I'm trying (with no good results) to prove the follwing:  Prop. Let $B^1\subset H$ be the unit ball in the Hilbert space $H$. If $(x_n)_{n\in\mathbb{N}}\subset B^1$ is a $r$-separated sequence, i.e. $\vert\vert x_j-x_k\vert\vert\geq r$, for any $j\neq k$, then $r\leq \sqrt{2}$.  Can you help me to prove the reult above?
We have a sequence $(x_{n})_{n=1}^\infty$ in a Hilbert space $H$ and we know that:   for every $n \in \mathbb{N}$ holds $\|x_{n}\|\le1$ for every $m\ne n$ holds $\|x_{n}-x_{m}\|\ge r &gt; 0$   How I can show that $r \le \sqrt2$ ?
$(12)$ and $(123\dots n)$ are generators of $S_n$ How to calculate $5^{3^{1000}}\bmod 101$? How can I wrap text around a bottom-right div? Every time I try to do something seemingly-simple in CSS, it doesn't work.  I have a content div that contains a 460x160 image. All I want to do is position the image at the bottom-right corner and wrap my text around it.  &lt;div id="contents"&gt;     &lt;img src="..." /&gt;     text text text text text text ... &lt;/div&gt;   So I want it to look like:  ------------------ | text text text | | text text text | | text text -----| | text text |    | ------------------   Doing it with a top-left or top-right image is cake:  #contents img { float:right; }  ------------------ | text text |    | | text text -----| | text text text | | text text text | ------------------   Now how do I push that to the bottom? The best I've come up with so far are:  #contents img { float:right; margin-top: 90%} // really needs to be 100%-160px  ------------------ | text text      | | text text      | | text text -----| | text text |    | ------------------   In this case the text does not print in the margin, so there is white space above the image.  #contents { position:relative; } #contents img { position:absolute; right:0; bottom:0; }  -or-  // move the img tag under the text in the html and: #contents img { float:right; margin-top:-160; }  ------------------ | text text text | | text text text | | text text -----| | text text | te | ------------------   In this case the text prints over or under the image.  So... how can I accomplish this? Sending a message to nil in Objective-C As a Java developer who is reading Apple's Objective-C 2.0 documentation: I wonder what "sending a message to nil" means - let alone how it is actually useful.  Taking an excerpt from the documentation:     There are several patterns in Cocoa   that take advantage of this fact. The   value returned from a message to nil   may also be valid:         If the method returns an object, any pointer type, any integer scalar   of size less than or equal to   sizeof(void*), a float, a double, a   long double, or a long long, then a   message sent to nil returns 0.   If the method returns a struct, as defined by the Mac OS X ABI Function   Call Guide to be returned in   registers, then a message sent to nil   returns 0.0 for every field in the   data structure. Other struct data   types will not be filled with zeros.   If the method returns anything other than the aforementioned value   types the return value of a message   sent to nil is undefined.      Has Java rendered my brain incapable of grokking the explanation above?  Or is there something that I am missing that would make this as clear as glass?  I do get the idea of messages/receivers in Objective-C, I am simply confused about a receiver that happens to be nil. I'm trying to identify fantasy book series I encountered some time ago. Here are things I remember:   it was a fantasy setting but with some s-f elements; with many worlds and possibility to travel between them; resembling  setting a little, the two main races were non-human and probably hostile to each other,  a labyrinth (?), which somehow played an important role in the plot, a protagonist was some kind of emissary and was sent on a mission to another worlds  time period: 80s - 90s    Does anyone know what it is the name of the series? Why are there $1 / 1.602176634 \times 10^{-19}$ electrons in a coulomb? Why that exact number of electrons in one coulomb? who decided it? there is nothing wrong with the number, it just seems slightly messy. Why didn't the scientific community just settle on an easier number, such as $1\times10^{-19}$ for example? How do I integrate the Poisson equation to determine the electric potential along a particular direction (e.g., $z$)? This question is a sequel of sorts to my  about .  In the paper, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes.  The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction.  In my previous question, I asked about how particle number density is computed.  In this question, I would like to ask about how the electric potential is computed, given the charge density distribution.  Recall that in CGS (Gaussian) units, the Poisson equation is   $$\nabla^2 \Phi = -4\pi \rho$$  where $\Phi$ is the electric potential and $\rho$ is the charge density.  So the charge density $\rho$ is proportional to the Laplacian of the potential.  Now suppose I want to find the potential $\Phi(z)$ along $z$, by integrating the Poisson equation.  How can I do this?  In the paper, on page 254, the authors write down the average charge density $\bar{\rho}_{\alpha}(z)$ at $z$:  $$\bar{\rho}_{\alpha}(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^{\prime} \; dy^{\prime} \; \rho_{\alpha}(x^{\prime}, y^{\prime}, z)$$  where $\rho_{\alpha}(x, y, z)$ is the local charge density arising from the atomic charge distribution of ionic species $\alpha$, $\bar{\rho}_{\alpha}(z)$ is the average charge density at $z$ obtained by averaging $\rho_{\alpha}(x, y, z)$ over $x$ and $y$, and $\sum_{\alpha}$ denotes sum over ionic species.  The authors then integrate the Poisson equation to obtain $\Phi(z)$:  $$\Phi(z) = -4\pi \sum_{\alpha} \int_{-z_0}^z (z - z^{\prime}) \bar{\rho}_{\alpha}(z^{\prime}) \; dz^{\prime} \; \; \; \; \textbf{(eq. 2)}$$  My question is, how do I "integrate the Poisson equation" to obtain equation (2)?  How do I go from  $\nabla^2 \Phi = -4\pi \rho$ to equation (2)?  In paricular, where does the $(z - z^{\prime})$ factor come from?   Thanks for your time. Algorithm for computing Hadamard product of two rational generating functions If I have two generating functions $A(x) = \sum_na_nx^n$ and $B(x) = \sum_n b_nx^n$   then the Hadamard product is $(A \star B)(x) = \sum_{n} a_nb_nx^n$.  Now when $A$ and $B$ are both rational functions ($P(x)/Q(x)$ with polynomials $P, Q$) I've seen non-constructive proofs that $A \star B$ is also rational.  Given two rational generating functions $A$, $B$, is there an algorithm that efficiently computes $A \star B$? Or at least a constructive proof you could use to write an algorithm? Turn off automatic preview in apple mail Is it possible to turn off automatic preview of attachments when composing an e-mail?  NOTE: Yes I know that I can manually ctr+select the attachment and disable preview, but I want this done automatically Rails Previous Sunday in Relation to Any DateTime Script to determine if files are Images I would like to create a shell script that will check to make sure all files in a directory are image files.  We recently had an issue where a hacker was able to generate a file in a directory and mask it as a .jpg file. I would like to create a shell script to check all files in the directory to make sure they are real jpg, gif or png files. How to re-install http windows service I replaced apache webserver with http(WINHTTP windows service) for use virtual hosts feature. But I was missing  http(WINHTTP windows service) which I previously removed.I  now want to  reinstall this service for use print spooler. Recently I paid a lot (almost one month of my salary) for repairing my car. I am wondering if there is a way to get deduction on federal or state taxes (USA, California). Fantasy novel with amnesiac witch, impotent soldier and cursed animals The story is set in a vaguely medieval world in which a young woman has been cursed by the evil witch who has enslaved her and caused her to lose her memory. With the aid of her talking animal friends who have also been cursed (toad has a huge stone under his tongue meaning he can't feed and a crow with a gem embedded in its wing and there's a little grey kitten too who never grows up) she sets out on a journey to break the curse for herself and her friends. Along the way she meets an injured unicorn and a horny-but-moral soldier rendered impotent by a curse after refusing the disguised witch's sexual advances.  After adventures there's a 'happy ever after' for all of the heroes and the witch meets a sticky end.    In the UK the book had a deep red cover with an illustration. I read it in either the 1990's or the early 2000's. Cracking book, definitely for adults given some of the situations, language and humour. How can I determine if a used game is pirated? Why does the classical path give the dominant contribution in the path integral? Where are the outliner toggles in 2.8? Update SQL Server table with local data I added this to /etc/crontab on a few different linux &amp; freebsd systems:  # monthly reboot: 3rd Tuesday of every month 56 07 15-21 * 2 root /sbin/shutdown -r now   I want a reboot on the 3rd TUESDAY of every month.  However, all the systems rebooted on the 3rd Wednesday of this month (the 19th).  What am I doing wrong?  Update: Thanks to Ranon's answer below, looks like the below revision will work, can anyone confirm or is there an even better way of doing it?  # monthly reboot: 3rd Tuesday of every month 56 07 15-21 * * root test $(date +\%u) -eq 2 &amp;&amp; /sbin/shutdown -r now Keep processes running after SSH session disconnects I sometimes have long running processes that I want to kick off before going home, so I create a SSH session to the server to start the process, but then I want to close my laptop and go home and later, after dinner, I want to check on the process that I started before leaving work. How can I do that with SSH? My understanding is that if you break your SSH connection you will also break your login session on the server, therefore killing the long running process.