# Tagged Questions

For requesting, clarifying, and comparing definitions of mathematical terms.

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### What is actually the standard definition for Radon measure?

I see that there are various definitions for Radon measure and they are NOT equivalent, but they are equivalent on locally compact Hausdorff spaces. I think this is the reason why Radon measure has ...
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### Why other definitions of convergence fail to be correct?

The following is an exercise from the book Advanced Calculus: Well obviously the b. is not correct since for $\epsilon= \frac19$, ${\{a_n}\}={\{\frac1n}\}$ and $N=2$ it fails to be correct. But the ...
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### Sublinear functions on a Riemannian manifold

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds. Thank you!
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### Some doubts on the definition of minimality of a graph. EDITED

If a graph $G$ is minimal with property $P$, does that specifically mean: Any proper subgraph of $G$ does not have that property $P$? Or, Any graph with less number of vertices or less number of ...
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### Does “sphere” denote the surface or the entirety of a solid ball?

In everyday English, the word "sphere" denotes a 3-dimensional object, including the points inside the surface and its center. However, I get the sense that in mathematics, the sphere is used ...
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### Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
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### Epsilon Delta Limit Intuition

I am having a hard time grasping the definition of a limit. I initially learned this loose definition of a limit: $\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from ...
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### Well-posed vs Well-conditioned

What's the difference between a well-posed (ill-posed) and well-conditioned (ill-conditioned) problem ?` Here is my finding up to now: "Even if a problem is well-posed, it may still be ill-...
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### Definition of Aut(G) in the graph theory and group theory

For a fixed group G, we define the collection of group automorphisms is the automorphism group Aut(G) in the group theory. (An automorphism: a permutation on the set G) In the graph theory, on the ...
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### Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
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### What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. ...
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### What is an adjoint operator?

The following conjeture is stated here: Every adjoint operator has a non-trivial closed invariant subspace. Reference 11 where adjoint is supposedly defined can be found here. But I don't have ...
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### What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
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### Is it possible to have an $a \times b \times c$ matrix?

The book Artificial Intelligence: A Modern Approach states that a certain variable is a $2 \times 2 \times 2$ matrix", but I thought that matrices could only be rectangular (i.e. $a \times b$). Is it ...
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### Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
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### What is a linear equation?

How do we define the linear equation? I mean, it looks like a polynomials with degree one but I'm not sure if $ax+by+c=0$ is a linear equation if $a=b=0$?
Is the angle between a vector and a line defined? The angle between two lines $a,b$ is defined as the smallest of the angles created. The angle between two vectors $\vec{a},\vec{b}$ is the smallest ...
### Family of “something very close to be a curve” over a curve $C$
Hartshorne (IMHO restrictive) definition of a curve: Definition of (complex) curve: A curve is an integral separated scheme of finite type over $\mathbb C$ of dimension $1$. (The definition of a ...