Edit toolbar and preview aren't loading correctly for tag wikis
Toolbar options are not displaying in Edit Profile - About me section
Yet another "inconsistent reputation history" The meaning of imaginary time What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics, why should imaginary (or complex) time make sense?    By imaginary I mean a multiple of $i$, and by complex I mean having a real and an imaginary part, i.e., $\alpha + i\beta$, where $\alpha, \beta \in {\mathbb R}$. $d_1(x_1,y_1)$ and  $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about proving the triangle inequality for $d$ ? Thank you for your help. Discrete Probability: Four dice are thrown, what's the probability that... Four dice are thrown, what's the probability that:  a) None of them fall higher than three?  b) None of them fall higher than four?  c) That four is the highest number thrown?  So for a, I wanna think about the denominator first. There are 6 possible outcomes for each dice, and we have four dice. So we're technically picking r out of n objects, aka picking 4 possible values out of 6. So the denominator/sample space should be $6^4$ right?  Now this is where I'm getting tripped up. Order technically shouldn't count, because we only care about the quadruplets that don't have a value higher than 3. I might be wrong, maybe I am, but could someone explain a bit more? My professor stated that typically in the sample space/denominator, we want order to count.   So I'm trying to think of the numerator now, so we want to find the probability that no values appear higher than 3, so some events that can occur are: $(1, 1, 2, 3), (1, 2, 2, 3), (1, 1, 1, 1)$ etc. Again, I can't see why order should count here, because if we only are concerned about what appears rather than how they appear, then we can say $(1, 1, 2, 3)$ is equal to $(1, 2, 1, 3)$? So if order does not count, and we have replacement, then does this lead the case where we use $$\binom{n-1+r}{r}$$to find out the probability?   I'm pretty sure if I can do a), I could prob do b), but if someone could maybe lead me in the right direction for c) I would appreciate that too! Adjective for someone who exaggerates emotions to get sympathy Can I cross the USA-Canada border with a birth certificate and a passport locator number? My US passport took too long to be shipped to me, and it will arrive after I need to leave. Will I be able to cross the border from the United States to Canada with a birth certificate (copy) and a passport locator number? When a star becomes a black hole, does its gravitational field become stronger? Simpler proof - Non atomic measures Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)&gt;0$ there exists $F \subset E$ measurable such that $0&lt;\mu(F) &lt;\mu(E)$.)  a) Prove that for every $ \varepsilon &gt;0$ there is a finite partition of $X$ in measurable subsets $X_1,..,X_n$ such that $\mu(X_i)\leq \varepsilon$.  b) Prove that for every $\alpha \in [0,\mu(X)]$ there exists $E \in \mathcal{E}$ with $\mu(E)=\alpha$.    I guess that a) is given to prove b) more easily. I have the following idea of solution for a) (inspired from , which proves b)  )  Denote  $$ \Gamma = \{ (X_1,..,X_n) : n &gt;&gt; \mu(X)/\varepsilon, X_i \text{ are disjoint }, \mu(X_i)\leq \varepsilon \}$$ ordered by componentwise inclusion. Totally ordered parts $(Y_\alpha)$ of $\Gamma$ have an upper bound element $(\bigcup Y_\alpha^i)_{i=1}^n$ which is still in $\Gamma$.   By Zorn's lemma $\Gamma$ has maximal elements. If a maximal $(X_1..X_n)$ element is not a partition, then we can replace it with something better, of the form $(X_1,..,X_n\cup A)$ where $A \subset X \setminus (X_1 \cup..\cup X_n)$ and $\mu(A)&gt;0$.     My questions are:  1) Is my solution of a) correct? I feel that the part with the upper bound element may not work, since there might be noncountable unions.  2) What is a simpler solution of a)? Separate long math text under sum symbol into different lines? Priority and timing between enchantments and instants If I have priority and cast an  on a just summoned 2/3 creature, then, passing priority, my opponent casts a  on that creature, does my creature survive? The first spell that resolves is obviously the bolt, that deals 3 point of damage to the creature; but immediately after, will the  Unholy Strength resolve (giving +2/+1), allowing the creature to survive? Elements of order 2 in $D_{2n}$ Im new at this abstract algebra stuff and im not comfortable with the proofs techniques yet, so I have a question related to the elements of order $2$ in $D_{2n}$.  Problem:   Prove that $\{x\in D_{2n}|x^{2}=1\}$ is not a subgroup of $D_{2n}$ for $n\geq 3$.   By now, I know that all the elements of the form $sr^{k}$ has order $2$ because I tested by brute force haha, but my real question is how do I prove prove that. How do I have to proceed to prove that $sr^k$ has order 2.  Any help will be really appreciated, thanks so much &lt;3 Possessive of a word that is already possessive If the cricket ground Lord's is a possessive, what if you want to describe something belonging to Lord's? Would you say:     I was very impressed by Lord's's customer services.   It doesn't look right, so what is the correct way of writing it? Google Chrome on a fresh CentOS 7.1 not starting up "All our X" vs. "all of our X" Is the word "of" optional in this instance? Is either of these considered preferable to the other?        Taste all our delicious treats.   Taste all of our delicious treats. How can I keep low resolution textures pixelated? I have a problem. I am going to make a minecraft animation in blender. But when I put a minecraft texture on a 3d cube that is 16x16x16 it is very low resolution is there a way to get the texture in a better resolution or is there a way to make blender make the texture better?  I hope you have an answer. Thanks in advance I'd like to add an invisible watermark to each PDF document that I produce so that in case it leaks, I know where it leaked from. Anyone knows what would a good way for an invisible watermark be in a PDF document and how to add one? Problems when adding a fillable PDF form if the page is rotated Why does Kylo Ren's lightsaber look different from normal lightsabers? Sometimes I hear sentences that sound conditional to me, just because of the (unassertive) ordering of the words. I don't know if I can come up with a proper example. Consider this:          I'd be a lot better, had she stayed with me.        And I guess it's equivalent to this conditional form: If she had stayed with me, I would have been a lot better.  So, this is my question: In what ways can you make a sentence conditional without using conditional conjunctions (if, unless, etc)? And what are their equivalent conditional forms? How does the computer know that the PC was not properly shut down? Actually, this question struck me because of power cuts in my house. When there is a power cut, there is a sudden loss of power from the computer.   How does the computer know that the shutdown was not properly done?