In the rank nullity theorem for $T \colon V \to W$, $V$ is finite dimensional. What if $V$ is infinite dimensional?
Does the rank-nullity theorem hold for infinite dimensional $V$?
$VAR vs ${VAR} and to quote or not to quote I'm trying to solve hard combinatorics that involve complicated factorials with large values.  In a simple case such as $8Pr = 336$, find the value of $r$, it is easy to say it equals to this: $$\frac{8!}{(8-r)!} = 336.$$  Then $(8-r)! = 336$ and by inspection, clearly $8-r = 5$ and $r = 3$.  Now this is all and good and I know an inverse function to a factorial doesn't exist as there is for functions like sin, cos and tan etc. but how would you possibly solve an equation that involves very large values compared to the above problem without the tedious guess and checking for right values.  Edit: For e.g. if you wanted to calculate a problem like this (it's simple I know but a good starting out problem) Let's say 10 colored marbles are placed in a row, what is the minimum number of colors needed to guarantee at least $10000$ different patterns? WITHOUT GUESS AND CHECKING  Any method or explanation is appreciated! I have a edge loop that has vertices on it. Is there a way I can separate the vertices on the the edge evenly without moving the end points? I would also like for the vertices to maintain the current shape. Javascript replace all "%20" with a space SE Android app for tablet has bad formatting on feed When my phone starts up there are several application/services running in the background that I would rather not.  And when I press the Home button while using an app, that app usually goes into the background without exiting.  How can I stop these apps from running? How to draw simple table How can I check if the Active Object has a specific modifier so that when I run my code, it doesn't add the modifier again to the object? It should add the modifier only to the object which does not use that modifier. How to make a file executable? How can I make my file so that I can double click on it, and it runs. It is a .sh script, but I also have files that say:  executable (application/x-executable)  in the description of what they are.  I can't run any of these from terminal, or by double clicking.  If possible, I would like a way using either the GUI or a Terminal, but not a combination of the two.  Here is a screenshot of what I see when I right click then go on properties. The file first:    And the shell script here:    NB: I accept that this is a duplicate (I was looking for it, and couldn't find it, so asked + answered it, hoping that I would find it) however, I don't think the question about .desktop files is a duplicate. If I play a spell with {X} in the mana cost, and I have a card out that reduces the casting cost by 2 generic mana, what happens to the X? Does it start at -2?  For example: I have  in play. If I play  for just 1 blue mana, do we draw 2 cards? I know that this question is a bit old or maybe overdone, but I'm running the latest edition of Blender and when I use the Cycles engine I get really fuzzy results and I can't find a simple, non-compromising solution to the quality.  I don't know exactly what I'm doing wrong here, but I'm sure others share my frustration.  Leaving everything in default settings, I started a new project and switched to the Cycles render.  I duplicated the cube and set colors for two Diffuse BSDF shaders, then went ahead and made walls and a floor all sharing the same diffuse shader.  I made two windows to emit blue light.  The screenshot has gaps between the walls because of the setup:  I'm aware that it's considered very incomplete and many have come to terms with some amount of noise, but is this normal?  Furthermore, with default settings or not, I can't change the amount of noise produced.  Supposedly, using "progressively refine" in the rendering tab should make it refine itself indefinitely, however Cycles only counts to 10 refines and then stops, creating a render which is a pixel-perfect match of the ordinary rendering method when it's completed.  I understand that there are tricks that can be used to reduce noise, but at the cost of extra processing time and usually small quality conflicts, like blurring.  My question essentially boils down to is this Cycles' current limit? Long ago someone demonstrated the following card trick with a standard 52-card deck:  (1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see.  (2) The assistant puts exactly 4 of those in a pile and passes it to the performer.  (3) The performer can then identify the 5th card.  There are a few ways to execute this trick, one of which goes as follows.  (1) The assistant finds a suit common to 2 of the 5 cards, with values $x$ and $y$ modulo $13$.  (2) He then removes one of them, say $x$, such that $x-y \in [1..6]$ modulo $13$.  (3) He then puts the other one at the top of the pile.  (4) And he puts the 3 other cards after that in a permutation corresponding to $x-y$ modulo $13$.  Since (1) works by pigeonhole, (2) works because $\frac{13-1}{2} = 6$, and (4) works because there are 6 possible permutations of 3 cards, the trick succeeds. My question is whether the same trick can work with more than 52 cards. It is not obvious to me how to do so, nor compute the maximum number of (pairwise distinct) cards possible. Also, it seems to me that even if the trick can be made to work for $n$ cards, it may not work for $n-1$ cards! Any ideas? tikz package causing error in IEEE access template I have been trying for a while to load the tikz package to the IEEE ACCESS template. Whenever I use the command \usepackage{tikz}, I am getting the error   /usr/local/texlive/2017/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex, line 33 Missing number, treated as zero.   It was working fine with the IEEEtran template and I was able to draw my circuit diagrams in that.   Anyone has any idea how to solve this issue? A lady and a monster A famous problem:  A lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is $v_m$, and the speed of the swimming lady is $v_l$. The goal of the lady is to come to the ground without meeting the monster, and the goal of the monster is to meet the lady.  Under some conditions on $v_m,v_l$ the lady can always win. What if these conditions are not satisfied?  Edited: the monster cannot swim. If the conditions are not satisfied, then monster can always perform a strategy such that the lady will not escape the lake. On the other hand this strategy is not desirable for both of them because they do not reach their goals. As there was mentioned, this deals with undecidability of the problem. On the other hand, if you imagine yourself to be this lady/monster, you can be interested in the strategy which is not optimal. What is it? If there are such strategies in the game theory?  Edited2: My question is more general in fact. If we have a game with one parameter $v$ when two players $P_1, P_2$ are enemies and if $v&gt;0$ then for any strategy of $P_2$ the player $P_1$ wins. If $v\leq 0$ then for any strategy of $P_2$ there is a strategy of $P_1$ such that $P_2$ does not win and vice versa. I am interested in this case. From the mathematical point of view as I have understood the problem is undecidable since there is no an ultimate strategy neither for $P_1$ nor for $P_2$. But we are solving somehow these problem IRL. Imagine that you are a lady in this game - then you would like to win anyway even while knowing that your strategy can be covered by the strategy of the monster. On the other hand, the monster knows that if he will cover all strategies of the lady she will never reach the shore and he will never catch her. I mean they have to develop some non-optimal strategies. I hope now it's more clear. If I am introducing someone to my daughter's husband's parents can I say "Hi, I'd like to you meet my ___". In-laws would not work here because they are my child's in-laws not my own. Is there a word for this relationship? Exam with 18pt main font Compute approximately the probability that more than 50 of the observations of the random sample are less than 3. Let $X_1,X_2,...,X_n$ be a random sample of size $n = 72$ from a distribution with probability density function $f(x) =\begin{cases} 1/x^2,&amp; 1 &lt; x &lt; infinity\\0,&amp;\text{otherwise}\end{cases}$ Compute approximately the probability that more than $50$ of the ob- servations of the random sample are less than $3$. What is the best way to clean the sensor on a interchangeable lens camera (mirrorless or digital SLR)? I've got a couple of spots of dust on the sensor of my Canon 20D. A rocket blower isn't shifting them, and I'm ridiculously scared of some of the more direct methods for cleaning sensors — what methods/systems have you tried, and how successful have they been? Is changing your passport a good strategy for removing a negative immigration history in Schengen? When certain nationalities of people want to travel in the Schengen zone, they are required to apply for a Schengen visa. This is a straight-forward process for most, but when an applicant has had problems with removals or refusals they may be reluctant to apply because they want to avoid the complexities of serial visa refusals. This reluctance may lead some into wondering how their history can be expunged, and this line of thought inevitably leads to the idea of getting their passport replaced with a new one.  This sounds like a great formula for people who have had prior refusals or removals.  In essence they can become 'reincarnated' as a first-time traveller and hence avoid the unpleasantness of revealing their prior interactions with the Schengen authorities.  The relevant parts of the application form are shown here...    Nothing asks about a prior removal or refusal, and the applicant is left with simple denials for 3 questions in order to present themselves as a first-time traveller.  Assuming the person is happy to set aside the moral issues associated with denying their history, are there any holes to this strategy?  This question is scoped to Schengen visa applications so that it will not be unmanageably broad, but answers from other regimes (particularly the affluent Commonwealth and the USA) are acceptable as long as the main points are relatively consistent with Schengen. Equation of a rectangle I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.