weird properties of complex exponentials: $e^{i 2 \pi f(x)} = 1$
What is wrong with this fake proof $e^i = 1$?
Windows 7 time issues update-alternatives --install says it needs link name path priority? I typed:  sudo update-alternatives --install "/usr/bin/java" "java" "/usr/lib/jvm/ jdk1.8.0/bin/java" 1   It's says alternatives --install needs link name path priority What do you call a person who started something that is later followed by everyone? For example, someone started to stay late at night till 8 p.m. and later everyone started to stay till 8 p.m.. Often while playing Pathfinder, my group will run into problems where rules for 3-dimensional play simply don't exist, so far as we can find. Arguments will come up over things like carrying-capacity, movement speed, area-effects, and a variety of other things.  Simply put, are there rules for 3D play for Pathfinder, and if so where can I find them? We always say that tree levels are classical but loop diagrams are quantum.  Let's talk about a concrete example： $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is source.   The equation of motion is $$\Box \phi=-g \phi^3+J$$  Let's do perturbation, $\phi=\sum \phi_{n}$ and $\phi_n \sim \mathcal{O}(g^n) $. And define Green function $G(x)$ as $$\Box G(x) =\delta^4(x)$$  Then   Zero order:  $\Box \phi_0 = J$   $\phi_0(x)=\int d^4y G(x-y) J(y) $  This solution corresponds to the following diagram:  First order:  $\Box \phi_1 = -g \phi_0^3   $  $\phi_1(x)=-g \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 G(x-x_1)G(x_1-x_2)G(x_1-x_3)G(x_1-x_4)J(x_2)J(x_3)J(x_4) $  This solution corresponds to the following diagram:  Second order:  $\Box \phi_2 = -3g \phi_0^2\phi_1  $  $\phi_2(x)= 3g^2 \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 d^4x_5 d^4yd^4z  G(x-y)G(y-x_1)G(y-x_2)G(y-z)G(z-x_3)G(z-x_4)G(z-x_5) J(x_1)J(x_2)J(x_3)J(x_4)J(x_5) $ This solution corresponds to the following diagram:  Therefore, we've proved in brute force that up to 2nd order, only tree level diagram make contribution.  However in principle the first order can have the loop diagram, such as  but it really does not occur in above classical calculation.  My question is:   What's the crucial point in classical calculation, which forbids the loop diagram to occur? Because the classical calculation seems similiar with quantum calculation. How to prove the general claim rigorously that loop diagram will not occur in above classical perturbative calculation. Solvability and reducibility of a polynomial in a "chain" of finite fields I read this set of stories in the 80's.  What I recall is this:  A planet where the ruling caste was made up of humanoid women.  These women would, as almost pubescent girls, go out into the wilderness with only a spear and attempt to kill an animal in a life or death struggle.  When it happened, they would burst open a DNA packet in themselves that caused them to rapidly mature, their skin to become bronze colored, and give them some kinds of other "powers", I think over fire.  They were called "Bahronas", I think.  The population of this planet was primitive, but they were in contact with an advanced humanoid species, I think of actual homo-sapiens, who were there observing and advising them.  The advanced species wasn't allowed to intervene, just advise and watch.  The main character was a castaway boy, crash landed on their planet and stranded.  He became friendly with the ruling family of Bahronas and entered their confidence.  We find out through the course of the story that he's been genetically engineered and is not a normal boy.  At some point, the advanced species notices him and explains that he is a clone of a famous explorer and a spy, and they explain that they won't do anything in front of him.  The central part of the story is something of a blur, but, the final part of the story is told in a dramatically different style and shifts to a planet populated by catlike creatures who live in boughs of silk scarves, some of which sing.  THey are recording devices which play back the songs.  On this planet, there is another silk scarf, a white one, which doesn't play a song at all.  The cat creatures are greatly disturbed by it when they hear it's message, which they don't understand.  The narrator then goes on to say that it's a spoken word recording, made by the explorer which the boy was a clone of, explaining that he is captured and held hostage on the planet which makes the scarves, and forced to help make them (I think.) I just learned about the C++ construct called "placement new".  It allows you to exactly control where a pointer points to in memory.  It looks like this:   #include &lt;new&gt;        // Must #include this to use "placement new"  #include "Fred.h"     // Declaration of class Fred   void someCode()  {    char memory[sizeof(Fred)];    void* place = memory;     Fred* f = new(place) Fred();   // Create a pointer to a Fred(),                                   // stored at "place"     // The pointers f and place will be equal     ...  }    (example from ) In this example, the this pointer of Fred will be equal to place.    I've seen it used in our team's code once or twice.  In your experience, what does this construct enable?  Do other pointer languages have similar constructs?  To me, it seems reminiscent of equivalence in FORTRAN, which allows disparate variables to occupy the same location in memory. For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$? I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you tell $N\subseteq M_i$ for some $i$?  I was first trying to show it just in the case $N\subseteq M_1\cup M_2$, and hoping to extend it to finitely many $M_i$. If either of the $M_i$ contains the other, the claim follows, so I suppose neither $M_i$ contains the other.   In hopes of a contradiction, I suppose $N\not\subseteq M_1$ and $N\not\subseteq M_2$, Picking $x_1\in N\setminus M_1$ and $x_2\in N\setminus M_2$, I'd have $x_1+x_2\in N\subseteq M_1\cup M_2$. The only thing I could conclude was that actually $x_1,x_2\notin M_1$ and $x_1,x_2\notin M_2$, which seems like a dead end. What's the right way to do this? How to find $ \int_0^\infty \dfrac x{1+e^x}\ dx$ Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. What causes these inconsistent dark bands in some of my photos from an indoor event? Yesterday I took photos of an event at school.  The lighting was poor so I increased the iso to 3200, shutter speed was between 80 and 250 and the aperture between 2.8 and 4.0.  Canon EOS500  about 10% of the photos got this weird horizontal black area. The position of the black area is not the same,  Sometimes it is in the middle, sometimes it is in the upper or lower part of the frame  Is this an artifact from the lighting frequency? or can it be a shutter problem? finding and replacing elements in a list How to create Unity indicators? I'm interested in Unity indicators and wonder if there is any tutorial on how to program them. I can use the sources of the existing ones as last resort, but I'd prefer a more friendly approach because my programming skills are quite limited. FTP File Upload with HTTP Proxy Is there a way to upload a file to a FTP server when behind an HTTP proxy ?  It seems that uploading a file is not supported behind an HTTP Proxy using .Net Webclient. ().  If there is no workaround ? If not, do you know a good and free FTP library I can use ?  Edit: Unfortunately, I don't have any FTP proxy to connect to. What is the exact difference between an autoregressive (AR) and vector autoregressive model (VAR)?  I always thought that VAR would just be for more than two variables, until I learned that AR can also have more than two variables. I.e. I had a misunderstanding(?) that AR models are not always bivariate.  Before I write too much potentially false assumptions about the difference between AR and VAR, it would rather be interested to see if anyone can make provide a quick and good summary. Distinct Sylow P subgroups Is there a way to "auto login" in PuTTY with a password? Is there a way to configure a password for a stored session in ?  I know there is the capability to specify an "auto-login username" (under Connection/Data), but is there a way to do the same with the password? How do I convert two lists into a dictionary? Prove: $\displaystyle\sum_{k=1}^n k k!=(n+1)!-1$ (preferably combinatorially)   It's pretty easy to think of a story for the RHS: arrange $n+1$ people in a row and remove the the option of everyone arranged to height from shortest to highest, but it doesn't hold up for the LHS.   Alternatively, trying to visualize the LHS, I noticed that it's like a right angle tetrahedra:  1  2!+2!  3!+3!+3!  ...  But it doesn't help to see a connection to the RHS.  Note: no integrals or gamma function nor use of other identities without proving them nor generating functions.