There exist an isomorphism of varieties? $$\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$$   I am considering $\mathbb{P}^n\times\mathbb{P}^m$ as the product in the category of varieties.
One can show that $\mathbb{P}^n\times \mathbb{P}^m$ is birational to $\mathbb{P}^{n+m}$ by making note of the Zariski topologies and the canonical isomorphism between affine spaces $\mathbb{A}^n\times \mathbb{A}^m\to \mathbb{A}^{n+m}$.   Question: Does it follow that there is isomorphism $\mathbb{P}^n\times \mathbb{P}^m\to \mathbb{P}^{n+m}$? If so, what is an example of such an isomorphism?   Unfortunately I am having a hard time coming up with one, and haven't seen too many examples thus far.
Sorting grid cells in fishnet grid by location and number using QGIS? I am using QGIS 3.2.0 on a Mac (and also have QGIS 2), and I have a fishnet grid (probably created in Arc) that comprises of nearly 2 million grid cells. As a result, every change I make to the attribute table takes hours, so I can't experiment as I would usually.  I need to label each grid cell such that the north-westernmost cell is "1", and the numbers proceed by rows, until the last number goes into the south-easternmost cell. Using the row-number doesn't work, as that's not how the table is sorted. I've done this before, but cannot remember how. I am currently studying  answer on the Moore-Penrose pseudoinverse and the Euclidean norm by the user &quot;Etienne dM&quot;. The first point of their answer proceeds as follows:  Let $x$ be $A^+y$.  Let me begin by the second point. For all $z$, we have: \begin{align} \lVert Az-b \rVert_2^2 &amp;= \lVert Ax-b \rVert_2^2 + \lVert A(z-x) \rVert_2^2 + 2 (z-x)^TA^T(Ax-b)\\ &amp; \geq  \lVert Ax-b \rVert_2^2 + 2 (z-x)^TA^T(Ax-b) \end{align} Moreover, because $(AA^+)^T = AA^+$, $$ A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$$ Thus, we prove that for all $z$, $\rVert Az-b \lVert_2^2 \geq\rVert Ax-b \lVert_2^2$, that is to say $A^+b$ is as close as possible to $y$ in term of the Euclidian norm $\lVert Ax-b\rVert_2$.   I realise that $x = A^+ y$, but I don't understand how any of this implies that &quot;$A^+ b$ is as close as possible to $y$ in terms of the Euclidean norm $\lVert Ax-b\rVert_2$&quot;. I would greatly appreciate it if people would please take the time to explain this to me. How did the AT-ATs get to the surface of Hoth? I'm curious how these beasts got down to the surface of Hoth.    Did the Galactic Empire land them in big ships, if so which ones? Assume you are given two objects whose exact locations are unknown, but are distributed according to normal distributions with known parameters (e.g. $a \sim N(m, s)$ and $b \sim N(v, t))$. We can assume these are both bivariate normals, such that the positions are described by a distribution over $(x,y)$ coordinates (i.e. $m$ and $v$ are vectors containing the expected $(x,y)$ coordinates for $a$ and $b$ respectively). We will also assume the objects are independent.  Does anyone know if the distribution of the squared Euclidean distance between these two objects is a known parametric distribution? Or how to derive the PDF / CDF for this function analytically? Transcendental numbers and irrationality measure What is the function of this circuit? Just wondering if someone can give me an explanation for this capacitor layout - what is it's function? The chip to the right is an ADC.  Circled in blue.    I'm guessing it is well known but i don't know how to search for it. Why does Type I error always occur in a NHST? Or, is NHST too weak to tell the truth? Recently I have noticed that Type I error of a test is too much persistent. Even if all the values are almost equal (variance is extremely low) and tested to that value (true mean), still there is nearly 5% Type I error of the test. I am frightened and curious to rethink, are the people who have banned (or skeptic about) NHST (null hypothesis significance testing) truly correct? Let's have a look at the following r code:       n=10000 t1err=0 set.seed(1235) for (i in 1:n){    x=rnorm(100, 10, 20)    if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)  } cat("Type I error rate in percentage is", (t1err/n)*100,"%")   It will produce roughly 5% as Type I error. Seems ok. But let's see this:      n=10000   t1err=0   set.seed(1235)   for (i in 1:n){      x=rnorm(100, 10, 0.000001)      if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)    }   cat("Type I error rate in percentage is", (t1err/n)*100,"%")   Each values are extremely close to 10. Still it produces about 5% Type I error. What conclusion we may draw from it? Or, am I missing something? Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$ I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - S$$ Therefore, $2S = 1$, so $S = \frac{1}{2}$. QED   I was under the impression that this sum doesn't converge, and therefore this step of the proof is invalid:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$   That is, it's not even valid to suppose that the sum is equal to some (implicitly real or complex) number $S$, and therefore the reasoning that follows is meaningless because $S$ doesn't exist and therefore has any properties we want it to have.  Is the reasoning I've given above correct? That is, is it correct for me to claim that the proof fails because $S$ doesn't exist?  (I've heard that there are techniques for evaluating divergent integrals by using complex analysis and Taylor series, so perhaps there's another way to prove that the summation is $\frac{1}{2}$; I just wanted to see whether my reasoning is sufficient to explain why this particular proof is incorrect. If the proof actually is correct, then I stand corrected!)  Thanks! How can I test the quality of a RNG? On electronics.stackexchange we had  about constructing a True Random Number Generator. Since the method relies on noise, which isn't deterministic, the only way to test the quality of the RNG seems to be empirical. Not being a statistician I suggested to test a long bit sequence for normality, but I have no idea what outcome is acceptable and what isn't. For instance counting single bits I guess we have a thumbs-up for a 499500/500500 distribution, but when is the ratio too much skewed to be acceptable? Same for 2-bit and longer sequences. Of course, if testing for normality is a Bad Idea™, I'd like to hear it, including better alternatives.    edit Diehard was mentioned a few times, but I'm not sure this answers my question. The normality test should give a normal distribution, with better bell curve approximations for longer sequences. Diehard seems also to have tests which either should result in normal or exponential distributions. But my question remains: how do I judge the results? Just by looking at the curve and discern a bell curve in it? To get back at my first example, a 499500/500500 distribution is definitely OK, and a 950000/50000 distribution definitely is a no-no, so where does the switch-over happen? The search parameter isanswered ignores acceptances What is the difference between Parse() and TryParse()?  int number = int.Parse(textBoxNumber.Text);  // The Try-Parse Method int.TryParse(textBoxNumber.Text, out number);   Is there some form of error-checking like a Try-Catch Block? Photoelectric effect in a a frequency range I know how to calculate the kinetic energy of the emitted electrons   $$KE=hf-\phi$$  And the number of the emitted electrons, but this is valid for a certain frequency  $$N=\frac{IA}{hf}=\frac{P}{hf}$$  What happens if I have a frequency range? for example between λ1 and λ2?  BTW: It's not homework, I'm an engineering student,  not a physics student What makes a field not groupable I'm trying to use an aggregate query, but one of the fields I want to group by is not groupable (saw it in the soql error and then verified groupable=false in the metadata).  It's just a number field that holds an integer (number 2,0 in salesforce field spec-speak).  Is there a trick to make it groupable?  Any way around this problem? I don't want to write hundreds of lines of code just to average a list by this field. Undefined control sequence and \chapter error I’m getting an “undefined control sequence” error with the following code, using TeXstudio:  \documentclass[11pt]{article} \begin{document} \begin{center} \chapter{CHAPTER 6} \end{center} \end{document} What methods can be used to micro-adjust autofocus of a camera body to a particular lens? I have a Canon 7D with a 50mm f/1.4 lens, and I think the auto-focus of the lens is off.  How can I test and adjust this reliably?  Will this approach work with all of my lenses? If I had a different camera body, would I have other/different options? White space below footer I have edited a script that creates some nodes, but the only problem is, it always created a new material. I want it to either add the nodes on the existing material, or a pre-defined material. Here is the script base I am using.    Thanks for the help in advance! :D I created a bitmap with transparent background but in the Windows Picture Viewer I see the white background and even if I reopen it in Photoshop CS3. how do I make sure the white colour in bitmap is really transparent or see if contain any transparent colour? 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. An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$