Submodules finitely generated if sum and intersection are finitely generated?
$M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences
Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)? I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity).   I'm wondering though if this can be generalized: if A is a nxm matrix and B a mxn matrix, then AB is a nxn matrix and BA a mxm matrix.  So my question is: will the eigenvalues of AB and BA, that differ from zero, have the same algebraïc multiplicity? How to convert mediawiki syntax to latex? Ubuntu 14.04 Wireless not working: no network interface. Atheros [168c:003e] device I just installed Ubuntu 14.04.3 LTS after installing windows 10 in my new VN7-571G and the wireless is not working. It never had: even during the installation it was not recognized. I run a script to gather the data , which is . lspci -knn | grep Net -A2 output 03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:003e] (rev 20)         Subsystem: Lite-On Communications Inc Device [11ad:0804]  The OS behaves like it has no wifi card. I do not see wifi settings in network settings I'm not a linux power user, so I do not know what to do to solve the issue. What should I do to make it work? What is the difference between \mathbin vs. \mathrel? These seem to be very similar as they both separate two operands. When would one be used over the other? Is there any difference in the math spacing surrounding them?  How would the spacing in these relate to to the spacing produced by \mathop? How does some of the black hole (BH) mass escape the event horizon (EH) of either BH, or the merged EH of two merged BHs? Host alias for ssh There are a few SSH destinations I will frequently be connecting to and I'm wondering if rather than remember the IP address I can create a host alias for them.  I'd like to be able to use...  ssh -p xx user@domain.ssh  I've tried establishing this host in /etc/hosts and unless there's a service I needed to restart it had no effect. What is the scope of variables in JavaScript? What is the scope of variables in javascript? Do they have the same scope inside as opposed to outside a function? Or does it even matter? Also, where are the variables stored if they are defined globally? XmlInclude : List and array Prove $5\mid2^{4n}-1$ by induction It looks like if I load dynamic content using $.get(), the result is cached in browser.  Adding some random string in QueryString seems to solve this issue (I use new Date().toString()), but this feels like a hack.   Is there any other way to achieve this? Or, if unique string is the only way to achieve this, any suggestions other than new Date()? How do you make div elements display inline? Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass?  Of course, $\pi$ is not constructible because it is transcendental and so is not a root of any polynomial with rational coefficients. But is there a simple direct proof that $\pi$ is not a root of polynomial of degree $2^n$ with rational coefficients?  The kind of proof I seek is one by induction on the height of a tower of quadratic extensions, one that ultimately relies on a proof that $\pi$ is not rational. Does any one know of a proof along these lines or any other direct proof?  I just want a direct proof that $\pi$ is not constructible without appealing to transcendence. Difference between "@id/" and "@+id/" in Android How to create an OD road distance matrix in QGIS? I'm looking for the steps required to produce an origin-destination (OD) cost matrix.  What I'm looking for is a matrix of road distances. The distance matrix tool only produces euclidian distances. The QGIS Roadgraph plugin produces the shortest road distance from one origin to N destinations. What I need is a file containing distances from all origins to the N closest destinations. I have a lot of origins so repeating the analyses with Roadgraph is not an option. Thanks again.  I'm looking for a step-by-step description, starting from scratch (ie, with my 3 shapefiles: origins, destinations, road network). An inequality for a sum of sine functions I want to show for $x\in\mathbb{R}$, the following inequality holds. $$ \left|\sin x\right|+\left|\sin (x+1)\right|+\left|\sin (x+2)\right|&gt;8/5. $$ This trigonometric inequality has been verified by Mathematica using the Plot commend. However, using the sum formula for the sine function seems doesn't work, and I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks! How to calculate Area Under the Curve (AUC), or the c-statistic, by hand For example, you're a fashion designer who wants to get the opinion of both a professional and someone who doesn't know much about fashion. What's a term for the latter person? Probability of sampling with and without replacement In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding probability is $\left[1- \left(\frac{1}{1-n}\right)^r\right]$.  Please help me show how this is proved. A new tab for a record Type! Hello Folks,              I have a question which might be more on the admin side of the force.com platform but involves development.  Question: If I create record types for an object, how can I create separate tabs for the same object but different record types. And, since they all are going to share the same fields, how can I distinguish the labeling?  I dont think this is currently possible, but I would like to know how can it be achieved?  What I have tried: I tried creating visualforce pages with enhanced lists but then in the enhanced list I can not specify the record type.  Please let me know how can I achieve this! Thank you for the help. I have the Nikon kit lens that came with a D5100 — the kit lens is  . I was thinking of getting a fast lens and I am looking to buy the .  Should I consider 35mm or instead buy a 55-200mm for better range?