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Number of solutions to $x^2-y^2 = n$ in some rings
I am interested in studying the number of solutions to $x^2-y^2 \equiv n$ in the ring $(\mathbb{Z}/{p^q}\mathbb{Z})[t] / (t^2-\delta)$. Can anyone please help me find resources that could be of great ...
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Prime ideal of infinite height in a Noetherian ring
Let $R$ be a ring. Is it possible that there is a chain of prime ideals in $R$, $$(*)\ \ \ \ \cdots\subsetneq\mathfrak p_i\subsetneq \cdots\subsetneq \mathfrak p_2\subsetneq \mathfrak p_1\subsetneq \...
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Div and curl operators on matrix multiplied vectors
Let $u(x)$ be an $n$-dimensional vector field and $M(x)$ be a matrix of dimension $m \times n$. Are there any elegant general identities for the divergence and curl of
$$w(x)= M(x)u(x).$$
Any identity ...
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Finding more closed form results for $\sum_{n=0}^{\infty}\frac{\alpha^n}{n!}\Gamma^2\bigl(\frac{2n+1}{4}\big)$
I have the following equation, which is valid for $|\alpha|\leq2$. $K(k)$ is the complete elliptic integral of the first kind.
$$\sum_{n=0}^{\infty}\frac{\alpha^n}{n!}\Gamma^2\biggl(\frac{2n+1}{4}\...
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Do universal properties like that of products hold in categories of categories?
I am trying to prove that certain expressions between product categories are functors, to this end I think it would be quite easy if we had something of universal propertiies, say of products in the ...
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Is topologist's sine curve contractible?
Topologist's sine curve $S$ is a good example of a 'wild' space in topology. It is connected but not path connected nor locally connected.
I wonder if it is contractible.
How I am reasoning on this ...
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Generalized “incenter map” is symmetric with respect to permutations of the four input lines
$ABCDEF$ is the complete quadrangle determined by the four lines $AB,BC,CD,DA$ so that $E=AD\cap BC$ and $F=AB\cap CD$.
Reflect $D$ across the line $AC$ to a point $D'$, and reflect $F$ across the ...
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Etale covers of finite type
An étale cover is usually defined as follows:
a jointly surjective family of étale morphisms.
This is short and clear.
However, on the $n$Lab (https://ncatlab.org/nlab/show/%C3%A9tale+cover), the `...
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Is there any intuition for continuous embeddings?
Let $X, Y$ be topological spaces and $X \subset Y$. $X$ is said to be continuously embedded in $Y$ if the inclusion map $i: X \rightarrow Y$, $x \mapsto x$, is continuous. The definition ...
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Fórmula for the parallel transport of a 1-form
Let $(\pi:E\rightarrow M,\nabla)$ be a vector bundle endowed with a linear connection. Let $\gamma:[0,1]\rightarrow M$ be a curve and let $\mathcal{P}_t$ be the parallel transport from $E_{\gamma(0)}$ ...
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On the Stability and Phase Dynamics of a Mono-monostatic Convex Body (Gömböc) with Inverted Optical Metric (negative metamaterial) under neutrino flux [closed]
I came up with a problem that might be interesting for topologists. Problem refer to Gömböc (a convex, mono-monostatic body) constructed from a metamaterial with a negative refractive index ($n < 0$...
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Area Identity of a Tangential Frustum
For a pentahedron with two parallel bases and three lateral faces, the two bases must be similar triangles.
Proof: The three side planes, necessarily intersect at a single point (the apex). Any two ...
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What else do we know about the 'Conway 99 iie' graph (if it exists)?
As per Wikipedia. 'Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common ...
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Area identities for a triangular "prism" circumscribed about a sphere
Consider a convex triangular "prism" in $\mathbb{R}^3$ circumscribed about a sphere.
The three lateral faces are planes tangent to the sphere whose outward normals lie in the horizontal ...
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Pure Imaginary Modified Bessel Functions of the Second Kind (exponential composition yields sine-wave?)
The pure imaginary modified Bessel functions of the second kind, are typically expressed, with $v,x\in\mathbb{R}$ such that,
$$K_{i v}(x)$$
It is known, that $K$ does not have real zeros unless, $v\in\...
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