The universe is a song, singing itself.

No, really. The solutions of the Schrödinger Equation are harmonics, just like musical notes.

Quantum state of an electron orbiting a hydrogen atom where n=6, l=4, and m=1 (spin doesn’t matter):

This is an equal superposition of the |3,2,1> and |3,1,-1> eigenstates:

This is an equal superposition of the |3,2,2> and |3,1,-1> eigenstates:

This is an equal superposition of the |4,3,3> and |4,1,0> eigenstates:

What is an eigenstate? It’s a convenient state to use as a basis. We get to decide which quantum states are “pure” and which “mixed”. There’s an easy way and a hard way; the easy way is to use eigenstates as the pure states.

More mathematically: the Schrödinger equation tells us what’s going on in an atom. The answers to the Schrödinger equation are complex and hard to compare. But phrasing the answers as combinations of eigenfunctions makes them comparable. One atom is 30% state A, 15% state B, 22% state C … and another atom is different percentages of the same states.

Just like vectors in 3-D space, where you can orient the axes differently — you can pick different directions for x, y, and z to point in. But now the vectors are abstract, representing states. Still addable so still vectors. Convex or linear combinations of those “pure” states describe the “mixed” states.

Related:

SOURCE: Atom in a Box

transformations on transformations on transformations

part 5 of linear algebra (toc)

stacks on stacks on stacks. racks on racks on racks. cats in hats on knox in box. Hey classy people, we’re rocking diagonalisation and markov chains today! (That means numbers. great.) Inherently included is matrix representation and the change of basis matrix mindboggle… I’ll briefly touch on those.

So we’ll be a bit numerical this time around. Let’s talk matrices. (I want to get through this stuff quickly so…speed time aiight guys?) How do we represent a transformation in Euclidean space with a matrix? It’s kind of an interesting question. For one, how do we know we can? For two, how do we know the matrix representation is unique?

Read More

plus ça change, plus c'est la même chose

part 4 on linear algebra (toc)

It’s been a while. whoops. So I’m going to do eigenvalues and eigenvectors today. These babies are an example of something that legitimately gets applied quite a lot outside of the realm of math, in (system? controls?) engineering. But…hey, we’re all math majors here. Who cares about practical application?

(Side note for anyone who cares: I will be flying back to the rainy west tomorrow, and school starts in a week for me. We’ll be hurling into complex analysis and abstract algebra territories soon. Can you barely contain your excitement? Because I can’t. It’s exploding out of me like acid reflux.)

Read More

alright, I have to be at work within 3 hours. I forgot about that. So I’m gonna close tumblr now and try to power through getting this homework assignment done.

I still am having issues with eigenvalues/eigenvectors and how they relate to residual error surfaces but w/e, I’ve got like a thousand illegally downloaded textbooks and no time, I’m sure I’ll figure something out.

My hive plot visualization of a non-profit organization’s partners, based on survey data. Each node is a partner, and the edges between two nodes are weighted by the number of responses the partners had in common; edge weights were normalized. Nodes’ colors and placements were determined by their eigenvectors, and their sizes by closeness centrality. Edges’ colors and widths corresponded to their normalized weights. Axes were assigned by hierarchical clusterings of average shortest paths (counter-clockwise from top). The R programming was poached heavy-handedly from Vessy’s Les Mis example. Next step: adding interactivity.

View this on github

I have no idea what an eigenvector is, but it sounds awesome.

I can’t wait to take linear algebra lol. and differential equations.

#RNA Graph Partitioning for the Discovery of #RNA Modularity: A Novel Application of Graph Partition Algorithm to Biology

by Namhee Kim, Zhe Zheng, Shereef Elmetwaly, Tamar Schlick Graph representations have been widely used to analyze and design various economic, social, military, political, and biological networks. In systems biology, networks of cells and organs are useful for understanding disease and medical treatments and, in structural biology, structures of molecules can be described, including #RNA structures. In our #RNA-As-Graphs (RAG) framework, we represent #RNA structures as tree graphs by translating unpaired regions into vertices and helices into edges. Here we explore the modularity of #RNA structures by applying graph partitioning known in graph theory to divide an #RNA graph into subgraphs. To our knowledge, this is the first application of graph partitioning to biology, and the results suggest a systematic approach for modular design in general. The graph partitioning algorithms utilize mathematical properties of the Laplacian eigenvector (µ2) corresponding to the second eigenvalues (λ2) associated with the topology matrix defining the graph: λ2 describes the overall topology, and the sum of µ2′s components is zero. The three types of algorithms, termed median, sign, and gap cuts, divide a graph by determining nodes of cut by median, zero, and largest gap of µ2′s components, respectively. We apply these algorithms to 45 graphs corresponding to all solved #RNA structures up through 11 vertices (∼220 nucleotides). While we observe that the median cut divides a graph into two similar-sized subgraphs, the sign and gap cuts partition a graph into two topologically-distinct subgraphs. We find that the gap cut produces the best biologically-relevant partitioning for #RNA because it divides #RNAs at less stable connections while maintaining junctions intact. The iterative gap cuts suggest basic modules and assembly protocols to design large #RNA structures. Our graph substructuring thus suggests a systematic approach to explore the modularity of biological networks. In our applications to #RNA structures, subgraphs also suggest design strategies for novel #RNA motifs.

http://bit.ly/1qhbYUi #PLoS
Text
Photo
Quote
Link
Chat
Audio
Video