# Questions tagged [definition]

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

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### If for every positive number $\varepsilon$ there exists a positive number $\delta$, why not the other way back? [duplicate]

Definition. Let $f(x)$ be a function of $x$. If for every positive number $\varepsilon$, however small it may be, there exists a number $\delta$ such that whenever 0 $<$ |x - a| $<$ $\delta$ we ...
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### Definition of a Lipschitz hypersurface

In the last days the expression Lipschitz hypersurface came up a lot. I didn't find a definition. I think a Lipschitz hypersurface is just a hypersurface which can locally be described by Lipschitz ...
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### Strange but simple question

Is there a name for the type of trick where we do something like $s=(1-p)r+pq$ where we are making a sum of complementary ratios of two numbers? This kind of reminds me of convolution but I thought ...
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### contradiction in the definition of exponents of congruence class in modular arithmatics

I know that if $a\equiv b\pmod n$ ,then $a^k\equiv b^k\pmod n$ where $k \in Z^{\geq0}$.However , i saw a general definition in Discrete mathematics and applications by Sussanna 's book. It says that ...
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### Why does a definition not have a proof? [duplicate]

Hi I have a question that in general in mathematics: Why does a definition not have a proof? I mean how did we reach it without proving it? I know that the proposition, or theorem, proof comes ...
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### Pedagogical arguments for/against the “defineorem” (simultaneously definition and theorem)

Often times, we like to define things in ways that are not obviously possible. For example, we may define something to be the unique object satisfying a certain property, or we may want to state many ...
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### Definition of transitive group

I need a definition of a transitive group that's accessible to someone who's just started learning group theory (so won't know about actions and orbits etc.). I've written the following: A ...
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### How might “a linear mapping is its own differential” be stated better?

This is Example II-2.3 of https://www.scribd.com/read/282634061/Advanced-Calculus-of-Several-Variables?mode=standard If $F:\mathbb{R}^n\to\mathbb{R}^m$ is linear, then $F$ is differentiable ...
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### Definition of characters in the context of twists of elliptic curves

In the Wikipedia article about Twists of curves, I encountered the term character several times and I do not know what the definition of it is (and there is no reference to another Wikipedia article ...
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### Definition of a formula in propositional logic

The set of formulae in propositional logic is characterised by ( at least in Derek Goldreis's book) Every propositional variable is a formula If $a$ is a formula then so is $\neg a$ If $a,b$ are ...
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Spivak's problem reads as follows: Given the horizontal line $L$ defined by $g(x)=\gamma$ and a point $P=(\alpha, \beta)$ not on $L$ so that $\gamma \neq \beta$, show that the set of all points $(x,y)... 0answers 62 views ### Ph.D. Reviewer complains: “(Syntax) Error in function definition” Next week I will defend my Ph.D. thesis in the field of Scientific Visualization. I have a question that is at the end related to syntax and personal preference, I think. I use bold letters for ... 1answer 63 views ### Is there a non-standard definition of an absolute value? So I was thinking about how we can define a number$x$such that$|x|=-1$. This topic was already talked about so many times as we can see here or there. So I wanted to consider it from a different ... 1answer 35 views ### differences between Ideals and principal ideals in math notational descriptions In the following post: (Confusion between principal ideal and ideal) on clarification between the concepts of ideals and principal ideals, @Yury stated: "... if$I$is a principal ideal then ... 1answer 81 views ### Law of composition in a group Most textbooks on group theory define the law of composition, say$\star$, as a function$\star: S \times S \to S$. The next part of the definition is sending$(a,b) \mapsto a \star b$. This confuses ... 2answers 36 views ### Empty function finite support If$I=\emptyset$is the empty set and$G$a group with trivial element 1. Then$G^I=\lbrace f:I\to G \rbrace$is the trivial group with the empty function as an element. Am I right that$G^{(I)}=\...
An algorithm $\tilde{f}$ of a problem $f$ is said to be backward stable if for any input $x$,  \tilde{f}(x) = f(\tilde{x}) \ \text{ for some } \tilde{x} \text{ such that } \frac{||\tilde{x} - x||}{||...
### Confused about the definition of $\mathcal O_{\operatorname{Spec}A}(U)$.
Consider $\operatorname{Spec} A$ for some ring $A$. When defining the structure sheaf, Hartshorne defines For an open subset $U \subset \operatorname{Spec}(A)$, we define $\mathcal O(U)$ to be the ...