Mathematics Stack Exchange
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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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Dependance and independace of relations

Let $G$ be a multiplicative group, and let $\prod_{i=1}^ng_i^{e_i}=1$, $\prod_{i=1}^mh_i^{f_i}=1$ be two multiplicative relations in $G$. What does it mean that these two relation are independent?
Or Shahar
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A formalization of circular characterizations of mathematical objects

One can define the algebraic numbers as the solutions to polynomials with integer, or indeed rational, coefficients. Also, it can be proven that algebraic numbers and only algebraic numbers are solutions to polynomials with algebraic coefficients.…
user107952
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How might we define "graph of an equation"?

Restrict attention to $\mathbb R^2$ (2D/cartesian plane). Attempted definition: Given an equation (in variables $x$ and $y$), define the graph of that equation to be the set of points $(x,y) \in \mathbb{R}^2$ for which the equation holds. (Now we…
user986614
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Formal Validation of Addition

This might be more suitable for Stack Overflow, but since I'm asking about the mathematical theory, rather than the implementation details, I'm initially posting the question here. Background The .NET 7 Framework is about to introduce an INumber…
ScottishTapWater
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Clarification on the definition of $C_b^\infty$

In Pierre-Louis Lions' famous work Axioms and fundamental equations of image processing, he defines $C_b^\infty$ on Page 9 as ...on $C_b^\infty$, i.e., the space of bounded functions having bounded derivatives at any order. And on the next page,…
Wang
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What is the Exact Definition of a Whole Number?

I have thought about this for a long time, since I have done some Grade 8 American Math League (A Competition) Past Papers, but when it asked some question like: How many whole numbers have their square less than or equal to $200$, I thought any…
Cheese Cake
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Name of the concept that you take a problem and transform it into a different scheme of thinking, which somehow magically makes it easier to solve

There are a few instances where converting some co-ordinates from cartesian to polar co-ordinates makes a problem far easier to solve. Something similar is done to improve computational complexity in the Fast Fourier Transform. We also could solve…
jonsploder
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Math Proofs: Definitions of Intersection and Union

I'm going through a book on math proofs and I'm struggling to understand these definitions: https://ibb.co/Lzrnt3J When I'm going through the definition of ⋂ F, what's going through my head is "For all sets A, if A is in the family of sets F, then x…
Faith Alone
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How to define path connected neighborhood

I notice the term path connected neighborhood. In my text,A First Course In Topology there is no mention of it I looked on MSE and all over and could not find it .This term isn’t in my text when dealing with path connectedness and it’s…
user960654
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What is the word for the last two digits of a number?

I'm working on a script to translate numbers to words. (e.g. 111 to "one hundred and eleven". It's very easy to create this for the English language since it has very clear rules. At this moment I have a function that converts the last two digits of…
Johan Vergeer
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Why the expectation of X is equal to sum(x.p(x))?

why the expectation of X is equal to $$sum(x.p(x))$$ , or the integral in the case of the continuous variable. I mean where this definition cames from?
St.artistics
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Definition of "with dense range"

it is written "T is a continuous map with dense range". What is the definition of map with dense range? Maybe it's simple but I am not native English speaker, thanks.
user570048
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What is denoted by an element + a space (i.e. $f+L^2$)

What does the definition of a space + an element mean, for example $f+L^2$. Is this the same as the direct sum $\{f\}+L^2$, e.g. $$\{f+g|g\in L^2\}$$
Tony
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What does it mean when a function is set to zero?

I am reading an article about the Reproduction numbers and I found the next phrase: If $\mathcal{F}(x)$ is set to zero, then all eigenvalues of $Df (x)$ have negative real parts. and my question is what does it mean that a function is set to zero.
Alex Pozo
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What does it mean if something is a solution for a particular state of another function?

I'm trying to learn how to calculate the Extended Internal (XIRR). One website says the following: "The XIRR is not an empirical formula though. It is a solution for a particular state of the the XNPV (eXtended Net Present Value)…
Userq49874
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