I have constructed a system of an arbitrary number of ordinary differential equations that describes my model at steady-state. There are ($i+1$) ODEs, with $i$ arbitrary.
Goal: I want to solve the resulting algebraic system (all equations set to $0$) and obtain an analytical expression that...
I need to show that the following expression,
$$a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,$$
is positive
given that:
$1.$ $\ a,b,c,x,y,z$ are positive real numbers
$2. \ \ a>b+x$
$3. \ \ c<b+y$
I...
Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers:
$$dX_1/dt = \ldots\\
dX_2/dt = \ldots \\
\ldots \\
dX_8/dt = \ldots$$
and given that $\lim\limits_{t \to \infty} K(t) \to 0$ for $K(t) = X_3(t)...
Motivation:
I am working with a code that minimizes the objective functional value in an optimal control problem. It takes $A_1,A_2,A_3,A_4$ (the balancing factors for various components of the objective functional) as inputs, and then outputs the values of the state variables, control...
Given $m,n \in \mathbb{N},$ how can I show that the polynomial $x^m+y^n-1$ is irreducible in $\mathbb C[x,y]$?
I'm given the following hint, but I don't follow. Note: I know Eisenstein's Criterion.
*Adapt Eisenstein's Criterion to work in $\mathbb C[x,y]$ by using irreducibles in $\mathbb...
I have this question:
Find all numbers $n\geq 1$ for which the polynomial $x^{n+1}+x^n+1$ is divisible by $x^2-x+1$. How do I even begin?
**So far I have that $x^{n+1}+x^n+1 = x^{n-1}(x^2-x+1)+2x^n-x^{n-1}+1,$ and so the problem is equivalent to finding $n$ such that $2x^n-x^{n-1}+1$ is...
A friend and I have been working on this problem for hours - how can the following expression be simplified analytically?
It equals $\frac{1}{2},$ and we have tried the following to no avail:
1. Substitution of $x = \sqrt{5}$
2. Substitution of $x = 2\sqrt{5}$
3. Substitution of $x =...
Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{b}^{a} f(t)dt \ \bigg| \leq \int_{b}^{a} \bigg|f(t)\bigg| dt,$$ where the first integral is a complex integral, and the second integral is a definite real integral...
Problem: Given $W = \{z: z=x+iy, \ y>0\}$ and $g(z) = e^{2 \pi i z},$ what does the set $g(W)$ look like, and is it simply connected?
Attempt: $W$ represents the upper-half complex plane. And $$g(z) = e^{2 \pi i (x+iy)} = \cdots = e^{-2\pi y}(\cos (2 \pi x) + i \sin (2 \pi x)).$$ (Am I on the...
How can I derive a contradiction from the following nasty statement:
Assume $\sqrt{5} = a + b\sqrt[4]{2} + c\sqrt[4]{4} + d\sqrt[4]{8},$ with $a,b,c,d \in \mathbb{Q}$?
*This is the last piece of an effort to prove that the polynomial $x^4-2$ is irreducible over $\mathbb{Q}(\sqrt{5}).$*
I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root.
How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?
I am working on proving that an equilibrium point of a two-dimensional dynamical system is globally asymptotically stable. The background and justifications are below. I have gotten to the final steps (in bold), but cannot justify it. It seems right intuitively. Can someone navigate the argument...
I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.)----------Proposition: Let $A$ be a countable subset of the open interval $(a,b).$ Then there is an increasing function on...
How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of the extensions?
This question has been crossposted here: abstract algebra - Finding the...
Here is the modified problem:
Let $x \in R - \{0\},$ where $R$ is a domain.
Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$
I need to show that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ or show that there...