lowdanie

So at this level of simulation it is currently only possible to simulate one medium size molecule or the interaction of a few small ones.

To simulate larger systems, it is necessary to work at a (semi-)classical level of abstraction that approximates quantum mechanics. For example using molecular dynamics to essentially simulate a fluid with a ball and springs model. In this case, electron level simulation can still be useful for deriving heuristics (conceptually, the spring tension).

I completely agree that it’s interesting to investigate how far the electron level simulation can be pushed.

I’m planning to add support for an alternative method called density functional theory which gives better results for molecular interaction.

In cached mode, it can currently jit compile the graph for molecules of around 10 atoms in ~5 minutes on one T4 gpu. Once the graph is compiled, the actual geometry optimization only takes a few seconds.

I’m working on optimizations that improve the scaling behavior (such as density fitting) with the goal of achieving similar or even better performance for molecules with ~50 atoms.

Of course, the true wave function is generally not a Slater determinant. In particular, electrons in a Slater determinant with different spins are uncorrelated.

The standard approach to resolving this is density functional theory. In that model, the main approximation is the choice of an “exchange correlation functional” which approximates the electron exchange and correlation energy. The choice of a functional is unfortunately a dark art in the sense that they can only be evaluated empirically rather than from first principles.

The classic reference for Hartree Fock is Modern Quantum Chemistry by Szabo and Ostland: https://books.google.com/books/about/Modern_Quantum_Chemistr...

It is very well written and I highly recommend it.

I also wrote up some notes here: https://www.daniellowengrub.com/blog/2025/07/26/scf

But for the colab demo I thought that sticking to nuclear positions (i.e atomic forces) would be easier to visualize.

For example, you could use an additively homomorphic scheme to compute a sum of encrypted values. This could then be converted into an average assuming you knew the number of values.

I completely agree that it is possible to homomorphically evaluate b - <a,s> much more easily using homomorphic addition and multiplication. However, wouldn't we still need some sort of blind rotate magic to bound the output error?

But finding a correlated gene is only the first step. One issue is that a single protein can participate in hundreds of seemingly unrelated chemical reactions throughout the body depending on the cell type and environment. So simply tweaking the genes expression will have many unintended consequences.

For instance, each cell is constantly maintaining a baffling complex balance between growing and dying. Any external perturbation has a good chance of either killing the cell or causing cancer.

In addition, B and T cells only detect fragments on the surface of active Dendric cells. Dendric cells become active in response to alternate and less specific indications of infection such as an unusually high amount of mRNA translation or families of protein that occur only in bacteria and viruses.

2. Yes, the mRNA is ultimately degraded. But part of the process of developing the vaccine is fine tuning the precise rna sequence in order to achieve the desired half life.

One issue I noticed: I looked up Chana Masala and the first recipe (Vegan Chana Masala) calls for “12 tsp salt” but the source calls for “1/2 tsp salt” :)

For example, you could write down axioms for the real numbers by specifying that there should be relations called + and x with the standard properties such as commutativity, as well as an ordering relation < such that for all elements x and y there is an element z for which: x < z < y.

Clearly the real numbers are a model for these axioms. But as it turns out the countable set of rational numbers is a model as well.

The general setting for this type of algorithm is that you have a large number of states and you want to find the state that maximizes a particular score. In the TSP, the states are paths and the score is the (negative of the) path length. The idea is to define a Markov Chain whose stationary distribution is proportional to the score. This means that if you jump from state to state according to the probabilities defined by the MC, eventually the probability of ending up at a given state is proportional to that states score.

In the case of TSP the states can be represented as permutations of the cities and the neighbors of a state are given by swapping the positions of two cities. The probability of moving to a neighbor is high if the length of the neighbor is shorter than the current path.

[1] https://en.wikipedia.org/wiki/Simulated_annealing

In contrast, since we know when each person in the treatment group got the vaccine, safety can be measured after a predetermined amount of time.

I mostly discuss interesting mathematics that has applications to engineering.

Regarding the question, it is true that the received pulse will be shorter or longer than the transmitted one (depending on whether the object is moving towards or away from the sensor) but since we are assuming that the speed of the object is significantly less than the speed of light, we're making the simplifying assumption that this difference is negligible. People sometimes refer to this as the "stop and hop" assumption since in practice we are assuming that the object "stops" at point x when the radio wave is emitted and then "hops" to x + dt*v once the wave has been received dt seconds later.

Regarding the number of cycles, you are again correct that the image represents a gross oversimplification. One reason for this is that since the carrier frequency is so much higher than the doppler frequency, the true graph of the received wave would not be noticeably different than the transmitted one. It is easier to plot a realistic graph of the demodulated wave which you can see in the plot titles "samples of the demodulated pulse".