Activity for The Amplitwist‭

Type	On...	Excerpt	Status	Date
Answer	—	A: What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface? The pathological embeddings arise because an "allowed graph" is a very general object. The goal of the assumption TM1 is to restrict attention to precisely the kind of allowed graphs that are described in the question as being > slightly more general than an undirected pseudograph in that it allow... (more)	—	6 days ago
Question	—	How do I unambiguously define the facial circuits in a $2$-cell embedding of a graph into a surface? Suppose that $\Gamma$ is a connected, locally finite graph that is embedded into a closed, connected surface $M$. The faces of this embedding are the connected components of $M - \Gamma$ (we choose to denote the image of the embedding also by $\Gamma$). Let us assume that the embedding is such that... (more)	—	17 days ago
Question	—	Is there a $\Delta$-complex structure on the sphere with less than three $0$-simplices? In Hatcher's Algebraic Topology, a $\Delta$-complex structure on a topological space $X$ is defined as follows. Here, $\Delta^n$ denotes the standard $n$-simplex in $\mathbb{R}^{n+1}$, and $\overset{\circ}{\Delta}{}^n$ denotes its interior. > A $\mathbf\Delta$-complex structure on a space $X$ is ... (more)	—	20 days ago
Question	—	Why does the method of separating variables work? One of the methods to solve a partial differential equation is to use separation of variables. For example, consider the heat equation: $$ ut - a^2 u{xx} = 0, \qquad 0 < x < L,\ 0 < t, $$ with the boundary conditions $$ u(0,t) = u(L,t) = 0, \qquad 0 < t, $$ and initial condition $$ u(x,0) ... (more)	—	about 1 month ago
Question	—	What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface? I am reading the following paper: G. A. Jones, and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307 (MR0505721, Zbl 0391.05024). The authors are interested in imbeddings of an allowed graph $(\mathcal{G},\mathcal{V})$ into a connected, oriente... (more)	—	about 2 months ago
Answer	—	A: If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$? > […] I would have commenced with defining $\dfrac{x}{n}$ as $\dfrac{1}{m}$, which is more intuitive than "let $m = n/x$". I confess I don't see the difference between the two statements. In changing the variable in the limit from $n$ to $m$, you keep $x$ fixed; after all, you are trying to prove ... (more)	—	about 3 years ago
Question	—	Does every divergence-free vector field arise as the curl of some vector field? I was introduced to the concepts of gradient $\nabla f$, curl $\nabla \times F$ and divergence $\nabla \cdot F$ in an introductory course on calculus during my undergraduate studies. There I learnt that for any scalar function $f$ on $\mathbb{R}^2$ or $\mathbb{R}^3$, we have $\nabla \times (\nabla f)... (more)	—	about 3 years ago
Question	—	What surface do I get by attaching $g$ handles as well as $k$ crosscaps to a sphere? I recently found out that there is a classification of compact connected surfaces that says that every such surface (or, $2$-manifold) is homeomorphic to either $Sg$, the sphere with $g \geq 0$ handles, or $Nk$, the sphere with $k \geq 1$ crosscaps. If my understanding is correct: - attaching a ... (more)	—	over 3 years ago
Question	—	Post revisions are displayed without rendered Markdown but with rendered MathJax When I view the post history on this question of mine, I notice that the Markdown is not rendered, but the MathJax is rendered. But, I guess I would expect neither Markdown nor MathJax to be rendered when viewing the post history? Even better would be if we could have options for viewing the post his... (more)	—	over 3 years ago
Question	—	Given a triangle with squares on two sides, the line segments joining the centres of the squares to the midpoint of the third side are equal and perpendicular I am reading Tristan Needham's Visual Complex Analysis (2012 reprint, OUP), and in $\S$1.III.3: Geometry, the author gives a geometric proof of the following fact, shown in Figure [12b] on page 16: >![Figure[12a] shows an arbitrary quadrilateral with squares constructed outward on each of the side... (more)	—	over 3 years ago
Answer	—	A: How can I deduce which operation removes redundacies? Let's take a look at the computation once more. The number of ways of arranging $3$ people in a line is $3! = 3 \cdot 2 \cdot 1 = 6$. That is, for any choice of $3$ people from the group of $8$ people, there are $3!$ ways to arrange them in a line. That is, there are $3!$ redundancies per choice o... (more)	—	over 3 years ago
Answer	—	A: Marketing Math Codidact One long-standing issue on Math StackExchange is the policy on "problem-statement questions", or PSQs in short. The current policy there is that questions must provide sufficient context, and there is a close reason specifically for questions that lack context. There are highly active users on Math S... (more)	—	over 3 years ago
Question	—	Product of empty set of elements vs. product over empty indexing set — is there any difference? I am reading Lang's Algebra (3rd ed., Pearson, 2003). In $\S$I.1 Monoids, on page 4 the author defines the meaning of and notations for products of finitely many elements of a monoid as follows: > Let $G$ be a monoid, and $x1, \dotsc, xn$ elements of $G$ (where $n$ is an integer $> 1$). We define... (more)	—	over 3 years ago
Question	—	Why does the decimal expansion of $1/(10n - 1)$ have this neat pattern? I was playing around with the reciprocals of some positive integers and found these interesting patterns: $$ \frac{1}{19} = 0.\overline{052631578947368421} $$ Now, this repeating decimal can also be obtained by "concatenating" the powers of $2$ successively to the left as follows: \begin{a... (more)	—	over 3 years ago