Questions tagged [algebraic-manipulation]

The art of manipulating an algebraic expression into the desired form.

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Simple Question: How to simplify what is inside the Sin function like below?

I have this code of Mathematica (1/(2 (fj - fk) π)) Cos[(fj - fk) π T + (fj - fk) π (T + δt)] Sin[(fj - fk) π T - (fj - fk) π (T + δt)] There are a few ...
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42 views

Solve the system using elimination What are a, b, and c in the quadratic equation, ax^2+bx+c=y given the [closed]

Solve the system using elimination What are a, b, and c in the quadratic equation, ax^2+bx+c=y given the points: (2,17), (-2,9), and (1,6), please solve the question using steps e.g. (step 1, step 2)?
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1answer
32 views

Changing initial conditions in a recursive polynomial family definition. Then, finding the generation function for this family

I'm following a paper trying to find a way to repeat the done computations using Mathematica. I'm beginner using Mathematica and I already read that: I could define functions by recursion; I could ...
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35 views

Testing Matrix inequalities

I was wondering how one might check whether matrix inequalities are true/false the same way it can be done for scalar variables (like in this post). Say I want to test the obvious inequality A,B are ...
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1answer
60 views

Counting terms in recursive operation

Suppose $X$ is an algebra and $T :X\to X$ is linear function. Let $L:X\to X$ be a function satisfying the following property $$L(a \cdot b)= T(a)\cdot b + a \cdot T(b)+b\cdot T(a)+T(b)\cdot a\tag{*}$$...
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87 views

Commutation, symmetrizer and duplication matrices

The following 3 matrices are useful when viewing matrices as vectors, known as commutation $K_n$, symmetrizer $N_n$ and duplication $G_n$. They are usually defined by their matrix relations below. $$ \...
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2answers
243 views

How to write a positive function as sum of positive terms only

A multivariate polynomial function can be written in a certain form (x^m (1-x)^n y^k + ... ) if the function is Positive in the whole domain and contains a finite ...
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4answers
291 views

How to generate all combination of multiplication of multiple variables (raised to some powers)

I want to generate all the possible combinations (commutative) of a few variables but also raised to some fixed powers. Lets take the following example: I have three variables ...
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2answers
85 views

How to simplify an equation corresponding to an ellipsoid? [duplicate]

The equation 29/36 - x/2 + x^2/4 + y/9 + y^2/36 - (4 z)/9 + z^2/9 == 1 is equivalent to ...
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57 views
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How to compute the factors in Bezout's Identity

If I have two polynomials $p(x),q(x)$, PolynomialGCD[p,q,x] can give me their greatest common divisor. But how to compute the polynomials $a(x),b(x)$ such that $d(x)...
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1answer
50 views

How do you convert an ordinary equation into a parametric equation? [duplicate]

It's a 3D linear equation {P1: 2x-y-3z+2=0,P1: x+2y-z-6=0} Are there any methods that can convert this linear equation to its parametric form. Ps:the parametric form is {x=7t, y=-t+14/5, z=5t}
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50 views

How to use Collect to group negative terms?

For example, I have: $-d^2-k^{d+1}+d k^d+d+k-1$. I want to get: $-\left(d^2+k^{d+1}+1\right)+d k^d+d+k$.
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69 views

Solving a coupled system of ODEs

I have two coupled ODEs for $T(x)$ and $t(x)$: $$\frac{d^2 T}{d x^2}-\beta (T-t)+K=0 \tag 1$$ $$\frac{d t}{dx}-\alpha(T-t)=0 \tag 2$$ $\alpha, \beta$ and $K$ are constants $>0$. Also, it is known ...
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1answer
57 views

How to get the result in terms of a previously defined expression?

Say I have a vector X[n_] := Table[Subscript[x, i], {i, 1, n}] and a matrix of connections: ...
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2answers
75 views

Trouble verifying a solution to a differential equation

I am reading a physics book which has discussed one approach towards solving the differential equation $$ \frac{d^2 x(t)}{dt^2} = Cx(t) $$ as follows: Using Mathematica to solve the equation. I tried ...
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1answer
94 views

Numerical integration of MeijerG function with variables

I applied the integration, How to solve the function which is mentioned below. here, 'theta' is the only variable. q13, q1,q2,p,qi1 are all variables will take value from the loop ranges from 0 to 2, ...
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Symbolic Set Difference

Suppose I have two symbolic set containing 2-points like A={(m,n)| for m,n belonging to whole numbers}. So m,n=0,1,2,... and I have another set containing 2-points ...
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1answer
138 views

Derivative of real antisymmetric matrix in mathematica

Is it possible to find the derivative of components of a real antisymmetric matrix using index notation? Eg: I have a very large real antisymmetric matrix. Then from Matrix Cookbook, we know the ...
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1answer
84 views

Converting an equation to the form $ax+by=c$

I have a linear equation that comes as a result of applying orthogonality on a non-homogeneous boundary condition with two unknowns $C_1$ and $C_2$. Can someone help me to express this equation in the ...
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1answer
136 views

Simplifying the final coefficient expression of the solution to a Laplacian

I have been trying to solve a boundary value problem analytically which involves a three-dimensional temperature Laplacian over a parallelepiped. In the final step of my solution, using the two non-...
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1answer
96 views

Solving a system of polynomial equations - can I be sure that the number of solutions, and the solutions are correct?

I need to solve a system of two polynomials with integer coefficients in two variables, $\{Q_1(w,z)=0,\,Q_2(w,z)=0\}$, and want to compute all real solutions. I'm able to run ...
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1answer
64 views

Counting real roots of bivariate polynomial

For a proof, I'm working with a polynomial $p(x,m)$ and I need to show that: For any $m\in (0,6)$, $p_m(x)\equiv p(x,m)$ has a unique real root in $(0,1/2)$. I find that this is true by using Reduce ...
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How implement Dual Quaternion algebra in Mathematica?

How can we manipulate and operate with dual quaternionic variables in Mathematica? Since a dual quaternion can be expressed like this $\hat q = a + \epsilon b $ where $a,b \in \mathbb{H}$, or ...
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1answer
108 views

How to express this radical in terms of Golden Ratio?

(1/3)*Sqrt[(1/11)*(-3 - 2*Sqrt[5] + 3*Sqrt[178 + 82*Sqrt[5]])] $$ r=\frac{1}{3} \sqrt{-2 \sqrt{5}+3 \sqrt{82 \sqrt{5}+178} \over 11} $$ I need to express this ...
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1answer
44 views

Define trace-like function (e.g. cyclically invariant) involving operators without built-in meaning

I am interested in defining a function, say $\mathrm{Tra}$, that imply operators without build-in meaning, say it involves products with the $\cdot$ (Esc . Esc), ...
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1answer
42 views

Derive arbitrary steps between two expressions

Can someone show me the syntax to find the algebraic transformations between: $\sqrt{1+(\frac{dy}{dx})^2}dx$ and $\sqrt{dx^2+dy^2}$ And more generally, how to get the algebraic derivation between ...
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1answer
67 views

Resolving a sum after making a substitution

Consider the following sum as expressed by $s3$: ...
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3answers
72 views

Mathematica inverting algebraic output order

This is a somewhat silly requirement, but would help a lot in my workflow. Whenever I output an algebraic expression which contains a number, mathematica always orders the number first whenever ...
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1answer
49 views

Replacing an indexed term

Consider the expression: s[i_, n_] := Sum[(e[c[j]] + e[z[j]])*(l[i, j] + m[i, j]), {j, 1, n}] I'm trying to replace e[c[3]] by c[3] and have attempted ...
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32 views

Expanding complex expression and replacing variables in the result

When I'm finished expanding my expression I would like the result to be expressed again in terms of the original variables (like α, S1, T1, T2). I've tried replace, ...
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1answer
62 views

Collecting terms from an indexed sum

How can I get Mathematica to collect terms from an indexed sum? My application is somewhat convoluted, but this minimum code captures the idea: ...
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2answers
115 views

Matrix vector multiplication of trigonometric functions [closed]

My problem is to show the expressions in the photo (from this book) are equivalent using Mathematica. The code below is not giving the desired results. What I have so far should reduce to the column ...
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1answer
55 views

Similar power merge

How can I tell Mathematica to write {x^z1*y^(-a1 - z1 - z4)*z^(-a1 - z1 - z4)} as (x/y)^(z1)(yz)^(-z4) I mean combining terms with similar power?
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1answer
108 views

Simplifying algebraically

Is there a way to derive an simplified expression from other equations. Or create an alternate form such that the main equation takes other equations and manipulates the main function. Here is what I ...
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68 views

Releasing multiple nested Hold in a single expression simultaneously

Trivia: I need to take a coefficient list of a long list of complicated expressions of the form $\prod c_{ij} \left( p_{i,j}(x_1,x_2) \right)^n$ where $n$ is positive integer, and $i,j$ label ...
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38 views

How do I substitute an expression into another?

I'm trying to substitute equation (1) to equation (2) in a way that mathematica chooses the best and simpliest form of a result. To clarify the problem and make it less generic, here is the specifics ...
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1answer
66 views

How to do a long division over a multivariate polynomial in Mathematica

Does anyone know how to do a long division of a multivariate polynomial over another multivariate polynomial to effectively find the remainder, hopefully with fewest terms and/or lowest degree, in ...
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1answer
53 views

Expression rewriting

i have the following expression ...
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29 views

How do I get the maximum the maximum degree of a polynomial? [duplicate]

For example applied to y^2*x^3 + z^2+1 it should give 5 due to the y^2*x^3component.
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2answers
74 views

Collect for nonlinear terms

Is there a way to expand Collect, so that it does not only separate equations by polynomial order, but by general nonlinear functions? For example ...
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2answers
193 views

Defining a*b=-b*a

I am fairly new to using Mathematica and was wondering if it's possible to define something along the lines of: $a*b=-b*a$ a sort of antisymmetry or skew symmetry if you will. I would like to do ...
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1answer
83 views

Finding Dual Plane Curve Using Eliminate

I want to find the dual curve of the projective plane curve $$F(x,y,z)=(x^2+y^2+z^2)x+t(x^3+y^3+z^3)=0$$ where $[x,y,z]$ is homogeneous coordinate in proejective 2-space $\mathbb P^2$. The dual ...
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1answer
53 views

cancel out the common terms and factor out the common terms and extract the coefficients

the following code fail to multiply each equation by dt, since dt does not seem to do any jobs. it remains as a common factor, I would like it to cancel out the 1/dt term and also multiply the rest of ...
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2answers
214 views

Denoting large recurring parts of an equation by a variable

Suppose I have formula like the following: (1 - x) (1 - y) + ((1 - x) (1 - y))/(z + 2) Obviously, (1 - x) (1 - y) is found a couple of places in the formula. ...
3
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1answer
48 views

How do I verify if an equation is correctly re-arranged?

Let's say for pen and paper re-arrangement of some equation, sometimes getting a single step wrong and then wasting time on subsequent steps can be a large time sink. My goal is to be able to verify ...
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1answer
29 views

Multiplying two expressions with a specific function, efficiently

Paraphrasing, the problem I am trying to solve is the following. I am given two "matrix valued" sums ...
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0answers
63 views

how to find derivative of an equation of matrices with element-wise multiplication or division

For an equation like this: $$ M= \left(\begin{array}{c} A\:x-B \otimes y\end{array}\right) \oslash C $$ where $\otimes$ is an element-wise multiplication and $\oslash$ is an element-wise division. ...
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0answers
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Manipulating a system of equalities into particular form

newbie here just trying to learn the Mathematica syntax. I have a list of equations, some of them definitions, others relationships, as follows: ...
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1answer
65 views

Leibniz rule with unknown orders of imput

I'm trying to find a simple way to define a bilinear and Leibniz functional/map "$B(\cdot,\cdot)$", that eats two functions $f(x),f(y)$ or products theirof and produces the following, $$ B(f(...

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