Preliminary definitions Let $C$ be a smooth algebraic curve. In Hulek's Elementary Algebraic Geometry, a differential form on an open subset $U \subset C$ is defined to be a section $$ \varphi : U \to \bigcup_{x \in U} m_x / m_x^2 , \ \ \ \ \ \varphi(x) \in m_x / m_x^2,$$ where for each $x \in U$, $m_x$ is the maximal ideal in the local ring at $x$. (This local ring is a DVR, since $C$ is a smooth curve.) $\varphi$ is a regular at $p$ if there exist functions $f_1, \dots, f_l, g_1, \dots, g_l$ regular in a neighbourhood $V$ of $p$ such that $\varphi = f_1 dg_1 + \dots + f_l dg_l $ on $V$. The ..read more

Let $f(x)\in \mathbb{Z}[x]$. The set content of $f(x)=a_{n}x^{n}+ \cdots +a_{0}$ is defined as the greatest common divisor of $a_{0}, ..., a_{n}$ and is denoted by cont$(f(x))$. I want to show that cont$(f(x)g(x))=$cont$(f(x))$cont$(g(x))$. I saw a proof of it, but I have some doubts. This is: We know that $$\text{mcd}\left( \frac{a}{\text{mcd}(a, b)}, \frac{b}{\text{mcd}(a, b)}\right)=1$$ without loss of generality let us cont$(f(x))=1$ and cont$(g(x))=1$ where: \begin{align*} f(x) = & a_{n}x^{n}+ \cdots +a_{0} \\ g(x) = & b_{n}x^{n}+ \cdots +b_{0} \end{align*} Supose that $f(x)g(x)\n ..read more

Maybe this is better suited for Physics Exchange, but I have been self studying line elements and I wanted to check my understanding against the community. I tried to derive the line element for some f(x,y): Consider the function of y = 3sin(x), I parameterized using x=(t) $$ y = 3sin(x) \to x=(t), y=3sin(t) $$ Plugging this into arc length I get: $$ S = \int_{0}^{1}\sqrt{(1)^2+(3cos(x))^2} dt $$ From that I got the line element: $$ ds^2 = (1 + 3cos(x))^2d x^2 $$ Is this right, or am I completely wrong here ..read more

Say we have $F(x+h) = F(x) + hf(x) + \frac{h^2}{2}f'(x) + \frac{h^3}{6}f''(x) + \frac{h^4}{24}f'''(\xi)$ and when we differentiate we get $f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x) + \frac{h^3}{6}f'''(x)$, but we expect a remainder for this aswell. Is there a correlation between the remainder $\xi$ from $F(x+h)$ and the missing remainder $f(x)$? Will $f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x) + \frac{h^3}{6}f'''(x) + \frac{h^4}{24}f''''(\xi)$ or will this be another number at the place for $\xi ..read more

Provide an interpretation that makes all of the sentences in the problem true. F(a), F(b), F(c), ∀x∀y(G(x,y) ↔ G(x,x)), ∀x(F(x) → G(x,x)) My answer is: enter image description here enter image description here Carnap says: Not all formulas are true in this model. Take another look at: ∀x∀y(G(x,y) ↔ G(x,x)). Because G(x,y) <-> G(x,x) , so I write G(,) when two values are the same or different And I make the domain as 0,1,2 because a,b,c have three values. Because F(x) so F(_) can take all the values in domain 0,1,2 For a,b,c I write a sample value from domain. How do I fix G(,)? How do I ..read more

Trying to evaluate these series of equations for graphing functions and which show some interesting relationships, and wondering if anyone has ideas as to the applications in physics, possibly in spherical harmonic expressions in Quantum Mechanics with electromagnetic interactions/fields: f(x) = [e^πR^2 + (Ryberg Unit of Energy in Joules * 10^17) * Sin [(π/(mass of proton in Kg * 10^28 - (√10 - 10^-1))X + [mass proton/mass electron * 10^-4) + (Hartree Energy in Joules*10^3)] I recognize the equation is multiplying the sum of the volume of even dimension n-balls + the Ryberg Unit Energy, which ..read more

If I am given a function $P = ax^3 + bx^2 + cx + d$ and following: $P(0) = P_1$, $P(1) = P_2$, $P^\prime(0) = v_0$, $P^\prime(1) = v_1$. How would I solve for $a$, $b$, $c$ and $d$? Edit: I have tried following: $P(0) = P_1$ => d = P1 $P^\prime(0) = v_1$ => c = v1 However, I am not sure how would I solve for $a$ and $b ..read more

I am looking at this answer to this question: Let $U \subseteq \mathbb{R}$ be open and let $x \in U$. Either $x$ is rational or irrational. If $x$ is rational, define \begin{align}I_x = \bigcup\limits_{\substack{I\text{ an open interval} \\ x~\in~I~\subseteq~U}} I,\end{align} which, as a union of non-disjoint open intervals (each $I$ contains $x$), is an open interval subset to $U$. If $x$ is irrational, by openness of $U$ there is $\varepsilon > 0$ such that $(x - \varepsilon, x + \varepsilon) \subseteq U$, and there exists rational $y \in (x - \varepsilon, x + \varepsilon) \subseteq I_y ..read more

I know the function of an archimedian spiral is r=theta. Is there a way to "squish" the spiral to make it thinner or taller? I'm trying to code something that requires an eccentric spiral ..read more

Here is the correct problem: By comparison with the geometric series, show that $$ e - \sum_{k=0}^{N}{\frac{1}{k!}}=\sum_{k=N+1}^{\infty}\frac{1}{k!}<\frac{1}{N\cdot{N!}} $$ But now I'm not entirely sure where to go. I have tried some things, but I can't see how to connect this with a comparison to the geometric series ..read more