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9

The source is likely referring to the English word "homeomorphic" and has nothing to do with mathematics. Notice how closely the definitions map to dictionary.com's homeomorphism: noun similarity in crystalline form but not necessarily in chemical composition. Mathematics. a function between two topological spaces that is continuous, ...


7

One of my favorite textbooks is Klaus Janich's Topology, and he has a nice motivation for compactness I feel, namely why we should care about. This is in addition to my comment about compact subsets of a Hausdorff space being essentially like finite point sets. But he writes: In compact spaces, the following generalization from "local" to "global" ...


6

Two topological spaces are homeomorphic iff there exists a mapping as described in (2) between them, a continuous 1-1 map $f$ from one onto the other space whose inverse $f^{-1}$ is also continuous. Because of this condition, both $f$ and $f^{-1}$ are called homeomorphisms, i.e. maps that preserve the underlying topological structure of a space. Thus the ...


6

It's not your fault, this is not a notation and it's not standard at all. I think the book is trying to tell you that $g$ is not defined at 8, by telling you that there's no value $g(8)$, showing with that vertical arrow where it would have been if it was defined.


4

The idea is that you first define the Fourier transform on $L^1$ and $L^2$. Then for $f\in L^p$ for some $p\in (1,2)$, write $$f(x)=\underbrace{1_{\left\{\left|f(x)\right|\geq 1\right\}}f(x)}_{f_1(x)}+\underbrace{1_{\left\{\left|f(x)\right|< 1\right\}}f(x)}_{f_2(x)}.$$ Then $f_1\in L^1$ is the tail part and $f_2\in L^2$ the body and you extend the ...


4

$\varphi(n)$ is the number of positive integers not larger than $n$ that are coprime to $n$. There are no positive integers not larger than $0$, so by definition, if we were to define $\varphi(0)$, we would want it to equal $0$. This agrees with $$\varphi(n)=n\prod_{\text{prime} \ p\lvert n} \left(1-\frac{1}{p} \right), $$ where the product runs over ...


3

The notion of a multi-set is not exact opposite of set. The fact is that in a multi-set, "the notion of importance to presence of multiplicity of same element" is exact opposite of the same notion in a set. Every set is a multi-set, with highest multiplicity 1. Here is a good example of importance of multiplicity in some natural ways.


3

$\phi$ is a vector-space homomorphism which is injective. That is, $\phi$ gives us a way of viewing elements of $V$ as elements of $S$.


2

When I arrived at university, my professor of mathematical analysis (twenty years ago, in Italy, the graduate program in mathematics used to have no calculus course at all, but directly mathematical analysis; we used Rudin's book) told us that the generic high school student believes that every function is of class $C^\infty$. Of course now you know that ...


2

Not every function is differentiable. Say, $f(x) = 0$ if $x<=0$ and $f(x) = 1$ if $x>0$. $f$ isn't differentiable at 0. Or, take a look at the Weierstrass function, which is continuous everywhere and differentiable nowhere.


2

In the article they borrow the term from, the authors use $t$-disjoint to mean the edges are at distance at least $t$. Two edges in this particular graph $G$ that induce a subgraph of two disjoint edges are actually at distance at least $3$ (as opposed to $2$, as you might think), because the graph is bipartite.


2

The first definition is bizarre and is not at all standard mathematical terminology. Maybe "homeomorphic" can be used with that informal meaning in a non-mathematical context (though I haven't found any such usage by googling; the only other usage I can find is a different technical meaning in chemistry). In short, don't worry about that supposed first ...


2

One could talk about $y=1/x$ being a solution of $y'=-y^2$ on $(-\infty,0)\cup (0,\infty)$; there is no problem with generalizing the concept of a solution in such a way. It's just not a useful generalization. What we want to eventually get is not a solution, but the solution: the one and only one that satisfies certain initial (or boundary) conditions. ...


2

In compositional data analysis, the distance function $d(x,y) = |\ln(x) - \ln(y)| = \left| \ln\left( \frac{x}{y} \right) \right|$ is known as the log-ratio distance, and it was introduced by J. Aitchison. But also the log-normal distribution involves this distance function. If parameterized with the geometric mean $m$, the probability density function of the ...


2

The way people make up words for things is far more haphazard than you seem to believe, both in math and in the broader world. There are many examples of terms that don't really make sense when interpreted too literally. For instance, an antlion bears little direct resemblance to either an ant or a lion. That said, I strongly disagree with your ...


2

I think the idea is that a particular player set contains all the nodes in which a given player makes a decision. There are $N+1$ of them because there are $N$ players and then we consider nature as an additional random player. In your diagram, if we call Nature player zero, then $g^0$ would include just the initial node. $g^1$ would include both of P1's ...


2

If it helps think about it like this - in concrete terms If $\lambda =\frac 1 2$ the point $\frac 12 x +\frac 12 y$ is half way between $x$ and $y$. If $\lambda =\frac 1 3$ the point $\frac 13 x +\frac 23 y$ is two thirds third of the way from $x$ and $y$. So as the values of $\lambda$ vary from $1$ to $0$ you get further from $x$ and closer to $y$ along ...


1

If $M$ is a vector space, then we refer $A$ as an endomorphism. There are differences when a function takes an element from one space to an element in the same space. Take for example: $f:C[0,1]_{\|\cdot\|_1}\to C[0,1]_{\|\cdot\|_2}$, where $\|\cdot\|_1 \to$ integral norm $\|\cdot\|_2 \to$ maximum norm Note that $C[0,1]$ is complete under maximum ...


1

Consider the trivial group $E = \{ e \}$. Forgetting about the "identity" axiom, we can let $E$ act on $\mathbb{R}$ by $e \cdot x = 0$ for all $x \in \mathbb{R}$. This satisfies the "compatibility" axiom, as clearly $ee \cdot x = 0$ always. But it is not an actual group action, and it is not a homomorphism to the group of bijections of $\mathbb{R}$.


1

The "homomorphism from a group $G$ into $\operatorname{Sym}(X)$" interpretation is equivalent to these two axioms; you need both axioms to imply the homomorphism interpretation and vice versa. The Wikipedia page that you linked states this pretty clearly: From these two axioms, it follows that for every $g$ in $G$, the function which maps $x$ in $X$ to ...


1

This is a "functional calculus". The point is that the function $f$, initially defined on complex numbers, can now be extended to suitable members of the Banach algebra. Moreover, this extension will turn out to have some useful properties. For a concrete example, consider the Banach algebra $\mathcal L(\mathbb C^n)$ of linear operators on $\mathbb C^n$ ...


1

For boundedness: You can give an exercise so that the students need to show that a bounded metric can induce the same topology as an unbounded one (at least you can easily show that for metric spaces with $d(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$). So, boundedness is not really a topological property. If each open covering has a finite subcovering (and using ...


1

An equivalent def'n is that if $ F$ is a no-empty family of closed sets with the F.I.P. (Finite Intersection Property) then $\cap F \not = \phi $ . This generalizes the idea of limits , and you can show that many results, e.g. on bounded closed subsets of $ R^n$ , using this property, so it is seen to be a useful tool that a space is compact. Once you ...


1

How does the following work. Let $A\subseteq \Bbb R$ be an interval and $c: A \rightarrow \Bbb{R}^2$ continuous such that $c(A)$ is unbounded in the sense of the usual metric $d$ on $\Bbb R^2$. We can certainly find a sequence $a_n$ in $A$ such that for each $r>0,\,\exists N \in \Bbb N$ such that for $n>N$ we have $c(a_n) \not\in B_{r}((0,0))$. Now ...


1

I think this may be more what you're interested in here. The author is Herbert Busemann, and the paper is on Local Metric Geometry, and he mentions asymptotes in metric spaces, where the distance is not necessarily symmetric, and according to the reference here on project Euclid, by Nasu, he says the concept of asymptotes was introduced. Check his references ...


1

Funny point of view: We know that not every function can be integrable, but as far as I know all functions is differentiable in the real domain. You know that not every function is integrable but you think that every function is differentiable? Have you every tried to differentiate a non-integrable function? You won't find any, because ...


1

"A is independent of C", $A\perp C$, means that: $\mathsf P(A\cap C) = \mathsf P(A)\;\mathsf P(C)$ Likewise, the meaning of "A conditioned on B is independent of C," is that "A is conditionally independent of C, when given B". $$A\perp C \mid B \;\iff\; \mathsf P(A\cap C\mid B) = \mathsf P(A\mid B)\;\mathsf P(C\mid B)$$ Given the definition above, ...


1

Wikipedia also says that orthogonality is generalizing that idea into $n$ dimensions. I only hear people say perpendicular when they are talking about 2-3 dimensions. In which case they are the same thing. I think the zero vector is usually considered orthogonal to every other vector. You could say that the zero vector is perpendicular to every other ...



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