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

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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
60 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
134 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
26 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
29 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
15 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 ...
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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 ...
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1answer
77 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
70 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
173 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
58 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, $\mathbf F^S$...
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0answers
43 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
55 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 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
24 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 $x$ ...
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1answer
131 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 $dU$...
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1answer
23 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
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 \...
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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 $(\Omega,\...
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3answers
31 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
14 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 $\tau^2$...
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1answer
19 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
60 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 ...
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2answers
25 views

Is there a connection between limit point of a subset of a metric space and the limit of a function?

Is there a connection between limit point of a subset of a metric space and the limit of a function, or limit of a sequence? I am not sure but I don't think there is because there can be more than ...
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2answers
49 views

Marsden's definition of Taylor Series

How does the following definition of Taylor polynomials: $f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ where $R_k(x_0,h)=\int^{x_0+h}...
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1answer
39 views

is it a vector space or not? [closed]

check if this is a vector space or not ? 1- let $v=R=\{(x,y):x,y \in \mathbb{R} \}$ check if $(V,+,.)$ where $(x,y)+(z,w)=(x,y)$ and $k.(x,y)=(k.x,k.y)$ is a vector space or not 2- ...
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2answers
25 views

An alternative definition of open sets using the following property?

Recently I have gone through the following problem. Let $E$ be a subset of a metric space with the following property: If a sequence $(x_n)_{n\in\mathbb{N}}\in X$ converges to $x\in E$ then there ...
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5answers
3k views

What does the symbol |_ mean?

For example, (6) The sequence of primes is endless. For, if $p$ is any prime, the number ${\begin{array}{|c}\color{red}p\\\hline\end{array} + 1}$ is greater than $p$ and is not divisible by $...
0
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1answer
47 views

Is there a difference between these two definitions of differentiable at a point?

I just recently learned the the definition of differentiable at a point in my multi-variable calculus class. The analogy between the multi-variable definition and that of the single variable uses the ...
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0answers
28 views

Short question about differential forms / exterior algebra

I am working towards understanding the wedge product of two vectors. Here is what I have so far: Let $V$ be an $n$-dimensional vector space over $\mathbb R$ (or $\mathbb C$, I'm not sure it makes ...
4
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0answers
84 views

Using the IVP definition of $\cos$ and $\sin$, how can we show that $\cos^2(x)+\sin^2(x) = 1$ without any “magic”?

One way to define the exponential, hyperbolic and circular functions is to assert that they're the unique solutions to certain IVP systems: The exponential function: $$\exp'(x) = \exp(x), \qquad \...
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0answers
27 views

Definition of two step centraliser

I was reading few topics related to p groups and pro p groups. Then I came across the term "two step centraliser". I could not find the definition of this in google search. It will be very helpful ...
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1answer
138 views

Are categories larger than classes?

In the definition of a category on Wikipedia, it is written that a category "consists of" a class of objects and a class of morphisms, as well as binary operations for compositions of morphisms. What ...
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1answer
31 views

Why isn't there equality in the definition of (upper) semicontinuity?

The "standard" definition of upper semicontinuity at a point $x_0$ in a metric space seems to be $\limsup_{x\to x_{0}} f(x)\le f(x_0)$. However, why is it this weaker condition instead of $\...
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4answers
33 views

$x = x' \pmod N$ iff $N$ divides $x - x'$

This is one of the first lines in one of my lecture notes, where they write: $x = x' \pmod N$ if and only if $N$ divides $x - x'$ I've taken a discrete maths course a while ago but this doesn't ...
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1answer
30 views

What's the formal definition of saying “$f$ is a function of $x$, $g$ is a function of $y$”?

I know the following definition of function (amongst others). Let $A,B$ be sets. A set $f\subseteq A\times B$ is said to be a function from $A$ to $B$ if the following two properties hold. ...
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10answers
3k views

Is there a math function to find an element in a vector?

I would like to write mathematically, if possible, the following statement: Given a vector $x=[1,4,5,3]$ and an integer $j=3$, find the position of $j$ in $x$? How to write this mathematically? ...
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0answers
57 views

How to define composition of distribution with a function correctly?

Recently I've been reading some notes on distribution theory and the author makes the following definition: Let $\zeta\in \mathcal{D}'(\mathbb{R})$ be a distribution and $f$ a $C^\infty$ function, ...
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2answers
42 views

Does the limit exist?

I had a question about the definition of a limit. I know for a limit to exist the right hand limit must equal the left hand limit but what if the graph of a specific function has the domain from [0,5] ...
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1answer
42 views

Recursive Definitions

I have two recursive definitions that I need to determine if they are correct. The first one is: The set S of all positive integer multipliers of 5 can be described by the following recursive ...
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1answer
95 views

What is the rank of a differential form

I've been searching the internet and books for a definition but none of the books on differential geometry and manifolds that I have contain the term rank in the index. While trying to find a ...
2
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1answer
46 views

Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
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3answers
63 views

Can limits be defined in a more algebraic way, instead of using the completely analytic $\delta$-$\epsilon$ definition?

Let $(X,d_X), (Y,d_Y)$ be metric spaces. Let $f:E\subseteq X\to Y$ and $a\in X$. We say that $\lim_{x\to a}f(x)=L$ if and only if for every $\epsilon >0$ there exists a $\delta>0$ such that ...
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2answers
53 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
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0answers
30 views

Étale morphisms definition?

Working over a commutative ring $R$, let $D= \left\{ d\in R : d^2 =0 \right\}$ A formally étale morphism $f:M\rightarrow N$ is one for which the square below is a pullback for every point $d:\mathbf{1}...