A vector space must satisfy closure under addition and multiplication. Sorry if this is obvious but does that mean that, assuming the normal rules of arithmetic and excluding the trivial examples like $V=\{0\}$  it is impossible for a vector space to have a finite number of elements? (since for every two elements $u,v \in V$ there must be another number $u+v \in V$)
I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space?
How do I add custom validations to Webform? I have created the country and state fields with the Webform module, and I want to add a field validation for the state field when user select US as country.   How can I achieve this? How to access the Keyboard Layout Options in 13.10+ Possible Duplicate:          There are a lot of really screwed up configurations out there, because people didn't take a minute to sit down and think about something as simple as the name for their Active Directory forest. I'm asking this question to self-answer so that this knowledge can be more readily available. If you think you have a better answer, by all means add your input (but be prepared to defend it!)   So, what should I name my new Active Directory? Low wifi signal in 14.04 LTS My site has 30,000 visits with 86,000 page views. If the bounce rate of 54%, then does it means only 15,000 users generated the 86,000 page views or 30,000 visits generated the 86,000 page views.  My website is www.cricandcric.com  Even though I have got 86,000 page views, my site's Alexa ranking is still getting worse day by day. How do I control that? Can there be more than one power series expansion for a function. I guess the answer is NO, for polynomials.  I know that there are more than one series expansion for every function.  But I am talking about power series here.  All Ideas are appreciated I need to delete some rows from one large table. The rows to delete shouldn't be in another table, example:  DELETE FROM LargeTable WHERE id IS NOT IN (SELECT DISTINCT foreign_id from EvenLargerTable)  but my server can't handle such blunt query, because there are almost a million records in the LargeTable and few million records in the EvenLargerTable  How can I solve it? How to draw a crisp, opaque hairline in JavaFX 2.2? Exploring ocean depths: is pressure an issue? Should it be? An upcoming adventure will have the party exploring the depths of the ocean. Breathing underwater shouldn't be an issue for them, as there are both items and rituals to take care of that, however, I was wondering what to do about about pressure, since none of these options mention it. I'll have to split my question into two, since they are closely related.   Is pressure an issue in 4e adventures? Is it mentioned in any of the published material? Has it been ruled anywhere that options that enable water breathing also grant immunity to pressure? Should it be an issue? This is somewhat subjective, but I really am wondering what other people have done. Would it be unresonable to assume that these options grant immunity to pressure? Should the players work to find another solution? $SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence?  I heard that this fact is related to the description of bosons and fermions. But how does it follow from the fact that $SU(2)$ is the double cover of $SO(3)$? How do I change file permissions on a FAT32 drive? Can I store all the content of my previous iPhone in a new iPhone using iTunes? For what natural numbers is $n^3 < 2^n$? Prove by induction Problem  For what natural numbers is $n^3 &lt; 2^n$?  Attempt @ Solution   For $n=1$, $1 &lt; 2$ Suppose $n^3 &lt; 2^n$ for some $n = k \ge 1$ It looks like the inequality is true for $n = 0$, $n = 1$ and $n\ge10$ But, how can I prove this through induction? Normality of subgroups and Sylow groups under homomorphisms... Prove or disprove: if $f:G \rightarrow H$ is a surjective group homomorphism, then   1) $f(A)$ normal implies $A$ normal.  2) $S$ is a p-Sylow subgroup of $G$ implies $f(S)$ is a Sylow-p subgroup of $H$.   I believe 1 is false since I believe we need injectivity, but I cannot think of a counterexample...  I believe 2 is true but do not know how to show it... Calculate E[X] from incomplete data? The exercise I'm doing describes the random variable $X$ as the following  | Number of cars |  0  |  1 |  2 |  3 |  4+ |  | % of families  |  15 | 45 | 25 | 13 |  2  |   Then it asks me to evaluate $E[X]$. But if there could be cases with 4+ cars, wouldn't that make it impossible to calculate the expectancy? SEO preference: does a site with WWW rank better than without WWW? After having read various articles on this site about WWW or Non-WWW website setup, ,  and , and after having setup a correct redirection rule in .Htaccess to at least eliminate duplicate contents (currently the non-www solution (e.g. http://example.com) I have a feeling that my site rank less since I changed this.   My confusion was strengthened by these two opponents:     &amp;     Could the change of www.example.com >> http://example.com have a factual impact on ranking?  say either is better or worse. Are they holding information back?  I am confused fellow folks...  As much as I would like to separate facts from fiction, I am aware that not all the good valid information is written. Not on google webmaster site and perhaps not elsewhere. If so it seems out of date!  Does or does not a WWW domain impact SEO/rank better than its NON-WWW version? (Given that you have setup a proper redirect to avoid duplicates and go with one) Passing more then 100 arguments to ColdFusion function throws an error Calculating roundness/compactness of polygon? I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.  The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.  So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.  But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.  For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?) How to get this only in small letters (lowercase)? I used the following code to give out the category name (without link) in a single post.  How would it be possible for the category name that is given out to only appears in small letters ?  I just think this has to be made with substring, but I dont get it :) Could somebody please help me :)  &lt;?php foreach((get_the_category()) as $category) { echo $category-&gt;cat_name . ' '; } ?&gt; Database design help. many-to-many relationship