Adding new classes when user changes type
How to add a class to a given element?
Lower bound for $\phi(n)$: Is $n/5  1$? Is it true that :  $\frac {n}{5} &lt; \phi (n) &lt; n$ for all $n &gt; 1$ where $\phi (n)$ is  .  Since $\phi(n)$ has maximum value when $n$ is a prime it follows that maximum value of $\phi(n)$ in term of $n$ is $n-1$ , therefore $\phi(n)&lt; n$ for all $n$. What is the best lower bound for $\phi(n)$? Show that a set is dense in $[-1,1]$ Show that $\{\cos\ n:n \in \mathbb{N}\}$ is dense in $[-1,1]$ by using the fact below:      Suppose $x$ is irrational. Then there exists $p_n,q_n \in \mathbb{Z}$   such that $\bigg|x -\frac{p_n}{q_n}\bigg| &lt; \frac{1}{q_n^2}$   I have no idea on how to apply the fact above to show the set is dense in $[-1,1]$. Can anyone help me? Let $f$ be a real valued and continuous function over $[a,b]$ with $a &lt;b$ such that $f(a)=f(b)$ , My question here is :     Question (claim): How to show that : there  exists  $c \in [a, \frac{a+b}{2} ]$ such that : $f(c)=f(c+\frac{b-a}{2})$.   Note: I have tried intermediate values but I didn't succeed. Shapefile Naming conventions in ArcPy? I have shapefile named `band-tailed_pigeon.shp which is fine and no problem on displaying or geoprocessing on Desktop but in ArcPy when I try to merge some layes to it I am getting this error   ExecuteError: ERROR 000354: The name contains invalid characters Failed to execute (Merge).  I tried to do the proccess with a simple name like sample.shp which it works but I HAVE TO keep the same name as we are using them in several projects. Here is the code which I have FinaloutLocation = &quot;E:\\GIS\\Data\\Final\\band-tailed_pigeon.shp&quot; inFeaturesToMerge = [dissolve_1, dissolve_2, dissolve_3, dissolve_4, dissolve_5] arcpy.Merge_management(inFeaturesToMerge, FinaloutLocation) 100 prisoners are imprisoned in solitary cells. Each cell is windowless and soundproof. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb from his or her own cell. Each day, the warden picks a prisoner equally at random, and that prisoner visits the central living room; at the end of the day the prisoner is returned to his cell. While in the living room, the prisoner can toggle the bulb if he or she wishes. Also, the prisoner has the option of asserting the claim that all 100 prisoners have been to the living room. If this assertion is false (that is, some prisoners still haven't been to the living room), all 100 prisoners will be shot for their stupidity. However, if it is indeed true, all prisoners are set free and inducted into MENSA, since the world can always use more smart people. Thus, the assertion should only be made if the prisoner is 100% certain of its validity.  Before this whole procedure begins, the prisoners are allowed to get together in the courtyard to discuss a plan. What is the optimal plan they can agree on, so that eventually, someone will make a correct assertion?    I was wondering does anyone know the solution for the 4,000 days solution. Also, anyone got any ideas on how to solve this?   Just wanted to know if anyone got any ideas how to solve this.  is offering prize money for a proof of a optimal solution for it. I suppose to make it clear the people want a plan that would mean on average it would take least amount of time to free the prisoner. The point is the average run time is as low as possible. Differential Geometry-Wedge product How can we prove the following relation for differentiating the wedge product of a p-form $\alpha_p$ and a q-form $\beta_q$$$d(\alpha_p\wedge\beta _q)=d\alpha_p\wedge\beta_q+(-1)^{p}\alpha_p\wedge d\beta_q$$ Let $A_n$ denote an $n \times n$ tridiagonal matrix.  $$A_n=\begin{pmatrix}2 &amp; -1 &amp; &amp; &amp; 0 \\ -1 &amp; 2 &amp; -1 &amp; &amp; \\ &amp; \ddots &amp; \ddots &amp; \ddots &amp; \\ &amp; &amp; -1 &amp; 2 &amp; -1 \\ 0 &amp; &amp; &amp; -1 &amp; 2 \end{pmatrix} \quad\text{for }n \ge 2$$  Set $D_n = \det(A_n)$  Prove that $D_n = 2D_{n-1} - D_{n-2}$ for $n \ge 4$. Which movie/TV show has an evil cyborg harvesting eyes from androids in a hidden town? I watched it maybe more than 8 to 12 years ago, but as far as I remember there was this place where androids (or maybe they were cyborgs) lived. (I think it was more like a hidden town). This guy finds them by accident (or at least I remember it that way) and tries to help them. I don't remember seeing any aliens or spaceships or anything like that. It was live-action, and I think US-based.  So, what I remember:   The bad guy is a human who has one red flashy robotic eye (not like terminators, it looks like a square eye patch above one of his eyes that functions like a second eye), he is collecting eyes from these androids/cyborgs. He also kills some of these androids/cyborgs but not all of them, some of them he just collects their eyes and doesn't kill (or get to kill). There was this woman who was a sexbot or something, she tried to give pleasure to the bad guy at some point, yet the bad guy added one of her eyes to his collection instead, but didn't/wasn't able to kill her. Towards the end she says something like "I used to give pleasure, now I only want to give pain" The human guy who found them tried to help them defend themselves from the bad guys, but they were in bad shape, many of them needed repair. To repair some of them, they needed spare parts they didn't seem to have, but one of the android/cyborg dudes said something like "we don't really bury our dead" and he brought a box of spare parts (or maybe android/cyborg body parts), they used these parts to repair some of the androids/cyborgs. There was this android/cyborg guy who was an assassin or something, but lost his fingers or hands, they couldn't find any hands compatible to his, but he said something like "I can still feel my trigger finger, just attach the weapons to my hands".   I remember it as a great movie/tv show, but I can't find anything about it. Can someone help me?  Please and thank you in advance... "register" keyword in C? What does the register keyword do in C language?  I have read that it is used for optimizing but is not clearly defined in any standard.  Is it still relevant and if so, when would you use it? What's the "official" explanation for Harry's (and everyone else's) absence from muggle school? So a bunch of kids all of a sudden "go missing" from school.  Harry Potter having attended school for a couple of years before starting at Hogwarts, one semester just doesn't show up.  How is this solved? What do parents say when a school teacher calls and wonders where their pupil is?  I assume that Hogwarts isn't an officially recognized school (though perhaps it is?), and due to education being compulsory, someone would do the math and find that low-hundreds of kids go missing in the transition from primary to secondary school every year. Is the wave function of a particle re-created after a measurement stops? "None is" or "none are" I was asked if the following sentences will create problems and what are the assumptions I have to make:  (d) There are grammarians in the room but none of them is an American. (e) There are grammarians in the room but none of them are Americans.  What is the problem in both sentences? They clearly have two main clauses which are connected by the coordinated conjunction 'but'. I know that grammar-wise, the 'is/are' don't really matter since both can be used with 'none', but would the tense in the first main clause 'are' affect the use of 'none'? In an alternating current, do electrons flow from the source to the device? If electrons in an alternating current periodically reverse their direction, do they really flow? Won't they always come back to the same position? It's $\mathbb{Z}\subseteq \mathbb{Z}[X]/(X^5-1)$ an integral extension? It's $\mathbb{Z}\subseteq \mathbb{Z}[X]/(X^5-1)$ an integral extension?  $\mathbb{Z}[X]/(X^5-1) \simeq \mathbb{Z}[x]$ where $x^5-1=0$.  So is an integral extension because $x$ is integral  over $\mathbb{Z}$. It's right? Problem 4.6, I. Martin Isaacs' Character Theory Number of ways to pair off $2n$ points such that no chords intersect For $n \geq 0$ evenly distribute $2n$ points on the circumference of a circle, and label these point cyclically with the numbers $1, 2 . . . , 2n$ Let $h_n$ be the number of ways in which these $2n$ points can be paired off as $n$ chords where no two chords intersect  I want to find a recursive formula for $h_n$.  First I find $h_1,h_2,h_3$ to see the recursive nature.    $h_1 = 1$ as their only one way to make one chord  $h_2 = 2$   Now $\require{enclose}      \enclose{horizontalstrike}{h_3=4}h_3=5$ ((1,4), (2,3), (5,6) case is missed in image)  I found a wikipedia link    and it is very close to my problem, But in this link you don't need to pair off $n$ chords.  But I can't get to come up with a recursive formula for my question.  If I would to guess I would say $h_n = 2 \times h_{n-1}$, And would this means that $h_4 = 8$ ?? A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm  dx = \frac \pi 2$$ Well, can anyone prove this without using Residue theory?  I actually thought of using the series representation of $\sin x$: $$\int\limits_0^\infty \frac{\sin x} x \, dx = \lim\limits_{n \to \infty} \int\limits_0^n \frac{1}{t} \left( t - \frac{t^3}{3!} + \frac{t^5}{5!} + \cdots \right) \,\mathrm  dt$$ but I don't see how $\pi$ comes here, since we need the answer to be equal to $\dfrac{\pi}{2}$. Why did Sirius Black's family live among Muggles? Multiple submit buttons in an HTML form Let's say you create a wizard in an HTML form. One button goes back, and one goes forward. Since the back button appears first in the markup when you press Enter, it will use that button to submit the form.  Example:    &lt;form&gt;   &lt;!-- Put your cursor in this field and press Enter --&gt;   &lt;input type="text" name="field1" /&gt;    &lt;!-- This is the button that will submit --&gt;   &lt;input type="submit" name="prev" value="Previous Page" /&gt;    &lt;!-- But this is the button that I WANT to submit --&gt;   &lt;input type="submit" name="next" value="Next Page" /&gt; &lt;/form&gt;     I would like to get to decide which button is used to submit the form when a user presses Enter. That way, when you press Enter the wizard will move to the next page, not the previous. Do you have to use tabindex to do this? App Store on Mojave: "We could not complete your purchase. cancelled" Whenever I try to Get then Install any app from the App Store, I get this message:     We could not complete your purchase.  cancelled   I did some searching online, but barely found any useful hits. What is this about?!