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

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0answers
53 views

Why is the area of a rectangle given as $l\times h$? [duplicate]

While this may seem like a simple question, I have found it very difficult to answer. My question is "Why is the area of a rectangle given as $l\times h$?" We define area as the "size of a two ...
1
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1answer
31 views

Is $\forall x \in U: f(x) \in V$ the same as $x \in U \implies f(x) \in V$?

As the title says, do the following two statements have the same meaning? $$\forall x \in U: f(x) \in V \text{ (for all $x \in U$, $f(x) \in V$)}$$ $$x \in U \implies f(x) \in V \text{ ($x \in U$ ...
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1answer
19 views

Definition of vertex-cut for digraph?

I am trying to understand vertex cut for digraph. I could find this for graphs Vertex cut is a vertex whose removal increases the number of components in a graph. (D67, Handbook of Graph Theory by ...
2
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1answer
31 views

Definition of component for a digraph?

I could find this in Wikipedia Component: A connected component of a graph is a maximal connected subgraph. The term is also used for maximal subgraphs or subsets of a graph's vertices that have ...
1
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0answers
46 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
2
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0answers
25 views

Showing that an intersection of indexed sets is a subset of every individual indexed set

I am required to show that for every $k \in I$, $\bigcap_{i\in I}A_{i}\subseteq A_{k}$ where $I$ is an index for a collection of subsets $A_{i}\subseteq S$, $i \in I$. This seems obvious to me from ...
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1answer
59 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something $\...
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0answers
27 views

w.r.t. which chain complex is $H^k_{sign}(M;R)$ computed?

This question is inspired by my previous question. People often write $H^k_{sign}(M;R)$ for the $k$th singular cohomology group with coefficients in $R$. However, I don't understand what this means. ...
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2answers
34 views

Interpretation of definitions and logical implication in Calculus - e.g. monotonic strictly increasing function

I read definitions in Calculus books that often confuse me from a logical perspective. For example, the definition of a monotonic function, e.g. a strictly increasing function, is defined as follows. ...
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2answers
452 views

What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
0
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1answer
10 views

How to show X is in the co-finite topology

This might be kind of a silly question but I can't fully grasp why the set X on which the cofinite topology is defined would be contained in the Topology. I know that the closure of all open sets U ...
1
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1answer
52 views

Coordinate charts vs. coordinates on manifolds

I just realised that I'm confused what coordinates really means in the context of manifolds. For example, say $M=S^2$. Then we can define smooth charts as follows: Let the open sets be $U = S^2$ ...
0
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1answer
45 views

What is the formal definition of a limit at infinity?

I keep coming across two different kinds of answers to this question. The first definition: We say that $$\lim_{x\to \infty} f(x) = L$$ if the following condition is satisfied: for every number $\...
0
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1answer
54 views

Is the Identity Type an Identity Function?

Definition 1. Given a set $S$, the identity function on $S$ is the function $id_S:S \to S$ that maps any element $x \in S$ to itself. Proposition. Given a type $S$, the identity type for $S$ is the ...
2
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1answer
41 views

What is the intuition behind covering spaces?

I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics. The definition of covering space is as ...
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1answer
38 views

Understanding Stokes' theorem and the fundamental theorem of calculus

I don't know a lot about differentials and boundaries, so it may be out of my grasp, but is there perhaps a simple way of understanding the Stokes' theorem for the FTC? $\displaystyle\int_a^b f(x)\...
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2answers
86 views

Why do authors make a point of $C^1$ functions being continuous?

I've just got a little question on why authors specify things the way they do. Is their some subtlety I'm missing or are they just being pedantic? I've encountered the function spaces $C^k[a,b]$ a ...
0
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0answers
32 views

Derivative of function of one variable with respect to function of two variables

I'm looking to find the derivative of a function of one variable with respect to a function of two variables: $$ \frac{df(x)}{dg(x,y)} $$ I'm not entirely sure whether this is possible in the first ...
0
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1answer
27 views

Taylor series for exponential function.

The Taylor series for $e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$. Then as $e^0 = 1$, if one evaluates the Taylor series at $x=0$ we find that $e^0 = \sum_{i=0}^\infty \frac{0^i}{i!} = \frac{0^0}{0!} + \...
1
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0answers
43 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
1
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1answer
24 views

Different versions of Markov's inequality

I have some doubts regarding some different versions of Markov's inequality. From Wikipedia's definition of Markov's inequality we have: If $X$ is a nonnegative random variable and $a>0$, then $\...
1
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1answer
31 views

Is assuming the gluing axiom for any pair of sections equivalent with gluing arbitrary collections of sections?

In Griffiths and Harris the gluing axiom for sheaves is given as For any pair of open set $U,V$ and sections $\sigma\in\mathcal{F}(U)$, $\tau\in\mathcal{F}(V)$ s.t. $\tau|_{U\cap V}=\sigma|_{U\cap ...
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1answer
109 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
1
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1answer
19 views

Definition of measure-preserving: why inverse image?

In the definition of measure-preserving dynamical system, the crucial equation is $$ \mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) . $$ Why is it not the seemingly more natural $$ \...
3
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1answer
37 views

Definition of convergence of a sequence

Can this be a valid definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Should I use "$<\epsilon$" or "$\le \epsilon$"?
2
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0answers
33 views

What is the structural hierarchy in mathematics?

By structural hierarchy, I mean the mental concept in which things are 'done' in mathematics. At the top, you have mathematics itself, which is a collection of systems, like arithmetic, algebra, ...
0
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1answer
35 views

Compactness and sequences in $\mathbb{R}^{n}$

Why is it that: If $A$ is a compact set and $\left ( a_{n} \right )$ a sequence in $A$, then there is a subsequence $\{a_{n_k}\}$ such that $\lim_{k\to\infty} a_{n_k}=a$ with $a\in A$. I get ...
2
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2answers
33 views

What is the definition of “prime ideal decomposition”?

I'm reading about Sunada's theorem in the book Geometry and Spectra of Compact Riemann Surfaces (Peter Buser) and I encountered this paragraph: If R is an algebraic number field and if $p \in \...
2
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1answer
31 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j \...
1
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1answer
13 views

What are those weighed graphs called?

Let $G = (V, E)$ be a directed graph, and define the weight function $f : V \sqcup E \to \mathbb{R}^+$ as follows: sum of weights of vertices is 1, if a vertex has edges coming out of it, their ...
4
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1answer
41 views

What definition of Reed-Solomon code is correct?

In the discuss with @Evinda we have the contradiction with the definition of Reed-Solomon Codes (not generalized case) over the finite field $\mathbb{F}_q$. We have two papers and http://kom.aau....
0
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1answer
34 views

How to lift an action along a covering

Took from Lawson-Michelsohn "Spin Geometry", pages 80-81. Let $E\to X$ be an oriented $n$-dimensional vector bundle over a manifold $X$, and let $P_{SO}E$ be the associated $SO(n)$-bundle. Assume $...
0
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1answer
24 views

Is this definition of a Euclidean frame well-defined?

Going through my lecture notes on geometry I find a definition of a Euclidean frame which doesn't seem to have been formed correctly (most likely written down wrong). So I've taken it upon myself to ...
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2answers
82 views

Is this definition of Mersenne Primes correct? [closed]

According to my understanding, the definition of Mersenne Prime is the following: A Mersenne Prime is a prime number that is obtained by using the formula $2^n-1$, where $n\in\mathbb{N}_+$
3
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2answers
49 views

Properties of sin(x) and cos(x) from definition as solution to differential equation y''=-y

I recently came across the interesting definition of the sine function as the unique solution to the Initial Value Problem $$y'' = -y$$ $$y(0) = 0, y'(0) = 1$$ (My first question would be why this ...
0
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0answers
21 views

Hatcher's notation and definition for the boundary of a singular chain complex

I have a hard time following Hatcher in general, but especially his notation in this case with the bar (page 108 in my pdf copy): For the boundary of a singular $n$-simplex, $\sigma$,he writes: $\...
2
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0answers
25 views

Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter $\...
1
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3answers
67 views

$0^0$ is undefined, but sometimes defined as $1$?

When typing a calculation to Wolfram Alpha it always takes $0^0$ as undefined, but for some calculations I need to preform, I specifically need it to be equal $1$. Is there a way to make it ...
1
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3answers
36 views

Extension of Up/Down/Right/Left in n dimensions

I'm wondering if there are extensions of the right/left/up/down concepts in n dimensions, i.e. "if $x_0$ points ahead of me, I'm looking LEFT/RIGHT along $x_1$, UP/DOWN along $x_2$, XXX/YYY along $x_3$...
0
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1answer
27 views

Definition of Polish Topology

Let $\xi$ such that $1 \leq \xi < \omega_1$ and $f$ be of Baire class $\xi$. In this paper (Section $5$), the author defined $$T_{f,\xi}=\left\{ \tau^{\prime} : \tau \subset \tau^{\prime} \text{ ...
0
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1answer
34 views

How to calculate $f'(0)$ from $f(x)=x|x|$ directly from the definition of derivatives as a limit?

How to calculate $f'(0)$ from $f(x)=x|x|$ directly from the definition of derivatives as a limit? So $\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim\limits_{h\to 0}\frac{(x+h)|x+h|-|x|x}{h}=\lim\...
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2answers
47 views

How to formally define $\lim\limits_{x\to\infty}f(x)=\infty$?

How to formally define $\lim\limits_{x\to\infty}f(x)=\infty$ I suppose it's $\forall K\in\Bbb{R},\ \exists N\in\Bbb{R},\ x\gt N\implies f(x)\gt K$. Could someone correct it?
2
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2answers
38 views

Definition of interior

An interior point is defined as the following in the Euclidean space. If $S$ is a subset of a Euclidean space, then $x$ is an interior point of $S$ if there exists an open ball centered at $x$ ...
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1answer
16 views

What does it mean for a boundary to be analytic in the context of a PDE?

I am reading a paper where they assume the boundary of a domain is "Analytic". They never define it. Is this a standard definition, and, if so, what is it?
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20answers
2k views

Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines ...
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0answers
21 views

Commutative rings and PI algebras

Any commutative ring $R$ with unity is a PI ring (polynomial identity ring). When could one take $R$ as a PI algebra? Essentially, what is the relation between an arbitrary commutative ring and a ...
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0answers
20 views

Is a finite dimensional vector also a covector?

I am reading something where it seems they have defined a covector/linear functional as a map $F: \mathbb{R}^n \to \mathbb{R}$ where $F$ is a vector For example, $F(x) = \begin{bmatrix} 1 & 2 \...
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1answer
26 views

An equivalent definition of connectedness in General Toplogy

I'm studying General Topology by the book of James Munkres and there he defined connectedness this way: "Let X be a topological space. A separation of X is a pair U, V of disjoint nonempty open ...
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2answers
32 views

the definition of random variable

If we supposed that X is a random variable, is X - X a random variable? Could the outcome of an event is only 1? Cause X-X has only one outcome, and the possibility of it is 1. How about X + X?
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0answers
32 views

Definition of an Integral Domain in the second edition of Herstein's Topics in Algebra

I think a definition in Herstein's Topics in Algebra needs to be modified and I am asking this question to make sure I am not missing something. Herstein defines an integral domain as a commutative ...