Questions tagged [symbolic-computation]
Numeric computation usually uses floating point numbers. Symbolic computations use symbols, and can give exact answers, such as $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$. Mathematica, Maple, and Geometry Expressions all use symbolic computation, when desired. An online source is WolframAlpha.
289 questions
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Is there a general criterion for when a function has an elementary inverse? [duplicate]
For some functions (like $f(x)=x+e^x$), we know an inverse exists by monotonicity, but that inverse is not expressible in elementary terms. Is there a general mathematical framework or theorem (beyond ...
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Eliminating all roots from expression doesn't change zeros?
I was reading the paper The Algebraic Degree of Geometric Optimization Problems and here they say that when given an expression like $$\frac{x-a_1}{\sqrt{d_1}}+\ldots+\frac{x-a_n}{\sqrt{d_n}}=0$$ we ...
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Constant field is preserved in elementary differential extension?
I'm studying symbolic integration. Liouville's theorem (the version I've learned) states that for an elementary function $f$, if $f'=g$ for some $g$ lying in some elementary differential extension $E=...
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Reference request: Symbolic integration and Complex integration are consistent?
Background:
I'm studying symbolic integration and I've covered Liouville's theorem and its proof in class. I'm now trying to prove for example $\exp(\exp(x))$ is not elementarily integrable.
The ...
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Viability of using a CAS for "exact" physics simulations
I've been analyzing a type of periodic physics simulation that looks like this:
By varying the angles of the two line segments, you can find configurations with different periods:
I've used high-...
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Symbolic computation of solution of determining equations in Lie symmetry
For higher order ODEs, the length of the prolongation and thus the determining equations in Lie symmetry scales with the order. So it is not convenient to determine/solve those equations manually. ...
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Expressing Integrals of the form $ x^n T(x)$ as a sum of derivatives (Need Advice)
I've been working on a method I call the J-Sum Method, which expresses integrals of the form
$$
\int x^n T(x)\, dx
$$
as finite sums involving derivatives of both $x^n$ and a transformation of $T(x)$, ...
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Solving a mixed system of 4 linear and 4 quadratic equations
I'm struggling to solve this system of equations by hand, even sympy can't solve it (it just keeps running for hours). I am not interested in a numeric solution (I already have that), I'd like an ...
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Proving $f(x_2) < \overline{l}$ for a Convex Function with Parameters $u, l, z$
I am working with the following setting in a proof involving convex functions and parameter-dependent inequalities. I would like to prove analytically that:
$$
f(x_2) < \overline{l} \quad \text{for ...
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Are there general methods to compare the largest real roots of two integer-coefficient polynomials using only their expressions?
Question
Are there general methods to compare the largest real roots of two integer-coefficient polynomials using only their expressions?
Example 1:
Consider:
$P_{1}\left ( x \right )= x^{5}- 20x^{...
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I have a system of $6$ matrix equations, where the matrices are $64 \times 64$. How can I approach this problem in a more efficient way?
I think this question is more of a mathematics question rather than a programming question, read below for further details.
Goal
I have a system of somwehat big ($64 \times 64$) matrix equations of ...
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Conditions under which a system of polynomial inequalities has exactly $18$ real solutions without explicitly solving it
In particular, I need to confirm (rather than simply conjecture) that the following parameterized system of polynomial inequalities has exactly $18$ distinct non-negative solutions:
$$\text{sys} ≔ \...
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Meaning of AlphaGeometry’s `eqangle` geometric property
This a basic question about Google DeepMind AlphaGeometry source code. This program uses a neural language model + a symbolic deduction engine to solve geometry problem. In particular, the symbolic ...
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Computing polynomial relation ideals (for model invariants)
I have a finite list of polynomials in several variables. For example, $L=\{ab,ac,cd,bd,\cdots\}.$
I am interested in finding model invariants, which I understand to be polynomial relations among the ...
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Computing the antiderivative of $\frac{1}{\sqrt{A^2+2 A B e^{C z}+B^2 e^{2Cz}-\beta^2}}$
Problem
I am attempting to compute the following antiderivative:
$$
I =\frac{A \sqrt{(A-\beta)(A+\beta)\left(C^2+K^2\right)}}{|K|} \int \frac{d z}{\sqrt{A^2+2 A B e^{C z}+B^2 e^{2Cz}-\beta^2}}
$$
—...
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