Wikipedia:Reference desk/Archives/Science/2017 October 31

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October 31

Math used in doctoral-level natural sciences

A question on the math desk about math in med school raised this question in my mind: What types of math are used in (a) astronomy, (b) physics, (c) chemistry, (d) biology, and (e) geology? (I presume that for physics the answer is “all of it except number theory”, so I’m more interested in the answers for the others.) Loraof (talk) 00:53, 31 October 2017 (UTC)

Stoichiometry and Statistics are special tools of choice in chemistry and biology. In Astronomy Geometry (yes that counts as math) has wide application but then the "included" Astrophysics is physics (and chemistry) too! Like in Astronomy physic science is used as base in every "natural" science, so what applies to physics applies to all the others you mentioned at some point. --Kharon (talk) 01:50, 31 October 2017 (UTC)

Some of the nastier math in biology is encountered in the context of Ronald Fisher regarding population genetics. But some serious computational issues go into things like protein structure prediction. Of course this is not an exhaustive list. Wnt (talk) 02:18, 31 October 2017 (UTC)

It depends on which specific subfield you are working in; for many fields of Chemistry, daily working knowledge of mathematics beyond basic algebra probably isn't necessary, though for some others being able to understand, calculus is essential, such as the fourier transform or partial differential equations involved in Lagrangian mechanics and Hamiltonian mechanics, especially in people who work in physical chemistry. --Jayron32 10:48, 31 October 2017 (UTC)

• Hah! Biology even uses number theory. (But so does physics, actually.) Looie496 (talk) 12:29, 31 October 2017 (UTC)

How is number theory used? Loraof (talk) 14:49, 31 October 2017 (UTC)

Can't speak for biology, but a good summary for physics would be Matilde Marcolli's paper Number Theory in Physics. A nice toy model would be the primon gas, and one can then look at the physics of the Riemann hypothesis. Double sharp (talk) 15:13, 31 October 2017 (UTC)

This speaks for biology. As does this. --Jayron32 15:37, 31 October 2017 (UTC)

Additionally for physics, aspects of number theory especially with renormalization of divergent series, show up all over physics. Perhaps most famously, the sum of all integers shows up in calculations involving quantum mechanics. The series sums to infinity, renormalization allows the use of -1/12 as the value of the function, and the use of that value in place of the infinite series produces experimentally verifiable results. --Jayron32 15:56, 31 October 2017 (UTC)

Thank you! I'd consider divergent series to be more analysis than number theory in general, but divergent Dirichlet series certainly have one foot in each. And of course, the renormalisation goes straight back to the ζ function, and the proof going through that and its relationship to the η function showing ζ(−1) = −1/12 is exactly where the grain of truth in the justly criticised Numberphile demonstration lies. Double sharp (talk) 07:11, 1 November 2017 (UTC)

Well, I think such criticisms are holding a pop-math youtube channel a bit too responsible for what isn't wrong if so much as oversimplified. Really it comes down to the fact that we have different defitions for what "equals" means, and usually (but not always) those definitions align in ways that make us forget that occasionally they don't. The sum of the integers is an example of that difference. When we say "f(x) = y" we mean both :

• performing function "f" as a task will produce result "y", AND:
• we can substitute "y" for "f(x)" into other situations and get valid results.

For most math, that works fine: for example, we can have two equations: y = x² AND the x+1 = 4. We can solve the second function to get x = 3, then substitute that value into the first to get y = 9, which is a perfectly valid result. For math involving infinite series, we get the paradoxical result that if we solve x = 1 + 2 + 3 + 4 + .... such that we get x = ∞, HOWEVER, if we substitute ∞ into a second equation we get nonsense that does not match, say, an actual observation of nature. The interesting thing about the -1/12 thing is that if we substitute -1/12 into equations where that particular infinite series shows up, we get observationally verified results, that is real physical process obey theoretical equations that use the infinite sum of integers if we substitute -1/12 for that series. Consistently. So, in one sense, it is equal to it because you can swap the one for the other and get true things. That's one thing that equals means. Unfortunately, it doesn't match the other part of being equals, which is to literally perform the operation on one side. So, it's not wrong. It's just a highly speciallized application. --Jayron32 11:27, 1 November 2017 (UTC)

Thanks for that clear explanation, Jayron. Now I finally understand what’s going on with that – there’s a difference between the two meanings of “f(x) = y”. Loraof (talk) 02:13, 2 November 2017 (UTC)

We're not quite replacing that series by −1/12. It's the finite part of the result that goes to −1/12. There's a good explanation on Math Stack Exchange here: basically, for stuff like the Casimir effect where this series comes up, there is an exponentially decreasing regulator making each term not n but ne^−εn, and the original series is obtained in the limit when ε approaches 0. If you sum this, you get e^ε/(e^ε − 1)², and a bit of Laurent series expansion leads to the result 1/ε² − 1/12 + ε²/240 + O(ε⁴). In the limit, the third and all succeeding terms go to zero, the −1/12 stays constant (of course), and crucially, the first term that causes the divergence is cancelled by a local counterterm representing vacuum energy – for the case of the Casimir effect, the sum of the zero-point energies if the region were not bounded by those metal plates. Since this is physics, ε has dimensions and the terms can be separated like that without affecting each other.

So it is not quite that we magically waive the usual meaning of "equals", but that for physics applications the divergent parts of the series are cancelled out and only −1⁄12 matters in the limit that we are interested in. Double sharp (talk) 12:46, 2 November 2017 (UTC)

That was a really good explanation. Thanks! --Jayron32 15:42, 2 November 2017 (UTC)

You're welcome, but the major credit goes to Luboš Motl from Stack Exchange: I merely summarised the important points and left out the details that were not pertinent to the main issue. Double sharp (talk) 16:28, 2 November 2017 (UTC)

Just for the sake of nitpicking: Since this is physics, ε has dimensions and the terms can be separated like that without affecting each other is not really true (epsilon is adimensional here) though it captures the idea. We are working with a perturbation series in a small exponent; the idea is that the results of our computations "work" as the exponent approaches zero, but in particular they must be true with arbitrary accuracy for infinitely many values of the small exponent; or formally, that for any given precision, you can find any number of values epsilon such that the result is within the precision (it is pretty much the definition of a limit (mathematics)). This is what authorizes us to say the different terms do not interfere, as if they were of different dimensions (but if they were, it would not be licit to sum them), because two powers series with the same form are equal iff. each pair of prefactors is equal (under certain mathematical assumptions that we don't care about here). Tigraan^{Click here to contact me} 16:56, 3 November 2017 (UTC)

Motl "cheated" here a little, and I followed him, since this wasn't quite the main point and as you say, the idea is captured: but it seems that thanks to you we are now building a very comprehensible series of explanations of this all-too-often touted as incomprehensible result! Thank you for that! Double sharp (talk) 04:20, 5 November 2017 (UTC)

My field of biology is ecology, and I study things like population dynamics and population ecology. Mostly the math I use is related to dynamical systems theory, including stochastic processes and statistical distributions, but also more pedestrian things like linear algebra and basic calculus. Some of my colleagues get heavily in to PDE, and various types of Matrix_population_models are also very popular. Some concepts from information theory get some play too, see e.g. species evenness. Many of the big names in my field (e.g. Simon Levin) have PhDs in math. Then there's a whole world of tools from statistics (e.g. Bayesian inference, General linear models, regression, curve fitting etc.) applied to ecology. Really all the fields you list make some use of mathematical modeling, broadly speaking. SemanticMantis (talk) 15:02, 31 October 2017 (UTC)

How about the use of mathematical logic and fuzzy logic in physics and biology?--82.137.10.247 (talk) 13:50, 2 November 2017 (UTC)

• Here is a silly conference in French with logical gates made of carnivorous plants. Strangely enough, the speaker does not put that area of research on his professional webpage. Tigraan^{Click here to contact me} 14:13, 2 November 2017 (UTC)

You might not call it silly when Dionaea muscipula realize their collective processing ability to calculate Cryptographic hash functions such as SHA-256 without electricity, and compete with humans at mining bitcoins. Blooteuth (talk) 15:00, 2 November 2017 (UTC)

Well, it is a serious conference about a silly subject. Right now, those are about 10^10 slower in FLOPS than a Raspberry Pi for about ten times the dimensions, so I wouldn't be too worried. Tigraan^{Click here to contact me} 15:32, 2 November 2017 (UTC)

I'm not sure that's ready for final implementation, but it's a SEED of an IDEA. DMacks (talk) 17:37, 2 November 2017 (UTC)

We have a whole Biological computing article. DMacks (talk) 17:37, 2 November 2017 (UTC)

Lots of mathematical logic at play in analyzing gene expression and gene regulation. Similar to how simple circuits embody logic, so can metabolic networks and other biological processes. See e.g. here [1] for some scholarly research on the logic and combinatorial control using mRNA. SemanticMantis (talk) 15:34, 2 November 2017 (UTC)

Molecular Summary Table

Dear all,

first of all I apologize for my bad English. I will improve it. I´m still new here at wikipedia and my instruction consists in publishing scientifical articles. Most of them contain chemical infoboxes (similar like drugboxes, enzymes)calles "molecular summary table". I tried everything to publish the following table but it doesn´t work.

It´s about the following infobox:

https://de.wikipedia.org/wiki/Wikipedia:Redaktion_Chemie#/media/File:Wikipedia_Support_2.jpg

The molecular summary table is almost published on a german wikipedia support page. I need urgent help in converting this table into wiki syntax. I would be very grateful for support!

Thanks and best regards

--Gunnar Römer (talk) 13:13, 31 October 2017 (UTC)

• The better place to bring this up may be Wikipedia talk:WikiProject Chemicals. They might be better suited towards helping you in this highly technical question. --Jayron32 13:25, 31 October 2017 (UTC)

• Thank you! — Preceding unsigned comment added by Gunnar Römer (talk • contribs) 14:00, 31 October 2017 (UTC)

Moved to Wikipedia talk:WikiProject Chemicals § Molecular Summary Table

IEC standards list

Hi! How can I find a complete list of IEC standards?

I searched in IEC website, but I couldn't find that list.--92.50.40.2 (talk) 15:54, 31 October 2017 (UTC)

Do you mean the List of International Electrotechnical Commission standards? --Jayron32 18:49, 31 October 2017 (UTC)

Our list is incomplete. These standard institutions usually forbid copying anything, maybe even a collective list, because they finance themselves by selling their standard definition papers or digital e-papers. In fact you usually already have to pay to just enter their "shop" aka subscribe to some access plan, so you may be able to see the list, if you start paying them! They have a monopoly and they act like it. Of course usually for breathtaking prices. The best chance to access, to the complete list and all standards, for private persons are usually Libraries in big cities or cities with a fitting university. They usually have all standard papers. Chances are high tho you may only read them in the Library but not lend them out. --Kharon (talk) 19:51, 31 October 2017 (UTC)

IEC though is typically one of the lowest priced sources for standards, and they often duplicate standards issued by other bodies (for example UL). AFAIK they don't charge any fee to browse the standards available from their web store. They even make some material, like the International Electrotechnical Vocabulary available gratis. I don't find any comprehensive list of standards on the website, but there is a of subcommittees with links to the publications available from each subcommittee. The Photon (talk) 05:11, 2 November 2017 (UTC)