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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Nspire cx CAS - Laplace inverse fails

I'm trying to calculate that easy integral but I get undef. When I replaced $\infty$ with $1000$, I got the right answer. ($e^{-1000}$ is zero roughly). Although this calculator knows that ...
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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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Interpolating transformation matrices

I read not to interpolate transformation matrices by linearly interpolating. Can someone explain to me why interpolating transformation matrices by linearly interpolating the matrix components is a ...
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14 views

Get the closed form of Taylor series with Maple

Is it possible to get the closed form of Taylor series with Maple? The series command can give any given number of terms, but the question is about the closed form ...
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27 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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39 views

Solve nolinear system of equaion with c/c++ [closed]

My system of equation is like this: (x-a1)^2 + (y-b1)^2 = c1 (x-a2)^2 + (y-b2)^2 = c2 I know it is simple using matlab: ...
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Verification of $F(m)^{d} \pmod n \equiv m$ with very large inputs, where $F(m)=m^e$

Does anyone have the computational power to check whether or not $F(m)^{d} \pmod n \equiv m$, where the values of the variables are found below. According to Wolfram Alpha, I found the result of the ...
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53 views

Any algorithm or theorem to decide whether two functions are equivalent? [duplicate]

Any algorithm or theorem to decide whether two functions that are polynomials,rationals and analytic over $\mathbb{N}$ or $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$ are equivalent ?
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25 views

MuPAD - complicated solve() solutions

I am trying to solve an equation for one variable as shown in the picture: Judging from the output of the second command, I should be able to get an expression for $\ddot{\theta}$. I tried ...
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32 views

MuPAD - rearrange an equation

I have a huge expression, say $f(x_1,x_2,x_3) = 0,$ and I want MuPAD to try to express $x_1 $ in terms of $x_2$ and $x_3$ Is this possible?
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65 views

How can I plot $y^x$?

How can I plot $y^x$? To keep things simple and to not have another $z$ variable on the other end of the equation, let's assume $y^x=10$. As long as that value is not $0$, the curve we get should look ...
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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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25 views

Variance of discrete probability distribution

I was wondering how I should calculate the variance of the following discrete probability distribution: $$P(y = 0|X) = w + (1-w)e^{-\mu}$$ $$P(y = j|X) = (1-w)e^{-\mu}\mu^{y}/y! \qquad j=1,2...$$ ...
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20 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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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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114 views

using wolfram alpha to solve a system of nonlinear differential equations

Will Wolfram Alpha solve a system of nonlinear differential equations with initial values and graph the solutions? Essentially, I want to hand Wolfram a first-order system with variables x_1, x_2, ...
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1answer
26 views

Decomposition of a polynomial over generators of an ideal

Let $f$ be a polynomial in six variables, say, over complex numbers, and $l_1$, $l_2$ are some linear forms in the same variables. If I know that polynomial $f$ belong to the ideal generated by $l_1$ ...
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55 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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28 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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58 views

how to use hilbert function in gap system with loadpackage singular

how to calculate hilbert function as it do in singular code singular code ...
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1answer
32 views

Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
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There is a closed form for $\sum _{n=1}^{\infty }{\frac {{{\it J}_{0}\left(\,\alpha\,n\right)} {{\it J}_{0}\left(\,\beta\,n\right)}}{{n}^{2}}}$?

Using the method showed here proposed by Olivier Oloa with simplifications proposed by Anastasiya-Romanova, it is possible to prove that $$\sum _{n=1}^{\infty }{\frac {{{\it ...
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226 views

What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?

Using Maple I am obtaining the numerical approximation $$0.5902373619$$ Please, let me know what is the closed form. Many thanks.
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What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\rm J_{0}}\left(2\,n\right)} ^{2}}{{n}^{2}}}$?

Using Maple I am obtaining $$\sum _{n=1}^{\infty }{\frac { {{\rm J_0}\left(2\,n\right)} ^{2}}{{n}^{2}}} = 0.09845497463 $$ Please check it. Many thanks.
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How to expand quadratic equations in Octave/Matlab?

I have some column vectors, and I put them in a row, so I have $[a \, b \, c]$. Now I want to get a matrix of the form $[a^2 \, b^2 \, c^2 \, ab \, ac \, bc]$. It is like expanding $(a+b+c)^2$, but ...
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1answer
55 views

Square-free factorization of polynomials over finite fields

For any $f\in\mathbb{F}_q[X]$, I want to derive an algorithm which computes a factorization $$f=\prod_{i=1}^kf_i^i\tag{1}$$ with square-free polynomials $f_i$. My Ideas: If $f'=0$, we're done ...
2
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1answer
49 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
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13 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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1answer
55 views

Chinese remainder theorem for polynomial evaluation

Let $R$ be a euclidean domain, $m_0,\ldots ,m_{k-1}\in R$ be pairwise coprime and $m:=m_0\cdots m_{k-1}$. The Chinese remainder theorem states: $$\varphi:R\to R/(m_0)\times\cdots \times ...
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37 views

CAS expression - Solve equations with $\sum$ and $\infty$

Can a TI89 or other CAS calculators solve this? I tried it on my classpad 330 did not work solve for p $$p = \sum_{j=0}^\infty p^j \frac{2^je^{-2}}{j!}$$ solve for p $$p = \lim_{z \to \infty} ...
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1answer
69 views

Programming PARI/GP to do a sum

I'm trying to compute the following sum in PARI/GP $C=\sum_{n=1}^{\infty} \frac{g(n)}{n^2}$ where $g(n)$ defined as $$g(n)=(-1)^r, \qquad r=\text{number of even indexed prime factors of $n$}$$ By ...
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94 views

Implementing a function in PARI/GP

I want to define a function: $$g(n)= \begin{cases} +1 & \text{if $n=1$},\\ +1 & \text{if $n$ is an odd indexed prime}, \\ -1 & \text{if $n$ is an even indexed prime},\\ (-1)^r & ...
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57 views

How to detect carry and overflow?

Let's say that 0011 + 0111 = 1010 How to detect whether the operation generate carry and overflow?
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how to solve two's complement question?

i tried 9 D + (-10 D) 9= 0000 1001 10= 0000 1010 Reverse 10 = 1111 0101 and add 1 become 1111 0110 after that add up 9 D + (-10 D) == 0000 1001 + 1111 0110 but the answer is equal to 1111 1111 ...
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54 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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1answer
28 views

Number conversion in decimal fraction

980.85D convert to hexadecimal number = 3D4 . ?? how to solve the answer after the decimal point? Thank you in advanced.
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1answer
65 views

Marking the roots of a quadratic function in Scilab

I have 2D plotted a simple quadratic function in Scilab and now have to mark the roots with an X. Is there any simple way of doing that? I have written a function that calculates the roots and ...
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0answers
31 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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39 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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1answer
46 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
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196 views

The J programming language: is it useful for mathematics?

I just stumbled upon the J programming language, which has the description: J is particularly strong in the mathematical, statistical, and logical analysis of data. It is a powerful tool in ...
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2answers
64 views

Analogue of Fermat's primality test for polynomials and irreducibility

We've got Fermat's primality test to test if a number is probable prime. Is there an analogous test for polynomials in $\mathbb{F}_{p^n}[X]$ and irreducibility?
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Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
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1answer
58 views

Number of monic irreducible polynomials over a finite field

Let $\mathbb{K}=\mathbb{F}_q$ and $\nu_n$ denote the number of monic irreducible polynomials over $\mathbb{K}$. It holds $$\nu_n=\frac{1}{n}\sum_{d\mid n}\mu\left(\frac{n}{d}\right)q^d$$ What I need ...
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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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1answer
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How many $\overline{a}\in\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$ pass the Fermat and Miller-Rabin primability tests?

Let $$\text{F}_{91}:=\left\{\overline{a}\in\left(\mathbb{Z}/n\mathbb{Z}\right)^\times:91\text { passes the Fermat primality test to base }a\right\}$$ and ...
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1answer
44 views

Risch differential equation algorithm by Bronstein

I was implementing the algorithm by Manuel Bronstein for solving the Risch differential equation. My question is: What does Bronstein mean by "Order" in the algorithm poly_DE (exponential case, ...
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2answers
73 views

Some questions about similar matrices

Two matrices A and B are similar, if and only if there exists an invertible matrix C with $A=C^{-1}BC$. A necessary condition for the similarity is, that the characteristic polynomials coincide. I ...
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If there is an $a\in\mathbb{Z}$ with $a^{n-1}\equiv 1\mod n$ but $a^{\frac{n-1}p}\not\equiv 1$ for all primes $p\mid n-1$, then $n$ is a prime

Let $n\in\mathbb{N}$ with $n\ge 3$ and $a\in\mathbb{Z}$ such that $$a^{n-1}\equiv1\text{ mod } n\;\;\;\wedge\;\;\;a^{\frac{n-1}{p}}\not\equiv1\text{ mod }n\;\;\;\forall p\in\mathbb{P}:p\mid n-1$$ ...
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60 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 ...