83
votes
Accepted
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
What are the state-of-the-art algorithms for long-integer multiplication?
First let me address the point you raised about the schoolbook algorithm having $\mathcal{O}(n^2)$ scaling, by saying that ...
- 33.6k
35
votes
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
This $O(n\ln n)$ integer multiplication algorithm is a galactic algorithm, meaning that it won't be used despite being "of lower complexity" because it only becomes more efficient than ...
- 451
27
votes
Accepted
What is a Padé approximant?
To complement mykd's already excellent answer, I will just add that the approximant we all learn in school (the Taylor approximant) is nice to teach and easy to help students learn the concept of ...
- 33.6k
24
votes
Accepted
Do we know for sure that all atomic and molecular wavefunctions decay exponentially as r goes to infinity?
I'll answer this question from the theoretical side. The exponential behavior follows simply from the Schrödinger equation. Consider the one-electron Schrödinger equation:
$$
(-\frac{1}{2}\nabla^2 + V(...
- 9,076
23
votes
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
To take a slight detour, we can also look at the progress of matrix multiplication algorithms. As mentioned in a few comments here, standard matrix multiplication is $O(n^{3})$ and any exact method ...
- 15.2k
20
votes
What is a Padé approximant?
PADE refers to the mathematician Henri Padé, who developed the Padé approximation, which can approximate a function using rational polynomial functions.
For ex:
$$
\sin(x)\approx {\frac {(12671/...
- 2,262
19
votes
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
Can someone explain in detail the impact of any of the multiplication algorithms scaling better than N2, for some practical application?
An actual application is right in front of our eyes: digital ...
- 291
18
votes
Accepted
Introduction to protein folding for mathematicians
Great question! Protein folding has been in open question for decades. Just recently, there's been a lot of discussion regarding DeepMind's AlphaFold project, which was discussed at length on our very ...
- 4,719
16
votes
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
Even the simplest better-than-schoolbook (O(n^2)) algorithms like Karatsuba are only useful in practice for large n. But what is ...
- 461
15
votes
Accepted
How are continued fractions related to quantum materials?
In the paper "A Continued-Fraction Representation of the Time-Correlation Functions", generalized susceptibilities and transport coefficients for materials are obtained using a continued-...
- 33.6k
15
votes
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
Practical importance: compactifying explanations
It is widely believed that $\mathcal{O}(n \log n)$ is the best possible result, and therefore we no longer have to say $\mathcal{O}(n\log n\cdot 2^{2\...
15
votes
Accepted
Artificial intelligence is a hot topic, but should I pursue it if I'm interested in Matter Modeling?
"I am a student now and should decide my major to research soon. AI/ML is a very hot research field and I am very interested in it, but I have some doubts before I study AI/ML."
AI is ...
- 33.6k
14
votes
Accepted
How is group theory used to deduce which of these integrals are equal to 0?
The question is too broad to be answered directly so I will provide a somewhat general scheme.
Basically in an integral like
$$
\int d\mu A B C
$$
one would seek to expand each part in irreducible ...
- 256
13
votes
Accepted
What are the applications of chemical graph theory?
This is a question that is answered by a straightforward literature search, here's e.g. two review papers from the 1980s:
J. Chem. Inf. Comput. Sci. 25, 334 (1985)
J. Chem. Educ. 65, 574 (1988)
as ...
- 18k
13
votes
How are continued fractions related to quantum materials?
You may take a look at the Method of Continued Fractions used in quantum scattering theory—this was only formed in 19831 so is rather recent. Related is the PhD thesis by Kónya (2000)2; §3.3 ...
- 1,607
13
votes
Accepted
What are the modelling techniques that can be used for simulating microstructure evolution in materials?
I could list the models that could be used for microstructural modeling as:
Phase-Field: It is constructed based on non-equilibrium thermodynamics and Onsager reciprocal relations to derive a ...
- 2,934
12
votes
Accepted
How do you calculate the "true" chemical potential in classical density functional theory?
Seeing that this question has gathered attention but no replies, I will give it a stab. Note that I am not an expert on DFT or functional calculus, so take this with a grain of salt. As usual, ...
- 1,918
12
votes
Introduction to protein folding for mathematicians
Someone more familiar with the problem might have a better suggestion, but I recently came across Daniel B. Dix' notes on Mathematical Models of Protein Folding. This is not my field, so I won't ...
- 4,711
11
votes
What are the applications of chemical graph theory?
I'd just briefly add that among other things, chemical graph theory is extensively used for computing so-called "molecular descriptors", which are used to capture some "common ...
11
votes
What are the advantages of the Davidson diagonalization method over other sparse matrix diagonalization methods?
Following Nike Dattani's suggestion, I now add some comparisons of the Davidson method with methods that are more "modern" than it, to supplement the existing answers which compared the ...
- 9,076
11
votes
Accepted
What does it mean when the first order correction energy is 0?
The first-order correction $\langle \psi_0|H|\psi_0\rangle$ vanishes since $H|\psi_0 \rangle$ is orthogonal to $|\psi_0\rangle$. You need to go higher in perturbation theory to get a correction that ...
- 18k
10
votes
Introduction to protein folding for mathematicians
Preamble
Since I don't know your specific background, this is a generic answer for any applied mathematician wishing to enter the field of protein folding. Not everything will apply specifically to ...
- 33.6k
10
votes
Accepted
Mathematical models for the plastic region in the tensile test
Plasticity is still an actively researched area. The Ludwik-Hollomon equation is one model that is used for the strain-hardening region since it captures the convex shape of the curve using a power ...
- 1,349
9
votes
What are the applications of chemical graph theory?
Some recent work that comes to my mind is generation and analysis of fullerenes done by Peter Schwerdtfeger's group and described for example in
J. Comput. Chem. 34 1508 (2013)
WIREs 5, 96 (2015)
- 766
9
votes
How is group theory used to deduce which of these integrals are equal to 0?
First, you have to transform all the basis functions into the irreducible representations (irreps) of the point group of the molecule. You can do this with standard projection formulas.
Once, you know ...
- 1,431
9
votes
What are the applications of chemical graph theory?
Too many references for me to elaborate on, but here's a recent example I like: "Graph Networks as a Universal Machine Learning Framework for Molecules and Crystals" in Chemistry of ...
- 7,281
9
votes
Accepted
Is it possible to get complex numbers as solutions for a secular determinant in simple Hückel method (SHM)?
In the Huckel method, you are just generating a very simplified version of the molecular Hamiltonian and determining it's eigenvalues. The molecular Hamiltonian will always be a Hermitian matrix (for ...
- 15.2k
9
votes
Accepted
What are the advantages of the Davidson diagonalization method over other sparse matrix diagonalization methods?
The Davidson-Method is the best algorithm when you want a comparatively small number of eigenvectors/eigenvalues from a sparse, diagonally dominant matrix.
MKL doesn’t have a sparse matrix ...
- 7,957
9
votes
Accepted
Given the adjacency matrix of a molecule, how can I get a graphical representation using only open source software?
NetworkX
This is a widely-used Python library for graph theory stuff. Open source (https://github.com/networkx/networkx) under a 3-clause BSD license.
Caveat: since this is a tool that knows nothing ...
- 2,027
9
votes
Inconsistencies in a famous point group table
After hours of conversation with three participants including me, I can make the following conclusions.
Answer to the question
It's likely an unintentional "error" in the publication. In ...
- 33.6k
Only top scored, non community-wiki answers of a minimum length are eligible