Questions tagged [computer-algebra-systems]
A computer algebra system (CAS) is a program which is able to carry out various symbolic manipulations with mathematical expressions. Some well-known computer-algebra systems are Mathematica, Maple, Wolfram Alpha, GAP, SAGE. Questions on this site about computer algebra systems should be mathematical in nature, and not just about the syntax or the mechanics of the CAS. Otherwise the question would be better suited for a different Stack Exchange site.
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Computer Algebra System for Homotopy Types of Finite Simplicial Sets
I hope this is the right place to ask the following question, if not please guide me towards the correct forum...
I want to show that two certain finite simplicial sets are not weakly homotopy ...
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Which CAS can do non-commutative differential algebra?
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute generic non-commutative differential expressions. Let me illustrate what I mean by two examples.
Let $(R,\partial)...
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Calculate $\int_{\mathbb{R}^3} x^2\exp(-a\sqrt{x^2+y^2+z^2})d\mathbf{x}$ without spherical coordinates
Which tool can I use to (numerically?) evaluate
$$
\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty x^2\exp\left(-a\sqrt{x^2+y^2+z^2}\right) dx ~ dy ~ dz,~a>0
$$
without having to ...
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Variant of Segre embedding
We work over a field $k$. We know that there is the Segre embedding $\def\P{\mathbb{P}} \P^2 \times \P^1 \to \P^5$.
Now I want an embedding of $\P^2 \times \def\A{\mathbb{A}}\A^1$ into some projective ...
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Multiplicity of a singular point, Ideals, and Maple/Algorithms
I am teaching myself about algebraic geometry, and the classification of singular points on algebraic curves $f(x,y)=0$, where $x,y\in\mathbb{C}$.
One way to classify these singular points (a set of $...
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Can you solve $X A X^T = B$ for $X$ over Gaussian rationals in GAP?
I have two symmetric matrices : $A$ is $a \times a$ and $B$ is $b \times b$. I'm trying to find solutions for $X A X^T = B$. $A$ and $B$ are constant and $X$ is the unknown $b \times a$ matrix. All ...
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Converting a group element to an ordered list of generators in GAP
Consider the free group on {a,b} which is gap is constructed with FreeGroup("a","b"). I want to cyclically reduce words. For example, given word $a^{-1}*b*a$, this would be ...
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Automatic Differentiation and Product Rule
There is an explanation for the algorithm for Automatic Differentiation on the MathWorks Website.
Their example equation is:
$$f(x) = x_1 \exp\left(-\frac{1}{2}(x_1^2 + x_2^2)\right)$$
Their ...
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Groebner basis over rational vs finite field
Some algorithms for calculating a Groebner basis are optimized for calculating with coefficients in a finite field. Having determined the basis over a finite field, I'd like to understand what ...
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Polynomial constraints for restricting: $a=0$ if and only if $b\neq 0$
For this discussion, I will be considering polynomials over multiple complex variables, and a system of polynomial constraints, where the constraints on the variables can be written as a set of ...
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Computer algebra system for Tensor Algebra
I am searching for a computer algebra system that allows to tackle the following problem:
Let $V = \mathbb{R}^3$. Consider the free commutative algebra $\mathrm{S}^\bullet V$, with $\mathrm{S}^kV$ ...
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Viewing monoid rings as rings with identity in GAP
I look at monoid algebra of finite monoids with GAP and want to force GAP to view them as algebras with one. But it seems it does not work:
...
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Problem about the Galois group elements in Sage
I am computing a Galois group as below in Sage.
K.<a> = NumberField(x^4-2)
L = K.galois_closure('b')
G=L.galois_group()
Here, $G=\text{Gal}(L/\mathbb{Q)} \...
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Computing the cohomology of a complex over $F[x,y]$
Is there a method for computing the generators of the cohomology of a chain complex over the ring $R=F[x,y]$?
When $R$ is a PID one can use Smith normal form and there are libraries in Sage that ...
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Pari GP/Magma: how to calculate the Galois group over arbitrary number fields? [closed]
Let's say I have a polynomial $f(x) = x^5 + 3x + 5$. You can check on Pari that the Galois Group over $\mathbb{Q}$ is $S_5$. $K$ be the intermediate field obtained by adjoining a root of $f(x)$ to $\...
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Question about automorphisms of the cyclotomic field and Sagemath
Let $K = \mathbb{Q}(\zeta_5)$ be the fifth cyclotomic field. I write the following code in sage
...
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Central Products in SageMath
I would like to compute the Central Product of two groups in SageMath. I cannot find any builtins and I'm not sure what group theory packages for Sage may exist. Is there anything out there or am I ...
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Index of an explicit subgroup of $\mathrm{GL}_4(\mathbb{Z})$
Let $H$ be the subgroup of $\mathrm{GL}_4(\mathbb{Z})$ generated by the $4!$ permutation matrices together with
$$
\begin{pmatrix}
1 & 0 & 0 & 0 \\
-1 & 0 & 1 & 1 \\
0 & 1 &...
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Best computer algebra system for non-commutative computations
I'm interested in simplifying expressions like
$$\int_{-\infty}^\infty dq_1\int_{-\infty}^\infty dq_2\int_{-\infty}^\infty dq_3f(q_1,q_2,q_3)\langle\ |b(p_1)b(p_1)a^+(q_1)b(q_2)b(q_3)b^+(r_1)b^+(r_2)a^...
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Algorithm to check whether rational polynomial is irreducible over $\mathbb{C}$?
Let $F\in\mathbb{Q}[x_1,\ldots,x_n]$ a multivariate polynomial. Is there an algorithm for checking whether $F$ is irreducible over $\mathbb{C}$? Is this implemented somewhere?
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How to calculate Hilbert polynomial
Consider the ideal $I=\langle x_0^2+x_1^2+x_2^2+x_3^2,x_0x_1+x_2x_3\rangle \subseteq k[x_0,x_1,x_2,x_3]$ for $k=\mathbb{F}_{32749}$. I am trying to calculate the dimension of the projective variety ...
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Unit group structure GAP code. [duplicate]
I want the structure of the unit group of the group algebra $F_{3^k}D_{30}$ using GAP, where $F_{3^k}$ is any finite field of characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
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Can one use the Berlekamp theorem repeatedly in the polynomial decomposition
Given a polynomial $f$ in the polynomial ring $\mathbb{F}_p[x]$, one may use the Berlekamp theorem to decompose it in irreducible polynomials. Let's suppose we run the algorithm once and we get as ...
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How to solve a trinomial with a rational exponent $x^{q}-x-c=0$
I require solutions to equations of the following form: $x^{q}-x-c = 0$ where $q$ is a rational number very close to 1 with $q$ such that: $$q=\frac{p^k+1}{p^k}$$ $$q=\frac{p^k}{p^k-1}$$ $p$ prime, ...
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Computing Tate cohomology using computer
$\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\coker}{coker}
\newcommand{\Z}{{\mathbb Z}}
$I want to use GAP in order to compute (on a computer) an example that I cannot compute by hand, see ...
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Groebner basis for system of integral quadratic forms
I have a system of quadratic forms (homogeneous polynomials of degree $ 2 $) with integer coefficients. Each quadratic form is the trace of a product of matrices. I'm solving for the matrix entries. ...
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How to count number of zeros in the character table of a finite group by GAP?
I am trying to write a function in GAP to count the number of zeros in the character table of a finite group. And I have a problem counting zeros.
...
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Looking for computer algebra systems for functional analysis
I'm looking for a CAS that is compatible with ideas from functional analysis. Specifically I want to work with Banach algebras of functions on finite groups or semigroups, so ideally there would also ...
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Checking whether two algebras are isomorphic with MAGMA
I want to use MAGMA to check whether to given finite dimensional algebras over a field are isomorphic.
Here my attempt:
...
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How did this paper obtain there result (Approximating the Conway-Maxwell-Poisson normalising constant)
I am currently reading Approximating the Conway-Maxwell-Poisson normalizing constant. On page 958 a numerical example is presented (I refer below to the third expression from the top of the page). I ...
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Inverting series with logs and W
You've all heard it: what does a drowning analytic number theorist say? Log log log log....
I very frequently deal with the sorts of functions that one comes across and want to invert them. Generally ...
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Algorithm to test isomorphism between two finitely presented modules
Given two finitely presented left (or right) $R$-modules, specified by free presentations, is there an algorithm to determine if they are isomorphic and produce an isomorphism? This is of course ...
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Open-source software to calculate quotient of multivariate polynomials
I need to divide one polynomial in two variables by another. I need to find the multivariate polynomial $\frac{Q}{P}$, where $Q,P\in\mathbb{R}[X_1,X_2]$, if indeed there is a polynomial solution for $\...
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Is it possible to describe extension of places of a function field in MAGMA?
I would like to describe, for example, the unramified places of an extension $F'/F$ of function field (in one variable) in MAGMA calculator.
More specifically, given $F'/F$ an extension of function ...
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Symbolic partial derivative on computer algebra systems
I'm working on calculating Lie groups of differential equations, and I am looking to calculate symbolic partial derivatives of expressions in the following way. For instance, if $f(x,y),g(x,y)$ are ...
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Coprime ideals and Dedekind zeta function over cyclotomic fields
For a positive integer $m$, the $m$-th cyclotomic ring is $R = \mathbb{Z}[\zeta_m]$, the ring extension of the integers $\mathbb{Z}$ obtained by adjoining an element $\zeta_m$ having multiplicative ...
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Group names in GAP character table library
Let $G = M_{12}.2$, automorphism group of the Mathieu group $M_{12}$.
In GAP, with DisplayAtlasInfo("M12.2");; we get information about G as listed here ...
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Automating modular arithmetic in local fields using MAGMA
Let $f(X) = X^4 + a_3X^3 + a_2X^2 + a_1X + a_0$ be an Eisenstein polynomial over the $2$-adic numbers $\mathbb{Q}_2$. Let $\mathbb{Q}_2(\pi)/\mathbb{Q}_2$ be the totally ramified extension defined by $...
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Finding x root of a Transformation Matrix
Let's say I have two boxes in 3D space. I know the Transformation of Box1 = $T^{world}_{box_0}$ and Box4 = $T^{box_0}_{box_4}$.
Our lovely designer want's me to write a tool that fills the the space ...
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Let $A(n)$ be the maximum number of intersections of $n$ lines. Show through induction that the following holds: [closed]
The following holds:
$$A(n)=\sum_{k=0}^{n-1}k$$
How to do the induction?
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Adding inverses of nilpotents as an extension of the "extended real numbers"
This is an idea that I had while playing with an automatic differentiation system built on dual numbers. This system, like most computer algebra systems built on floating point arithmetic, has the ...
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Algorithm for writing an image of a polynomial in a quotient ring in terms of a given basis of the quotient ring.
I need to calculate the equivariant Chern classes of certain vector
bundles on the classifying spaces of complex algebraic groups. In order
to do this I am looking for a way to do the following ...
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0
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Matlab svds function incompatible with symbolic data
I am currently working with very small numbers and hence in order to increase Matlab's precision I have made use of Matlab's vpa function which turns a $n\times n$ ...
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how to find reduced words for each element of reflection (Coxeter) group in GAP
I have a finite reflection (or Coxeter) group defined abstractly through the standard presentation
$$(s_i s_j)^{c_{ij}}=1$$
For each of its elements I want to find the number of reduced words equal to ...
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I don't understand how to properly use runge kutta
So I am trying to use runge kutta 4, to more acurately calculate forces. I am using code from github and I used their example of rabits and wolves populations. In IntegratorLSODE.cs, in lotkaVolterra()...
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A question about singular system [closed]
How can I define the formal power series ring $k[[x,y,z]]$, where $k$ is a field, in the Singular system?
Is it possible?
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compute group automorphisms, part 2
This is continuation of my previous question about computing automorphism groups using MAGMA. My new question goes in a different direction so I start a new thread. If this breaks the forum rules, I'...
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Are there strict superset Fields of the rational numbers where each element has a canonical finite representation?
The Field $\mathbb{Q}$ has the useful property that you can represent each element in a canonical way (in the sense that the represenation is equal if and only if the element is the same) using a ...
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compute group automorphisms using magma
I'm learning how to use MAGMA to compute automorphism groups, and I have difficulty interpreting the output. Concrete (and functional) example:
...
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Restricting Characters on Sage
I am using Sage to obtain the character table of different permutation groups using the command G.character_table(). Is there any implemented command in Sage that restricts an irreducible character (...
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