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sympy/sympy

A computer algebra system written in pure Python http://sympy.org/ . To get started to with contributing https://github.com/sympy/sympy/wiki/Introduction-to-contributing

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Jun 29 10:28

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Added expand to log rewrites (compare)
Jun 29 09:23

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

updated test for asin,atan,acot… (compare)
Jun 27 12:01

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

removed wrong test (compare)
Jun 27 11:21

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Modified test (compare)
Jun 26 00:22

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Fixed leading term for acos/ase… (compare)
Jun 25 09:43

anutosh491 on series_for_arg

Updated code (compare)
Jun 24 13:33

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Improved code quality (compare)
Jun 22 14:25

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Fixed code quality (compare)
Jun 22 14:17

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Added test for 21721 (compare)
Jun 22 10:02

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Fixed Failing tests (compare)
Jun 20 13:32

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Code Cleanup (compare)
Jun 18 02:50

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Fixed error in test_sums_produc… (compare)
Jun 17 14:28

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

Fixed all failing tests (compare)
Jun 17 13:41

anutosh491 on GSOC_Pr1_Refactoring_Log_leading_term_method

prevent unnecessary hashing add type hints to printing/late… Merge pull request #23588 from … and 34 more (compare)
Jun 10 13:16

anutosh491 commented #22373
Jun 10 13:15

anutosh491 commented #22373
Jun 10 12:42

anutosh491 commented #22806
Jun 10 12:39

anutosh491 commented #22806
Jun 10 12:21

github-actions[bot] commented #23614
Jun 10 12:15

github-actions[bot] commented #23541

Micah

@micahscopes

Hi there, I'd like to understand how best to compose multilinear maps of symbolic elements in sympy

so far I've been doing it the hard way

which is to make lists of multivariate polynomials, iterating over them and substituting, then sorting

I'm still a kindergartner at tensor algebra

but I'm learning

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 \times V \rightarrow V

so in sympy, I'm representing these as lists of multivariate expressions

Micah

@micahscopes

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

M

, whose vectors represent matrices in some finite dimensional real matrix algebra, and a (real) vector space

C

, whose members represent complex numbers, then I have a map

f:M \times M \rightarrow M

, the matrix algebra product, and another map

g: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 \times M \to M

Bjorn

@bjodah

@moorepants if (when?) we use pytest there's also: https://docs.pytest.org/en/latest/skipping.html#skipping-on-a-missing-import-dependency

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)

Micah

@micahscopes

Ah... I found a clue! https://math.stackexchange.com/questions/2889140/can-a-rank-three-tensor-act-as-a-trilinear-bilinear-or-linear-map

Kalevi Suominen

@jksuom

A bilinear map like

f: 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

a_{ijk}

of the tensor associated with the bilinear product in an algebra can be found as follows. If

(e_i)

is a basis of the algebra, then the components are the coefficients of the pairwise products

e_i e_j = \sum_k a_{ijk} e_k

Kalevi Suominen

@jksuom

For example, if

\Bbb{C}

is considered as an

\Bbb{R}

-algebra with basis

(1, i)

, then

a_{110}=-1

is the coefficient of

1

ii

and

a_{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?

https://matiasmorant.wordpress.com/2017/12/29/dual-numbers/

Kalevi Suominen

@jksuom

It is a standard construction in algebra. To any ring

A

, one can associate the "dual ring"

A[\delta]

where

\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

A

\delta

Its elements are polynomials in

\delta

with coefficients in

A

. In this case, only linear polynomials.

Micah

@micahscopes

that makes sense

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