entire function with constant modulus on circle -Schwarz reflection principle
Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$
Low-res texture blurry I unwrapped a simple plane and applied the following 80x80 pixel picture:    Yes, it's quite small. The problem is, the result is all blurry, like so:    How do I make it accurate to the pixel? Does Earth's Rotation Affect Its Shape? The question I am working on is, &quot;Consider the following. (a) Find the angular speed of Earth's rotation about its axis. rad/s (b) How does this rotation affect the shape of Earth?&quot;  I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. I tried to search my textbook for the answer, but could not find anything. Is there an actual effect on the shape? I never dealt with Android devices before so I'm completely newbie at it, please comment if something is too simple.  Say I purchase , that has got Android Froyo 2.2 installed in it.  If there is a new version of Android released tomorrow or next month, will I be able to update it (like I will be able to update my iPod Touch 4 to iOS 5)? How to sum a variable by group libsensors5 or libsensors-config upgrade problem Proof by induction on a rooted tree with even number of vertices Let $T = (V, E)$ be a rooted tree with an even number of vertices. Prove by induction that there exists a set of edges $S \subseteq E$ such that every $v \in V$ is incident with an odd number of edges in $S$.  I was wondering if my argument is sufficient to prove the statement above:  Base case: $|V|=2$, there is only 1 edge and both vertices are incident with 1 edge $-&gt;$ $S \subseteq E$.  Inductive step:  Let $|V|&gt;2$, implying that $T$ has at least 2 external vertices. Let us choose two arbitrary external vertices $w,u \in V$. Such vertices are only incident to 1 edge each, $e_w$ and $e_u$. We can say that $e_w,e_u \in S$.   By induction, there exists a set $S$ in $T'=T-\{w,u\}=(V',E')$ such that every $v \in V'$ is incident with an odd number of edges in $S$ because removing 2 vertices from a tree with an even number of vertices maintains the property $|V|=2n$. Provide an easily discoverable way to get the full URL to an answer In  when Ambassador Sarek became critically ill, why didn't they send him home or send for a Vulcan medical expert? Since they were hosting many prominent political figures from many races, it seems strange that they did not have much medical support for a race so common as Vulcans (who are part of the Federation). Remove duplicates from a List in C# Anyone have a quick method for de-duplicating a generic List in C#? We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$.   I'm now trying to say something about the possible cardinalities of subfields of $F$. I can see that there is a subfield of cardinality $p^s$ for each $s$ that divides $r$, given by the fixed field of the group generated by $\phi^s$, where $\phi$ is the Frobenius automorphism.  Now suppose $K$ is a subfield of $F$. Then (since both are additive groups), Lagrange gives us that $|K|$ divides $|F|$, so $|K| = p^t$ for some $1 \leq t \leq r $ (alternatively, $K$ contains $\mathbb{F}_p$ and so is a vector space over $\mathbb{F}_p$ and is thus isomorphic to $\mathbb{F}_p^t$, where $t = [K:\mathbb{F}_p]$). By considering the multiplicative group of units of $K$ and $F$ respectively, we get that $ p^t - 1$ divides $p^r -1$. I want to make the leap to $t|r$, but I'm failing to see why this needs to be true. Any help would be appreciated. What is the meaning of the density of a distribution at a point? I know how to use PDFs to calculate probabilities, but I don't think I understand them. For example, at $X=0$ the PDF of the standard normal distribution is $\approx 0.4$. Does this have any useful meaning? Spanning Trees: Prove if $e_1$ is an edge in $X$ that isn't in $Y$, then there exists an edge ... Say $X$ and $Y$ are two different spanning trees of a simple and connected and undirected graph $G$. Prove that if $e_1$ is an edge in $X$ that isn't in $Y$, then there exists an edge $e_2$ in $Y$ that isn't in $X$... so the graph $(X − e_1) + e_2$ (obtained from $X$ on replacing $e_1$ by $e_2$) is also a spanning tree of the original graph, $G$. I can't figure out how to turn on the Bluetooth on my Nikon D5600. I have updated the firmware to the latest 1.01 and still the same problem.   Bluetooth is dimmed out on the camera and will not turn on. I get an error message saying, "This option is not available at current settings or in the camera's current state".  I have tried everything I can think of and no go. I can't find help for this problem anywhere on the internet and still no reply from Nikon.  How do I turn on my camera's Bluetooth? The number of solutions to an $n^{th}$ order differential equation. For an $n$th order differential equation, why are there always $n$ solutions? Why exactly $n$, not $n - 1, n+1$ or infinite many?    Addendum by LePressentiment :   This is motivated by P176 on Strang's Intro to Lin Alg, 4th Ed. An $n$th order differential equation, how many basis functions does it have? I'd guess $n$, because the diffl eqn have $n$ linearly independent solutions (predicated on Julián Aguirre's answer), all of which look to span the nullspace/solution space of the homogeneous ODE. I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.  Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.  Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.   I have not been able to get the converse part and would like to see the proof for the converse part. Use of would in sentences Saleforce Too many future calls: 11 Redefine default color of todonotes How to run applications as root? I am having some strange issue with Kate and Kwrite. When I click on Open File, it crashes with segmentation fault.   I am a complete newbie to Linux, and I think the issue is that I am not running the application as root.   How do I run applications as root in Ubuntu? Is it bad practice to do this? What is the purpose of the whole root thing, where even though we need to use root so frequently, it is not utilized as default? Calculating Mean for Multiple Rows in ArcGIS for Desktop? I am trying to calculate a mean across a multiple number of attribute rows and have this value added to a new field that I have entered in the attribute table. I have an attribute table which has a number of different ecoregion types associated with them.  For example:  If I select "Arid Canyonlands" there are 3 attribute rows that are selected, each with their own "mean". I am trying to add a new field to the attribute table which will take the "mean" of these three and populate that value in the new field created. I could manually do it through an editing session and using a calculator but just wanted to know if there is an easier way to actually do it through field calculator?