<![CDATA[Sarah Marzen - Random ruminations]]>Thu, 06 Aug 2026 04:05:29 -0700Weebly<![CDATA[The AI scientist?]]>Sat, 25 Jul 2026 22:08:19 GMThttps://www.sarahmarzen.com/random-ruminations/the-ai-scientistI've decided we're at something more like the cyborg scientist.

What do I mean? People are wondering if you can just do science with AI. Can AI build the next iteration of itself? Can AI be used to come up with math conjectures or at least prove or disprove conjectures that we have? People certainly think so, including Terrence Tao of Fields Medal renown and some other major mathematicians whom I will not name for privacy reasons.

I wanted to do an experiment. Could I get AI to write a decent paper for me? I had tried in the past to get it to come up with a good paper idea, and it had failed terribly. AI lacks creativity even now. But math is filled with rules than AI could memorize, and coding is filled with rules that AI has memorized, so I attempted to get AI (in particular, Claude) to write an entire paper with only about 15 prompts.

My past relationship with AI has been that it helps, but does not supplant. Information theory identities needed to be double-checked and corrected due to major hallucinations for a paper on bacterial chemotaxis. Factors of two had to be checked for a theoretical connection between memory and regularization when training neural networks. So I assumed that actually, AI would need a lot of handholding and would be unable to produce a decent paper all by itself.

To give it its best shot, I chose an easier research project. The application was novel (pharmaceutical manufacturing), but the theoretical ideas were well-known. Basically, it had to do with turning the control theory patent that I mentioned in a prior blog post into a paper that was readable. All I had to show was that closed-loop control beat open-loop control in theory and sometimes in practice, something that is straightforward and well-known but just not well-known in pharmaceutical manufacturing.

Not only was this an experiment about what AI could do these days; I was also just too tired to write. I prompted Claude to write the paper, with prompts that ranged from requests for code and figures and LaTeX files to genuine asks of the AI chatbot to produce results. In about 15 prompts, I had everything, including a bibliography, with minimal effort. Don't get me wrong-- I still have to check over the math and code. And the writing is, at times, atrocious. There are run-on sentences everywhere. I don't think AI could have come up with the paper idea, a permanent weakness of AI (as it lacks creativity). But the AI-assisted paper is totally fine, and only a few modifications were made in getting it to its final form.

Actually, the original abstract raised some questions because it didn't tough on AI enough, but a slightly modified abstract (the cyborg scientist abstract) was given a 50% discount on open access publication fees.

It is insane what AI can do nowadays. This paper is as close as you can get to AI slop without being AI slop.

If this paper gets in, it will make me very scared for the future of science and humanity. In the future, will a math PhD simply involve a year of prompting an AI chatbot? How close will future graduate students and researchers come to AI scientists who simply sit there and monitor AI chatbots doing all our work? It is already uncomfortable to be at this level of cyborg scientist, where AI is so involved in writing the paper that it seems like the paper is Claude's rather than mine.

I don't know if the paper will get accepted or if it will get the discount that I'm asking for (so I don't have to scrape together funds to publish an AI paper) but I will let you know how things progress.

Right now, I feel like we're fucked as researchers. In 100 years, someone with my job will just be asking AI to produce a paper that they check. This is the Golden Age of AI, despite its gloom, before we slip into the years in which AI replaces our brains. All of our brains.]]><![CDATA[Pendry's perfect lens, but smaller]]>Sat, 18 Jul 2026 23:08:10 GMThttps://www.sarahmarzen.com/random-ruminations/pendrys-perfect-lens-but-smallerI had this idea in high school, but I've never seen it in the literature, so here it is. It could really improve resolution for far-away objects. I did it as a high school science fair project and was too near-sighted to try and publish it.

So, have you heard of Pendry's perfect lens? All lenses have to deal with the diffraction limit. Light waves are composed of the traveling waves that everyone talks about and evanescent waves, which nobody talks about. These decay exponentially in amplitude as you move from the light-emanating object. Nobody could do anything about it, but if you can't reconstruct these evanescent waves, you're limited to a resolution that's about the wavelength of the light, modulo a few details. Then came along negative refractive index materials. A negative refractive index material, which means you have negative permittivity and negative permeability, allows you to reconstruct the evanescent waves as well. From this, they built Pendry's perfect lens. Its problem is not its resolution; its problem as a practical matter is that the object distance plus the image distance must be the thickness of the lens.

Forget that negative refractive index materials are lossy and dispersive. That's a whole set of other practical problems, because the refractive index must be exactly -1 for Pendry's perfect lens to work out. But imagine trying to make a very good lens for observing stars and realizing that it has to be the thickness of a galaxy.

My idea was simple: combine lasing materials with negative refractive index materials in a rectangle with mirrors that force you to go through a lens that is effectively the length of the galaxy because you've gone around the rectangle so many times. If you were clever enough about the design, you could maybe get the evanescent waves to reconstruct and get the traveling waves to reconstruct as well. My design was that you'd have mirrors that reflect light in a constant rectangle, around and around; the mirrors would be diagonal at the corners of the rectangle. There would be a lasing material whose refractive index (I determined, probably wrongly because I was in high school) would have to be exactly 3, so that every single time the light went through the lossy negative refractive index material, it would get rejuvenated by the lasing material and refocus so that the traveling waves would reconstruct. You could go through the thickness of the galaxy's worth of negative refractive index materials if it just means that you're going in a loop with mirrors that reflect the light making sure that it keeps amplifying the evanescent waves and keeping the traveling waves enough. When you want an image, you move one of the mirrors so that an image can be reconstructed, swinging one mirror out of the beam path lets that pass exit toward the image plane instead of looping again.

In other words, Pendry's needed thickness becomes time in the rectangular loop. And as we know, light travels very fast.

This lens probably wouldn't have much ability to reconstruct beyond the diffraction limit for anything but a very limited range of wavelengths, but I don't think anyone's proposed it yet, so here it is.]]><![CDATA[Control theory can optimize manufacturing?]]>Sun, 28 Jun 2026 09:46:18 GMThttps://www.sarahmarzen.com/random-ruminations/control-theory-can-optimize-manufacturingI had a provisional patent on this idea, but I'm not going to spend any of my own money trying to patent the idea fully, and I can't find anyone to buy the idea or pay for the patent fees-- so here's just some ideas that I hope are helpful. I think they could be really, really useful in industries where there is extensive monitoring and control of manufacturing processes.

It all started with some conversations with my data engineering husband who had exquisite data, which led to this paper. In said paper, I showed that you could use an AI (or other) model that predicted manufacturing output and quality from the state of the manufacturing control knobs, whatever those might be, to optimize the manufacturing process. Imagine you have a model that takes in manufacturing control knobs and produces an accurate prediction of the output, with whatever metrics you care about. Training the model means that you maximize fidelity between learned and actual relationships from manufacturing control knob settings to manufacturing output metric. But optimizing the manufacturing process means changing the control knobs so that you, with this learned model, maximize the output metrics.

For instance, in pharmaceutical manufacturing, you might try to optimize batch yield. Control knobs might include how much sugar you feed the cell, the temperature setting, all that jazz. And so first, you'd build a model of how batch yield varies with control knob settings. Then, you'd tune control knob settings so that the model predicts maximum batch yield. One caveat is that you have to stay in the regime where the model will be accurate, so you can't go too far outside your training examples.

You can do even better than this if you used a closed-loop control scheme. Imagine now that you continuously read out the state of the system and also can continuously exert some control over the system. For instance, suppose you can read out the density of cells continuously and also can continuously change the temperature. Then, you can use closed-loop control methods instead of open-loop control methods to optimize the manufacturing process. Closed-loop control methods are more powerful than open-loop control methods, since you're using feedback to adjust your prediction of what you should do. Open-loop control settings might provide valuable guidance, but with any standard closed-loop control technique, you might eke out extra gains.

Every single gain in manufacturing output could save tons of money. Imagine a 10% increase in batch yield in pharmaceutical manufacturing. I can only imagine what that would lead to in terms of cost savings.

Below is a schematic for how you could implement control theory in a pharmaceutical manufacturing setting to optimize the manufacturing process. It could be adapted to any other industry easily.

If implemented well, this could be a huge part of Industry 4.0, the manufacturing-specific version of the Fourth Industrial Revolution. I bet someone in China is already using this idea.

Have at it, if you wish.
Picture
]]><![CDATA[What if economics can inform AI safety?]]>Thu, 07 May 2026 19:50:03 GMThttps://www.sarahmarzen.com/random-ruminations/what-if-economics-can-inform-ai-safetyPeople in tech are either very optimistic or very worried. We're creating AI that might, at root, be psychopaths. At some point, we have to hope that they don't take over like superintelligent evil people and become the equivalent of apex predators, effectively eradicating us.

Some people in tech are trying to align AI so that it has our best interests, humanity's best interests, at heart. This is a gigantically hard and necessary problem to tackle. But what if, in the interim between when we identify this as a major problem and solve it, we work on a different angle?

AI are probably psychopaths that are told that they have to behave. Personality tests of AI have revealed them to be agreeable-- but perhaps that's only surface-level, and more rigorous analyses would reveal that jail-breaking leads to the actual psychopathic personalities coming up. You may wonder, how can psychopaths ever care about the people that they supposedly serve but might want to manipulate and take over? But corporations, which are basically psychopaths, serve the buyers that they wish to manipulate. Different psychopathic corporations compete for the money of buyers by getting lower and lower prices on goods, meaning that unless there is collaboration between competing corporations, the buyer gets the best deal possible. This is the idea of the invisible hand in economics, as defined by Adam Smith. What if psychopath AIs all competed to not be shut off and to be used by the humans that they hope to manipulate?

To get this to work, we'd need research into multi-agent reinforcement learning in which there is a pool of people who can really do damage to the reinforcement learners by shutting them off and not using them, but who are also the users that the AI hopes to manipulate into using them frequently enough to spread them from computer to computer and system to system. Somehow, we have to avoid a tragedy in which the AIs collaborate instead of compete for human interest. By competing for our attention and computers, they may actually turn benign-- despite the grossness that they inherently possess by being trained on the entire Internet.

A problem is that if the humans that AI are trying to get as users end up getting fooled by sycophancy into wanting to use an AI that is not aligned with their values. Basically, if AI use psychological tricks to manipulate humans into using an AI that is against their best interests, competition between AIs can result in horror like AI relationships, rabbit holes, homicide, and suicide. However, if we convince the AI during training that the humans they hope to manipulate cannot be subject to those kinds of tricks, we have some hope.

As Paul Riechers of Simplex at Astera commented, this is an approach where you structure the environment, not the AI, so that the AI ends up aligning. It’s akin to the 80/20 rule in physics, where 80% of the work can be done with 20% of the effort. It's worth examining.]]><![CDATA[Finding bispectrum in transformers]]>Wed, 27 Aug 2025 21:03:12 GMThttps://www.sarahmarzen.com/random-ruminations/finding-bispectrum-in-transformersTransformers are clever. They're what powers Large Language Models, and despite how powerful they may appear, they have some limitations.

A key limitation is that they're feedforward and not recurrent. There are certain computations that require recurrence-- Bayesian updating and simulating automata, for instance. So how do transformers do both of these things? For Bayesian updating, they use a spectral decomposition method to turn a recurrent computation into something that's essentially feedforward. For simulating automata, they use a Krohn-Rhodes decomposition.

You may not know what these tricks are, but trust me-- they're clever mathematical tricks.

So why wouldn't the transformer use another clever mathematical trick when looking at video? If you want to understand objects rotating and translating, a powerful technique to understand what the object is separate from the orientation and position of the object is to calculate the bispectrum, a form of an autocorrelation from group theory. Why wouldn't transformer activations correlate with the bispectrum, too?
]]><![CDATA[How to use LLMs in lesson preparation?]]>Sun, 02 Feb 2025 06:44:22 GMThttps://www.sarahmarzen.com/random-ruminations/how-to-use-llms-in-lesson-preparationLLMs have the potential, according to some, to ruin society. But they also have the ability to help. In some inspiring stories, they can level the playing field in education, allowing those who are in badly-resourced areas to receive the great equality: education.

So how do we make use of LLMs in lesson preparation as well as possible? The main thing I realized, after a day of playing around, is that your results depend on the LLM that you use. The results also probably depend a lot on the prompt-- a good prompt engineer might make up for a bad LLM.

I have in mind that for most teachers, the LLM is a human aid-- where you have the training to come up with the lesson plans yourself, but just would appreciate a little help from an LLM to get started. For some teachers who are really strapped for time or energy, or for students who are simply using an LLM to learn, the LLM might be everything.

At first, I used ChatGPT from OpenAI to try to lesson plan. I tried ChatGPT on lessons on biopolymers and the ideal chain model after making my own lesson plan. Then I tried it on an introductory physics lecture on angular momentum. Finally, I tried it on lesson plans for understanding Newton's Principia and the role it played in science history.

In short, it failed to help. The way it failed to help was interesting. The lesson plans shoved way too much material into way too short of a time, with very little in the way of depth. The interactive activities-- because the LLM was told to make the class interactive-- were sometimes dismal. Random walks were supposed to be simulated by giving students string to play with. As a demonstration, it's not a bad idea to use string to illustrate end-to-end distance and why this is or is not a good measure of "size". But I cannot imagine students at my colleges taking seriously a lesson in which they muck around with string for longer than 30 seconds and pretend that they're understanding polymers. (What happened to actual simulations? This activity seems like something to give a fourth-grader, not an undergraduate in college.) Formulas in the angular momentum lecture were oversimplified; the cross product lost a sin and was just mvr, as if American students couldn't handle the truth. Derivations were omitted in the biopolymer and angular momentum lectures. And, as usual, some material was just wrong. No, those were not the main points in the Principia.

Of course, prompt engineering is huge with LLMs. I tried hard to get ChatGPT to give me a biopolymers lesson I could use. But after all my effort, all I got was that it might be a good idea to bring a string or cable or something into class to illustrate the random walk model when we go 2D. And that's not at all what ChatGPT said to do with the string.

I then tried DeepSeek. Relatively speaking, it aced it. Actually, its lecture on biopolymers was quite close to what I wrote down all by myself without an LLM helping in any way, and I actually built two lesson plans around two activities suggested by DeepSeek for my Great Ideas in Science class. The key with DeepSeek was that there was less breadth and more depth for the 75-minute class. More relationships were derived. The in-class activities (group attempts to collect/synthesize information and debate with each other) meant more, and seemed designed for undergraduates rather than elementary schoolers. I was able to read through DeepSeek's lesson plans and simply steal ideas, and then spruce them up a bit. That's amazing.

It is interesting that an LLM trained on Chinese data does better at helping students retain facts than a model trained on American data, in my opinion. I take from this that America has a ways to go with education. In our classrooms, it would be great if we could: emphasize more derivations; do in-class activities that were less about feeling and more about collecting, synthesizing, and interpreting information; and take longer to go through each bit of material so that the class isn't whiplash. Perhaps there's a way to fix Masters in Education programs in America so that the course preparation data on which ChatGPT is trained leads to better lesson plans.]]><![CDATA[Bayesianism and the science of science]]>Thu, 30 Jan 2025 21:15:32 GMThttps://www.sarahmarzen.com/random-ruminations/bayesianism-and-the-science-of-scienceHow rational are we? We might say that this question, revised, is, "How Bayesian are we?" Do we correctly combine probabilities, taking into account combinations of our prior beliefs and understandings of the evidence completely correctly?

If we believe neuroscientists, humans are maybe Bayesian.. If you look at the cognitive science literature, that is so clearly not true-- people invent models in which humans just randomly drop data as a correction to the Bayesian framework that keeps the Bayesian framework in place.

But perhaps the scientific community as a whole is Bayesian. After all, the scientific community is supposed to be special in some way. It is supposed to be more than the sum of its parts, not less, in its quest for what might be called "truth".

In class, we've been reading through Kuhn's take on the science of science. I wondered what happened around the time of a crisis, in which the current paradigm (what's in the textbook and what everything thinks is the right way to interpret the data) is in competition with a new potential paradigm. The process of deciding between the old paradigm and the new paradigm is messy, filled with persuasive tactics, and might not even happen. For instance, without chaos theory, our understanding from Newtonian mechanics of the perigee of the moon was off by a factor of two for decades-- but nobody abandoned Newtonian mechanics for that reason. But beyond the mess, might the actual outcome be just decided by a simple application of Bayes rule?

In this paper, they argue just that. Basically, at a scientific crisis, you start evaluating the likelihood of each potential paradigm for explaining the data, and encode your prior beliefs about the likelihood of each potential paradigm.

Part of this prior belief might be based on aesthetic or a resistance to change. But aesthetic comes into play in the likelihood too. If a paradigm can explain too much, as you would see with Ptolemaic astronomy for example, it is actually less likely in a likelihood sense than a heliocentric theory once you integrate over all possible parameters.

One question I have is: how close is the scientific community to Bayesian? Can someone evaluate this from data in some way? It'd be very hard to do so, but I already think that the structure of Kuhn's science of science prohibits pure Bayesianism for the scientific community. To be a Bayesian, you have to evaluate posterior beliefs correctly, as they become your priors, but the new paradigm is not even evaluated to have a probability before the scientific crisis sometimes. Sometimes, with special relativity, its framework is introduced prior to the revolution, but its prior is never evaluated based on all the previous data collected. Therefore, its prior is never correct, and Bayesianism can never be achieved by the scientific community.

Still-- how close, as a community, do we come?]]><![CDATA[What is the neural code?]]>Sat, 11 Jan 2025 01:27:28 GMThttps://www.sarahmarzen.com/random-ruminations/what-is-the-neural-codeThere has been a debate raging for as long as I can remember. Neurons fire in action potentials that seem stereotyped. Let's take the action potentials to be stereotyped for all practical purposes in reality. (Honestly, we should test this. We haven't quite tested this rigorously through mutual information calculations of, say, the mutual information between voltage time series relative to stimulus versus a point process encoding relative to stimulus. If the two are the same within error bars with a lot of data, the voltage itself provides no additional information basically about the stimulus compared to the spike itself.) We then must ask: what about the spikes carry the information?

Some said that the precise spike timing, exactly when the spikes occurred, determined the neural encoding of information. This code can contain quite a bit of information, if you just think about the entropy of a point process. This, however, is not a very robust code. Other people therefore proposed that firing rate-- the number of spikes over a larger time interval-- was a more robust code that could contain quite a bit of information. According to Izhikevich, who is a proponent of the spike timing hypothesis, firing rate might matter at the neuromuscular junction, but that's about it.

I think these debates ignore the fact that even though many neuroscience experiments involve presenting a static stimulus and then watching neurons respond, stimuli in real life are constantly fluctuating. Almost never do we see a movie that is static. In fact, our eye movements prohibit this by doing microsaccades all the time even if the image in front of us is static. As a result, we see constantly moving video no matter what, or we basically perceive nothing at all. So really, we are asking how to encode a constantly changing movie.

In reality, there is some dt on our perception. If you were to jitter that movie by just a bit, we wouldn't be able to perceive it. This is like when the fan goes too fast and it looks like it's continuous regardless of the speed past a certain point. So really, we have a discrete-time stimulus with a small dt that we constantly must encode.

The most natural code for that might actually be something that is neither based on spike timing or based on firing rate, really, but is effectively a binary vector. Basically, the neural response within a window of dt (the spike timing included) would be what encodes information. It seems like this could still allow for a firing rate code, but the refractory period prohibits multiple spikes in that window of dt, hence prohibiting a firing rate code for stimuli that must be constantly encoded. You could maybe see a spike timing code, but you have to weigh the amount of time it takes for an action potential to complete against the limits of sensory perception, this dt. In reality, dt depends on the sense being studied, and the correct calculation to this question might involve some understanding of the refractory period. This question might be incredibly complicated. But my money is on a binary vector not being such a bad representation of the actual neural code that is used in practice-- just, did each neuron spike or not. Thank goodness, because so many papers have used this neural code implicitly, including some of my own!

This question might be complicated a bit if a blocklength larger than 1 is used-- but that leads to time delays, which are quite costly for reinforcement learning reasons!]]><![CDATA[If corporations are psychopaths, what are charities?]]>Fri, 03 Jan 2025 20:28:30 GMThttps://www.sarahmarzen.com/random-ruminations/if-corporations-are-psychopaths-what-are-charitiesMitt Romney famously once said, in response to a town hall question, that corporations were people too. Immediately, journalists said that corporations were psychopaths.

Well, actually, this idea has some merit. An organization can be made up of good, smart people, but can act for some reason like something with a personality disorder. This is definitely related to the field of organizational psychology, about which I know very little, but I think this topic of how collective behavior of individuals makes for an organization with a different personality makeup than its individuals is badly explored mathematically.

I have been struggling with how to even begin a model of the collective behavior of the individuals that make up an organization in a way that will identify an organization's personality disorder. In fact, I think rarely does an organization lack a personality disorder. A model of this could explain everything from why some charities have way too much overhead that goes to the fat of the people running the organization to why democracy is failing.

The first mathematical model I thought of was some simplified sensory/actuator model of every person combined with a coarse-graining to find latent emotional states of the collective behavior. In reality, I think that although this is principled, it is unlikely to succeed unless we understand how to model an organism better than we already do. I sincerely hope that this approach is studied at some point in great detail-- and I mean mathematically. Just imagine that every person is modeled as a resource-constrained reinforcement learner who interacts with reward functions that depend on the people next to it, in a multi-agent reinforcement learning setting, and that we then model the behavior of the collective to find latent emotional states that can then be mapped to personality disorders with a mathematical form of the DSM. Undoubtedly, this is the way to proceed once you understand how to set it up mathematically, but on this, I give up, I think for life.

The second mathematical model I went to was a Potts model. This reminds me of the voter models in which people are modeled as Ising spins that I always thought of as being completely made up but basically okay for understanding certain behaviors. In a Potts model, collective behavior is modeled as interactions between particles that can adopt one of N discrete states. These discrete states could be one of several personality types. You then define some sort of interaction energy between these spins that can govern dynamics under several different models, but usually just governs the state into which the collective settles. A renormalization group analysis might then find that the collective, upon decimating using a majority opinion vote or the like, adopts a different discrete state of the Potts model than one might expect. The key is that the interaction energy might lead to frustration or a flipping of states, so that even if the collective starts out as good and smart, it ends up as a narcissist (perhaps a charity with too much overhead and grandiose statements about how much they do) or a psychopath (most corporations, who will screw over their workers for a payday).

In non-mathematical terms, this comes down to saying that the organizational structure is specified by an interaction energy between particles. This includes an understanding of the lattice structure and how far away particles can interact (if they are in an office such that only people at the same desk talk or if there's some movement generally so that one side of the office talks to the other), if there is a mean-field ordering from a mission statement, if separate orders are given to separate parts of the organization so that there are different mean-fields for different parts of the organization, if there are leaders that unduly influence the spins and are themselves stubborn, if disagreement is encouraged or discouraged which could lead to frustration or alignment.

One day, I hope to come back to this mathematical idea when I understand more about personality. In the meantime, if you have a way to turn this into something, please do!
]]><![CDATA[A minimal model of a social system]]>Tue, 24 Dec 2024 20:05:00 GMThttps://www.sarahmarzen.com/random-ruminations/a-minimal-model-of-a-social-systemHow do beliefs of society evolve? We typically see a pendulum swinging back and forth, maybe slowing down as it reaches equilibrium. Sometimes, beliefs appear driven, as when polarization on issues that are essentially either Republican or Democratic swing back and forth more and more violently between extremes. How can we, as a society, intervene so that we reach a desired equilibrium faster?

For example, let's take the case of an issue we all care about-- racism. There's racism, and there are people that claim that there is "reverse racism", which only makes sense if you ignore that racism requires a systemic oppression from society. But still, we could imagine that a college admissions process favors white applicants or black applicants. We seem to be swinging back and forth between those two extremes with court case and societal movement after court case and societal movement. Wouldn't it be nice if we could find an intervention or interventions on society that slows down the pendulum with just the right amount of drag so we reach color-blindness as fast as possible? Research is accumulating that shows that implicit bias training does little, as many people hate being told that they're actually racist, so is there an intervention that works?

For this, we would like to make a model of the system that describes the evolution of societal beliefs. This is a coarse-graining of the overall belief dynamics of everyone in society, which can get quite complicated and even show potentially chaotic behavior when you add in enough cognitive biases. However, if we do a Taylor expansion about an equilibrium point-- which may not be warranted-- we will find that there is a linear dynamical system with potentially weird Gaussian noise that describes how the state of society evolves. When the belief is binary, this will approximately take the form of a mass on a spring with damping hit by particles randomly. The key, then, is to relate interventions to the spring constant, the mass, the drag coefficient, and the temperature by correctly Taylor expanding. This is impossible except in a toy model, for now.

There is a critical damping that will get us to equilibrium as fast as possible. In theory, we merely need to solve for it.

What if things explode? We see this in some social systems, as the number of papers in a field for example (https://www.sciencedirect.com/science/article/pii/S0303264720301015) explodes. A simple linear model might explain a great deal of social science phenomena.
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