Newest Questions
1,698,687 questions
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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
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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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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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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Find a simple polyhedron where every face is a triangle and is combinatorially inequivalent to any deltahedra and has all vertex degrees less than 6
Any polyhedron where every face is a triangle with a vertex of degree $≥6$
(for instance, a triangulation obtained by refining near a vertex)
cannot be realized with all faces equilateral.
While high-...
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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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Please prove this question [closed]
Given $𝑎_1 = 3, 𝑎_{𝑛+1} = 3𝑎_{𝑛} + 2$. Prove by induction that $𝑎_𝑛 = 2 \cdot 3^{𝑛−1} + 1$
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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 ...
- 6,605
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Definitions of "inside" and "outside" in topology
I was watching this video from the legendary Cliff Stoll:
https://youtu.be/6Qpfv5y-7WU
Note: this is an "extras" video for which the original is: https://youtu.be/k8Rxep2Mkp8 but I do not ...
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Smooth Urysohn's Lemma [duplicate]
Let $E$ and $F$ be disjoint closed subsets of $\mathbb{R}^n$.
Is there a smooth function $f\colon \mathbb{R}^n \rightarrow [0,1]$ such that $f(x) = 0$ on $E$, and $f(x)=1$ on $F$ ?
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How Do Multiple Lottery Tickets Change the Odds? [closed]
I’m trying to understand how people think about lottery odds when they buy tickets regularly. Some players say buying weekly improves your chances a little, others say it barely makes a difference, ...
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A mathematician want to extend the domain of $\cos$ to $\mathbb{R}$. The mathematician does not feel any particular need to extend the domain of $f$.
I am reading Calculus Fourth Edition by Michael Spivak.
Problem 28 on p.322:
This problem gives a treatment of the trigonometric functions in terms of length, and uses Problem 13-25. Let $f(x)=\sqrt{...
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The point equidistant to all of ($n+2$) hyperplanes in $\Bbb R^n$ (if exists) is a rational function of the hyperplane coefficients?
Motivated by Question 5117602 regarding the rational construction of the incenter for a tangential quadrilateral, I am interested in how this property extends to higher dimensions.
The point ...
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The incenter of a quadrilateral is a rational function of the vertex coordinates
Given only the coordinates of $A,B,C,D$, the incenter can be recovered purely by reflections across diagonals and perpendicular constructions, without ever touching angle bisectors.
Take diagonal $AC$...
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Every subring of a noetherian ring is the global sections of a noetherian scheme?
As shown in this question, the ring of global sections of a Noetherian scheme may not be Noetherian. My question is, let $A$ be a Noetherian ring and $R\subseteq A$ a subring. Does there exist a ...
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