Can you construct a 1-1, onto map from [0,1] to (0,1]? I know there are similar questions being asked but I have not found any help or advice for this specific 1-1, onto map and am rather stuck. Thank you.
How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective.
How can tangential acceleration from a radial force be explained? A mass is attached to a rope, and put into a circular motion. If I pull the string from the center, the tangential speed of the mass will increase (by conservation of angular momentum).   I am applying a force only in the radial direction, so how can the tangential velocity increase if there is no tangential force? The  lists various combinatorial objects that are described by them. I posit that there might be bijections between these various combinatorial objects. For some of them (like Dyck paths, correctly matched parentheses and paths from the bottom left to top right of a $2n \times 2n$ grid) they are quite obvious.   I was then trying to find a bijection between the number of Dyck words and the number of ways of associating $n$ applications of a binary operator to $n+1$ items (third one on the list). I attempted to do this for a simple case ($n=3$ which is the example provided in the Wikipedia article). However, couldn't find one after multiple hours. Is it reasonable to expect such a bijection will exist? If so, how do we go about finding it?    EDIT: In addition to the very nice answer by @Marc, the following page also helped me see the bijection:   "Let $P$ be the Dyck path and $f(P)$ be the binary tree. If you go up in the Dyck path, create a left child. Otherwise, go up one vertex until creating a new right child is possible and create one."    Here is one of my attempts:   Number of Dyck words with length $2 \times 3$ is $\frac{6 \choose 3}{4} = 5$. They are:  hhhttt; hhthtt; hhttht; hthhtt; hththt  And the number of applications of a binary operator among $3+1=4$ factors is:  ((ab)c)d;     (a(bc))d;     (ab)(cd);     a((bc)d);     a(b(cd))  Both combinatorial objects have been arranged in a way that there is some kind of ordering between them. For the Dyck words for example, if h equals +1 and t equals -1, then the order is lexicographical from the left to the right of the cumulative score along the sequence.   Now, the first and last characters of the Dyck words are always h and t respectively. So, we can ignore them. We are left with:  hhtt; htht; htth; thht; thth  I tried to start from the left of the sequence abcd and if I see 'h', merge the character with the one to its right. This approach didn't produce a valid mapping from the third Dyck word to the third binary operator precedence order. Is it possible to use Matcap materials as an actual material for a model so that I can render with it? There are certain parts that I want to apply specific Matcaps textures to but I want to actually use them to render and not just for sculpting purposes.  Is this at all do-able? Looks like there is something wrong with reputation calculation:        +2*5 -6*2 = -2...   why there is +6 in reputation popup?  As @Catija said in comment calculation is OK in the profile:        Perhaps the issue is in the top-bar. Is globbing a feature of the shell? Object Positions Reset On Blender Load When a vehicle makes a turn, the two front wheels trace out two arcs as shown in the figure below. The wheel facing towards the inside of the turn has a steering angle that is greater than that of the outer wheel. This is necessary to ensure that both front wheels smoothly trace out two arcs, which have the same center, otherwise the front wheels will skid on the ground during the turn.   During a turn, do the rear wheels necessarily trace out the same arcs as the front wheels? Can't redirect cut output Is Autopilot Openstack 16.04 available? For a 2x2 matrix A satisfying $A^k=I$, compute $e^A$  Oh, the exponential of a matrix is: $e^A=\sum_{i=0}^\infty\frac{1}{i!}A^i$    I thought I'd solved the $e^A$ form but I actually did something really silly, now I'm a little stuck.  If $A^k=I$ then $A^{k+1}=A$ and we have a cycle forming, so we will get:  $$\sum^{k-1}_{i=0}\sum^{\infty}_{j=1}\frac{1}{(kj+i)!}A^i$$  (Or something like this form, I don't have paper to hand and just spotted that now)  Is this what the question wants? It doesn't use the 2x2 property.  I believe the answer lies in finding an expression for the inner summation, I can do this for two terms (think of $e+e^{-1}$ all the odd power leave) When I am in texture mode one of my objects looks like it is in solid mode. The model is textured like all other models in the scene, but it doesn't look like it is in texture mode.  Screenshot will explain better:      What this model looks like. What all models should look like in texture mode. Allow converting comments to answers I have a Schengen visa valid from 19/07/2016 to 10/08/2016. But the duration of stay is 7 days only.  I want to stay for 6 days more, for a total of 14 days. And my flight includes travel from a Schengen state to another. SWR surname that in fact describes or applies to the human in question? There are (in real life) many amazing examples of this (Bill Gates, "gates" being the fundament of computers, Tiger "Woods", the greatest ever user of "woods" golf-clubs, etc) and it is a staple of fiction.  Is there a single-word for this, or a frequently-used term?  Note that in fictional usage, I observe there is literal "X", so, for example I just saw a children's movie where "Madame Work" was a taskmaster. However, authors often employ near-literal "X": for example, most of the minor character surnames in Harry Potter are 'kind of" "X".  So, is there an "X" for Bill "Gates", Tiger "Woods", an undertaken named Mr. Death, etc?  Sorry if this has been asked already, I couldn't find it.    Note - I've just found there's a "wikipedia" page claiming 'aptonym' or 'euonym' is sometimes used.  Wikipedia is often valueless, so I do not know if this is a meaningful claim. What does 1x1 convolution mean in a neural network? I am currently doing the Udacity Deep Learning Tutorial. In Lesson 3, they talk about a 1x1 convolution. This 1x1 convolution is used in Google Inception Module. I'm having trouble understanding what is a 1x1 convolution.  I have also seen  by Yann Lecun.  Could someone kindly explain this to me? When I place a multipage pdf file into InDesign CC, the pages go outside of my document settings, making this function useless.  What do I have to do to make the placed  pdf pages go within my document settings, which in this case is 11 in. Length X 8.5 in. width, .5 in. margins on all sides, with two columns per page. I would greatly appreciate an answer to this, Thank you. Let's take the numbers 0-10.  Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) deviation from the mean is $30/11 \approx 2.72$.  However, this is not the standard deviation.  The standard deviation is $\sqrt{10} \approx 3.16$.  The first mean-deviation is a simpler and by far more intuitive definition of the "standard-deviation", so I'm sure it's the first definition statisticians worked with.  However, for some reason they decided to adopt the second definition instead.  What is the reasoning behind that decision? How do I install a different Python version using apt-get? What exactly is the Box-Jenkins method for ARIMA processes? The  page says that Box-Jenkins is a method of fitting an ARIMA model to a time series. Now, if I want to fit an ARIMA model to a time series, I will open up SAS, call proc ARIMA, supply the parameters $p,d,q$ and SAS will give me AR and MA coefficients. Now, I can try different combinations of $p,d,q$ and SAS will give me a set of coefficients in each case. I select the set with the lowest Akaike information criterion.  My question is: where in the above procedure did I use Box-Jenkins? Am I supposed to use Box-Jenkins to come up with initial estimates of $p,d,q$? Or did SAS use it internally somehow? Please provide a way to download SEDE data via an API I recently suggested on the Stack Apps site that , and it was suggested I reference the feature request here, for visibility.  My use case is a query that the API does not support, so I thought it would be interesting to get it programmatically from SEDE instead. I wonder if this feature might be of interest to others here?  outlines the current ways that SE data is available via various APIs and UIs.  I wonder if my use case (retrieving revisions filtered by user) might have been supported by OData,  since April of this year.  This question has sort-of , though that was not an explicit feature request, and is now rather out of date. It would be good to find out what the current views are on this topic, and indeed whether there is much demand.  My use-case does not urgently require this, since I can use public SE screen scraping or SEDE CSV downloading. From reading around the SE Meta network, both of these approaches seem to be tolerated, as long as it is done with careful regard to frequency and bandwidth.