How to type theta as shown below?  Hope image is clear
I know what my symbol or character looks like, but I don't know what the command is or which math alphabet it came from. How do I go about finding this out?
Given a recurrence relation:   $$ x_{n+2} = px_{n+1} + qx_n + r $$   Find the general term $x_n$ given initial conditions $x_1 = a$ and $x_2 = b$, where $a,b,p,q,r$ are some given numbers, $n \in \mathbb N$.   Since this is a non-homogeneous linear recurrence relation i've started with solving for $x_n^h$ which is a solution for homogeneous relation:  $$ x_{n+2} - px_{n+1} - qx_n = 0 \\ \lambda^2-p\lambda - q = 0 $$  I'm assuming here that two distinct roots $\lambda_{1,2}$ exist for this equation (single root case is handled similarly). So the general term is in the form:  $$ x_n = C_1\lambda_1^{n-1}+C_2\lambda_2^{n-1} $$  With the initial condition it's possible to find the values for $C_1$ and $C_2$:  $$ C_1 + C_2 = a \\ C_1\lambda_1 + C_2\lambda_2 = b $$  After solving the system of equation with respect to $C_1$ and $C_2$:  $$ C_1 = \frac{b-a\lambda_2}{\lambda_1 - \lambda_2} \\ C_2 = \frac{b-a\lambda_1}{\lambda_2 - \lambda_1} \\ $$  Which eventually results in:  $$ x_n^h = \frac{(a\lambda_2 - b)\lambda_1^{n-1} - (a\lambda_1-b)\lambda_2^{n-1}}{\lambda_2 - \lambda_1} $$  Final solution consists of a sum of particular and homogeneous solutions, but that is where I got stuck. What is the way to find the particular solution for the given problem? I'm pretty new to such kind of problems so any details are very much appreciated.  upd:  Based on the technique in @rtybase answer here are my findings:  $$ \begin{align} x_1 &amp;= a \\ x_2 &amp;= b \\ x_3 &amp;= pb - qa + r \end{align} $$  Writing the system for $c_1, c_2, c_3$:  $$ \begin{align} a &amp;= c_1 + c_2 + c_3 \\ b &amp;= c_1\lambda_1 + c_2\lambda_2 + c_3 \\ pb+qa+r &amp;= c_1\lambda_1^2 + c_2\lambda_2^2 + c_3 \end{align} $$  Now solving for the coefficients gives a huge formula. The answer to this problem suggests that:  $$ x_n = \frac{(\lambda_2(a+\gamma) - b -\gamma)\lambda_1^{n-1} - (\lambda_1(a + \gamma) - (b-\gamma))\lambda_2^{n-1}}{\lambda_2 - \lambda_1} - \gamma $$  Where:  $$ \gamma = \frac{r}{p+q-1} $$  Unfortunately I was not able to arrive at exactly the same result, but testing my solution (not posting since it's monstrously big) using SameQ in Mathematica with the given answer shows that they are equivalent. Those transformations are not very obvious to me, so i skipped them since the answer matches the given one (yet the symbolic expressions are different) How to solve differential equations of the form $f'(x) = f(x + a)$ What could one do to find analytic solutions for $f'(x) = f(x + a)$ for various values of $a$?  I know that $c_1\sin(x + c_2)$ is solution when $a = \frac{1}{2}\pi$, and of course $c_1e^x$ when $a = 0$.  For instance, is there a function satisfying $f'(x) = f(x + 1)$?  What about negative or imaginary $a$?  Is there possibly a generalization? Backstory: I've been developing a game with LibGDX for the past two years, and I've seen it refactored more times than I remember.  I jumped on the iOS iKVM port as soon as it was released. But I've been struggling with it ever since in regards to cross platform networking.  Kryonet will not port over to iOS unfortunately. So it's either I ditch Kryonet, or lose a large chunk of my target audience (iOS Users)  My question for you is:  I noticed LibGDX has cross platform classes ready for sockets and HTTP Requests, but no documentation exists for these sockets at all.  Has anyone used these sockets at all in the past, and does anyone know how they work? Suppress black box at end of line Under TeX, when I write a line too long, it prints a fat black box at the end of the line.  That is probably a feature, but I dislike it.  How to turn that OFF? Is there a command-line option or something else?   I usually use LaTeX, but I have a complicated file which happens to be plain TeX. I have a realy strange problem in Ubuntu 11.10. I have by accident my password set to empty. If I wan't to install software now the system asks for a password, but I haven't one anymore. If I enter my old password it sais it's wrong and if I leave it empty it also does. Now I wan't to set my password back, but the system asks for the old password before I can change it. That password doesn't exist, so I can't do anything anymore! Can someone help me please? How to change radius/weight of curve bezier? try scaling, change radius, but its dont work Vertex of a pentagon-Does the algorithm always stop? To each vertex of a pentagon,we assign an integer $x_{i}$ with sum $s=\sum x_{i}&gt;0$. If x,y,z are the numbers assigned to 3 successive vertices and if $y&lt;0$,then we replace $(x,y,z)$ by $(x+y,-y,y+z)$.This step is repeated as long as there is a y&lt;0.Decide if the algorithm always stops. Is there a way to see who voted on your posts? How to subtract a day from a date? Castellation metal loop connector Prior to the spell cast on Mjolnir, did Thor control lightning on his own? In the film Thor, Odin casts a spell on Mjolnir and strips Thor of his power.  He then banishes Thor to Midgard (Earth).  On Earth, Thor is seemingly just a strong and durable human.  Later in the film, he becomes worthy, and regains control of Mjolnir.  This then grants him his full power set and control over Mjolnir.  My question is, prior to this, could Thor control lightning all on his own?  Or is this purely an ability of Mjolnir? $f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$ I have question about ArcGIS and python script. There are following sample code, So I would like to change Another Field , not OIDFieldName. Please let me know how to change Another Field.  import arcpy import itertools  arcpy.env.workspace = r"C:\path\to\workspace"  base_shapefile = "some_shapefile.shp"  oid_field = arcpy.Describe(base_shapefile).OIDFieldName feature_count = int(arcpy.GetCount_management(base_shapefile).getOutput(0))  for (i, j) in itertools.combinations(range(feature_count), 2):     arcpy.Select_analysis(base_shapefile , "combination_%i_%i.shp" % (i, j), '"%s" in (%i, %i)' % (oid_field, i, j))   These sample code is combination for 5C2. I want to create all combiantions 5C3 , 5C4 , 5C5. I noticed that the paper size invoked by a4paper is not working right in my examples:  \documentclass[a4paper,11pt,oneside]{article} \begin{document} Test \end{document}   I checked with Preview.app and it tells me the paper size is 21.59 cm × 27.94 cm. A4 should be 21.0 cm × 29.7 cm.  When I use \usepackage[a4paper]{geometry} I notice a different size of the paper (it gets narrower) and Preview.app says: 21.01 cm × 29.71 cm, which seems right. I can reproduce this exact behavior with every class: article, book etc., scrartcl, scrreprt, etc., g-brief2 except memoir which gives me 21.01 cm x 29.71 cm    Is there something wrong or am I missing something? Dirac delta function and volume charge density I just got introduced to the Dirac delta function and one of the questions was to express volume charge density $\rho({\bf r})$ of a point charge $q$ at origin. I saw that the answer is related to Dirac delta function as: $$\rho({\bf r}) = q \delta^3({\bf r})$$ where $\delta^3({\bf r})$ is the 3-dimensional Dirac Delta function. Why is it so? I would really like to see all of my questions and answers on the profile page, even if some of them were deleted, and I don't have enough rep to see them on the site. (Note that some questions are  after 30 days or 1 year, and the author might be oblivious about what happened.)   , the &quot;deleted recent [questions/answers]&quot; links in one's own user profile now list all respective posts that were deleted in the past 60 days, rather than just those that were originally posted in the last 60 days. , users with the &quot;access to moderator tools&quot; privilege (10k+ on designed sites, 2k+ on beta and non-designed sites) can use the search operator deleted:1 to see all their own deleted posts. , deleted questions and answers that were posted in the past 60 days, can be seen using the &quot;deleted recent questions&quot; and &quot;deleted recent answers&quot; links on the questions and answers tabs (at the bottom of the tab) in your user profile. For older posts: as of 2013-04-23 you can  if you already have a link to them, but they still aren't linked from the user profiles (not even just for you). Nor do you see inbox notifications for comments on them if you come back a few hours later. You have to have thought to bookmark your question, or you have to go digging for it in your browser history. Can a shadow move faster than the speed of light? I'm just a layman with no real science background  (a) I'm not really sure what a shadow is. Some people are saying that it's the absence of something (a photon?) and therefore it is not subject to laws of physics. But I know when I'm in a shadow and when in sunlight... that's "information", and I thought that information flow can't break the speed of light limit.  (b) Some others use the laser pointer/flicking wrist example to talk about laser dots moving faster than light speed. But that can't be right, can it? The photons from the laser point have to travel from A to B and (after wrist flick) from A to C ... but the photons are all moving at the speed of light - there's no law-breaking in moving the laser dot from point B to point C because the photons are moving either from A to B or A to C ... the distance from B to C is not material (is it?).  (c) Also, can a shadow be thought of as a gap in a photon flow? E.g. if I interrupt a beam of single photons then can I think of it as e.g. PPPSSSPPPSSSPPPSSSPPP, etc where P = photons and S(shadow) = no photon &amp;, if so, the "SSS" is effectively travelling at the speed of light ... no faster. Strange folder in my home folder after a failed run of BleachBit how to stop working for other people's thesis? Last semester, I took a course with my supervisor and he assigned me a project. This project is part of one of his student's theses. I did the project and passed the course.   Later, I was asked to upload my code for use in a paper. It turned out that the data the grad student had generated was not appropriate, and he needed to do other experiments. It later turned out there were other problems with the underlying model, and I redid a lot of that work as well.  The problem is that this other student is not really contributing at all.    Sometimes he does not come to work at all On one occasion the professor assigned us to do something, and he immediately told me he had no idea how to do it In one of our meetings with the professor, he talked 2 minutes with the prof with no slides; I talked for the other 58 with slides.    What should I do in this situation? I could not work like this, I have my own project and my thesis and stuff. I cannot dedicate my self for another student's thesis.  Question 1: how should I make this student understand this is his thesis and he should work more than me?  Question 2: How can I get off this project? I am taking another course with my supervisor, he again assigned me a term project with the same system and same grad student. I do not want to keep going like this. What should I tell to my supervisor Due to the problems we found, I think this current system is a bit of a dead end. But, I do not want to lose my connection with my supervisor; he is a good man. I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite tetration as follows $$x^{x^{x^{.^{.^.}}}} \, = \, a$$ With a given $a$ find $x$. The next part of the problem was to discuss the convergence radius of a. If a is too big or too small the tetration does not converge. Below is my humble stab at the problem.  My friend said you would have to treat the tetration as a infinite series, and therefore could not perform algebraic manipulations on it before it is know whether it converges or diverges. However my attempt is to first do some algebraic steps, then discuss the convergence radius. I) Initial discussion At the start it is obvious that the tetration converges when $a=1$ (just set $x=1$) Now after some computer hardwork it seems that the tetration fails to converge when a is roughly larger than 3. II) Algebraic manipulation $$ x^{x^{x^{.^{.^.}}}} \, = \, a$$ This is the same as $$ x^a \, = \, a$$ $$ \log_a(x^a) \, = \, \log_a(a)$$ $$ \log_a(x) \, = \, \frac{1}{a}$$ $$ x \, = \, a^{\frac{1}{a}}$$ Now, if we let $a=2$ then $x = \sqrt{2}$. After some more computational work, this seems to be correct, which makes me believe this formula is correct. III) Discsussion about convergence By looking at the derivative of $ \displaystyle \large a^{\frac{1}{a}} $ we see that the maxima occurs when $a=e$, which also seems to correspond to the inital computational work. Now I think, that the minima of $\displaystyle \large a^{1/a}$ is zero by looking at its graph, studying its derivative and the endpoints. So that my &quot;guess&quot; or work shows that it converges when $$ a \in [0 \, , \,  1/e] $$ VI) My questions Can my algebraic manipulations be justified? They seem rather sketchy taking the a`th logarithm and so on . (Although they seem to &quot;magically&quot; give out the right answer) By looking at Wikipedia it seems that the tetration converge when $$ a \in \left[ 1/e \, , \, e \right] $$ This is almost what I have, why is my lower bound wrong? How can  I find the correct lower bound?