Questions tagged [algebraic-manipulation]
The art of manipulating an algebraic expression into the desired form.
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Redundancy in Solve
Solve[{
a[1] a[2] == a[3] a[4],
a[1]^2 == a[4]^2,
a[1] a[3] == a[2] a[4],
a[2]^2 == a[3]^2
}]
returns
...
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1
answer
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Finding Hamiltonian from a perturbative Action
I am given a perturbative action $$\frac{S}{\mathcal{T}}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+7.11 c_0^3+35.3 c_0 c_1^2+4.66 c_0 \dot{c_0}{}^2+1.32 c_0 \dot{c_1}{}^2-7.57 \dot{c_0} ...
- 181
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2
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Rewriting algebraic expressions as combinations of other expressions
I would like to write x^5 + 1/x^5 in terms of x^2 + 1/x^2, x^3 + 1/x^3 and ...
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How to use System`Private`AssumptionAccess and Assumptions`*?
I see this function is used by some symbols in Assumptions`*, and it HasDownCode and does not ...
- 1,463
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Eliminate works on small system of Equations, but fails on big ones
I am trying to calculate the Hamiltonian of a system in terms of momenta. When applying the Eliminate function on a small toy system it works as expected
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Division by 0 Error when checking whether something is equal to zero
I was running a program and suddenly got Divide::infy messages coming from the following (rather ugly) expression.
...
- 928
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Simplifying RegionUnion call after CylindricalDecomposition
I want to simplify unions of output from CylindricalDecomposition calls.
Consider:
...
- 121
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2
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Efficient algorithm for calculating polynomial that has roots of a certain form
I have looked into this specific question on Math.SE concerning a more "mechanical" approach to finding a polynomial $p \in \mathbb{Q}[x]$ satisfying $p(\sqrt{2}+\sqrt{3}) = 0$. The user MJD ...
- 183
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recognize overall factor in a variable
I have an expression (not necessarily a polynomial) which I know is proportional to an overall power of a symbolic constant, so something like $x^a (...)$. What is the best way to get $a$?
- 415
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Factorize expression by specific term [closed]
I am new to Mathematica but I would like to know how to force Mathematica to factorize an expression by a specific term.
Exemple, imagine I have a term like this:
And I have an expression like this:
...
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Easy upper and lower bounds for curve genus in Mathematica
I am writing a Mathematica code in which I need, at some point, to compute genus of a lot of curves given by polynomial equations in 2 variables in affine coordinates. Unfortunately, this is possible ...
- 141
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2
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Expression of the infinite summation result of a series
The infinite sum of a series should be
$\sum_{n=1}^{\infty} \frac{x^{4 n+1}}{4 n+1}$ $=\frac{1}{4} \ln \frac{1+x}{1-x}+\frac{1}{2} \arctan x-x \quad(-1<x<1)$
However,
...
- 1,365
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showing that CylindricalDecomposition returns partition the space
I have 24 objects returned from eight calls to CylindricalDecomposition, say $exp_1$ ... $exp_24$. The objects are in the three dimensional reals, $(a, b, c)$, so ...
- 121
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3
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Which level of complexity of equations that Mathematica Solve or Reduce functions cannot deal with?
Here I have a system of equations which takes $B$ matrix as input, and output the solutions. The code works very well for 4-rows matrices $B$ that I tried, however it is hard to give solutions in ...
- 245
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1
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How to get this expression in a wanted form
This is the endresult of integration calculation
(I*Log[(b - I*z)/(b + I*z)])/(2*b)
See the question
integral calculation post
This endresult must transformed to a ...
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handling roots in mathematica [duplicate]
I am looking for the solution of:
$\text{Solve}\left[-\frac{\alpha }{\sqrt{1-L}}-\frac{L t}{2}=-\alpha -2,L,Reals,\text{Assumptions}\to 0\leq \alpha \&\& 0<t\right]$
Mathematica is giving ...
- 101
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3
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Factoring a quadratic form associated with a $4 \times 4$ matrix into a sum of four squares
The matrix W = $\left(
\begin{array}{cccc}
5 & 7 & 6 & 5 \\
7 & 10 & 8 & 7 \\
6 & 8 & 10 & 9 \\
5 & 7 & 9 & 10 \\
\end{array}
\right)$
has a ...
- 485
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3
answers
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Transforming implicit solutions of an ODE involving InverseFunction to an explicit form
When I DSolve a nonlinear differential equation:
...
- 97
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Defining a non-commutative algebra
I'm new to Mathematica, and I'm trying to learn the ropes.
I'm trying to write a little boson algebra engine, with basic useful functions such as non-commutative algebra, normal-ordering and vacuum ...
- 11
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1
answer
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Third/Fourth derivative of cross-entropy loss
I need a nice formula for the third (or fourth derivative if it's easier) of cross-entropy loss $\frac{\partial^3 J}{\partial z^3}$ where
$$J(p(z)) = -\sum_i q_i\log p(z)_i$$
$$p(z)_i=\frac{\exp z_i}{\...
- 5,679
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answers
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A remarkable elimination involving the basic reproduction number R0
I have a polynomial pol of third order, whose coefficients depend on 5 parameters, and whose free coefficient is proportional to R0-1, where R0 is a certain rational function of the parameters (the ...
- 896
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Implementing einsum in Mathematica
I've been using Carl Woll's einsum function, but it's missing the ability to handle repeated indices.
For instance np.einsum('i,i->i',w, w) will perform a ...
- 5,679
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1
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Spotting and condensing expressions in output
I am running some symbolic scripts to generate a model problem. One of my outputs is
This is generated by the Mathematica code
or, in Mathematica format (owing to the fancy lettering and formatting ...
- 155
2
votes
2
answers
87
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Speeding up ComplexExpand
I have a code where I have to take the dot product of two lists, that contain some symbols, and then use ComplexExpand. For a long list it take a very long time and ...
- 14.5k
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Fully expanding and distributing a symbolic sum
I have quite a large expression I want to simplify. For example, this can be generated by a recursive definition:
...
- 323
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1
answer
58
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Expand tensor expressions with matrix powers and outer products
I would like to have TensorExpand distribute across repeated matrices involving outer products. This question notes that ...
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1
answer
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Stability analysis of coupled ODE's
I'd like to reproduce Panel A of Figure 7 (red curve) of this paper, which represents the steady-state solutions of four coupled ODE's, and verify this claim that the diagonal branch in the Z-shape ...
user82988
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Solving a system of differential-algebraic equations
I am trying to solve a coupled system of differential-algebraic equations as follows.
First define a list of 6 equations
...
- 313
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3
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How to help Eliminate simplify simultaneous equations
I have a set of three moderately simple simultaneous equations that I'd like to simplify and eliminate a set of variables for (this is a simple example of a more general class of problem - but I'd ...
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2
answers
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Compute real and imaginary part of parametrized complex rational expression
I would like to compute the real and imaginary part of a complex valued rational function $Z_b=\frac{1}{\frac{1}{\frac{1}{C_1 \text{i$\omega $}}+L \text{i$\omega $}+R}+C_2 \text{i$\omega $}}$ that ...
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Collecting coefficients of non-commutative products from an expression
I am trying to manipulate a Hamiltonian with non-commutative operators, $a_j,a_j^\dagger$. After some algebra using NCAlgebra package in Mathematica, I have an expression like, $$H = C_1 (b**a) + C_2 (...
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Wrong limit appears [closed]
Consider the following limit
$$
\lim_{x\to-\infty}x-\sqrt{x^2+7x}
$$
Going through some algebra leads to
$$\begin{align}
\lim_{x\to-\infty}x-\sqrt{x^2+7x}&=\lim_{x\to-\infty}\frac{(x-\sqrt{x^2+7x})...
- 4,138
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Simplify radical expression (a + b - Sqrt[a^2 - b^2])/Sqrt[a + b]
I have the following expansions:
$\frac{a+b-\sqrt{a^2-b^2}}{\sqrt{a+b}}$ and $a>b>0$
Obviously we can simplify it further by hand $\sqrt{a+b}-\sqrt{a-b}$, but Mathematica doesn't know how to ...
- 4,856
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Ordinary differential equation with variable (polynomial) coefficients -form of solution
If you have a differential equation with polynomial coefficients then quite remarkably Mathematica can find a solution. For example for the second order differential equation
$ y''(t)+b y'(t)+\left(\...
- 14.5k
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2
answers
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Get rid of a certain variable in a fraction's numerator
Consider expression $\frac{a - b}{a + b}$. When I apply $FullSimplify[\frac{a - b}{a + b}]$, I get $-1 + \frac{2a}{a+b}$, effectively getting rid of $b$ in the numerator. However, I want to get rid of ...
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Trigonometric simplification .Correct syntax? [closed]
I cannot simplify this expression any further, which should result in 7.
Could you please provide me with the correct syntax, I have tried several without success.
$$Assumptions[A + B + C == 0 ,
...
- 480
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Rewrite a variable in terms of another variable
I have two variables
t = 1/Sqrt[2] (Sqrt[1 + Cos[a]] + Sqrt[1 - Cos[a]] +
Sqrt[1 + Cos[b]] + Sqrt[1 - Cos[b]] + Sqrt[1 + Cos[c]] +
Sqrt[1 - Cos[c]])
and
<...
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1
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Grouping together same exponent terms in a multivariate monomial
This is a rather simple question but for some reason, I am spending too much time on this.
Suppose I have a monomial in 4 variables, each raised to some linear combination of 3 exponent variables: <...
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What is the ComplexityFunction used by FullSimplify in Mathematica 12? [duplicate]
I was surprised to see FullSimplify make my expressions numerically unstable. The result can be summarized below:
...
- 5,679
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Conjugate transpose of noncommutative matrix multiplication
I have:
$$p=iab$$
$p,a,b$ are operators. So in mathematica I am writing this as:
p=I*a**b
Conjugate of above is $$-iba$$
In mathematica I tried ...
- 1,163
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Telling Mathematica to treat square roots as real valued
I often deal with problems which I know are real-valued, but Mathematica skips some simplifications because it doesn't know which branch of Sqrt to take. Is there ...
- 5,679
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Simplifying quantum operators (ladder operators with multiple modes)
Similar to this question, I am interested in working with an implementation of quantum ladder operators.
I've been trying to use this "Quantum Mathematica Package" to solve some lengthy ...
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2
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Formal substitution of exponential factors in an expression
Assume that we have some complex algebraic expression, like
Exp[(a t + b s)/w] ( t Exp[ q t] w + q Exp[q s] Exp[-2 s] + a^2 Exp[3 s] + s t + 1 )
Now, this is just ...
- 143
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Commutator of QFT Operators $[\hat{a}_p, \hat{a}^\dagger_k]$ and 4-Momentum Integrals
I just recently started looking into CAS (i.e. i have very little experience in Mathematica), specifically for doing basic calculations in Quantum Field Theory.
For that i would like to define $\hat{a}...
- 131
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40
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Issues in expanding expressions with formal sums
I have a very long expression involving formal sums to infinity, unfixed functions and several variables.
In this very long expression, I need a typical term, like
$\frac{6 A \sigma \sum _{n=0}^{\...
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How to rearrange terms in an expression to a particular form (e.g. A/w-> A*(1/w)
I have a huge expression for which the numerators are a big mess, involving numbers, variables, formal sums and unevaluted function, let's call the numerators Ai, with i=1,2,...n
This "beast"...
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1
answer
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Expression should vanish under assumptions. How can I use Mathematica to simplify this type of of expression (and similar)?
This is maybe the worst type of question because it is super specific, but I am hoping that some Mathematica function or rule exists (built-in or custom) that can be helpful for simplifying a somewhat ...
- 33
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How to simplify for a ratio of two variables
Say I have an equation like
$$2 z \ln[\frac{a v_{2}^{2}}{b_{2}}]+(v_{1}-2v_{2}) - z \ln[\frac{a v_{1}^{2}}{b_{1}}]\geq0$$
...
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