I'm interested in understanding compositum of general fields better. Assume we have $\Omega/K/F$ and $\Omega/L/F$ field extensions, and consider the composite $KL$.   It seems to me that every $a \in KL$, exists $n_1,n_2 \in \mathbb{N}$, $ \left \lbrace k_i \right \rbrace_{i=1}^{n_1}, \left \lbrace k'_i \right \rbrace_{i=1}^{n_2} \subseteq K$ and $ \left \lbrace l_i \right \rbrace_{i=1}^{n_1}, \left \lbrace l'_i \right \rbrace_{i=1}^{n_2} \subseteq L$ such that: $$ a = \frac{\sum_{i=1}^{n_1} k_i l_i}{\sum_{j=1}^{n_2} k'_j l'_j} $$  (That is since the family of all such sums is closed under addition, multiplication, and distribution because of the abelian nature of fields, therefore all numbers of this form are subfields that contain both K and L)  Is it a correct way to refer to a compositum of fields? When is it correct that the general item of an extension is $a = \sum_{i=1}^{n} k_i l_i$?
Where $F$,$K$ are two fields. What does the element in the composite field $FK$ look like? All the elements are generated by the elements of $F$ and $K$? (combination of the elements of $F$ and $K$) I think $FK$ is pretty close to the free product of $F$ and $K$; is $FK \subsetneq F*K$? Thank you!
Continuous and preserves measurability $\implies$ preserves null sets. 19.10: Deactivate Alt Key From Triggering Applications' Menu Bar How can I stop the Alt key from triggering the menu bar of applications in Ubuntu 19.10?  I know that there are these , , and , as well as  but those solutions are no longer applicable. I did try searching for a similar option to the one in the answer of those questions but with no success. Why is my "Committed" memory so much higher than my actual RAM space? I have run two virus scans and neither of them have found anything so far (Malware Bytes and Avira Rescue). During gaming I will get Windows popups saying that my system is low of memory and recommends closing the game. This can also happen while browsing. My computer seems to become sluggish a lot and will hang up often, such as when clicking a link from a messenger program to open in my browser.   I am unsure what is causing this. My memory usage on the processes tab doesn't add up to such a high percentage of my actual memory.    As a side note, is my Disk Usage supposed to spike up to 100% sometimes? Research seems to say so. I am wondering if the explanations correct and whether what is the difference between different &quot;seems&quot;.   He seems happy. He seems a happy fellow. He seems to be a happy fellow. He seems to work hard. He seemed to be happy   &quot;Seem&quot; also enters into an 'impersonal' construction which employs 1) the 'dummy' subject it + an active tensed form with &quot;seem&quot;, followed by 2) a that clause (sometimes called a 'content' clause) which expresses the assertion which is hedged or qualified.   It seems that he is happy (but he may not be).  It seemed that he was a happy fellow (but he wasn't).  It will seem that he is working hard (but he will in fact be spending his time answering questions on ELL instead of working).  He is seemed to be something. df says I have 20G more disk space used than du. Why? From time to time I come up with ideas of actions I'd like to have available in the context menu of Nautilus. How do I add them there? Is there something like a Thunar UCA plugin for Nautilus? Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups How can edge rings be selected in a predictable way? When trying to select multiple edge-rings Shift-Alt-RMB sometimes loops get selected in the one (green) way or the other (red) way.  I couldn't derive a general rule.   How can two or more face-rings following one direction be selected ?  (Easy to reproduce with a default torus mesh.) In the late 1970s I bought a book at a school book fair (likely published by Scholastic) about a young boy who lived on a generational starship.  The ship was failing.  I remember some bit about an icey cold airlock.  One of the constellations (Cygnus) somehow took on actual fleshy form and manifested as a talking swan, which somehow helped the starships reach their destination.  Pointers? Question  I want to write a book in Arabic (I have an Arabic keyboard, so I don't need a converter). By Arabic book I mean literally Arabic:    title, table of contents  etc    all in Arabic.   The equations, however, I want to write them in English/Latin.  Related    This link helped a lot but it was for Arabic texts in an English main document.    I already checked these two (and plenty other sites). Minecraft Test If A Chest Contains An Item I've seen a lot of questions for finding specific items in chests using commands using testforblock but how would you test if any block/item has been placed in a chest?  Thanks Google Chrome will take up my memory to the point where it causes my computer to freeze to a near halt. What can I do to prevent this? It gets so horrible that I can not even switch to TTY1 to force a reboot. I have to hard reset, and I lose work.  Ideally, I would like to be able to use the chrome browser on Ubuntu without having to keep reminding myself to close tabs.   Are there any extensions for Chrome, or tools for Ubuntu that can help prevent me going past the point where Ubuntu eats up all the RAM and becomes incapacitated? My Y axis is inverted! How do I fix that? When I play Black Ops 2, it seems that my xbox controller is inverted. Example: if I press up, my character looks down and vice versa. It was not like this before. Does anyone know how to reset the controller back to its original settings? How to use an optimization solver to get t-stats and p-values for the estimates? Evaluate the integral $\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$ This integral looks a lot like an elliptic integral, but with cubes instead of squares: $$I(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$$ Let's consider $a,b&gt;0$ for now. $$I(a,a)=\int_0^\infty \frac{dx}{x^3+a^3}=\frac{2 \pi}{3 \sqrt{3} a^2}$$ I obtained the general series solution the following way. Choose $a,b$ such that $a \geq b$, then: $$I(a,b)=\frac{1}{a^2} \int_0^\infty \frac{dt}{\sqrt{(t^3+1)(t^3+b^3/a^3)}}=\frac{1}{a^2} I \left(1, \frac{b}{a} \right)$$ $$\frac{b^3}{a^3}=p, \qquad I \left(1, \frac{b}{a} \right)=I_1(p)$$ $$I_1(p)=\int_0^\infty\frac{dt}{\sqrt{(t^3+1)(t^3+p)}}=2 \frac{d}{dp} J(p)$$ $$J(p)=\int_0^\infty\sqrt{\frac{t^3+p}{t^3+1}}dt=\int_0^\infty\sqrt{1+\frac{p-1}{t^3+1}}dt=$$ $$|p-1| \leq 1$$ $$=\sum_{k=0}^\infty \left( \begin{array}( 1/2 \\ ~k \end{array} \right) (p-1)^k \int_0^\infty \frac{dt}{(t^3+1)^k}$$ Now this is the most problematic part. The first integral of this series diverges. However, it's a constant in $p$, so if we differentiate, it formally disappears: $$I_1(p)=2 \sum_{k=1}^\infty \left( \begin{array}( 1/2 \\ ~k \end{array} \right) k (p-1)^{k-1} \int_0^\infty \frac{dt}{(t^3+1)^k}$$ Now, every integral in this series converges. The integtals can be computed using the Beta function, if we substitute: $$t^3=\frac{1}{u}-1$$ Finally, we rewrite: $$I_1(p)=\frac{\Gamma (1/3)}{3 \sqrt{\pi}} \sum_{k=1}^\infty \frac{k^2}{k!^2} \Gamma \left(k- \frac{1}{2}\right) \Gamma \left(k- \frac{1}{3}\right) (1-p)^{k-1}$$ Or, using the Pochhammer symbol:  $$I_1(p)=\frac{2 \pi}{3 \sqrt{3}} \sum_{k=0}^\infty \frac{(k+1)^2}{(k+1)!^2} \left(\frac{1}{2}\right)_k \left(\frac{2}{3}\right)_k (1-p)^k$$  My questions are:  Is the method I used valid (see the 'problematic part')? How to get this series into a Hypergeometric function form? Is there any 'arithmetic-geometric mean'-like transformation (Landen's transformation) for this integral? How to go about finding it?  If the method I used is correct, it can be used for any integral of the form ($m \geq 2$): $$I_m(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^m+a^m)(x^m+b^m)}}$$ In music, when two or more pitches are played together at the same time, they form a chord. If each pitch has a corresponding wave frequency (a pure, or fundamental, tone), the pitches played together make a , which is obtained by simple addition. This wave is no longer a pure sinusoidal wave.  For example, when you play a low note and a high note on a piano, the resulting sound has a wave that is the mathematical sum of the waves of each note. The same is true for light: when you shine a 500nm wavelength (green light) and a 700nm wavelength (red light) at the same spot on a white surface, the reflection will be a superposition waveform that is the sum of green and red.  My question is about our perception of these combinations. When we hear a chord on a piano, we’re able to discern the pitches that comprise that chord. We’re able to “pick out” that there are two (or three, etc) notes in the chord, and some of us who are musically inclined are even able to sing back each note, and even name it. It could be said that we’re able to decompose a Fourier Series of sound.  But it seems we cannot do this with light. When you shine green and red light together, the reflection appears to be yellow, a “pure hue” of 600nm, rather than an overlay of red and green. We can’t “pick out” the individual colors that were combined. Why is this?  Why can’t we see two hues of light in the same way we’re able to hear two pitches of sound? Is this a characteristic of human psychology? Animal physiology? Or is this due to a fundamental characteristic of electromagnetism? Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected?  This is not homework but is in response to working several problems where countable or uncountable many points are removed and the pattern always holds.  Thank you. Showing directional line symbol in ArcGIS for Desktop? Is this possible in ArcGIS to have variable number of arrows on the line to show direction?  This is an ArcGIS example, where I can have fixed (1,2,3,4,5...) number of arrows and it stays the same doesn't matter what the line length is. First and last always stick to the end, doesn't help with network mapping, where there is usually the node to show. Having just one in the middle is never enough, check line at the right...   No such issues with ArcView 3: Consider equations defined in $D_x = (0,\infty) \times (0,\infty)$  1) Find the maximal solution to  $$ \dot y = \frac{3t^2}{5y^4} $$ satisfying the initial value problem $y(1) = 2^{\frac{1}{5}}$  I have found the maximum solution to be $((0,\infty), (t^3+1)^\frac{1}{5})$  2) Find the maximal solution to   $$ \dot y = -\frac{2y}{5t}+\frac{3}{5y^4}$$  satisfying the IVP $y(1) = 2^{\frac{1}{5}}$.  (HINT: Use a transformation of the form $t^{\alpha}y(t)$.   I hope someone can explain how to use the transformation to find the maximal solution. Can I have multiple primary keys in a single table? Can I have multiple primary keys in a single table?