Mathematics

Ted Shifrin

12:23 AM

@nbro: How do you define a function?

nbro

$f : A \rightarrow B$

So I guess that's an evaluation

Ted Shifrin

You define a function by giving its values at points of its domain, right?

nbro

Yeah

Ultradark

12:55 AM

can a closed form used to approximate another number include a transcendental number

would this be useful though because you'd have to compute the transcendental number to approximate the other number

Silent

1:26 AM

Let $f(w,x,y,z) = wz − xy$. The set of points $(a,b,c,d)$ where the total derivative of $f$ is zero is finite.

How to see this?

Secret

in The h Bar, 34 secs ago, by Secret

This paper is interesting to me not only because it derive an inverse correspondence to go from quantum operators to classical phase space distributions, but also the various change of variables used that made the integral to be separable. May study more on these rules to see what substitution can lead to separation given an integrand of n variables

Semiclassical

@Secret saw this earlier and thought you might find it interesting: arxiv.org/abs/1801.04781

Secret

Should have a look after I finish reading two other quantum papers

Semiclassical

@Secret Figure 1.3 on page 31 seems neat

Secret

1:41 AM

Huh, so you can probe reaction pathways and transitions states with Bohm, hmm...

Semiclassical

yeah

i mean, setting aside any interpretational questions, I find pilot wave stuff neat insofar as it suggests the possibility of visualizations which employ trajectories and nevertheless are intrinsically quantum

and that seems valuable to me, even if one ultimately concludes that those trajectories shouldn't be given any further significance than that

Secret

Visualising the trajectory profiles may be easier to follow dynamics compared to just having a blob of electron density map which doe not contain enough information on where the electrons are flowing

Also, for those who do Bohm by focusing on the quantum potential, those trajectories highlight which area the quantum potential is largest, and then stuff may be done about those regions to fine tune reaction dynamics and hence controlling the chemical reaction

Semiclassical

aye

Secret

The explicit non locality is what attract me the most along with the pretty trajectory diagrams in Bohm, because it give us the idea on directly do engineering on the wavefunction

Semiclassical

my own view of the 'quantum potential' is that it has rather the status of the 'effective potential' you get when talking about rotating systems. it's not as fundamental as the guidance equation, but it's nevertheless effective in understanding what influence the wavefunction has

Secret

1:54 AM

In contrast in the orthodox framework, the wavefunction can only be implied by the distribution of observables obtained, thus it provide less insight on the direct control of its shape

Currently, the most imaginative idea that Bohmian mechanics inspires me from the context of the slow pointer experiments, is the possibility to get the environment to work with you on the quantum stuff

Since Bohmian mechanics is explicitly nonlocal, for a noisy enough system, the wavefunction will have a lot of dependence on nonlocal factors from the environment. Thus in theory if there is a way to repeatedly perform experiments with different configuration of devices and etc. to modulate the wav function, some background wavefunction contributed only by the environment may be extracted. Then the trajectory diagram of that could be useful to identify where to place devices in order to shape

the wavefunction at strategic locations taking account of the environment contributions to get a much greater control on the probability distributions, thus in effect "controls the predestination itself"

Of course, one likely possibly that such scheme will fail is the environment may decohered the system so much that it loses superposition, thus its wavefunction if any, will look like classical phase space and not interfering

But in theory, changing the experiment setup should leave whatever nonlocal factors from the environment unchanged, because you cannot signal with nonlocality even in Bohm, thus no way a local change in the experiment setup will be able to influence the environmental factors far away unless they are already entangled, thus it should be possible to probe the background wavefunction contributed entirely by the nonlocal factors

Semiclassical

i like the idea that the Bohm picture, though it can't enable any behavior which QM didn't allow you before, could nevertheless be useful in designing better experimental systems

Crosby

Hi

Semiclassical

I don't know if that's realistic, but it's a nice little dream.

Secret

yeah I am not entirely sure either, as said, if the wavefunction becomes so decohered that it consists only of pointer states, then such background wavefunction will not be able to do anything useful

Another thing I am not sure is what will the orthodox description of the notion of "controlling predestination" be like, since the wavefunction is not necessary to be real in the orthodox's interpretations

What I knew so far is that the implementation of the "slow pointers" scenario requires weak measurement and post selection, as that surreal trajectory paper showed, and weak measurement itself is a complicated setup both in the theory and its experimental implementations

thus it could be possible that the setup is so complicated that the background cannot be resolved easily, if exist

And I do personally think that Bohm could contribute a lot in computational chemistry, though we still need to find a protocol such that we only need to compute some of the trajectories instead of all of them to describe the phenomena as otherwise it is more expensive than the orthodox

Semiclassical

2:16 AM

yeah, that kind of thing makes me wonder about the 'synthetic' approach to Bohmian trajectories

Ultradark

2:26 AM

How do you solve $\zeta(s)=x$

Secret

7

Q: Inverse of the Riemann zeta function

I'm wondering if there is any information on the inverse of the Riemann zeta function (not it's reciprocal, but its functional inverse). This would obviously be a multi-valued function.

Partial results only

Ultradark

I put it into wolfram alpha and it gave me s=-2n and s=nontrivial zeros

that has to be wrong though

surprised that got 7 upvotes on mathoverflow

Secret

2:44 AM

@Ultradark I think that is pretty common. Just look at how many infinite series representations of some numbers contains $\pi$

List of formulae involving π

The following is a list of significant formulae involving the mathematical constant π. The list contains only formulae whose significance is established either in the article on the formula itself, the article Pi, or the article Approximations of π. == Euclidean geometry == π = C d {\displaystyle \pi ={\frac {C}{d}}} where C is the circumference of a circle, d is the diameter. A = π r 2 ...

1

Q: Infinite series with e and pi

Please, help me to prove that $$ \sum_{n=1}^\infty \frac{\cos(\pi n-\sqrt{\pi^2n^2-9})}{n^2}=-\frac{\pi^2}{12 e^3} $$ I found this fact here https://en.wikipedia.org/wiki/List_of_representations_of_e but there's no reference.

reference-request summation

Semiclassical

Note that, if it was easy to invert the Riemann zeta function, then the Riemann hypothesis wouldn’t be a big deal: just invert $\zeta(s)=0$

Secret

lol yes

Semiclassical

So, I wouldn’t hold my breathe

Secret

In other news, I don't know if there a rules that governs how to deal with bare transcendentals in an infinite sum e.g.:

$$\sum_{k=1}^{\infty} \frac{\pi}{a_k} + s a_k$$

I think such sums will be pretty hard to compute since one does not know in general whether the sum is a transcendental

where s is some known transcendental number. Sums where the transcendental is not wrapped by a transcendental function are very rare though

Then of course, one can always turn it into a multiple sum by expanding the transcendental into its infinite series representation

but that is more brute force and less illuminating on the rules that governs transcendentals

Tbh, $\pi$ is such a ubiquitous number that it is unthinkable that there are no nice formulas of the form:

$$f(\pi,s) = g(\pi)$$

where $f,g$ are algebraic functions, which can be used to simplify some summands involving $\pi$ and other transcendental numbers $s$

$0 = df = f_x dx + f_y dy + f_z dz + f_w dw = (wz-y) dx + (wz-x) dy + (w-xy) dz + (z-xy) dw$

$= wzdx -ydx + wzdy - xdy + wdz -xydz + zdw - xydw$

$= w(zdx+zdy) + (zdw + wdz) - (ydx + xdy) -y(xdz - xdw)$

$= wz(dx+dy) + d(wz) - d(xy) -xy(dz - dw)$

Simplex

4:27 AM

A while back one of my professors mentioned a theorem in topology and described it as follows:

If you take two identical sheets of paper and lay one on top of the other, each point on the top sheet is directly above a corresponding point on the bottom sheet.

If you crumple the top sheet and place it anywhere above the bottom sheet, there is at least one pair of corresponding points still lined up vertically.

Does anyone know the name of this result?

Secret

Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f {\displaystyle f} mapping a compact convex set to itself there is a point x 0 {\displaystyle x_{0}} such that f ( x 0 ) = x 0 ...

Érico Melo Silva

@Simplex brouwer's fixed point theorem

Buddhini Angelika

5:27 AM

Hi, for a graph G, "if complement of G is not connected, then G is connected", right?

Transcript for

$Mathematics$ Mathematics