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Questions tagged [computational-algebra]

Computational algebra is an area of algebra that seeks efficient algorithms to answer fundamental problems concerning basic algebraic objects (groups, rings, fields, etc.). For questions about generic computer algebra systems, use [tag:computer-algebra-systems].

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Polynomial ring a module over ring of invariants

Let $\mathbf{k}$ be a field of characteristic 0 and $S=\mathbf{k}[X_1,\cdots,X_n]$ be a polynomial algebra over $\mathbf{k}$. Let $G\subset GL_n(\mathbf{k})$ act linearly on $S$ and $R = S^G$ be the ...
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2answers
84 views

Subgroups of $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$

Is there some program or a location which would allow me to work and calculate with the subgroups of the group $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$?
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18 views

How to compute a unitary representation of finite group isomorphic to a given rep?

Suppose I am given some representation of a finite group: $\rho : G \to \text{GL}(n, \mathbb{C})$. I want to compute a unitary representation $\tau$ which is isomorphic to $\rho$. I know about Weyl's ...
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2answers
56 views

Algebraically independent polynomials iff linearly independent differentials

This is an exercise question in Appendix A of Introduction of Algebraic Geometry, Justin R Smith. I am looking for an intuition for the solution. if $k \rightarrow K$ is an extension of fields of ...
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64 views

Finding the numerical listed string to the center algorithm.

Suppose we have a string that occupies a length of seven characters, and the number of strings altogether is ten. We include the hamming distance of three characters. If we calculate the exact amount ...
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11 views

Proof Using Strong Reducibility

I read the following proof so many times and I could not, for the life of me, figure out how the professor went from (1) to (2). I know to prove $\overline{K} \le A$ , where $\overline{K}=\{x:\...
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26 views

Schur Index for Quaternion Algebra

I learned form this question and this answer that Schur index in GAP can be found using LoadPackage("wedderga") the functions "SchurIndex". But I am working on the field $K=\mathbb Q (\sqrt{-39})$ ...
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1answer
41 views

Cartesian Product over a list of objects in MAGMA

I'm currently trying to create the Cartesian Product of certain objects (namely: Character Tables of different finite groups). However, it seems like I don't really understand the car<$...$> ...
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56 views

Ideas for Undergrad Research Project

I'm in my second year of my maths undergrad course and hoping to do a 6 week research project this summer. I'm interested in doing a project wherein I apply a maths topic, (i.e Manifolds, Metric ...
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22 views

How to compute rational points on a projective variety in Macaulay2

I know that the package "rationalPoints" will compute rational points on an affine variety over a finite field, but I would like to do the same for varieties in projective space. Is there a built-in ...
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42 views

How to compute the number of equivalence classes under the relation $a\sim b\iff a=x^m b x^n$?

Let $G$ be a finite group. Fix an element $x\in G$, and denote by $\sim$ the equivalence relation on $G$ given by $a\sim b \iff \exists m,n\text{ such that }a=x^m b x^n$. Example: Let $G=\langle(123),...
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Is there a program able to compute index of quotient sets and the size of the orbits?

I am interested to compute Hurwitz-Kronecker class numbers and this asks to compute a sum over representatives of a quotient set. Do you know any implemented program that does the job for any quotient ...
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21 views

Computation with infinite Weyl algebra

My question here is very computational. My problem is in mathematical physics, so I want to ask the community what kind of software they use to do the following computation if there is any? Let $$L_{...
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0answers
10 views

For what values of alpha is the matrix diagonally dominant ? Does the jacobi method converge?

For what values of alpha is the matrix diagonally dominant ? Does the jacobi method converge? (A = {{1, $\alpha$, 0}; {$\alpha$,1, 0}; {0 , $\alpha$ , 1}})
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1answer
41 views

How can I find an isomorphism between two representations?

Suppose I have two representations (over $\mathbb{C}$) of a finite group $G$, $\rho : G \to GL(V)$ and $\tau : G \to GL(W)$. If I am given that these representations are isomorphic, i.e. there exists ...
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1answer
27 views

Library for visualizing computation graph

Does anyone know a tool or library (preferable JavaScript) that can visualize an equation as a computational graph, such that for example the sigmoid function with inputs $\mathbf{w}$, $\mathbf{x}$ ...
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21 views

Near-Near-MDS codes

I am trying to understand the codes which are not maximum distance separable but are at a distance of 2 from being Maximum Distance Separable. I have trying to find articles specifically related to ...
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46 views

Computing geodesics on pseudo-riemannian manifolds

Consider a pseudo-riemannian manifold $M$ with a metric tensor $g$. Now, given two points $p_1, p_2$ in $M$, how do I compute (as in, programatically compute) the geodesic between these two points? ...
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1answer
86 views

Is there a group with $2$ generators having exactly $17$ subgroups of index three?

I recently saw a fun problem from a past qualifying exam from Stanford. It is Problem 10, part (b) in this document. I will screenshot the problem and its solution here: My question is the following....
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2answers
67 views

What does it mean that field $\mathbb{F}_{p^n}$ “contains” the prime field $\mathbb{Z}_p$?

I have read in few books (example Computational Number Theory, page 77) that any extension field $\mathbb{F}_{p^n}$ "contains" as a subfield the prime field $\mathbb{Z}_p$? What exactly does "...
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2answers
71 views

Probability of ending up with a particular card on top when placing random cards in random spots

This is a probability problem I encountered today. What algorithm or equation solves the situation below generically? There is an unlimited supply of playing cards. There are 10 spots on the ...
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2answers
40 views

Random subgroup of a group

Given a finite group $G$, Is there any known algorithm which gives a random subgroup of $G$?
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0answers
43 views

Computer program for the conjugacy problem of $GL_n(\mathbb{Z})$

In "Solution of the conjugacy problem in certain arithmetic groups" Grunewald provided an algorithm for solving the conjugacy problem of $GL_n(\mathbb{Z})$. My question is: Is there some computer ...
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2answers
41 views

Normalized coordinate of point on 4-sided concave polygon

Considering I have concave polygon made up of 4 points: P1, P2, P3, P4, and a point M which I already know is inside this polygon. How would I go about determining its "normalized" position in an ...
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1answer
46 views

Computing quotients of group by elements of its lower exponent$-p$ central series

Let $G$ be a finite $p−$group of number of generators $d$ and exponent$−p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G)=1$ in the series $$ G=P_0(G)≥...≥P_{i−1}(G)≥P_i(G)≥... $$ ...
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Why is the $p-$multiplicator of a $p-$group is an elementary abelian $p -$group?

Let $G$ be a $p-$group that have the finite presentation $F/R$($F$ is a free group of rank $d$); The $p-$multiplicator of $G$ is defined by $$ G^* = R/[F,R]R^p $$ Why $G^*$ in an elementary abelian $p ...
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1answer
31 views

Can every finite group be presented as an Fp group?

If we have a finite group $G$, then we know that $G$ is a quotient of some free group, say $F$, on some free set, say $X$. Now, let $N$ have this: $G\cong F/N$, where $N$ is the normal closure of $R$...
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1answer
272 views

What is Vector Offset?

I am reading a paper on computational linguistics. Here I have problems in some basic terms of the paper. I searched internet about them, but I could not find an appropriate answer about the ...
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1answer
41 views

Network analysis of category of groups

I am taking a course in Network Analysis at my university and I would like to do something related to category theory in my home project. To be more precise, I would like to take some amount of finite ...
2
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1answer
63 views

How to find prime elements

Let $K$ be an arbitrary number field and $\mathcal{O}_K$ its ring of integers. I have seen many concrete examples about finding prime elements. For example I calculated the prime elements of $\mathbb{...
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1answer
76 views

List of path arrows in GAP

I am using the QPA package with GAP and have the following problem: I want to, given a path $p$, obtain a list (preferably ordered) of the arrows which make up $p$. For example, if I have arrows $...
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28 views

Computing locus of points with positive dimensional fibers

I became interested in the following problems by studying kinematic configuration spaces: Let $C=V(f_1,\ldots,f_r)$ where $f_i\in \mathbb{R}[\vec{x},\vec{y}]$, $J=V(g_1,\ldots,g_s)$ where ...
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1answer
49 views

Computing orbits of a $S_5$ group action

Consider the group action of $S_5$ on $(\mathbb{Z}/5\mathbb{Z})^6$ given by $$(12345)\colon (a,b,c,d,e,f)\mapsto(b,c,d,e,a,f)$$ $$(12)\colon (a,b,c,d,e,f)\mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$ Do we ...
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1answer
52 views

Choosing different term orders for Groebner basis calculations in GAP

I am using the QPA and GBNP packages in GAP to analyze path algebra quotients. I use the GBNP package for computing Groebner bases for the ideals in the path algebras, and the term orders for these ...
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47 views

Conjugate gradient method for a Big Coefficient Matrix

Well I want to solve a 2D homogeneous Laplace equation (Dirichlet BC.) in a domain with 256*256 grid point with CG algorithm. common CG Algorithm I'm using MATLAB and it's so obvious that my ...
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1answer
63 views

What is the relation between Gröbner basis and syzygies?

Let $I=(f_1,f_2,f_3,f_4,f_5) $ an ideal in $S=\mathbb {C}[w,x,y,z ]$. $f_1=w^2-xz $ $f_2=wx-yz$ $f_3=x^2-wy$ $f_4=xy-z^2$ $f_5=y^2-wz $ I want to compute the syzygies of $S/I $ using Gröbner ...
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1answer
44 views

Problem with hom-spaces and their dimensions in GAP

I am running the following example in GAP using the QPA package: ...
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0answers
24 views

Compute equalizer of map of polynomial rings, perhaps using Gröbner bases?

Suppose that $k$ is a field and I have two ring homomorphisms $f, g: k[x_1, ..., x_m] \to k[y_1, ..., y_n]$. How can I use Gröbner bases (or other computational tools) to compute the subring of ...
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1answer
78 views

Full projective resolutions for path algebras in GAP

I am trying to do various computations on path algebras using GAP. One such computation is obtaining projective resolutions of simples modules (and possibly other modules). The command in the GAP ...
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47 views

Find H such as |Aut(ZxG)| = |Aut(ZxH)|, where G is a given finite abelian group

I am really new in GAP, please accept my apologize if my question is a simple one. I want to implement an algorithm in GAP for the next problem: Given an abelian finite group G, find the group H such ...
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0answers
39 views

How to build a simple Mathematical formula with matching condition

I'm trying to write a mathematical equation that sums a set of variables on the condition that they match variables in an original array. Example: List A: A1 = Dog, A2 = Cat, A3 = Monkey List B: B1 =...
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1answer
75 views

Grobner basis: Basis for K-vector space.

This is a problem from my Groner basis class. I am unsure about how to do this problem: Let $A$ be an ideal of $R=K\left[x_{1},...,x_{n}\right]$, and let $G=\{g_{1},...,g_{t}\}$ be a Grobner basis ...
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Efficient way to calculate solution to the Von Neumann equation for time evolution

I am trying to calculate the following equation efficiently: $\rho(t) =exp(-iHt/\hbar)\rho(0)exp(+iHt/\hbar)$ where $H$ and $\rho$ are some matrices of varying size (from 2x2 up to 128x128). It isn'...
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37 views

Basis for the vector space over $\mathbb{Q}$

I have a question about the following problem from the Grobner bases class: Given the polynomial ring $\mathbb{Q}[x, y, z, t]$ and $A=\langle f_{1},f_{2},f_{3},f_{4} \rangle$ where $f_{1}=x-2y+z+t,...
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1answer
74 views

computer program-software for galois

I need a reference for a good algebra program-software, especially for Galois theory. What I have found so far is PARI which calculates the galois group over $\mathbb Q$ of a polynomial up to degree 8,...
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1answer
61 views

How can I use GAP to collect words into a normal form?

I would like to be able to give GAP some relations and a word, and have it collect that word into a normal form (say with collection from the left). For instance, I would tell GAP $ [x_{2},x_{1}]=x_{...
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5answers
51 views

The most efficient way to solve $2^{2017} \mod 9$

I have just started my Computational Algebra course and I have to solve this relatively easy problem: $$2^{2017} \mod 9$$ The first thing that I noticed was that $2$ and $9$ are coprime, therefore, I ...
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0answers
102 views

How to use a stabilizer chain (Schreier-Sims) to prune a centralizer search?

I've implemented the Schreier-Sims algorithm. My sources indicate that a next useful step is to use this structure to prune a backtracking search of group elements, which should yield an improvement ...
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1answer
50 views

How to create a group action on some group with GAP

Given group $G$ is generated by some elements $g_1,g_2,\cdots$ with some relations. Group $A$ is generated by some elements $a_1,a_2,\cdots$ with some relations. I want to define a group action $\rho: ...
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0answers
18 views

Method for measuring distance of a irregular curved profile

I am trying to measure a curved profile of a surface(2D) to determine the surface availability at different rate of testing. I have attached an image for a rough picture.enter image description here ...