# Tagged Questions

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

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### What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does "...
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### Questions which have false conditions

There are many "questions" on the internet like If $$1=5$$ $$2=6$$ $$3=7$$ $$4=8$$ then how many is $5$? With one "logic" answer is $9$ because $n=n+4$, then $5=9$. With other "logic" ...
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### Is “algebraic system” the same as “algebraic structure”?

1. algebraic system | planetmath.org: http://planetmath.org/algebraicsystem It seems that algebraic system is only a set on which some operations are defined. Is it necessary that some additional ...
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### Singular Points on Algebraic Curves

What does it mean for a point on an algebraic curve (over any field) to be singular? Over $\mathbb{R}$ or $\mathbb{C}$ is means that all derivatives vanish, but how does this definition generalise to ...
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### Why $1$ isn't a prime? [duplicate]

I was wondering the reason behind defining the Prime Numbers in a manner of which $1$ isn't an example. I read in Rotman's A First Course in Abstract Algebra that one reason that $1$ is not called a ...
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### What's the definition of a “local property”?

Is a property called local if and only if for every point there exists a neighbourhood for which the property is true? For example: Let $X,Y$ be topological spaces. Then $f: X \to Y$ is continuous if ...
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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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### Is there a difference in definition between an increasing sequence and a monotonically increasing sequence?

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence? In some places I see they call it just inc/decreasing and some call it monotonically inc/...
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### Does the word scalar still apply if it's not a vector?

If I take the value of something, say $50$g and I multiply that value by something else, so perhaps $50\text{g} \times 3$ what would that $3$ be called? It acts like a scalar but I'm not sure that ...
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### Question about proving $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence

Prove $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence. What I did: Suppose by contra position that $n$ tends to a finite real limit $L$, so from the ...
I'm attempting to learn abstract algebra, so I've been reading these notes by John Perry. Monomials are defined (p. 23) as  \mathbb{M} = \{x^a : a \in \mathbb{N}\} \hspace{10mm} \text{or} \hspace{...
In Shigeyuki Morita's Geometry of Differential Forms, orientability is defined in the following way: If we can assign an orientation to each point on a manifold $M$ in such a way that the ...