Questions tagged [algebra]
Abstract manipulation of symbols. Transforming an algebraic expression into the desired form.
291
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How can I check whether a long rational expression contains a minus?
I have some monster expressions, but for simplicity, consider
m = -(-3 + a+3 b-5 c)/(-d -5);
How can I check if "undisplayable" expressions contain ...
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0
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How to find the solutions to simulteneous linear equations in mathematica, in case if the solution to them is not unique? [closed]
I have these three simultaneous equations
b+c-a-d=0; 2b-a+c=0 and c-e=0
I entered the command
Solve[{b+c-a-d==0, 2b-a+c==0, c-e==0},{b,d}]
But I got {}
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4
answers
361
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PolynomialQ behaviour
I am crafting some functions on polynomials that must be in x. But I checked that it is always True whatever variable I use:
...
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2
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Evaluating square roots of quartic powers
I wonder why Mathematica does not act on the powers inside square roots. See the following example:
Sqrt[a^4 b^4 c^4 d^4] does not give
...
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3
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How to get the analytical form of a solution to an algebraic equation?
The real analytical solution of the algebraic equation $x^5 + 10 x^3 + 20 x == 4$ is $x=-2^{2/5} + 2^{3/5}$, how to get it with Mathematica?
I've tried with ...
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Solving algebraic equations perturbatively (using function series)
I have linearised some equations and trying to solve them perturbatively in powers of small parameter $e$. Here is my script
...
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2
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Simplification of quantum ladder operators (ideally using Using NCAlgebra)
I want to do math that involves a series of ladder operators. In the past I've tried to get it working with a 20 year old Mathematica package. But my feeling is that this is a very unused package and ...
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Modern way of simplifying non-commutative algebraic expressions in 2022
I have been asking about how to do modern quantum optics calculations in 2022, and one potential solution is to use a package that uses noncommutative algebra.
There are currently very old questions ...
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Modern way to do quantum optics calculations in 2024?
Quantum Optics can often require messy algebraic simplifications of noncommuting operators (usually ladder operators and commutators). I've asked here about how to do it before.
What are some modern ...
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Solving high dimensional nonlinear algebraic equation numerically
I would like to solve an algebraic equation A=f(A) whose argument A is a large 3D matrix. Ideally, I would not like to flatten the system of equations, before feeding into FindRoot because f(A) ...
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Finding coefficient of vector expression when using SuperLie package
I'm using the SuperLie package and using that have defined a Lie superalgebra with an ordered basis:
...
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2
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Graded commutativity
As we all know, Mathematica has extensive built-in support for computations in free commutative algebras, AKA polynomials algebras. It's also not that hard to make computations in exterior algebras ...
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Expression for an integral
I am facing a problem obtaining an expression for my integral. My code goes in a time-consuming loop whenever I execute this integral. The code is:
...
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3
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Why is the result of TrueQ[(x^n)^m == x^(n*m)] False?
Why is the result of the code below False ?
TrueQ[(x^n)^m == x^(n*m)]
How can I get it to be ...
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Rearranging a simple algebraic expression
I have a polynomial of variables $x,y$, where $|x|<1$ and $|y|<1$. When I apply the Simplify function to this expression, I get an expression of the form
$(x-...
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A script for finding a basis of positive vectors in the kernel of a matrix which is minimal in some sense?
The kernel to the left of the matrix
...
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Transform a list of delayed rules with conditions to a new list of delayed rules
I have a list of rules that implement some abstract algebras, for instance, that of the Pauli matrices,
...
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2
answers
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If x^x^x=y^y, express y in terms of x [closed]
This question is one of my friend think out suddenly and asked me how to do, but I have no idea how to do it, anyone have ideas? (even my friend don't know how to do it, he just think up the question)
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5
votes
1
answer
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Present a logarithm in terms of two logarithms
If $\log _8 3=p$ and $\log _3 5=q$ then, in terms of $p$ and $q$, what does $\log _{10} 5$ equal? I tried by my hand, I get the answer is $\dfrac{3pq}{1+3pq}$. How can I tell Mathematica solve ...
- 1,118
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1
answer
134
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Solve[] isn't producing a correct result
I want to find the exact algebraic roots of the following function in terms of Z, P, Q, <...
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0
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How do I efficiently define rules for non-commutative algebras, (like differential forms)?
Defining rules seems to be quite troublesome, because whatever rule I make has to have the exact syntax ordering of the expression I want to modify.
For example,
take this differential form
...
- 1,055
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0
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How to solve quartic equation modulo a composite? [closed]
I have an univariate polynomial equation over a composite moduli.
Namely the composite is of for $q=(2p)^{2}-1$ where $p$ is odd and $2p-1$ and $2p+1$ are distinct primes.
The modular equation is
$$ax^...
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Move variables to one side of equation
I have a bunch of linear equations of the form
x[1] == 5012 - 5x[3] - 2x[4] + 5x[7]
etc. Specifically there is one linear variable on the left, and the right-hand ...
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How to determine the shape of real quadric surfaces? Use discriminants?
Let $a x^2+2 b x y+c y^2+2 d x+2 e y+f=0$ be the implicit equation that defines
a conic section , where $a,b,c,d,e,f$ are real numbers.
I have known from wikipedia how to use 2 discriminants to ...
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How to define an infinite dimensional algebra with a known basis and multiplication rule?
My goal is to do symbolic calculations in the noncommutative associative algebra generated by two elements $x,y$ satisfying the relation $x.y=q~y.x$. This infinite dimensional algebra has the basis $...
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1
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Validate equalities among Square roots for a simple expression
I am trying to validate this property:
$\sqrt{1-z}=\sqrt{-(-1+z)}=(\text{}\pm i) \sqrt{-1+z}$
FullSimplify[Sqrt[1 - z] == I Sqrt[(-1 + z)]]
I have seen how we need ...
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1
answer
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ParallelTable and NC polynomial simplification
I use NC algebra to apply simplification rules on non commutative polynomials.
I would like to treat a huge number of polynomials, ~200, so I would like to run these computations in parallel.
However, ...
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0
answers
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NoncommutativeMultiply, wedge product and exterior algebra
I would like want to automate some calculations involving wedge products of differential form of different order. Is it possible to define a NoncommutativeMultiply function that has the properties of ...
- 11
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vote
3
answers
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Product involving Kronecker Deltas
I am trying to do some products involving some objects made out of Kronecker deltas. For example, taking an object like $x_{abcd}=\delta_{ab}\delta_{cd}$, where all the indices run from $1$ to $N$, I ...
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Can we ask Mathematica if a reduction of a system to one scalar equation in one variable+ rational representations of the other variables exists?
Nonlinear determined systems have typically several solutions involving square or higher order roots. Instead of solving them, it may be more profitable to reduce them to smaller systems with fewer ...
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How can I solve/verify the following matrix?
Given $ZFZ’ = C$, I have the following set of symbolic matrices:
$Z$ of order $m\times q$
$Z’$ is the transpose of matrix $Z$ of order $q\times m$
$F$ of order $q\times q$
$C$ of order $m\times m$
...
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votes
1
answer
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Resubstituting variables back into the calculated
I have lengthy second order derivative of a function that is defined with multiple lines of variables. When I compute the derivative how can I put back into the computed result the auxilary variables ...
- 1,620
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0
answers
108
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Plug in values in symbolic expressions
With the help of answers on this post: new rule for noncommutative multiplication, defined new noncommutative multiplication:
...
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0
answers
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Standard Monomials
I'm trying to find the number of standard monomials of an ideal. The footprint bound says (1) that the number of solutions to a system of polynomials is bounded by the number of standard monomials of ...
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0
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Mathematica reacts to a wrong condition in a weird way
With apologies, I evaluated the eigenvalues of a matrix under a condition which included wrongly some Falses. The weird result was a failure to obtain the eigenvalues?
...
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Reducing roots of power sums
Often I encounter expressions like for example
Root[ 1 - # + #^2 - #^3 + #^4 - #^5 + #^6 &, 3 ] or
...
- 1,620
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votes
2
answers
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ReplaceAll for expressions involving negative power [closed]
I have an expression:
$exp=1+\frac{1}{x}+\frac{2}{x^{2}}-\frac{3}{x^{3}}$
I am trying to make the following replacement:
$\frac{1}{x^{i}}\rightarrow f(i)$, where f is any arbitrary function.
The ...
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votes
1
answer
367
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Zassenhaus formula in Mathematica
I'm looking for Mathematica implementation of Zassenhaus formula -- given two matrices $A,B$, truncate the expansion of $\exp(A+B)$ from this paper:
$$e^{t(A+B)}= e^{tA}e^{tB}\prod_{n=2}^\infty e^{t^n ...
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votes
2
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Series expansion using binomial theorem in Mathematica
The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by
$$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
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votes
2
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Precise control of fraction expression
It seems that expression for fraction differs
when the number of terms in the numerator is more than 2.
...
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1
answer
59
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How to Simplify expression into partial Trignometric form?
I have an expression with thousands of terms each of them has the form
(1- Exp[-2 a x]) . I want to be able to write it as
2 Exp[- a x] Sinh[ a x] . How do I do this without doing
it for each term ...
- 1,708
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Expressing n in terms of x, given x in terms of n [closed]
Using Wolfram Alpha, I tried finding solution of this:
If n(n - 1)/2 = x,then find value of n. Unfortunately, not getting value for n.
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0
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Polynomial GCD over a ring with composite modulus [duplicate]
I need the gcd of two polynomials,
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0
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0
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58
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Cells stuck at running after running after finding symbolic eigenvalues of a 4x4 matrix
I used the code below to calculate the eigenvalues of a 4x4 matrix symbolically.
...
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0
answers
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Expanding a large dynamic expression involving roots of a degree 4 polynomial
I am trying to find the eigenvalues of a 4x4 Matrix symbolically. Below is the code I am using,
...
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2
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305
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Function to write an algebraic expression in terms of another
So I'm working on a physics problem. I have two functions:
$\rho = \frac{F}{(\beta F^\alpha+1)^\frac{1}{\alpha}}$, $p = - \frac{F}{(\beta F^\alpha+1)^\frac{1}{\alpha}} + \frac{4}{3}\frac{F}{(\beta F^\...
0
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1
answer
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Conditional Expectation of an autoregressive (AR1) Process
Suppose X is a time series that follows the process:
$X[t]=\rho X[t-1] + \varepsilon[t]$
where $\varepsilon[t] \sim \mathcal{N}(0.0, \sigma)$, $\varepsilon[t]$ is IID and $X[0] \sim \mathcal{N}(0.0, \...
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1
answer
77
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Simple Algebra (solving for an unknown)
I am trying to solve this equation for kss with mathematical but the system is failing and so I am hoping to get some pointers here.
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2
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1
answer
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How to get the real part of a complex function with some real coefficients?
I want to get the real part of the function n3[t] (see below) which is the solution to a differential equation.
Solve the differential equation. (Suppose ...
