# Tagged Questions

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

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### How is the set of all even numbers definable from $\omega$?

This Set Theory textbook (page 89) defines definable sets as follows: Definition 6.8. Given a set $a$ and a formula $\Phi$ we define the formula $\Phi^a$ to be the formula derived from $\Phi$ by ...
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### Definition of extrema in calculus of variations

I am reading Gelfand and Fomin about Calculus of Variations and in page 12 they say: ' Analogously, we say that the functional $J[y]$ has a (relative) extremum for $y=\hat{y}$ if $J[y]-J[\hat{y}]$ ...
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### What exactly are the meaning of the followings in the definition of a category?

In Awodey's Category Theory a category is defined as follows. A category consists of the following data, Objects: $A, B, C,\ldots$ Arrows: $f,g,h,\ldots$ For each arrow $f$ there are ...
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### What is necessity for integral to be well-defined in defining solutions?

When trying to give a notion of solutions for differential equations with non-local terms, e.g., integral of unknown functions, to guarantee that the integral is well-defined, i.e., finitely ...
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### What is the difference between hyperreal numbers and dual numbers

Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number I cannot stop seeing ...
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### Definition Of Polynomial Equation

I need some clarification on the definition of polynomial equation. Let $f(x) = \displaystyle\sum_{i=1}^n f_i \,x^i ,\; g(x) =\displaystyle\sum_{i=1}^m g_i \, x^i\in \mathbb{F}[x]$, where $\mathbb{F}$...
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### Is there a more rigorous definition of “natural”? [duplicate]

I keep coming across this word both in textbooks and in lectures and it always strikes me that all that is meant by "natural" is that for example a natural mapping is one that is "obvious from the ...
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### Give an example of an affine space that is not a vector space

We know that any vector space is an affine space, but can you give an example of an affine space which is not a vector space? I don't know any such examples. This is an interview question, not ...
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### Can someone please offer a simple definition of “derived net”?

I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" ...
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### On the definition of a reductive group.

Wikipedia defines a reductive group $G$ as an algebraic group with trivial unipotent radical. The radical is the connected component of identity in the maximal normal solvable subgroup of $G$. The ...
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### What should I call an “injective” algebra?

Given rings $A,B$, we say that $B$ is an $A$-algebra if there is a ring homomorphism $f:A\rightarrow B$. This homomorphism give the structure of the algebra. Various properties of algebras can either ...
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### What are the components and quasicomponents?

I'm trying to understand the difference between components and quasicomponents. I'm using the following definitions: $x\sim y$ iff $x$ and $y$ lie together in some connected set. The component of ...