Large tables going beyond margins
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dynamic does not contain a definition for a property from a project reference If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable Proof that $\inf A = -\sup(-A)$ How to switch window controls to the left (Gnome Shell)? Is there a way to the switch gnome-shell window buttons to the left? I've gotten so used to them being on the left that them being on the right has thrown me way off.  (gnome shell has them defaulted to the right corner) Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following : Firstly, how do I read it? Is below right? With braces   $$p \vdash (q \Rightarrow p)$$  The given proof is:    ${p}$, premise ${q}$, assumption  $p$ by (1) // what???  ${q} \Rightarrow p$ by implication introduction with 2 and 2.1 QED...    I can't really get the link between $p$ and $q$ in 2.1 ... "This one is" OR "This is" I have seen people using "This one is Daddy's" with the word "one" before a form of the verb be. I am confused about which is more accurate, to use:     This is Daddy's.   or     This one is Daddy's.   Hope you can address my concerns, thank you. First Question for me.  I have benefited from other answers here but have not fully resolved my problem.  Here's my problem...  The LM311 Comparator output does not drive the N channel MOSFET fully on.  The LM311 has a 110mv ref voltage on the inverting input. The non-inverting input is driven by an opto-isolator output. The Opto-isolator is driven by 1 of 4, AC 60Hz voltages.  &lt;1VAC, 9VAC, 55VAC and 115VAC.  The goal of the design is to have the FET fully turn on the 12VDC fan when there is appreciable AC voltage.  So fan fully On for the 9, 55, and 115VAC voltages.  It seems it should work because I measure voltages on the non inverting input of 1.6, 2.2 and 3VDC respectively for the 3 appreciable AC voltages.  And my ref voltage on the inverting input is 110mv.  So it "almost" works because the fan is off at the &lt;1VAC, but not fully "On" for the other AC voltages.  I believe this is because Vgs is 167mv, 6.1V, 7.7V and 9.9 Volts for the 4 AC voltages listed above.  And the fan is only getting 7, 8 and 10VDC instead of a desired 12VDC. Would really like to be able to fully turn this FET all the way on.  So Close!  Any input or guidance is appreciated.  Ps-Things I think I've done right? Flyback diode across the fan. Putting N channel FET to switch to ground on low side of fan. LM311 has Open collector output using correctly?  PSS-I realize that the pinouts for the LM311 are not right in my schematic.  I have them right in my protoboard circuit.  Its a single device.  Pin 2 is noninveritng and pin 3 in Inverting  Thanks Again.      The suggested duplicate question and answer doesn't help for the following reasons: Both diagram and schematic are"No file found" making it pretty difficult to follow the answers. I am not using any positive feedback in my circuit as I don't think I need one. I have a simple pull up resistor to my 12V supply, not a voltage divider to the power supply as is mentioned in the answer. Thanks.  Just for the record I wanted to post a more accurate schematic and thanks for the response on using a Cap to hold up the opto isolator output. I actually have a full wave input to the phototransistor but I'm sure the cap will help anyway. To answer the last question, the Pin 7 comparator output has a pullup resistor to 12v. The issue is still that Vgs is only 10v when the comparator has 3v on pin 2 and 110mv on pin 3. I would have expected the LM311 output to go much closer to 12v (Vcc) with those inputs. The intent is for the FET to go completely on for pin 2 voltage > pin 3. I am reading through Folland's Real Analysis: Modern Techniques and Their Applications, and they have the following proposition and proof:     Proposition: If $\{f_{j}\}$ is a sequence of $\bar{\mathbb{R}}$- valued measurable functions on $(X, \mathcal{M})$, then the functions:      $$g_{1}(x)=\sup_{j} f_{j}(x)$$ $$g_{2}(x)=\inf_{j} f_{j}(x)$$ are   measurable.       Proof:      We have       $$g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}  f_{j}^{-1}((a,\infty])$$ $$g_{2}^{-1}([-\infty,a))=  \bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$$      Where the result follows from the fact that $g_{1}^{-1}((a,\infty])\in  \mathcal{M}$ for all $a \in \mathbb{R} \iff g$ is measurable, and   $g_{2}^{-1}([-\infty,a))\in \mathcal{M}$ for all $a \in \mathbb{R}  \iff g$ is measurable.   I understand the last part of the proof, but how do we know $g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}f_{j}^{-1}((a,\infty])$ and $g_{2}^{-1}([-\infty,a))=\bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$? I would appreciate any help with this. Following are other questions which I think I need to know:   From a non X Session? (meaning root isn't logged into X) If multiple people were logged in on X, could I auto-detect who was on which screen, and thus programmatically detect which screen I need to launch the app on? Can I launch the app as the user? ( ok I'm 99.999% sure this is a yes ) Can I detect if users of group X are logged in to X? Array of zero length I am working on refactoring some old code and have found few structs containing zero length arrays (below). Warnings depressed by pragma, of course, but I've failed to create by "new" structures containing such structures (error 2233). Array 'byData' used as pointer, but why not to use pointer instead? or array of length 1? And of course, no comments were added to make me enjoy the process...  Any causes to use such thing? Any advice in refactoring those?  struct someData {    int nData;    BYTE byData[0]; }   NB It's C++, Windows XP, VS 2003 Show that    $$\frac{\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}}{\binom{M}{n}}=\frac{k}{M},$$ where $M-k&gt;n$ and $k&gt;n$   From Vandermonde's identity, I get $\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}=\binom{M}{n}$.   But what about the sum of the product in the numerator? Any hint will be helpful. How do I boot into a root shell? When booting to the graphical desktop fails we sometimes are recommended to boot into a "root shell" or to boot in "recovery mode".  How do I do that? What bizarrities lurk within the decimal expansion of an irrational number? i.e. can we write  $\pi = 3.14159\dots X\dots$  where $X$ consists of (say) $10^{100}$ consecutive zeroes?  [Originally asked on reddit without response :-( ] Are there any tools in Blender that take irregular polygons and make them regular (approximate a circle)? I'm specifically looking for a solution that works with preexisting polygons. Also, it would be nice if it keeps the overall position after making it regular.    Irregular Polygons    Regular Polygons Egyptian fractions with prime power denominators summing to 1? Inspired by        Can we solve the equation $$1=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots +\frac{1}{a_n}$$ where $a_1,a_2,a_3,\cdots ,a_n$ are distinct prime powers (primes are allowed) ?   I tried a modified greedy-algorithm with different starting vectors and brute force with $6$ entris with limit $200$, but did not find a representation. During iPhone update to iOS 10.3.1, I was asked for my password using the correct machine name for my Mac mini. To be clear, the request was not for my iCloud password, my actual local user account password for macOS. This can also happen in the other direction - setting up a Mac may ask for verification for you to enter your iPhone / iPad passcode (and not your iCloud password)  I could not find any information about this kind of access. I was thinking it was security for iCloud and possibly related to two step authorization or two factor authorization. This is the first time I can think of that I entered a desktop password outside of that actual desktop.  I don't have a screen shot but I recall an implication that some files in my Mac are encrypted and need my password. '... need to enter the password for .'   Are there settings I can I control this assuming it's a legitimate part of iCloud setup process or iOS setup process? App keeps crashing & rendering error since update to 1.0.68 I can reproduce that every time now, I go to a site, open an article, then to the next site, open another article. The Feed page pops up for a second, the article page lays over this with some some transparent areas (see picture). The same app-version is ok on a Smartphone with Android 4.4.2. I also sent in some bug reports.    Edit  1.0.71 isn't the solution yet. I go to Meta SE, then to an article. When I want back to Meta SE again, it crashes. Much earlier than before. I already sent in a bug report for this.  Another Edit  I can describe it better now. From the questions-page of a site I go to a post. When I use the back button, everything is ok, I can go to the next site. When I try to go to the next site directly from the post, it crashes. This only happens on the Nexus 10 with Android 5.1.1, App version 1.0.71. Hope this helps.  Edit3  This is fixed in 1.0.73 now. Thanks again. If $a$ and $b$ are relatively prime, prove that $a + 2b$ and $2a+b$ are also relatively prime or have a $\gcd$ of $3$.  I'm very new to number theory so please don't assume I am familiar with some of the terminology. How do I enable Scroll Lock?