I'm sure some (if not most) have seen this by now and since I'm fairly new to real deep mathematical explorations I'm stumped as to why this would be true.   For class I was asked to cut out an 8" x 8" square and then asked to cut the square into 2 right triangles each of base height 3" and base 8" and two trapezoids each of base1 3" and base2 5" with height 5".   I was then asked to rearrange the shapes to make an 5" x 13" rectangle and sure enough the pieces fit snuggly. Now 5" x 13" will not give us the same area that we initially had. Measuring the pieces separately one would get that each trapezoid has an area of 20 inches squared and each triangle has an area of 12 inches squared.    Any explanations please, thank you!
how is this possible? I know there is some trick, should someone please explain?!
Epson LX-310 not working after Cumulative Update 2017-11 Windows 10 I have Windows 10 Pro x32 and an Epson LX-310 printer that was working. Yesterday, Windows 10 automatically updated with 2017-11 Cumulative Update for Windows 10.   Today, I cannot print. The printer status is on. When I click "Print a test page", it quickly says the test page has been sent but the printer does not print it.   My printer is connected to a USB port. What wrong with my Windows 10? Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$ A former student of mine was TA-ing an algorithms class last quarter and asked students to solve this famous recurrence relation:  $$T(n) = \sqrt n T(\sqrt{n}) + n$$  There are  and this question has already been asked here.  I was talking to my student about this problem and mentioned that it's quite hard to solve and is pretty dependent on the particular choices being made. As an example, I suggested this variant of a recurrence relation that was less simple to solve:  $$T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$$  The TA and I tried solving this recurrence for about an hour and a half without making any progress. Here are a few things we tried:   My approach to solving the initial recurrence relation was to draw out a recursion tree and notice that each level of the tree contributes $n$ to the total and that there are $O(\log \log n)$ layers, so the recurrence solves to $O(n \log \log n)$. When I tried doing this here, I noticed that the work per layer was no longer constant; instead, the top layer sums to $\sqrt{n}$, the second layer to $n^{3/4}$, the third to $n^{7/8}$, etc. We got stuck working with the summation $n^{1/2} + n^{3/4} + n^{7/8} + ...$. We tried using the iteration method to unroll the recurrence. With the original recurrence relation, this works out nicely; here, we got stuck at the same summation given above.   I'm completely stuck trying to figure out how to solve this recurrence relation. There's nothing riding on it per se - I don't need to solve it for any particular reason - but the fact that we arrived at it by a straightforward modification of a common algorithms problem set question makes it all the more enticing.  Any idea how to solve this recurrence?  Thanks! For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$ For a polynomial $p(z)=a_0+\ldots+a_nz^n$, prove there exist $R&gt;0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$.  I get $|p(z)|\geq |a_n|R^n$, which makes that factor of $1/2$ useless. So I must have done something wrong.  $$|\frac{\partial^n{f}}{\partial z^n}(P)|\leq \frac{\sup_{z\in \overline{D}(P,r)} |f(z)|k!}{r^k}$$. This is the Cauchy estimate.   $$|\frac{\partial^n{p}}{\partial z^n}(z)| = |a_n n!|$$ $$|\frac{\partial^n{p}}{\partial z^n}(0)|\leq \sup_{\zeta\in \overline{D}(0,|z|)} |p(\zeta)|n!/R^n$$ $$|a_n n!|\leq \sup_{\zeta\in \overline{D}(0,|z|)} |p(\zeta)|n!/R^n$$ $$R^n |a_n|\leq \sup_{\zeta\in \overline{D}(0,|z|)} |p(\zeta)|$$  There exist a $z_0$, such that $|z_0|&gt;0$ and $$\sup_{\zeta\in \overline{D}(0,|z_0|)} |p(\zeta)| = |p(z_0)|$$.  This is saying there exist a disk centered at the origin, such that $|p|$ is maximum on some point on the boundary.   Let $R=|z_0|$, we have $R^n |a_n|\leq |p(z_0)|$  Edit: I know what I got wrong. $z_0$ is just one point on the circle $|z| = R$. One can chose another point $z_1$ on the circle, such that $p(z_1)&lt;p(z_0)$. Difference between Can/Could: "Can (could) you sell your old computer to me?" What is difference between  Could you sell your old computer to me, seeing (that) you have no use for it. Can you sell your old computer to me, seeing (that) you have no use for it.  Can I use either one of them? Which one is grammatically correct or if both are, what is the difference in meaning? Vertical placing of wide accents in superscript The following code  $\widetilde{aaa} \quad L^{\widetilde{aaa}}$  $\tilde{aaa} \quad L^{\tilde{aaa}}$  $\widehat{aaa} \quad L^{\widehat{aaa}}$  $\hat{aaa} \quad L^{\hat{aaa}}$  $\overline{aaa} \quad L^{\overline{aaa}}$  $\bar{aaa} \quad L^{\bar{aaa}}$   produces the following output:    What is the reason for additional vertical spacing for \widetidle and \widehat in superscript and what can one do about it? Seasonal Loan Amortization Hi I can compute a Basic Loan Amortization Schedule, But now I'm trying to Calculate a Seasonal Loan Amortization Schedule . So for example A loan starts at January And Will go on for 3 years, with Monthly Payments. Interest is at 3% Annually and is compounded Annually. During the months of September, October, November, December It'll only accumulate Interest But no payments and principal. Principal is 100000. How Do I calculate Payments and interest. How can I make things link url references, my index, table of contents, and cross references clickable links in documents produced by LaTeX? I generally use pdflatex to produce my documents. What is the difference between sector and cluster? I have gone through wikipedia and other sites and here's the information I have obtained.  Sector: Smallest Addressable Unit (mostly 512 bytes) Cluster: Smallest Allocation Unit (equals to 1 or more sectors and the number of clusters depends on the file system).  Let's say in the below example:  sector =512 bytes and a cluster =8 sectors=4096 bytes. Now, I am having a hard time figuring out, when I am trying to create a file what does the file system do? Does it consider 4096 bytes as the smallest write/read unit or 512 bytes?  If the answer is 4096 bytes: When the manufacturer of the disk drives has provided 512 bytes as the smallest addressable unit, why does the file system consider cluster as the smallest unit? I mean, what are the advantages in doing so? Two git repositories in one directory? Is it possible to have 2 git repositories in one directory?  I'd think not, but thought I'd ask.  Basically, I'd like to check in my home directory config files (e.g. .emacs) which should be common across all of the machines I work on, but have a second repository for local files (e.g. .emacs.local), which contains machine-specific configurations.  The only way I can think of to do that is to have the local config in a subdirectory and ignore that subdirectory from the main git repository.  Any other ideas? What should a main game loop do? Generally, what are the core things that one should do in the game loop, and what are some things that one shouldn't do in the game loop? I'm currently working on a web application which has a page which displays a single chart (a .png image).  On another part of this page there are a set of links which, when clicked, the entire page reloads and looks exactly the same as before except for the chart in the middle of the page.  What I want to do is when a link is clicked on a page just the chart on the page is changed.  This will speed things up tremendously as the page is roughly 100kb large, and don't really want to reload the entire page just to display this.    I've been doing this via JavaScript, which works so far, using the following code  document.getElementById('chart').src = '/charts/10.png';   The problem is that when the user clicks on the link, it may take a couple of seconds before the chart changes.  This makes the user think that their click hasn't done anything, or that the system is slow to respond.  What I want to happen is display a spinner / throbber / status indicator, in place of where the image is while it is loading, so when the user clicks the link they know at least the system has taken their input and is doing something about it.    I've tried a few suggestions, even using a psudo time out to show a spinner, and then flick back to the image.    A good suggestion I've had is to use the following  &lt;img src="/charts/10.png" lowsrc="/spinner.gif"/&gt;   Which would be ideal, except the spinner is significantly smaller than the chart which is being displayed.  Any other ideas? This problem is from Discrete Mathematics and its Applications  My question is on 9b. I know that the sign represents an element is a member of. (from book)  I know that the O with a slash across it is the empty set which &quot;is a special set that has no elements&quot;. From , I got that the empty set is a subset of all sets, meaning that every member of the empty set(nothing) is also a member of any other set. Based on all of this, for 9b, would {0} contain the empty set because it fundamentally has the elements that consist of the empty set(nothing) or does it physically have to have the empty set? Create model from XYZ data points I'm trying to create a 3D heat map model out of data that I have from a csv file. I have a few thousand data points with XYZ coordinates.  I'm able to import the data into blender as a point cloud with , however I would like to create a model with solid faces. Is there a way when importing the data to somehow join the points to create faces instead of generating thousands of separate points?  Currently, this what I have in terms of point cloud  The end product would look something like this (minus the texture) and obviously the model wouldn't be as regular as this one  Any help is greatly appreciated! Multinomial logistic regression analysis or binary logistic regression How to design a product table for many kinds of product where each product has many parameters Upgrade to the latest PHP version in Ubuntu A well-known application of the MVT is to prove that any $f: \mathbb{R} \to \mathbb{R}$ with $f'= 0$ is constant. But of course, the MVT relies fundamentally on the properties of $\mathbb{R}$, and in particular does not hold in $\mathbb{Q}$. A standard counter-example is the function $f :\mathbb{Q} \to \mathbb{Q}$ defined by $ f(x)=0 $ if $ x &lt;\sqrt{2}$, and $f(x)=1$ if $ x &gt; \sqrt{2}$, for this function is locally constant. There are several ways of thinking about the difference between $\mathbb{R}$ and $\mathbb{Q}$ : $\mathbb{R}$ is connected while $\mathbb{Q}$ is totally disconnected, closed bounded intervals are compact in $\mathbb{R}$, but not in $\mathbb{Q}$, $\mathbb{R}$ satisfies the least upper bound axiom, $\mathbb{Q}$ does not.   Now, I wanted to see how far one could push this counterexample - is there a function $f :\mathbb{Q} \to \mathbb{Q}$ such that $f'=0$ and $f$ is strictly increasing? Clearly, any slight modification of the usual counterexample where $f$ is locally constant will not do. Also, one can show that if $f$ is uniformly differentiable with derivative zero, it must be constant, even in $\mathbb{Q}$, so I knew I was looking for non-uniformly differentiable but still differentiable $f$.   Eventually, I came up with a construction, albeit not a very explicit one: the idea is that, since $\mathbb{Q}$ is countable, one can enumerate it $q_{1}, q_{2}, q_{3}, \cdots$, and then define $f: \mathbb{Q} \to \mathbb{Q}$ inductively and recursively. If one could define $f$ in such a way that $ \forall \ q_{n} \in \mathbb{Q} \ \exists \ c_{n} \in \mathbb{Q} : \forall \ x \in \mathbb{Q} \ |f(x)-f(q_{n})| \leq c_{n}|x-q_{n}|^2$, and $f$ is strictly increasing, we would be done. The recursive definition essentially defines $f(q_{n+1})$, after having defined $f(q_{1}),f(q_{2}),f(q_{3}), \cdots , f(q_{n})$, in such a way that these inequalities hold and $f$ is strictly increasing, and at each step the new $c_{n}$ is chosen large enough so that these inequalities do not become incompatible at subsequent steps ( I am willing to provide more details of this construction if asked).  Note that, as $n \to \infty$, we must have $c_{n} \to \infty$, so that $f$ is not uniformly differentiable. Interestingly, the only key fact about $\mathbb{Q}$ used in the construction is that it is countable, so in a sense the construction provides a proof that $\mathbb{R}$ is uncountable, given the MVT.  While the construction works, it is not very simple nor explicit, which is perhaps to be expected, and the function must be evaluated point by point in the order given by the enumeration of $\mathbb{Q}$. So I was wondering if anyone else knows or can come up with a simpler construction of a function $ f : \mathbb{Q} \to \mathbb{Q}$ such that $f'=0$ and $f$ is strictly increasing. Any thoughts and ideas are welcome.  Edit: I've asked the same question on MO: Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$ How should I type author names in a bib file? Can someone please explain in a clear and concise way how am I supposed to enter the names of authors in a BibTeX file? I have entries with something like   @article{someid,    author = {Juan A. Navarro P\'erez, Ra\'ul de la Garza and Andr\'es Espinoza},    ...   but somehow the output seems kind of wrong! Auto-run scripts security options