Concertina wire by Ciro Santilli 37 Updated +Created
Complete metric space by Ciro Santilli 37 Updated +Created
In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.
One notable example where completeness matters: Lebesgue integral of is complete but Riemann isn't.
Atwood machine by Ciro Santilli 37 Updated +Created
Continuous mechanics problem by Ciro Santilli 37 Updated +Created
Matrix inverse by Ciro Santilli 37 Updated +Created
When it exists, which is not for all matrices, only invertible matrix, the inverse is denoted:
Eric W. Weisstein by Ciro Santilli 37 Updated +Created
Ahh, this dude is just like Ciro Santilli, trying to create the ultimate natural sciences encyclopedia!
In 1995, Weisstein converted a Microsoft Word document of over 200 pages to hypertext format and uploaded it to his webspace at Caltech under the title Eric's Treasure Trove of Sciences.
List of markup languages by Ciro Santilli 37 Updated +Created
Artificial intelligence by Ciro Santilli 37 Updated +Created
Natural language processing by Ciro Santilli 37 Updated +Created
An impossible AI-complete dream!
It is impossible to understand speech, and take meaningful actions from it, if you don't understand what is being talked about.
And without doubt, "understanding what is being talked about" comes down to understanding (efficiently representing) the geometry of the 3D world with a time component.
Not from hearing sounds alone.
Image generation by Ciro Santilli 37 Updated +Created
Milky Way by Ciro Santilli 37 Updated +Created
Figure 1.
Arms of the Milky Way
. Source.
Scientology by Ciro Santilli 37 Updated +Created
Dual space by Ciro Santilli 37 Updated +Created
The dual space of a vector space , sometimes denoted , is the vector space of all linear forms over with the obvious addition and scalar multiplication operations defined.
Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the , and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.
One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space .
Infinitesimal generator by Ciro Santilli 37 Updated +Created
Elements of a Lie algebra can (should!) be seen a continuous analogue to the generating set of a group in finite groups.
For continuous groups however, we can't have a finite generating set in the strict sense, as a finite set won't ever cover every possible point.
But the generator of a Lie algebra can be finite.
And just like in finite groups, where you can specify the full group by specifying only the relationships between generating elements, in the Lie algebra you can almost specify the full group by specifying the relationships between the elements of a generator of the Lie algebra.
This "specification of a relation" is done by defining the Lie bracket.
The reason why the algebra works out well for continuous stuff is that by definition an algebra over a field is a vector space with some extra structure, and we know very well how to make infinitesimal elements in a vector space: just multiply its vectors by a constant that cana be arbitrarily small.
Generator of a Lie algebra by Ciro Santilli 37 Updated +Created
Software license by Ciro Santilli 37 Updated +Created
Real world applications of the Lebesgue integral by Ciro Santilli 37 Updated +Created
In "practice" it is likely "useless", because the functions that it can integrate that Riemann can't are just too funky to appear in practice :-)
Its value is much more indirect and subtle, as in "it serves as a solid basis of quantum mechanics" due to the definition of Hilbert spaces.
German learned society by Ciro Santilli 37 Updated +Created

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