Prove: a closed ball is a closed set
A closed ball in a metric space is a closed set
Yes, I'm really sure I want to delete that post Can we continue to show the "Take a tour" banner after a user has created an account? When you view a Stack Exchange site without being logged in, you see a friendly banner, prompting you to take a tour and understand the features of the site:    This links off to a high quality resource that can help new users understand how the site works. However, the moment a user signs up to the site, this banner disappears from immediate view and hides itself as a link under the help menu.  My proposal is to continue showing this banner until one of the following conditions are met;   The user gets the "Informed" badge The user reaches a set amount of reputation (250 or 500 should be sufficient) The user chooses to hide the banner   This should increase the exposure of the tour to new users, who are the people who need the tour, rather than hiding it the moment a user signs up. SQL Server CE in a multi user scenario - is this really, really stupid? Check a string to see if all characters are hexadecimal values 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. Parse HTML in Android I'm looking for a way to reinterpret an array of type byte[] as a different type, say short[]. In C++ this would be achieved by a simple cast but in C# I haven't found a way to achieve this without resorting to duplicating the entire buffer.  Any ideas? I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help! How do you express that your understanding of something is completely changed? When something blows your mind and changes your understanding of something that you've believed in for years, how do you express it (formally or informally)? In the series finale, Admiral Janeway travels back to 2378 to take Voyager back to Earth. Out-of-universe, of course, this point in time was chosen to fit in with the continuity of the rest of the series. In-universe, however, it seems that Janeway could have chosen any point in time, since she had the ability to travel to and from the Delta Quadrant at will. Is there a reason given, in-universe, for why she chose that particular incident to travel back to? I am trying to remember the title of a children's SF book I once read, maybe 20 to 25 odd years ago.   I don't have a great number of details (and some of what I recall may be completely wrong), but the details (such as they are):   A war between two alien groups ended, and some technology that should have been destroyed was instead imprinted into a human baby. A few years later, both groups find this out and attempt to retrieve it. The main characters are the child, and an alien that I believe was called "Yacob" (or similar) who was sent to protect the child/retrieve the technology, and the antagonists were the other group of aliens that had previously lost the war and were trying to get the technology to start it up again (again, I don't remember the exact details, but I think it was a fairly simple black &amp; white/good vs. bad scenario, being a children's book and all). There was a scene in which Yacob/the alien gives the child blue food, who complains that it's blue, and the alien states "close your eyes and it's just food". The alien had a space shuttle/some other piece of tech disguised as a van. I vaguely recall a scene in which it is under attack and it flies or something. Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$ Let $a&gt;0$,$b&gt;0$ and $c&gt;0$. Prove that:    $$\dfrac{a^2+b^2}{a+b}+\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}\le \dfrac{3\left(a^2+b^2+c^2\right)}{a+b+c}.$$     $$\Leftrightarrow \frac{(a^2+b^2)(a+b+c)}{a+b}+\frac{(b^2+c^2)(a+b+c)}{b+c}+\frac{(c^2+a^2)(a+b+c)}{c+a}\leq 3(a^2+b^2+c^2)$$  $\Leftrightarrow 2(a^2+b^2+c^2)+\frac{c(a^2+b^2)}{a+b}+\frac{a(b^2+c^2)}{b+c}+\frac{b(a^2+c^2)}{a+c}\leq 3(a^2+b^2+c^2)$  $\Leftrightarrow \frac{c(a^2+b^2)}{a+b}+\frac{a(b^2+c^2)}{b+c}+\frac{b(a^2+c^2)}{a+c}\leq a^2+b^2+c^2$  $\Leftrightarrow \frac{c(a+b)^2-2abc}{a+b}+\frac{a(b+c)^2-2abc}{b+c}+\frac{b(a+c)^2-2abc}{a+c}\leq a^2+b^2+c^2$  $\Leftrightarrow 2(ab+bc+ac)\leq a^2+b^2+c^2+2abc\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\right)$  I can't continue. Help me Working with objects created with Activator.CreateInstance How to plot and visualize a single linear vector in 3D? How can you make an armor that has infinite protection armor? When I was using the new attribute command, I saw an attribute called &quot;armor.&quot; I don't know what it actually does, but I think it has to do with protection. I was wondering if I could make an armor that has infinite protection so I tried to make a command: give @s netherite_helmet{AttributeModifiers:[{AttributeName:&quot;generic.armor&quot;,Name:&quot;generic.armor&quot;,Amount:1.79769313486232E+308d,Operation:0,UUID:[I;439,684,724,99]},Slot:&quot;head&quot;}]}  I thought it would give me infinite protection, but it only gives me a helmet with the empty &quot;protection&quot; attribute. It works fine on older versions, but not on the 1.17 snapshots. Shuffling questions in exam "Pandora's Box" metaphors The majority of definitions give the same meaning - "Pandora's box" is a synonym for "a source of extensive but unforeseen troubles or problems."  Are there any other metaphors with the same meaning? What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$? I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be greatly appreciated!  (i) $\displaystyle\sum_{n=0}^\infty \bigg(\dfrac{1}{2n}\bigg)z^{3n}$.  (ii) $\displaystyle\sum_{n=0}^\infty(2^n+i)(z-i)^n$.  (iii) Show that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$ and that there are infinitely many $z \in \mathbb{C}$ with $|z|=1$ for which the series diverges. Is there a single word or very short phrase to describe the act of contemptuously referring to someone by using a diminutive "nickname", with which that person is not normally associated?  For example, referring to Henry Wadsworth Longfellow as Hank Longfellow? Or William Penn as Billy Penn?