Since it is very much possible to have functors with $F_0 \dashv F_1 \dashv F_2$ or even $F_0 \dashv F_1 \dashv F_2 \dashv F_3$, it begs the question:     Can there be an example of infinitely many, non-repeating mutually adjoint   functors?   Obviously, "non-repeating" is important, since $\operatorname{id} \dashv \operatorname{id} \dashv \dots$ would otherwise of course be a boring example (and every other pair of pseudo-invers functors would be too).
Suppose we have (some categories and some functors such that) $F_1$ is left adjoint to $G_1$, $G_1$ left adjoint to $F_2$, $F_2$ left adjoint to $G_2$. Will $F_1$ then be equal to $F_2$ (and $G_1$ to $G_2$)?  Or can there exist arbitrarily long 'chains' of adjoint functors that do not trivialize?
When umpire change the cricket ball, was the old ball thrown as waste or keep for reuse? Count number of increasing functions, nondecreasing functions $f: \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, m\}$, with $m \geq n$. I stumbled upon a question given like:  Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.   Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following two types:  (a) strictly increasing; i.e., whenever $x &lt; y, f(x) &lt; f(y)$, and  (b) non-decreasing; i.e., whenever $x &lt; y, f(x) \leq f(y)$.  I tried in the following way.  For (a). We can say $f(n)&gt;f(n-1)$ AND $f(n)&gt;f(n-2) \ldots f(n)&gt;f(1)$ (total $n-1$ elements)  Similarly for $f(n-1)&gt;f(n-2), f(n-1)&gt;f(n-3) \ldots f(n-1)&gt;f(1)$ (total $n-2$ elements)...  and $f(2)&gt;1$  Like this $$(n-1) + (n-2) + \ldots + (1) = \frac{n(n-1)}{2}$$  Is this correct?  For (b). Since there's equality, it will be $$n + (n-1) + ... (1) = \frac{n(n+1)}{2}$$  Is my approach correct? Any help is appreciated. Thanks in advance. Hi im new to Blender and i want to know when im in edit mode if i can copy or cut a selections of polygons faces and paste to another mesh.  Thanks in advance for any help Show that $$\int_{0}^{\infty} \frac{x - \sin(x)}{x^3} \, dx = \frac{\pi}{4}$$   My attempt is as follows: Let $$f(z) = \frac{z - i e^{iz}}{z^3}$$ and consider the contour on $[\epsilon, R]$ followed by a semicircular arc in the counter clockwise direction, then on $[-R, -\epsilon]$, then the semicircular clockwise contour avoiding the origin. We have, then, that  $$0 = \int_{\Gamma} f(z) dz = \int_{[\epsilon, R]} f(t) dt + \int_{C_R}f(Re^{it})Rie^{it}dt + \int_{[-R, -\epsilon]}f(t)dt + \int_{C_{\epsilon}}f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  Then the first and third integrals ( $I_1$ and $I_3$) combine so that  $$I_1 + I_3 = 2\int_{\epsilon}^R \frac{t - \sin{t}}{t^3}\,dt$$  Further,   $$|I_{C_R}| \leq \int_0^\pi \left|\frac{Re^{it} - ie^{-R\sin{t}}e^{iRcos{t}}}{R^2 e^{2it}} \right|dt \rightarrow 0 \text{ as } R\rightarrow \infty$$  (I've omitted the details, it isn't too bad to bound)  However, I'm having trouble computing the limit  $$\lim_{\epsilon \rightarrow 0}\int_{C_{\epsilon}} f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  No matter which way I look at it, it seems like this limit does not exist. Perhaps I'm seeing something wrong or have I chosen a bad $f(z)$? Java generics type erasure: when and what happens? I read about Java's type erasure .  When does type erasure occur? At compile time or runtime? When the class is loaded? When the class is instantiated?  A lot of sites (including the official tutorial mentioned above) say type erasure occurs at compile time. If the type information is completely removed at compile time, how does the JDK check type compatibility when a method using generics is invoked with no type information or wrong type information?  Consider the following example: Say class A has a method, empty(Box&lt;?&nbsp;extends&nbsp;Number&gt;&nbsp;b). We compile A.java and get the class file A.class.  public class A {     public static void empty(Box&lt;? extends Number&gt; b) {} }     public class Box&lt;T&gt; {}   Now we create another class B which invokes the method empty with a non-parameterized argument (raw type): empty(new Box()). If we compile B.java with A.class in the classpath, javac is smart enough to raise a warning. So A.class has some type information stored in it.  public class B {     public static void invoke() {         // java: unchecked method invocation:         //  method empty in class A is applied to given types         //  required: Box&lt;? extends java.lang.Number&gt;         //  found:    Box         // java: unchecked conversion         //  required: Box&lt;? extends java.lang.Number&gt;         //  found:    Box         A.empty(new Box());     } }   My guess would be that type erasure occurs when the class is loaded, but it is just a guess. So when does it happen? Given the following code, what does the if __name__ == "__main__": do?  # Threading example import time, thread  def myfunction(string, sleeptime, lock, *args):     while True:         lock.acquire()         time.sleep(sleeptime)         lock.release()         time.sleep(sleeptime)  if __name__ == "__main__":     lock = thread.allocate_lock()     thread.start_new_thread(myfunction, ("Thread #: 1", 2, lock))     thread.start_new_thread(myfunction, ("Thread #: 2", 2, lock)) I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In honesty I've been working on this paper a while now and I'm a bit tired so I'm probably giving up earlier than I usually would.  I won't write out the full question, only the last part:     Let $$S_r(n) = \sum_{m=0}^n\ (-1)^m m^r {n \choose m}$$ where r is a   non-negative integer . Show that $S_r(n)=0$ for $r&lt;n$. Evaluate   $S_n(n)$.   I have shown that $S_r(n)=0$ for $r&lt;n$ by taking $(1+z)^n= \sum_{m=0}^n\ z^m {n \choose m}$, letting $D_r(f(z))=z\frac d{dz}(z\frac d{dz}...(\frac d{dz}(f(z)))...)$ where $z\frac d{dz}$ is applied $r$ times and applied it to both sides.  The left hand side gives a polynomial, degree n which has factor $(1+z)$ for all $r&lt;n$ and the right hand side gives $\sum_{m=0}^n\ z^m (m)^r {n \choose m}$. Setting $z=-1$ yields the required result.   There is some build up to this, so I'm fairly certain that this was the intended method.  I'm stuck however on the very last part. I have tried finding a form for $D_r((1+z)^n)$, but I'm fairly sure that this isn't the correct approach, as the wording of the question implies that $S_n(n)$ needs to be considered separately.  I'm surprised that I didn't find that this question had already been asked, so apologies if it has been.  Thank you. What is covered by CDIC savings deposit insurance? How to install PHP from source with latest cURL package on Linux? I simply have to use the latest cURL version (7.21.3) with PHP on my webserver. (Has to do with specific bug fixes)  It is not available by default on most current Linux variant repositories. (Natty Narwhal by Ubuntu is the only one and it's still in beta stages) - so I cannot simply run "aptitude install php5-curl" or similar to get it (the very latest version of cURL) installed.  How would one go about installing the very latest PHP + cURL packages?  Is it as simple as changing the to-download file for cURL in a repository somewhere? (ie. changing a text file's entry somewhere from  to  or something..?)  Basically I need the very latest cURL library running with PHP on Linux. How would you personally go about doing that? I have HTML code like this :  &lt;div&gt;        &lt;a&gt;Link A1&lt;/a&gt;        &lt;a&gt;Link A2&lt;/a&gt;        &lt;a&gt;Link A3&lt;/a&gt; &lt;/div&gt;  &lt;div&gt;        &lt;a&gt;Link B1&lt;/a&gt;        &lt;a&gt;Link B2&lt;/a&gt;        &lt;a&gt;Link B3&lt;/a&gt; &lt;/div&gt;   When user clicks a link from above HTML, I want to get the jQuery object of the corresponding &lt;a&gt; element, and then manipulate its sibling. I can't think of any way other than creating an ID for each &lt;a&gt; element, and passing that ID to an onclick event handler. I really don't want to use IDs.   Any suggestions? How do I create a very long delay with Redstone? The only things I've found so far are a cart method (which ceased working in 1.6), a dispenser method (utilizing the decay on dropped items, but this requires occasional feeding), and a binary counter (which won't work for a single input but rather a fluctuating one).  I want to have a circuit that when I apply an input signal, the output signal doesn't change for 2 to 5 minutes.  When the input turns off, I again want a 2 to 5 minute delay.  I could accomplish this with (roughly) 240 to 600 redstone repeaters, but this is far in excess of being useful.  The only method I can think of, but would require a XOR gate (on the button-input and final output) to start and stop a clock attached to a binary counter.  While I can make room for a device of this size, I was hoping for something simpler. Unable to Retrieve Metadata Retrieve globalpicklist/globalvalueset without SOQL Is it possible to retrieve the values of a Global Picklist/GlobalValueSet using Schema class without a SOQL ? Set normal orientation of origin I want to change the orientation of my object to a specified direction.  Say I have an axle of a train whose normal orientation is set off by 90° from the global orientation:    The green arrow (Y-axis) points to the back of the train (along the global X-axis). But I want it to point to the sides of the train, so all I want is to rotate the orientation by 90° without rotating the object itself.  In other words, I want the normal and global orientation of the axle to be the identical:    Is this possible and if so, how?  Background: The game engine to use this model will move the bogies and the axles according to their orientations. After importing this train into the game the axles rotated 90° off. For almost a year now, I have been in the uncomfortable position of having an idea.  However, there is one nice thing about this idea.  It makes a concrete, exact and relatively easy to test physical prediction.  The idea predicts that there is a 11,187 m/s (Earth’s escape velocity) aethereal wind directly into the surface of the Earth at its surface.  I believe it would be possible to test this by performing a vertical variation of modern versions of the Michelson-Morley experiment (MMX) with one arm pointed in the vertical direction. ()  In 2003, Müller et al. performed a normal (2 horizontal orthogonal arms) modern MMX using cryogenic optical resonators that found a “possible anisotropy of the speed of light c, (of) 2.6 +/- 1.7 parts in 10^15” (  )  In a brief conversation with Holger Müller a Professor at Berkeley and the lead author of that paper, he stated that to this knowledge no one had ever performed a variation of the experiment using a vertical arm.  He also mentioned that such an experiment would be complicated by the fact that gravity would slightly compress the length of the vertical bar making two equal length bars no longer equal in length.  “They haven't been done as far as i know. The problem is that any interesting physics signal would be hard to tell from a large signal from stretching of the arms under their own weight.” - H. Müller  I am interested in attempting to run this experiment myself.  To that end, I have the following questions:   Given current Physics understanding, is there any reason to expect that such a vertical variation of the MMX wouldn’t return the exact same results as all other MMXs, namely that there is no anisotropy in the speed of light? Given the complications mentioned by Professor Müller, are there reasonable methods available to overcome them?  Especially considering the size of the effect (c + 11,187m/s vs c) is substantially larger than the accuracy obtained in his and similar modern MMXs. What is the order of magnitude cost of such an experiment?  If I am to fund this personally, would such a project cost \$10,000?  \$100,000? \$1,000,000? more?   Any insight offered on this topic will be greatly appreciated.  EDIT: For what its worth, for people looking at this years later.  As I originally alluded to in a comment to [WetSavannaAnimal aka Rod Vance]'s much appreciated and comprehensive answer.  While, I've become increasingly comfortable with the idea that there is a flow of aether or spacetime or whatever you want to call it into the Earth at 11km/s; I strongly believe that a vertical MMX would return the same null result as the horizontal one.  And I believe it will do so because of length contraction; that length contraction exactly masks any anisotropy of the speed of light by ensuring that the roundtrip time of light in any direction is a constant; that length contraction ensures Lorentz co-variance. I would like to compare the contents of a couple of collections in my Equals method.  I have a Dictionary and an IList.  Is there a built-in method to do this?  Edited: I want to compare two Dictionaries and two ILists, so I think what equality means is clear - if the two dictionaries contain the same keys mapped to the same values, then they're equal. unable to reset root password of mysql Probability of complete coupon set in $K$ boxes where a box has $N$ distinct coupons? Say there is a contest to collect a full set of $M$ coupons.  Each box of product has $N$ distinct coupons of the $M$ possible coupons in it, selected uniformly without replacement from the $M$ possible coupons, the same $N$ in number for all boxes.  I'm interested in the probability that after opening $K$ boxes of product I have at least one complete set of the $M$ coupons.  My attempt has been to take the probability of a coupon not being in a box, use that to determine the probability of that coupon not being in any of the $K$ boxes, and then... I get flummoxed, and don't think this is the correct approach.  I searched the site for coupon collector problems, there are a couple that deal with similar situations (multiple in a box), but with the box having coupons selected with replacement, so duplication in a box is possible.  Right now I'm using a CAS to get probability of sums of hypergeometric samples having all bins covered to check my ideas, but that's utterly impractical for other than tiny cases.   Is this not as trivial as I think it is, or is there a direct way to calculate this? Trying to install Atom. Unable to start Clangd language server In the book I'm reading at the moment () it says that if you don't declare a default constructor for a class, the compiler supplies one for you, which "zeroes out each data member". I've experimented with this, and I'm not seeing any zeroing -out behaviour. I also can't find anything that mentions this on Google. Is this just an error or a quirk of a specific compiler?