I'm using Rudin, and in its proof of the implicit function theorem, it uses the inverse function theorem. I've heard that you can prove the inverse function theorem as a consequence of the implicit function theorem, but I can't figure out how to do this. Does anyone have some ideas how to start?
I can prove the converse of it, but I cannot do this one. Here is the problem:     Prove that the implicit function theorem implies the inverse function theorem.
Why is HMAC-SHA1 still considered secure? This Q &amp; A   says that the security of HMAC-SHA1 does not depend on resistance to collisions? Are they are saying specifically with respect to HOTP or for HMAC-SHA1 for any use?     If HMAC-SHA1 is used for verifying the integrity of a document and you can find a collision, you can make changes in the document, right?      Even the RFC says that resistance to collisions is relevant       The security of the message authentication mechanism presented here depends on cryptographic properties of the hash function H: the resistance to collision finding (limited to the case where the initial value is secret and random, and where the output of the function is not explicitly available to the attacker), and the message authentication property of the compression function of H when applied to single blocks    What am I missing here? Proof that no polynomial with integer coefficients can only produce primes Doing a discrete math review and am trying to solve problem 1.6 in the text found here:  - I believe I've gotten parts (a) and (b) correctly, but (c) is a bit tricky for me. Would appreciate a review of a/b and a hint for c if possible. Restatement here:  For n = 40, the value of polynomial $p(n)::=n^2 + n + 41$ is not prime, as noted in Section 1.1. But we could have predicted based on general principles that no nonconstant polynomial can generate only prime numbers.  In particular, let $q(n)$ be a polynomial with integer coefﬁcients, and let $c::=q(0)$ be the constant term of q.  (a) Verify that $q(cm)$ is a multiple of c for all $m \in Z$  Proof: Since q is a polynomial, it will be of the form: $q(x) = a_nx^n+a_{n-1}x^{n-1} + ... + a_1x + c$  If $x = cm$, then notice that cm will be inside every term of the polynomial, and since the final term is c, c can be factored out of every term. Hence, q(cm) is a multiple of c for all $m \in Z$.  b) Show that if q is nonconstant and c > 1, then as n ranges over the nonnegative integers, $N$, there are infinitely many $q(n) \in Z$ that are not primes.  Proof: Continuing from the result found in (a), it is easy to see that for all $m \in Z$, there will be an infinite number of multiples of c that will be generated by q. Since c > 1, these multiples are guaranteed not to be prime.  c) Conclude that for every nonconstant polynomial, q, there must be an $n \in N$ such that $q(n)$ is not prime. Hint: Only one easy case remains.  c = 0 is trivially easy to prove using the (b) above.  I assume the "easy" case is referring to c = 1, but in this case I'm not sure how to continue, since the result of (a) and (b) don't apply: I can't use them since adding 1 to any even number may make it prime. If the result of the terms not including c is odd, then adding 1 to that result makes it even and it is not prime. However, if the terms add up to an even number, I don't see a way of using the knowledge I have so far to prove conclusively that that number + 1 will NOT in fact be prime. Why is super.super.method(); not allowed in Java? I read  and thought that would easily be solved (not that it isn't solvable without) if one could write:   @Override public String toString() {     return super.super.toString(); }   I'm not sure if it is useful in many cases, but I wonder why it isn't and if something like this exists in other languages.  What do you guys think?  EDIT: To clarify: yes I know, that's impossible in Java and I don't really miss it. This is nothing I expected to work and was surprised getting a compiler error. I just had the idea and like to discuss it. Managing constructors with many parameters in Java In some of our projects, there's an class hierarchy that adds more parameters as it goes down the chain. At the bottom, some of the classes can have up to 30 parameters, 28 of which are just being passed into the super constructor.  I'll acknowledge that using automated DI through something like Guice would be nice, but because of some technical reasons, these specific projects are constrained to Java.   A convention of organizing the arguments alphabetically by type doesn't work because if a type is refactored (the Circle you were passing in for argument 2 is now a Shape) it can suddenly be out of order.  This question might be to specific and fraught with "If that's your problem, you're doing it wrong at a design level" criticisms, but I'm just looking for any viewpoints. The words in the node of tree I want to plot this graph, but I don't know how to code the words in the nodes.     Here is my code:  \documentclass{article} \usepackage{amsmath} \usepackage{titletoc} \usepackage{tikz} \usetikzlibrary{trees,positioning,fit}  \begin{document}{\allowdisplaybreaks[4]}  \tikzstyle{bag} = [text width=2em, text centered] \tikzstyle{end} = []  \begin{center} \begin{tikzpicture}[sloped] \node (a) at ( 0,0) [bag] {$\ln r_0$}; \node (b1) at ( 3,1.5) [bag] {$\ln r_{10} + \sigma_1\sqrt{\Delta t_1}$}; \node (b2) at ( 3,-1.5) [bag] {$\ln r_{10}$}; \node (c1) at ( 6,3) [bag] {$\ln r_{20} + 2\sigma_2\sqrt{\Delta t_2}$}; \node (c2) at ( 6,1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c3) at ( 6,0) [bag] {$\ln r_{20} + \sigma_2\sqrt{\Delta t_2}$}; \node (c4) at ( 6,-1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c5) at ( 6,-3) [bag] {$\ln r_{20}$}; \draw [-&gt;] (a) to node [below] {} (b1); \draw [-&gt;] (a) to node [above] {} (b2); \draw [-&gt;] (b1) to node [below] {} (c1); \draw [-&gt;] (b1) to node [below] {} (c2); \draw [-&gt;] (b1) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c4); \draw [-&gt;] (b2) to node [below] {} (c5); \end{tikzpicture} \end{center}  \end{document}       Update code use  \draw [-&gt;] (a) to node [below] {} (b1.south east); \draw [-&gt;] (a) to node [above] {} (b2.north east); \draw [-&gt;] (b1.south east) to node [below] {} (c1.west); \draw [-&gt;] (b1.south east) to node [below] {} (c2.west); \draw [-&gt;] (b1.south east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c4.west); \draw [-&gt;] (b2.north east) to node [below] {} (c5.west); Function overloading in C How to bypass "Verify your account" (Factory Reset Protection) I followed the beginner steps for creating a .  The child theme was created and it appears as an option on my Themes page. But unlike the other Themes on the page (including its parent), there is no preview.  The other themes ("Twenty Twelve," "Twenty Thirteen," and "Twenty Fourteen") are all listed with an image preview.  My child theme ("Twenty Fourteen Child") does not have any image. The child theme can be applied without issue.  This issue arises when I create child themes from other themes as well. Any ideas why this is happening and if I can resolve? Windows keeps making these random "device connect" and "device disconnect" sounds Throughout the day I notice that my computer makes apparently random device-connect and/or device-disconnect ("boink") sounds.    I suppose it is the same sound you hear when connecting or disconnecting a USB device such as a thumb drive.  I've noticed that this happens on each of three computers I work with at home, my wife's computer, and my machine at work.  It happens without any user action at all - i.e. I'll be just sitting there (hands off my mouse and keyboard), and the computer will make the sound.  There is no visual queue or anything.  Just the sound.  I have sometimes gone in pursuit of the sound - running virus scans, examining event logs and such, and observing task manager - but have never had any luck tracking this thing down, but have not had any luck.  I've been noticing this since Windows Vista and now the sounds have followed me through to Windows 7.   Surely someone else out there must be experiencing this and can explain what is happening. I'm a physics teacher and I would like to find a simulation software that could be used to give exact numerical answers to high school level problems in Newtonian mechanics.  I want the students to first calcultate a numerical answer to a physics problem and then use the simulator to visualize the problem in order to understand it better and verify that the simlator gives the same answer that they got when doing the calculation themselves.  Here are some examples of what kind of problems I want the students to be able to solve with the simulator:   A slope with a specific angle of inclination and a ball starts rolling at a certain point. Determine the speed, angular momentum and total mechanical energy of the ball after a specific time t. A ladder of known mass and length is leaning against the wall at a specific angle. The contact between the ladder and the wall is frictionless. Calcultate the x and y components of the force between the floor and the ladder.   I would like the software to have the following features:  - Simulation in 2D. No 3D simulations are necessary and the software should at least have the option of doing simulations only in 2D.  - Position all objects exactly by writing numbers (just positioning the objects using the mouse is not precise enough).  - Show animations of the time evolution of the system.  - Show forces, velocities and accelerations as arrows.  - Give exact numerical results so that the students can use the software to verify their calculation.  Ideally the software would have a graphical user interface, but if no suitable software with graphical interface exixts, I could also consider a software with text-based inputs. I would consider both free and commercial software.  My question is: Is there any software that would meet these requirements and if there is, where could I get it? Why is "union" an exception to the "a/an" rule? Lipschitz Condition and Optimization I am reading paper on . I am trying to understand the following statement:     I would like to ask how the Ball doesnot contain the minimum of f? I would appreciate if this could be graphically shown something like following taken from Bayesian Optimization and Data Science There are two syntaxes for command substitution: with dollar-parentheses and with backticks. Running top -p $(pidof init) and top -p `pidof init` gives the same output. Are these two ways of doing the same thing, or are there differences? How to check if a 2D point is in a polygon? Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ I have an assignment, that I cannot seem to find a starting point for. I am not looking for the specific answer, but only where to begin. Any pointers are appreciated.  I have to prove the inequality $\vert \tanh(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ where $x-\frac{x^3}{3}$ is the 3rd order Taylor expansion.  I've tried with the triangle inequalities, Lagrange remainders, and a lot of other stuff, but I am no where near a correct argumentation. Does anyone have any suggestions? Canadian visa rules for flight connections I know that in the US foreigners must have a US visa if they have a connection in the US, even if they fly from a different country to a different country. That is because in the US you have to go through passport control regardless of your final destination.  What is the situation in Canada? Does a person who would need a visa to enter Canada need a visa to connect through Canada without exiting the airport? "Member for: today" Very new members will have this text displayed in their profile:    Should this not be changed to say "1 day" or "3 hours", etc? Especially as the "last seen" also works in this way.  Note that members who joined 24 hours ago have been members for "2 days", etc.  You'd say "x has been a member for 1 day" over "x has been a member for today"; saying "member for today" just doesn't feel right to me. Change the default-view of Media Library in 3.5? When inserting media into a post, is there a way to change the default-view of the Media-Library from "All media items" to "Uploaded to this post"?    There is another thread where this question was extracted from: How to see data from .RData file? I'm facing this frustrating "feature" that I can't seem to remedy; when I duplicate an object (and now, even when I create two completely new objects), their transformations/scaling is synchronized. This is extremely annoying when trying to use any sort of transformation keyboard shortcut.  It should be noted by going to the properties menu (N) and manually setting the properties works just fine.  For instance, here is just the camera selected (trust me, I made sure):   And when I try to transform by clicking+dragging on the X axis OR by hitting G->X, this happens:   They are not grouped (verified by going to the Groups list up in the Scene panel) and they do not have parents (verified by going to Object->Relations).  I have tried the unlink command as well, to no avail.  I have seriously run out of options; what did I change to have my G->X shortcuts transform multiple objects? It seems completely at random, too; certain objects seemed to be "linked" to other random objects, and this "linking" is guaranteed when duplicating even when I do a non-linking duplicate (Shift+D).  What gives?