I read a paper and was inspired to give this advice to my son: "Keep your math muscles working."

[NNadir]

My day to day life is involved peripherally in chromatography, analytical chromatography as opposed to industrial chromatography, but I really don't think on a very profound level about it. When someone from the lab comes to me to describe what they have done and to get my comments or advice, and I ask about the chromatographic conditions, I often have to look up the chemistry of the columns in order to make suggestions. As I've aged and advanced in my career, I've moved further and further away from the theory into a lazy realm of automatic reference to experience.

It's been a long time, too long probably, since I sat down to think much about solving a differential equation, and my mathematical muscles have atrophied, something that struck me when I came across this beautiful paper, by Pakistani scientists, about the theory behind what we do so lazily day to day, without much thought. The paper is this one: Discontinuous Galerkin Scheme for Solving a Lumped Kinetic Model of Non-isothermal Liquid Chromatography with Bi-Langmuir Isotherms Ambreen Khan, Sadia Perveen, and Shamsul Qamar Industrial & Engineering Chemistry Research 2021 60 (34), 12592-12601.

From the text, which starts with the sophomore chemistry discussion of chromatography and quickly moves beyond:

Chromatography is widely used for purification, separation, and identification of the mixture’s components for quantitative and qualitative inspection. It is successfully practiced in laboratories and industries to carry out various complex separations. This technique is regularly used in the pharmaceutical sector to detect the unknown compounds and purity of the mixture, in the chemical industry to test water samples and to check the air quality, and in the food industry to determine the nutritional quality of food. It also plays a vital role in forensic science, fingerprinting, protein separation (such as insulin purification), plasma fractionation, and enzyme purification. This technique is also used in different sectors such as fuel industry, biotechnology, and biochemical processes.(1−5)

Chromatography has several types, such as liquid chromatography, gas chromatography, and ion-exchange chromatography. However, all of them follow the same basic principles. In chromatography, separation of the mixture components takes place between two phases, called as mobile and stationary phases. Here, we consider a liquid chromatography setup in which the adsorbent is solid and the solvent is liquid. The mobile phase (solvent), carrying the components of the mixture, is injected into the column containing the stationary phase (adsorbent). As a result, different chemical and physical interactions take place between the mixture’s component and the particles of the column. Those components of the mixture which are interacting strongly with the stationary phase are slowing down, while weakly interacting components are moving faster. Thus, the separated mixture components can be collected at the other end of the column.

The temperature of the column has an increasingly important role in the development and optimization of high-performance liquid chromatography.(6−10) Variations in the column’s temperature can enhance its performance, such as shortening of analysis and separation times, sharpening of elution profiles, reduction in the use of organic solvents, and allowing faster conversion of the reactant into products in reactive chromatography.(11−16) The efficiency of the column and the stability of the stationary phase and analytes could be improved under controlled temperature operation. Several experimental studies have been carried out in the literature on non-isothermal chromatography to demonstrate thermal effects on the retention behaviors of the elution profiles, variations in the concentrations and volumes of the injected pulses, packing materials, and ion-exchange chromatography.(17−25)
Mechanistic modeling is an extremely important section of chromatographic theory to understand and visualize the dynamics inside the column, to hypothetically examine the procedure of chromatography, and to streamline the product quality. Different multifaceted models have been introduced in the literature for simulating mass exchange and partitioning phenomena inside the chromatographic columns, for instance, equilibrium dispersive model (EDM), the lumped kinetic model (LKM), the general rate model, and many more.(1−5) All these models consist of a system of partial differential equations (PDEs) of advection–diffusion type coupled with some algebraic or differential equations. The linearity and non-linearity of these models are determined by the adsorption isotherm related to them. Analytical solutions of these models are possible for linear adsorption isotherms.(26−29) However, to study different types of thermodynamic adsorption equilibria, such as the generalized bi-Langmuir isotherm, accurate, stable, and efficient numerical techniques are the only available tool to obtain solutions.

The literature provides a wide range of numerical methods for solving the convection-dominated PDEs. Some of them are the non-oscillatory finite difference (FD) methods such as TVB (total variation bounded), TVD (total variation diminishing), ENO (essentially non-oscillatory), and weighted ENO schemes. These methods have the ability to avoid oscillations due to sharp discontinuities in the solution profiles.(30−47) Despite having simple coding, the FD methods are generally difficult to apply if complicated boundary conditions or/and complex geometries are involved.(48) The adaptive stencil idea of the ENO scheme provides high-order accuracy by implementing large stencil but gives low-order accuracy in the elements near the boundaries of the domain.

On the other hand, the finite element (FE) methods have the capability to handle complicated boundary conditions and complex geometries. FE methods, such as classical artificial viscosity and standard Galerkin methods, have low-order accuracy, are unstable, and can generate non-physical solutions. Streamline diffusion of the FE method, developed in refs (49−52), significantly reduces oscillations by considering the L∞ bound of the numerical solution. This method is implicit in time, whereas hyperbolic problems are functioning more naturally for explicit techniques due to the existence of strong shocks.(53−55) Therefore, the DG method was introduced and implemented to address these issues.

The DG method was first presented by Reed and Hill(56) to solve the steady-state linear hyperbolic equations and later in 1974 Lesaint and Raviart(57) proposed the mathematical analysis of this method by demonstrating the importance of applying the DG method to the neutron transport problem. Hulme(58,59) has suggested a similar approach for the solutions of ordinary differential equations (ODEs) by considering continuous, rather than a discontinuous, approximation of the solutions in their weak form. In 1988, the local discontinuous Galerkin (LDG) method was introduced by Cockburn, and afterward, the method was studied in detail for one-dimensional (1D) and multidimensional problems by Cockburn and other researchers.(60−64) Further development was made by Bassi and Rebay(65) in 1997 by considering DG space discretization in a different manner and by choosing numerical fluxes to solve compressible Navier–Stokes equations.

Boris Grigoryevich Galerkin was an interesting guy, a political radical in his youth, sentenced to a Czarist prison - where one was apparently allowed to do lots of math - he gave up political activism to focus on engineering mathematics, although he did become a Menshevik at one point; being a former Menshevik in Stalinist times was often fatal. He then was instrumental in developing a branch of mathematics that is well suited in modern times, given the development of modern computational power, to numerical solutions to differential equations. Irrespective of the repressive time and place in which he lived, Galerkin went on to become a world leading scientist, and during World War II was actually commissioned as a General in the Soviet Military, even though he knew nothing of military life and was embarrassed whenever people saluted him. He died in 1945, apparently of exhaustion.

Anyway.

I sometimes find myself pondering Langmuir adsorption coefficients, and at other times the related but different BET (Brunauer-Emmett-Teller) adsorption conceptions, but again, lazily.

Some text in a graphics format owing to the lack of an equation editor here:





...and so on.

The models by the way, produce some very ugly chromatography. Here's figure 6:



The caption:

Figure 6. Comparison of RKLDG and HR-FVS for two-component non-isothermal elution considering a non-linear isotherm with ce/cf = 1, k = 10 min–1, and k = 100 min–1. However, b1Iref = b2Iref = 1.0, b1IIref = b2IIref = 2.5.

My son has honored me by proposing to steal all of my ideas after pursing a Ph.D. in nuclear engineering. My "ideas" such as they are, do involve chemical separations of the valuable components of used nuclear fuel. I'm a Charles Forsberg fan.

Nuclear Engineering texts involve a considerable amount of mathematical sophistication, but in recent years, when I go through them - if I go through them, and sometimes I do - it's in a lazy fashion, a kind of mental note that lazily says "it's there if I want to spend the time to go deeper" but I never do.

I suppose in an nuclear engineer's career, a time comes that one just calls up the software without looking under the hood. This is my habit; I trust the mass spectrometry software, but seldom think about what lies beneath. (This can, albeit under rare circumstances, be dangerous.)

It is better, I think to really live in the mathematics. I sent this paper to my son, not because I expect him to read it or need it but as a warning to be better than his father throughout his life, something he is well on his way to achieving.

Have a nice weekend.

[hwmnbn]

1. EUREKA!!!!

I think I almost nearly understood about < 1% of the above.

I feel confident now to take on the day.

In all seriousness, NNadir, I marvel at the elegant complexity of your mind.

[NNadir]

6. Thanks for your kind words. n/t.

[NH Ethylene]

2. As a Chemistry Teacher, I struggled with students' overreliance on technology constantly.

It seems to be established in society today that machines will do the math for us, so we don't need to. I had students who, when confronted with the need to solve the simplest arithmetic, literally 2 + 2, would pull out their calculator to do it for them.

More than once I have chided a class about their lack of effort involving math, saying that if they don't use their brains they will get Alzheimer's before me! (and I'm old)

I think of handing over our basic daily math tasks to a computer or a calculator or a cash register as giving away our intelligence.

Congrats on your son surpassing his dad. Yours was a job well done.

[eppur_se_muova]

3. Are you familiar with this story ?

https://archive.org/details/1958-02_IF/page/n5/mode/2up?view=theater

[NH Ethylene]

5. What a delightful short story!

It's just rich with irony.

This is what I mean by 'giving away our intelligence'.

Of course we need to use computers now for complicated, time-consuming computations, and a calculator in high school Chemistry class is a required tool. But many of the kids of high school age now were never asked to memorize addition or multiplication tables in elementary school and up to one half are incapable of doing any kind of math in their head. That means estimating and predicting answers to problems is difficult, so that any answer that pops out of the calculator is okay.

Thank-you for the link.

[NNadir]

4. I think it's probably a mixed bag.

I do agree that people should be able to do basic calculations and make certain estimations in their head, but on the other hand, one can cut to the chase more quickly as one advances into high school - assuming a decent elementary school education - and will benefit in particular if they are required to program their own software.

The experience of programming is a pretty good way to demonstrate the limits of software.

I will say that when my youngest son - the possible nuclear engineer (he still has time to change his mind) - was eight and I taught him binary notation, the little brat refused to learn how to do it on Excel. He sat there with a pencil and paper and wrote out all the powers of two up to around 8192 if I remember correctly. I was pretty impressed with that. I assume it was a good exercise to do it that way.

But I did buy him access to Mathematica when he was in high school, and challenged him to write software for Peng Robinson calculations, which he never did, but subsequently he was required to learn to do lots of programming for his Materials Science Engineering degree and was able to cover a lot of material that would have been impossible with hand calculations.

He still writes code - I guess all the kids and many grown ups use Python these days - and using it in his classes. He's finishing up a one year 30 credit Masters in materials science on a scholarship. Recently for one of his classes, he wrote some kind of program that he was running on his laptop when we visited him. He said he expected it to take 24 hours to complete the computational run. His older brother, the artist, who never took a programming course in his life but somehow has developed considerable knowledge autodidactically, after some discussions, made all kinds of suggestions that were way over my head. They might as well have been speaking Estonian. Sure enough, the program my younger son was running crashed, and he took his big brother's advice for streamlining the code and ran the whole thing about 5 hours.

These are the tools for the times in which these kids live. They made, in my son's case, him far more powerful than I could have even imagined being at his age.

It is the structure of math, I think, looking for the elegant solution for the least cost, that is valuable, because computer time is precious, and we don't want to waste it and we certainly don't want it spitting out the wrong results.

In the text of the paper I discussed in the OP, this was reported as the issue for various numerical schemes. Of course, to be capable of making decisions about what to program, one has to know the mathematical structures to include, which is the point.

I'm OK with young people using technology - it is essential to do so in their era - but only if they really understand how the technology works. That's the issue, and I feel for educators, such as yourself, who need to get that across. I suppose it isn't easy.

[Igel]

7. Rec. n/t

[hunter]

8. Living in the mathematics isn't for everyone...

... but it would be nice to live in a world where ordinary people knew enough math to recognize complete and utter bullshit.

Innumeracy is not a problem exclusive to the political "left" or the "right."

Do mathematical "muscles" atrophy?

It seems to me some of the mathematical experience gets transferred to more autonomous parts of the human mind, the same parts of our brain that allow us to think about something else while riding a bicycle or driving a car. Usually that's enough to take us safely where we want to go even without giving our full attention to the problem. (This can, albeit under rare circumstances, be dangerous...)

Maybe we can read a scientific paper and grok most of it without a deep dive into the equations.

If we don't frequently practice bringing that math back up to the surface, to write it down for example, this ability to manipulate math symbolically seems to fade quite rapidly compared to that intuitive feeling for the math that remains. But it's the intuitive feeling that tells us the numbers coming out of the computer are somehow not right.

Many people have little feeling for math, any math, and have a lot of trouble dealing with very large or very small numbers. Thus they can be easily convinced that any tritium released in the cleanup of Fukushima is a horror beyond reckoning, or that keeping a national economy stable is similar to managing a household budget. (Some of them might also choose to believe that a cruel and capricious god created the earth less than 10,000 years ago. Apparently a 13.77 billion year old universe frightens them, as does the 541 million years of evolution beginning with the Cambrian radiation.)

Sometimes I imagine I could dredge back up enough math to pass a graduate level course in population biology (now generally referred to as population ecology), but honestly I'm too easily distracted and lazy.

As for the paper, I think many people who know what chromatography is from high school or college science labs may not know it's used in industrial scale manufacturing processes as well.

Wikipedia is still saying "Chromatography is a laboratory technique for the separation of a mixture. The mixture is dissolved in a fluid (gas or solvent) called the mobile phase, which carries it through a system (a column, a capillary tube, a plate, or a sheet) on which a material called the stationary phase is fixed."

[NNadir]

9. Certainly I can look at the equations in this paper and grasp the meaning, but my math "muscles..."

...what I mean is the capability to utilize them to do useful work.

In his case, useful work would be mean being able to actually decide on an algorithm that might, for example, to have a systematic way choose the general chemistry of a stationary phase based on what is called here, "Henry's law" constants, which generally are applied in introductory courses for gas liquid interfaces rather than, as here, solid liquid interfaces.

What bugs me is that I have no insight whatsoever to the general mathematics of discontinuous Galerkin approaches, for example, and many other techniques in the solution of differential equations. It turns out, on further investigation, that this particular branch of mathematics, Galerkin finite element methodology is very important in a completely different area than Langmuir adsorption, specifically, neutron transport, and many other only loosely related areas as well.

For example: Discontinuous Galerkin spatial discretisation of the neutron transport equation with pyramid finite elements and a discrete ordinate (SN) angular approximation...

...and...

Sensitivity Analysis of the Galerkin Finite Element Method Neutron Diffusion Solver to the Shape of the Elements

What I'd like for my son would be to be able to look at these titles and say..."that makes sense..." or "why didn't they just use..."

There was a time in my life I was there myself, but I feel it's gone and I miss the feeling, the connectedness.

But you're right, most people don't need that level of math, although your other point is well taken, innumeracy drives the world to some very, very, very bad places, the Fukushima tritium terror being only one such example of the triumph of ignorance.

[hunter]

10. My children are smarter than I am...

... maybe smarter than their mom and her high power, sometimes highly lucrative, education as well.

I'm an incredibly fortunate animal. My own grasp of the harder math has always been tenuous.

Looking back in the family tree none of us were ever victims of anti-intellectual religions or ideologies.

"Doing the math" was, and continues to be, celebrated.

One of my nieces is highly proficient in statistics. She's been accepted to several graduate schools but is currently having too much fun making money. Beats the waitressing that paid for a substantial portion of her university education.

That's more than enough evidence of progress in our society.

In the bad old days women who could do the hard math often had to attach themselves to men who would take all the credit for it.

I'm just a university educated simpleton who happened to have a lot of skills as a homemaker. Couldn't help it. I was doing laundry and changing diapers when I was ten years old.

[caraher]

11. Getting the right balance is tricky

In my graduate quantum optics course my professor one day asked the room full of physics graduate students, after writing a differential equation, "How do you solve a differential equation?" (Read the question in a New York accent tinged with exasperation.)

Everyone in that room had solved many differential equations over the years, but we were stunned into silence. Is there some universal trick that we'd missed? After the silence had become unbearable, he continued...

"You write down the answer."

The real moral of that story is that, when it comes to analytic solutions, the most efficient approach is to classify the equation before you and call to mind (or re-invent if you're way smarter than me!) the special method of solution, or the general solution itself.

In any case, when you have a system of coupled partial differential equations, analytic solutions are largely out of the question - just go straight to numerical methods. Yet even then there's an element of "look up the answer," since there are often substantial differences in the suitability of one algorithm over another that depend on exactly what equations you have.

Weirdly, in COVID I'm finding myself actively shunning computer math tools when possible in my teaching, simply to give students a chance not to stare at screens. It's meant going fairly "old school" in the way I taught linear algebra, differential equations and vector calculus last year (less emphasis on numerical methods, more emphasis on analytic solutions).

This year, however, I'm co-teaching a combined computer science, physics and calculus course, and am leveraging the computer science component to support Python physics simulations. Yes, Python is the current favorite for general purpose programming in physical sciences. I'm still in the "keeping one step ahead of the students" phase in my Python learning (which usually means I have a student or two teach me the things I didn't learn yet).

[NNadir]

12. Well in a sense, it's somewhat easier for you to live in the math.

I have always felt that the best way to stay current with something is to be required to teach it.

I suppose part of my job is to teach the younger staff about issues in their work, but the reality is that in instrumental chemical analysis, all calculations are carried out by sophisticated software usually featuring and audit trail. Thus the conversation is general more about molecular biology and organic chemistry than it is about the theoretical underpinnings of either the instrumental set up..

As for "looking it up" that's easier than ever before. There are many reference books that I could throw away were it not for nostalgia. Does anyone ever open a table of integrals anymore?

Surprisingly, my son did take course in Mathematica from the guy who ended up being his advisor. (Nobody really uses it at his institution. ) The tenor f the course was "Everything is there somewhere if you want to find it." As it happens, my knowledge of statistics was embedded in other courses part of a formal course. Thus when I am confronting with a concept like false discovery rate in proteomic software spitting out the location of post translational modifications, I have to dig down deeper to be sure of the results.

I'm encouraging my son to TA, irrespective of his intense time constraints. Teaching is good for learning and knowing I think.