What is $\mathbb{R}^0$? More specifically, I assume, the Cartesian product: $$\prod_\emptyset \mathbb{R}.$$ Is there a convention here?  I ask because I've stumbled across a definition in topology where the $0$ dimensional closed unit ball and interior of the $0$ dimensional closed unit ball are equal and seem to be a point (although I'm just assuming the last part by context, hence my confusion).
(I'm going to link to this Stackexchange post concerning Nullary Operations: )  In general an  is a function $f:S^n \to S$, where $n$ denotes the arity of the operation, and a  is a function $f:S^0 \to S$.  It is clear that $S^n=S \times S \times \cdots \times S$ denotes the Cartesian product of a set with itself ($n$ times), however $S^0$ makes little sense to me.  One might define $S^0=\emptyset$. At first glance, that seems like a good choice. However then a nullary operation makes no sense, since it must be a function $f: A \to B$ where $A$ and $B$ are two sets representing the domain and codomain of the function. And note a function is defined in terms of relation, where a relation is a subset of the Cartesian product $A \times B$. So when we think of $A=S^0=\emptyset$ and $B=S$, so that we have a nullary operation $f:\emptyset \to S$, what is a a subset of $\emptyset \times S$? What even is $\emptyset \times S$? By definition of the Cartesian product, \begin{equation} \emptyset \times S = \{(a,b):a \in \emptyset\textrm{ and }b \in S\}. \end{equation} But there exists no $a \in \emptyset$! Therefore there exist no ordered pairs in our Cartesian product! Therefore $\emptyset \times S = \emptyset !$. Remember a function is a relation which is a subset of the Cartesian product. Therefore our relation is the empty set, and therefore also $f=\emptyset$.    Edit: We got the definition sorted out in the comments. Thanks especially to Hayden and Christoff for their great help. In summary, we figured out that $S^0=\{()\}$, the set containing only one element, the "empty $n$-tuple". Then if taking $S=\{1,2,...,k\}$, then a nullary operation $f:S^0 \to S$ is a function, i.e. some subset of $S^0 \times S = \{()\} \times S$. Looking at $\{()\} \times S$, this equals $\{((),1),((),2),...,((),k)\}$. Then a function is a special subset of this, so that each element of the domain (and there is only one element, namely $()$) is paired with exactly one element of the codomain. Therefore a function would have the form $f=\{((),a)\}$ where $a$ is one of $1,2,...,k$. Thank you for your help.
Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then (UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES)        $f$ maps a bounded sequence to a bounded sequence.   $f$ maps a Cauchy sequence to a Cauchy sequence.   $f$ maps a convergent sequence to a convergent sequence.   $f$ is uniformly continuous.      I choose all of the options as possible answers because the condition $\sup_{x\in \Bbb R}|f'(x)| \lt \infty$ forces $f$ to be uniformly continuous.(Because $f$ becomes Lipschitz and Lipschitz condition implies uniform continuity)  i.e. $\frac {|f(x)-f(y)|}{|x-y|} \le \sup_{x\in \Bbb R}|f'(x)|$ $ \forall x,y$.  Hence all other options are bound to be true.  Am I correct? How can I separate multiple quote blocks? I have a series of quotes which I want to display in two separate quote blocks.  &gt; Are questions about Microsoft Word Online on-topic here? I don't &gt; really see any posted here, but perhaps I am not searching correctly. &gt; - mweiss &gt;  &gt; Yes, they are. We even have a specific tag: [tag:word-online]. -ale &gt;  &gt; Thanks! I'm sorry I missed that tag. -mweiss   When I try to add these to my post, they display as a single combined quote.     Are questions about Microsoft Word Online on-topic here? I don't   really see any posted here, but perhaps I am not searching correctly.   - mweiss      Yes, they are. We even have a specific tag: . -ale      Thanks! I'm sorry I missed that tag. -mweiss   How can I separate quotes so that they aren't combined into a single quote block? Having problems with listings and UTF-8. Can it be fixed? Prove that for any positive integer $n$,       $$n^{1/n}  &lt; 1 + \sqrt{\frac{2}{n}}.$$   This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29.   Hint:      Use the binomial theorem. How many research interests should be included in a statement of purpose/objective I am wondering how many research interest should be included in a statement of purpose. Is it better to include just one core interest or to include up to three as in my case? How do I migrate my Keychain's "Local Items" to a new Mac? After preforming a Mavericks-Mavericks migration using Migration Assistant, I have a "Local Items" Keychain on my new machine in which many (perhaps most) items are out of date (using old passwords), and some are missing altogether. While I have access to a copy of the entire contents of the ~/Library/Keychains/ directory from my old machine, which presumably contains all the information I'm missing, I see no way of importing that information into my new machine.  How can I import the "Local Items" from my old Keychain into the Keychain on my new machine? I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The  contains a delta function in the formula, but  gives only the formula without the delta function,  $$\bar{E} = -\nabla \phi = \frac{-p}{4\pi \epsilon_{0} r^{3}} ( \hat{e}_{z}-3\hat{u}_{r}(\hat{u}_{r} \cdot \hat{u}_{z}) ) $$  Is it intentionally left out?  My notes mention that the dipole is with a circular thing $P=Q d$ where $d$ is distance and $Q$ is charge. What are the minimum root filesystem applications that are required to fully boot linux? Check if url has domain name and not an ip How can I split an equation over two (or more) lines list all suggested packages for currently installed packages How can I recover files from partitioned disk? Toolbar options are not displaying in Edit Profile - About me section Unreasonable RAM usage in Windows 7 I have a fresh install of Windows 7, and my RAM in normal startup goes up to 2.5GB. I looked at Task Manager, and there's at least 1.5GB missing from there and in Resource Monitor as well. There's no indication of what's using that RAM. Can someone tell me how I can find out what's consuming my memory? I believe it's a virus, but none of the free antivirus programs tracked it down.   I added the pictures below. If this is normal usage of Windows 7 as some answers imply, then I don't get this at all.   P.S. My Windows 8 Installation was running below 2.5GB, actually it was 1. something, until recently that I transferred some files from another computer which had the same problem that mine has now. I deleted Windows 8 and installed Windows 7, since I would do that anyway, and now I'm getting high memory usage in Windows 7 as well.   (Click images to enlarge) Dublin transit visa requirements for Indian citizens In Book 1, Artemis et al escaped from the time stop by drugging themselves to sleep.  In the same time stop, Julius Root "accidentally" plugged Briar Cudgeon with a tranquilizer dart.  Cudgeon fell asleep, but did not disappear from the time stop.  Why not? How to kill applications I have been playing around with Ubuntu installing new themes and and docks. Sometimes during the installation, whole screen just hangs. What is the best way in Ubuntu to Kill a process/application? Anything similar to Ctrl + Alt + Del of windows? U.s. boiler / Burnham boiler alterations i have a 203ncl-tei2 boiler installed in danvers Massachusetts. Regular elevation. This boiler also has a failed tejurland (universal control) powerventer motor on it that is only humming, not spinning. Looked for a dead animal, or nest and checked for free spinning of the squirrel cage it was fine. Its installed 3 feet from outside wall on the verticle, (installed wrong also has 2 seperate 90° bends after the venter, not my issue but may have caused the motor to fail). Ontop of the boiler we have a factory automatic damper assy.   So the 2nd part of the problem is I noticed the tei in the model number and assumed this ment i should expect to see a tejurland "universal" power venter installed and i do. I assume because its a universal control unit thats where they went wrong. This unit is atleast 200$ cheaper here than the model burnham uses.  I call burnham and ask what the original power venter model number is and they tell me what some pros may already know, (i didnt), that the tei2 is actually (t)taco, (el)elivation, (2) 2 is 2,000 ft  .Note:he was not.100% sure about the el. He was definetly sure PV would be in the model number and it was not tejurland. There is no chimney on this house, never was. Power venter wall thimble outside is very old definetly older than the 1 year old motor that failed. Burnham also said this is not a boiler that can be converted to a powerventer which is sometging i do not understand. He said if it does not have pv in the model number it should not have one. I have in the past converted a boiler with a factory reccomended power venter after a customer lost the chimney. I called and got lucky they had a option for it. I do not remember manufacturer but do remember also seeing it in the install manual as an addendum or seperate manual.  Is this a fact?   Anyone have experience with this?   What are the ramifications of an elevated boiler installed at sea level?   If i can not find another awnser, i will havemto tell customer they need a new boiler or i will have to walk away.   Why do they not just make an orifice kit for elevated boilers to work properly at higher elevations? Is it because they have to add another section to the boiler so it can put out the same  b.t.u.?  I know there is a reason. Seems like smaller orifice would fix the air fuel ratio along with a regulator adjustment. Its to simple of an awnser knowing that manufacturers make a complete different boiler for higher elevations.   ! Proving which QM interpretation is correct Let's assume that the existence of gravitons is theoretically proven or they are detected by LHC could one interpretaion be proven correct? By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.)