Questions tagged [systems-of-equations]

This tag indicates that several equations (of some type) must all hold. Do not use alone! Use in conjunction with (linear-algebra), (polynomials), (pde), (differential-equations), (inequalities) or another tag that describes the nature of the equations being considered.

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417 views

Four squares such that the difference of any two is a square?

I. This post asks to find $4$ integers $a,b,c,d$ such that the difference between any two is a square. As mentioned by my answer, it is equivalent to finding $3$ squares such that the difference of ...
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How do you solve linear least-squares modulo $2 \pi$?

I have an overdetermined system of $m$ equations ($i = 1, 2, \dots, m$) $$\sum_{j=1}^n A_{ij} \, x_j = y_i \pmod{2\pi}$$ where the $x$ coefficients are unknown, and $m > n$. This is, essentially, ...
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Finding a minimal set of equations that determine a variable.

I have a system of $m$ linear equations on $n$ variables, which I'm representing as $Ax=b$, with $A$ an $m\times n$ matrix representing the equations and $b$ an $\mathbb R^m$ vector representing the ...
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How can I guarantee the existence of a solution to this quadratic system of equations?

I have $n$ real quadratic equations and $n$ real variables, $x_i$, of the following form: $$\sum_{i\neq j} a_{ijk}x_ix_j+\sum_ib_{ik}x_i+c_k=0 \ \forall k$$ for $i,j,k\in\{1,\dots n\}$; all ...
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Unifying abstraction of duality between $A - B$ and $A + B$

I'm wondering whether there's an abstraction that unifies the special cases of dual or complementary equations of the form $A - B$ and $A + B$ that I've seen in math. Here are some examples: 1: Even ...
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How to prove existence of a solution of this determinant equation?

Let $D\in\mathbb{R}^{n\times n}$ be a real diagonal matrix where $\sum_i D_{ii}<0$. Let also $R\in\mathbb{R}^{n\times n}$ and $L\in\mathbb{R}^{n\times n}$ be real (possibly) non-symmetric (...
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Matrix permutation-similarity invariants

https://en.wikipedia.org/wiki/Matrix_similarity https://en.wikipedia.org/wiki/Permutation_matrix The determinant and trace (and characteristic polynomial coefficients) are well-known similarity ...
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6 linear PDE for only 3 unknowns?

Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
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Reducing to reduced echelon form in linear algebra seems to do so many useful things - what gives? Why?

What's the underlying thing about reducing a matrix to reduced echelon form that solves so many things for us? Some determinations that can be from reducing to reduced echelon form: Testing linear ...
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Probability to obtain no solutions for a linear system

Suppose $Ax=b$ is a linear system and A is a $n \times n$ matrix and vector $b \neq 0$. Suppose all numbers $a_i$ in $A$ and $b_i$ in $b$ belong to $\mathbb{Z}$ and suppose they are in a fixed range ...
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Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, $b\in\mathbb{R}^n$, $k>0$, and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive function. Consider the system of nonlinear ...
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System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy$ where $p$ is a real constant. Thank you very much cheers
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A method to evaluate functional roots of $e^x$

I've an idea to find exact function $f(x)$ such that $f(f(x))=e^x$. But it involves solving complicated systems of non-linear equations, the skills for which I don't have. Here's how I intend to do ...
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How to combine equations that are describing related physical phenomena?

A quick word about what is going on to put this question in perspective. I am trying to dock a robot onto a charger. Sometimes the robot center is not in line with the charger center. Let's call this ...
Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where $N=S+I$ is the total population. By substituting ...