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Math

### Calculating presentations for the normal subgroup of a semidirect product

Let $G$ be a group with a finite presentation $\langle S \cup S' | R \cup R'\rangle$. Assume that I happen to know that $G$ is the semidirect product of a normal subgroup $N$ and another subgroup $H$. Furthermore, assume that $H$ is the subgroup generated by $S'$ and that ....

Waylon: 1 hour ago

Physics

### Differentiability of electric field due to bounded volume charge distribution

In books on electromagnetism, one often sees expressions of Maxwell's equations like $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$. These expressions make sense if $\mathbf{E}$ (which is due to bounded volume charge distribution) is differentiable. I ask this question because in all the textbooks on electromagnetism which I have seen, ....

Hazel: 1 hour ago

Biology

### Melatonin production, sleep, and "cyan light"; how might this finding be possible?

The BBC News article Cyan colour hidden ingredient in sleep (https://www.bbc.com/news/education-44565320) describes research that suggests melatonin levels as measured in saliva could be affected by the presence or absence of cyan color in a displayed screen, even when the color balance was adjusted so that the difference was not visibly ....

Weston: 1 hour ago

Physics

### Does the recent announcement of information transmission via entanglement really indicate superluminal information transfer?

Given that various answers here at Physics assert that information isn't transferred (such as this Phys.SE post (https://physics.stackexchange.com/questions/61126/does-entanglement-not-immediately-contradict-the-theory-of-special-relativity)), and given recent announcements (http://www.sciencealert.com/quantum-teleportation-was-just-achieved-over-7-km-of-cable), does this invalidate our understanding of what's happening? Is information actually translated superluminally? ....

Bella: 1 hour ago

Wolfram Mathematica

### convert MeijerG to form Standard Functions in Mathematica

I'd like to convert MeijerG to its Standard Functions (For example Bessel function or ...). Any suggestion? ....

Grayson: 1 hour ago

Physics

### Is there an argument for using the $\theta$-vacuum for a Yang-Mills theory that works regardless of the presence of fermions?

Consider a Yang-Mills theory, possibly including fermions. It has many possible vacua $\{|n\rangle\}$ labelled by integer winding number $n$, defined as the Maurer-Cartan topological invariant: for the gauge element $g_{(n)}$ and corresponding unitary large gauge transformation $U(g_{(n)})$ we have  |n\rangle = U(g_{(n)})|0\rangle, \quad n = \frac{i}{24\pi^{2}}\int \limits_{S^{3}} d^{3}\theta \epsilon^{ijk}\text{tr}\big ....

Riley: 1 hour ago

Physics

### Bounds on Matrix Elements of Unbounded Operators

Let $H$ be a separable Hilbert space und $T$ a possibly unbounded densely defined linear operator. (One could probably assume that it's ess. self-adjoint but I would like to avoid this assumption.) Let $\{e_n\}$ be an orthonormal basis of $H$. The matrix elements of $T$ w.r.t. this basis are ....

Olivia: 1 hour ago

Physics

### Could it be that at the big bang equal amounts of matter and antimatter were created, but the antimatter got stuck in black holes?

I recently watched PBS spacetime's video series on the black hole information paradox. Where they first discussed the no hair conjecture and then later also the true destruction of information by Hawking radiation. This led me to wonder, if information is apparentely unavailable to the outside universe after it has ....

Emily: 1 hour ago

Math

### Interdefinability of two expansions of the Real Field

I was asked the following question two days ago, but I couldn't completely resolve it. Here is the claim: $\mathcal R = (\mathbb R,+,\cdot)$ is the real field. Let $I$ be an open interval (perhaps unbounded) in $\mathbb R$ and let $f: I \rightarrow \mathbb R$ be $C^1$ and such ....

Landon: 1 hour ago

Physics

### What are the benefits and reasons behind considering a probability density distribution for electrons as opposed to a mere density distribution?

Before now, I had never questioned this matter. Why must we assume a probability density distribution (pdd) as opposed to a general density distribution (gdd)? Perhaps I have a misunderstanding. My current understanding is that a pdd for an electron would suggest that the electron is never in one place, ....

Emilia: 1 hour ago