The concepts of **Hilbert series** and **Hilbert polynomial** arise primarily in algebraic geometry and commutative algebra, particularly in the study of graded algebras and projective varieties. ### Hilbert Series The **Hilbert series** of a graded algebra (or a graded module) is a generating function that encodes the dimensions of its graded components.
In the context of mathematics, particularly in abstract algebra, a **perfect ideal** is a concept that can arise in the theory of rings. However, the term "perfect ideal" is not standard and could be used in various contexts with slightly different meanings depending on the specific area of study.
A quasi-homogeneous polynomial is a type of polynomial that exhibits a certain kind of symmetry in terms of its variable degrees. Specifically, a polynomial \( f(x_1, x_2, \ldots, x_n) \) is called quasi-homogeneous of degree \( d \) if it can be expressed as a sum of terms, each of which has the same "weighted degree".
She's truly passionate about health research and keeping healthy, almost obsessed by it. Also she's strong willed, and energetic. Good traits for founding 23andMe.
- www.vanityfair.com/style/2014/04/sergey-brin-amanda-rosenberg-affair Fantastic painting of the people.
As www.nytimes.com/2017/11/18/style/anne-wojcicki-23andme-genetics.html puts it well:
The Wojcickis grew into Silicon Valley royalty. It’s the sort of family, Anne jokes, where “you’re only a viable fetus once you have your Ph.D.
Anne Wojcicki interview by Talks at Google (2018)
Source. She's athletic! As mentioned at: www.vanityfair.com/style/2014/04/sergey-brin-amanda-rosenberg-affair. And despite the name, and unlike Sergey, she's completely american as seen from her perfect accent!- youtu.be/pDoALM0q1LA?t=173 coding on garage while they do dishes and burritos
- youtu.be/pDoALM0q1LA?t=331 why she's obsessed with healthcare. Also mentioned at: www.vanityfair.com/style/2014/04/sergey-brin-amanda-rosenberg-affair how she was trying to save Sergei from some of his genetic predispositions
- youtu.be/pDoALM0q1LA?t=571 she really cared about 23andMe, but the public didn't as much as her. She's truly passionate about mining genetic data. Maybe she came a bit early.
- youtu.be/pDoALM0q1LA?t=1038 doc in a box workaround
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) shows that this is a tensor that represents the volume of a parallelepiped.
It takes as input three vectors, and outputs one real number, the volume. And it is linear on each vector. This perfectly satisfied the definition of a tensor of order (3,0).
Run output is placed under
out/:Some of the output data is stored as
.cpickle files. To observe those files, you need the original Python classes, and therefore you have to be inside Docker, from the host it won't work.We can list all the plots that have been produced under Plots are also available in SVG and PDF formats, e.g.:
out/ withfind -name '*.png'The output directory has a hierarchical structure of type:where:
./out/manual/wildtype_000000/000000/generation_000000/000000/wildtype_000000: variant conditions.wildtypeis a human readable label, and000000is an index amongst the possiblewildtypeconditions. For example, we can have different simulations with different nutrients, or different DNA sequences. An example of this is shown at run variants.000000: initial random seed for the initial cell, likely fed to NumPy'snp.random.seedgenereation_000000: this will increase with generations if we simulate multiple cells, which is supported by the model000000: this will presumably contain the cell index within a generation
We also understand that some of the top level directories contain summaries over all cells, e.g. the
massFractionSummary.pdf plot exists at several levels of the hierarchy:./out/manual/plotOut/massFractionSummary.pdf
./out/manual/wildtype_000000/plotOut/massFractionSummary.pdf
./out/manual/wildtype_000000/000000/plotOut/massFractionSummary.pdf
./out/manual/wildtype_000000/000000/generation_000000/000000/plotOut/massFractionSummary.pdfEach of thoes four levels of
plotOut is generated by a different one of the analysis scripts:./out/manual/plotOut: generated bypython runscripts/manual/analysisVariant.py. Contains comparisons of different variant conditions. We confirm this by looking at the results of run variants../out/manual/wildtype_000000/plotOut: generated bypython runscripts/manual/analysisCohort.py --variant_index 0. TODO not sure how to differentiate between two different labels e.g.wildtype_000000andsomethingElse_000000. If-vis not given, a it just picks the first one alphabetically. TODO not sure how to automatically generate all of those plots without inspecting the directories../out/manual/wildtype_000000/000000/plotOut: generated bypython runscripts/manual/analysisMultigen.py --variant_index 0 --seed 0./out/manual/wildtype_000000/000000/generation_000000/000000/plotOut: generated bypython runscripts/manual/analysisSingle.py --variant_index 0 --seed 0 --generation 0 --daughter 0. Contains information about a single specific cell.
E.g. with our qiskit/hello.py, we obtain the Bell state circuit:
OPENQASM 2.0;
include "qelib1.inc";
qreg q[2];
creg c[2];
h q[0];
cx q[0],q[1];
measure q[0] -> c[0];
measure q[1] -> c[1];Requires entangled particles, unlike BB84 which does not.
These people are cool.
They use optical tweezers to place individual atoms floating in midair, and then do stuff to entangle their nuclear spins.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact