If $R \subseteq k[X]$, then $R$ is a finitely generated $k$-algebra.
Must $k$-subalgebra of $k[x]$ be finitely generated?
mysql_fetch_array, mysql_fetch_assoc, mysql_fetch_object Why is angular velocity of any point about any other point of a rigid body always the same? Can any one help? I purchased a solar powered PIR security light, with the intention of putting it inside my kitchen window. The PIR would be inside the window, but this does not work.  When I go outside to activate the light, it does not work. When I go inside to run my hand across the sensor, it works.  Will a PIR unit penetrate glass, or am I barking up the wrong tree? I have encountered this question while tring to figure out why every sequential space $X$, is a quotient spaces of a metric space. If I understand correctly, given a sequential space, every sequence (including it's limit) can be identified with the space $Y=\{0\} \cup \{\frac1{n+1}|n \in N\}$. So The space $X$ can viewed as a quotient space of $\bigoplus_{(x_n) \in C} \{(x_n) \} \times Y$ where $C$ is the set of all converges sequences in $X$. My question is, How can I show that this space is metrizable?  Thank you! Shir I have upgraded from Ubuntu 16.04 LTS to Ubuntu 18.04 LTS. I was surprised to see that the context menu entry "Make Link" (keyboard shortcut Ctrl+M) in Nautilus doesn't seem to exist anymore. It used to appear upon right mouse click on a file or folder and allowed the user to create a symbolic link. In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product.  I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$.   From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much.   Are there any categories of higher interest? I am new to compact sets, and I had some hard time trying to solve this:  Let $X$ be an infinite set; $T$ the discrete topology on $X$ .Prove that $(X,T)$ is not compact.  I know that I may consider the open cover $\{\{x\},x \in X\}$ and prove that it doesn't have a finite subcover,but I don't know how to write it exactly.  Moreover, I seriously can't understant why $X$ itself cannot be considered as a finite subcover?  Any help is appreciated. How to implement an abstract class in ruby? I know there is no concept of abstract class in ruby. But if at all it needs to be implemented, how to go about it? I tried something like...  class A   def self.new     raise 'Doh! You are trying to write Java in Ruby!'   end end  class B &lt; A   ...   ... end   But when I try to instantiate B, it is internally going to call A.new which is going to raise the exception.  Also, modules cannot be instantiated but they cannot be inherited too. making the new method private will also not work. Any pointers? I can't get grub menu to show up during boot Is there a way to set app-specific volume? A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$ I know that a perpetual motion machine (a device that produces work without an energy source) would violate several physical laws, the first that comes to mind being the . I also understand that a claim that perpetual motion is possible would probably not be congruent with the scientific method, occum's razor, etc.  What I am interested in is whether there exists some logical and intuitively meaningful (to they lay person) demonstration that perpetual motion is universally impossible, based only on axioms rooted in observable physical phenomena, rather than the obvious reasons why it is merely improbable and incompatible with some of our physical laws.  Certain violations of physical laws, such as travelling faster than the speed of light, can be shown to be unlikely as they would cause paradoxes or logical impossibilities, such as a . In that instance, the axioms needed to establish a paradox (time dilation) are empirically observable, and a counter-claim would require the existence of new paradox-resolving laws that are not established, or the permission of paradoxes which is difficult to imagine.  Conversely, the idea that some invention is impossible on the basis of its violation of a physical law alone is difficult because it requires a .  So, without relying on the assertion that a physical law is omnipresent (e.g. conservation of energy), is there some impossibility or logical paradox that would arise from the existence of a machine that could produce work without indefinitely without an energy source?  Finally, I'm very aware that perpetual motion is a bit of a dirty word and will likely touch a nerve with people, so please take the time to let me know how I can make this question less offensive and avoid the inevitable down-votes. Determine number of pages in a PDF file Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$ In the sentence: "Drinking and driving IS dangerous", why does the verb "to be" need to be singular? In the sentence: "Drinking and driving is dangerous". Drinking and driving are both subjects and the word dangerous is the object (I think) but why do I have to use the verb to be in singular? Find $P(n+1)$ for a polynomial P Fix a nonnegative integer $n$. Let $P(x)$ be a polynomial of degree $n$ (over the real numbers) such that for all $k\in\left\{0,1,\ldots,n\right\}$, we have $P(k)=\dfrac{k}{k+1}$. Find $P(n+1)$.   My attempt:  I consider a new polynomial $Q(x)=(x+1)P(x)-x$. This $Q$ is a polynomial of degree $\leq n+1$ satisfying: $Q(k)=0\quad\forall k\in\{0,1,2,\ldots,n\}$. Therefore $Q(x)=\lambda x(x-1)\cdots(x-n)$ for a certain real $\lambda$. Hence, $Q(-1)=\lambda \cdot (-1)^{n+1}(n+1)!$, but also $Q(-1)=(-1+1)P(-1)+1=1$. Comparing these, we get $\lambda \cdot (-1)^{n+1} (n+1)! = 1$, thus $\lambda=\dfrac{(-1)^{n+1}}{(n+1)!}$. Since $Q(n+1)=\lambda(n+1)!$ then $Q(n+1)=(-1)^{n+1} $, and from $P(x) = \dfrac{Q(x)+x}{x+1}$ we deduce: $$P(n+1)=\dfrac{(-1)^{n+1}+n+1}{n+2}.$$ I doubted that the answer were $P(n+1)=\frac{n+1}{n+2}$ but the term $(-1)^{n+1}$ wasn’t expected. Is there any mistake here? Does someone have an alternative proof? Set the QGIS display to show decimal degrees? Using QGIS 2.6.1 Brighton. I am trying to set the CRS for the project to display Decimal Degrees, where it is defaulting to Feet. I can change it in the Properties Dialog, but it does not change it, it is just staying on Feet. How do I change it?  Edit: this layer was imported from a shapefile where it was internally set to Feet. I was able to get the vendor to supply one with Decimal Degrees so my problem is gone. Still would love to know if there was a way to do it myself from within QGIS. R code to solve for probability of normal distributions? I don't understand which R code I am supposed to be using to figure these problems out. I brief explanation of what the code is doing would be amazing. The problems below are two different applications I'm expected to understand and I don't get either. Thank you!  Problem 1. Let Z~N(0,1) For each of the following, find c such that the equation is true.   a) P(Z&lt;c) = 0.8413  b) P(0&lt;Z&lt;c) = 0.3051  c) P(-c&lt;Z&lt;c) = 0.8664   Problem 2. X~N(3,4) compute  a) P(X&gt;3)  b) P(1&lt;X&lt;8)  c) P(-1.5&lt;X&lt;1) Does increasing voltage increase the torque of a brushless motor? 1.) I have a brushless motor with testing data at 12V. I want to know if I increase the voltage supply, will this increase the torque?   In my research I keep seeing "current controls torque, and voltage controls speed." But that doesn't seem quite right. Let's say I have a theoretical electrically powered unicycle that is weighted. With a 5V source and 2A, the motor can't move the unicycle. But now I increase the voltage to 10V, and I believe current will also increase by x2 if resistance is constant (V = IR). So now the torque increases and the unicycle can move itself. It that a reasonable sequence of events? Can I say that increasing voltage increases torque?   2.) A similar question: if P = VI = tau*omega, then P/omega = tau, so:  2*(P/omega) = 2*tau, which means increasing power will increase torque, correct? And since P = VI, increasing V will increase P, which will in turn increase torque. Is this accurate at all? Are \( and \) preferable to dollar signs for math mode?