Gradient of a matrix expression
How to take the gradient of the quadratic form?
Are there any published details of Kenobi's communication with Lars family on Tatooine before ANH? Appropriate data transformation Do you pluralize the singular possessions of / items or people associated with individual members of a plural group? Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? The book "A First Course in Abstract Algebra" by Fraleigh says      If $G$ is a finite subgroup of the multiplicative group $\langle F^*,\cdot\rangle$ of a field $F$, then $G$ is cyclic. In particular, the multiplicative group of all nonzero elements of a finite field is cyclic.    I wonder why that is. Take $a,b\in F$ such that $a^2=b^2=1$. Then $\{1,a,b,ab\}$ is a non-cyclic subgroup of $F$. Where am I going wrong?  Thanks in advance! This looks like a question asked , but it isn't.  $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad &amp; \text{ if } n&gt;1 \\  1 &amp; \text{ if }n=1\end{cases}$$  My professor gave this to me in class yesterday. This is where I'm stuck:   $$T(n) =  T(n^{1/2})+1 =  T(n^{1/4})+2 = T(n^{1/8})+3 = \dots =   T(n^{1/ 2 ^k})+k$$  Now this will continue till  $$n^{1/ 2 ^k}=1$$  How do I proceed ahead? If I take log on both sides, then RHS becomes zero. How do I solve the relation?  I don't want to use the Master theorem, I want to know where and why am I stuck. Why do Oreo crumbs float to a single glob at the very center in a glass of milk? I had Oreos and milk a while ago and left my half-full cup of milk out on the counter. Afterwards I noticed that the crumbs had surfaced in a circular coin-sized glob, and just now I looked again to see all of the crumbs seem to have compressed into a more dense glob at the very center of the cup.   Why does this happen? ! LaTeX Error: Undefined color `BLACK' i have written all the text in black color, but i get this error several times in the same page:  ! LaTeX Error: Undefined color `BLACK'.   i tried to solve it by writting this: \definecolor{black}{gray}{0,0,0} but it didn´t work.   what should i do? Is there a mass spectrometer software that can measure the mass of a proton How to create my own list of things Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational? Suppose we wish to prove that the square root of two is irrational. We begin by assuming that it is rational.  Namely, where both $a$ and $b$ are integers $$\frac{a}{b} = \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GH9aqpdaGcaaqaaiaaikdaaSqabaaaaa!31D3! $$  Why is it so important that both $a$ and $b$ be coprime and the fraction be irreducible?  That is, if $a$ and $b$ are not coprime and the fraction is indeed reducible, then$$\frac{a}{b} \ne \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GHGjsUdaGcaaqaaiaaikdaaSqabaaaaa!3294! $$ Do Israeli security agents interview all those flying to Israel or is it restricted to El Al flights? The one thing I hate about flying to Israel are the security checks at the origin airports. I don't mind X-rays and deep luggage searches, but there's also an annoying interview you have to "pass". The security folks can't really turn you down (especially since I'm a citizen), but they always make it a pain in the butt, because for some reason I fit their profile of a suspicious person.  Can I avoid the personal interviews if I book on a non-Israeli airline? Or perhaps a low-cost airline? Rayleigh-Jeans Law My question is simple, why do we believe  to be absurd? Is the Ultra-violet catastrophe incorrect or is it only because we can not create or know of a perfect emitter? I am a bit confused as to why Quanta was necessary. Since the Denoise feature was added to Blender I only use the default setting with it and it really does the work. But sometimes the denoise is noticable on some of my renders. How can I overcome the blurred effect? Denoising for me is a big help because my system is quite aged. Under what conditions does loot despawn? Is there any difference between "offense" and "offence"? "Offense" vs. "offence", which is more correct? If both are correct, are there any differences in shades of meaning and/or usage? "Didn't use to get" or "Didn't used to get"? I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function. Show that     $$ \int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)&gt;t\right\})\, dt. $$     (The right integral is to be read as an Lebesgue-integral.)      Hello, my idea is to approximate $f$ by the functions $$ f_n:=2^{-n}\sum\limits_{k=1}^{\infty}\chi_{\left\{x\in X: f(x)&gt; k/2^n\right\}}. $$  With this approximation by simple functions I got $$ \int\limits_X f\, d\mu=\lim\limits_{n\to\infty}\int\limits_X f_n\, d\mu=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{\infty}\frac{k}{2^n}\cdot\mu\left(\left\{x\in X: f_n(x)=k/2^n\right\}\right)\\=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{\infty}\frac{k}{2^n}\cdot\mu\left(\left\{x\in X:f(x)&gt;k/2^n\right\}\right) $$  This is the point where I do not come along anymore... is that right to this point and if yes: How can I continue in order to get the desired right side?  With regards Does the equivalence $$\neg\neg (P \rightarrow Q) \leftrightarrow (\neg\neg P \rightarrow \neg\neg Q)$$ hold in propositional intuitionistic logic? In propositional classical logic the equivalence holds obviously since $P \leftrightarrow \neg\neg P$. However, the similar-looking implication $\neg\neg (P \vee Q) \rightarrow (\neg\neg P \vee \neg\neg Q)$ is . The validity of the backward implication comes up in  as well. This was a 'young-adult' SF novel I read in the late 1970's, about the same time I was reading Voorloper by Andre Norton, Madeleine L'engle's Wrinkle In Time books and the Dragonfall 5 young adult novels. so I am not sure if it is actually one of the Dragonfall 5 stories, but I can't confirm it is or isn't by reading available synposes online.   As I recall, the story went something like this...    * SOME SPOILERS BELOW *  A young person and sibling - I think female protag and younger brother, but not sure - is scheduled to visit their aunt and uncle on another planet but to make the voyage in a timely fashion, must utilize this new consciousness transference technology to transmit their consciousness across the cosmos to a host replica body at the destination. After arriving at the destination the protagonist is impressed by their new body and is aware of the only way to tell they are in a replicant body is a small seam, scar or patch on their back, I think near the top of their spine, just below the collar of whatever clothing they wore.  The mystery begins when the protagonist realizes their hosts also sport the same scar when they should legitimately be in their real bodies.  The plot thickens when the protag or sibling finds a room of people in stasis when their replicant counterparts are roaming the same planet, and is subsequently caught in the act.  Sadly, I do not remember anything else, including the reasoning behind the deception.  I was quite young when I read this novel, so I probably got some of the details mixed up, but would love to know the name and author of the novel if anyone remembers.