How do we know that the charges in an electron and a proton are equal?
Why do electron and proton have the same but opposite electric charge?
Is it possible to make my laptop's screen act as a second monitor for my desktop computer? Question on the coefficients of $(1+x+x^2+x^3+x^4)^{496}$ Consider the expansion $$(1+x+x^2+x^3+x^4)^{496} = a_0+a_1x+\cdots+a_{1984}x^{1984}.$$   $\quad$ (a) Determine the greatest common divisor of the coefficients $a_3,a_8,a_{13},\ldots,a_{1983}$.      $\quad$ (b) Prove that $10^{340} &lt; a_{992} &lt; 10^{347}$.   Is there an easier way to solve this and is there a formula for the multisection for just the sum of the coefficients and not including $x^k$?  I thought of using the Multisection formula to prove (a). That is, $$\sum_{k \equiv r \pmod{m}}a_kx^k = \dfrac{1}{m} \sum_{s=0}^{m-1} \epsilon^{-rs} f(\epsilon^s x)$$ where $\epsilon$ is a primitive $m$th root of unity and $\displaystyle f(x) = (1+x+x^2+x^3+x^4)^{496}$.  Thus we have $r = 3, m = 5$ and so \begin{align*}\sum_{k \equiv 3 \pmod{5}}a_kx^k &amp;= \frac{1}{5} \sum_{s=0}^{4} w^{-3s} (1+w^s x+(w^{s}x)^2+(w^{s} x)^3+(w^{s}x)^4)^{496}\\&amp;=\dfrac{1}{5}\sum_{s=0}^4 w^{-3s}\left(\dfrac{(w^s x)^{5}-1}{w^s x-1}\right)^{496}\\&amp;= \dfrac{1}{5}\sum_{s=0}^4 w^{-3s} \left(\dfrac{x^{5}-1}{w^s x-1}\right)^{496}.\end{align*} How can I visualize or simulate the effect of different focal lengths? I have seen many camera review sites illustrate the effect of using different focal lengths on the same photograph using frames. There are also similar illustrations to compare different sensors (). I find this kind of visualization to be very useful in comparing the actual effect of the longer focal length\different sensor size. Of course the ability to do a wider focal length visualization would also be equally useful, but that's not possible.  Is there any ready-made software or plugin that enables this? If not, is there any simple technique to visualize the effect of longer focal lengths on an image (one would of course have to use the zoom information in the image EXIF data to accurately represent this)? All I can think of is to use some trigonometry to do the necessary calculations for the frame sizes for cropping. The final state of 1000 light bulbs switched on/off by 1000 people passing by Catenatives followed by infinitives and gerunds What is the difference in meaning when the catenative verb “like” is followed by an infinitive, or by a gerund? For example:      Do you like ski jumping?    vs.      Do you like to ski jump?   Also, what is the difference between:      My brother taught me to read and write.    vs.       My brother taught me reading and writing. Replace and Association Correct use of "is" or "are" Possible Duplicate:          Would I use is or are in the following sentence?      The Pohutakawa Coast, Hauraki Gulf, and Waiheke Island is/are the backdrop for our day.   If I rewrote it so the subject (backdrop) starts the sentence, then it seems to me that as it is singular, the verb should be the singular is. But as it is written above, are seems less clumsy. Turn off close Facebook friend notification Using package nomencl Book about classical mechanics I am using longtable and I cannot find a way to adjust its column spacing. I want a table like this:    But I am getting this:    This is what I am doing for defining table size:   \begin{longtable}{p{.1\textwidth} p{.2\textwidth} p{.2\textwidth}p{.3\textwidth}}     xxxxx &amp; xxxxx &amp; xxx &amp; xxxxx\\     \end{longtable} I was reading Brian Greene's "Hidden Reality" and came to the part about Hawking Radiation. Quantum jitters that occur near the event horizon of a black hole, which create both positive-energy particles and negative-energy particles, produce the so-called Hawking radiation. However, I do not understand why only the negative-energy particles are being absorbed into the black hole, while the positive-energy particles shoot outward. Shouldn't there be 50/50 chance that each type of particle is being absorbed by the black hole? Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why? It is quite easy to show that (by using residue theorem) $$\int_0^{\infty} \frac{1}{1+x^n}dx = \frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)} $$ for $$n \ge 2$$  Is there any possibility to simplify $$\frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)}$$ or it is best result?  Thanks in advance! How to collapse consecutive numbers into ranges? Interpretation of interaction term I have a model: $$ \ln({\rm earnings}) = a+b_1{\rm female}+b_2{\rm white}+b_3{\rm female}\times{\rm white} $$ ${\rm female}$ and ${\rm white}$ are dummy variables.  I have interpreted $b_1$ and $b_2$:    $b_1$ = change in female earnings comparing to male given you are non white $b_2$ = change in white earnings comparing to non white given you are male   But I am unable to interpret the coefficient of the interaction term ($b_3$). Please help me with this.  Let me make it more clear what I need out of this regression $$ \ln({\rm earnings}) = 2.618656-.0899657{\rm female}+.382019{\rm white}-.2754126 {\rm female}\times{\rm white} $$ Now i know there is gender pay difference with b1, i also know there is race pay difference with b2. Now with b3 i need to know is their a gender pay gap for whites only. How can i figure that out with regression above and without test. Which preposition is correct in the phrase "proficient in/at/with English"? The minimum system requirements for 18.04 are "2 GHz dual core processor or better"; would a quad-core at 1.8 GHz meet (or approximate) this requirement? force long text inside \item[{\texttt{}] to wrap Both  (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and  (given a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table) are not theorems in a mathematical sense, but they work quite good in the real life.   Does anyone have an idea why this happens?  (see also ) Intersection of powers of an ideal in a Noetherian ring Given a Noetherian ring $R$ and a proper ideal $I$ of it.  Is it true that $$\bigcap_{n\ge 1} I^n=0$$ as $n$ varies over all natural numbers? If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ is the maximal ideal of a local ring $R$? If not, is it true under additional assumptions on $R$ (like $R$ is regular)?