I am wondering if someone could help me shed light on why the following bound doesn't work.  Suppose that $X_1, \ldots, X_n \sim N(0,1)$ are independent random variables. I am interested in finding a constant C that satisfies:  $$ E\left[\max_{1\leq i\leq n}|X_i|\right] \leq C \sqrt{log\  n} $$  My Method:  Let $Y = \max_{1\leq i\leq n}|X_i|$  $$ \begin{align} E\left[Y\right] &amp; = \int_{0}^{\infty}P(Y &gt;y)dy \\ &amp;\leq\int_{0}^{\infty}\frac{1}{y^2}\sum_{i=1}^{n}Var(X_i) dy \\ &amp; \leq \int_{0}^{\infty}\frac{n}{y^2}dy \\ \end{align} $$  I used Kolmogorov's inequality in the second step above, but don't know the exact mechanics of why this fails and am unsure why my integral in the end doesn't converge. Is there a specific reason this bounding doesn't work? Am I failing to take into account tail activity? thanks.
Suppose that $X_1, \ldots, X_n \sim N(0,1)$ are independent random variables. I am interested in finding a constant C that satisfies:  $$ E\left[\max_{1\leq i\leq n}|X_i|\right] \leq C \sqrt{log\  n} $$  I know one method is to employ the moment generating function trick, then take logs of both sides. However, I was wondering if there exists a more direct method. thanks!
Counting surjective functions We have $r$ objects and $n$ boxes. I have to count all the combinations possible if the objects and boxes could be both different. If that happens I can count the number of variations from $n$ to $r$ (number of functions between objects and boxes). I mean, $n^r$.  But if I have to count all the combinations possible if anyone of the boxes is empty ($r\ge n$). Now I have to count the surjective functions.  For example if I have $r$ objects and 1 box. The number of surjective functions is 1. Then, if I have $r$ objects and 2 boxes, the number of surjective functions is $2^r-2$. And finally if I have $r$ objects and 3 boxes, I will count $3^r-2\cdot[3\cdot2^r-3]$. How do I get the formula for $r$ objects and $n$ boxes? Can photons be detected without being absorbed? I am thinking about a detector that would beep if light passes through it. Is it possible? If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that would enable mathematicians to continue pursuing mathematics, should the unthinkable occur? I want to make a long shadow just like in this image.  I can create this shadow with shapes and by deleting the unwanted shadows after rasterizing the layer. But it's not very reusable. I can't resize the layer after rasterizing it.  Is there any easy way to make a long shadow like in the image below? Why is Modus Ponens valid? I am having trouble understanding what defines Entailment operator. On Mathoverflow I posted this question on what I perceive to be .  Consider: :     P therefore Q  P            Therefore, Q   My question is could we have arbitrarily defined it such as:     P therefore Q  P            Therefore, P   However, if an advanced civilization receives information in the form of following nested structures/containers or packets, how will they interpret Modus Ponens if in an alien land Modus Ponens is not defined?    Therefore, it raises the validity of MP. How do we define the implication or entailment operator? Because if we use an argument that relies on implication or entailment operator ipso facto, then it is trivial that logic is circular.  Is there a flaw in my rationale? I am trying to find a good rough guesstimate of how much of the maximum wattage usage of a graphics card is turned into hot air.  Say a graphics card is rated at using X watts at maximum usage. How much of that is released as heat into the air via the cooling setup, and how much is not (If not where does it go? To ground?)  Yes I know there are a lot of variables here, I am just looking for a guesstimate from someone who would understand better than me. I'm specifically using  (Radeon 6990 375 watts max) How do you run your own code alongside Tkinter's event loop? My little brother is just getting into programming, and for his Science Fair project, he's doing a simulation of a flock of birds in the sky. He's gotten most of his code written, and it works nicely, but the birds need to move every moment.  Tkinter, however, hogs the time for its own event loop, and so his code won't run. Doing root.mainloop() runs, runs, and keeps running, and the only thing it runs is the event handlers.  Is there a way to have his code run alongside the mainloop (without multithreading, it's confusing and this should be kept simple), and if so, what is it?  Right now, he came up with an ugly hack, tying his move() function to &lt;b1-motion&gt;, so that as long as he holds the button down and wiggles the mouse, it works. But there's got to be a better way. I tried many ways to make a bigger screen in VirtualBox because I do not like the small size of the guest window? Is there is a way to make it big? Whenever the Doppler effect is mentioned, it's typically in the context of sound waves or electromagnetic radiation. On the cosmological scale, red-shifting is also important because of the enormous speed of receding galaxies, thanks to the expansion of the universe.   Yet, red-shift is always discussed as the red-shifting of electromagnetic waves. Can gravitational waves be red-shifted? If so, could observations of them be used like red-shifted electromagnetic waves from distant sources are; that is, to figure out how fast an object is receding? Real-life examples of common distributions What is the cardinality of the set of the empty set? Given set $ A = \{  \emptyset   \}$ what is $|A|$? I just learned about sets today in class and I'm unsure how to answer this. I am a Chinese citizen married to a US citizen last year. I am going to apply for a UK tourist visa. The name shows on my passport is my maiden name, while any other documents the UK visa needed, such as bank statement, is with my married name. I can provide my marriage license to prove the link between my maiden name and married mane.   Can I apply for this UK visa with my situation? Is it possible to use an Ubuntu Live CD and run GParted to partition a my HDD (including the partition that is usually mounted on a given system)? I'm encountering a problem with my HDD's partitions and can't resize it. I've described my problem and my system here: .  Now, from a more general perspective, I was thinking on whether it is possible to use an Ubuntu Live CD and use GParted in a live session to possibly go around that problem. Is that possible? I made a separate question as I'm interested in the possibility independently from my current problem.  You see, I only have one partition and that's mounted. When I use GParted through Ubuntu I don't get any error reports (unlike the case reported in the question linked above).  Thanks beforehand. How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $? In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$. I'd like construct a non-negative sequence $\{a_{k}\}_{k=0}^{\infty}$ with $a_{0}=0$ such that the Cesaro sum $\frac{1}{n}\sum_{k=0}^{n-1}a_{k}\longrightarrow 0$ but $a_{n}$ does not converge to $0$.   I have a really bad attempt:  Define $a_{0}=a_{1}=0$, and $a_{k}:=\log(\log(k))$ for $k\geq 2$, then $a_{k}\longrightarrow\infty$. But $$\dfrac{1}{n}\sum_{k=0}^{n-1}a_{k}=\dfrac{\log\Big(\prod_{k=2}^{n-1}\log(k)\Big)}{n}$$ and Wolframalpha told me that this series converges to $0$ as $n\longrightarrow\infty$.  However, I have no idea about how to show this convergence to $0$. Also, I wish to have a sequence as simple as possible, since at some stage I need too how $f(k):=a_{k}$ is positive semi-definite.   Is there any simpler example?   Thank you! Does a car use friction to move? When a car's engine injects fuel into the cylinder chambers, the reaction creates a force that generates rotational momentum to the shaft and over the transmission, it translates that power to the wheels, right?  But, something's bothering me, how does the car actually move, is it dependent on the force of friction? Wheels have a contact with the road in one exact point and when the drive gets to rotate them, a force is exerted on the road in the direction of movement and there is friction that is counteracting it.  Is the key to get a car properly moving to have the drive force equal or less than the force of friction to keep that point locked to the ground and use the rolling of the wheel for translational displacement ie. moving forward? If it's bigger than the force of friction, that would cause the wheels to spin in place?  Am I completely off or am I getting something right? If there's a knowledgeable person on this topic, I'd greatly appreciate some insight, perhaps even a bigger response with some basic vehicle physics. When voting to migrate an off-topic question, why is the list of sites limited? Why are there two ticker symbols for Google — GOOGL and GOOG?  They seem to be mostly identical, but their market values are different: currently, 551.64 for  and 548.79 for . Are humans the only animal that requires "clean water"? Closing a Java FileInputStream