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  • Aug 17 11:56

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Fixed limits where expressions … (compare)

  • Aug 14 11:01

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Removed simplify call for sign(… (compare)

  • Aug 14 02:48

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Added tests for all asymptotic… (compare)

  • Aug 14 01:27

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Refactored limits for dealing w… (compare)

  • Aug 13 06:02

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Fixed errors arising for limits… (compare)

  • Aug 12 03:51

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    added bottom_up simplification … (compare)

  • Aug 12 03:05

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    added testing for changes adde… (compare)

  • Aug 12 02:57

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    removed condition from leading … (compare)

  • Aug 09 10:29

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    minor changes (compare)

  • Aug 09 05:13

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Fixed failing test (compare)

  • Aug 09 01:30

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Fixed code quality failure (compare)

  • Aug 05 12:45

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    included exp forms for bessel f… (compare)

  • Aug 05 12:05

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Minor change (compare)

  • Aug 05 11:59

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Refactored mrv_leadterm in grun… (compare)

  • Aug 01 03:22

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Added e.is_negative case for be… (compare)

  • Aug 01 02:43

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    hanled e.is_negative case for b… (compare)

  • Aug 01 02:03

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Replacing exp rewrite with trig… (compare)

  • Jul 31 11:41

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Minor changes (compare)

  • Jul 31 11:30

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    added support when exponent of … (compare)

  • Jul 31 11:00

    anutosh491 on GSoC_Pr4.1_Implementing_few_series_methods_for_bessel_functions

    Fixed leading term method for b… (compare)

Micah
@micahscopes
it seems like some kind of tensor representation would be ideal for this
so the specific objects I'm interested in currently are finite k-algebras, which I'm "harvesting" from sage
the core of each finite k-algebra is its product, which is a "bilinear map" V×VVV \times V \rightarrow V
so in sympy, I'm representing these as lists of multivariate expressions
this is a very simple way of representing these products, but when I want to compose these functions, it makes things difficult
Micah
@micahscopes
An example: say I have a (real) vector space MM, whose vectors represent matrices in some finite dimensional real matrix algebra, and a (real) vector space CC, whose members represent complex numbers, then I have a map f:M×MMf:M \times M \rightarrow M, the matrix algebra product, and another map g:C×CCg:C\times C \rightarrow C, the product of complex numbers.
Now, I'd like to compose these products to make a matrix algebra over the complex numbers
Micah
@micahscopes
so I know from wikipedia that formally, what I'm trying to do is some kind of tensor product... I think a "tensor product of algebras" (https://en.wikipedia.org/wiki/Tensor_product_of_algebras). but while it's easy enough to comprehend a tensor product of vector spaces, I'm still trying to understand how to apply this in sympy
Micah
@micahscopes
ahh, I'm reading now that this is also called "base change"
Kalevi Suominen
@jksuom
@micahscopes How are you planning to represent the (bilinear?) map f:M×MMf: M \times M \to M?
Bjorn
@bjodah
Micah
@micahscopes
@jksuom currently I've just been making a list of symbolic multivariate polynomial expressions. The symbols are the components of the two input vectors, E. G. f([u_1, u_2,... , u_n], [v_1, v_2,... , v_n]) would return a list of polynomials of terms from the u and v lists.
In Sage, you can construct algebras over a SymbolicRing, then take the product of two symbolic vectors and convert the result to a list of sympy expressions
(after that I'm using the glsl code printer in sympy)
Kalevi Suominen
@jksuom
A bilinear map like f:M×MMf: M\times M \to M corresponds naturally to a rank three tensor.
A (1, 2) -tensor actually, I think.
Micah
@micahscopes
@jksuom that makes sense... so if I figure out how to represent my algebras as (1, 2)-tensors, I should also be able to find a way, using tensor algebra, to more easily compose them
Kalevi Suominen
@jksuom
I think that is true though I'm not sure what you mean by 'compose'.
Micah
@micahscopes
I'm not sure either, I don't think "compose" is the mathematically correct term, but I'd like to do multiple "base changes"
I'd like to be able to construct some of these division algebras: https://core.ac.uk/download/pdf/25253839.pdf
Kalevi Suominen
@jksuom
Those tensor products of algebras can be represented by products of tensors (taking proper care of the indices).
Ayushman Koul
@ayushmankoul
@jksuom As you pointed if I am not wrong we need to amend subclasses of BaseSeries containing 2D lines series.Please can you tell me how should I proceed further on this issue ?
Kalevi Suominen
@jksuom
You could probably start by adding methods that would create instances of List2DSeries to some geometric classes such as Segment and Triangle, and also Polygon. (Those could then be appended to a suitable Plot object for showing.)
Micah
@micahscopes
@jksuom I think I'm starting to get some intuition for this idea
Kalevi Suominen
@jksuom
The components aijka_{ijk} of the tensor associated with the bilinear product in an algebra can be found as follows. If (ei)(e_i) is a basis of the algebra, then the components are the coefficients of the pairwise products eiej=kaijkeke_i e_j = \sum_k a_{ijk} e_k.
Kalevi Suominen
@jksuom
For example, if C\Bbb{C} is considered as an R\Bbb{R}-algebra with basis (1,i)(1, i), then a110=1a_{110}=-1 is the coefficient of 11 in iiii and a111=0a_{111}=0.
Micah
@micahscopes
@jksuom !thank you so much!
basically then, to construct the tensor of a finite k-algebra, I just need to get the coefficients in k of pairwise products of basis elements of the algebra
Micah
@micahscopes
I think I need to really play with the summation notation
(and the concept of summation/contraction)
Kalevi Suominen
@jksuom
It seems to me that tensor.array might be the best module for handling algebras and their tensor products.
Micah
@micahscopes
yeah, that's my intuition too. I didn't realize how malleable tensor contraction could be... basically it's very important to keep track of the indices
the indices aren't inherently ordered, so it's gonna be up to me to keep them structured
I had been getting caught up on thinking about duality and covariance/contravariance, but for this problem it's maybe not so necessary to worry about that aspect for now
Micah
@micahscopes
by the way... one of my motivations for this is to be able to do automatic differentiation with "dual numbers". have you heard of this?
Kalevi Suominen
@jksuom
It is a standard construction in algebra. To any ring AA, one can associate the "dual ring" A[δ]A[\delta] where δ2=0\delta^2 = 0.
Micah
@micahscopes
awesome
I didn't realize that
what is that square bracket notation?
is that a "ring extension"?
Kalevi Suominen
@jksuom
Yes, extension of AA by δ\delta.
Its elements are polynomials in δ\delta with coefficients in AA. In this case, only linear polynomials.
Micah
@micahscopes
that makes sense
you can use the dual ring to do automatic differentiation
in my case, I'd like to use it to render fractals made out of clifford algebras and other interesting algebras
with automatic differentiation, you can create a "distance estimator" that can be used to render the surface of a fractal set
(theoretically)
so far I've only been able to render these fractals using a "brute force" approach, which actually goes and tests each point to see if it's in the fractal