You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta which I denote a $\vec{x}=(x,y)$. Consider the following propagator  $$ P(\vec{r}) = \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} \,\, \gamma_i $$ Then we want to calculate $$ I(x) = \int \frac{\,dx \,dy\, dz}{(2\pi)^3} \text{Tr} \left( \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,}\gamma_x \gamma_x  \right) $$ Where Tr$(\gamma_i \gamma_j)=2\delta_{ij}$. Therefore  I am trying to resolve the following integral where $x,y,z \in \mathbb{R}$ and $a,b,c$ are constants (internal self-energies if you wish) in $\mathbb{R}$.  Then, after simplifying the propagator $P(x,y)$ we end up with the following expression for the $x$ component:  $$ I = \frac{1}{2i^2} \int \frac{dxdy}{(2\pi)^2} (x-ia) \int \frac{dz}{2\pi} \frac{1}{[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} $$ but it can be shown that  $$ \int \frac{dq}{(q+iv)^2+w^2} = \frac{\pi}{|w|}\theta(|w|-|v|).  $$ [e.g. see here  ] Thus we get that for $|(y-ib)^2+(x-ia)^2|&gt;|a|$, $\theta=1$ (for this is what we are interested about) and then we are left with $$ I = \frac{1}{4i^2} \int \frac{dxdy}{(2\pi)^2} \frac{(x-ia)}{|(x-ia)^2+(y-ib)^2|}   $$  The question is wether I can make the transformation $$ (x-ia) = q \cos \theta  $$ $$ (y-ib) = q \sin \theta $$ in polar coordinates. Then $dx dy = qdqd\theta$ and the previous integral becomes, assuming $|q|&gt;0$, $$ I = \int_0^{\infty} dq \int_0^{2\pi} d\theta \frac{q \cos \theta}{|q|} =0 $$ Is this correct? I am confused since one can find complex poles in the $I$. If not, what would the correct way to resolve this be? I have a suspicion that this integral should indeed give zero but how can we find this?
You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta which I denote a $\vec{x}=(x,y)$. Consider the following propagator  $$ P(\vec{r}) = \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} \,\, \gamma_i. $$ Then we want to calculate $$ I(x) = \int \frac{\,dx \,dy\, dz}{(2\pi)^3} \text{Tr} \left( \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,}\gamma_x \gamma_x  \right), $$ where Tr$(\gamma_i \gamma_j)=2\delta_{ij}$. Therefore  I am trying to resolve the following integral where $x,y,z \in \mathbb{R}$ and $a,b,c$ are constants (internal self-energies if you wish) in $\mathbb{R}$.  Then, after simplifying the propagator $P(x,y)$ we end up with the following expression for the $x$ component:  $$ I = \frac{1}{2i^2} \int \frac{dxdy}{(2\pi)^2} (x-ia) \int \frac{dz}{2\pi} \frac{1}{[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} $$ but it can be shown that  $$ \int \frac{dq}{(q+iv)^2+w^2} = \frac{\pi}{|w|}\theta(|w|-|v|).  $$ (e.g. see here ) Thus we get that for $|(y-ib)^2+(x-ia)^2|&gt;|a|$, $\theta=1$ (for this is what we are interested about) and then we are left with $$ I = \frac{1}{4i^2} \int \frac{dxdy}{(2\pi)^2} \frac{(x-ia)}{|(x-ia)^2+(y-ib)^2|}.   $$ The question is wether I can make the transformation $$ (x-ia) = q \cos \theta  $$ $$ (y-ib) = q \sin \theta $$ in polar coordinates. Then $dx dy = qdqd\theta$ and the previous integral becomes, assuming $|q|&gt;0$, $$ I = \int_0^{\infty} dq \int_0^{2\pi} d\theta \frac{q \cos \theta}{|q|} =0. $$ Is this correct? I am confused since one can find complex poles in the $I$. If not, what would the correct way to resolve this be? I have a suspicion that this integral should indeed give zero but how can we find this?
I am a PhD student in mechE. After paying the first year tuition out of pocket, I accepted a PhD project due to desperation for funding. However, after an year I realized that I hate the project. Here I made a mistake of not leaving/changing the project, I just waited out hoping I would like the project. I also asked my advisor a couple for a different project and he did not want to change my project due to sponsor obligations. For some reason I couldn’t assert myself and stayed on. However my hate for the project only grew with time and now I am in my fifth year. I have reached a stage where I absolutely hate the project and don’t know what to do. I can hardly do any work. I feel like quitting everyday and still feel it is wrong to quit this late. I have a very substandard thesis. I am lost and feel stuck, please advise on what to do. Thanks If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$. Calculating Image boundary / footprint of satellite images using open source tools? How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?  The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at most $M$, is given by the generating function: $$  \binom{M+k}{k}_{x} = \prod_{j=1}^{k}\frac{1-x^{M+k-j+1}}{1-x^j}= \sum_{n=0}^{kM} p_{k}(\leq M,n) x^n $$  For the number of the partitions $p_k(\mathcal{D},n)$ of $n$ into at most $k$ parts there is the recurrence relationship: $$   p_{k}(\mathcal{D},n) = p_{k}(\mathcal{D},n-k) + p_{k-1}(\mathcal{D},n) $$  But what, if I want to count only the partitions with distinct parts and restricted number of parts and restricted part size?  Update: Now I know the generating function for the number of distinct restricted partitions $p_k(\leq M, \mathcal{D},n)$ of $n$ into exactly $k$ distinct parts, all at most $M$ is $$    \prod_{j=1}^{M} (1+xq^{j}) = \sum_{k,n=0}^{\infty}p_k(\leq M, \mathcal{D},n)x^{k}q^{n} $$ and there is also a recurrence relation $$    p_k(\leq M, \mathcal{D},n) =         p_{k-1}(\leq M-1, \mathcal{D},n-k) +         p_k(\leq M-1, \mathcal{D},n-k) $$ How can I prove this? Could you recommend a book, where I could read about this? Difference between a space and a "wave" symbol (tilde/"~")? How to show that Klein four-group is a normal subgroup of the alternating group $A_4$ Making an elastic waistband on clothing? Proof by induction: $\sum\limits_{i=1}^{n} \frac{1}{n+i} = \sum\limits_{i=1}^{n} \left(\frac{1}{2i-1} - \frac{1}{2i}\right)$ How can the following be proved by induction?  $$\sum\limits_{i=1}^{n} \frac{1}{n+i} = \sum\limits_{i=1}^{n} \left(\frac{1}{2i-1} - \frac{1}{2i}\right)$$ I am out of ideas after practicing for a while:  $$\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}=\left(1-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\dots+\left(\frac{1}{2n-1}-\frac{1}{2n}\right) $$  Does this involve telescoping series? I have Dubai work permit mailed to me by my employer. It contains my old passport number which has ECR stamp. I then got a new passport issued without ECR stamping, but with different passport number. I have both passports.  Can I travel to Dubai now? Cannot update/get any packages. gedit not installed my problem is that my sources.list is out of date for some reason. I already found the fix in another port, but it doesn't work. It says to use gedit to edit sources.list and replace everything with the proper sources. Well, for some reason I don't have it installed (Ubuntu Server 14.04) and I can't install it or else I get the main problem again.  Failed to fetch http:// some ubuntu site Could not resolve hostname "us.archive.ubuntu.com"  Is there any other way to edit sources.list? Or do I need to do a re-install of the OS? What and with what was Luthor doing in the ship at the end of the movie? After Doomsday was defeated, a spec-ops team sent to the Kryptionian ship in Metropolis found Luthor submerged in water, in front of a... thing, which "held" three metal cubes. The thing looked up, roars, and dissipates into thin air.    What the hell was that all about? I am really confused. What was that creature and what was Luthor doing with it? How do I make a Villager Trade a Spawn Egg I am trying to make a Custom Trade on a villager. I need to make him buy 8 gold nuggets and sell a Cow Spawn Egg.  basically you pay 8 nuggets for a Cow Spawner.  The ID of the Cow Spawn Egg is 92 but I don't know where to put it in the command /summon Villager ~ ~2 ~ {Invulnerable:1,CustomName:Spawner_Vendor,Profession:2,Career:1,Offers:{Recipes:[{buy:{id:gold_nugget,Count:8},sell:{id:spawn_egg,Count:1}}]}} Let $f:(a,b)\to \mathbb{R}$ be a strictly increasing function. Does the limit $\lim_{x\to a^+}f(x)$ necessarily exist and is a real number or $-\infty$?  If so, is it true that $\ell=\lim_{x\to a}f(x)\le f(x) \ \ \forall x\in (a,b)$? Please provide proofs. It appears that Unity on Raring only comes with a single virtual desktop (aka workspaces) by default, and CCSM's Wall plugin has no way to add more. Is this configurable at all? Old fantasy book with a party finding coloured stones having powers over fire, water, mood I'm trying to remember the name of a book series of which I remember only the following, with Google yielding no results.   the name of the author starts with a Z   the main character is a knight in an household with a name sounding like Malachai who goes on a quest with six others guys and girls (a bard, another knight, a priest, etc.) and each of them finds a stone of colour which has a power on various things (like fire, their mood or water) and then they set toward a quest to beat some ancient evil. I used to find the book at the local library 10 years ago in France (but it was translated). Let $p\in[1,\infty]$, $a\in\ell^\infty$ and $$T:\ell^p\to\ell^p\;,\;\;\;x\mapsto ax.$$ Note that $\left\|T\right\|=\left\|a\right\|_{\ell^\infty}$. It's easy to see that if $a\in\ell^1$, then $T$ is nuclear and hence compact.  Can we more generally conclude that $T$ is compact?  Assume $p&lt;\infty$. The intuitive approach would be to define $$T_n:\ell^p\to\ell^p\;,\;\;\;x\mapsto(a_1x_1,\ldots,a_nx_n,0,\ldots)$$ for $n\in\mathbb N$ and to show that $\left\|T_n-T\right\|\to0$. Now, $$\left\|(T_n-T)x\right\|^p=\sum_{i=n+1}^\infty|a_ix_i|^p\tag1$$ for all $x\in\ell^p$ and $n\in\mathbb N$. The right-hand side of $(1)$ clearly converges to $0$ as $n\to\infty$, but this only yields convergence in the strong operator topology. On the other hand, if $(a_n)_{n\in\mathbb N}$ would be nonincreasing and $a_n\to0$ (e.g. $a_n=\frac1n$), then $$\left\|(T_n-T)x\right\|^p\le|a_{n+1}|^p\left\|x\right\|_{\ell^p}^p\tag2$$ for all $x\in\ell^p$ and $n\in\mathbb N$ and we could immediately conclude $\left\|T_n-T\right\|\to0$. EDIT: Maybe the general result can be obtained from the convergence in strong operator topology (which holds by $(1)$) by noting that $\ell^p$ has the approximation property; but I'm not sure whether this is true. Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $ Given $$\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$$ evaluate:   $$\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx. $$   I can find that  $$\left(ax+\frac{b}{x}\right)^2 = a^2x^2+2ab+\frac{b^2}{x^2}$$  therefore: $$\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx = e^{2ab}\int_0^{\infty}e^{-\left(ax+\frac{b}{x}\right)^2}dx$$  but I can't find any clue then. So I have a Palm Pre (original P100EWW) model that I enabled developer mode on, and installed a Debian Squeeze chroot.  Works great.  I have plans to use this for ANYTHING (bittorrent peer, web server) but a phone.  I noticed if I do a cat /dev/urandom &gt; /dev/fb0 it actually writes random pixels to the screen until a No space left on device error is generated.  Awesome, now I can use the display.  So what kind of utilites are there that will either A) let me use /dev/fb0 as a console I can output text to or B) render text on /dev/fb0 from the command line.  I don't know about recompiling the kernel for this yet (I'd love to eventually strip WebOS off entirely and turn this into a minimal ARM server) so userspace tools if they exist is what I'm asking about.  Also would prefer to render directly to /dev/fb0 and not use X. What are good advanced LaTeX books? What are the most important things I need to do to encourage Google Sitelinks? My website used to have sitelinks and now it doesn't. It's very possible that it's due to changing the website to a sidebar design instead of having an "interstitial" type landing page which limited the number of choices, but I'm not sure.  Here is how sitelinks might look for a site:    What are some things that I can do to improve my chances of getting sitelinks?