I have a hard time solving the following equation.  $$z^{2} + (1 - i) \cdot z - i = 0$$  I tried factorization and got $z = -1$ or $z = i$, which I suppose are correct.  However, when I try the quadratic formula for solving second degree equations, the results I get don't match with the solutions I get above.
Let $a$, $b$, and $c$ be complex numbers with $a\neq0$. Show that the solutions of $az^2+bz+c=0$ are $z_1,z_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, just as they are in the case when $a$, $b$, and $c$ are real numbers.
$$  \lim_{n\to\infty} n^{1/2} \int_{0}^{1} \frac{1}{(1+x^2)^n}\mathrm{d}x=0 $$  Is my answer correct? But I am not sure of method by which I have done. Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$? Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$.   I know that   $\mathrm{char}(R) =2$  $R$ has $2^k$ elements for some $k$  (this is because, if $p\neq 2$ is a prime with $p\mid|R|$, then there exists a non-zero $x \in R$ s.t. $px=0$ and $p $ is odd $\implies x=0$, contradiction).   Does this suffice to answer my question? Congruent Modulo $n$: definition In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$,  and $9 \equiv -1 \pmod 5$. But when I first learned about the modulo operation my junior year, I would have told you that $13 \equiv 1 \pmod 4$, and that $9 \equiv 4 \pmod 5$.     Is this just a difference in the definition of modulo? Or is this pretty typical (to not take it to the largest factor, or one over)? How to preserve detail when downscaling scanned photos? I'm mixing analog with digital photography — I tend to take better photos when I can't look at a small digital screen, unconsciously forcing me to take the picture before I shoot instead of checking afterwards if it succeeded. I also like the grain film gives me.  However, most shots aren't worth spending much time in the darkroom, and scanning them is a lot faster. I'm wondering how best to convert the film to digital.  Scanning at lower than maximum (non-interpolated) resolution of my image scanner tends to exaggerate the contrast of the grain and screw up the mid-tones because of that.1 On the other hand, the resulting size of the scans are major overkill in most cases (I don't print them at A0 size), so I want to downscale the pictures. However, the grain confuses most methods and results in loss of detail and/or loss of the characteristic grain. For example, bicubic sharper tends to result in a "bigger" grain than the original. Now, the best compromise I've found so far is simply using bilinear interpolation, but I doubt that this is the best method to preserve both overall details and the look of the grain.  In short: when processing the scanned negative, how do I "best" preserve both image quality and the characteristic look of the grain?   As Stan Rogers pointed out below, there's a difference between truly scanning the grain of the film, and scanning the "grainy" character of film at lower resolutions. This question is about the latter; Stan Rogers explains how to do the first.  I'm looking for alternatives to the standard Photoshop options to try out, and reasons for each. Is the fact that every film has a different grain characteristic a problem? Are customized advanced noise reduction methods perhaps an option?  1. I'm using a Minolta DiMAGE Scan Elite 5400, using the software provided by Minolta.  As requested, here is a scanned picture with different resize methods. Just scanned this, scanning a B&amp;W negative to get better tonality range. I turned off grain dissolver, automatic dust and scratch removal, etc, for obvious reasons. On the right the pictures are upscaled again with Nearest Neighbor. I set the 8 bit scan to 16 bits before downscaling, assuming it reduces risks of banding if I choose to mess with curves, channel mixers, etc. No other modifications (usually I first retouch dust &amp; scratches, and in the end channel mix the picture to B&amp;W, then switch to 8 bits again). How to change maximum number of cores? It says in  that there is a maximum number of 8 cores supported by default in Ubuntu 12.04 Desktop/Server 32-Bit. What is the way to increase the maximum number of cores from the default of 8 to a higher value, like 12?  The reason I ask is because I have just built a machine with 6 hyper threading cores.  There were issues installing the 64-bit version of the software from a CD (it "panicked" and crashed several times).  The 32-bit installed with no problem, but I only see 8 of the 12 cores.  The BIOS is fine.  If I turn hyper-threading off and reboot, I see 6 cores as expected.  I am nervous to try and compile a kernal, I am still new to all of this.  Is there a config setting somewhere that I should change? A man enters a comatose author's dreams to wake him up This is a short story, I think, but I might be wrong.  A man is recruited to enter an authors' dreams to try &amp; bring him out of a coma &amp; finds himself in a hackneyed barsoom type world where the coma victim is a muscle bound hero, it's a desert world with two suns, the damsel in distress is the typical scantily clad princess trope.  The "hero" basically spoils the authors' dreams by manifesting a pistol &amp; simply shooting the monsters dead at one point &amp; the author tries to write him out of the "story" by simply jumping forward several chapters &amp; is both surprised &amp; peeved to find him still there.  The author eventually comes out of his coma &amp; immediately assaults the man when he recognizes him from his dreams, he tries to strangle him if I remember right.  He enters the authors' dreams with some kind of experimental dream imaging tech, basically a two ECG sort of set up linked through a computer with him put to sleep with drugs I think. Is the Death Star’s gravitational force strong enough to hold an atmosphere? The Death Star is the size of a small moon. Endor being a moon (not sure if it's small or not) has a rich atmosphere.  Could the Death Star hold a nitrogen/oxygen based atmosphere with its gravity?  (Now re-reading my question, I suppose this sounds a lot like the plot of SpaceBalls... that was not my intention) When I press the power button a dialog opens, but I want to shutdown the system without receiving "Do you really really want to shutdown?". How can I choose a direct shutdown action on a pressed power button?   I had the same problem with the 11.04 version, but I don't remember the solution and yesterday I upgraded to Ubuntu 11.10. Error with square bracket in table how to hide mesh but keep hair particles "Whether" Vs. "If" Could someone please help me with these sentences? I need to know  Which one doesn’t sound natural? Which one is formal and which one is not?   A - I don’t know whether this word fits in the sentence. B - I don’t know if this word fits in the sentence.   C - I don’t know whether or not this word fits in the sentence. D - I don’t know if or not this word fits in the sentence.   E - I don’t know whether this word fits in the sentence or not. F - I don’t know if this word fits in the sentence or not.  For me only D doesn't work and sounds not idiomatic. Meanwhile for me all of these sentences can be used in both written and spoken languages. Do you confirm me? Where to start learning Linear Algebra? I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because I wanted to create games.  I'm told that Linear Algebra is the best place to start. Where should I go? I have an internet connection that I'd like to be enabled only during the day. With multiple household internet devices, this would need to be something disabled at the level of the router.  I don't believe the router offers this sort of functionality, but I'm reasonably familiar with setting up web servers and the like, and I'm wondering if I can set up some sort of proxy server on the internet that the router must go through and have that proxy only active at certain times of day.  It's a bit vague I know, but is something like this even possible?  If not, are there other solutions to the problem? I'm very new to the GIS world, so my problem could be very stupid, but I'm going to try anyway.  Objective  Given a list of coordinates in longitude/latitude, stored as a .csv file, I want to create a buffer with distance of x km around those coordinates. (In case this is relevant, these coordinates are locations of the societies in the . These societies scatter all around the world.)  Problem  I'm only able to create buffers with distance in radial degrees, but I want to do them in (kilo)meters. I'm aware of the following Q&amp;A's, but following what's suggested therein doesn't seem to solve my problem:          What I have tried...   I start QGIS (v2.4), and Add delimited text layer  choose my .csv file and the x/y fields, click OK select WGS 84 as my CRS (also tried NAD83 but didn't make a difference), click OK right-click on layer and select Save As..., in the dialog box, choose the following and click OK:  add sccs_meter.shp as layer to the current project Select Vector > Geoprocessing Tools > Buffers > set buffer distance as 10   The result is a buffer with radius of approximately 1068km, which suggests that buffer distance is 10 degrees rather than 10 meters.  What did I do wrong here? Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $ How to insert text into a root-owned file using sudo? Is there a good resource or tool to help build a palette/color scheme around colors I select? No matter how nice the composition, flow, and lines of my design are, if I am not copying directly from nature, I have a very difficult time picking a decent color scheme.  It either looks too drab, or too garish.  Is there a good resource that will help me to choose a color scheme that has 2 or 3 colors, and the colors do not clash?  I'm not looking for specific colors or color schemes.  What I need is a resource that will help me choose a color (or colors) that go nicely with, or provide a good accent to, whatever main color I am working with on the design. View history of commands run in terminal Is there a way to save all my typed terminal commands and view it like history in a log book? How do some Ws on transcripts affect PhD admissions in mathematics? Can I backup my entire Ubuntu installation to ISO DVD?