Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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3
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2answers
113 views

Is every triangle a quadrilateral?

I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral? More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?
1
vote
4answers
977 views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
1
vote
1answer
259 views

$\epsilon - \delta$ definition of a limit

Where can I find a good explanation of the $\epsilon - \delta$ definition of a limit. I have tried looking at my textbook and it doesn't make much sense, and I have also looked on Google as well ...
1
vote
3answers
151 views

Taylor series of a modulus argument

What is the definition of a Taylor series of the function $F(|\vec a -\vec x|)$ about the point $\vec a$ in $\vec x$?
1
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4answers
75 views

Acummulation point

Which one of these points is accumulation point, which not and why? I read the definition x-times but I'm quite confused :-/ I also found this post which is relevant to my question but it seems to me ...
3
votes
1answer
183 views

Minimal set of trig identities to define all the trig functions

What are a minimal set of trig identities that can uniquely define the trig functions? I know that you can define, for example, $\sin(x)$ as the unique solution to the differential equation $f''(x) = ...
0
votes
0answers
125 views

What is the correct definition for an imaginary number?

The following is taken from Wikipedia's definition. An imaginary number is a number whose square is less than or equal to zero. But I also heard that An imaginary number is a number whose ...
3
votes
1answer
111 views

What does “overdetermined” mean

When we say a problem is an overdetermined system, what do we mean by that in a rigorous fashion? Thanks.
2
votes
0answers
86 views

Where can I find a description of math language symbols?

I am reading math articles. I meet math symbols. For example $\exists$ or $\forall$. For example for "For any a exist e that" can be rewriten as: $\forall a \exists e$ Where can I find full ...
1
vote
1answer
58 views

Suppose I said “$X$ spans $W$”…

So I've seen two definitions of this: Let $V$ be a vector space with subspace $W$. We say that $X \subseteq V$ spans $W$ if and only if (Definition 1): Every $\vec{w} \in W$ can be written as ...
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0answers
139 views

How to prove the definition of arctangent by G. H. Hardy through integral?

From introduction to analysis,by Arthur P. Mattuck,problem 20-1. I am stuck in the sub-problem (d) of this problem,especially the magic number 2.5,please help,thanks. Problems 20-1 One way of ...
0
votes
2answers
77 views

Exponentation vs Power

What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power? Is any difference between them?
0
votes
1answer
332 views

Epsilon delta definition: $\lim _{x\to-2} (2x^2+5x+3)=1$

I'm trying to use the epsilon delta definition to prove that $$\lim _{x\to-2} (2x^2+5x+3)=1$$ evaluating: $|(2x^2+5x+3)-1|\lt \epsilon$ under the condition: $0\lt |x-(-2)|\lt\delta$ I arrived at: ...
2
votes
1answer
154 views

What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ? I heard it while I studied Galois theory and it was defined as $K/F$ is called cyclic ...
1
vote
1answer
708 views

Bounded Set: definition

I'm having some trouble with the definition of "Bounded Set". I have a pretty good idea of what "Limited" means: a Set with a Upper and a Lower bounds. Now i have a quiz in which I must choose the ...
0
votes
0answers
76 views

Definition of matrix derivative

Let $A$ be an $m \times n$ matrix. Let $a_{ij}$ be an element of $A$. What does the notation $\frac{\partial A}{\partial a_{ij}}$ mean?
4
votes
1answer
220 views

Question about definition of regular projection of a knot

One can define a knot in two ways: (1) A knot is a closed polygonal curve in $\mathbb R^3$ (2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$ And perhaps also: (3) ...
0
votes
5answers
563 views

Question about the definition of a category

I am confused about the definition of a category given in the Wikipedia article on Category theory: It seems to me that the structure being described (the "arrows" between objects in some class) is ...
7
votes
2answers
211 views

Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?

What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
0
votes
1answer
52 views

Legal actions on a PDA and terminology

I am unsure about the following so I would like to verify if my statements are true: We can remove at most a single character ($Z\in\Gamma$) from the PDA (top ?) of the stack with one step of the ...
1
vote
2answers
215 views

Measure and Outer Measure Definition

I would like to find exemples to show and demonstrate that each of the statements of the definition of: -measure $\mu\left(\emptyset \right)=0$ $\mu \left( \bigcup A_n\right)=\sum \mu \left( ...
3
votes
1answer
77 views

Formalizing the idea of a set $A$ *together* with the operations $+,\cdot$''.

I will illustrate my question in the case of the definition of vector spaces. It is custom to define a vector space in the following way: "Let $K$ be a field. Then a $K$-vector space is a set $V$ ...
2
votes
1answer
92 views

Definition of a deterministic Pushdown automaton

According to my book the definition of a deterministic Pushdown automaton allows for $\delta(q,\epsilon,Z)$ to be non-empty if $$\forall\sigma\in\Sigma:\,\delta(q,\sigma,Z)\neq\emptyset$$ Can someone ...
1
vote
2answers
97 views

Pushdown automata - definition and definition of $\vdash$

I am reading about pushdown automata and I don't understand the definition of $\vdash$. My book writes that $$(q,aw,Z\alpha)\vdash(p,w,\beta\alpha)$$ if $$(p,\beta)\in\delta(q,a,Z)$$ Can someone ...
5
votes
2answers
204 views

I need to disprove an alternate definition of an ordered pair. Why is $\langle a,b\rangle = \{a,\{b\}\}$ incorrect?

So we know that the an ordered pair $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$. And we know the Kuratowski definition of an ordered pair is: $(a,b) = \{\{a\},\{a,b\}\}$ ...
5
votes
4answers
295 views

What is the correct definition of the absolute value of $x$, $|x|$?

What is the correct definition of the absolute value of $x$, $|x|$? Option A $$ |x|= \begin{cases} -x&\text{if } x < 0\\ 0& \text{if } x=0\\ x&\text{if } x>0 \end{cases} $$ Option ...
0
votes
0answers
90 views

Definition of “eventually dominates”

What is the definition of the term "eventually dominates"? I guess it's either $f$ eventually dominates $g$ if for large enough $n$, $f(n) > g(n)$ or the same with $\ge$. A quick Google search ...
3
votes
3answers
154 views

Intuition behind independence & conditional probability

This is a basic question. I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$. ...
0
votes
1answer
103 views

About the positivity of the inner product on $L^2[0,1]$

My textbook on Hilbert space theory claims that the map $$\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}~\mathrm{d}x$$ is an inner product on $C[0,1]$. But I am not sure whether ...
6
votes
2answers
74 views

Alternate definition for boundedness in a TVS

Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if for any open neighborhood $N$ of $0$ there is a number $\lambda>0$ ...
1
vote
1answer
205 views

Definition of tautological section.

Reading Barth, Peters: Compact complex surfaces, i stumbled across the following: Let $Y$ be an algebraic surface over $k =\mathbb{C}$, and $\mathcal{L}$ an invertible sheaf on $Y$. Denote by $p: L ...
9
votes
2answers
621 views

Check my workings: Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.

Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition. Precalculations My goal is to show that for all $\epsilon >0$, there exist a $\delta > 0$, ...
3
votes
1answer
137 views

What is meant by `element $x\in H$ of minimal norm'

I do not seek a proof of the following exercise. I just want to understand this question in order to solve it myself. Let $H$ be a Hilbert space over $\mathbb R$ and let $a, b\in H$ be such that ...
1
vote
1answer
92 views

Question about piecewise linear paths

I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it. Consider a finite family of hyperplanes in a finite-dimension real ...
0
votes
1answer
63 views

What is the “spectrum of $L^1(G)$”?

If $G$ is a locally compact abelian group, what does "the spectrum of $L^1(G)$ mean?" This comes from Folland's A Course in Abstract Harmonic Analysis. As I understand it, $L^1(G)$ is the integrable ...
2
votes
1answer
83 views

Why is $X^- = -\min\{X, 0\}$?

Why is $X^- = -\min\{X, 0\}$ ? This is how it is defined in a probability book I am self-studying. For the function $X:\Omega \to \mathcal{R}$, The positive part of $X$ is the function $X^+ = \max ...
2
votes
1answer
382 views

What is a rational character?

Let $G$ be the group of $F$-points of a connected, reductive group over a $p$-adic field $F$. The unramified character of $G$ are $\chi\circ\psi$ where $\chi$ is an unramified character of ...
288
votes
18answers
53k 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 ...
15
votes
4answers
615 views

What's wrong with this “backwards” definition of limit?

Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?: $\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if ...
2
votes
1answer
264 views

Definition of a universal example

I'm not sure how the term is being used here: Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$. Given a ring homomorphism $\phi: R ...
2
votes
2answers
147 views

Motivation behind definitions of the Integral without reference to Derivatives

If a (definite) Integral can simply be calculated as the difference of two Antiderivatives and Antiderivatives are simply the "reverse process" of Differentiation. Then it seems to me that the ...
1
vote
1answer
132 views

Meaning of Point Evaluation

I read in some general measure theory books and there is always like "define measure $x$ to be the point evaluation at $y$..." but when I look around online and some other books there is no mention on ...
2
votes
1answer
177 views

Discussion: Differing definitions for the rank of a set

I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.). $\text{rank}(A)=\min\{\alpha\mid ...
1
vote
1answer
91 views

What's the meaning of $C$-embedded?

What's the meaning of $C$-embedded? It is a topological notion. Thanks ahead.
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2answers
212 views

visualisation of pointwise boundedness

A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$ ...
2
votes
2answers
109 views

What is a non-degenerate module?

I know what a non-degenerate bi-linear form is, but what does it mean for say a left $R$-module $M$ to be non-degenerate? (Here $R$ is a ring without unit$) I came across a module being called ...
7
votes
1answer
142 views

When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …

One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
4
votes
3answers
270 views

What does $H(\kappa)$ mean?

As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence: I've heard that if ...
1
vote
3answers
206 views

Time independence in SISO systems

I started learning dynamical system a couple of weeks ago and the lecture tried to define what is a SISO system in the last lecture. The lecture wasn't very clear and did not give a formal definition ...
8
votes
6answers
452 views

What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?

I thought I'd bring this question to math.SE, as it could spark some interesting discussion. When I first learned vectors - a long time ago and in high school - the textbook and teachers would always ...