Exhaustion by compact sets eventually include fixed compact set.
Mathematics Stack Exchange
by slowlight
4h ago
Suppose that $\Omega \subseteq \mathbb R^n$ non-empty, open, and $n \ge 1$. We have an increasing collection $(K_i)_{i \in \mathbb N}$ of compact subsets of $\Omega$ with union $\Omega$. For $K \subset \Omega$ fixed and compact, is it necessarily the case that $K \subseteq K_i$ for $i$ sufficiently large? I know that this is true when we have the constraint that $K_i \subseteq \mathrm{interior}(K_{i+1})$ for each $i \in \mathbb N$. I am not convinced that this will be the case without this constraint, but I have had some trouble cooking up a counterexample. I think one could arise if we took ..read more
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Does this type of tensor appear anywhere?
Mathematics Stack Exchange
by Nikita
4h ago
Antisymmetric, or skew-symmetric tensors on a subset of indices are those that get multiplied by $-1$ when any of the indices from the subset are transposed. This type of tensor is widely used in physics and mathematics. Now imagine I have, say, a tensor of type $(3, 0)$ that gets multiplied by $e^{2\pi i\over3}$ when I apply a cyclic permutation to the indices. More generally, let $h$ be an element of order $k$ in the multiplicative group of the underlying field, and say there is a tensor that gets multiplied by $h$ when a certain permutation of order $k$ is applied to its indices. This prope ..read more
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What does a parametric equation mean?
Mathematics Stack Exchange
by desert_ranger
4h ago
I am following the last module of Differential Calculus on Khan Academy, that deals with Parameteric equations. Here are the parametric equations described in the lecture. $x(t) = 5t + 10$ $y(t) = 50 - 5t^2/2$ However, I really don't understand what parametric equations really mean. How do they differ from normal equations. According to Wikipedia: "In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters." I really don't understand what this definition is trying to convey. From what I observed, if two functions shar ..read more
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In an art museum, there are $n$ paintings, $n ge 33$, ...
Mathematics Stack Exchange
by Unknowduck
4h ago
In an art museum, there are $n$ paintings, $n \ge 33$, for which there are used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values ​​of $n \ge 33 $ so that anyway we color the paintings with the above properties we can choose four distinct paintings $T_1$, $T_2$, $T_3$ and $T_4$, so that any color that is used in both $T_1$ and $T_2$, it can be found in $T_3$ or $T_4$. I've been trying to solve this combinatorics problem for some time, but I can't think of what ..read more
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Maximizing $(x-y)^2(y-z)^2(z-x)^2$ where $x^2+y^2+z^2=1$
Mathematics Stack Exchange
by mathhello
16h ago
The problem is to maximize $(x-y)^2(y-z)^2(z-x)^2$ over all $(x,y,z)\in \mathbb{R}^3$ satisfying $x^2+y^2+z^2=1$. Since the domain is compact and the function is continuous, it is certain that extrema must exist. Plus I figured out that the maximum is $\frac{1}{2}$ where $(x,y,z) = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$ by Lagrange multiplier. However, the calculation for Lagrange multiplier was quite long. This problem is from past problems of regional math contest which do not assume calculus knowledge. Is there an elementary way to get the maximum? Thanks in advance for any form of he ..read more
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Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same?
Mathematics Stack Exchange
by agv-code
16h ago
I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student who has just read the definition of free group as a set of words over an alphabet" would know. See for example the answers to this question Is there a simple proof of the fact that if free groups $F(S)$ and $F(S')$ are isomorphic, then $\operatorname{card}(S)=\operatorname{card}(S')?$. I think I have come with such a proof, but I would like to know if it is valid. The proof goes as follows. If A and B have the sa ..read more
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Topological properties vs homeomorphisms
Mathematics Stack Exchange
by Amanda Wealth
16h ago
I'm studying general topology and a question has come to my mind. We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it. I can see the ambiguity of this definition lying in its lacking to specify the language in which the properties are expressed. Anyway, I was wondering if, in some appropriate language, topological properties are enough to capture the notion of homeomorphisms. More precisely, is it true that, if two spaces satisfy the s ..read more
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How to represent $x^n$ as a sum of $P_k:= (x)(x-1)dots(x-k+1)$?
Mathematics Stack Exchange
by pie
16h ago
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$. Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have $$x^n = \sum\limits_{k=1}^n C_{(n,k)} P_k$$ where $C_{(n,k)}$ is just some constant that depends on $n, \ k$, This also give us a way to calculate the first $C_{(n,k)}$ for $1\le n\le 6$. $$x=P_1$$ $$x^2 = P_1+ P_2 $$ $$x^3 = P_1 + 3P_2 +P_3 $$ $$x^4=P_1 +7 P_2 + 6P_3 +P_4$$ $$x^5=P_1+15P_2 +25P_3+10P_4 +P_5$$ $$x^6 ..read more
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Why is this function almost Lipschitz?
Mathematics Stack Exchange
by Kadmos
1d ago
We are still in the saga of solving the 2002 qualifier. This question 6b has stumped me and I am mostly clueless about it: Say $f:\mathbb{R}\rightarrow \mathbb{R}$ is bounded with a finite constant $B$ such that: $$\frac{|f(x+y)+f(x-y)-2f(x)|}{|y|}\leq B$$ Prove there exists $M(\lVert f \rVert_\infty, B)$ such that for all $x\not=y$: $$|f(x)-f(y)|\leq M |x-y|\left(1+\ln_+(\frac{1}{|x-y|})\right)$$ Where $\ln_+(x)=\max \{0,\ln(x)\}$ Intuitively this means that away from $y=x$, $f(x+y)\rightarrow f(x)$ linearly. Close to $x$, it is still true $f(x+y)\rightarrow f(x)$ but it is slightly perturb ..read more
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Maximizing area of the triangle in a quarter circle
Mathematics Stack Exchange
by Etemon
2d ago
The radius of the quarter circle is $6\sqrt 5$ and we assume that $OA= 5$ and $OC=10$. What is the maximum area of the blue triangle? Interpreting the problem statement, I believe that points $A$ and $C$ are fixed and point $B$ can move on the arc. To solve this problem, I assumed that the coordinate of $O$ is $(0,0)$ and then assigned coordinates for each vertex of the triangle: $A(5,0), C(0,10), B(x,\sqrt{180-x^2})$ where $x \in [0, 6\sqrt5]$. Then I applied the formula for the area of the triangle given its vertices, and the problem is reduced to maximizing $$A(x)= \left|25-(\frac52\sqrt ..read more
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