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

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1answer
20 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
81 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 $n\in\mathbb{N}_+$
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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 ...
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0answers
20 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: ...
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0answers
21 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 ...
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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 ...
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3answers
35 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 ...
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1answer
26 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{ ...
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1answer
33 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 ...
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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?
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2answers
31 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
15 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
31 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 ...
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1answer
38 views

What is the catch when introducing measure theory using $\sigma$-ring instead of $\sigma$-algebra?

I am currently using Matthew A Pons book Real Anaysis for Undergrad for introduction of measure theory In my opinion this book is unbelievably clear for almost all the sections EXCEPT the section ...
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1answer
39 views

What is the definition of binary sequence?

Can I write an infinite binary sequence like so: ...0111001001, ...10010 because I saw some people write infinite binary set from left to right like so: 1011000... , 101111... But I was not sure if ...
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1answer
28 views

Combinatorial Geometry explanation

I do not understand what is going on in $(4)$: for every flat $E \in \mathcal F$, $E \ne X$, the flats that cover $E$ in $\mathcal F$ partition the remaining parts. What is meant by "the flats ...
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1answer
28 views

Definition of matrix transformation

I really understand the definition of linear transformation, but I'm not sure about the definition of matrix transformation. Could it be that a matrix transformation is defined as a linear ...
0
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1answer
24 views

Is a function, in part, defined by it's domain?

I have read the definition of a function from two sources.Both sources state that a function defines a relationship between the input and the output. However, the first source states that it is the ...
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2answers
59 views

Is there a symbol for always less than (or just always?)

For e.g, the quotient of $\frac{1}{n}$, $q$, where $n \gt 1$, $q$ will always be less than $1$. $$\frac qn\le n$$ etc. I can't really write $\frac {q}{n} < n$, because whilst true, it doesn't ...
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2answers
129 views

A Pure Maths Approach to Thinking About Vectors

Introduction Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no ...
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5answers
63 views

Does excluding or including zero from the definitions of “positive” and “negative” make any consequential difference in mathematics?

I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative. But today I made a few Google searches, and they all say the same ...
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1answer
25 views

Explanation of defintion of ${[n] \choose k}$

I am reading a definition in a paper, but am not sure of how to interpret the following definition: If $K \in {[n] \choose k}$, then let $\operatorname{Path}(K)$ denote the set $$\{S: S \text{ is ...
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1answer
28 views

Condition on symplectic form: $(d\alpha)^n \neq 0$?

I started to read about contact and symplectic forms and I came across this answer here. It seems to state that the definition of symplectic form is that $d\alpha$ is non-degenerate if and only if ...
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0answers
12 views

Terminology for a Complex Semicircle

A while back I read about a special type of number system termed (if I remember correctly) as "degrees of sign". The idea was that numbers sat on a series of 0 to 180 degree rays. Positive ways the 0 ...
0
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1answer
16 views

Cocycle condition on bundles as some equivalence relation?

The three cocycle conditions on the transition maps of a fiber bundle look alot like the three condition defining equivalence relation - refelxivity, symmetry, and transitivity. I was wondering ...
4
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1answer
72 views

Is this a valid definition of “self-similar fractal”?

I have always been fascinated by self-similarity, particularly in fractals. I was always wanted to find a simple definition of a self-similar fractal. Of course, saying "is self-similar, and is a ...
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0answers
33 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
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2answers
68 views

Does this definition of $e$ even make sense?

This sprung from a conversation here. In Stewart's Calculus textbook, he defined $e$ as the unique solution to $\lim\limits_{h\to 0}\frac{x^h-1}{h}=1$. Ahmed asked how do you define $x^h$ is not by ...
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1answer
38 views

Elementary Definition of Differentiability: What's Correct?

In most if not all elementary calculus books, a function $f : A \rightarrow \mathbb{R}$ with $A \subseteq \mathbb{R}$ is said to be "differentiable'' at a point $a \in A$ if and only if the limit $$ ...
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0answers
169 views

Is an étale morphism of algebraic stacks locally quasi-finite?

An étale morphism of schemes is unramified, and an unramified morphism is locally quasi-finite. Does the same hold for étale morphisms of algebraic stacks? Let us recall the definitions, following ...
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2answers
55 views

Why is $S$ specified to be nonempty in Axler’s definition of function spaces $\mathbf F^S$?

In Example 1.24 of S. Axler’s Linear Algebra Done Right, the following statement is made: If $S$ is a nonempty set, then $\mathbf F^S$ […] is a vector space over $\mathbf F$. (Here, $\mathbf ...
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0answers
42 views

different definitions of a group

A group is classically defined as a set with a binary operation (the group product) which is associative, such that there is a unit and for every element there is an inverse. I know we can define a ...
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0answers
53 views

How are neighbouring sequences defined? (Metric spaces)

What does it mean for sequences to be neighbours in a metric space? My attempt is: In a metric space $(X,d)$, $(x_n)$ and $(y_n)$ are neighbouring sequences iff $$\forall_{\epsilon>0}\exists_N ...
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0answers
41 views

Categories of defintion for sites on spaces and sites on schemes

Starting with example 2.26 in Vistoli's Notes he defines topological sites on $\mathsf{Top}$, while sites on schemes are always defined on a slice category $\mathsf{Sch}/X$. Why is this?
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3answers
19 views

What does it mean for simple functions to have finite range

In Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics By Dan Simovici, Chabane Djeraba, it says: A simple function is a function $f: S \to \mathbb{R}$ that has finite ...
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1answer
53 views

Non-geometer friendly definition of $\cos$ and $\sin$.

Let $S=\{(x,y)\in \Bbb R^2: x^2+y^2=1\}$ be the unit circle. From here, I want to define the $\sin$ and $\cos$ functions. I've seen some sites saying things like "$\cos\theta$ is the $x$ ...
0
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1answer
130 views

Help to prove numerically the given equation below?

Consider a spectral decomposition of a unitary matrix $U$ given by $WAW^*$ where $A$ is diagonal matrix of eigen-values of $U$ and the symbol $^*$ means transconjugate. An infinitesimal shift ...
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1answer
22 views

Existence of asymptotic variance for an estimator when it doesn't converge to normal distribution.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,\ldots)$, where: $U_i=U_i(X_1,\ldots,X_i)$, if for a sequence of constants $\{k_n\}$: ...
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0answers
34 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: $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b \mid c$, ...
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1answer
37 views

Why do we have to use pre-image in the formal definition of random variable?

There is this definition of random variable: Let $(\Omega,\mathcal{F}), (\Omega',\mathcal{F}')$ be two event spaces. We say that a function $X:\Omega\to\Omega'$ is a random variable from ...
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3answers
30 views

What does “within $x$% of $y$” mean?

I cannot find a specific definition for "within $x$% of $y$". If I want a number within $10$% of $100$, am I looking for numbers in the set $[90,110]$? Or does it mean percentage difference? Or ...
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0answers
13 views

Comparing definitions of limiting and asymptotic variances - what is the intuition behind?

In Casella's inference, it says: Definition 10.1.7: For an estimator $T_n$, if $\lim_{n\to \infty} k_n Var T_n = \tau^2 < \infty$, where $\{k_n\}$ is a sequence of constants, then ...
0
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1answer
17 views

Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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3answers
39 views

Difference between a continuous function and an isometry? Is a continuous function a homomorphism?

Definition of continuous function on a set: Suppose $X$ and $Y$ are metric spaces. Let $f: X \to Y$. We say $f$ is continuous on $X$ if for every $\varepsilon >0$, $\exists \delta >0$ such that ...
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1answer
38 views

What does exhaustive, non exhaustive and mutually exclusive mean in probability

I am doing some work on probability. I have done some background reading on the definitions on exhaustive, non exhaustive and mutually exclusive but the definitions that I found do not make any sense ...