Questions tagged [definition]

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

Filter by
Sorted by
Tagged with
0
votes
1answer
24 views

what $H^1_q$ is?

I don't know the meaning of the notation $H^1_q$ (a Sobolev space)? Where I can find the exact definition of that notation? Thank you in advance.
0
votes
1answer
33 views

What does “smallest” mean in this definition?

In first-order logic, a set $\tau$ of terms is the smallest set of strings (finite sequences of symbols) over the sets of variables, constants, and function symbols such that: Any variable is a term. ...
-1
votes
1answer
51 views

Why is it that $a\equiv b \pmod{m} \iff m|(a-b)$

Why is it that $a\equiv b \pmod m \iff m\mid(a-b)$? I know this is a definition, but it's a definition I don't understand. I know that if $m\mid(a-b)$ then there exists some $q\in\mathbb Z$ such that $...
0
votes
1answer
22 views

What is the formal defintion of $\sigma(X_1,X_2,\ldots)$ for a sequence of random variables $X_1,X_2,\ldots$?

In probability theory one often seeks to construct the $\sigma$-field $ \sigma(X_{1},X_{2},\ldots) $ for a sequence of R.V.'s $\{X_n\}_{n\in \mathbb{N}}$ (Assumption: $X_n:(\Omega, \mathcal{F}) \...
1
vote
0answers
30 views

Looking for an intuition of the definition of Generalized Eigenspaces

The eigenspace of (a square matrix) $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x}=\lambda\mathbf{x}$, or equivalently, $(A-\lambda I)\mathbf{x}=...
0
votes
1answer
29 views

Intuitive vs Formal (model-theoretic) definition of consistency of a theory

A theory $T$ of $L$-sentences is said to be consistent if there exists a model of $T$. More precisely, there exists an $L$-structure $\mathcal{M}$ such that $M$ satisfies $\sigma$ for each $\sigma \in ...
1
vote
0answers
26 views

Is the root of a number defined as a single number or a collection of numbers?

I always had trouble understanding one thing about roots: Imagine that we want to find the solutions for the following equation: $$x^2 = 4$$ It's easy to see that the solutions are $2$ and $-2$. But ...
1
vote
1answer
32 views

Why are there no cycles of length 2?

A graph $G$ is an ordered pair $(V,E)$ of disjoint sets where $$E\subset V^{\underline{2}}:=\left\{\{x,y\}\ |\ x,y\in V\land x\ne y\right\}.$$ Let $G=(V,E)$ be a graph. A path $P$ in $G$ is an ...
1
vote
0answers
36 views

Definition of normal subgroups

One of the equivalent definitions for normal subgroups that I have studied is: "Let $G$ be a group and let $H \leq G$. We say that $H$ is normal in $G$ if $\forall g \in G$ we have $gHg^{-1} \...
0
votes
2answers
31 views

Helping me understand the formal definition of ODE's

Currently I am writing a project at Uni, where we define Ordinary differential equations (ODE's). However, I seem to have some trouble comprehending the definition. It's taken from Lecture notes on ...
0
votes
1answer
18 views

Quick question about the definition of a random variable in a baseball pitches example

I'm just wondering, in the last sentence of the third paragraph, why can X not be larger than 6? Is that because the number of innings is 9? I'm really confused, I thought X goes from 3,4,5,6,...to ...
-1
votes
0answers
38 views

What is an inclusion of a subgroup?

I have to find all the inclusions of the subgroups of $\mathbb{Z}_{48}$ but I don´t know what is that. What is an inclusion?
2
votes
3answers
41 views

Notation for fields and the elements of the sets used to define them.

Context: While trying to figure out how to prove that the characteristic of a field $\mathcal{F}$ is either $0$ or prime, I realized that I was conflating the elements of the field with integers, ...
1
vote
1answer
67 views

Can't understand definition of system $P$ given by Kurt Godel in his “On formally undecidable propositions…”

I am reading On formally undecidable propositions of Principia Mathematica and related systems by Kurt Godel. And I am not quite sure I am getting his definitions right. See the screenshot beneath: .....
0
votes
2answers
28 views

About finding the closure of a set.

I want to learn how should I find the closure of a set $S$. It's a long that I was not worked with topological concepts, and I am afraid if I am mistaken about some easy things. Here I will list some ...
1
vote
2answers
72 views

Why $0$ is not irreducible?

We took the following definition of irreducible: If $R$ is a commutative ring, $r \in R$ is irreducible if $r = ab$ for $a,b \in R$ implies $a \in R^{*}$ or $b \in R^{*}.$ Then my professor said: &...
2
votes
1answer
60 views

What is an “induction principle”?

In section 4 Empirical Risk Minimzation of the paper Principles of Risk Minimization for Learning Theory by V. Vapnik, the author says the following: In order to solve this problem, the following ...
9
votes
5answers
521 views

If sum of triangle angles is $180$ degrees, how $\sin(270)$ is possible?

I'm not new to trigonometry, but this question always bothers me. As it is in Wolfram MathWorld- $$ \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} $$ We know that the sum of the angles in a ...
0
votes
1answer
14 views

Definition of linear dependence

My linear algebra teacher stated the following definition: Let $V$ be a vector space and $v_1,...,v_n \in V$ vectors. We say that $v_1,...,v_n$ are linearly dependent if $\exists \alpha_1,...,\...
0
votes
0answers
29 views

How to quantify “rare”

When you consider the word unusual, it can be mathematically defined as being two standard deviations outside of the mean. Does the word rare have a definition or commonly agreed upon quantifiable ...
1
vote
1answer
35 views

Definition of $k$-times differentiable on $S$?

I'm trying to come up with a good definition of being $k$-times differentiable on a subset $S$. Here's the definitions I'm working with. Let $f$ be a function from $X\subseteq\mathbb{R}$ to $\mathbb{R}...
10
votes
1answer
136 views

Codomain as part of the definition of a function

For most practical purposes, the codomain is part of the definition of a function. The codomain is specified at the moment we write $f:A \to B$, and is exactly what makes the word "surjective&...
0
votes
0answers
25 views

What does it mean for an automorphism of sheaves to cover an automorphism of schemes?

Let $X$ be a scheme, $G$ a group scheme and $\mu \colon G \times X \rightarrow X$ an action. On p. 104 of Mumford's Abelian Varieties 2nd ed. (111 of the 1st ed.) Mumford gives the following ...
0
votes
0answers
20 views

Linear subspace tangent to a manifold

What does it mean for a vector space/linear subspace to be tangent to a manifold (e.g. in $\mathbb{R}^n$ or $\mathbb{C}^n$)? Does it mean that it is contained in the tangent plane to some point? If ...
1
vote
0answers
27 views

Definition of Finite Regular Graph

Massey makes the following definition in his book Singular Homology Theory: A finite, regular graph is a pair consisting of a Hausdorff space $X$ and a finite subspace $X^0$ (points of $X^0$ are ...
3
votes
2answers
39 views

little-oh notation

Assume I have a function $ f:\mathbb{R}^{m}\to\mathbb{R}^n $, and let $ h\in \mathbb{R}^m $. Does the notation $ f\left(x_0+h\right)=o\left(h\right) $ (for fixed $x_0 $) mean that $$ \lim_{h\to0}\frac{...
3
votes
1answer
63 views

What does it mean to some logical statement to be provable and decidable?

I am not a mathematician and trying to get deeper insight into modern logic. It happens all the time, that statements like statement P is unprovable arise, or, more ...
2
votes
1answer
93 views

Definition of Cell Decomposition?

In Chapter 5 of Lee's Intro to Topological Manifolds (page 130), he defines a cell decomposition as follows: I've been struggling to properly unpack this characterization. I have the two following ...
-1
votes
1answer
75 views

Calculate $ \lim_{x\to +2} \frac{x+2}{x-4}$ by definition [closed]

Calculate $ \lim_{x\to +2} \frac{x+2}{x-4}$ by definition. It is clear that the limit is equal to $-2$ , but I have to use the definition , and I don't know how to choose the $\delta$ correctly. ...
0
votes
1answer
57 views

Dartboard paradox and understanding independence

By definition, events $A$ and $B$ are independent if $$P(A \cap B) = P(A)\:P(B).$$ Therefore if an event $A$ happens almost never $\left(P(A)=0\right)$ or almost surely $\left(P(A)=1\right)$, then ...
1
vote
2answers
49 views

What does “$A \leq B : \Longleftrightarrow A \subseteq B$ is an order relation of $\mathcal{P}(N)$” mean?

I have read the following in some exercise for discrete mathematics. Let $N$ be a set and $\mathcal{P}(N)$ be its power set. Then $A \leq B : \Longleftrightarrow A \subseteq B$ is an order relation of ...
1
vote
1answer
32 views

Which notion of equivalence is being used here?

I'm reading Homotopy associativiy of H-spaces I by Stasheff and in the proof of Proposition 2 he says that any map is equivalent to a fibring (in modern day language I think this is either a fibration ...
1
vote
1answer
27 views

Definiton of inner product on $L^2$ with positive definite symmetric matrix

Consider a measure space $\left(\Omega,\mathcal A,\mu\right)$. Now one can define an inner product $$\langle f,g\rangle_Q:=\int_\Omega\int_\Omega f(x)g(y)q(x,y)\mu\left(dx\right)\mu\left(dy\right),$$ ...
0
votes
0answers
39 views

Show that $\mathbb{Z}[\sqrt{3}]$ satisfies the following division property:

Here is the question I want to solve: Show that $\mathbb{Z}[\sqrt{3}]$ satisfies the following division property: Given $a,b \in \mathbb{Z}[\sqrt{3}]$ there exist $q,r \in \mathbb{Z}[\sqrt{3}]$ such ...
3
votes
0answers
66 views

Different definitions of the same object in math and how they are related

While reading I came across in a book with two different definitions about the same mathematical object. This kinda make feel anxious. Let me explain precisely what I mean. Cartesian product can be ...
0
votes
1answer
46 views

Two equivalent definitions of continuity

In the book Topology: A Geometric Approach, two equivalent definitions of continuity. The first one is pretty basic. It is this: Definition $1.1.1$. $f$ is continuous at $x \in X$ if, given $\epsilon&...
0
votes
1answer
31 views

Open ball of the Discrete Metric

Let $(X,d)$ be the discrete metric space, such that for $x,y\in X$, $$d(x,y)=\cases{0 & $x=y$\\ 1 & $x\ne y$}$$ The open ball is defined as $$B_r(x)=\cases{\{x\} & $0<\varepsilon \le 1$\...
2
votes
0answers
18 views

Is there anyone defining asymptotic dominance so that $f \precsim g \text{ and } f \succsim g \implies f \sim g$?

I came to understand that asymptotic negligibility, dominance and equivalence form together a partial order of real-valued functions, analogous to $=, \leqslant, <$ for the real numbers. In the ...
0
votes
0answers
25 views

Are all recurrence relations sequences?

Definitions of recurrence relations online say: Equations that define sequences of values using recursion and initial values. (WolframAlpha) An equation that uses recursion to relate terms in a ...
2
votes
3answers
41 views

Determine whether ${\bigcap_{n=1}^\infty \left(0,\frac{1}{n}\right)}$ is open

I want to determine whether ${\bigcap_{n=1}^\infty \left(0,\frac{1}{n}\right)}$ is open. Here is my attempt at the problem, but I am having some trouble with this idea so please point out any errors. ...
1
vote
0answers
38 views

Determine whether $(-1, 2) \cup [3, \infty)$ is open.

I want to determine whether the set $(-1, 2) \cup [3, \infty)$ is open. I'm having trouble wrapping my head around the definition, but here is my attempt at the problem... Please let me know if this ...
0
votes
1answer
30 views

Is the statement of this question correct?

Here is the question I want to solve: Let $f: R \rightarrow S$ be a map of commutative rings. Show that, if $\mathfrak{p}$ is a prime ideal of $S,$ then the inverse image $f^{-1}(\mathfrak{p})$ is a ...
0
votes
1answer
35 views

What is a difference between Elementary Math and the Fundamentals of Mathematics? [closed]

I recently came across the phrase Elementary Math. Is it the same thing with the Fundamentals of Mathematics?
2
votes
1answer
46 views

What does $\leq ^{-1}$ mean?

Just came across a question on my homework assignment. The only thing it says is: What is $\leq ^{-1}$? It is on chapter for relations. I'm guessing it means: $R \leq R^{-1}$, for $R$ being a ...
1
vote
1answer
26 views

Equivalence between elements of different sets - how to formally define the “equivalence classes”?

What would be a good formal definition for the set $\mathcal T$ constructed below? I will try and give a simple example. Consider the set $[\;n\;]=\{1,2,\ldots,n\}$, as well the family $\mathcal{C}$ ...
0
votes
0answers
31 views

Is there a definition for similar matrices that aren't square?

The definition of similar matrices as I am aware is only for square matrices, and the motivation is to find a transformation matrix for a different basis. Isn't it possible to have two non-square ...
1
vote
1answer
53 views

If we define $x := y$, is it true that $P(x) \iff P(y)$ for any property $P$?

The question should be trivial, but I still can't get my head around it: Question. Suppose we have an object $y$ belonging to the type $T$ (e.g $y$ may be an integer, matrix, set etc). Now suppose we ...
0
votes
1answer
29 views

“$X$ is an ordered set in the order topology.” What is the meaning of this sentence? (James R. Munkres “Topology 2nd Edition”)

I am reading "Topology 2nd Edition" by James R. Munkres. There is the following sentence on p.90 in this book: Now let $X$ be an ordered set in the order topology, and let $Y$ be a subset ...
1
vote
0answers
22 views

Propositional Recursion in van Dalen's Logic and Structure

In Logic and Structure, Van Dalen writes, Theorem 1.1.6 (Defintion by Recursion) Let mappings $H_\square:A^2\rightarrow A$ and $H_\neg:A\rightarrow A$ be given and let $H_{at}$ be a mapping from the ...
0
votes
2answers
53 views

Justification of the tangent name for the goniometric operator $\tan$

When we give a proof that the tangent is the sine to cosine ratio of an oriented angle, $$\bbox[5px,border:2px solid #C0A000]{\tan \alpha=\frac{\sin\alpha}{\cos \alpha}}$$ with $\cos \alpha \neq 0$, ...

1
2 3 4 5
123