Need help identifying frames to make sure make model and possibly year
We often get questions on bicycles.stackexchange about identifying a bicycles manufacturer, model, make, and year.  Often these questions are downvoted or closed -- usually with a comment attached to them that it's not necessary to know the exact model/make/year of a bicycle if you want to fix, repair, or ride your bike. Why should owners not care about the model/make/year of their bicycle? Especially when it comes to less expensive bikes (aka BSOs) and BMX bicycles? This is intended as a canonical question that we can point closed questions to. Also see      (generally not helpful for identifying a brand/model)
Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized Compute $\int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx$ How can I add an application to the GNOME window manager? The GNOME on-board documentation covers only the things anyone can easily guess. I have an application for that I can only start from the command line. Not that I mind using a terminal but ...  Anyway, how do I add the command (and preferably a nice logo) to GNOME WM? Are there ways to determine if a wall is load bearing? Other than looking up blue prints, which many homeowners may not have, are there ways to determine if a wall is load bearing?  Methods I can think of might include:   Going up in the attic to check if ceiling trusses run perpendicular to the wall If the wall is an exterior wall   Any others? Reprojecting raster from lat/lon to UTM in R? The "" formulas often come in handy, but it's not immediately obvious that they would be true.  \begin{align} \sin(\alpha \pm \beta) &amp;= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &amp;= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{align}  So what I want to know is,    How can I prove that these formulas are correct? More importantly, how can I understand these formulas intuitively?   Ideally, I'm looking for answers that make no reference to Calculus, or to , although such answers are still encouraged, for completeness. Script Tag Error - Aura Lightning Component How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$ Given the integral $(1)$     $$\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3\tag1$$   This integral seems to be quite tough, I can't even know where to start. I have try substitution $u=1+\cos^3 x$ and $u=\cos^3 x$, can seem to remove the x variables.  We have the following identities   $$\sin^5a={5\over 8}\sin a-{5\over 16}\sin 3a+{1\over 16}\sin 5a\tag2$$  $$\cos^3 a={3\over 4}\cos a+{1\over 4}\cos 3a\tag3$$  We may write $(1)$ as  $$\int_{0}^{\pi}{\sin x(1-\cos^2 x)^2\over 1+\cos x(1-\sin ^2 x)}\mathrm dx\tag4$$  $u=1-\sin^2 x\implies \mathrm du=-2\sin x\cos x\mathrm dx$ then $(4)$ becomes  $${1\over 2}\int_{0}^{1}{(1-u)^2\over \sqrt{u}+u^2}\mathrm du\tag5$$  $u=v^2\implies du=2vdv$ then $(5)$ becomes  $$\int_{0}^{1}{(1-v^2)^2\over v+v^4}\cdot{v\mathrm dv}\tag6$$ Simplify to  $$\int_{0}^{1}{(1-v)(1-v^2)\over 1-v+v^2}\mathrm dv\tag7$$  I guess using partial decomposition of fraction and integrate term by term, seem lengthy process  How else can we prove $(1)$? How can I add left aligned text to an equation? I have a document I am trying to copy to learn TeX. Here is what I have encountered: How can I have the "or" in the equation. This is what I have right now:  \[ f = ma; \]  But $a$ is the change in velocity,  \[ f = m \frac{dv}{dt};\] \[ f = m  \frac{d^2y}{dt^2};\]   Edit: Trying out Stefan's answer Running a caching DNS server on my LAN MAXDOP not working? I wanted to test out MAXDOP on my computer. So I set MAXDOP to 2 for a specific query. However, when I looked at my logical processors in Task Manager as I ran the query, it looked like they were all being used. I thought that it would only be using 2 logical processors if MAXDOP was set to 2? Does anyone know what is happening? Please see image below.  Another issue is, the DOP returned by the execution plan said 1. Now I know that setting MAXDOP does not mean SQL Server will actually use the number set. However, given that it seemed all my 4 logical processors were being used to process the query, it made it even more strange to see DOP 1.  This is the query I ran:   This was what happened as I ran it (i.e. it looks like all 4 logical processors are being used to run the query): I'm a teacher and I teach at/in school OR at/in a school There are a lot of posts on this topic, but I can't find any answers to my question.  I came across the following two definitions of 'school':        [uncountable] (used without the or a) the process of learning in a   school; the time during your life when you go to a school (British   English)   [uncountable] (used without the or a) the time during the day when   children are working in a school      Source:   I know there are better ways to convey the message, but which sentence would a teacher utter if he was a school teacher teaching students, adults, etc.? What about a janitor, director, and secretary?     I teach at/in school OR I teach at/in a school   The above definitions concern students, and not working individuals like teachers or janitors.  I would appreciate if you also stated whether you're American, Canadian, British, etc. Is there any standard notation for specifying dimension of a matrix after the matrix symbol? I want to explicitly specify dimension of matrices in some expressions, something like   $$\boldsymbol{A}_{m \times n} \boldsymbol{B}_{n \times m} = \boldsymbol{C}_{m \times m} \, .$$  Is there any more or less standard notation for this?    While this notation is generally unambiguous, I think it become ambiguous, for example, in the following case. Suppose that I have some column or row vector which is conjugate transpose of correspondingly row or column vector. If I write it in the following way  $$\boldsymbol{A}_{n \times 1}^{\dagger} \, ,$$ we can have 2 different interpretations:   matrix $\boldsymbol{A}_{n \times 1}$ is $n \times 1$ column vector and by $\boldsymbol{A}_{n \times 1}^{\dagger}$ I'm referring to its conjugate transpose which is $1 \times n$ row vector; matrix $\boldsymbol{A}_{n \times 1}^{\dagger}$ itself is $n \times 1$ column vector which is conjugate transpose of $1 \times n$ row vector $\boldsymbol{A}$. Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$ Let $\mathbb R$ denote the set of all real numbers. Find all function $f: R\to \ R$ such that   $$f\left(x^2+f(y)\right)=y+(f(x))^2$$   It is the problem. I tried to it by putting many at the place of $x$ and $y$ but I can't proceed. Please somebody help me. Two rotating discs with same angular momentum when brought in contact completely stop. Why is the angular momentum not conserved in this case? Two discs mounted on different thin, lightweight rods oriented through their centres are made to rotate about their axes seperately such that the angular momentum of the two about their respective axes is same in both magnitude and direction. When both are brought in contact, they stop due to force of friction. Why is the angular momentum not conserved about their axes in this case?(Was positive before contact, but zero after the process completes, no external force present either) I am little confused with the applicability of reinterpret_cast vs static_cast. From what I have read the general rules are to use static cast when the types can be interpreted at compile time hence the word static. This is the cast the C++ compiler uses internally for implicit casts also.  reinterpret_casts are applicable in two scenarios:   convert integer types to pointer types and vice versa  convert one pointer type to another. The general idea I get is this is unportable and should be avoided.     Where I am a little confused is one usage which I need, I am calling C++ from C and the C code needs to hold on to the C++ object so basically it holds a void*. What cast should be used to convert between the void * and the Class type?  I have seen usage of both static_cast and reinterpret_cast? Though from what I have been reading it appears static is better as the cast can happen at compile time? Though it says to use reinterpret_cast to convert from one pointer type to another? What is the difference between Unix, Linux, BSD and GNU? What is the very fundamental difference between Unix, Linux, BSD and GNU?  Unix was the earliest OS, so the term 'Unix like' is understandable, since they have kernel, file system structure, most of the commands, users etc are same as Unix.  Still why are they different? What made them set apart? Is it the kernel? How can you find and replace text in a file using the Windows command-line environment? [Salutations]  (check one)  [ ] Well trained professional, [ ] Casual reader, [ ] Hapless wanderer,   I have a (check all that apply)  [ ] query [ ] stored procedure [ ] database thing maybe     that was running fine (if applicable)   [ ] yesterday [ ] in recent memory [ ] at some point    but is suddenly slower now.  I've already checked to make sure it's not being blocked, and that it's not the victim of some long running maintenance task, report, or other out of band process.  What is the problem, what should I do, and what information can I provide to get some help?  [*Insert appropriate closing remarks*] Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$ Prove the inequality $$\frac{1}{\sqrt{1}+ \sqrt{3}}  +\frac{1}{\sqrt{5}+ \sqrt{7} }+......... +\frac{1}{\sqrt{9997}+\sqrt{9999}} &gt; 24$$   My work:   Rationalizing the denominator gives   $$\frac{\sqrt{3}-1}{2}  +\frac{\sqrt{7}-\sqrt{5}}{2}+......+\frac{\sqrt{9999}-\sqrt{9997}}{2} .$$  Now by taking two as common and separating the positive and negative terms gives  $$\frac{1}{2} [ \{\sqrt{3} +\sqrt{7}+\dots +\sqrt{9999}\} - \{1+\sqrt{5} +\dots+\sqrt{9997}\}].$$  Can we do like this please suggest. Thanks.