For a system of nonlinear algebraic equations, how to find the number of solutions to this system? Any related degree theory can be used to determine the number of solutions? Are there any recommended references?

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline b_i & B_i \end{array} \right ] \quad$ where $\quad \exists (\| \cdot \|_\triangle). \forall i. \| B_i \|_\triangle < 1$. Here, $\Omega: \mathbb{R}^n \rightarrow \{1,2,\dots,K\}$ has the […]

According to Derrick’s theorem we can write \begin{align} E &= \frac{1}{2} \int d^Dx \frac{1}{\lambda^2}\left( \nabla \phi_i (\frac{x}{\lambda})\right)^2 + \int d^Dx V(\phi_i(\frac{x}{\lambda})),\\ &=\lambda^{D-2} I_K +\lambda^{D} I_V. \tag{1} \end{align} $\lambda = 1$ must be a stationary point of $E(\lambda)$, which implies that, \begin{equation} 0=(D-2) I_K[\bar\phi]+D I_V[\bar\phi]. \end{equation} I don’t understand is For $D \geq 3$, ($1$) can […]

I’m trying to solve the following system over $\Bbb R$: $\begin{cases}{x^2 +y^2 −z(x+y)=2\\ y^2 +z^2 −x(y+z)=4\\ z^2 +x^2 −y(z+x)=8}\end{cases}$ Adding all the equations gives $2(x^2+y^2+z^2-xz-yz-xy)=14$. This doesn’t look like $(x+y+z)^2$… Do you have some hints?

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 &= x_2 – 314.2 \\ \dot{x}_2 &= 122\sin{x_1}\cos{x_1} – 154.8x_3\sin{x_1} – 0.5x_2 + 201.2 \\ \dot{x}_3 &= 0.64\cos{x_1} – 0.8x_3 + 0.2u \end{split} \end{equation} where $\mathbf{x} = \mathbf{x}(t) = [x_1(t), x_2(t), x_3(t)]^T \in X […]

One of my teachers told me that a fundamental problem in Computer Graphics is that the rotations are not linear. The transations and scalar are linear, but not the rotations. He tried to explain to me why rotations were not linear, but I didn’t understand at all. Not to mention my undertanding about the linearity […]

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What’s an approach to solve this kind of equation?

Do there exist integers $x,y,z,w$ that satisfy \begin{align*}(x+1)^2+y^2 &= (x+2)^2+z^2\\(x+2)^2+z^2 &= (x+3)^2+w^2?\end{align*} I was thinking about trying to show by contradiction that no such integers exist. The first equation gives $y^2 = 2x+3+z^2$ while the second gives $z^2 = 2x+5+w^2$. How can we find a contradiction from here?

I’m working on a prey-predator model. I’m using the following system of differential equations for it: \begin{align} x’&=-a_1x+a_2xy+a_3xz\\ y’&=b_1y-b_2xy\\ z’&=c_1z-c_2xz \end{align} Where $a_i, b_i, c_i >0$. One of the stationary points is $P=(\frac{b_1}{b_2},\frac{a_1}{a_2},0)$. Question: How can I determine the stability of this point $P$? Attempt: First I wrote the equation as: \begin{align} \frac{\mathrm d \underline{v}}{\mathrm […]

I have five points in 3D space which I want to use to form a cylinder in 3D. I assume that these points are exactly on the cylinder. I know the equation for a cylinder is given by an origin and a normal to define the center axis of the cylinder, along with the cylinder’s […]

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