I am finding problem in evaluating the flowing function using Rouché's theorem. I have looked out for examples but it lacks the explanation and steps to reach the required output. I need to determine the number of zeros of the function f(y) = y^5+14y+1 for 1 &lt; |y| &lt; 2 I have understood that i need to break it into 2 parts. If you could help me out. Thank you !
Find the number of zeroes (counting multiplicity) of $p(z)=z^6+z^3+10z^2+4z+3$ inside the annulus $1&lt;|z|&lt;2$. I think this can be solved using Rouché’s theorem. First consider $|z|\leq 1$ on the boundary $|z|=1$ and $|z^6+z^3|&lt;|10z^2+4z+3|$ which can be shown using triangle inequalities, the left side is at most $2$ while right side is at least $3$. We could also consider $|z|\leq 1+ \epsilon$ and our proof still works. Thus $p(z)$ has 2 roots inside $|z|\leq 1$. Also for $|z|= 2$ we have $|z^6+z^3|&gt;|10z^2+4z+3|$. Thus we have 6 zeroes of $p(z)$ inside $|z|&lt;2$ and therefore 4 of them are in $1&lt;|z|&lt;2$. Is this correct? The only &quot;tricky&quot; part is ruling out that there are zeros on $|z|=1$.
I am using ring modulator for DSB-SC  am modulation in a transmitter circuit. Can anyone please tell me about more DSB SC modulator for am modulation? I want to know which one to use where? I only ask questions. Am I unwanted? Possible Duplicate:          I've been  for maybe a year to get answers to questions I couldn't find on my own.  I always make an attempt first to find the answer on my own.  As of this posting, I've asked 14 questions.  I only once gave an answer, and it was to a non-technical question, and only because I was really bored and had too much coffee to drink.  I wasn't intending to answer any other questions in the future, because on on my best days I'm only a mediocre programmer, and there are some real geniuses here on StackOverflow who can answer other peoples' questions faster and more accurately than I can.  It actually is my nature.  I am an Aquarian.  We are asker-listeners.  Today I discovered Meta Stack Overflow and started browsing around.  I became familiar with the concept of the "Help Vampire".  I seem to qualify.  I continued reading more posts on the topic and got the distinct impression that people who only ask, and never answer, are less than welcome here, rather despised.  So, should I just go away?  Or should I start contributing answers just to have a answer-to-question ratio of 3-to-1 (I read somewhere that's the average) even though I am less than one eighth as knowledgeable as the 10K+ guys and gals?    Or should I continue as I have begun?  I mean, doesn't a 0-questions-asked user like Jon Skeet need a Blankman?  Yin and yang, you know? How can I stand on the ground? EM or/and Pauli? How can I convert a std::string to int? If you try to execute a program that is not installed you will get a message  The program 'x' is currently not installed. You can install it by typing:  sudo apt-get install x   and instead of offering you to install the program for you, you have to type the actual command. How can this behavior be changed? Overstaying Schengen Visa for 1 day I was in Germany last September, my visa said I had 14 days to stay in the Schengen area.  When I was leaving back for Egypt the passport control officer noticed that I was leaving on the 15th day. I didn't even realise that as I had written previously in my visa application that I was staying for 15 days and the embassy in Cairo had my complete airline ticket with specific dates.  Anyway the officer just filed a report on the incident but didn't stamp my passport with my exit date from the Schengen area.  Do any of you guys know if that would make it harder for me to be granted a visa in the future? No help from ?-command Does anyone know of a .NET date/time parser similar to  (handles stuff like "tomorrow" or "3pm next thursday")?  Note: I do write Ruby (which is how I know about Chronic) but this project must use .NET. Smooth option sometimes produces incorrect arrow tips in PGFplots It seems that the smooth option applied to lines sometimes effects the arrow tips and produces incorrect results.  No problem with vertical lines, but any other seems to have a problem. Is this a known bug, or is there a reason why the smooth should not be applied to straight lines?  Updated: to show problem does NOT occur with TikZ, and problem also occurs with curves if smooth option is applied.  \documentclass{article} \usepackage{pgfplots} \begin{document}  \tikzstyle{MyStyle}      =[-&gt;, black, ultra thick,       ] \tikzstyle{MyStyleSmooth}=[-&gt;, black, ultra thick, smooth]  \begin{tikzpicture} \begin{axis}     % 'smooth' option seems to have no effect on lines '1' and '2'     \addplot [MyStyle      , green ] coordinates{(0,0) (0,1)} node [above] {1};     \addplot [MyStyleSmooth, blue  ] coordinates{(1,0) (1,1)} node [below] {2};     %     % But, can't seem to have the 'smooth' on lines, '4', '5' and '6'     \addplot [MyStyle      , orange] coordinates{(2,0)   (3,0)   } node [right] {3};     \addplot [MyStyleSmooth, red   ] coordinates{(4,0)   (2,0.5) } node [below] {4};     \addplot [MyStyleSmooth, red   ] coordinates{(2,1)   (3,1)   } node [left ] {5};     \addplot [MyStyleSmooth, red   ] coordinates{(3,0.8) (2,0.8) } node [right] {6};      % Similar problem with curves: "non-smooth" is ok, "smooth" is not     \addplot[MyStyle,       mark=none, domain=0:1,samples=50, blue] (x,x*x+1);     \addplot[MyStyleSmooth, mark=none, domain=0:1,samples=50, red ] (x,x*x+2);  \end{axis} \end{tikzpicture}  \begin{tikzpicture} % These are all fine     \draw [MyStyle      , green ] (0,0)   -- (0,1)    node [above] {1};     \draw [MyStyleSmooth, blue  ] (1,0)   -- (1,1)    node [above] {2};     %     \draw [MyStyle      , orange] (2,0)   -- (3,0)    node [right] {3};     \draw [MyStyleSmooth, red   ] (4,0)   -- (2,0.5)  node [below] {4};     \draw [MyStyleSmooth, red   ] (2,1)   -- (3,1)    node [right] {5};     \draw [MyStyleSmooth, red   ] (3,0.8) -- (2,0.8)  node [left ] {6}; \end{tikzpicture} \end{document} What is the exact difference between fluid and responsive design? What is the exact difference between fluid and responsive design?  I am bit confused about fluid and responsive design. Fluid is the one where we give width in percentages so that design will look good on browsers even if shrink or expand the browser. But what's different about responsive? Alien invasion movie, giant robot aliens with long legs capturing people So I saw this movie about 10 years ago, when I was about 5 years old. Since I was really young, my memory of it isn't really that great.   I think it may have been an alien invasion movie, but the only scene I remember is that there were these giant robot-alien things:   I think they had several long legs. They had a big glass dome on top of them They would grab civilians and drop them in their glass dome.   That's all I remember. If you could please help me out, that would be great.   Again, I was about 5 years old when I watched it, so my memory of it isn't that great, but I hope you know what I'm talking about. Thanks so much :) How to draw circle in html page? CSS two divs next to each other Find the limit $ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}} $ Find the limit $$ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}}. $$  I know that the limit is of indeterminate form type $\frac{\infty}{\infty}$, but it seems using L'Hopital's rule directly does not help here.  Do I somehow use the fact that $\lim_{x \to \infty} (1+\frac{1}{x})^{x}=e$? First chapter appearing before first line in table of contents Is it possible to use the startup.blend UI layout in older files? Is there a way to open files without changing the UI layout?  Sometimes in older files the layout will be in the old 2.4x style, which can be annoying.  One way would seem to be appending everything in, but that gets tedious with having to do it over again and again for objects, materials, textures, etc. In the example  $\cup^{i=1}_{\infty}F_i$   How do you get the i=1 directly below the union symbol and the infinity directly above? How do I check if a given string is a legal/valid file name under Windows? I want to include a batch file rename functionality in my application. A user can type a destination filename pattern and (after replacing some wildcards in the pattern) I need to check if it's going to be a legal filename under Windows. I've tried to use regular expression like [a-zA-Z0-9_]+ but it doesn't include many national-specific characters from various languages (e.g. umlauts and so on). What is the best way to do such a check? Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent? As I was just checking this 'child prodigy' out on Youtube, I stumbled upon  video, in which Glenn Beck asks the kid to do the following proof:    Further on, the kid starts sketching a proof (without a shadow of a doubt regarding the accuracy of his solution ) including the Integral Test. I don't know much about improper integrals since I just finished highschool, but this integral approach seemed, intuitively, pretty inaccurate to me since this is not a strictly decreasing function. Then I found this out from Wikipedia !   Furthermore, the blunt assessment that the series are convergent seems dubious as well..Upon a few computations of my own(mostly partial sums) , I tend to believe that the series are, in fact, divergent .  Can anyone suggest a rigurous take on this problem (easy as it may seem to some amongst you) ? How do I move all files from one folder to another using the command line?