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

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28 views

What is the name of this homotopy?

Here is my definition for homotopy: Let $X,Y$ be topological spaces. Let $f,g:X\rightarrow Y$ be continuous functions. If there is a function $F:X\times[0,1]\rightarrow Y$ such that $F(x,...
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1answer
187 views

Measures: Atom Definitions

Let $\Omega$ be a measure space with measure $\mu$. (Here, a measure is only meant to be countable additive!) Consider a subset $A\in\Sigma$. Then according to the wikipedia article it is an atom if:...
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42 views

What's the best way to define $\mathbb{H}$?

I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that $\...
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1answer
26 views

What is the definition of finitely generated?

To be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of $...
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34 views

Integrable Systems-Description

I am relatively new to the topic of integrable systems and I came across the following description: "Integrable systems share a fundamental property which is related to the geometry of the initial ...
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86 views

Definition of “one element normalizes another element”

I understand that someone has asked the definition of "an element of a group normalizes a subgroup". Let $G$ be a group and $a,b\in G$. Then what does "$a$ normalizes $b$" mean? Does it make sense at ...
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1answer
55 views

What is an exact differential?

My book says "A differential expression $M(x, y)dx+N(x, y)dy$ is an exact differential in a region $R$ of the $xy$-plane if it corresponds to the differential of some function $f(x, y)$ defined on $R$"...
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49 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, Q:=M\...
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348 views

What is $R$-algebra?

(To be clear, I mean a ring by a ring with unity.) Here is a definition in Wikipedia: Let $R$ be a commutative ring. Let $(M,+,\cdot)$ be an $R$-module. Let $\ast$ be a binary operation ...
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2answers
130 views

Complex Measures: Integrability

Problem On the one hand, a complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to the integrability condition: $$...
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1answer
101 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: $$\mathcal{S}:=\{s:\Omega\to\mathbb{C}:s=\sum_{k=1}^{K<\infty}s_k\chi_{A_k:\lambda(A_k)<\infty}\}...
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40 views

Linear independence and grammar

Let $A$ be a commutative ring. Normally, the linear independence is a property of a subset or a family elements of an $A$-module. But one often sees statements like "$v$ and $w$ are linearly ...
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2answers
216 views

(Are there) subtleties in the definition of 'biproduct'

I always thought that a biproduct of two objects $A_1,A_2$ in some category $\mathcal{C}$ is an object $P$ with two maps $p_i:P\to A_i$ making it a product and two maps $j_i:A_i\to P$ making it a ...
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1answer
251 views

What makes a line “straight”?

In Euclidean space, there can be several definitions that makes a straght line: Line of shortest distance between two points Line that is linear, i.e with the points satisfying a linear equation ...
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2answers
154 views

Precise definition of affine, smooth, and irreducible

A book which I'm reading now says that "the Drinfeld curve $$ \mathbf{Y} = \{\, (x, y) \in \mathbf{A}^2(\mathbb{F}) \mid xy^q - yx^q = 1 \,\}$$ is affine, smooth, and irreducible." Here $p$ is an odd ...
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1answer
207 views

Two definitions of left-invariant vector fields of a Lie group

I am reading these lines from a text which shows why the bracket of two left-invariant vector fields is also a left-invariant vector field. But cannot easily get one of the lines. Let $L_a$ be the ...
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2answers
188 views

Algebraic Structures: Does Order Matter?

(I want to link to similar question with a very good answer: Question about Algebraic structure?) An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and ...
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3answers
521 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
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22 views

Weierstrass conditions, what does strong mean, and are both conditions required?

I have the Weierstrass condition: In order that the extremal $\bf{C^*}: x = x^*(t)$ give a strong local minimum to $\bf{J[x]}$ it is sufficient that: # $C^*...
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73 views

Rotational invariant curve definition

I have a question about the definition of rotational invariant curve. The definition I have is the following "By an invariant curve for a twist map $F$ we mean a simple closed curve that is invariant ...
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1answer
34 views

Can a linear functional be infinite at a point?

On a Banach (or Hilbert) space $X$, when we define a linear functional (not necessarily bounded), we define it to be a linear function from the elements of $X$ to the field $\Bbb F$. (Say, $\Bbb R$). ...
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2answers
362 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) \...
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3answers
114 views

What is free product?

I have searched for it, but I found there are several many different definitions. Even wikipedia states just free product of $2$ sets, not an infinite product. I know what exactly free group of a ...
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1answer
35 views

Is my understanding of free group correct?

Let $(G,*)$ be a group. Let $S$ be a subset of $G$. Then, construct the free group $(F(S),*')$ on $S$. If there exists an isomorphism $\phi:(G,*)\rightarrow (F(S),*')$ such that $\phi(s)=(s)$ on $S$...
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2answers
64 views

Cauchy property of a series

Are these two definitions equivalent, even though the first one has an extra term: If we consider the series $\sum_{n=1}^{\infty}x_{n}$ and the formal definition of a Cauchy property defined in terms ...
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1answer
699 views

Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every ...
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39 views

Definition question.

Suppose we have $x \in \mathbb{R} / [0,1)$. We call $\lfloor x \rfloor$ the integer part of $x$. What do we call $\bar{x}=\frac{x}{\lfloor x \rfloor}$??? I would call it fractional part but it ...
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1answer
38 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
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2answers
29 views

Give the definition of S $\subseteq$ T for general sets S and T.

The answer I can come up with is; S is a subset of T, denoted by S $\subseteq$ T, or equivalently, T is a superset of S, denoted by T $\supseteq$ S. Can someone correct me if I am wrong, or provide ...
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1answer
40 views

Why is the structure group for lengths $\mathbb{R}^+$ and not automorphisms in $\mathbb{R}^+$?

While reading what is a gauge to understand what a gauge is, I got stuck at a point where Terry Tao wrote: This isomorphism group is called the structure group (or gauge group) of the class of ...
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3answers
507 views

Understanding of the formal and intuitive definition of a limit

The intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$ . I can easily ...
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1answer
35 views

Concise description of Lebesgue measure in $\mathbb{R}^{n}$

I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$. The outer Lebesgue measure $E \subset \mathbb{R}^{n}$: $$\lambda^{*}(E) = \inf\...
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2answers
36 views

Regarding retractions of $X$ onto subspaces

Let $A \subset X$ be a subspace of $X$. Recall that a retraction of $X$ onto $A$ is a continuous map $r: X \to A$ such that $r(a) = a$ for every $a \in A$. Let $X = \bf R$ endowed with the standard ...
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2answers
42 views

Product over a vector space

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms. However, none of those axioms talk about the product of two vectors. Is ...
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1answer
42 views

The definition of a submanifold

I am wondering why it is insufficient to define a submanifold of a manifold $M$ as a subset $S\subset M$ such that $S$ itself is a manifold. Why do we need the notions of embedded submanifolds or ...
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1answer
170 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
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2answers
124 views

Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ [proof reading]

Here was the question asked to me :: Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ I spontaneously said that it was because of their very ...
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0answers
82 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
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35 views

Different definitions of an affine algebraic set

Hartshorne assumes we are working over an algebraically closed field $k$ and we simply define algebraic sets to be any set of the form $V(I)\subset k^n$, where $I\subset k[X_1,\ldots,X_n]$ is an ideal....
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2answers
47 views

Is every set a subset of a vector space?

I was taught that a functional is a map from a subset (not subspace) of a vector space into the reals, $F: D\subset V \to \mathbb{R}$. I know there are other definitions, but is there any reason to ...
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1answer
95 views

What does $R[[X]]$ and $R(X)$ stands for?

I'm reviewing Linear Algebra these days and I saw these two notations in my notes without definition. Those are, $R[[X]]$ and $R(X)$ where $R$ is a commutative ring with unity. I remember that one ...
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1answer
25 views

What do $A \upharpoonright x$ and $\mu s \ge x$ denote?

I am reading Computability Theory by Cooper and I do not understand the notation in the definition on the page 230: Let $\{A^s\}_{s \ge 0}$ be a $\Delta_2$-approximating sequence for $A \in \...
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1answer
539 views

Maximum/Maximal set

Maximum or maximal set with property $P$ When I was reading some textbooks, I noticed that I do not get the meaning of the following two phrases. ($P1$) $\quad$ maximum set with property $P$ ($P2$) ...
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32 views

Changing the zero product property and defining division by zero [duplicate]

I know that defining division by zero is not possible because it violates the zero product property we define, that is, $0\times a=0$ for every $a$. I wonder whether we can somewhat circumvent and ...
2
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1answer
112 views

How would you describe category the $\mathsf{Rel}$?

I encountered two definitions for a category denoted by $\mathsf{Rel}$: Objects are pairs $\left(A,R\right)$ where $A$ is a set and $R$ a relation on $A$. Arrows in $\mathsf{Rel}\left(\left(A,R\...
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60 views

Calculating the argument of a complex number… something tends towards infinity?

A simple question, but I like to be clinical with my choice of words: I have a complex number, $z=-i$. If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) \...
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4answers
256 views

Why is the expression $\underbrace{n\cdot n\cdot \ldots n}_{k \text{ times}}$ bad?

For writing a (german) article about the power with natural degree I have the following question: In school one defines the power with natural degree via $$n^k = \underbrace{n\cdot n\cdot \ldots \...
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1answer
32 views

What is the usual definition of a zero divisor?

Let $R$ be a ring. I found there are two distinct definitions: Wikipedia Definition $a\in R$ is a zero divisor iff there exists nonzero $b\in R$ such that $ab=0$ or $ba=0$. Another: ...
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1answer
43 views

Definition of Random Sample in Estimation

In my statistics class, we're just beginning to talk about (point) estimation. I understand the basics for the most part, but I have a small question that might actually be due more to notation than ...
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1answer
2k views

Recurrence vs Recursive

Say team 1 is studying the recursive characteristics of a function. Team 2 is studying the recurrent characteristics of the same function. Are the 2 teams studying the same thing? I have found for ...