Self-inverse element in a field of prime characteristic
Proving that an integral domain has at most two elements that satisfy the equation $x^2 = 1$.
How is dark energy consistent with conservation of mass and energy? I initially thought that dark energy must in some way violate conservation of mass and energy since the component of the energy density of space that comes from dark energy is constant, and space is expanding. Therefore, as space expands the energy in the universe that comes from dark energy would increase. I presumed the source of this energy was not coming from the conversion of other types of energy to dark energy, so it must violate conservation.  I decided to Google this and came upon this article:    It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992):  "…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent."  Is there a way to explain this in layman's terms? (The blog attempted to do this, but it was very unclear to me.) More specifically, can you explain where my initial train of thought described above fails when I erroneously concluded that dark energy violates conservation?  Thank you. Unknown Column In Where Clause Let's say I fire a bus through space at (almost) the speed of light in vacuum. If I'm inside the bus (sitting on the back seat) and I run up the aisle of the bus toward the front, does that mean I'm traveling faster than the speed of light? (Relative to Earth that I just took off from.) Why gravity is an attractive force? Why gravity is an attractive force?     One may say that it is because of space time curvature but General Relativity is built on this law: $\displaystyle G \frac{m_1 \times m_2}{r^2}$ (To be more precise, it is  from it's potentiel form known as  that can be written like this: $\nabla \cdot g = - 4\pi Gp  $). So GR can't explain why gravity is attractive. General Relativity only explains how gravity occurs in terms of space time.    Another may stick with the Quantum theory stating that gravitons are responsible for the force of gravity, but why a massless particle with spin 2 will exchange positive momentum? Comparison theorem for ODE I am trying to recreate a penne pasta. I am creating a cylinder removing the two end faces and then I solidify the mesh.  However I do not know how to extrude (or cut away) the two ends to make them pointy like in the picture of the penne below. I have tried to use bisect and mirror, but without successful results.  Any idea? The collection of all subsets of $\mathbb{Z}$ is uncountable, due to   But how can I prove that the collection of all finite subsets of $\mathbb{Z}$ is countable? Implementing the decorator pattern in Python No Dialect mapping for JDBC type: -9 Why does N[1.000 01, 10] return 1.00001, but N[1.000 001, 10] returns only 1.? How to evaluate the integral $\int_0^{\pi}\frac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx$? Evaluate the integral $$\int_0^{\pi}\frac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx.$$   I have no idea.   $$\int_0^{\pi}\dfrac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx=\int_0^{\pi/2}\dfrac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx=\int_0^{\pi/2}\dfrac{a^n\tan^2x+b^n}{a^{2n}\tan^2x+b^{2n}}dx$$  I try the substitution $\tan{x}=t$, but it doesn’t work. Different and opposing voltage sources? &ndash; Schematic created using   The diagram above, the voltage sources are equal and wired parallel, they would increase the current(2x) flow to the load(R), but what about this case:          Are the two voltage sources in series now? I'm a European citizen living in the US on a work visa. Due to some unfortunate circumstances, I need to urgently visit my home country. However I would like to get back to the US after my visit is complete. Flying from the US to Europe is not a problem as there are still daily flights between some city pairs. However going back could be tricky, as  anyone who has been in the Schengen area in the past 14 days from re-entering the country.   Is there a flight route that would let me from Europe to some country, stay there for 14 days and then fly to the US? Canada is out, as they've banned international visitors. Mexico seems to keep their border open, but I  any direct European flights from their airports. Almost every other country either suspended flights or doesn't allow foreign nationals to enter. "Deleted by {owner}" vs "Deleted by owner" in self deleted answers If $\lambda_{max} = n$, then $n\times n$ positive, reciprocal matrix A is consistent At the end of . the author states that  Suppose we have $n \times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=1/a_{ji}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $\lambda_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij} a_{jk} = a_{ik}$ where $i,j,k=1,2,...,n$.  I'm not convinced this theorem is true. Could anyone help? Windows 10 won't let me set default apps for file extensions Is there a limit as to how fast a black hole can grow? Is there a one-liner that allows me to create a directory and move into it at the same time? I find myself repeating a lot of:  mkdir longtitleproject cd longtitleproject   Is there a way of doing it in one line without repeating the directory name? I'm on bash here. Proof of linear independence of $e^{at}$ Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$,  form a linearly independent set of functions.   Any tips on how to go about this proof. I tried a working from the definition of an exponential and combining sums but that didn't seem to get me anywhere. I saw a tip on the internet that said write it in the form  $\mu_{1}e^{a_{1}t}+\dots+\mu_{n}e^{a_{n}t}=0$ to try to show $\mu_{1}=\dots=\mu_{n}=0$ considering each term of the left hand side must be positive, but I can't get my head around that because while I understand $e^{x}&gt;0\forall x\in\mathbb{R}$ I cannot see why $\mu_{i}$ must be positive in any case. I have thought about differentiating but that doesn't seem to help. The question did originally ask for a "rigourous" proof but I'll take any hints right now and the provided the solution of 'is obvious' is most unhelpful to me.  Any input would be fantastic. Thank you. Suppose I have a Python function as defined below:  def foo(arg1,arg2):     #do something with args     a = arg1 + arg2     return a   I can get the name of the function using foo.func_name. How can I programmatically get its source code, as I typed above?