Questions tagged [symbolic-computation]
The symbolic-computation tag has no summary.
37 questions
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Indefinite integral of the product of two Chebyshev polynomials
Can someone tell me if the following two integrals have generic formulas?
(1) $\int T_m(ax+b)*U_n(cx+d)\ dx$;
(2) $\int x*T_m(ax+b)*U_n(cx+d)\ dx$;
where $T_m()$ and $U_n()$ are the m-th and n-th ...
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2
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1k
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How to evaluate $\int_0^{\frac{\pi}{2}} \ln\left(x^2+\ln^2(a \cos x)\right) dx$ for arbitrary $a>0$?
This question has also been posted on MSE.
I tried to find generalization for the integral for $a>0$
$$\Omega\left(a\right)=\int_0^{\frac{\pi}{2}} \ln\left(x^2+\ln^2(a \cos x)\right) dx$$
here we ...
- 711
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Modern directions in Computational Algebraic Geometry
I've been having a good time working through Cox, Little, and O'Shea's Using Algebraic Geometry. In general I am interested in computational aspects of algebraic geometry. I am wondering what new big ...
1
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49
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Multipoint evaluation in Lagrange basis on subset smaller than degree
Setup. Let $\mathbb{F}$ be a finite field with a multiplicative subgroup $E = \{e_1, \dots, e_k\}$ of order $k$. Given a list $y = y_1, \dots, y_k\in \mathbb{F}$ let $p$ be the unique polynomial of ...
3
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237
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Admissibility condition of wavelet functions
After a badly formulated question, I decided to make a new post searching for help.
The basic problem is the follows: I have a wavelet function $\psi(t)$ (real or complex) and would like to compute (a)...
3
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163
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Solving an underdetermined system of linear equations among the rationals, in the neighbourhood of an approximate solution
I have a system of linear equations $Ax=b$. Extremely underdetermined, for concreteness $x \in \mathbb{R}^{17,000}, b \in \mathbb{Z}^{156}$. $A$ is sparse, integer, full rank. I have a very precise ...
- 31
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Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]
Let $a>b>0$.
Suppose we want to minimize
$$
f(x)=(x-a)^2+(1/x-b)^2,
$$
over $x>0$.
Equating $f'(x)=0$ leads to the quartic equation
$$
g(x)=x^4-ax^3+bx-1=0. \tag{1}
$$
Question:
Is the ...
- 6,861
1
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1
answer
219
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Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$
Disclaimer. Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE.
Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) :=...
- 7,043
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1
answer
183
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Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$
Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set
$$
\alpha := \sup_{(x,y) \in C} ax + b y.
$$
Question. ...
- 7,043
11
votes
1
answer
694
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Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?
Consider the quartic system in four variables $a,b,c,d\in\mathbb R$:
$$-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac).$$
Does this system admit rational solution with $$abcd(c^2-d^2)(a^2-b^2)(a^2-c^2)(b^2-d^2)\...
- 1
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55
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Computer verification for hyperbolic trigonometry
I am currently writing a paper that requires some lengthy computations using basic hyperbolic trigonometry. So, several hyperbolic figures appear, and we apply the law of sines and so on in order to ...
- 541
3
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1
answer
393
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Symbolic powers of a prime ideal of height one
Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal ...
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225
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Advice on how to deal with an elementary "long-to-prove" statement
I am not entirely sure if this question totally fits here. If it doesn't, I apologise in advance.
In a paper I've been working on, we have a very elegant result which, when forgetting about the ...
- 2,288
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0
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184
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How to decide if an algebraic number is a root of a given polynomial?
Let $p$ be a polynomial with rational coefficients and $\alpha = \sqrt[n]{q}e^{i2k\pi/m}$ for some natural numbers $n,m,k$ and a rational number $q > 0$. Is there an effective algorithm for ...
- 147
4
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251
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Effective bounds for a Bertini-type result
Suppose $X$ is a projective subvariety of $\mathbb{P}^n$ of codimension $r$ over $\mathbb{C}$, defined set-theoretically by $r$ homogeneous polynomials $P_1,\dots,P_r$ of degree at most $d$. By ...
- 537
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460
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Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative
I vaguely remember a book/some lecture notes which introduce integration algorithms such as Risch algorithm by first giving a list of quasi-algorithmic way of evaluating symbolic integrals. (For ...
- 1,795
2
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1
answer
294
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Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional ideals
This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...
- 707
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Differences between GAP and MAGMA [closed]
GAP and MAGMA are computer algebra systems. What are the objective differences between the two?
Which capabilities are not shared?
How do they compare on facilities for working with character tables?...
- 147
2
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1
answer
117
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CAS implementing free algebras with involution
Is there any software that easily allows to make symbolic computations with involutions and homomorphisms? I need to define a product in an associative algebra with an (abstract) involution and ...
- 3,242
27
votes
5
answers
7k
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Minimal polynomial of cos(π/n)
I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
- 1,501
5
votes
1
answer
354
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speeding up Gosper and WZ algorithms
In our ongoing work to speed up symbolic summation and other similar algorithms in Sagemath, we notice that naive implementations of Gosper and Wilf-Zeilberger (a.k.a. WZ) algorithms are usually quite ...
- 14.7k
8
votes
2
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3k
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Fast Symbolic Linear Algebra CAS?
I am a regular user of Mathematica, Julia, and MATLAB but I am looking for something different. The problem I am trying to solve in Mathematica only requires (dense) linear algebra to specify but is ...
- 464
6
votes
1
answer
504
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Finite field analogue of Chebotaryov theorem on roots of unity?
Chebotarev's theorem on roots of unity says that all the minors of a prime-length DFT matrix over the complex numbers are nonzero. I was wondering if there was an analogue for finite fields.
More ...
- 115
4
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148
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Algorithmic quantifier elimination over p-adic fields
It is known that the first-order theory of p-adic fields is decidable, and that the p-adics admit elimination of quantifiers. What is the state of the art in algorithmic aspects of quantifier ...
- 1,051
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967
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How to compute the normals to Costa's minimal surface?
I am trying to draw Costa's minimal surface in high resolution using the PovRay raytracer. For this I need to compute points on the surface as well as the normals. It is relatively easy to compute the ...
- 51.6k
1
vote
0
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Testing functional equivalence
We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
- 11
2
votes
1
answer
1k
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Finding zeros of a multi-variable nonlinear trigonometric function
I am trying to calculate analytic solution (or locus) of zeros of a very large multi-variable function which is consisted of thousands of nonlinear trigonometric terms. All the variables are real ...
- 123
5
votes
1
answer
759
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The existential theory of the reals
Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...
- 53
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0
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120
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Tools for "bound guessing"
I have a somewhat complicated symbolic expression of the form $\frac{J-a+\frac{q}{a}}{J(J-a)+q}$, where $J,a$ and $q$ are themselves affine functions of four other variables $d,r,c,s$, and I want to ...
- 7,092
2
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1
answer
466
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Computer algebra system (CAS) with good re-presenting or transformation support
Such heavy-weight transformations as expanding or factoring are provided by most of CAS-es, but what about light-weight, but a useful transformations, like "reorder some terms to make expression more ...
- 29
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291
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What became of PoSSo and FRISCO
I know PoSSo and FRISCO were pretty big projects involving many European universities.
Interestingly, I couldn't find much information about these projects
(the the top of the PoSSo homepage says "...
- 1,151
2
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2
answers
2k
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Solving a system of equations/inequalities that have trigonometric functions on the left-hand side
Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?
Ex) Find $x,y,\theta \in \mathbb{R}$ that ...
- 23
5
votes
1
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871
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Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ?
Please give suggestions about soft to make symbolic computations with NON-commutative variables.
Typical examples I am interesting - Capelli identities
http://en.wikipedia.org/wiki/Capelli'...
- 26.4k
1
vote
0
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symbolic diagonalization of a matrix
Hi,
I am looking for algorithms that can perform a diagonalization, in a symbolic way,
of a given matrix. I need to find a similarity transformation, if it exists. Desired features of the algorithms ...
- 11
3
votes
2
answers
600
views
What are some resources discussing mathematical notation?
I'm looking for resources discussing mathematical notation, the theory, the philosophy, the distinct advantages of various notations. Stuff about notation for computer algebra systems is interesting ...
- 31
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votes
2
answers
2k
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The easiest symbolic integration method to try implementing.
Hello! I wonder how hard is it to implement more or less general symbolic integration algorithm (number of lines in a certain language)? Maybe someone here did this or knows some good blog posts ...
- 451
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0
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Why is mechanical differentiation so hard to get right?
This question is related to this question on differentiation/integration which asks why differentiation is mechanical but integration is an art. The answers given all make a huge assumption: that one ...
- 12k