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: Mathematica, Maple, Wolfram Alpha, GAP. For questions about Mathematica please see the ...

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Is there any Software package to confirm correctness of your derivation steps?

Suppose I made a very complicated (not necessarily difficult) derivations in which it is very likely for anyone to make some silly mistakes, e.g., incorrect sign, overlooking a term, and so on. I ...
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36 views

Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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174 views

How to integrate $\left(1+\ln(x)\right)\sqrt{1+(x\ln(x))^2}$ with Risch algorithm?

How would you integrate $\left(1 + \ln\left(x\right)\right)\, \sqrt{1 + \left(x\ln\left(x\right)\right)^{2}\,}$ using the Risch algorithm? I want to know this because Mathematica is using the Risch ...
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240 views

Test for equivalence of algebraic expressions

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
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38 views

Tool for construction of graph with specified properties

Is there a tool (class of algorithms for graph generation) that can construct graph with specified properties. E.g. construct graph who is homeomorphic with both to K5 and K3,3. Construct planar graph ...
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76 views

Adjoint action of a Lie algebra in MAGMA

Let's say I have a Lie algebra $L$ and an element $x$ in $L$. I need to compute $(\mathrm{ad} \ x)^n(y)$ for several values of $y$ and a particular value of $n$. I thought the best way to do this ...
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78 views

Computing the decomposition of a representation of $S_n$

I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily? The best approach I have so far is as follows: Find a ...
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22 views

Can SAGE or othe software compute or guess growth rates of infinite discrete groups?

I am interested in the growth rate of some finitely generated (infinite, non-abelian) discrete groups. Knowing very little about geometric group theory, I am wondering if I can plug them into sage and ...
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29 views

signature function of Weyl group element in LieArt

I am currently using LieArt Mathematica package for some calculations in Lie algebra, I am wondering if there is a way to know what is the signature of a Weyl group element, it seems the package can ...
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26 views

Improvement of Buchberger's Algorithm (second part)

Suppose $S_j$ is a homogeneous syzygy of multidegree $\gamma_j$ in $S(G)$, where $G=\{g_1,\dots,g_t\}$. Show that $S_j G=\Sigma_{i=1}^{t} c_ix^{\alpha(i)}g_i$ has multidegree $< \gamma_j$. Now, I ...
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95 views

Algorithm to reduce expressions to canonical form

I'm writing a small computer algebra system that only knows rational numbers and all expressions that you can get from them by using basic arithmetic operations and powers. So the expressions are ...
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94 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
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72 views

Are there high performance computing applications for symbolic integration?

Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite ...
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37 views

Muirhead's Inequality (software?)

I just started learning about inequalities: Schur's, Karamata's, Muirhead's, etc... They are beautiful but it seems that in the case of more than two variables, some of the computations become a ...
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89 views

Todd-Coxeter algorithm: coincidences

I'm trying to understand the Todd-Coxeter algorithm with the help of a multiplication and relator table, but there is one thing about coincidences that is not really clear. For some small groups (for ...
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204 views

How does one solve equations over finite fields in SAGE?

Sage has the method solve (or function, I'm not sure what's the correct terminology) that finds solutions to 'symbolic expressions'. In particular, if one wants to find solutions for a given set of ...
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50 views

Is there a known algorithm for approximating all the real and imaginary zeros of any well behaved equation of a single variable?

Does there currently exist a general algorithm (or set of algorithms used together) that will approximate all the zeros of any well behaved non-differential equation of a single variable which has a ...
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526 views

Solving large, sparse system of linear equations

I have a system of linear equations as follows: $$(A+I)x=B$$ where $I$ is the $n\times n$ identity matrix, $A$ is a $n\times n$ matrix such that the first and last rows are blank, and, for every ...
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61 views

Minimal set of algebraically independent numbers

Suppose we have a set of polynomials $f_1, f_2, \ldots, f_n \in \mathbb{Q}[x]$. Consider the set $$S := \{\alpha \in \mathbb{C} \; | \; f_i(\alpha) = 0 \text{ for some } i \}$$ of complex roots of ...
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84 views

Free software for expresing a resolvent as function of coefficients

This relates to question "Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?" I would like to try absolute resolvent for group $C_5$ in $S_5$. For example ...
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162 views

How do mathematicians handle functions of functions that may change?

E.g. let $f(x) = $ some function. Now define $h(x) = f(g(x))$. Now suppose the definition of $g(x)$ changes around in a discussion. Do we still refer to $h(x)$ as the original $h(x)$ or only when ...
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177 views

What is a good software package for ( assisted ) theorem proving and documenting?

Background: An issue in my math study is that I haven't found a good way of storing the theorems ( mostly abstract algebra ) I studied and want to (re-)use in proofs. At the moment I use a personal ...
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17 views

Determining equations for Lie symmetries

Am I right that using methods of the 1-forms (that usually implemented in system of computer algebra for Lie Symmetries), we can always generate determining equations for ODE, that solved for highest ...
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35 views

Maxima CAS and programmatically defining a function with a variable number of arguments. -?

This is a very simplified question of what I had asked. In Maxima, how can I include a for-loop counter in the left-hand side of an assignment, e.g. ...
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48 views

Understanding BlowUp Computation in Singular

Many of us might know that "Singular" is a computer algebra system for Algebraic Geometry, Commutative Algebra and Non-commutative algebra. This is a procedure in "Singular" for computing blowups. ...
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31 views

Is there some database or software to look for patterns in polynomials?

Like if I am looking at these polynomials, $$x^8-8x^6+20x^4-16x^2+3$$ $$x^{10}-12x^8+48x^6-72x^4+33x^2$$ $$x^{12}-16x^{10}+88x^8-192x^6+138x^4$$ And I want to know if they are members of some ...
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36 views

Lower bounds on possible integer relations from the PSLQ algorithm

For the equation: $$ \sum_{i=1}^na_ix_i=0 $$ where all $x_i$ are real numbers and all $a_i$ are integers, the PSLQ algorithm can either find an integer relation or give lower bounds on the norm of ...
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59 views

Is there any efficient progam or software to calculate the fractional chromatic number?

The fractional chromatic number $\chi_f(G)$ is a generation of the chromatic number of a graph $G$. It can be formulated as a linear programming question: Let $\mathcal{I}(G)$ be the set of all ...
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38 views

Recent benchmarks of CAS programs? (HNF, Groebner)

I've been trying to find recent benchmarks of CAS systems for computing the Hermite Normal Form as well as Groebner basis over both $\mathbb{Q}$ and $\mathbb{C}$. I've been unable to find anything ...
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123 views

Revealing output slowly in Magma

When working in Magma, is there a command to gradually reveal the output. For example, if you have a finite group $G$ and you want to calculate SubgroupLattice(G), then the output may be thousands of ...
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16 views

Tools for optimizing asymptotic bounds.

Is there any tool for this task ? Given the asymptotic bound in term of $n$ and other paramaters $t_1,\dots,t_r$, then return the value for each $t_i$ which optimizes the expression in term of $n$, ...
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99 views

Flatten kronecker product in CAS?

I am a new user of Maxima, and I need to trace the elements of a big messy kronecker product of symbolic matrices. I tried the following to get my feet wet, but I don't get a simple, flat matrix -- ...
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84 views

Looking for a binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\ ...
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0answers
43 views

CAS for counting points of varieties over finite fields

I am looking for a computer algebra system, which is able to some of the following (in theory equivalent) things for a smooth projective variety defined over a finite field: -Count the number of ...
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867 views

Simple example application of Karush-Kuhn-Tucker conditions to minimization problem

I am wondering if there is a simple example application of the Karush-Kuhn-Tucker conditions to show that a minimum exists for a multivariate minimization/optimization problem. Could anyone suggest a ...
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0answers
61 views

Computing relations on the columns of a matrix

Given an $m\times n$ (with $n>m)$ matrix $M$ over a polynomial ring $R=k[x_1,...,x_n]$, suppose that every column of $M$ is an $R$-linear combination of $m$ specified columns. I would like to ...
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20 views

factor out of an expression, a couple principal ideals, software?

I have an expression, $f$, consisting of a few rational fractions of large multivariate numerators, $n1,\,n2,\ldots \in \mathbb{Q}[a1,a2,b1,b2;Q]$ and large multivariate denominators, $d1,\,d2,\ldots ...
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48 views

How does the factor command on the TI-89 works?

So to put my question in context, I am working on the following problem. Let $N=1291233941$. Eve's magic box tells her the following three encryption/decryption pairs for $N$: $$(1103927639, ...
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0answers
23 views

Derivative with respect to matrix X of Tr(A Exp(-iX)) where A in another matrix

How could I numerically evaluate $\nabla_\mathbf{X} \textrm{Tr}\left(\mathbf{A} \exp(-i \mathbf{X})\right)$? Is there a simple expression for it?
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24 views

3rd order non linear differential equation

$2f'''+f.f''=0$ The boundary conditions are given as $f(0)=r, f'(0)=0, f'(∞)=1$ , where $r$ is a constant . How can I solve this differential equation using Matlab or Mathematica ??
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What is the “cost” of computation of two special CAS algorithms

Suppose I have an integer $n$ with e.g. a large number of say decimal digits. I would like to get some information about the runtime "cost" of standard CAS algorithm which factors $n$ into primes ...
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20 views

defining an upper bound for n-tuple

Let we have $n-tuple ~~like ~ a=(a_1,...a_n) \in N_0^n ~and ~every ~a_j < (2^{m_j} -1). $ and let suppose we have another n-tuple like $b:=(b_1,...,b_n)$ such that $ b_i \leq (2^{m_j} -1)$. I ...
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34 views

Improvement of Buchberger's Algorithm

1) Let $S(F)$ be the subset of $(k[x_1,\dots,x_n])^s$ consisting of all syzygies on the leading terms of $F=\{f_1,\dots,f_s\}$. Then every element of $S(F)$ can be written uniquely as a sum of ...
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38 views

Question about multiple solutions to a polynomial

Assume that $f(X,Y,Z,V,W)\in \mathbb{Z}[X,Y,Z,V,W]$ is some polynomial and assume that $f(x,y,z,v,w)=0$. I would like to know if there is some way to figure out if there are non-trivial constants in ...
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19 views

Gram-Schmidt-procedure with PARI/GP

Can the Gram-Schmidt-procedure to find an orthogonal basis of a vector space spanned by given linear independent vectors be easily done in PARI /GP or do I have to program the procedure ?
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66 views

Can a Computer Algebra System 'experiment' with expressions?

I have recently been reading about software for symbolic manipulation, and I can see its use as a tool for performing large calculations that would be unfeasible otherwise. Given that these systems ...
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42 views

How to reduce a graph via decomposition?

Is there a Java / C# library that can be used to reduce a graph via decomposition? Or could someone point me to a good tutorial where I can learn all these? E.g.
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60 views

Why are these CAS calculated integrals equal

I let Wolfram evaluate this integral: $$\int\frac{-2\sin(x)\cos(x)}{2\cos^2(x)+\cos(x)-1}dx$$ 1st result from WolframAlpha online was $$\frac43 \log(\cos(\frac{x}2))+\frac{1}{3\log(1-2 \cos(x))}+C$$ ...
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38 views

Can Maple solve recurrence relations when n is only certain numbers?

If you have an equation $a(n)=a_{n-1}+2a_{n-2}$ only defined when n is powers of 2, how would convey that to Maple that n has a restriction? What if you wanted only $n \ge 2$?
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How to solve an equation in three variables fixing two of the variables?

Also, I have the following equation, I want to solve it for $b$ keeping $a$ and $c$ fixed. $5b^5+(60-5a)b^4+(125+50c-80a)b^3+(594c-445a-775)b^2+(2324c-1005a-3270)b+3000c-750a-3000=0.$ Also how to ...