Let $$0 \overset{d_{n+1}}{\rightarrow} G_n \overset{d_n}{\rightarrow} G_{n-1}\overset{d_{n-1}}{\rightarrow}\ldots\overset{d_2}{\rightarrow}G_1 \overset{d_1}{\rightarrow} 0$$ be a sequence of modules and homomorphisms over the commuative ring $R$ such that $d_i \circ d_{i+1}=0$ for all $i=1,..,n-1$, and suppose that $G_i$ has finite length for all $i$. Set $H_i=\ker d_i / \text{im} \, d_{i+1}$. Show that $H_i$ has finite length for all $i$ and that $$\sum_{i=0}^{n} (-1)^i l(H_i)=\sum_{i=0}^{n} (-1)^i l(G_i)$$  It's obvious to see that $H_i$ has finite length by the exact sequence $$0 \to Im \; d_{i+1}\to Ker \; d_i \to Ker \; d_i / Im \; d_{i+1}\to 0$$ But I can indicate that $\sum_{i=0}^{n} (-1)^i l(H_i)=\sum_{i=0}^{n} (-1)^i l(G_i)$. Can anyone help me?
Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i&lt;0$. Suppose that $C_i=0$ for all $i$ is large enough. The Euler characteristic of $C_*$ chain complex is defined as $$\chi(C_*)=\sum_{i\geq 0}(-1)^iRank(C_i)$$ Prove that $$\chi(C_*)=\sum_{i\geq 0}(-1)^iRank(H_i(C_*))$$  I have to prove that $\sum_{i\geq 0}(-1)^iRank(C_i)=\sum_{i\geq 0}(-1)^iRank(H_i(C_*))$, I think that one way to do this is by showing that $Rank(C_i)=Rank(H_i(C_*))$ but I do not know if this is true in general, in a nutshell, I want to show that the cardinality of the base of any $C_i$ is the same as the cardinality of the basis of the corresponding homology, how can I do this? Thank you
biblatex: Cite commands with the same author prints authors name twice I'm using biblatex 1.4b, and Biber 0.9. However, I think the problem is with my settings, not with the versions of those programs.  I'm trying to cite a single author, and produce their name, followed by the dates of two papers, using the authoryear-ibid style, and natbib compatability. Here's my biblatex preamble:  \usepackage[style=authoryear-ibid,natbib=true,backend=biber]{biblatex}   What I want is for the following command: The works by \citet{Blau81, Blau88} ... to produce: The works by Blau (1981; 1988) ... however, what I'm getting is: The works by Blau (1981); Blau (1988)...  I know that biblatex knows they're the same author, since in the bibliography they fall under the same name (i.e. after the first instance of the name, there's just a dash instead of the name)  What am I doing wrong here? I've also tried it with the commands \cite and \cites, and neither produces what I want.   (crossposted with comp.text.tex) How to sort lines with >> double angle brackets? It seems that sort behaves strangely on lines such as &gt;&gt;b    $ cat test a &gt;&gt;b b c &gt; &gt;&gt;  $ sort test &gt; &gt;&gt; a b &gt;&gt;b c   I would expect the &gt;&gt;b line to be the third line output by sort but it is the fifth. Why does it do this and is there a way to make sort give my expected output?    I'm using GNU/Linux Ubuntu 16.04. How is it possible in Eclipse JDT to convert a multiline selection to String. Like the following  From:  xxxx yyyy zzz   To:  "xxxx " + "yyyy " + "zzz"   I tried the following template   "${line_selection}${cursor}"+   but that way I only get the whole block surrounded not each line separately. How can I achieve a multiline processing like commenting the selected block? I have to give low and high bounds for the following:  $$ \sum_{n=1}^\infty \frac{1}{2^n - 3^n } $$  How do I determine an upper bound? How can I show this sum exists?  edit: removed erroneous conclusion. How to Add Background Images in 2.8 and later 2.81a here. I am trying to add a "background image" similar to how , so that I see a screen that looks something like this:    However I can't figure out how/where to access this screen in 2.81a as the UI is wildly different. Any ideas?  I am new to Blender so if you can provide a click-by-click list of instructions it would be enormously appreciated! Why haven't I received the "Curious" badge? Why do I see a saddle in this picture of a computer screen? I am not entirely sure this is an appropriate question for PSE, however since many of you have such diverse backgrounds, I'll give it a shot.  I have noticed that when one takes a picture of a computer, TV screen with a phone, you can "see" what looks like a saddle phase portrait.     Why does this happen? Does it have any relevance to differential equations or other areas of math? Appendix in LaTeX 14.04 --> 16.04 failed; apt (>= 1.0.1ubuntu2.13)' is not installed Why does the zoom sometimes stop at a point? I was using the amspset document class for generating exam practice problems. I had wanted to add some referencing to the problems so I started a new counter like this (adapted from some Tex Exchange post)  \newcounter{P} \newcommand{\plb}[1]{\refstepcounter{P}\label{#1} \textbf{P-\ref{#1}.}}  \begin{problem} \plb{p1} A parallel-wire transmission line is constructed of \#6 AWG copper wire (diameter $d=0.162$ in., conductivity $\sigma_c=5.8\times 10^7$ S/m) with a 12-inch separation in air. Assuming no leakage between the two wires, find $R'$, $L'$, $G'$, and $C'$. Assume a working frequency of 1 MHz. \end{problem}  \begin{solution} Using Table.~\ref{tab:tlparam} and Fig.~\ref{fig:tlempl},  \[ Rs = \sqrt{\pi f \mu_c/\sigma_c} = 2.61\times10^{-4}~ \Omega\] \[ R' = \frac{2R_s}{\pi d} = 4.04\times10^{-2} ~\Omega\text{/m} \] \[ L' = \frac{\mu_0}{\pi} \ln{\frac{2D}{d}}=2.0~\mu \text{H/m} \] \[ C' = \frac{\pi \epsilon}{\displaystyle\ln{\frac{2D}{d}}}=5.56 ~\text{pF/m} \] $G'=0$ because the problem states that there is no leakage between the two wires. \end{solution}  \begin{figure}[ht] \begin{centering}     \includegraphics{fig24}     \caption{(Fig.2-4 from FAE) A few examples of transmission lines.} \end{centering} \label{fig:tlempl} \end{figure}  \begin{table}[ht] \caption{(Table 2-1 from FAE)Transmission-line parameters $R'$, $L'$, $G'$, and $C'$ for three types of lines.}     \begin{tabular}{c}         \hspace{3cm}\includegraphics{tb21}     \end{tabular} \label{tab:tlparam} \end{table} ... and more   Now all the figure references stop working, I get "Fig. " instead of "Fig. 1". The table references are alright though. Any idea what the problem could be? I have been trying to make a short clip of a bullet hitting a wine glass with wine in it, but I keep running into the same problem with the add-on. Every time I have used it the result has left me with at least a partial box being generated as several of the fractured pieces.  I have tried subdividing and/or adding a sub-surf modifier but the result only changes where the large distorted fragments end up. I have been having trouble with this for quite some time and I would greatly appreciate any help. I'm currently typing a mathematical document which counts many sum symbols. I frequently use the align environment to render equations, although the default behaviour of this environment is to render the equations in displaystyle, which in my opinion wastes vertical space. I therefore have added \textstyle at the beginning of each line of math in this environment, but I'm looking for a more convenient solution.   So is there an environment that is similar to align with the exception it displays textstyle math by default ? Thank you. Existence of subgroup of order six in $A_4$ Show that  the  alternating  group  $A_4$  of all even permutations of $S_4$ does not contain a  subgroup of order $6$.   For me am thinking to write all elements of $A_4$ and trying to find every cyclic subgroup generated by each element of $A_4$, then I have to check whether there exist such a subgroup or not! This is a long procedure for me, I ask if there is a short way to do this. I'm totally new to Ubuntu/Linux, using Ubuntu Server at the moment. Just trying to  figure out something basic.   How can you tell where you are installing a program. For example I just installed Sphinx search engine by placing the tarball that I downloaded from their site to my:  /home/sphinx   directory. I created the sphinx directory to place that tarball in. Then I ran these commands:  tar xvzf sphinx-0.9.8.1.tar.gz cd sphinx-0.9.8.1/ ./configure --with-mysql-includes=/usr/include/mysql --with-mysql-libs=/usr/lib/mysql   and then these:  make sudo make install   Now I have a lot of files sitting in the directory where I ran these commands. Is this my Spynx installation or did it install somewhere else?  In windows if you run an installer (.exe file) anywhere the program will still install in your C:\Program Files directory. Does something similar apply to linux where all programs are installed in a central place, or can you install programs anywhere on the system.  Questions   I would prefer to keep all my installed programs in one place so what is the right place for this in terms of best practice. In other words what is the Linux equivalent of C:\Program Files? And how does one always install at this location, is it just a matter of placing the tarball and running the install commands from this location? What about if I use sudo apt-get to install a package. How can I point to this location to tell apt-get to always install there? I gave a quick answer to part of  which had not been covered by previous answers, trying to clarify the reason you would say time of decoding but not decoding’s time. I said it was ’s usually indicates possession, but of course there were several counterexamples that would have occurred to me after a moment’s consideration, and these where helpfully supplied:   Britain’s climate two days’ time  a day’s work  the sun’s rays   I am still of a mind to say that possession of some sort is what allows the ’s. Even though the sun does not have title to its rays, they do belong to the sun. Now, at the risk of duplicating the original question and/or being pigheaded, I am curious as to why time of decoding but not decoding’s time is correct, if not for the reason I gave. Docker ppa package for Ubuntu 19.10 Does someone knows if it exists some ppa packages source for the newest Ubuntu version 19.10 i had problem with that when i upgraded from Ubuntu 19.04 to 19.10 i didn't work? How did Bill and Percy get 12 OWLs? In Prisoner of Azkaban, Hermione took 12 subjects, and needed a Time-Turner to do so. However, it is mentioned in the series that both Bill and Percy Weasley got 12 OWLs, so would have taken 12 subjects as well, so both would have needed Time-Turners for three years to do this. Doesn't this seem a little unusual, given how strict the Ministry is over the use of Time-Turners,  that they would give them to several Hogwarts students? Eclipse is giving me a warning of the following form:     Type safety: Unchecked cast from Object to HashMap   This is from a call to an API that I have no control over which returns Object:  HashMap&lt;String, String&gt; getItems(javax.servlet.http.HttpSession session) {   HashMap&lt;String, String&gt; theHash = (HashMap&lt;String, String&gt;)session.getAttribute("attributeKey");   return theHash; }   I'd like to avoid Eclipse warnings, if possible, since theoretically they indicate at least a potential code problem.  I haven't found a good way to eliminate this one yet, though.  I can extract the single line involved out to a method by itself and add @SuppressWarnings("unchecked") to that method, thus limiting the impact of having a block of code where I ignore warnings.  Any better options?  I don't want to turn these warnings off in Eclipse.  Before I came to the code, it was simpler, but still provoked warnings:  HashMap getItems(javax.servlet.http.HttpSession session) {   HashMap theHash = (HashMap)session.getAttribute("attributeKey");   return theHash; }   Problem was elsewhere when you tried to use the hash you'd get warnings:  HashMap items = getItems(session); items.put("this", "that");  Type safety: The method put(Object, Object) belongs to the raw type HashMap.  References to generic type HashMap&lt;K,V&gt; should be parameterized. Why are some genes dominant over others? What is the mechanism behind it? If I have a brown eye gene which encodes the protein that is responsible for the brown color and have a blue eye gene as well, what is the reason that my eye color is brown? How does one gene maintain dominance over the other?