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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Implement using switches the following logic functions

Implement using switches the following logic functions: F(a,b,c,d)= ab + cd F(a,b,c,d)=(a + bc)d' how can i do it ? Digital System
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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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44 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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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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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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1answer
71 views

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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42 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 ...
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41 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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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
46 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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1answer
28 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
46 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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67 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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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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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
24 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
37 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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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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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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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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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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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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65 views

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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49 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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33 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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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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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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1answer
23 views

Relationship between the Carmichael function and Euler's totient function

Let $\lambda$ denote the Carmichael function and $\varphi$ Euler's totient function. Furthermore, let $p$ denote any prime number and $k\in\mathbb{N}$. The wikipedia article about $\lambda$ states: ...
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Does MAPLE not simplify correctly or am I goofing?

While attempting to answer this question with help of MAPLE, something very strange happened, at least according to me. We have the following function: $$ \phi(x,y) = ...
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What “meta-dimension” do algebraic numbers have?

actually what I am asking for is "how many ways do there exist to create a real number out of a sequence of coefficients?" there is the solution of polynomials through radicals, some polynomials can ...
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Proving $n^{97}\equiv n\text{ mod }4501770$

How do we show $$n^{97}\equiv n\text{ mod }4501770$$ for all integer $n$? First of all, I thought I could use Fermat's little theorem or Euler's theorem, but I'm not sure if they are applicable here.
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Changing the “Type” of an object in Magma

The computer algebra system Magma can store groups as one of many different types. For example, using AbelianGroup([2,2]) we obtain the Klein $4$-group stored as type GrpAb. However, we can also ...
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Why $face_w(New(f))=New(in_w(f))$?

Is there anyone who can help me with this problem? Any hint to the solution would be appreciated! Let $f\in \Bbb K[x_1,\ldots,x_n], w\in \Bbb R^n$ and $New(f)$ be the Newton polytope of $f$. Why ...
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60 views

Evaluating the real and imaginary parts of a nasty complex number

This seems like an elementary question, but I was unable to find an clear answer to it. Generally, the real and imaginary parts of a complex number comprised of radicals are not expressible by ...
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1answer
77 views

Computing 2d radially symmetric Fourier transforms (with Wolfram Alpha)

Let's assume that I "know" $\mathcal{F}\{\operatorname{circ}(r)\}(\rho)=\frac{J_1(2\pi\rho)}{\rho}$ $\mathcal{F}\{(1-r^2)\operatorname{circ}(r)\}(\rho)=\frac{J_2(2\pi\rho)}{\pi\rho^2}$ because I ...
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Boolean Expression simplification help

Hi I am new to the board. Taking a computer architecture course and I am having trouble understanding further simplification on a question I got on a previous quiz. When I type in the expression ...
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60 views

Prevent Maple to evaluate before simplify a function

I have been trying to find a domain of $f(x)=\frac{x}{\frac{(x+2)}{(x-3)}}$ using different kind of software ( its clear the domain of this function is $\mathbb{R}\backslash \{-2,3\}$ ). When I tried ...
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Computational complexity of expanding a MacLaurin/Taylor Series

What methods exist to computationally determine the first $k$ coefficients of a function (possibly polynomial or rational polynomial function)? How do Mathematica/MatLab/Maple/etc. solve this ...
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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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In SAGE, what function factors a polynomial whose coefficients are parameters?

In SAGE the function "factor" will factorize elementary polynomials with coefficients in $\Bbb Q$. For example: x,y = var('x,y') poly = x^3-y^2*x factor(poly) SAGE: x*(x-y)*(x+y) ...
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Symbolic computations in finite fields of unspecified order

The general setting is that I want to multiply some matrices (to many to do it by hands) over a finite field. The problem is that these matrices depend on certain parameters taken from the field and ...
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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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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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2answers
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Simple algebra in a differential equation.

I have the differential equation: $$\frac{dy}{dx}=\sin (x-y).$$ Substituting $v=x-y$ and $dy=dx-dv$, I got down to the equation:$$\frac{dv}{1-\sin(v)}=dx.$$ Multiplying the LHS by $\dfrac{1+\sin ...