Can i use 19.5V 4.62A charger for a 19V 2A laptop?
Cann I use a 19.5V 4.62A laptop charger for 19V 4.74A laptop?
Disable autostart on boot and restrict background execution How to prove those "curious identities"? How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$ Why do I get more satisfaction out of participating in SO than out of my job? Alternatively, are there good ways to get more satisfaction out of my job, possibly by leveraging my participation in SO? Show $\text{Hom}_{Ab}(A,\mathbb{Q}) = \{0\}$ for finite $A$ I have a finite abelian group $A$ and I have shown that $A\otimes\mathbb{Q} \cong \{0\}$, the trivial group.  Now I want to show that $\text{Hom}_{\textbf{Ab}}(A,\mathbb{Q}) = \{0\}$, i.e we only have the trivial zero group homomorphism from $A$ to $\mathbb{Q}$.  The question I'm trying to solve is phrased in such a way that $\text{Hom}_{\textbf{Ab}}(A,\mathbb{Q}) = \{0\}$ should fall out almost automagically once you have that the tensor product collapses but I'm really stuck on this even though some people I talked to said it was really easy.  Some thoughts:   I've tried to assume the existence of a nontrivial homorphism $A \to    \mathbb{Q}$ to generate a contradiction but with no success. We have the Tensor-hom adjunction theorem stating that for abelian groups $A,B,C$ we have a natural bijection  $$\text{Hom}_{\textbf{Ab}}(A\otimes B,C) \cong \text{Hom}_{\textbf{Ab}}(B ,\text{Hom}_{Ab}(A,C))$$ and by plugging in our particular groups we get that:   $$\text{Hom}_{\textbf{Ab}}(A\otimes \mathbb{Q},\mathbb{Q})\cong\text{Hom}_{\textbf{Ab}}(\mathbb{Q} ,\text{Hom}_{Ab}(A,\mathbb{Q}))$$  Since $A\otimes\mathbb{Q} \cong \{0\} \implies \text{Hom}_{\textbf{Ab}}(\mathbb{Q} ,\text{Hom}_{Ab}(A,\mathbb{Q})) = \{0\}$ so we only have the trivial morphism from $\mathbb{Q} \to \text{Hom}_{Ab}(A,\mathbb{Q})$ but I don't think this implies that $\text{Hom}_{Ab}(A,\mathbb{Q}) = \{0\}$ and I'm not sure if this reasoning helps me at all. Many of the various items have a list of enchantments that would be ideal. For instance, I might want Infinity, Power, Flame, and Punch all on a bow... and at the highest levels. Barring incredible luck, I will never get those with an enchantment table.  I can enchant books until I find those spells, and add them to the bow with an anvil. However, at some point (I haven't yet figured out how many times) it will no longer allow you to add enchantments to an item, saying that it's "too expensive". There is a hidden counter that is incremented every time, and once this goes past a threshold you can never add more magic (though, I've read in places you can continue to repair the item at great cost indefinitely... not clear if that's true or not).  This counter is cumulative, such that if you combine two items' magic into a single item, that won't sidestep the limit.  However, I've only just caught on that one can also combine two books. I can take two Power I books, and make a single Power II book. And I can take two of those, and get it up to Power III, and so on, but also have two spells combined in a book.  If I do this until I get all the magic I need onto a single book (or even just two or three books, supposing the books max out), will I get to the point where the books will no longer combine with the item because it's too expensive? Or are books exempt from the cumulative counter thing? It would suck to spend so much time and effort to get a powerful book and discover that it can't actually be used on anything.  Also, is it true that you can indefinitely repair, as long as you have the repair material and are at level 39? Or is that a misunderstanding of the rules (or maybe just an earlier version of the game and it no longer works like that... I'm on 1.8.9)? Does that only work for those items that can be repaired with diamond/iron, or will it work for fishing rods and bows as well? I thought I remembered hearing the food agency  say that you should never reheat chicken meat after the initial cooking. However, I cannot find the statement of it anymore. Now I'm starting to suspect my memory.  Could this statement be true?  Is it unsafe to reheat chicken? I did some mathematical induction problems on divisibility   $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5.   Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?  I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic? The question is based on the following exercise in real analysis:     Assume that $A,B\subset{\Bbb R}$ are both bounded and $x&gt;0$ for all $x\in A\cup B$. Show that $$ \sup(AB)=\sup A\sup B $$   where    $$ AB:=\{ab\in{\Bbb R}:a\in A, b\in B\}. $$   Since $0&lt;a\leq\sup A$ and $0&lt;b\leq\sup B$ for all $a\in A$ and $b\in B$, we have $$ ab\leq\sup A\sup B $$ for all $ab\in AB$ which implies that $\sup AB\leq\sup A\sup B$. I have trouble with another direction: $$ \sup AB\geq\sup A\sup B $$ I was trying to show that for every $\epsilon &gt;0$, $\sup AB-\epsilon \geq \sup A\sup B$. If one uses the definition of supremum, one has the estimates that for every $\epsilon&gt;0$, $$ \sup A-\epsilon\leq a, \quad \sup B-\epsilon\leq b $$ for some $a\in A,\ b\in B$. It follows that $$ \sup A\sup B\leq (a+\epsilon)(b+\epsilon)=ab+\epsilon(a+b)+\epsilon^2\leq \sup AB+\epsilon (a+b)+\epsilon^2 $$ which seems quite close to what I want. How can I go on? How can I find unused functions in a PHP project Expressing an opinion: to me or for me? I would like to add a new cable outlet to an outside wall and wanted to find out what kind of drill bit is needed to drill through and feed the coax cable as well as anything else to be mindful of. Comcast wanted $70 for a new outlet installation ($30 install, $40 tech visit), so I figured I could do it myself. The part that I'm most unfamiliar with is how to properly weather seal it. The outside just has vinyl siding and there's no electrical outlets nearby.  So far, I think I need the following   Coax Cable Coax Cable Wall Plate Long drill bit Add a 3-way Cable Splitter to cable box Weather sealer? Animating Dynamic Paint with noise texture which of the following is NOT a possible value of $(e^{f})''(0)$?? Let $f$ be an analytic function on $\bar{𝐷} = \{z \in \mathbb{C}: |z| \le 1\}$. Assume that $|𝑓(𝑧)| ≤ 1$ for each $z\in \bar{D}$. Then, which of the following is NOT a possible value of $(e^{f})''(0)$??  $(A) 2$  $(B) 6$  $(C) \frac{7}{9}e^{\frac{1}{9}}$  $(D) √2 + 𝑖√2$  So I take a look at Cauchy Integral Formula and then $$(e^{f})''(0)=\frac{1}{\pi}\int_{|z|=1}\frac{e^{f(z)}}{z^3}$$  Taking modulus on both sides we have $$|(e^{f})''(0)| \le \frac{e}{\pi}2\pi=2e$$  Hence (b) cannot be the choice.   Is this alright!!  Thanks for the help! What are SPF records, and how do I configure them? Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not? Does the force of kinetic friction increase with the relative speed of the objects involved?    I have heard and read that the answer is no.  This is counter intuitive, and is a big part of why the "Airplane on a Treadmill" question is so interesting.  What phenomena are at work with kinetic friction, and why does it not increase with relative speed? Deleting comments under posts in the mod queue should be easier Ever since mods gained the ability to undelete comments, deleting them has become easier as the confirmation dialog was removed.  However, this is not the case in the mod queue &mdash; I still get a confirmation dialog for each comment I want to delete in the queue under a custom-flagged post (single-comment flags can be deleted just fine). And since this is the place from where I delete comments the most, it can be annoying.  In addition, the "purge comments" button, when invoked from the queue, refreshes it, so I have to scroll back down, remember which post I just purged, and mark the flag as helpful. Usually not a big issue but if there are multiple "obsolete comments" flags, then it can get confusing.  Could the workflow for comment thread moderation from within the queue be improved? Ideally, make comment deletion work like everywhere else on the site, and make the purge button AJAX-based like the comment deletes. In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop. Aware abbreviating those commands need not always be helpful, I would like to know how to abbreviate them? The following is what I tried, which does not work:  \documentclass[11pt,handout]{beamer} \newcommand{\bframe}{\begin{frame}} \newcommand{\eframe}{\end{frame}} \newcommand{\ftitle}{\frametitle}   \author{Author} \title{Title}   \begin{document} \maketitle  \begin{frame} \frametitle{Outline}  \end{frame}   \bframe \ftitle{Introduction}  \eframe    \end{document}   At first I thought it might be the case that commands such as \bframe was already defined. But using \def instead still gives no desired result. The Fourier transform of a "comb function" is a comb function? Let $f(x) = \sum_{n=-\infty}^{\infty} \delta(x - n)$, where $\delta$ is the Dirac delta function. This function $f$ (a "comb function") is important in signal processing because evenly sampling a function $g$ can be viewed as multiplying $g$ pointwise with $f$. This idea is the first step towards understanding how to approximate the Fourier transform of $g$, given evenly spaced samples of $g$. The next step is to note that $\hat{gf} = \hat{g}*\hat{f}$, where $\hat{f}$ is the Fourier transform of $f$ and $*$ denotes convolution.  Is it true that $\hat f$ is also a comb function? If that statement isn't quite right, what is the correct statement?  I would like to see: 1) an informal, intuitive, nonrigorous but easy derivation of the Fourier transform of $f$ . 2) A rigorous version of the same calculation.  (How would you even define $f$ rigorously, is it a distribution?)  (I'd also be interested in recommendations of math textbooks that cover this topic, including the Nyquist sampling theorem, even if it's only an exercise or series of exercises in an analysis textbook.) In which cases is a comma/period placed inside or outside of parentheses? Are there cases where a period is placed inside parentheses? In which cases the period (or a comma) should be placed inside the parentheses?