What is common practice w.r.t. providing signatures on printed photographs?
Should I sign, watermark or otherwise label a framed print? And if so how?
I have a midterm coming up and on the past exams the hard question(s) usually involve some form of $\lim_{x\to0^{+}} x \ln x$. However, we're not allowed to use l'Hopital's rule, on this year's exam anyways.  So how can I evaluate said limit without l'Hopital's rule? I got somewhere with another approach, don't know if it's useful:   $\lim_{x\to0^{+}} x \ln x = \lim_{x\to0^{+}} x^2 \ln (x^2) = L$ $= (\lim_{x\to0^{+}} 2x)(\lim_{x\to0^{+}} x \ln x)$ $= 0 * L$   Then I just need to prove that L is finite/exists (which means it must be 0) Question 1: Why are irrational numbers uncountable and rationals countable?  I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers there will always be a irrational number. But now it has come to my attention that the same holds for rational numbers. So I can't really see the difference between them. What is it that makes the difference? Why can we line up all rational numbers one-to-one with the natural numbers, but not the irrationals with the naturals?  Sorry for a perhaps basic question, but I really, really struggle with it. Thanks. Sufficient Identification for Intra Schengen flights If one takes a direct flight from one Schengen member state to another, what's the minimum sufficient ID that needs to be provided to board the flight? Why are there two tensions on my free body diagram? Estimate the population variance from a set of means What commands are there for horizontal spacing? I know that \: in LaTeX produces a space when rendered.  Are there any alternatives, because my LaTeX renderer doesn't support \: (it renders it as text), and there is no help / FAQ that I can find. It is will known that increasing cooling of a metal casting will lead to finer micro-structure which mean higher mechanical properties. If I have 2 molds one is thick copper die ,&amp; second is just thin-walled copper cup or mold cooled by water. which one of them excepted to cool the molten metal i.e. aluminium faster ?   by calculation it seems that copper die have higher cooling power as it has higher thermal diffusivity , but in real life it seems water have more cooling power. I am so confused. Sometimes chicken breast can be thick and I never know how to tell when it's cooked! How can I be certain it is cooked all the way through? Let $X$ be obtained by removing finitely many lines from $\mathbb{R}^3$. How do I prove $X$ is connected? Let $X$ be obtained by removing finitely many lines from $\mathbb{R}^3$. How do I prove $X$ is connected?  I think the way to go about this is to show $X$ is path-connected but I'm not quite sure. Deriving Schrodinger equation from Klein-Gordon QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$ In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, , the position space wavefunction is defined as   $$\psi(x)=\langle 0|\phi(x)|\psi\rangle, \tag{2.82+2.83}$$  where $|\psi\rangle$ is any state in the Fock space. Then he uses the equations (i) $\partial_t^2\phi_0=(\nabla^2-m^2)\phi_0$ (i.e., the Klein-Gordon equation for the free massive scalar field $\phi_0(\textbf{x},t)$) and (ii) $[H,\phi_0]=-i\partial_t\phi_0$ to derive Eq. 2.85, in following 3 steps:  $$i\langle 0|\partial_t\phi_0(\textbf{x},t)|\psi\rangle=\langle 0|\int \frac{d^3\textbf{p}}{(2\pi)^3}\frac{\sqrt{\textbf{p}^2+m^2}}{\sqrt{2\omega_\textbf{p}}}(a_p e^{-ip\cdot x}-a^\dagger_p e^{ip\cdot x})|\psi\rangle\\ =\langle 0|\sqrt{m^2-\nabla^2}\phi_0(\textbf{x},t)|\psi\rangle.\tag{2.85}$$  This equation is used to successfully derive the Schrodinger equation in quantum mechanics for the state $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$.   The first equality follows from differentiation of $\phi_0(x)$ w.r.t $t$. How does the second equality follow from the first? How are the inputs (i) and (ii) are utilized to derive the second equality from the first? Is it true that the operator $$i\partial_t\sim\sqrt{m^2-\nabla^2}~?$$ If yes, what is the use of doing the middle step? Story about lost colonized planet sent into orbit around "free" star Logistic regression with binary dependent and independent variables Validation Error: Value is not valid Why is $L^{\infty}$ not separable? In his 1984 paper , Paul Holland raised one of the most fundamental questions in statistics:     What can a statistical model say about   causation?   This led to his motto:     NO CAUSATION WITHOUT MANIPULATION   which emphasized the importance of restrictions around experiments that consider causation.  Andrew Gelman makes :     "To find out what happens when you change something, it is necessary to change it."...There are things you learn from perturbing a system that you'll never find out from any amount of passive observation.   His ideas are summarized in .  What considerations should be made when making a causal inference from a statistical model? Thermodynamics of supercooled water Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.  My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this?  $(0)$.  $(f(X))$, where $f(X)$ is an irreducible polynomial.  $(p)$, where $p$ is a prime number.  $(p, f(X))$, where $p$ is a prime number and $f(X)$ is an irreducible polynomial modulo $p$. How do favored classes work in Pathfinder? I'm working on creating a half-elf paladin character for a Pathfinder game.  Full disclosure: this is the first time I've tried creating a character; I've used ones other people created before.  In the text, it mentions something about "favored classes" I believe.  What is this? Eclipse is giving me a warning of the following form:     Type safety: Unchecked cast from Object to HashMap   This is from a call to an API that I have no control over which returns Object:  HashMap&lt;String, String&gt; getItems(javax.servlet.http.HttpSession session) {   HashMap&lt;String, String&gt; theHash = (HashMap&lt;String, String&gt;)session.getAttribute("attributeKey");   return theHash; }   I'd like to avoid Eclipse warnings, if possible, since theoretically they indicate at least a potential code problem.  I haven't found a good way to eliminate this one yet, though.  I can extract the single line involved out to a method by itself and add @SuppressWarnings("unchecked") to that method, thus limiting the impact of having a block of code where I ignore warnings.  Any better options?  I don't want to turn these warnings off in Eclipse.  Before I came to the code, it was simpler, but still provoked warnings:  HashMap getItems(javax.servlet.http.HttpSession session) {   HashMap theHash = (HashMap)session.getAttribute("attributeKey");   return theHash; }   Problem was elsewhere when you tried to use the hash you'd get warnings:  HashMap items = getItems(session); items.put("this", "that");  Type safety: The method put(Object, Object) belongs to the raw type HashMap.  References to generic type HashMap&lt;K,V&gt; should be parameterized. Many C++ books contain example code like this...  std::cout &lt;&lt; "Test line" &lt;&lt; std::endl;   ...so I've always done that too. But I've seen a lot of code from working developers like this instead:  std::cout &lt;&lt; "Test line\n";   Is there a technical reason to prefer one over the other, or is it just a matter of coding style?