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.

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21 views

Identifying Hamiltonian dynamics in 2D

While studying a physical system I've encountered an two dimensional autonomous ODE defined on a torus. The vector field describing it is a messy expression involving rational functions of ...
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Partial derivative of a heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can ...
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15 views

Solving first-order linear recurrences with polynomial coefficients in terms of roots

Suppose $K$ is a characteristic zero field with decidable equality, and $a(n), b(n), c(n) \in K[n]$ are polynomials with all roots in $K$ (i.e., these polynomials split over $K$). I am trying to ...
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35 views

Symbolic points in an elliptic curve over $\mathbb{Q}$ of the form $(u/e^{2},v/e^{3})$

Suppose I have an elliptic curve $E : y^{2} = x^{3} + D$ over $\mathbb{Q}$ , where $D$ is a symbolic constant (an integer). I want to define two points in the curve of the type $P = (u_{1}/e_{1}^{2}, ...
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61 views

Symbolic point in an elliptic curve over $\mathbb{Q}$ in SageMath

I want to study the behavior of polynomials sitting in the co-ordinates of multiples of a rational point of an elliptic curve over $\mathbb{Q}, $ $E(\mathbb{Q}).$ Suppose if we take a point $P = (s,t) ...
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102 views

How to use Octave with real numbers and syms at one time

I want to use GNU Octave to solve my equations. Equations contain syms variables and vectors with decimal numbers in them. How properly solve these equations without getting warnings that im doing ...
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58 views

Integral of Squared Sum of Sine [closed]

Let's say we have the following function $$ \int \left( \sum_{i = 1}^\infty a_i \sin(b_ix)\right)^2$$ Assuming that this series converges (it is actually a Fourier Series of the Lagrangian of the Wave ...
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23 views

Coordinates of the largest small decagon

I've been trying to compute the abcissa $x_1$ of the first point of the largest small decagon in this slide show (p.13) using Maple. I let $x_1=\frac{2p}{1+p^2}, x_2=x_1+\frac{2q}{1+q^2}, x_3=x_2+\...
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26 views

How to prove the following optimization result from Mathematica?

Currently, I am working on a decision problem in Telecommunication. This problem is reformulated under a form of Optimization problem that is: $$\begin{gathered} {\text{Minimize}}\,\,\,A \hfill \\ ...
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57 views

Algorithm for expressing Gröbner basis in terms of ideal generators

Given a polynomial ring and an ideal $$A \supset I = (f_1, ..., f_m)$$ there are plenty of implementations of an algorithm (e.g. Buchberger's) that produces a Gröbner basis $$G = (g_1, ..., g_n)$$ and ...
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46 views

How to prove a transcendental equation is never zero or sometimes zero?

I am an engineer by profession. Recently, while working on beams, I obtained the following transcendental equation: $$1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$. I was wondering if this expression can ...
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28 views

How can you show that $(p \rightarrow q) \land (q \rightarrow r) \leftrightarrow (p \rightarrow r)$?

Obviously, $(p \rightarrow q) \land (q \rightarrow r) \rightarrow (p \rightarrow r)$ due to Hypothetical Syllogism, but how about the converse?
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Negative multinomial notation doubt

I am working with the negative multinomial and I have a doubt about a parameter. The probability mass of a $d$ dimensional count vector $y = (y_1, . . . , y_d)'$ under a NegMN distribution with ...
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58 views

Effectively decidable vs effectively computable functions on the real numbers

Let $f : \mathbb{N}^n \to \mathbb{N}$ a function on the natural numbers. We call such a function effectively computable if there exists an effective algorithm that can compute $f(x_1,...,x_n)$ given $...
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Problem evaluating Wolfram Symbolic Integration Result

****** Note: In response to the comment below, I've modified the problem ****** I'm trying to evaluate the following indefinite integral $$ \Large\int x^{-\frac{3}{2}} e^{-\frac{K}{x}\Big[(x-a)^2+(x-b)...
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Handling elementary but messy computations in proofs

Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious ...
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1answer
51 views

Why does the function $\frac{1}{x^{2222}}$ seem to have *two* asymptotes at around $\pm0.744$?

When screwing around with the Dirac-delta function in Desmos i noticed that when i multiply it by a really low-degree rational function $$f(x)=\frac{1}{x^i}$$ , the resulting function has the domain $$...
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Compute closed-form expression for the integral of $-\frac{2x}{x+s}+\frac{x^2}{(x+s)^2}$ against Marchenko-Pastur density

Let $\gamma \in [0,1]$, $s \ge 0$, and define $a:=(1-\sqrt{\gamma})^2$ and $b := (1+\sqrt{\gamma})^2$. For $a \le x \le b$, define $g(x) := -\dfrac{2x}{x+s}+\dfrac{x^2}{(x+s)^2}$ and $f(x) := \dfrac{\...
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Computation of $\int_{t_-}^{t_+}\dfrac{\sqrt{\left(t_+^2-t\right) \left(t-t_-^2\right)}}{2 \pi {\gamma} (t+\lambda)^2}\,dt$ using the Maxima tool.

Let $\gamma \in [0, 1)$ and $\lambda \ge 0$. Define $ I(\gamma,\lambda)=\int_{t_-}^{t_+}\dfrac{\sqrt{\left(t_+^2-t\right) \left(t-t_-^2\right)}}{2 \pi {\gamma} (t+\lambda)^2}\,dt$, wheere $t_{\pm} :=...
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How to write that the only integers that have a multiplicative inverse are $1$ and $-1$, in symbolic form

Could it be something like for $(1,-1) \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$, such that $xy = 1$?
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what is the meaning of this element $\varphi(f|_Y)?$

I have some confusion in symbolics . My confusion is given below marked in red colour what is the meaning of this element $\varphi(f|_Y)?$ My main confusion about the symbolics is $(f|_Y)$?
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What is this hybrid symbolic-numeric algorithm for solving non-linear ODEs called?

I'm an engineer so please forgive my lack of knowledge in exact mathematical terminology: For linear differential equations, it is possible to use the principle of superposition or convolution ...
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90 views

Factor SYMBOLIC third degree polynomial

I apology in advance for the long post... Say I have a very general 3rd-degree polynomial $$x^3 + a_1x^2 + a_2x+a_3$$ I would really like to factor that polynomial as a product of a 1st order and a ...
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68 views

Finding $b$ such that $y=mx+c$ is tangential to $y=b^x$.

I was thinking about the following problem: Find $b$ such that $y=mx+c$ is tangential to $y=b^x$. My work so far: To satisfy the condition we need to find $x$ such that $\frac d{dx}\left[b^x\right]=\...
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1answer
66 views

How to find solve for second order pde with initial conditions using Wolfram Mathematica?

I have next task: $$ \frac{\partial^2 u}{\partial x \partial y} = 0,~ u(x,x^2) = 0,~ \frac{\partial u}{\partial x}(x, x^2) = \sqrt{|x|},~|x| < 1 $$ I write this, but it don't work: ...
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57 views

How to use the Wolfram Language or another tool to find a second order pde solution with initial conditions?

I want to find a solution the Cauchy problem using the Wolfram Language or some other tool. I have next task: $$ 3\frac{\partial^2 u}{\partial x^2} + 8\frac{\partial^2 u}{\partial x \partial y} - 3\...
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pipe (vertical bar) symbol in function

could you please explain me what pipe symbols | means in bellow's function?! $$ force = -R_B * D(v_a^B|v_a^B|)*c_D $$ where we have introduced a notation $ D(b) = diag(b_x; b_y; b_z) $for a vector $...
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129 views

A very heavy computation

I'm doing calculations about elliptic curve, which involve the sum of two rational points and birational transformation. I need help regarding the symbolic substitution below, where $u,d,x,y$ are ...
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1answer
37 views

Relation between symbolic computation and symbolic dynamics

From what I read about the fields symbolic computation and symbolic dynamics , I expect these two fields to be having vivid connections . I would like to know what are some of the prominent topics ...
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2answers
59 views

What are $c$'s in antiderivative of double integral in Wolfram Alpha

I would like to solve the following double integral: $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x y)^{-\theta - 1} dx \hspace{1mm} dy$$ Using Wolfram Alpha for symbolic algebra, it gives a ...
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81 views

What is parametric equation of a cylinder with an arbitrary norm in Cartesian system?

I just need to intersect a plane and a cylinder for an industry application and latter we would fit the intersection curve on to some sample data we have and that supposed to provide us with some ...
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2answers
61 views

Pull `t` out of trig functions and solve it

I'm trying to come up with an analytical solution for inverse kinematics for my robot; but some of my trig has left my brain! The robot is very simple; with one rotary and one prismatic joint. I'm ...
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61 views

Symbolic sum similar to Zeta function or harmonic number

(I have no idea why this was closed, nor did I receive any information/email regarding what details or clarity were missing. I have rendered the Mathematica into mathematics.) copied from mathematica....
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How do I extend SymPy pretty printing for new structures in Jupyter notebook?

Note 1: I am asking this on Math Stack Exchange rather than Stack Overflow, where most SymPy questions are, because StackOverflow doesn't have MathJax, making it very difficult to pose the question. ...
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Construction of the Steinhaus probability space

Q: Please let me know if the following sketch of the build-up of the Steinhaus probability space $(\Omega,{\cal A},P) = ((0,1), {\cal B}((0,1)), \lambda\vert_{[0,1]})$ is correct, as far as it goes: ...
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277 views

Open source software for calculation of eigenvalues of symbolic matrix

I have following matrix \begin{bmatrix} -\alpha & 0 & \beta & \gamma\cdot\omega_m \\ 0 & -\alpha & -\gamma\cdot\omega_m & \beta \\ R_r\frac{L_h}{L_r} & 0 & -\frac{R_r}{...
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81 views

Exact SVD algorithm of matrix with symbolic entries

I am making a library with symbolic computations which supports matrices. A matrix may have symbolic entries (eg be over the ring of polynomials of a variable $x$ ie $\mathbb{Q}[x]$). I have ...
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51 views

Evaluating $\int_0^{\infty} x^r \exp(e^{-s x}) e^{-ax^b}dx$

Can you analytically evaluate the integral $$ \int_0^{\infty} x^r \exp(e^{-s x}) e^{-ax^b}dx, $$ where $s,r,a,b>0$ are constants, $e$ is the exponential?
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Simplifying $\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}$

Simplify $$\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}\,.$$ I tried very hard but I am not being able to solve it easily I opened up everything and ...
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1answer
78 views

GAP - Is it possible to work with symbolic expressions?

Is GAP capable of symbolic calculations? For example, I would like to be able to expand and simplify long algebraic expressions such as $(ab+c)^4(a+3d)-bd+11$, define a matrix $\begin{pmatrix} a & ...
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How to simplify the polynomial $T=(a+t_1)+(a+t_1)(a+t_2)+\cdots\prod_{i=1}^n(a+t_i)-(a+a^2+\cdots a^n)$

Is it possible to simplify the following polynomial? $$T=(a+t_1)+(a+t_1)(a+t_2)+\cdots\prod_{i=1}^n(a+t_i)-(a+a^2+\cdots a^n)$$ What I think: We can first consider a special case when $t_1=t_2=\cdots=...
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1answer
30 views

Simple upper bound for $t(2\Phi(\alpha/t) - 1)$, where $\alpha > 0$ and $t \in (0, 1)$

Let $\alpha > 0$ and $ t \in (0, 1)$. For simplicity, take $\alpha=1$. Let $\Phi$ be the normal cumulative distribution function. Of course, the core of the problem is the term $t\Phi(\alpha/t)$. ...
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2answers
105 views

Closed form solution of Integral with many "free variables"

I have a pretty complicated expression that I'm interested in integrating. There's a lot of parameters, so it looks pretty involved: $$ \int_{-\infty}^{\infty}d\Delta\frac{W \sqrt{\frac{\log (2)}{\pi ...
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Inconsistent formulas when solving recurrent equations using sympy.rsolve_poly

I am getting inconsistent formulas when using sympy.rsolve_poly to solve recurrent equations. This first piece of code solves $a(n+1) = a(n) + 4n^3$ with $a(0) = 0$...
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Are there hybrid (discrete-continuous) combinatorial structures? Semantic parsing of multimedia data.

Are there hybrid (discrete-continuous) combinatorial structures, which can grow (develop) according to the discrete and continuous rules at the same time: i.e. some part for some time can grow ...
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180 views

Expression simplification in sympy -- symbolic integration

I am trying to integrate $$p_1=\int_0^1\int_z^1\int_y^1\frac{6x^2}{(x+y)(x+y+z)}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ which arose in connection with an earlier question. When I plug it into sympy,...
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3answers
32 views

How to solve for unknown symbols in a base-3 number system?

Imagine that someone has a base-3 number system which is represented by A, B, and C. A, B, and C correspond to our usual 0, 1, and 2, but you do not know which is 0, which is 1, and which is 2. ...
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1answer
21 views

F5 algorithm for non-regular sequences

Faugère proves in his first Paper about the F5 algorithm the termination of the algorithm for regular sequences and mentions that some slight changes can be done to adapt this algorithm for non-...
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1answer
13 views

How can I read the multinominal naive bayers in simple english?

I trying to studying about text classifier but I have some problems to understand the math representations of scikit.learn implementation: How can I describe this using simple english?
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38 views

Is there anything online like the old "calca"?

Calca was a tool for Windows/Mac where you could type formulas with markdown text interspersed, like: # My work a=7 b=2+a^2 b=> 51 Where it would dynamically ...

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