Consider a succession $\{f_{n,k}\}_{n,k}$ s.t. for any fixed $k$, $\lim_{n\rightarrow\infty}f_{n,k}=g_k$. Which are the conditions s.t. \begin{equation} \lim_{n\rightarrow\infty}\sum_{k}f_{n,k}=\sum_{k}g_k \; \;? \end{equation}
Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks!
How can I create an AddIn to ArcGIS Pro and where can I find it from ArcGIS Pro application? What category does the 'the recent perfect' belong to? A scholar has said that there are 4 usages of the present past tense:   The Universal sense of the Perfect, used to indicate that a state of affairs prevailed throughout some interval stretching from the past into the present       I've known Max since 1960.  The Existential sense of the Perfect, used to indicate the existence of past events     I have read Principial Mathematica five times.  The Stative/Resultative sense of the Perfect, used to indicate that the direct effect of a past event still continues     I can't come to your party tonight - I've caught the flu.  The Hot News sense of the Perfect, used to report hot news     Malcolm X has just been assassinated.    To what category does the recent past belong? Surjective morphism of affine varieties and dimension I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry:     Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ is at least as large as the dimension of $Y$.   Could anyone give me a hint? I'm looking for a proof that avoids commutative algebra since it is not mentionned in the book at this point. How to Select Hyperlink Text in Google Chrome? I am running Google Chrome 5. How can I select hyper link text using the mouse with, for example, CTRL or ALT keys? Does existence of $\gcd$ implies PID? Flight reservation for Schengen visa One of the necessary documents for a Schengen visa for non EU citizens is proof of your flights, but it doesn't state the actual (e-)tickets (for which I already paid), or the reservation of flights.  So my question is, which option would be more acceptable (if I can provide any of them) to provide printed tickets to an EU member embassy or reservation of flights (reservation does not expect payment)? Root drive is running out of disk space. How can I free up space? What blocks can Endermen pick up and what blocks they can't? Origins of Tension First of all, I have a confusion about the definition and idea of Tension. For example, in my Physics Textbook, the idea of tension is written like this: &quot;Let's say there is a wire with a cross-section area of A and its length is $L_o$. If I hang the wire from a point and hang a weight of W on the other end of the wire, it will generate a repulsive force of T. This T is called the Tension Force.&quot; For me, this only explains to me what force will we call tension force, but it doesn't quite explain to me what is exactly this Tension force or where it originates from. Also, I have been doing pulley problems (Figuring out the acceleration of weights hanging from pulleys)  by first labeling all of the tension forces by $T_1$, $T_2$, and then doing some math with them to get my answer. But It bugs me that I am using these tension force values without even knowing where they came from or what they are. My idea was, maybe Tension force comes from the intermolecular bonds between the molecules of the rope, rubber or wire, etc. But even then, I have some different situational questions. For example:  Why you can't push a rope, copper wire against a wall?  What happens when I pull on two ends of a wire with 2 different forces?  Does the force increase proportionally to the applied force? Why does the wire break after a certain amount of force is exceeded.?  Does the idea of tension apply to everyday objects,(for example, a coffee mug, glass pane, etc). If it does, can I use something like a heat map or 3D equation to describe the tension in every point of the object? [Edit: more concisely, is there a mathematical way to show the various amounts of tension forces on a 2D (like a metal sheet) and 3D objects (like a football ) instead of a 1D object(like a string)].  If I take a metallic object like a paperclip or a thin wire. I can bend It multiple times and it eventually breaks into two pieces. I used to do this to pull of capacitors from old circuit boards. But does it happen with things like ropes or paper? If not, why it doesn't happen? ( Maybe it has something to do with intermolecular forces and the various internal stresses?)  Let's say I have an iron bar and I have thrown it out into space. Does that iron bar have equal stress distribution or still uneven or will the stress on all points become zero? (Is there even such thing as zero stress?) I have a number of friends on my battle.net account. However, sometimes I don't want to appear online to some (or all) of them. Is there a way to be hidden from your friends on battle.net? This has been puzzling me for a few days now, and I asked a graphic designer I knew before trying here and haven't been able to solve it yet. It's quite weird so please bear with me for a bit.  I am using Adobe Illustrator to create a logo, and I have something that can potentially work. I have used only the pen tool with strokes (no fill), but the issue is, when I view the logo at 100%, it looks pixellated! I know vectors should not even be able to appear pixellated, but they are. Here is the screenshot from AI, without exporting to anything:     That was just done using the snipping tool within Windows, and it's at 100% zoom.  I have anti-aliasing turned on from Edit > Preferences > General, and the document resolution is at 300 ppi (I get the same result with 72 ppi).   Could anyone help me with this?   If relevant: Using CS6, and have a 24 inch Dell U2412M at 1920x1200 resolution. I do have a 21.6" Samsung at 1680x1050 and I get the same result there as well. Let me know if you need any other settings, or if you would like the file, I'd be happy to provide it to be able to solve this problem. In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not?  I tried by assuming a set is equal to its interior points and contains its limit points.  A bounded set will not do since stuff like $[1,4]$ and $\{5\}$ will not work, though that is not really proof. Help please?  Anyway, it must then be unbounded.  If $a$ is a real number then $(a,\infty)$, $(-\infty,a)$, $[a,\infty)$ and $(-\infty,a]$ don't seem to cut it so it must be $\mathbb{R}$. How does the Deposit-User work? On SU there's the  which is suspended for quite some time. And I also know that it is used for spammers, but how does it work? Does the account of the spammer get 'linked' to Deposit? Do the answers get moved there? From axiom of regularity, how can I prove $x \notin y \lor y \notin x$ Ubuntu 19.04 show applications box Is there a benchmark tool for Ubuntu? Is there a benchmark tool to measure computer performance for Ubuntu like SuperPI, 3DMark, or PCMark? For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.  When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?  Update:  Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.  We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on. How do I merge 2 VMware disks in to a single one? I have installed an Ubuntu server on a VMWare workstation with a space of 20GB storage as seen below:    After the installation I've attached another 40GB worth storage to the existing Ubuntu server installation (seen below):    Now I want to add this 40GB to my / (root) directory it mean's  I want that my / directory became to 60G , like it will be resize from 20G to 60G (20G + 40G), i don't know what should i do, can any body help me ?  , my df -h result shows;  root@Userver:~# df -h Filesystem      Size  Used Avail Use% Mounted on /dev/sda1        18G  900M   16G   6% / udev            990M  4.0K  990M   1% /dev tmpfs           400M  308K  399M   1% /run none            5.0M     0  5.0M   0% /run/lock none            999M     0  999M   0% /run/shm none            100M     0  100M   0% /run/user root@Userver:~#   it must be 60G !  i have no graphic mode, then i can't use Gparted tools, i'm just have ssh access ! "It was me" or "It was I" It was ... salted the earth around your flower deck.    I wrote the second form "It was me". I have chosen the most "appropriate" answer to this question as somebody who is learning English.  This decision was made on a phonetic basis so it's for sure I made a mistake.   Surely you have a rational explanation to this exercise, maybe a grammar rule, or is it an English expression? Whenever I try to install anything with the Ubuntu Software Center, the following error appears.     Failed to download repository information      Check your Internet connection.      E:Unable to parse package file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_precise_multiverse_i18n_Index (1)