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

learn more… | top users | synonyms

0
votes
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 ...
1
vote
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
votes
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 ...
1
vote
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 $\{k_n\}$: ...
1
vote
0answers
40 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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
2answers
24 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 ...
2
votes
2answers
48 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 ...
-1
votes
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- ...
0
votes
2answers
22 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 ...
38
votes
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
votes
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 ...
1
vote
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
votes
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 ...
0
votes
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 ...
8
votes
1answer
126 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 ...
1
vote
1answer
29 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 ...
0
votes
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 ...
1
vote
1answer
28 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. ...
16
votes
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? ...
2
votes
0answers
54 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, ...
1
vote
2answers
40 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] ...
0
votes
1answer
40 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 ...
0
votes
1answer
84 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
votes
1answer
36 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 ...
2
votes
3answers
62 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 ...
1
vote
2answers
44 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, ...
1
vote
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 ...
0
votes
1answer
38 views

What are “members” of an event in probability theory?

My homework asks a question along the lines of "Suppose a fair six-sided die is rolled and a fair coin is flipped. List the members of the event 'the die lands on an even number.'" I can only guess ...
1
vote
1answer
22 views

Definition of a monotone class

The way I learned it, a monotone class on a set $X \neq \varnothing$ is a collection of subsets of $X$ that is closed under monotone countable unions and intersections. According to this, $X$ is not ...
2
votes
1answer
29 views

What is the closure of $A = \{x| 1<\|x\|<3\} \cup \{(0,0)\}$ and why am I wrong?

Given $A = \{x| 1<\|x\|<3\} \cup \{(0,0)\}$ Find $\bar A$ My hunch is $\overline A = \{x| 1 \leq x \leq 3\} \cup \{0,0\}$, but my friend says I am wrong, the closure of $A$ must ...
0
votes
1answer
28 views

Nothing on the web; What is a Ruffini Radical

Surprisingly, it's not clearly defined online. The first thing that comes up is Abel-Ruffini theorem, which only refers to "radicals" and not RUFFINI radicals. Ian Stewart's book has it appear out of ...
1
vote
1answer
32 views

finite presentations of algebraic structures

I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators. I'm ...
6
votes
0answers
38 views

Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
0
votes
1answer
51 views

Why is $\Delta x \approx \left(\dfrac{\partial x}{\partial u}\right)_0\Delta u$?

I'm trying to understand the proof for variable substitution of multivariable integrals here, but I'm not really sure what is meant by $\Delta x \approx \left(\dfrac{\partial x}{\partial ...
0
votes
1answer
21 views

Mathematical definition of congruent sets

I cant really find a proper definition for this. If two sets are congruent, then what does that mean. I heard that it can be defined in terms of isometries... This is with respect to the banach ...
3
votes
0answers
35 views

A Cartesian coordinate system is a mathematical or physical thing?

I'm convinced that if I ask what of the coordinates systems in the figure is a Cartesian system almost all say that it is the system $O_1$. This answer comes immediately from our habit and ...
3
votes
0answers
46 views

С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
1
vote
2answers
52 views

Is a function differentiable in $x$ if $\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\infty$?

All the definitions of differentiability I found (Wolfram Mathworld for instance) only require this limit to exist, but say nothing about the domain in which that has to happen. So what if that ...
1
vote
0answers
40 views

Is this how we define “limit in the distributional sense”?

Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over ...
2
votes
0answers
37 views

Does the metric in the Theory of Relativity actually satisfy the definition of a metric?

Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ...
2
votes
2answers
277 views

Are these two definitions of differentiability equal?

A function $f$ is differentiable in $x$ iff the limit $~\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}~~\text{exists}$ ("normal" definition) $|f(x+h)-f(x)|<C|h|~$ holds for small $h$ ...
2
votes
0answers
32 views

Is the generalized mandelbrot set a fractal in the $d$ dimension?

The $d$-mandelbrot set is defined as the set of $c$ such that the iterations of $$z \mapsto z^d + c$$ starting with $z=0$ is bounded in absolute value. Here is a picture of the mandelbrot sets from ...
2
votes
1answer
31 views

Quick clarification on the definition of vector field

I am having a class on differential geometry and another on ODEs In the ODE class, if we were given something of the type $$\dot x = x^2$$ The professor refers to $x^2$ as the vector field. In ...
5
votes
2answers
75 views

Probability: mathematically what does it mean to say “let $X$ be a random variable WITH a cdf/pdf”

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and ...