Questions about Schrodinger's cat
What is an observer in quantum mechanics?
Compactness of $\operatorname{Spec}(A)$ In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. Network reputation graph becomes nervous when clicking quickly, backs away slowly $dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is. How to get Beamer Math to look like Article Math In a LaTeX article, the math characters are more "curvy" than the text characters:  \documentclass[12pt]{article} \begin{document}  $x^2 + y^2 = z^2$ \end{document}     In a beamer presentation, the math characters do not have the same curves:  \documentclass{beamer} \begin{document} \begin{frame}  $x^2 + y^2 = z^2$ \end{frame} \end{document}     Question  How can I get the beamer math to look just like the article math?  I would prefer if the solution was a single line (that I could place at the top of my LaTeX file) as opposed to something that needs to appear next to every piece of math. Understanding MCMC and the Metropolis-Hastings algorithm Over the past few days I have been trying to understand how Markov Chain Monte Carlo (MCMC) works. In particular I have been trying to understand and implement the Metropolis-Hastings algorithm. So far I think I have an overall understanding of the algorithm but there are a couple of things that are not clear to me yet. I want to use MCMC to fit some models to data. Because of this I will describe my understanding of Metropolis-Hastings algorithm for fitting a straight line $f(x)=ax$ to some observed data $D$:  1) Make an initial guess for $a$. Set this $a$ as our current $a$ ($a_0$). Also add $a$ at the end of the Markov Chain ($C$).  2) Repeat the steps bellow a number of times.  3) Evaluate current likelihood (${\cal L_0}$) given $a_0$ and $D$.  4) Propose a new $a$ ($a_1$) by sampling from a normal distribution with $\mu=a_0$ and $\sigma=stepsize$. For now, $stepsize$ is constant.  5) Evaluate new likelihood (${\cal L_1}$) given $a_1$ and $D$.  6) If ${\cal L_1}$ is bigger than ${\cal L_0}$, accept $a_1$ as the new $a_0$, append it at the end of $C$ and go to step 2.  7) If ${\cal L_1}$ is smaller than ${\cal L_0}$ generate a number ($U$) in range [0,1] from a uniform distribution  8) If $U$ is smaller than the difference between the two likelihoods (${\cal L_1}$ - ${\cal L_0}$), accept $a_1$ as the new $a_0$, append it at the end of $C$ and go to step 2.  9) If $U$ is bigger than the difference between the two likelihoods (${\cal L_1}$ - ${\cal L_0}$), append the $a_0$ at the end of $C$, keep using the same $a_0$, go to step 2.  10) End of Repeat.  11) Remove some elements from the start of $C$ (burn-in phase).  12) Now take the average of the values in $C$. This average is the estimated $a$.  Now I have some questions regarding the above steps:   How do I construct the likelihood function for $f(x)=ax$ but also for any arbitrary function? Is this a correct implementation of Metropolis-Hastings algorithm? How the selection of the random number generation method at Step 7 can change the results? How is this algorithm going to change if I have multiple model parameters? For example, if I had the model $f(x)=ax+b$.   Notes/Credits: The main structure of the algorithm described above is based on the code from an MPIA Python Workshop. Can I open Microsoft Word and Excel files? Can I open Microsoft Word and Excel files with Ubuntu? Ubuntu 12.04 comes with Gimp 2.6, so I installed Gimp 2.8 using the instructions below:   sudo apt-get purge gimp*   sudo add-apt-repository ppa:otto-kesselgulasch/gimp   sudo apt-get update   sudo apt-get install gimp    When try to open help, I get the message below. How do I install and configure the help for 2.8 on Ubuntu? SSH in shell script with password I've often wondered why Stack Exchange has opted to keep the upvote/downvote buttons available for clicking on question/answers posted by the user. For instance:    Even though these buttons have no use for the user, why are they still placed on the post's left hand column? Moreover, the comments have no controls for flagging/+1'ing the user's own comment. What is the reason for this discrepancy? What benefit do the +1/-1 buttons provide to the user, even though they can't be used for anything? How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals? I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly appreciated. Can you clone a larger drive onto a smaller drive (used space fits on the smaller) I have a 500GB drive which is about 50GB full. I got a 80GB SSD, which has 74.5GB of available space. I want to clone the system drive onto the SSD.   DriveImage XML will not let me copy a larger drive to a smaller one. Is it possible for moderators to review more than the daily limited count? Do I have to peel red potatoes before baking them? I recently made some potato wedges in the oven. I shared this recipe with my relative, who admonished me for not properly peeling the red potatoes before baking them.  She claims that there are disease and viruses that lie on the skin, so peeling it will get rid of it.  I bake it, so I assume that even the high heat in the oven would kill bacteria and germs. And besides that, I make sure to wash potatoes thoroughly before cooking them.  Am I in the wrong here? Should I have properly peeled the potato wedges before baking them? Personally I feel they add more flavor, and from all the pictures on the recipes online, it seems they also do not peel it. Wizard sickness I remember in an interview, or during a Q&amp;A, Rowling stating that Wizards live longer than Muggles. This got me thinking about Wizard deaths in general, and then deaths being attributed to illness.  So, I ask, has there ever been a Wizard, that has died from something like cancer? Is there anything, even briefly mentioning a Wizard becoming ill?  And finally, are there any illnesses that are exclusive to Wizards? A toplogical space $X$ is said to be second countable   if there exists a countable basis for the topology.      $X$ is separable if there exist a countable dense subset.   Show that a second countable space always is separable.      Give an example of a space that is separable but not    second countable   The first part was easy and i managed to solve it on my own  Assume that $X$ is second countable and let $B_1, B_2, B_3...$ be a countable basis. Construct the set $A$ by taking one element $x_n$ out of every $B_n$. Then $A$ becomes dense because every open set $U$ contain a $B_n$ which in turn contain the element $x_n$ in $A$.    Again, like in the question i posted yesterday, i had trouble to come up with an example but our professor gave an example during the last lecture but i dont fully understand it:  "For an example of a separable but not second countable space let $X$ be the real line provided with the cofinite topology, i.e. where a set is open if its either empty or if its complement is finite.   Then every infinite set is dense and hence $X$ is separable.  But for an arbitrary countable family of open sets $U_1, U_2...$ write $U_i = X$\ $A_i$, where $A_i$ is finite, there is an element $x \in X$\ $(\cup_i Ai)$, because $A_i$ is countable,  such that the open set $X$\ $\{x\}$ is then not able to contain any $U_i$. Consequently $X$ is not second countable."  We asked him to explain it in more detail but his english is poor and nobody understood anything...:)       Def: A subset $A$ of a topological space $X$ is dense in $X$ if for any point $x$ in $X$, any neighborhood of $x$ contains at least one point from $A$.   $1.$ "Then every infinite set is dense"  Take $A$ to be one of these infinite sets. If $x \in A \subset X$ then it trivially fulfills the definition. But if we take a $x \not\in A$ why is it not possible to find a neighborhood in these finite sets that does not contain an element in $A$?  edit: its obvious if it is only open neighborhoods, open sets in the topology, but that is not specified in the definition.  $2.$ I am not fully understanding this either  "But for an arbitrary countable family of open sets $U_1, U_2...$ write $U_i = X$\ $A_i$, where $A_i$ is finite, there is an element $x \in X$\ $(\cup_i Ai)$, because $A_i$ is countable,  such that the open set $X$\ $\{x\}$ is then not able to contain any $U_i$. Consequently $X$ is not second countable."  Is it possible to express this in a simpler way? Bind application to a specific network interface I have tried ForceBindIP, but it has a significant downside - it doesn't affect the children of the application I'm trying to bind, it only affects the application itself. It also can't force an application to always run through a specified interface, it has to be run through forcebindip.exe every time. It becomes a problem with applications like League of Legends where the process tree looks like that:    The launcher runs the patcher, the patcher run the client, etc. I can only affect the parent of all these processes in the tree, so the actual game is not bound to the interface I want, making this whole venture pointless.  Is there a more modern alternative to ForceBindIP for Windows 7? There are many questions similar to this one on this site, but they are mostly old. Maybe there is now a better way to solve this problem?  My current idea is to do the following:   Set up local 3proxy server bound to the desired interface. Run the game through Proxifier or similar software configured to run through that local proxy.   I'm not sure if that will work, but even if it will, it seems like a sub-optimal solution. Do you guys have any better ideas?  Edit: My idea didn't work :(  Edit 2: Basically, what I'm trying to achieve is bind a few applications to a regular interface, while VPN is running. The reason is that I need to connect through VPN most of the time, but some applications (such as games) don't work properly this way, because of higher ping and other issues. How to increase difficulty in Dungeon World I've been GMing for a couple of sessions now and I'm still struggling at making encounters or situations more difficult. I will describe dangerous situations and encounters in fiction but the mechanics make it too easy for my players to succeed.  For instance no matter how big I describe the Troll, if the fighter of the party charges him and swing his sword, roll+2 for Hack and slash, he'll easily hit the troll. Almost 40% chance to hit without consequences. I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why   $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$   isn't a counter-example. Any ideas? Thanks! Building a graph from OSM? I spent days looking in all questions here and trying tutorials. I need to build a graph for a region (It will be my input in simulation) I downloaded Postgresql and Postgis and osm2pgsql. Now i can load .osm files into a database. I want to have a complete database with roads ids so i can generate a table of edges and nodes then build my graph  nodes = intersections and edges = roads for a city using python.  Please any detailed help with steps.I saw a lot of keywords (Postgresql/postgis, Networkx, QGIS ..) but i couldn't manage to assemble a good tutorial  I am a very beginner. Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says it is to formalize. As far as I can tell, it sort of generalizes a bunch of fields in mathematics, like in topology, graphs and groups we have isomorphisms and automorphisms and homomorphisms, it seems to me like it just finds a bunch of results that are true to these functions in a bunch of disciplines. But why would generalizing that be of use? I mean, wouldn't it be easier to prove them specifically in each of the important disciplines?