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May 4

Flatness of a transfer function

In terms of a transfer function, what conditions must be satisfied to ensure the output amplitude is maximally flat. Not a homework question . I'm just curious.86.8.201.210 (talk) 23:28, 4 May 2019 (UTC)

Our Butterworth filter article, particularly the maximal flatness section, might (or might not) give some clues.catslash (talk) 14:53, 5 May 2019 (UTC)

May 7

Are there "interesting" theorems, that only use the addition operation, and need induction for their provability?

Besides the well known "interesting" theorem, stating that every natural number - not being any sum of two identical natural numbers - is followed by a successor which is such a sum (along with analogous theorems about a sum of more than two identical natural numbers, e.g. the theorem stating that every natural number not being any sum of three identical natural numbers - is followed by a successor which is either such a sum or followed by a successor which is such a sum). 185.46.78.64 (talk) 20:28, 7 May 2019 (UTC)

The commutative law of addition comes to mind. 67.164.113.165 (talk) 19:22, 8 May 2019 (UTC)

Möbius—Lorentz group isomorphism ?

Several (usually) reliable sources have asserted an isomorphism between the Möbius group and the Lorentz group: they both have six real parameters as Lie groups and they both contain copies of SO(3). But there are parabolic transformations in the Mobius group. The Galilean transformations are parabolic on space-time, but they are not included in the Lorentz group. An editor has exposed the null rotations which should be in the Lorentz group if indeed the isomorphism held with the Mobius group. Such null rotations fail to exist as seen in coverage of Lorentz transformation. Short of a WP:reliable source calling out the absurd null rotations, is there a way to halt the perpetuation of the harmful identification of the Riemann sphere with the celestial sphere ? Details of some findings are listed at Talk: Lorentz group#Null rotations ?. — Rgdboer (talk) 21:35, 7 May 2019 (UTC)

This is more mathematical physics than mathematics proper, so some of the terminology is unfamiliar to me; maybe someone will correct me if me if I go wrong somewhere. You're right that the articles are poorly referenced in places, but the section Lorentz group#Relation to the Möbius group does construct a homomorphism from the Möbius group to the Lorentz group. The image is not the full Lorentz group but the restricted Lorentz group (the connected component containing the identity), however I think your issue is in the other direction, meaning that parabolic Möbius transformations don't seem to be represented in the Lorentz group. The section Lorentz group#Parabolic describes the Lorentz transformations corresponding parabolic and they appear to be a bit more complex than Galilean transformations (which, as you say, are not Lorentz transformations). In any case, while I don't claim to understand everything in the articles, I'm not convinced there is anything factually incorrect in them and the isomorphism does not seem to be controversial in the literature; see for example [1], [2] and [3] (section 3.7.3). --RDBury (talk) 09:00, 8 May 2019 (UTC)

May 8

Open quantum system tending towards equilibrium

This question uses some of the terminology of quantum mechanics, but it is fundamentally a maths question. If you recognize the physics, I am trying to justify a system tending towards the same unique equilibrium regardless of the initial state.

Let finite sequences of 0s and 1s form the basis of a vector space. For example, $|0010\rangle$ is a element of this basis.

Let $U$ be a unitary operator on this vector space. This operator essentially "appends a 0 or 1", but to a linear combination. One example is:

$U|s0\rangle =\alpha |s00\rangle +\beta |s01\rangle$

$U|s1\rangle =-\beta |s10\rangle +\alpha |s11\rangle$

For any sequence s. Think of U as the next-time-step operator.

Let $f_{x}(v)$ pick all the coefficients of basis vectors that form v that end with x, and takes the sum of their norm-squared. For example,

$f_{0}(c_{1}|00\rangle +c_{2}|01\rangle +c_{3}|10\rangle +c_{4}|11\rangle )=|c_{1}|^{2}+|c_{3}|^{2}$ and $f_{1}(c_{1}|00\rangle +c_{2}|01\rangle +c_{3}|10\rangle +c_{4}|11\rangle )=|c_{2}|^{2}+|c_{4}|^{2}$.

Now, consider $\lim _{n\rightarrow \infty }f_{x}(U^{n}v)$ for some initial v. It can be shown that $\lim _{n\rightarrow \infty }f_{x}(U^{n}v)={\frac {1}{2}}$ for any initial v and either x.

Furthermore, if you consider sequences of some finite element space with m elements (beyond just 2 elements as in this example) then $\lim _{n\rightarrow \infty }f_{x}(U^{n}v)={\frac {1}{m}}$. So the equilibrium distribution is uniform over the element space, and is independent of the initial state.

What I am interested in is when the element space becomes infinite. I suspect that the equilibrium distribution is not uniform over the element space anymore in that case. Is the quantity $\lim _{n\rightarrow \infty }f_{x}(U^{n}v)$ still independent of the initial v?

--49.183.55.236 (talk) 05:30, 8 May 2019 (UTC)

I'm guessing, without getting into details, that the problem you're having is that while the f_x(Uⁿv) all seem to converge to 0 (the equilibrium), their sum remains constant (≠0). Tannery's theorem gives conditions under which limit and infinite sum can be exchanged, and these conditions aren't met in this case. The article is a bit sparse in terms of examples, but it's not hard to come up with simpler cases where limit and sum can't be exchanged; one is a_k(n) = δ_kn. I don't know how this is interpreted with bras and kets, but presumably you need some quantum mechanical version of the conditions in Tannery to hold before you can assume the equilibrium state behaves as expected. --RDBury (talk) 16:26, 8 May 2019 (UTC)

May 10

Solving Inequality with Polygarithms

I need some help with solving the inequality $\log ^{k}(b/a)+\log ^{k}({\frac {c}{d-a}})+\log ^{k}({\frac {e-bc}{d}})\geq \log ^{k}(d/e)$ for the variables $a,b,c,d,e$ (where $k>0$ is fixed).

In particular, I'm intersted to know if $\min _{a,b,c}\{\log ^{k}(b/a)+\log ^{k}({\frac {c}{d-a}})+\log ^{k}({\frac {e-bc}{d}})\}=\Omega (\log ^{k}(d/e))$ (I assume that $d=\omega (e^{k})$ if that helps).

Any help would be highly appreciated! David (talk) 13:55, 10 May 2019 (UTC)