If I have a noun of cardinality 1, I am taught that I use the singular form.   E.g. I have 1 Apple    If the value is still 1, but represented as 1.0, do I still use the singular form?  E.g. I have 1.0 Apple  It seems laymen would say I have 1.0 Apples; is this just a trick of the brain but not valid English, or is plural correct even though the value is still 1?
Possible Duplicate:          I'm working on a website that displays distance to various locations to the nearest 1/10 mile. One of the developers (a non-native speaker) asked me an interesting question: If the distance is 1.0 mile(s), is that singular or plural? Clearly, anything other that 1.0 is plural, but is it:     1.0 miles away   or     1.0 mile away   The first option sounds more correct to my ear, but the voice of my elementary school language arts tells me that the latter option is correct.  Edit: Added the word 'away' to the example. It doesn't change the question, but might provide clarity for the answer. Also, regarding numbers between 0 and 1, after more thought, I'm not sure that the answer is as clear as I originally thought. See .
Use variable with TOP in select statement in SQL Server without making it dynamic Is there anyway to remove denied permissions from a group for a folder if I dont have access? Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism? Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w \mapsto \sum_i f_i(v) \cdot g_i(w)$ for all $(v,w) \in V \times W$. In the finite dimensional case it is easy to see this is an isomorphism by comparing dimensions on both sides. I'm aware that this is not an isomorphism in the infinite-dimensional case (see Relation with the dual space in the  on tensor products) but I must admit that I'm not sure why. Cardinal arithmetic does not seem help here and, even if it did, I would be much happier to see an explicit example of a functional in $(V \otimes W)^*$ outside the range of this map. I'm having trouble cooking one up myself.  Added: Let me elaborate on my comment about cardinal arithmetic. Suppose that $X$ and $Y$ are infinite dimensional vector spaces over a field $k$. I'm confident that $\dim (X \otimes Y) = \dim X \cdot \dim Y$ holds. It is clear that $|X^*| = |k|^{\dim X}$ since we may identify a functional on $X$ with a function from a basis for $X$ to $k$. Also I think I've convinced myself that if $\dim X \geq |k|$ then in fact $|X| = \dim X$ ie. the cardinality and dimension of a vector space agree when the dimension is larger than the cardinality of the ground field. Consequently, assuming say that $|k| \leq \dim X \leq \dim Y$ it seems we have $$ \dim(X \otimes Y)^* = |k|^{ \dim X \cdot \dim Y} = |k|^{ \dim Y} = \dim Y^* = \dim X^* \cdot \dim Y^* = \dim (X^* \otimes Y^*)$$  which implies that in fact $(X \otimes Y)^*$ and $X^* \otimes Y^*$ are isomorphic but, rather frustratingly, the obvious map does not do the job. Did I make a mistake here? If not, is it generally true (ie without making assumptions about $|k|$) that there exists some isomorphism $(X \otimes Y)^* \to X^* \otimes Y^*$? Word to describe the fear of unknown future Website logo text - best tag for SEO purposes? I'm trying to develop a new website in HTML5 adhering to SEO standards where possible and I'm having a little trouble trying to figure out what tag my site name and slogan should live in.  Originally I had my site name in a &lt;h1&gt; tag and my sloagan in a &lt;h2&gt;, but soon realised this wasn't fitting with the "each page is a separate document/article" approach.  The site name is the largest text on the page (sometimes the same as my article heading), but I've currently got the text in a &lt;p&gt; tag with CSS doing all the styling work.  My questions is, what should I do with the site name and slogan? Is the &lt;h1&gt; appropriate, then use another &lt;h1&gt; tag for each page title or should I continue using the styled &lt;p&gt; tag? Or should I convert this into an image and use that for the site name/slogan?  (Side note: on my index page I want the site name and slogan to be &lt;h1&gt; and &lt;h2&gt;respectively as this makes sense with the homepage...) Computer unstable at lower temperatures My computer is playing some strange joke at me, and I'm at a loss.  Everybody knows that hot temperatures causes many components to become unstable and even cause damage. When I built this machine, I was careful to do a bunch of experiments with different airflow configurations, with CPU-and-GPU intensive benchmarks running, and measuring the temperature at different points to find the best configuration.  Generally, the internal temperature of this machine is very close to ambient temperature (with 2 exceptions: the GPU always runs ~10°C above ambient, and the CPU goes to at most ~12°C above ambient when running CPU-intensive applications).  Now comes the problem: at really cold mornings. At cold mornings (~10°C (50°F) or below), the system shows really strange general instability. Beeps from the BIOS, the operating system displays strange messages about memory corruption and PCI devices that failed to initialize, and then, everything hangs. The computer hangs to the point that all USB devices turn of. Hitting the Reset button blanks the screen but the mainboard fails to proceed to the boot.  I never thought that low temperatures could cause computer instability. And 10°C is not that cold, it is still within operating temperatures of all components as far as I checked.  I'm quite sure it is related with the temperature. There were too many cold and warm mornings that demonstrated that the problem only showed in the cold ones. I'm working around this problem by letting the computer sit at the GRUB menu for some 5-10 minutes before letting the OS start, or blowing hot air inside the case if I'm in a hurry.  I don't know what should I approach next to find the source of this problem. Since it is difficult to reproduce (I need a cold morning, and the computer must be cooled down (powered off for many hours)), I need to plan ahead the experiment, since it stabilizes after a few minutes on.  Machine is an ASUS P5VDC-X equipped with Intel P4 64-bit 3GHz, 2GB of DDR2 RAM, a 500W PSU and an nVidia GeForce 9600GT. Everything else was already disconnected/removed before and the problem persisted, so the problem should be in one of these remaining parts.  Any help is appreciated! How to obtain the sum of the following series? $\sum_{n=1}^\infty{\frac{n^2}{2^n}}$ For each positive even integer $n$, set   $$P_n = \displaystyle {{n} \choose {\frac{n}{2}}}\frac{1}{2^n}.$$  Show that $\displaystyle \lim_{n \to \infty} P_n$ exists and determine its value.   Here's what I have so far: each $P_n$ can be thought of as the probability of tossing a fair coin $n$ times, and obtaining exactly $\frac{n}{2}$ heads. My expectation is that $P_n \to 1$, since in any large trial, one would expect to record just about as many heads as tails. But I'm not sure how to mathematically justify this hunch.  Hints or solutions are greatly appreciated. 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Not sure if this option ever existed, but the new QGIS version hides a lot of things somewhere else and this seems so simple that I'm kind of convinced it was possible to do this before. I have quite a large data set with ship tracks. I now want to analyse individual ship tracks (made from point layer with  Plugin). However, it is so crowded that it is hard to look at individual tracks. I don't want to save each single track as a new layer, so I wonder if it is possible to select one track in the Attribute table and display only that one on the map/hide the others? Fantasy book with shape shifting dragon warriors Set in a fantasy world with shape shifting Dragon kings and their warriors. They terrorize the humans, and kill those with magic.   The protagonist is the great-grandson of a powerful warlock. He finds out that he has magic, and fights with it to save his new friends, the Griffin, Blackhorse (I think that's his name), some creepy necromancer who is immortal, and Gwen (fellow sorceress), along with the kingdom that the Griffin rules in.   They fight the protagonist's father, the leader of the Dragon kings, the king's Duke, and the creepy sorcerer, who has a curse (when he dies, he changes side from good to bad, or bad to good). I have no clue what this is called, but I want to find out, since I'm out of things to read. How to enable root login again after changing password? I had enabled root login before and tried to change its password using this . But after successfully changing the password and locking it via sudo passwd -dl root. I cannot login back to SSH, it says Access is denied.  I have two users on my machine. root and admin, admin does not have su permission. When I try to use sudo on admin user, it says that admin is not in the sudoers file. That is why i need my su user back. Received an email intended for another person I'm not really sure how email routing works, but someone ordered something on Amazon and I received the email instead of them.  Or maybe we both got it, I don't know.  The order doesn't show up in my account, so I'm certain I wasn't charged for it, but I shouldn't be getting other peoples' emails.  We'll say that my email is johndoe1@gmail.com, and somebody who's email is john.doe1@gmail.com places an order on Amazon.  The confirmation email is sent to me at johndoe1@gmail.com.  I checked the email header, and it did say To: john.doe1@gmail.com which is not my email address.  At first I thought that Google ignores periods in email addresses, but I tested the account setup and it doesn't give any error when you put a period in the address.  I didn't create the account; I just used the "check availability" function and the address I chose with a period was fine.  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I am trying to learn CLI but I would like to retain access to the GUI for now. And, how can I switch between CLI and GUI anytime?