I want to know when we use &quot;be going to&quot; + be + ing?
"Heard me [infinitive]" vs. "heard me [present participle]"             At that time, you wouldn't have heard me talk about it.   At that time, you wouldn't have heard me talking about it.         At that time, you wouldn't have heard me condemn it.   At that time, you wouldn't have heard me condemning it.         Which one of the above sentences are incorrect and why?
In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$.  But what is being measured with complex line integration, when you go from a point $z_1$ to a point $z_2$ in the complex plane?  With regular line integration I can see $f(x,y)$ maps $(x,y)$ to a point on the $z$ axis directly above above/below $(x,y)$.  But in the complex case, when you map from the domain $Z$ to the image $W$, you are mapping from $\mathbb{R^2}$ to $\mathbb{R^2}$ ...it is not mapping a point to 'directly above/below'...so I don't have any intuition of what is happening with complex line integration. I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that  (i) $\bar{U} \oplus U^\perp \neq H$  (ii) $ \bar{U} \neq U^{\perp \perp}$  Since finite and closed subspaces will probably not do it, I am not good at inventing nice examples. Does anybody here have a few at hand? Characteristic polynomial of companion matrix I want to switch to French locale. So I tried the below command:  myUbundu@myUbundu-desktop:~$ export LC_ALL=fr_FR   But I am getting the warning  -bash: warning: setlocale: LC_ALL: cannot change locale (fr_FR)   How to set the locale to French? Do I need to install additional packages? Explaining two-tailed tests I am looking for various ways of explaining to my students (in an elementary statistics course) what is a two tailed test, and how its P value is calculated.  How do you explain to your students the two- vs one- tailed test? When is Gauss' law not applicable? $\lim_{h\to 0}\int \frac{f(x+h)-f(x)}{h}\,\mathrm{d}x = f(b)-f(a)$ Let $f:[a,b]\to  R$ be integrable and continuous on $(a,b)$. Prove that: $$\lim_{h\to 0} \int_a^b\frac{f(x+h)-f(x)}{h} \,\mathrm{d}x = f(b)-f(a).$$  What I've tried so far: make a change of variables, which didn't help. I also thought of using the intermediate value theorem inside the integral, but the function being differentiable is not specified. Any help, tips? Thanks in advance! Why can't a single photon produce an electron-positron pair? In reading through old course material, I found the assignment (my translation):     Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.   My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.  How to solve this task? Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? Why do snap-packages exist - is there a real need? Assumptions  Honestly, I do not know much about  snap-packages - but this is irrelevant for this question - see below. I assume the system is significantly different from the existing one.  Does the change make sense?  Is there an actual need, strong enough? That is - is there a new use case, that is important enough to develop a new format - and the associated infrastructure?  Was it not feasible to change the current method to cover the new use cases too?  Or am I missing the point?  It is possible that what I see is mostly marketing - new names and presentation for minimal technical change, to get a chance that any body assumes it as "new and better" and it may be actually used. Also, it may be that the new packages are so closely based on the existing format that it is mostly a change in presentation to the user. That could be a good solution, of course. In this case, this question would have not much relevance.  Then, I would hope that is still useful enough to answer to side aspects.  Just let me know if the question is not helpful or confusing to new users, I'm happy to delete it.  So, why do they exist?    Background  My first reaction was "That makes no sense!"  This is similar to a situation where, on the physics site, someone asked, unhappy, why nobody discusses his brilliant new ideas in answers. It looked strongly like crackpot ideas; So far from actual physical knowledge that it was hard to find a point to even start. I wrote an answer that did not touch his ideas with a single word, but explained why one would just not discuss assuming crackpot ideas - not the first case of that. The answer was actually hitting the point, I think.    If my assumptions were right, this case is similar.    But then, maybe not - let's see. Can X.org Load Two Different Drivers? (Triple-Head Setup with Intel and NVidia Card) We want to set up a 3 screens with an onboard intel graphics adapter and an nvidia card.  Both cards are recognized in lspci   01:00.0 VGA compatible controller: NVIDIA Corporation GT218 [GeForce 210] (rev a2) 00:02.0 Display controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09)   And bascially work: If I switch to the console, I see output on the Intel card, X is only running on the Nvidia card.  But when I type xrandr | grep conn I get only one connected port:   DVI-I-0 disconnected (normal left inverted right x axis y axis) VGA-0 disconnected (normal left inverted right x axis y axis) DVI-I-1 connected 1920x1200+0+0 (normal left inverted right x axis y axis) 518mm x 324mm HDMI-0 disconnected (normal left inverted right x axis y axis)   I am on Ubuntu 12.10 (and XFCE) and installed the proprietary nvidia driver.  I was assuming that X maybe has a problem to autodetect both cards, so I thought the following xorg.conf could help:   Section "ServerLayout"     Identifier     "Layout0"     Screen      0  "Screen0"     Screen      1  "Screen1"     InputDevice    "Keyboard0" "CoreKeyboard"     InputDevice    "Mouse0" "CorePointer" EndSection  Section "Files" EndSection  Section "InputDevice"     # generated from default     Identifier     "Mouse0"     Driver         "mouse"     Option         "Protocol" "auto"     Option         "Device" "/dev/psaux"     Option         "Emulate3Buttons" "no"     Option         "ZAxisMapping" "4 5" EndSection  Section "InputDevice"     # generated from default     Identifier     "Keyboard0"     Driver         "kbd" EndSection  Section "Monitor"     Identifier     "Monitor0"     VendorName     "Unknown"     ModelName      "Unknown"     HorizSync       28.0 - 33.0     VertRefresh     43.0 - 72.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device0"     Driver         "nvidia"     VendorName     "NVIDIA Corporation"     BoardName      "GeForce 210" EndSection  Section "Screen"     Identifier     "Screen0"     Device         "Device0"     Monitor        "Monitor0"     DefaultDepth    24     SubSection     "Display"         Depth       24     EndSubSection EndSection   Section "Monitor"     Identifier     "Monitor1"     VendorName     "Unknown"     ModelName      "Unknown"     HorizSync       28.0 - 33.0     VertRefresh     43.0 - 72.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device1"     Driver         "intel"     VendorName     "Intel"     BoardName      "Internal Intel Graphics" EndSection  Section "Screen"     Identifier     "Screen1"     Device         "Device1"     Monitor        "Monitor1"     DefaultDepth    24     SubSection     "Display"         Depth       24     EndSubSection EndSection   But it is still not working.   Is this generally possible to use two different drivers in X ? Is it necessary to create an xorg.conf manually then? (because Ubuntu 12.10 normally works without the xorg.conf) Are there any errors in my xorg.conf? LaTeX conditional expression I would like to be able to set a flag at the beginning of the main flag (say output) that tells if I want to produce a paper document or an electronic document.   Then I should check this flag to define or not empty pages, hypperref, oneside or twosides, etc.  How could I do that ? Problem on Sum of Digits of a Number I was given this problem to Prove if $Q(n)=Q(2n)$ implies $9∣n$. Where $Q(n)$ be the sum of digits of $n$. I tried to use the fact that sum of digits is congruent to the number modulo $9$ but I had no luck. Any help is appreciated. How can I switch a 32-bit installation to a 64-bit one? I've got a 32-bit Ubuntu installation running on 64-bit hardware. Now that , I would like to switch to 64-bit without having to reinstall the OS.  This is one of the :  Shawn installed his system using the 32-bit version of Ubuntu, but his hardware is 64-bit and he wants to switch over. He manually installs the amd64 versions of dpkg and apt, replacing the i386 versions and changing which architecture is used as the default; then he installs the amd64 ubuntu-minimal package; then he installs the amd64 ubuntu-desktop package. Over time the remaining i386 packages are replaced automatically on upgrade.  However, when trying to follow the instructions in there, I cannot find any 64-bit version of dpkg or apt.  Did this user story got implemented in a different way in the final spec, or do I need to do something differently?  In short, how can I switch my 32-bit installation to 64-bit? The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.  We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in {\cal L}(V), TS = ST$. Here, ${\cal L}(V)$ denotes the set of all linear operators over $V$.  One direction is easy to prove. If $T$ is a scalar multiple of the identity, then there exists a scalar $a$ such that $Tv = av$, $\forall v \in V$. Hence, given an arbitrary vector $w$, $$TS(w) = T(Sw) = a(Sw) = S(aw) = S(Tw) = ST(w)$$ where the third equality is possible because $S$ is a linear operator. Then, it follows that $TS = ST$, as required.  I am, however, at a loss as to how to tackle the other direction. I thought that a proof by contradiction,  ultimately constructing a linear operator $S$ for which $TS \neq ST$, might be the way to go, but haven't made much progress.   Thanks in advance! Does lower resolution result in better signal-to-noise ratio, given the same sensor size? With the release of the Canon EOS R5/R6, there have been many saying that they prefer the 20MP R6 camera over the 45MP R5 camera, due to its lower price but also because of &quot;better low-light performance.&quot; In camera comparison sites, I've also seen the sentiment that bigger pixels (lower resolution) = better low light performance. However, is this a myth? Here is my reasoning for this. Let's say that there are two full-frame sensors, one 20MP sensor and a 80MP sensor. According to many people, the 80MP sensor would be terrible for low-light photography since the pixels are smaller. However, each 2x2 block in the 80MP sensor could be mapped, and in post averaged to a 1x1 pixel, effectively making the image into a 20MP image. This could also be done in-camera as an option. Would this image have just the same amount of usable information as the image from the 20MP sensor? In other terms, since both sensors are 35mm sensors (i.e. equal area), the amount of light falling onto the sensor is the same. So, at the end of the day, there will be no difference between a 20MP image and a 20MP image that has been downsampled from 80MP, right? So their low-light performance would be the same. You even get an advantage from having the option to get 80MP photos if you wanted. Why do people say that a lower-resolution camera has better low-light performance, when it should theoretically be the same as one with higher resolution (all other factors equal)? Note: this question is about sensors with the same size but with different pixel density. Hence, it is different from  asking about different sensor sizes. I am aware that a full-frame camera will have better low-light signal due to its bigger area than an APS-C sensor, but what about the similarly sized sensors? As linguistics student, I have to put more than one thing on a character every now and then. For example, I need to type ā̆. I thought I could simply do that by typing \={\u{a}}, but then the macron is placed above an empty space in front of the character.  How can I solve this? Thanks  P.S. I'm using MiKTeX 2.9, running on Windows 8; in case it might be of use. Why and how do "bigger pixels" make a better picture? This is going off of today's iPhone 5S announcement, when Phil Schiller said that "bigger pixels" is the key to a better picture.      What does this mean? Is it true or just a marketing ploy? My computer name in System Preferences &gt; Sharing is set to &quot;archos&quot;, but is showing as &quot;iphone&quot; on Terminal. It just started doing this after I loaded Xcode for doing iPhone development: Last login: Mon Nov  7 14:46:55 on ttys001 iphone:~ travis$   Any ideas what could be causing this? Geiger and Marsden's experiment led Rutherford to believe that the positive charge and most of the mass of the atom was concentrated in a small region. I understand what led him to conclude the way the positive charge is positioned in the atom. But how did he conclude that most of the mass was in a small region (the nucleus)?  How did the distribution of the mass matter after all? Given that the electric force is greater than the gravitational force by many magnitudes, the force between the positice charge and the electrons was predominantly electric.   So how did Rutherford conclude that most of the mass is in the nucleus?