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

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1answer
25 views

Definition of Torsion Submodule

If $R$ is an integral domain one defines an $R$-module $M$ to be torsion-free if the torsion submodule $T(M)=\{x\in M\mid l_R(x)≠ 0\} $ is zero, where $ l_R(x)$ is the left annihilator of $x$ in $R$. ...
2
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1answer
34 views

Definition of length of a sequence

In definition $3.11$, the author defined the following rank: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of ...
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1answer
33 views

What are pullbacks of finite-coproduct injections along arbitrary morphisms?

I am studying a definition of an extensive category: An extensive category is a category $E$ with finite coproducts such that pullbacks of finite-coproduct injections along arbitrary morphisms exist ...
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0answers
44 views

What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to formulate the definition but succeeded only ...
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0answers
43 views

Relations, Ordered Pairs, Naive set theory by Halmos

I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?"
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4answers
168 views

Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
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1answer
102 views

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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0answers
18 views

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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4answers
117 views

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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0answers
14 views

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 ...
2
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1answer
48 views

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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1answer
31 views

definition of derivative

Definition: A mapping $f:U\to \mathbb{R}^n$ from an open set $U\subset \mathbb{R}^m$ into $\mathbb{R}^n$ is differentiable at a point $a\in U$ if there is a linear mapping $A:\mathbb{R}^m\to \mathbb{R}...
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1answer
26 views

Confusion about difference between Normal and Perfectly Normal

I am working with the following definitions: A topological space is normal if and only if every pair of disjoint, nonempty closed sets can be separated by a continuous function. A ...
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0answers
15 views

Equivalent definition of Lebesgue measurability in terms of additivity?

When introducing measurability, we noted that we wanted the following property to hold for $A, B \in \mathcal{P}(\mathbb{R})$ $m(A \cup B) = m(A)+m(B)$ (additivity) We then defined a set A to be ...
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3answers
76 views

What does it mean to say “a divides b”

I am not a number theorist and I am learning about relations. I encountered a relation that says $a \leq b$ if $a$ divides $b$ Can someone clarify what it means to a number to divide another ...
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3answers
44 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
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2answers
37 views

Can we define component of a matrix which is orthogonal to another matrix?

Given two vectors $A$ and $B$ one can easily find component of $A$ along $B$ and component of $A$ perpendicular/orthogonal to $B$ and vice versa. This is possible as we can define dot product of two ...
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2answers
63 views

Definition of an Ordered Field

A text I'm looking at has the following definition of an ordered field: DEFINITION A field ($F$, $+$, $\cdot$) is ordered iff there is a relation $\lt$ on $F$ such that for all $\quad\quad\quad\...
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1answer
25 views

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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0answers
54 views

supporting function and halfspace (definition)

we've defined the following: supporting function: Let $P$ be a convex polygon in $E^d$ (euclidean vector space). Then the supporting function is defined as $h_P: S^{d-1} \to \mathbb{R}$ by $h_P(u) :=...
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0answers
21 views

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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2answers
51 views

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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1answer
30 views

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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1answer
69 views

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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0answers
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Definition of the left and right derivative.

The definition of the derivative is $$g'(a)=\lim \limits_{\delta \rightarrow 0} \frac{g(a+\delta) - g(a)}{\delta}$$ also the left derivative is $$ \lim \limits_{\delta \rightarrow 0^-} \frac{g(a+\...
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0answers
24 views

Arc-bases and Point-bases: when are they different for finite digraphs?

Definitions Given a digraph $D=(X,U)$, a point-basis of a digraph $D$ is a minimal vertex set from which a dipath exist to every vertex in $D$. An arc-basis is a minimal arc-reaching set. A subset ...
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2answers
74 views

Defining “Countably Infinite”

I was reading about countably infinite sets and the definition goes as, "A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers"(Source: ...
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2answers
63 views

May I use comma to avoid redundancy in expressions?

Is it generally possible to write $$1\leq k,l\leq 8$$ instead of $$1\leq k\leq8\quad \mathrm{and} \quad 1\leq l\leq 8$$ to avoid redundancy?
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5answers
2k views

Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
3
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1answer
43 views

About definition of UFD

On Wikipedia, UFD is defined as an integral domain in which every element can be uniquely factored as product of primes (irreducibles), up to multiplication by units and arrangement. My question is ...
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1answer
45 views

How would you define a 3d angle?

Today I somehow was wondering how could you define a 3d angle. First what I could think of was volume of a cone with side a=1. Then if it wasn't round, volume of a tetrahedron with side 1. Is any of ...
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1answer
40 views

What is the difference between accumulation point and $\omega$ accumulation point?

The title says it all. Accumulation point has a widely known definition: a point in $X$ is accumulation point if every open set containing $x$ contains infinitely many points of $X$ Sometimes I ...
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1answer
30 views

Open vs Closed Immersions of Locally Ringed Spaces

I'm reading Qing Liu's book at the moment and I'm trying to figure out why open immersions of locally ringed spaces are required to be isomorphisms on stalks, but closed immersions are only required ...
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2answers
31 views

Why can projection function on $X \times S$ be regarded as a local homeomorphism?

I am studying some properties of local homeomorphism I am in particular trying to find a local homeomorphism that is not a homeomorphism and the projection function seems to be the perfect candidate ...
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1answer
22 views

Could anybody please clarify the relationship between numerical stability and accuracy?

I was reading a paper and came up with this statement. Stability merely avoids uncontrolled error growth but cannot guarantee actual numerical accuracy. From what I understood from the concept of ...
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1answer
21 views

Properties of Infinite set on co-finite topology and Countable set on co-countable topology

I am trying to verify some of the properties of infinite set on co-finite topology and countable set on co-countable topology but it is proven to be very tricky because I cannot visualize the spaces. ...
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1answer
52 views

Difference between NP-hard and NP-complete

I am struggling to tell the difference between the definitions of NP-hard and NP-complete problems. I know that NP-complete problems are NP-hard, so this tells me that $$\text{$P_1$ polynomially ...
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1answer
32 views

On the definition of free product of groups.

Let $G$ and $H$ be groups. Their free product is a group $G*H$ with homomorphisms $\iota_G:G\rightarrow G*H$ and $\iota_H:H\rightarrow G*H$ so that given any other group $X$ with homomorphisms $f_G:G\...
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2answers
38 views

Is this the correct definiton of $T_1$ space?

I found this in a handwritten note: Def: A topological space $X$ is $T_1$ if $\forall x \neq y \in X$ there exist a neighborhood of $y$ such that s.t. $x \not\in V$. I was almost certain that ...
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0answers
20 views

definition of a traveling wave (solution)

today I've got a problem with understanding what is meant by the term "traveling wave solution" of a PDE in a more general sense. To be more precise, I've got a following Poisson equation \begin{...
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0answers
37 views

Relevancy of standard deviation

I'm a maths student but I'm a bit of a retard when it comes to probability. I have this very basic question in mind : Why is standard deviation defined as the square root of the variance, when surely ...
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3answers
52 views

Why other definitions of convergence fail to be correct?

The following is an exercise from the book Advanced Calculus: Well obviously the b. is not correct since for $\epsilon= \frac19$, ${\{a_n}\}={\{\frac1n}\}$ and $N=2$ it fails to be correct. But the ...
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1answer
21 views

Sublinear functions on a Riemannian manifold

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds. Thank you!
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1answer
24 views

What's the definition of being zero to infinite order at some point?

I'm reading a paper and it says The differential form $\psi$ is zero to infinite order at all points of $F$.That is,all coefficients of $\psi$ are zero to infinite order at points of $F$.(Here $\psi$...
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1answer
57 views

Can someone reconcile the two definition of Suslin's condition?

I am given two definitions of the so called "Suslin's condition" and I need to reconcile them. I am an undergrad and this is just for exploration. Definition 1. A partially ordered set $X$ is said ...
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3answers
33 views

Problems with the definition of vectors as directed line segments in $\mathbb{R}^3$

First I'll say where I'm working: The vectorial spaces $\mathbb{R}^2$ and $\mathbb{R}^3$. Then I'll define a vector of this spaces as the following: $\textbf{Definition. }$ A vector $\vec{v}$ is the ...
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2answers
123 views

When are quantities outside of the real numbers considered equal, and when do they exist?

I know of the complex number $i$ and it's existence as the result of invalid square rooting (the square root of negative one does not exist inside the real numbers), but other than complex numbers, ...
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10answers
5k views

Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
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1answer
25 views

Connection between complementarity problem and optimization problem?

I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ...
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1answer
36 views

Topology: What is a quick way to check whether a subset $D$ is dense in $(X, \mathcal{T})$?

Def $1$: Let $(X, \mathcal{T})$ be a topological space, then $D \subseteq X$ is dense if $\overline {D} = X$ Def $2$: $x \in \overline D$ iff for all $U \in \mathcal{T}, x \in U \implies D \cap U \...