If two pieces of the same type of metal touch in space, will they bond and be permanently stuck together?
Why don't metals bond when touched together?
If $M^n$ is a compact topological manifold (not necessarily with additional structure), is there an upper bound on the number of charts needed to cover $M$ ? Does this bound depend on the dimension of $M$ ?  Thanks in advance...  Cheers Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$ I want to solve:  Show that, in $\mathbb Q[x]$ (rationals), the polynomial $(x-1)^2$ divides a $zx^{z+1} – (z+1)x^{z}+1$, where $z \geq 1$ is any integer  I am really lost with this one, don't know how to aproach it JSF 2.0 File upload Calculate $E(X^{2n})$ where $X$ is normal (0,1) I need help proving the following:  Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that:  $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$  I've tried the change of variables $Y=X^2$ and then calculating the integral, but got nowhere. Any hints?  Thanks in advance! Community FAQ For sites in the Stack Exchange 2.0 network To see a list of commonly used words and phrases, see the . For official guidance from Stack Exchange, visit the . Asking questions                            Answering questions                 Accepting answers        Working with posts  What does it mean when a question or answer is...  ?  ?  ?  ?                                       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Link back to this index. Paste this at the end of the question:  [Return to FAQ index](/q/7931)   Tag as  and .  When a post seems to have hit a mature state, it will have  added by a moderator.  Flag for a moderator to mark community wiki.  Any FAQ post not yet community wiki should be made into a community wiki.  After the post has the  tag added to it, add a link in the above . Can't see a glass object through another glass object in Eevee I usually am able to render successfully to the c:\tmp folder, but when I came into work today, I rendered out the image on my work laptop, and everything seems to work, but I'm not getting an output to the tmp directory.  Does anyone know how to make this work? SF Story about human adapting to alien environment We have a situation: someone saw the boy break the window. Can I make this passive sentence?     The boy was seen to break the window.   I use Murphy's Grammar and this structure is never discussed. Why do we use this definition of "algebraic integer"? A number is an &quot;algebraic integer&quot; if it is the root to a monic polynomial with integer coefficients. Artin says (Algebra, p. 411):  The concept of algebraic integer was one of the most important discoveries of number theory. It is not easy to explain quickly why it is the right definition to use, but roughly speaking, we can think of the leading coefficient of the primitive irreducible polynomials $f(x)$ as a &quot;denominator.&quot; If $\alpha$ is the root of an integer polynomial $f(x)=dx^n+a_{n-1}x^{n-1}...$ then $d\alpha$ is an algebraic integer, because it is a root of the monic integer polynomial $x^n + a_{n-1}x^{n-1} + ... + d^{n-1}a_0$. Thus we can &quot;clear the denominator&quot; in any algebraic number by multiplying it with a suitable integer to get an algebraic integer.  When I first learned of algebraic integers, I looked online and saw some hints that maybe they were used to prove the Abel-Ruffini theorem. So I put off questioning their usage for a while; I now think I understand one proof of this theorem (the one at the end of Artin's Algebra) and it has nothing to do with algebraic integers (that I can tell). So basically: why is it important if a number is an algebraic integer? I think I understand what he's saying about the relationship between roots of integer polynomials and algebraic integers, but I fail to see why this is &quot;one of the most important discoveries of number theory.&quot; "that which" used together When I read essays from Eliot, I find him using "that which" frequently, e.g.        the combination which is the murder of Agamemnon is probably as complex as that which is the voyage of Ulysses.   A very small part of acting is that which takes place on the stage!   They belong to a different race. Their crudity is that which was of the Roman, as compared with the Greek, in real life.      I can kind of guess its usage, but I want to know more about this grammar structure. Searching on Google mostly gave me the simple difference between "that" and "which", and some examples using "that which":        that which we call a rose (from "Romeo and Juliet")   that which we persist in doing      It is a pity that Google search does not direct me to any useful page about "that which".  Can someone explicate its grammar for me? There's a book I borrowed from my grade 8 teacher's lending shelf (in Ontario, Canada) that I've been trying to identify for years. The main character was a man in a futuristic setting who... I think... had the secret to an improved version of the lifespan-extension serum everyone used. If I remember correctly, the flawed version caused people's skin to turn clear over the course of their extended lives. I think he was revived from some sort of stasis, but I can't be sure of that. I just vaguely remember a &quot;stranger in a strange land&quot; sort of cultural disconnect, him being cared for in some kind of hospital-like setting, and a sense that maybe the formula they want from him was invented by a now-dead family member of his. The most distinctive detail I remember was the love interest, a cloned nurse named &quot;Eliza Tertia&quot; who he encourages to break out of her role. (All the cloned nurses were named Eliza, and I remember &quot;Eliza Duodecima&quot; being the name of another one who is briefly seen.) I also remember the secret eventually being gotten out of him, only for it to be discovered that it's flawed, in that, while it does replenish the compounds that go missing in extended old age, it doesn't moderate their production. I don't remember if it was long enough to be a novel or if it was just a novella. My vague memories of the cover design and material suggest it was a 1970s printing or earlier, which would also fit with what I remember of the story's atmosphere. Restrictive vs Non-restrictive Relative Clause Can you tell me the difference in the meaning of the two sentences below?  As a defining relative clause.     The location which was called Central Park was a park in New York.    As a non-defining relative clause.     The location, which was called Central Park, was a park in New York. It is my understanding that I can disable my touchscreen by adding Option "Ignore" "on" to this file                                /usr/share/X11/xorg.conf.d/10-evdev.conf   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "on"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           EndSection   so it would look like this   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "on"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           Option "Ignore" "on"           EndSection   What would happen if I simplied changed MatchIsTouchscreen to "off"?   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "off"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           EndSection   Would this disable my touchscreen? Under what circumstances can the word "be" be used directly without any change? Possible Duplicate:          2 examples first:  1.It is announced that the accounting class of the 3rd and 4th hours on the morning of this Wednesday be cancelled  2.It is expected that everybody and each household be on guard  Someone told me this kind of examples above appears in the textbook as strict sentences to be taught in the classroom, so is that right for the usage of be  Meantime I looked up the word be at thefreedictionary.com and found one piece of interpretation says :      Archaic Used with the past participle of certain intransitive verbs to   form the perfect tense: "Where be those roses gone which sweetened so   our eyes?" (Philip Sidney).   So is there any connection between the archaic usage &amp; the examples above? BTW, how to understand the short phrase "the 3rd and 4th hours". Many thanks! Replacing strings in files based on certain search criteria is a very common task. How can I    replace string foo with bar in all files in the current directory? do the same recursively for sub directories? replace only if the file name matches another string? replace only if the string is found in a certain context? replace if the string is on a certain line number? replace multiple strings with the same replacement replace multiple strings with different replacements Midpoint-Convex and Continuous Implies Convex A week ago I  people on this site what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I have some questions.    How is this going to be applicable to quantum physics? I have learned about matrices (addition, subtraction, multiplication and inversion) and about how to solve multiple equations with 3 unknowns using matrices, and now I am starting to learn about vectors. I am just 1 week in, so this is probably not even the tip of the iceberg, but I want to know how this is going to help me.  Also, say I master Linear Algebra in general in half a year (I'm in high school but I'm extremely fast with maths), what other 'types' of math would I need to self-study before being able to understand rudimentary quantum physics mathematically? Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$ Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows. The algebraic closure of a finite field and its Galois group