As is typical in this kind of thing I have a node tree filled with tangled curvy node connections going every which way. Even using reroutes it is somewhat difficult to read.  I've seen a number of people who have the node connections appear far more like a straight line than a bezier curve...how would I replicate this feature?
Unitl now I haven't found the option to enable the curved representation of connectors between nodes. The  doesn't mention such an  option. Is there a keyboard shortcut?
What is the relationship between symmetry and degeneracy in quantum mechanics? Doubly stochastic matrix proof A transition matrix $P$ is said to be doubly stochastic if the sum   over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$.   If such a chain is irreducible and aperiodic and consists of $M+1$   states $0,1,\dots,M$ show that the limiting probabilities are given by   $$\pi_j=\frac{1}{1+M},j=0,1,\dots,M$$   I have no idea how to prove it but   If a chain is irreducible then all states communicate i.e $$P_{ij}&gt;0\space \text{and}\space P_{ji}&gt;0\space\forall i,j$$ If $d$ denotes the period of any state, if a chain is irreducible aperiodic, then $d(i)=1\forall i$   If $P_{(M+1)\times (M+1)}$ matrix and $\pi$ is the stationary distribution  $$\pi_j=\sum_iP_{ij}\pi_i\space j=0,1,\dots,M+1$$  but how I can get this expression? I'm rather interested in getting my feet wet at the interface of  and . Can anybody give me references to some good books on these topics? I'm looking for very introductory technical books. My proof is very different from my reference, hence I am wondering is I got this right?  Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of continuous function on closed set is closed.  Let $D$ be a closed set, Consider a sequence $x_n \to x_0$ in which $x_n \in f^{-1}(D)$, and we will show that $x_0 \in f^{-1}(D)$.  Since $f$ is continuous, we have a convergent sequence $$\lim_{n\to \infty} f(x_n) = f(x_0) = y.$$  But we know $y$ is in the range, hence, $x_0$ is in the domain. So the preimage is also closed since it contains all the limit points.  Thank you. Why don't all Jedi stick around as force ghosts? In many of the Star Wars movies, Jedi come back as Force ghosts.    Why don't they stick around forever, giving advice to new generations, as well as countless knowledge from thousands of years? Is there a way of conditionally triggering a compile-time error in verilog? What effect does the Barr body have, in relation to female Turner syndrome? Why do persons with  have  developmental abnormalities, when normal XX-females do not, even though they only have 1 active X chromosome? From what I know, one X-chromosome is inactivated into a Barr body in normal XX-females, but these obviously don't show the symptoms of Turner syndrome. What function does the inactive chromosome have? How Do I get into the castle walls? I already have the pink and red leather gloves, the armor, the crown, and the monkey's staff (and they're all enchanted). Regardless of how many health potions I use, I always die. Help! What happens when a creature with "Whenever a creature dies" dies simultaneously with multiple other creatures? has the following ability:     Whenever a creature you control with a +1/+1 counter on it dies, return that card to the battlefield under your control at the beginning of the next end step.   I had a number of creatures on the battlefield, including Marchesa herself, with one or more +1/+1 counters on them when  was played, destroying each creature on the board.  When Marchesa and several other creatures would be killed simultaneously,   Does Marchesa's ability trigger for herself? Does Marchesa's ability trigger for the other creatures under my control with +1/+1 counters?   In attempting to resolve the issue, we tried to look things up, leading to , which cites Rule 603.6d as proving that Marchesa's ability should trigger both for herself and for each other creature with +1/+1 counters.  Unfortunately, none of us could actually understand Rule 603.6d, aside from this example:     Two creatures are on the battlefield along with an artifact that has the ability Whenever a creature dies, you gain 1 life. Someone plays a spell that destroys all artifacts, creatures, and enchantments. The artifacts ability triggers twice, even though the artifact goes to its owners graveyard at the same time as the creatures.   The example certainly seems analogous to me, but without understanding the actual rule, rather than just the example, we weren't too sure.  A complete answer has to not only answer the question, but back up that answer with citation and explanation of the official rules on the matter. Extremely difficult logic question using conditional statement In this problem, we will only consider (real-valued) functions with domain $\mathbb{R}$. We define two new concepts. Let $f$, and $g$ be two functions.  We say $f$ is a rival of $g$ if   &emsp;&emsp;&emsp;&emsp; $\exists x\in \mathbb{R} \ \text{s.t.} \ \forall y\in \mathbb{R}, x \lt y \implies \lvert f(x)- g(x)\rvert\lt\lvert f(y)-g(y)\rvert$    We say $f$ is a frenemy of $g$ if  &emsp;&emsp;&emsp;&emsp; $\forall x\in\mathbb{R} \ \text{s.t.} \ \  x \lt y \ \ \text{AND} \ \ \lvert f(x)-g(x)\rvert \lt \lvert f(y)-g(y)\rvert$  Below are four claims. Which ones are true and which ones are false? If a claim is true prove it. If a claim is false, show it with a counterexample.  a) If $f$ and $g$ are any two functions and $f$ is a rival of $g$ then $f$ is a frenemy of $g$. b) If $f$ and $g$ are any two functions and $f$ is a frenemy of $g$ then $f$ is a rival of $g$. If $f$ and $g$ are any two functions and $f$ is a rival of $g$, then $g$ is a rival of $f$. Let $f$, $g$, and $h$ be any three functions. If $f$ is a frenemy of $g$ and $g$ is a frenemy of $h$ then $f$ is a frenemy of $h$  What are the four claims stating? Especially for c), the concept is not even present I am having a misunderstanding on how to solve them? If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of $n$ proper subspaces of $V$ How to get scroll position from GridView? Limit of a decreasing sequence of measurable sets. Let $(X,\mathcal{A})$ be a measurable space, with measure $\mu$. Let $\{E_n\}_{n \in \mathbb{N}} \subseteq \mathcal{A}$ be a sequence of measurable sets, with $E_{n+1} \subseteq E_n, \ \forall n \in \mathbb{N}$, that is a decreasing sequence, and $\mu(E_n)=+ \infty, \ \forall n \in \mathbb{N}$. Let $E=\bigcap_{n \in \mathbb{N}} E_n$ the limit set of the sequence, $E$ is measurable for definition. Is true that $\mu(E)=+\infty$? If not in all cases, is true with $X=\mathbb{R}^{N}$, $\mathcal{A}$ the collection of Lebesgue-measurable sets, and $\mu$ the Lebesgue measure on $\mathbb{R}^N$? How to hook into the CPT's title placeholder? Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$ Limit of the geometric sequence $\{r^n\}$, with $|r| < 1$, is $0$? Prove that the $\lim_{n\to \infty} r^n = 0$ for $|r|\lt 1$.   I can't think of a sequence to compare this to that'll work.  L'Hopital's rule doesn't apply.  I know there's some simple way of doing this, but it just isn't coming to me. :( I've already see a  by Marko Riedel which I list it follows:     The standard way to treat these sums is to integrate   $$ f(z) = \frac{1}{(z+\alpha)^2} \pi \cot(\pi z)$$   along a contour consisting of a circle of radius $R$ and with $R$ going to infinity and hence being larger than $\alpha$, where the circle does not pass through the poles on the real axis.   Now along the semicircle in the upper half plane we have   $$|f(z)| \le \frac{1}{(R-|\alpha|)^2}\pi \left|\frac{e^{i\pi R\exp(i\theta)} + e^{-i\pi R\exp(i\theta)}} {e^{i\pi R\exp(i\theta)} - e^{-i\pi R\exp(i\theta)}}\right|= \frac{1}{(R-|\alpha|)^2} \pi  \left|\frac{e^{2i\pi R\exp(i\theta)}+1}{e^{2i\pi R\exp(i\theta)}-1}\right| &lt; \frac{1}{(R-|\alpha|)^2} \pi   \frac{1+e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}} {1-e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}}$$   This last term is clearly $O(1/R^2)$ as $R$ goes to infinity as the quotient of the two exponentials goes to one since $\exp(-R)$ vanishes and there is no singularity when $\theta = 0$ or $\theta = \pi$ as $R\cos\theta = \pm R$, which is not an integer by the assumption that the circle avoids the poles and hence cannot be one.     My question: It is clear that for each $z=Re^{i \theta}$ , $f(z)=O(1/R^2)$ ，but can this guarantee that the last term on RHS is uniform bounded ? Since $\theta$ varies on a compact interval $[0, \pi]$ , I want to show that for each $ \theta \in [0, \pi]$ , and a fixed positive number $M \gt 1 $ there exist an open ball contains $\theta$ and a fixed $R_{\theta}\gt 0$ , for every $R \ge R_{\theta}$ and $x \in $ the open ball $$\frac{1+e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} {1-e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} \le M$$  For $\theta \neq 0,\pi$ , it is easy to find the desired $R_{\theta}$ and the open ball , but if $\theta = 0$ or $ \theta = \pi$ , for every open ball containing $\theta$ , it behave erratically near $\theta$ , I have no idea how to deal with this . Javascript add js libraries by code I had applied for a PLAB (Doctors) Visa for the UK to appear in PLAB test. I got refused on the basis of two recent large deposits in the bank statement I provided (my mothers). In reality, these are my own savings which I had deposited in her account to make it more strong. The only objection I got was that I must provide evidence for the source of these deposits. Other than this, I have previously travelled to the UK and USA as well as other parts of the world and my travel history is fine. Can someone please advise on how to proceed and re-apply? I have already read        and many others   But I still have the following problem:   On one hand I've been told that if pure rotation takes place then the object goes on forever. This must imply no frictional force in any form acting on the body. On the second hand I've been told that during pure rotation(no external forces like wind) on a plane surface friction acts opposite to velocity of center of mass (accompanied by change of normal) and slows it down.     Both seem logical enough: The point of contact doesn't slip therefore no friction is required; So no retarding force and it must go on. Things don't go on forever; Friction must be present to slow it down.    So, which hand is correct? Or is there a third hand altogether?