I am using the IEEE Transaction class in LyX. With a figure float, I was able to put a long equation spanning two columns at the top of the next page, as shown in the pictures below     My questions are: 1. How do I place the long equation at the bottom of the page  and 2. How do I let it appear in the same page (not the next page) where the equation is defined?   Do I need to load some packages? I prefer to still use the figure float in LyX where the labeling and numbering features still work for this long equation than a solution completely using ERT. In that case, I think the numbering and labeling of equations become more cumbersome.
Is there a convenient way to place a figure* at the end of the current page instead of taking up the entire next page in the twocolumn environment? Placement modifiers [Hb] do not work.
When I open a window to air out the room, how does the smell disperse? Let's say I'm in a room with some kind of noxious stink, possibly of flatulent nature.  The quickest way to right the world that comes to mind is to open a window.  When I open a window, how do the stank particles leave the room? Induction proof (bitstring length) Theorem : The number of bitstrings with the length $x$ that begin with $1$ and/or end with $0$ is  $3 \times 2^{x-2}$.  I know there are easier ways to prove this but I must figure out how to do it with induction. Blender 2.78 file size grows and does never shrink I made a huge project, wanted to reduce file size from 1.3GB to below 1GB. Compressing did not help. I removed objects from scene. Also textures. I then realized, the file size even was the same when I delete ALL objects. Why?  Download of sample:   Before deleting all: 1.2GB, after: 1.1GB. I'm a C++ newbie, but I wasn't able to find the answer to this (most likely trivial) question online.  I am having some trouble compiling some code where two classes include each other.  To begin, should my #include statements go inside or outside of my macros?  In practice, this hasn't seemed to matter.  However, in this particular case, I am having trouble.  Putting the #include statements outside of the macros causes the compiler to recurse and gives me "#include nested too deeply" errors.  This seems to makes sense to me since neither class has been fully defined before #include has been invoked.  However, strangely, when I try to put them inside, I am unable to declare a type of one of the classes, for it is not recognized.  Here is, in essence, what I'm trying to compile:  A.h  #ifndef A_H_ #define A_H_  #include "B.h"  class A {     private:         B b;      public:         A() : b(*this) {} };  #endif /*A_H_*/   B.h  #ifndef B_H_ #define B_H_  #include "A.h"  class B {     private:             A&amp; a;      public:         B(A&amp; a) : a(a) {}  };  #endif /*B_H_*/   main.cpp  #include "A.h"  int main() {     A a; }   If it makes a difference, I am using g++ 4.3.2.  And just to be clear, in general, where should #include statements go?  I have always seen them go outside of the macros, but the scenario I described clearly seems to break this principle.  Thanks to any helpers in advance!  Please allow me to clarify my intent if I have made any silly mistakes! I have a MacBook Pro 2009 running MacOS El Capitan. Recently the Wi-Fi has stopped working.   When I click on the Wi-Fi icon it displays the message "No hardware installed", but Ethernet still works.   Is there any way of fixing this by using a Wi-Fi dongle or something of the sort, if so do I need a specific dongle and how do I install it? How to calculate the summation $(\sum_{p = k}^{n} \binom{n}{p}) / 2^n$ quickly? I was solving a question which technically reduces to the following     Given $N$ items, it is equiprobable for each item to be good or bad, what is the probability that number of good items in the set are greater than or equal to $K$.   This can be reduced to $\dfrac{x}{2^n}$ where $\displaystyle x = \sum_{p = k}^{n} \binom{n}{p}$. Is there a more simplified form which is easier to calculate for large values of $N, K$?  Note: It may be safe to assume that we do not require extremely high precision while calculating(first 5-10 digits only).  Thanks! Find the greatest value of $a^2\cdot b^3$ where $a+b=10$ Find the greatest value of $a^2\cdot b^3$ where $a,b$ are positive real numbers satisfying $a+b=10$. Determine the values of $a,b$ for which the greatest value is attained.  At first I tried to find the value by putting $a=10-b$, I found a expression, but I got not way to do anything.   If we apply A.M-G.M. inequality, I think we found the minimum value or something else. It is a pre-Olympiad level problem. So I need some answers or some good hints. Is the integral $\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$ convergent? Problem Statement  Let $f(x) = x^3 + ax^2 + bx + c$ and $g(x) = x^3 + bx^2 + cx + a$ where $a,b,c$ are integers with $c\not=0$   Suppose that the following conditions hold:   $f(1)=0$ the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$      $$\text{Find the value of} \: \: a^{2013} + b^{2013} + c^{2013}$$   My attempt  Let $f(x) = (x-p)(x-q)(x-r) \: \:$ and $g(x) = (x - p^2)(x-q^2)(x-r^2)$  Expanding we get $$f(x) = x^3 - (p + q + r)x^2 + (pq + pr + qr)x - pqr$$  $$g(x) = x^3 - (p^2 + q^2 + r^2)x^2 + (p^2q^2 + p^2r^2 + r^2q^2)x - p^2q^2r^2$$  And so we have $$a = -(p+q+r) = -p^2q^2r^2$$ $$b = pq + pr + qr = p^2 + q^2 + r^2$$ $$c = -pqr = p^2q^2 + p^2r^2 + r^2q^2$$  Then we get the following relations  $$a = -c^2$$ $$b = a^2 - 2b \implies 3b = a^2 \implies b = \frac{c^4}{3}$$  From $f(1) = 0$ we have $a + b + c = -1 \implies \dfrac{c^4}{3} + c - c^2 = -1$  However, upon graphing this function, I saw that there are no integer solutions for $c$. Have I done something wrong?  Edit  For future readers, my approach can simplified alot if you replace $r$ with $1$ since $f(1) = g(1) = 0$ Battery Ampere-hour rating vs Battery Amps (not an experienced user) I have almost no experience whatsoever with the technicals of electronics so this is probably a very easy question for someone who does.  Do you know if a battery with a 200Ah rating can put out 200A for one hour or are there limitations? According to this website's third paragraph (), you can.     For example, an average automotive battery might have a capacity of about 70 amp-hours, specified at a current of 3.5 amps. This means that the amount of time this battery could continuously supply a current of 3.5 amps to a load would be 20 hours (70 amp-hours / 3.5 amps). But let’s suppose that a lower-resistance load were connected to that battery, drawing 70 amps continuously. Our amp-hour equation tells us that the battery should hold out for exactly 1 hour (70 amp-hours / 70 amps), but this might not be true in real life. With higher currents, the battery will dissipate more heat across its internal resistance, which has the effect of altering the chemical reactions taking place within. Chances are, the battery would fully discharge some time before the calculated time of 1 hour under this greater load.   But wouldn't that mean you could hook up a 200Ah battery and ask it to put out 12000A for one minute or 720000A for a second?  That seems very unrealistic, lol.  I'm trying to find the proper kind of off grid battery that can power a microwave through a 3000 watt power inverter.  The microwave needs 1800 watts and the battery needs to be 12 volts so that would mean I need about 150 amps, I'm wondering if a battery with a 200Ah rating could do it? Difference between logit and probit models Force newlines with cat wildcard printing I want to use cat with wildcards in bash in order to print multiple small files (every file is one sentence) to standard output. However, the separate file contents are not separated by a newline, which I'd like to for ease of reading.  How can I add a file delimiter of some sort to this command? I'm looking for some sample code that will sort the list items in an HTML list by alphabetical order.  Can anyone help?  Here is a sample list for people to work with:  &lt;ul class="alphaList"&gt;     &lt;li&gt;apples&lt;/li&gt;     &lt;li&gt;cats&lt;/li&gt;     &lt;li&gt;bears&lt;/li&gt; &lt;/ul&gt; I need to prove that the center of $A_n$ is trivial for $n \geq 4$.  $Z(A_3) = A_3$, since $A_3 = \mathbb{Z}/3\mathbb{Z}$ is commutative.  One idea is two use "counting" technique. First of, all we count cojugacy classes of even permutations in $S_n$. It's the classes correspodning to the types $[\lambda_1, ..., \lambda_r]$ where $n \equiv r \mod 2$  Now, the conjugacy class in $S_n$ correspoding to the type $[\lambda_1, ..., \lambda_r]$ splits into two equal-sized classes in $A_n$ if $\lambda_1, ..., \lambda_r$ are distinct odd numbers. If they aren't then it is preserved in $A_n$.  Then we can use "the counting formula"(  ) of elements in each "even" class of $S_n$ and divide one by $2$ if needed.  Still, I'm not sure if it can be done. Seems to be a lot of work there. Maybe there are other, he easier ways? Or, maybe, it is the way, in this case, I wouild appreciate any advices. Search field with or without a button? I can see two broad patterns as far as the design of the search field is concerned. One like we see on Google has a button that the user can hit to initiate Search. (They could also hit enter to do so)  The other pattern much like what is on this site (stackexchange) do not have a button. So the only way the user can initiate search is by hitting enter.   Has anyone done any research or analysis of this pattern ? What I have noticed is that search fields without the explicit search button (that normally is a magnifying glass) have "Search" as placeholder text, left aligned, whereas when there is no button the placeholder is usually a "Magnifying glass icon" with "Search" all left aligned. Is it true that for each set $M$ a given injective function $f: M \rightarrow M$ is surjective, too?  Can someone explain why it is true or not and give an example? Maybe the underlying question is what the Voltage-Current curve looks like. Can I drive it from a voltage source (like you drive a heater) or from a current source (like you drive an LED)? Or even different than those two options?  ADDITIONAL1: Say (hypothetically) I have two commercially available identical Peltier's, they are spec'd 6V/3A. Can I connect these in series to a 12Vdc power supply without any worries?  CONCLUSION1: Current/Voltage load curve is reasonably linear, both driving from current or voltage source will do fine as long as the device is operated within its spec. (Olin Lathrop, Russell McMahon)  CONCLUSION2: Don't drive a Peltier from PWM, power loss due to current increase, grows more rapidly than cooling power. (Olin Lathrop)  CONCLUSION3: Beware the mechanical wear of the device with continues cycling. Eg. don't use a thermostat on/off controller. (Russell McMahon) I have ran into some issues when it went down to install Steam on Debian. The console as well as Apper comes up with missing dependencies  and the package isn't going to install. If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p. If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.  I suppose this would not be true if p $\equiv$ 1 (modulo 4).  To prove something is a non-square I find to be tricky. It's difficult to see any straightforward way to do this using only the definition of congruence for example. How to edit 'etc/hosts' file in non-rooted phone?