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Questions tagged [definition]

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

Definitions are at the core of mathematical precision; they answer the question "What is X?" in mathematics. Into this category fit questions regarding equivalence of definitions, clarification of complicated definitions, or proposed new definitions for mathematical notions, with requests for improvements or comments.

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How can an axiom ever be dependent

An axiom is defined, officially, as 'a statement or proposition which is regarded as being established, accepted, or self-evidently true.' Yet an Independent axiom is one where it is not derived from other axioms within an axiomatic system, meaning…
user2901512
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When 2 functions are equal?

Are 2 functions equal when they have same domain, same codomain and same law ? EXAMPLE 1 $f: \mathbb{R} \to \mathbb{R}$ $x \to x^2$ and $g: \mathbb{R} \to \mathbb{R^+_0}$ (set of positive reals with zero) $x \to x^2$ are equal ? EXAMPLE 2 $f:…
halfpog
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Explanation of definition of ${[n] \choose k}$

I am reading a definition in a paper, but am not sure of how to interpret the following definition: If $K \in {[n] \choose k}$, then let $\operatorname{Path}(K)$ denote the set $$\{S: S \text{ is a set of pairwise vertex disjoint paths from}…
urpi
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Definition for not divergent

What is the definition of real numbers $\{x_n\}$ does not diverge to -$\infty$? Would it be $x_n$ does not go to infinity if and only if there exist $M>0$ for all $N \in \mathbb{N}$ numbers such that $n$ greater than equal to $N$?
ematth7
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Is this backslash a typo here in showing a function?

"Then there exists a homogeneous function $\rho\in C^\infty(R^n \backslash\{0\},R^n)$ with negative degree of homogeneity which specifies the following properties:" - Pablo Monzon (2006), Almost global stability of dynamical systems, page 27.,…
Mehdi Haghgoo
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What is this No thing?

What exactly is this No? Is there any other use of it other than graphs? Thank you so much. I am not trying to cram or anything it's just that I took a course online and a lot of the time it focused on concepts rather than notation. Sometimes when…
user253055
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General meaning of the term "Functional"

In the most general is a "functional" simply a function which can accept a function as input? So, is it natural to describe: $f: \mathbb{N} \rightarrow \mathbb{N}$ as a function. Whereas it is more natural to describe: $F: (\mathbb{N} \rightarrow…
John
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Meaning and Underlying idea of a definition or a theorem

What does it mean by 'explain the meaning and underlying idea of a definition or a theorem'? For example, if we are asked to explain the Fundamental Theorem of Algebra, how should we explain its meaning and idea? I can only think of saying: The…
Idonknow
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Minimal planar domain

I am recently studing minimal surfaces on my own. I have meet in many places the fallowing statement: The only connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ are a plane, a helicoid, a catenoid or one of the Riemann minimal…
J.E.M.S
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Correct understanding of col and diag operators

In a scientific paper I am currently working with, a definition of $Col$ and $Diag$ operator is introduced: We use the operator $Col_{k\in K}(x_k)$ which stacks up its vector (or matrix) arguments $x_k$ for all $k$ in the index set $K$ into a…
Liglo App
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Radical function in two (or more) variables

I'm reading a paper where the author uses the word radical function for a function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$. I understand the definition of a radical function if $n=1$, but what if $n>1$?
fromnowon
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Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same height equidimensional==all associated primes…
Wayne S
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What do these definitions of conjugacy have in common?

Here are four (seemingly) different uses of the word conjugate: Complex conjugates are a concrete instance of the idea of conjugacy in field extensions. In group theory, there's the idea of conjugacy classes In probability theory, there are…
Mike Izbicki
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What is a "nonparametric function"?

I am looking for a formal definition of the term "nonparametric function." I understand the term and I use nonparametric regressions http://en.wikipedia.org/wiki/Nonparametric_regression but I haven't been able to find a formal definition of the…
user103828
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Is there a term for something equally distributed around zero?

Let's say X is uniformly distributed in [-1 , 1] Then what can we call the distribution of X³ ? It is not uniform, but it "mirrors" around 0 as well. Is there a simple word describing X³ that would also apply to X⁵ and to 3X, for example? I…
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