Multivariate polynomial where each term has degree 2, e.g.:is a quadratic form because each term has degree 2:but e.g.:is not because the term has degree 3.
There is a 1-to-1 relationship between quadratic forms and symmetric bilinear forms. In matrix representation, this can be written as:where contains each of the variabes of the form, e.g. for 2 variables:
Strictly speaking, the associated bilinear form would not need to be a symmetric bilinear form, at least for the real numbers or complex numbers which are commutative. E.g.:But that same matrix could also be written in symmetric form as:so why not I guess, its simpler/more restricted.
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).
Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.
Examples:
- In this case, the normal derivative at the boundary is proportional to the difference between the temperature of the boundary and the fixed temperature of the external environment.The result as time tends to infinity is that the temperature of the plaque tends to that of the environment.
www.cbpp.org/wealth-concentration-has-been-rising-toward-early-20th-century-levels-2 shows historical for top 1% and 0.5% from 1920 to 2010.
TODO why is it so hard to find a proper cumulative distribution function-like curve? OMG. This appears to be also called a Lorenz curve.
Wealth Inequality in America by politizane
. Source. Every invertible matrix can be written as:where:Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.
- is a diagonal matrix containing the eigenvalues of
- columns of are eigenvectors of
Intuitively, Note that this is just the change of basis formula, and so:
- changes basis to align to the eigenvectors
- multiplies eigenvectors simply by eigenvalues
- changes back to the original basis
This basically adds one more ingredient to partial differential equations: a function that we can select.
And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?
Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.
The literal Chinese name says it all: "Fake Mountain". The stones evoke the feeling of the beautiful rock mountains of China.
The term "奇石假山" (qi2 shi2 jia3 shan1, lit. "weird shaped stone fake mountain") is also used, almost as a synonym by many people, since the stones are often chose in interesting shapes. Choosing the right stone is basically an art form in itself.
The stones used are generally limestone, which as a sedimentary rock is weaker, and more likely to be eroded into interesting shapes.
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 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . 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. 
- 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