NumPy for Python 2.7 on 64 bit Windows
Installing Numpy on 64bit Windows 7 with Python 2.7.3
Show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$. The problem is:  Suppose that $f(z)$ is continuous on a domain $D$ that contains the unit circle, and that $f(z)$ satisfies: $$\vert f(e^{i\theta})\vert \leq M \; \forall \theta \, \in [0, 2\pi )$$ and $$\bigg \vert \int _{\vert z \vert=1} f(z) \text{d}z \bigg \vert =2\pi M $$ show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$.  I've tried lots of different approaches such as defining a function $g(z)=\frac{f(z)}{\bar{z}}$ and trying to show that $g'(z)=0$ so that $g$ is constant on the circle, or trying to use some bounding theorems for analytic functions to show that $g$ is constant but this requires $g$ to be analytic. Some other methods were using the Cauchy-Riemann equations but I couldn't find any reason why $g$ should be analytic.  Any help would be greatly appreciated, thanks! $N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic. The following is a problem from the Berkeley Problems book.      Let $G$ be a group with three normal subgroups $N_1$ , $N_2$, and   $N_3$. Suppose $N_i \cap N_j = \{ e\}$ and $N_iN_j  = G$ for all   $(i,j )$ with $i \ne j$ . Show that $G$ is abelian and $N_i$, is   isomorphic to $N_j$ for all $i, j$ .   Attempt:   I have not got very far. The only thing that comes to my mind is that as $|HK|=\frac{|H||K|}{|H \cap K|}$, $|N_1|=|N_2|=|N_3|=m$, and order of $|G|=m^2$. I have made other futile attempts of trying to show m prime, finding an isomorphism between the $N's$. I think the fact that the equation $N_i N_j=G$, could be seen as, left coset of $N_j$ with the multiplying elements restricted to only $N_i$ could provide me an answer, but I haven't got anywhere with this.   Thanks in advance. If a sequence $(u_n)$ be such that its every subsequence has a subsequence that converges to $0$, then $\lim u_n= 0$ A No-Nonsense Introduction to Quantum Field Theory How to get the list of properties of a class? I have an image with a woman, and I need to make an ad (size 260mmx125mm). I want to put the woman to the left side, but then right side becomes empty. So, how I should solve the problem? I have picked the color from image, but it is definitely different than in the actual picture.  Should I use the Clone Stamp Tool in Photoshop, or there is a better way to do this?  By the way, it will be used in print, so I need a good quality. GPU Rendering causing strange lines When I render using my r9 390 graphics card, weird lines are appearing over the render. This problem has just started happening recently, as a few days ago I was rendering the same scene without the lines appearing. I don't believe I've changed any settings in Blender and the below scene is from a new file, so render settings should be default.   Rendering using my CPU is fine:   The only thing I can think that has changed is I recently updated my card to the latest drivers. Playing games with it is still fine and the card is quite new so I wouldn't think it's a problem with the card itself. How do you reset timers? If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$. Why does a cordless circular saw have the blade on the left? At the moment considering the purchase of a circular saw for light work. I'm looking at a corded model and a cordless model. But the question I have is not about which one is better.   My question today is why the corded one has the blade on the right of my hand and the cordless one has the blade on the left of my hand. See both models below.     Now, I know some manufacturers make left- and right-handed versions of their circular saw. And I'm also aware that 'worm drive circular saws' usually have the blade to the left of my hand. But it seems odd to me. Especially as it seems to apply to most manufacturers too.   Was wondering if somebody has an explanation for this. I stumbled upon a  with the following paragraph:     MBIE has launched an investigation into claims Chinese galvanised   steel coil imports have been dumped on the local market, causing   "material injury" to New Zealand industry, and the case will end up on   the table of newly appointed Commerce Minister Jacqui Dean.   I thought that sentence was grammatically wrong because the author should put the word "that" (as a conjunction) right after the word "claim". I don't see why the author could drop the conjunction. Is this a prevailing phenomenon in American or Australian English? Is there a reason the British omit the article when they "go to hospital"? Why do British speakers omit the article in constructions like "go to hospital" or "go on holiday"? Pretty much all American speakers would rephrase those as "go to the hospital" and "go on a holiday", I think. Is there any good reason, or forgotten sense behind those words that might explain why the articles are ommitted?  Are there other common constructions other than those two that the British use that drop the article?  EDIT: I just realized per Kosmonaut's comment that Americans do much the same thing with a few nouns, so this isn't all that special.   Do grammaticists designate nouns that can have their article dropped with anything, i.e. do they have anything in common? What's the difference between real and effective aperture? I understand that the simple idea that f-stop is focal length divided by aperture really applies to a simple single-glass lens, not the compound lens systems used to provide good image quality in modern lenses.  I've heard that in this case, what matters is the ratio of the aperture as it appears through the front element. Why the front element, and not the back (which would be the "point of view" of the sensor or film)?  What about for effects such as diffraction? Does the literal physical aperture size matter, or is it the effective size? We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$.   I'm now trying to say something about the possible cardinalities of subfields of $F$. I can see that there is a subfield of cardinality $p^s$ for each $s$ that divides $r$, given by the fixed field of the group generated by $\phi^s$, where $\phi$ is the Frobenius automorphism.  Now suppose $K$ is a subfield of $F$. Then (since both are additive groups), Lagrange gives us that $|K|$ divides $|F|$, so $|K| = p^t$ for some $1 \leq t \leq r $ (alternatively, $K$ contains $\mathbb{F}_p$ and so is a vector space over $\mathbb{F}_p$ and is thus isomorphic to $\mathbb{F}_p^t$, where $t = [K:\mathbb{F}_p]$). By considering the multiplicative group of units of $K$ and $F$ respectively, we get that $ p^t - 1$ divides $p^r -1$. I want to make the leap to $t|r$, but I'm failing to see why this needs to be true. Any help would be appreciated. I opened terminal windows and I got a message   your screen size %d%d is bogus. Expect trouble   I closed the terminal and the next time I opened terminal, the message didn't show up.   What is this message and why did I receive it? How to install latest HPLIP on my Ubuntu to support my HP printer and/or scanner? Let's assume that one has some modern HP printer and/or scanner which is not supported by HPLIP package from official repository. Currently Ubuntu versions have :  trusty (14.04LTS): 3.14.3-0ubuntu3.4 xenial (16.04LTS): 3.16.3+repack0-1 bionic (18.04LTS): 3.17.10+repack0-5 focal (20.04LTS): 3.20.3+dfsg0-2 groovy (20.10): 3.20.5+dfsg0-3build1 hirsute (21.04): 3.21.2+dfsg1-2   What can one do if printer is supported in newer version of HPLIP (checked this in )? Tagging best practices I am wondering what I should do when tagging a question that fits multiple tags, in the particular case were the question might belong to general as well as more specific tags.  Let me give you an example to illustrate this:  Let's say I have a question about ASP.NET MVC 3.0 and I use C# as my main language.  The following tags could apply:   c# .net .net-4.0 asp.net asp.net-4.0 asp.net-mvc asp.net-mvc-3   Tagging with C# might make .NET redundant (I don't think C# is used anywhere else outside of the .NET scope), and tagging with  certainly makes  and  redundant. However, my question would get a better visibility with tags such as ,  and , as probably no one browses a tag as specific as .  How shall I tag my questions? Using more general or more specific tags, or a mix of the two? Abstract algebra book recommendations for beginners. I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved problems and exercises? The lecturer said in this semester, he will cover group theory, ring theory and a bit of field theory. Which book contains extensive information in these three fields? Control a property with no socket (color ramp swatch, etc) from a Group Node input? Bash string range and replace To print part of the string, replace , with . I use command:  echo "${q:16:6}" | sed 's/,/./'   Is it possible to use something like:  echo "${q:16:6/,/.}"   because it does not work?