Questions tagged [algebra]
Abstract manipulation of symbols. Transforming an algebraic expression into the desired form.
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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
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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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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 ...
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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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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
...
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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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2
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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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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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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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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 ...
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3
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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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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 ...
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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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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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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
...
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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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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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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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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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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 ...
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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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Polynomial GCD over a ring with composite modulus [duplicate]
I need the gcd of two polynomials,
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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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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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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^\...
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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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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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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 ...
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2
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How to get the algebraic form of cellular automata
In atlas.wolfram.com (see here http://atlas.wolfram.com/01/01/views/173/TableView.html, for example) we can get the algebraic form of any given elementary cellular automaton in 1D.
Is there a way to ...
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FullSimplify not working when simplifying a complex number
Here is a snippet of the code I am working on, which requires me to get the Real part of the Harmonic Oscillator wavefunction.
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FullSimplify command not showing any output
I am trying to FullSimplify an expression and in the end I am not getting any output displayed. The process runs for sometime and stops after making a sound. The ...
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2
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How to define a new algebra using TagSet?
For example, I would like to define the algebra of quaternions. I coded something like
...
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How to define an operator like ⊕？
For example, I define the circle plus as
a_\[CirclePlus]b_ := a + b^2;
a__\[CirclePlus]b_ := (a[[1 ;;-2]]\[CirclePlus]a[[-1]])\[CirclePlus]b;
But when I try
...
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How to define functions using $Z_2$ coefficients and $Z_N$ variables, e.g, realizing $1+1=0$?
A minimal example is to realize the following:
$x+x = (1+1)x = 0$, and $x^{N+1}y^{N+1}=xy$,
where $N$ is a given positive integer.
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Separating the Numerator & Denominator
It's been years since I last programmed in Wolfram & for the first few days, I felt like I've been feeling around the dark; now that I've mostly forgotten how to code it.
Again, I could be ...
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Reducing Polynomial Functions over a Finite Field
Over $\mathbb{F}_{4}$, for example, I am looking for a function that will, for example, reduce the polynomial function (not an element of the finite field itself) $$x^5 + 6x^4 +x^3 + 1 \rightarrow x^2 ...
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How to understand the very long confusing output of a command?
I have a complicated expression with many parameters
...
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Tropical Operation Conversions: Multiple Operations
I apologize if my syntax is bad. I looked around, but didn't see anything on this in the stack exchange.
I would like to be able to transform a classical real-valued polynomial to a tropical ...
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How to get the extension degree
The real number field cannot express $\sqrt{2}$, so we extend $2$ times to get $Q(\sqrt{2})=a+b\sqrt(2)$. But it cannot express $\sqrt{3}$ still. So we are going to extend it $2$ more times to get $Q(\...
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How to speed up Resultant?
I am experimenting with the polynomials Y and T:
i1 = 9; i2 = 4;
Y = Sum[ x^i y[i], {i, 0, i1}];
T = Sum[ x^i t[i], {i, 0, i2}];
Timing[Resultant[Y, T, x]]
Whith ...
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Solving System of six Non-Linear Equations in Six Unknowns
I've been trying to solve the following system of equations using NSolve and the code has been running for close to 24 hours already. Is this normal? Is there a quicker way to solve these equations? ...
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How to solve this equation analytically?
Why can't Mathematica 13 solve this equation in radicals?
Solve[x^5 - 5*x^4 - 10*x^3 - 10*x^2 - 5*x - 1 == 0, x] // ToRadicals
I also tried to use ...
user86117
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Define a new rule for multiplication
I am trying to implement a new rule for noncommutative multiplication:
$$ x * q^n * y \to q^n * x * y, $$ where $\ast$ stands for noncommutative multiplication and x and y are random products in ...
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Simple algebra question
Am I missing something here? This does not resolve to True.
Clear[x,y]
$Assumptions={x>0,y>0}
FullSimplify[(Sqrt[x] + Sqrt[y]) == Sqrt[x + y + 2 Sqrt[x y]]]
...
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