How can I show that a sequence of regular polygons with n sides becomes more and more like a circle as n→∞? In which fields this concept is applied?
124255
What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided polygon." for example parallax method .
51492
Assuming a series of points in 2d space that do not self-intersect, what is an efficient method of determining the area of the resulting polygon?  As a side note, this is not homework and I am not looking for code.  I am looking for a description I can use to implement my own method.  I have my ideas about pulling a sequence of triangles from the list of points, but I know there are a bunch of edge cases regarding convex and concave polygons that I probably won't catch.
43686
Does there exist a decreasing sequence  of sets ($A_{n+1}\subseteq A_n$) ; $\{A_n\}$ of real numbers such that   $m^*(A_n)&lt;\infty , \forall n \in \mathbb N$ and $m^*(\cap_{n\in \mathbb N} A_n) &lt; \lim_{n\to \infty} m^*(A_n)$ ? Here $m^*$ is the Lebesgue outer measure on real line
100935
The question is - Let $k \geq 1$ be an integer and consider a sequence $n_1,n_2,\cdots,n_k$ of positive integers. Use a combinatorial proof to show that  $\binom{n_1}{2} + \binom{n_2}{2} +\cdots+ \binom{n_k}{2} \leq \binom{n_1+n_2+\cdots+n_k}{2}$  How can i prove this without using induction?  Additionally, For each $i$ with $1 \leq i \leq k$,consider the complete graph on $n_i$ vertices. How many edges does this graph have?  I thought of making graphs and figuring out a relation but still wasnt able to.
22061
If hexagons are formed by joining the vertices of a 20 sided convex polygon such that no side of  hexagon is common with side of the polygon, then how many possible ways are there?   I have no idea where to start. I actually have difficulties when I deal with geometric applications of permutations and combinations.   My approach: Choose 1 point (20 ways), then leave two adjacent points and choose 1 out of the remaining 17 (17 ways) and so on. But I think I can't use the multiplication principle without double counting and even if I double count, I can't find the method to remove those cases. Would someone please help me to solve this question?
122040
Given a convex $n$-gon. The circumcircle is constructed for every triple of consecutive vertices of the polygon. We get the $n$ circles. Select the circle with the largest radius. Prove that the circle contains the polygon.  My work so far: $n=3 -$ triangle - obviously. $n=4 -$ If $\angle B = \max \left\{A,B,C,D \right\}$ then $ABCD \in \omega_{ABC}$  $n\ge 5$. I need help here.
41619
I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one.      For each of the following properties, give an example consisting of two subsets $X, Y \subset \mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \rightarrow Y$ that has the indicated property.         Open but not closed/continuous   Closed but not open/continuous    Continuous but not open/closed         etc... (all cases)   I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces).   I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved.   Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context.
122250
This is exercise 1.9 in Lie Groups, Lie Algebras and Representations - Hall.     Suppose $a$ is an irrational real number.  Show that the set $E_a$ of the numbers of the form $e^{2\pi i n a}$, $n \in \mathbb{Z}$, is dense in the unit circle $S_1$.  Hint: Show that if we divided $S^1$ into $N$ equally sized "bins" of length $2\pi/N$, there is at least one bin that contains infinitely many elements of $E_a$.  Then use the fact that $E_a$ is a subgroup of $S^1$.   My proof of this proposition is as follows.  Since $a$ is irrational, you can determine that the set of rotations $E_a$ is infinite.  Since $S^1$ is compact we can find two $r_1, r_2$ that are within $\epsilon$ of each other.  Then $r_1^{-1}r_2$ is a small rotation of size $\epsilon$.  Now, $r_1^{-1}r_2$ generates rotations that are within $\epsilon$ distance of any point of $S^1$.    That said, I don't believe that the hint suggested in the problem uses that technique.  My knowledge of algebra is not all that strong so I was hoping someone could shed some light on what is being suggested there.
19806
How to show that the sequence $$x_n, n \geq 0, x_{n+2}=|x_{n+1}-x_{n-1}|, n \geq 1$$ with $x_0, x_1, x_2$ positive integers, not all null,  is periodic?  I tried to pick up the square but obtained equalities not helped.  Thanks so much for any suggestion.
2377
Suppose we have an $2^n\times 2^n$ board. Prove you can use any rotation of L shaped trominoes and a monomino to fill the board completely.  You can mix different rotations in the same tililng.
42323
Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that    $$\lim_{n\to\infty}x_n=\sup S.$$   I'm struggling with creating a formal proof for this problem. I understand that this has to be true, but I'm unsure of how to go about proving it.  I said that $\sup S = C$ and $x_n= C - y/n$ where $y$ is dependent on the size of $S$, but this is just a bunch of jumbled ideas and I can't really think of a way to formalize a proof. Any idea, even totally different from mine would be appreciated.
17410
Let $k$ be a field. Let $R$ be the formal power series ring $k[[x]]$. Define $N$ on $R \setminus \{0\}$ as follows: $N(f)$ is the smallest $n$ of which the coefficient of $x^n$ in $f$ is nonzero.  (a) Prove that $R$ is a Euclidean domain with Euclidean norm $N$.  (b) For $a$, $b$, $a+b$ nonzero elements of $R$, prove that $N(a+b)$ cannot be bounded as a function of $N(a)$ and $N(b)$.  (c) Prove that if $a$ and $b$ are two power series such that $b$ does not divide $a$ (and $b \neq 0$), there are infinitely many pairs $(q, r)$ for which $a = bq + r$ and $N(r) &lt; N(b)$.
75801
"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that the manifold boundary of each is equal to its topological boundary as a subset of $\mathbb{R}^n$, namely a sphere. Hint: for the unit ball in Rn, consider the map  $\pi \circ \sigma^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\sigma$ is the stereographic projection and $\pi$ is a projection from $\mathbb{R}^{n + 1}$ to $\mathbb{R}^n$ that omits some coordinate other than the last."  So, I've got (the first part, anyways) of the question done, but my technique was a little different than what Prof. Lee suggested: I considered the ball as infinitely many foliated spheres, mapped each one to a plane using the stereographic projection, and then put the last coordinate as a function of the distance from the north/south pole. However, my solution seemed to neatly avoid any use of $\pi$ as mentioned, and I'm curious if anyone knows how that solution runs.
86753
So I am studying for an exam on Discrete Mathematics coming up in a few weeks, when looking past papers it seems that my lecturer enjoys asking questions of the form   "Find all n-regular graphs on m vertices. Justify your answer."  Where he will replace m and n with actual numbers rather than having us deal with the arbitrary case. However, these questions take me ages since I create, for a fixed m and n, all the n-regular graphs on m vertices that I can think of, check that they are not isomorphic and hope i have found all of them.  Is there a better way of approaching this question?  For example;  'Find all 5-regular graphs on 8 vertices.'  Any help is appreciated.
14842
Well if one looks at the formula of circular permutations $Pc = (n-1)!$  But as we come to that formula, I need a concrete example and an explanation.  Another thing, I have seen that when working with a bracelet, the formula changes. A concrete example please along with an explanation  Finally how the formula is worked when there are repeated elements that are permuted.  Thank you
69583
Assume $C_n$ is the number of triangulations of a polygon with $n+2$ sides. Using a combinatorial proof, show that $(4n+2)C_n=(n+2)C_{n+1}$.  I don't even know where to start with this one. I understand the triangulation after doing some googling, but I don't know where to go from here to prove the above using just a combinatorial formula.
64746
For which $n \geq 3$ is it possible to draw a regular $n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have integer coordinates? I figured out that $n=3$ is not possible. Is $n=4$ the only possibility?