Trace morphism in Lipman's "Dualizing sheaves, differentials and residues on algebraic varieties".
Trace map from finite surjective morphism of normal irreducible varieties
What is a "closed" or "duplicate" question? Questions can be closed or marked as duplicates.  What does it mean for a question to be &quot;closed&quot;? What does it mean for a question to be marked as a &quot;duplicate&quot;? Who can close a question? What are the reasons for closing a question ? Is closure the end of the road for a question? When are closed questions eligible for deletion?   Related        For more information, see &quot;&quot; in the . Simple one really.  In SQL, if I want to search a text field for a couple of characters, I can do:  SELECT blah FROM blah WHERE blah LIKE '%text%'   The documentation for App Engine makes no mention of how to achieve this, but surely it's a common enough problem? Failed to download repository information due to missing CDROM When updating I got this error message:  W:Failed to fetch cdrom://Ubuntu 11.10 _Oneiric Ocelot_ - Release amd64 (20111012)/dists/oneiric/main/binary-i386/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs W:Failed to fetch http://ppa.launchpad.net/synce/ubuntu/dists/your_ubuntu_version/main/source/Sources  404  Not Found   What to do? Limit of Series with differences of Floor function Infinite Series Convergence Verification I'm currently tackling the following infinite series in order to determine if it converges or not:  $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{n}$$  What I've done is rearranged this to make use of the comparison test, shown below:  $$= \frac{\sqrt{n+1}-\sqrt{n}}{n} * (\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}})$$  $$= \frac{(n+1)-(n)}{n(\sqrt{n+1}+\sqrt{n})}$$  $$= \frac{1}{n(\sqrt{n+1}+\sqrt{n})}$$  $$= \frac{1}{n^{3/2}+\sqrt{n+1}} &lt; \frac{1}{\sqrt{n+1}} &lt; \frac{1}{\sqrt{n}} $$  However, $\frac{1}{n}$ is smaller than $\frac{1}{\sqrt{n}}$, therefore:  $$  \frac{1}{n}   &lt; \frac{1}{\sqrt{n}}$$   This implies divergence by the comparison test due to the fact $\frac{1}{n}$ diverges, and if a smaller series diverges any larger series must as well.  Is this right? If I've made any mistakes in any part, please tell me! I am doing an animation. The images I have attached show 2 different render layer view node results.    Image one shows a cube on layer 1, and 2 icosphere on layer 2. I have layer 2 set to fast blur. I have to use transparent film and therefore I use the Alpha over nodes in composition.  Image two shows a cube and 1 icosphere. Image 1 should not be showing 2 icospheres, as the left most one is behind the cube.    Image 2 is from the same render result, and is correcting showing only 1 icosphere. However, that icosphere should be blurred as it is on layer 2. It seems for whatever reason I can not have both the correct composition effect and also have every object's visibility render correctly. I can only either have everything work fine, with no composition, or composition with every emission object showing through every other object in the scene.   I am aware that masks would solve this problem, but because of the complicated setup of object materials and particles, masking produces these very strange and suddenly appearing artifacts. Material Indexing also produces odd artifacts and does not consistently effect the appropriate materials.   This is also a new problem, I did not have this issue in the last blender version. So, material indexing is out, masking is out, object indexing is out.       I want to use the compositor to blur certain layers, but every emission object in the scene shows through. It shows through as though it was on x-ray, regardless if it is behind another or inside another object. The first image is an example of this, the left icosphere is behind the cube, and should not be visible. That is the problem. Why is it visible when it shouldn't be? How to find the index of an element in an array in Java? Distance between footnote and the line above it I am looking to replicate black &amp; white maps in QGis, using hatching patterns and greyscales to indicate different categories of a continuous variable (example below).  It is possible/straightforward to do so in QGis? At first sight I only see color gradients and symbol fills.   ). What is the difference between lands and mana? What is the relationship between lands and mana in Magic and what do each of them do?  A good answer should especially include how a new player should think about them.    Note:  This is an attempt to create a canonical question about the relationship between lands and mana in Magic.  Beginners are often confused about this given the same symbols are used to mean different things across basic lands, non-basic lands, and mana costs.  This can lead to confusion about , about , or even about . Unable to install Cisco Packet Tracer 7.3 on Ubuntu 20.04 I am trying to install Packet Tracer 7.3 (latest version) on Ubuntu 20.04 but do not succeed to so so because of two dependencies which are not installable ("libdouble-conversion1" and "qt-at-spi").  However I successfully installed the previous version (7.2.2) which was available in an Ubuntu installer format (.run file) but I nothing happens when I run it from my gnome desktop.  I am quite a newbie in Linux so I appologies if answers are obvious...  Thanks in advance ;-)  Denis Travelling with two different passports I was just wondering if you have two passports (a Chinese one and an Australian one), and you booked your airline ticket with the Chinese one, will you have to show both of your passports or just one?  The thing is, on my Chinese passport, it has my birth name, but then when I became an Australian citizen, I legally removed my Chinese name and replaced it with my English name. So I don't know whether I should just show the passport officials the one that I booked the airline ticket with. Difficulty in creating macro '\newcommand' for the beginning and end of frames in Beamer My interest in combinatorics was recently sparked when I read about the many things that the Catalan numbers count, as found by Richard Stanley. I picked up a copy of Brualdi's Combinatorics, and while browsing the section on counting sequences I found a nice little puzzle that has definitely puzzled me.     Let $m$ and $n$ be nonnegative integers with $n\geq m$. There are $m+n$ people in line to get into a theater for which admission if $50$ cents. Of the $m+n$ people, $n$ have a $50$-cent piece and $m$ have a $\$ 1$ dollar bill. The box offices opens with an empty cash register. Show that the number of ways the people can line up so that change is available when needed is   $$ \frac{n-m+1}{n+1}\binom{m+n}{m}. $$     I first noted that the first person to enter must be one of the $n$ with a half-dollar. Now the register has a half-dollar change. The second person can be either a person with a half-dollar or a dollar. In the first case, the register will now have two half-dollars, in the second case, the register will now have one dollar bill. So it seems to me that when one of the $n$ people with a half-dollar enters, the number of half-dollars in the register increases by $1$, and when one of the $m$ people with a bill enters, the number of half-dollars decreases by $1$ but the number of bills increases by $1$.   I tried to model this by looking at paths in $\mathbb{Z}^2$. The $x$-axis is like the number of half-dollars, and the $y$-axis is the number of bills. You start at $(0,0)$, and you can take steps forward $(1,0)$ or backwards diagonally $(-1,1)$ corresponding to who enters, but you must always stay in the first quadrant of the plane without crossing over the axes. The goal is to make $m+n$ moves, and I figured maybe the number of such paths is counted by $\frac{n-m+1}{n+1}\binom{m+n}{m}$, but I'm not sure how to show this. I don't know if this observation simplifies the problem at all, as I don't know how to finish up. I'd be happy to see how this problem is done, thank you. I'm looking for the title of a Sci-Fi story (probably pre-1990) where a group is sent off into deep space and I believe unbeknownst to them it's a one-way journey.  In the course of the journey they all become very intelligent, they make psychoactive drugs from the plants on board, I recall that they discovered it's a one-way journey to test the affects of space travel, and they redesign/refit their engines and return to Earth.  Lame recall, sorry, that's all I got. Any help appreciated. What is wrong with my mob spawner? Erf as term for plot of land in a town or city. Is this word only used in southern Africa? The Free Dictionary website states and so does Wikipedia:     erf [ɜːf]   n pl erven [ˈɜːvən]   (Engineering / Civil Engineering) South African a plot of land, usually urban, marked off for building purposes   [Afrikaans]   The term erf is used in legal documentation and technical English throughout South Africa and Namibia, maybe even further afield in the region.  Is it used anywhere else, except maybe the Dutch/Flemish regions in Europe, where I assume the term originated?  An example:      The client purchased Erf 132, Windhoek for a large sum of money. He intends to build a house on this erf.   The capitalisation of "Erf 132" is commonplace, since it is a definition of a location. This is similar to capitalisation of towns or cities. I find it slightly awkward, but that is how it is done. How to install the previous version of a .deb package and pin it? I am trying to install the latest version of subversion on Sid, and because it has a bug I receive a warning and I abort the installation. How do I get to locate the previous version version, install it and pin until the bug is resolved?  root@server01:~# apt-get install subversion   Reading package lists... Done                                                                                                                                                                                  Building dependency tree                                                                                                                                                                                       Reading state information... Done                                                                                                                                                                              Suggested packages:                                                                                                                                                                                              db5.3-util subversion-tools                                                                                                                                                                                  The following NEW packages will be installed:                                                                                                                                                                    subversion                                                                                                                                                                                                   0 upgraded, 1 newly installed, 0 to remove and 205 not upgraded.                                                                                                                                               Need to get 0 B/981 kB of archives.                                                                                                                                                                            After this operation, 4,844 kB of additional disk space will be used.                                                                                                                                          Retrieving bug reports... Done                                                                                                                                                                                 Parsing Found/Fixed information... Done                                                                                                                                                                        critical bugs of subversion (-&gt; 1.9.2-2) &lt;Outstanding&gt;                                                                                                                                                          b1 - #803725 - subversion: dump-load of a repository modifies verbose log output: M line lost serious bugs of subversion (-&gt; 1.9.2-2) &lt;Outstanding&gt;  b2 - #803589 - FTBFS with ruby2.2 (only) Summary:  subversion(2 bugs) Are you sure you want to install/upgrade the above packages? [Y/n/?/...] n ********************************************************************** ****** Exiting with an error in order to stop the installation. ****** ********************************************************************** E: Sub-process /usr/sbin/apt-listbugs apt returned an error code (10) E: Failure running script /usr/sbin/apt-listbugs apt How to divide aleph numbers Summation equation for $2^{x-1}$