# Tagged Questions

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

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### $(+\infty,+\infty,\cdots,+\infty)$ exists in $\mathbb{R^{n}}$?

$(\mathbb{R}^{n},d)$ is a metric space and $d$ is the standard metric on $\mathbb{R^{n}}.$ Let $(\mathbb{R^{n},\tau_{d}})$ is the topology space induced by metric space $(\mathbb{R}^{n},d)$ .We can ...
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### How to formalize that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0 \implies$ $g$ “grows faster” than $f$?

I understand that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0$ implies that, for sufficiently large values of $x$, $f(x)<g(x)$, as a direct consequence of the definition of limit to $\infty$....
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### Categorical definition of subdomain

Let $R$ be an integral domain and $S$ a subrng of $R$. Definition 1. $S$ is a subdomain of $R$ iff $S$ is an integral domain Definition 2. $S$ is a subdomain of $R$ iff $S$ is an ...
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### Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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### Closed vector space and a subspace of a vector space [duplicate]

What is a closed operation in a vector space? I don't see any difference between a closed operation in some vector Space R$^n$ and the open operation. What I mean by the closed operation is addition ...
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### Interpretation of the radius of convergence

What interpretation should one give to the radius of convergence of a series $\sum a_nz^n$ ? I do know how it is mathematically defined and what it implies for convergence/divergence, but I'm having ...
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### E is closed if every limit point of E is a point of E?

E is closed if every limit point of E is a point of E? Should that be "E is closed if every point of E, is a limit point"? I don't understand. Limit points are essentially points that hug other ...
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### Understand the definition of convex metric spaces

I am trying to understand the following definition: We call a set $E\subset \Bbb R^k$ convex if>$$\lambda x+(1-\lambda)y\in E$$ Whenever $x\in E, y\in E$ and $0\lt \lambda \lt 1$ Clearly ...
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### Ordinary differential equations of order zero?

Is $x+y+2=0$ a differential equation without derivatives of order $n$, $n>0$? Could it be called a differential equation (for unknown $y(x)$) of order $0$? If not, can we define differential ...
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### Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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### Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. Shafarevich definition (pg 128) - A variety is non-singular in codimension one if the singular locus has codimension $> 1$. ...
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Alright, so I am reading "Computability and Logic" by Boolos and Jeffrey, specifically I'm on chapter 9 "A Precis of First-Order Logic: Syntax. There has been more than a handful of definitions in ...
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### What is the oscillation of a function?

Define the oscillation of a function at a point $x$ to be (for an open interval $I$): $$\omega_f(x)=\inf_{x\in I}\sup_{s,t\in I}|f(t)-f(s)|$$ I am a bit confused about the definition above. How am I ...
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### Theory of definitions

I am reading "Introduction to Logic" by P Suppes at the moment. In the Chapter 8 - Theory of definitions of it, I 've some confusion, actually about the Conditional Definition. The brief explanation ...
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### Why do some authors use terms “non-ascending” and “non-descending” instead of ascending and descending? [duplicate]

In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with ...
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### Definition of monomial

I thought the definition of a monomial is an algebraic term that has no subtraction or addition. I saw on my online college homework that 2/x is not a monomial. Why?
What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...