Scalars, Vectors, Matrices,Tensors, Holors....

[billtodd]

Is there something beyond Holors?

 

[FactChecker]

There are different mathematical properties associated with a your list of objects, so it is difficult to know what direction of "beyond" you are asking about. I have been interested lately in the multivectors and n-vectors of Geometric Algebra.

 

[fresh_42]

Is there something beyond Holors?

I have no idea what holors are, but the rest of your list is all tensors. So your question is actually: Is there something beyond tensors? Simple answer: everything that isn't linear.

 

[billtodd]

There are different mathematical properties associated with a your list of objects, so it is difficult to know what direction of "beyond" you are asking about. I have been interested lately in the multivectors and n-vectors of Geometric Algebra.

I heard of the book by Lasenby, but never found the time to read it.
https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521715954?tag=pfamazon01-20

The bot suggests
Holors are a broad concept encompassing scalars, vectors, matrices, and tensors—essentially, they cover multi-dimensional arrays of any rank. The idea of a holor is already quite general, but there are a few concepts that could be considered "beyond" holors in the sense of extending or generalizing their framework:

1. Multilinear Maps and Tensors in Infinite Dimensions:​

• Multilinear Maps: While tensors (or holors) are finite-dimensional objects, multilinear maps can be considered an extension into infinite dimensions. These are functions that take multiple vectors from potentially infinite-dimensional vector spaces and produce a scalar. This is common in functional analysis and can be seen in the study of Hilbert spaces or Banach spaces.
• Infinite-Dimensional Tensors: In some advanced contexts, such as quantum field theory, one might deal with objects that have properties similar to tensors but live in infinite-dimensional spaces.

2. Hypermatrices:​

• Hypermatrix: A hypermatrix is a higher-dimensional generalization of a matrix, which could be considered as a holor of rank 3 or more. However, the term hypermatrix is sometimes used to emphasize specific properties or operations on these higher-dimensional objects that go beyond standard tensor operations.
• Operations on Hypermatrices: Research in this area involves generalizing matrix operations like multiplication, eigenvalues, and decompositions to hypermatrices, which can involve more complex algebraic structures.

3. Category Theory and Functors:​

• Category Theory: In category theory, objects and morphisms (functions) are abstracted further, and one can define structures that generalize holors. Functors in category theory map between categories, and higher-order structures like natural transformations generalize morphisms between functors. This is more abstract and deals with relationships between different mathematical structures, which can extend beyond the concept of holors.
• Higher Categories: In higher category theory, the concept of morphisms between objects can be extended to morphisms between morphisms (and so on), leading to very abstract and generalized structures that go beyond the traditional concept of holors.

4. Non-Linear Generalizations:​

• Nonlinear Tensors: In some advanced mathematical frameworks, there are generalizations where the linear structure is relaxed. This leads to nonlinear analogs of tensors, where the underlying spaces are not necessarily vector spaces, but more general topological spaces or manifolds.
• Tensor Networks: In quantum computing and machine learning, tensor networks represent complex, interconnected systems of tensors. These networks can encode higher-order interactions and correlations beyond traditional holors.

5. Algebraic Structures:​

• Grassmannians and Flag Varieties: These are spaces that parametrize all linear subspaces of a vector space. They involve complex algebraic and geometric structures that generalize the concept of vector spaces and thus indirectly extend the idea of holors.
• Tensor Categories: These are categories where objects are tensors and morphisms respect the tensor product structure. Tensor categories are used in algebraic geometry, representation theory, and quantum field theory to generalize the concept of holors.

Summary​

While holors are already a very general concept encompassing a wide range of mathematical objects, there are indeed more abstract and complex structures in mathematics and theoretical physics that extend or generalize the idea of holors. These include infinite-dimensional analogs, higher-dimensional categories, nonlinear generalizations, and advanced algebraic structures that go beyond the traditional framework of holors. These areas of study often require sophisticated mathematical tools and are applied in fields like quantum mechanics, advanced geometry, and category theory.

Garssmannians I heard from my Algebraic Topology course I took a decade ago.
Well I guess one can always get higher and higherer in pure mathematics.

 

[fresh_42]

The bot suggests

AI is one of the worst things you can ask when it comes to science. It sells you the illusion of an answer without actually answering anything. Even Wikipedia is a far better reference. AIs make terribly lousy mistakes and don't understand a thing. I really recommend not using them.

There is a reason why we use textbooks!

And by the way require them as valid references versus "something on the internet".

 

[FactChecker]

AI is one of the worst things you can ask when it comes to science. It sells you the illusion of an answer without actually answering anything. Even Wikipedia is a far better reference. AIs make terribly lousy mistakes and don't understand a thing. I really recommend not using them.

There is a reason why we use textbooks!

And by the way require them as valid references versus "something on the internet".

And AI can not even weed out other AI-generated results on the internet. AI results will become like the secret that gets passed around in a circle and ends up vastly distorted.

 

[fresh_42]

And AI can not even weed out other AI-generated results on the internet. AI results will become like the secret that gets passed around in a circle and ends up vastly distorted.

Have you heard of holors before?

 

[berkeman]

Thread closed for Moderation...

 

[berkeman]

OP has been reminded that AI chatbots are not acceptable references in the technical forums at PF. For a number of reasons, the OP is on a temporary vacation from PF, and this thread will remain closed.