One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of .
Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.
As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.
The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.
For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.
The questions are: who is this Mark, and why does he have to go down?
And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.
They are the finite projective special linear groups.
This was the first infinite family of simple groups discovered after the simple cyclic groups and alternating groups. The first case discovered was by Galois. You should understand that one first.
For the typical case of a linear form over , the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.
Examples:
- a 2x2 matrix can represent a linear map from to , so which is a linear operator
- the derivative is a linear map from to , so which is also a linear operator
Ciro Santilli has felt that perhaps what is missing in 2020's AGI research is:
- the interface between:The key question is somewhat how to extract symbols out of the space-time continuous experiences.
- the continuous/noisy level (now well developed under artificial neural network techniques of the 2010's)
- and symbolic AI level AI
- more specialized accelerators that somehow interface with more generic artificial neural networks. Notably some kind of speialized processing of spacial elements is obviously hardcoded into the brain, see e.g. Section "Grid cell"
Forcing these boundaries to be tested was one of the main design goals of Ciro's 2D reinforcement learning games.
In those games, for example:Therefore, those continuous objects would also have "magic" effects that could not be explained by "simple" "what is touching what" ideas.
- when you press a button here, a door opens somewhere far away
- when you touch certain types of objects, a chemical reaction may happen, but not other types of objects
Bibliography:
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2026-08: turns on to black screen: Start Up Failed: Your device didn't start up successfully.
2023-10: unable to connect to Giffgaff. "Network unavailable". Same SIM works in other phones. So annoying. Update APN to match: www.giffgaff.com/help/articles/internet-apn-settings-guide. Went next to a tower and then got signal. So the receiver is much worse than the pixel one. Vibration appears to be broken.
Analogous to a linear form, a bilinear form is a Bilinear map where the image is the underlying field of the vector space, e.g. .
Some definitions require both of the input spaces to be the same, e.g. , but it doesn't make much different in general.
The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.
This section is about ideas that are thought to be part of an AGI system.
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