My Erdős #

I only found out last year (at the Water Chair 3oth reunion) that I had one.

An Erdős number is a measure of someone's degree of separation from the prolific mathematician Paul Erdős. Coauthorship on the same peer-reviewed paper counts as a link. A related concept is the "degrees of Kevin Bacon" in the domain of film, linking actors that have appeared in the same movies. Small world networks and graph theory are topics that I've written about before on this blog.

Last summer I attended the 30th year anniversary of the research group where I did my master's degree. There was a guest speaker there, Dr. Bruce Rittmann, who is a very prolific scholar in the field of water and wastewater treatment. He's collaborated with my supervisor before. On a whim, I decided to check if he has a defined Erdős number. He does: 5. That gives my supervisor a number of 6 and me (and any other students of Dr. Huck who have co-authored papers with him and don't have a shorter collaboration path to Erdős) a number of 7.

The path of linkages is thus: Paul Erdős, Persi W. Diaconis, Ofer Zeitouni, Sigurd B. Angenent, David L. Chopp, Bruce E. Rittmann, Peter M. Huck, Daniel B. Scott.

(While I was looking stuff up for this post, I found that my Dad has an Erdős number of 6. This is because he co-authored a paper with his supervisor from grad school, who has a number of 5.)

I decided to track down an article from each link in the chain and comment on them here. For some co-authors, there were multiple papers and I chose one (or two) that looked interesting. Some of these papers only had the abstract available, whereas for others I was able to read the whole thing.

To start with, I've previously noted on this blog when a couple of papers with Dr. Huck as one of the authors were published:

• Scott et al., 2021.
• Scott et al., 2015.

Next is the collaboration between Dr. Huck and Dr. Rittmann:

Bruce E. Rittmann , Peter M. Huck & Edward J. Bouwer (1989) Biological treatment of public water supplies, Critical Reviews in Environmental Control, 19:2, 119-184, DOI: 10.1080/10643388909388362

This is a long review paper. Unfortunately, I only have access to the abstract; I wish I could read the whole thing to see how far my field has progressed in my lifetime. It's about strategies to avoid or minimize regrowth in drinking water via biological treatment. It seems to mainly deal with biofilm-related methods (as opposed to suspended growth). They also review methods for measuring low concentrations of biodegradable organic matter (BOM). Finally, they examine some trends in technology and regulations—I'd like to see how accurate their vision of the future was.

I read two papers with Dr. Rittmann and Dr. Chopp as coauthors since those works were squarely in my current professional interests:

Brian Merkey, Bruce E. Rittmann, David L. Chopp (2006). "A Multi-Species, Multi-Component Biofilm Model".

• Extracellular polymeric substances (EPS) holds biofilms together. It is considered as a solid. Hydrolysis of EPS, along with other processes, releases soluble microbial products (SMP).
• SMP from nitrifiers can help feed heterotrophs in biofilm communities when the C:N ratio is low.
• The goal of this paper was to use a model to identify conditions where nitrifiers will be suppressed in biofilms.
• Their model assumed that each solid species had a volume fraction (adding up to 1) while dissolved species had concentration profiles that were diffusion-limited and assumed to be in a quasi-steady-state.
• The biofilm thickness was modelled to grow in proportion to the sum of growth rates of all solid species. At the same time, detachment (taken to be proportional in rate to the thickness squared) balances out this growth.
• The model shows that layers develop: inert biomass accumulates near the surface, then heterotrophs, then autotrophs.
• Shear/sloughing at surface helps prevent nitrifiers from getting buried too deeply.

B.V. Merkey, B.E. Rittmann, D.L. Chopp (2009). Modelling how SMP support heterotrophic bacteria in autotroph-based biofilms. Journal of Theoretical Biology. 259:4, 670.

• A continuation of the modelling from the previous paper. Compared to that previous work, this one uses two dimensions instead of just one. They also investigated varying values of kinetic parameters for SMP utilization. Additionally, they focused on situations where there was no external substrate for heterotrophs, only the SMP released by autotrophs (i.e. nitrifiers).
• Heterotrophs grow 10-20x faster than autotrophs. But in this case, they examined situations where byproducts from autotrophs were the only available substrate for heterotrophs.
• The model included two different types of SMP: utilization-associated products (UAP) and hydrolyzed biomass-associated products (BAP).
• As with the model in the preceding paper, solid species were tracked with a density and a volume fraction. Soluble species could diffuse to/from the bulk liquid via a boundary layer.
• The thickness of the biofilm was modelled as a dynamic balance between growth, decay, and detachment.
• UAP was found to be more important to heterotrophic growth than BAP.

Getting into the papers that are more purely about mathematics, I have the following notes from reading the abstracts or papers. Some of them were fairly far outside my previous experience, so I might have only gotten the bare basics of what they were about:

S. Argenent, T. Ilmanen, D.L. Chopp (1995). A computed example of non-uniqueness of mean curvature flow in R³. Commun. in Partial Diff. Eq. 20:11&12: 1937-1958.

This paper only had a 1-page preview (not even an abstract) available. Apparently, they're trying to answer a question about non-uniqueness (thus a single example will suffice) of mean curvature flow. From the title, it's in a 3-dimensional space of real numbers. Mean curvature flow follows the differential equation,

∂x / ∂t = H(x)

Here, x is a point on a manifold/surface in a family of surfaces moving in a defined way and H(x) is the mean curvature vector of the surface at point x. H(x) is the sum of the product of each eigenvalue (2 for a surface in 3-space) with the unit normal vector at x.

S. Angenent, A. Tannebaum, A. Yezzi Jr., O. Zeitouni (2006). Curve shortening and interacting particle systems. Statistics and Analysis of Shapes, p. 303-311.

This is a book chapter, and only the first 2 pages were available as a free preview. Per the abstract, applications of "curvature driven flows" of the type they were studying include crystal growth and flame propagation. Apparently, they use semilinear diffusion equations and present a stochastic approximation, but that's about all I was able to glean. Between the previous paper and this one, I was exposed to the concept of curve-shortening flow.

Persi Diaconis, E. Mayer-Wolf, Ofer Zeitouni, M.P.W. Zerner (2004). The Poisson-Dirichlet Law is the Unique Invariant Distribution for Uniform Split-Merge Transformations. Annals of Probability. 32:18: 915.

• This paper is a proof of a conjecture about the continuous coagulation-fragmentation (CCF) model. They also mention the discrete version (DCF).
• It was the most unfamiliar territory for me (at least of the papers where I could read more than the abstract—the previous two are hard to evaluate for sure). Lots of unfamiliar terms and notation.
• They were partitioning the interval [0,1] with a Markov process.
• The Poisson-Dirichlet measure is described as "the law of the non-increasing rearrangement of the uniform stickbreaking process" (by non-increasing they mean it can decrease or stay the same from one element to the next).
• The most accessible part of the paper had some Young Tableaux. These look like an understandable introduction to partitions in combinatorics/group theory. Boxes are placed in rows to depict ways in which a group of sub-totals can be selected to reach an overall total. The overall total is the number of boxes in the diagram and each sub-total is represented by a row. The rows are arranged in non-increasing order so the top rows overhang the lower ones (or are the same length). One of these diagrams represents a partition, which is concerned with how a number can be broken down into a sum of smaller numbers, but isn't concerned with the order in which the sum is written (so having a standard order to draw the Young Tableau means that if the only difference between two sums is their order then they'll have the same diagram because they're considered the same partition).

Persi Diaconis and Paul Erdős (1977). Technical Report #12: On the distribution of the greatest common divisor.

• This technical report is about the greatest common divisor (GCD) and least common multiple (LCM) of two randomly-selected positive integers.
• It contains some complicated proofs with notation I wasn't fully following, but something I did get out of it was a useful application: how to back-calculate the population n of a survey with k categories when only the proportion of each category pᵢ is given. Let b be a vector with k elements, each an integer between +/- (k-1), and not all zeroes. Calculate Σᵢ pᵢ bᵢ for all possible b. Keep the result with the minimum absolute value. Its reciprocal is an estimate for n.
• This method works because the probability that the GCD of the number of individuals in each category equals 1 increases rapidly with even a modest number of categories.
• Specifically, this probability (i.e. that k randomly-selected integers are coprime) is 1/ζ(k), the reciprocal of the zeta function.
• Doing some reading on Wikipedia to learn more about the previous point, I found that 1/ζ(k) = Σₙ Mobius(n)/n^k , for n from 1 to ∞. However, since these terms get rapidly smaller, the first seven terms (1,-1,-1,0,-1,1,-1) of the Mobius function should be enough to calculate the reciprocal of the Zeta function to three decimal places.

As my notes surely make clear, I was getting far out of my wheelhouse as these papers got away from water and wastewater treatment. However, it was an interesting exercise tracing the chain of papers back to Paul Erdős and I got exposed to a few new equations and concepts along the way.

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