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

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2
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0answers
19 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 ...
39
votes
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 ...
1
vote
4answers
68 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 ...
2
votes
1answer
25 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}...
0
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1answer
25 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 ...
0
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0answers
13 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 ...
1
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1answer
1k views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
2
votes
3answers
43 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 ...
384
votes
20answers
66k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
2
votes
1answer
68 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 ...
1
vote
2answers
34 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 ...
0
votes
2answers
122 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, ...
2
votes
1answer
531 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
0
votes
0answers
32 views
+50

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) :=...
0
votes
2answers
25 views

A small confusing part in the definition of initial value problem

Assume $U\subset\mathbb{R}\times\mathbb{R}^{n}=\mathbb{R}^{n+1}$, $U$ is open and $(t_0, \bf{x}$$_0)\in U$. Assume ${\bf f} (= {\bf f}(t,{\bf x})) : U \to \mathbb{R}$ is continuous. Then the following ...
2
votes
2answers
61 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\...
0
votes
1answer
24 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 $\mathbb{F}$...
0
votes
2answers
50 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 ...
0
votes
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 ...
1
vote
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" ...
70
votes
6answers
8k views
1
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0answers
18 views

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+\...
1
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0answers
23 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
73 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: ...
3
votes
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?
3
votes
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 ...
3
votes
1answer
463 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
0
votes
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 ...
-1
votes
0answers
36 views

What is the difference between an adherent point and a limit point?

What is the difference between an adherent point and a limit point? Please explain this with a proper example.
1
vote
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 ...
0
votes
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 ...
1
vote
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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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 ...
-1
votes
2answers
106 views

Can't understand some definitions of abstract algebra, can you help me please?

I was reading different definitions of vector space but still I have some dark places without meaning... I can't understand exactly what type of relation is defined between the vector space and the ...
1
vote
0answers
53 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b \...
0
votes
1answer
21 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 ...
1
vote
1answer
18 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. ...
8
votes
1answer
4k views

Formal definition of big-O when multiple variables are involved?

(My apologies if this is a duplicate; I did some searching but didn't turn up anything else like this on the site. Please let me know if it's a duplicate and I'll gladly delete it.) I was reading up ...
51
votes
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,"...
1
vote
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\...
1
vote
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 ...
5
votes
1answer
127 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
1
vote
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{...
1
vote
0answers
35 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 ...
2
votes
3answers
51 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 ...
2
votes
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!
0
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1answer
23 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 $\psi$...
1
vote
1answer
56 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 ...
1
vote
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 ...