViewContext.HttpContext.Response;     response.Flush();     var oldFilter = response.Filter;      try     {         //Put a new filter into the response         filter = new MemoryStream();         response.Filter = filter;          //Now render the view into the memorystream and flush the response         viewPage.ViewContext.View.Render(viewPage.ViewContext, viewPage.ViewContext.HttpContext.Response.Output);         response.Flush();          //Now read the rendered view.         filter.Position = 0;         var reader = new StreamReader(filter, response.ContentEncoding);         return reader.ReadToEnd();     }     finally     {         //Clean up.         if (filter != null)         {             filter.Dispose();         }          //Now replace the response filter         response.Filter = oldFilter;     } }   Example usage  Assuming a call from the controller to get the order confirmation email, passing the Site.Master location.  string myString = RenderViewToString(this.ControllerContext, "~/Views/Order/OrderResultEmail.aspx", "~/Views/Shared/Site.Master", this.ViewData, this.TempData); Effect of space time relativity on the age of the universe? So we all heard about the twins paradox to explain einstein's time space relativity.  Wikipedia Quote :" In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as traveling, and so, according to a naive application of time dilation, each should paradoxically find the other to have aged more slowly. "  So what if the universe has been travelling at varying speeds (increasing or descreasing), wouldn't this effect our measurements on age of the universe? Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant. While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated.  Let $U ⊂ \mathbb{C}$ be a domain and $f_1, . . . , f_n : U \rightarrow \mathbb{C}$ holomorphic functions, such that $\sum_{k=1}^n \left| f_k(z) \right|$ is constant on $U$. Show that $f_1, . . . , f_n$ are constant. I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this:  $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$  $g_{ii}$ are given, from metric, but that's not important. I need to assume that the solution (since I'm looking for components of $\xi^\mu$, which is a vector with components $\xi^t,\xi^r,\xi^\phi$) is given with power series of the form:  $\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$, and this is to be seen as expansion around $1/r$ (expansion around $r=\infty$).  Now when I plug this in the ODE I get this  $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where   $\xi^\mu_{n,i}$   is the derivative with the respect to i-th component.   What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my $\mathcal{O}(r)$ terms cancel each other out? Or?  I'm kinda stuck :\ Unity remove logo in splash screen I have the message:   Error Broken count &gt;0 Check if you are using third party repositories. If so disable them, since they are a common source of problems. Furthermore run the following command in a Terminal: apt-get install -f Details: The following packages have unmet dependencies: firefox-globalmenu: Depends: firefox (= 19.0+build1-0ubuntu0.12.04.2) but 19.0+build1-0ubuntu0.12.04.1 is installed   Any suggestions? The system L0 is defined as follows:  Axioms: A1 (α → (β → α))  A2 2. (α → (β → γ) → ((α → β) → (α → γ))  A3 ((¬β → ¬α) → ((¬β → α) → β))  In one of my problem sheets, I am told that I am allowed to use the following theorem: if   ⊢ (α → α)    ⊢ (α → α) → (α → α)  My attempt to solve/prove the second one  ⊢ (α → α) → (α → α) choose 2 axioms which match.  α →α   β→α  γ→α  (α → (α → α) → ((α → α) → (α → α))  I'm struggling to see an obvious way of proving the theorem.  Is it provable? And if so, can you point me in the right direction?  Thanks a lot! How do I run two commands in one line in Windows CMD? What exactly is tested when a fire extinguisher is tagged? I am just curious: the obvious is that the  pressure mass   can be inspected /measured to ensure that the device has not been discharged. Is there anything else that is measured \ checked? Using processing.runandload in QGIS 3 Why was Sirius ever convicted? How does Hextech Crafting work? How do you earn free loot?  What's the deal with essences and why are they so important?  What do the different options mean when dealing with Redeeming Rerolling Disenchanting and Upgrading?  How does it all work? Question:      Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$.    Attempt:   Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, \ldots, \alpha_{p^2 - 1}\}$ and considering $$\begin{align}\tau_{jk} : \mathbb Z_p \times \mathbb Z_p&amp;\to \mathrm {Aut} (\mathbb Z_p)\,\\e &amp;\mapsto id\\\alpha_j &amp;\mapsto id \\\alpha_i &amp;\mapsto \rho_k  \end{align}$$  $\forall i \neq j$ where $\rho_k : \mathbb Z_p \to \mathbb Z_p$ is such that $\rho_k (1) = k$, with $k \in \{2, \ldots, p-1\}$. Then we take the group $(\mathbb Z_p \times \mathbb Z_p) \ltimes_{\tau_{jk}} \mathbb Z_p$.   Then $$(\alpha_i , 2) \ltimes_{\tau_{jk}} (\alpha_i, 1) = (2\alpha_i, 1  + \tau_{jk} (\alpha_i)(2)) = (2\alpha_i , 1  +2 k)$$  and $$(\alpha_i , 1) \ltimes_{\tau_{jk}} (\alpha_i, 2) = (2\alpha_i, 1  + \tau_{jk} (\alpha_i)(2)) = (2\alpha_i , 1  +k)$$  taking $k = 1$ for example we have that the group $(\mathbb Z_p \times \mathbb Z_p) \ltimes_{\tau_{jk}} \mathbb Z_p$ is not abelian.    I was wondering if this is a good aproach. If not, is there an easier way to reach the result?  If this is the "best" way then would the higher cases ($n &gt; 3$) be more of a notation play game? How to choose between learning algorithms I need to implement a program that will classify records into 2 categories (true/false) based on some training data, and I was wondering at which algorithm/methodology I should be looking at. There seem to be a lot of them to choose from -- Artificial Neural Network, Genetic Algorithm, Machine Learning, Bayesian Optimization etc. etc., and I wasn't sure where to start. So, my questions is:  How should I choose a learning algorithm I should use for my problem?  If this helps, here is the problem I need to solve.    The training data: The training data consists of many rows like this:    Precursor1, Precursor2, Boolean (true/false)   The run I will be given a bunch of precursors. Then,     I choose an algorithm A from different algorithms (or dynamically generate an algorithm), and apply it on every possible combinations of these precursors and collect the "record"s  that are emitted. The "record" consists of several key-value pairs*. I apply some awesome algorithm and classify these records in to 2 categories (true/false). I will generate a table that has the same format as the train data: Precursor1, Precursor2, Boolean   And the whole program is scored based on how many true/false I got right.  *:"Record"s will look like this (hope this makes sense)  Record         [1...*] Score -Precursor1             -Key -Precursor2             -Value   There are only a finite number of possible Keys. Records contain different subset of these keys (some records have key1, key2, key3... other records have key3, key4... etc.).  I actually need 2 learning. One is for step 1. I need to have a module that looks at the Precursor pairs etc. and decide what algorithm to apply in order to emit a record for the comparison. Another is for step 2. I need a module that analyzes the collection of records and categorize them into the 2 categories (true/false).    Thank you in advance! java.io.IOException: An existing connection was forcibly closed by the remote host Upon entering any given minecraft server, after a minute or two I am forced out of the server, and I'm given the following error:  java.io.IOException: An existing connection was forcibly closed by the remote host.  What does this mean, and how can I prevent it from happening again?