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

learn more… | top users | synonyms

6
votes
2answers
101 views

Collecting definitions of continuity.

Let $X$ and $Y$ denote topological spaces and consider a function $f : X \rightarrow Y$. I'm collecting possible definitions/characterizations of the statement "$f$ is continuous." Here's two to get ...
7
votes
3answers
239 views

What's Geometry?

I am a grad student. I am writing an article on geometry and relativity theory and trying to start with discussing basic ideas of topology. In my article I tried very hard to motivate the idea of ...
2
votes
1answer
474 views

What is the definition of a geometric progression?

If the first term in our geometric progression (GP) is $k$, and the common ratio is 0, then our sequence is $\{k, 0, 0, 0, 0,\ldots\}$. Is there anything wrong with this statement? So, is $\{0, 0, ...
1
vote
5answers
1k views

Definition Of Symmetric Difference

The definition of a symmetric difference of two sets, that my book provides, is: Set containing those elements in either $A$or $B$, but not in both $A$ and $B$. So, in set builder notation, I figured ...
2
votes
1answer
69 views

Why Can't we define the differentiation of vector fields in the same way as in $\mathbb{R^{n}}$

In $\mathbb{R^{n}}$, if $X$ is a vector field on $\mathbb{R^{n}}$, and $X=$$\sum_{i=1}^{i=n}$ $X^{i}$ $\frac{\partial}{\partial x^{i}}$, $X^{i}$ $\in$$C^{\infty}(p)$. Then 1.The differentiation of ...
1
vote
4answers
202 views

Definition of a metric

I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along ...
0
votes
1answer
1k views

How to find Df in functions

Well, I do understand what Df is and how you find it in simple equations, however, I am kinda confused in "complex" functions. For example, the following functions: 1* f(x)=x^3+x^2-x-1 , Df=R ...
9
votes
4answers
668 views

A question on definition of field of fractions

Wikipedia defines the field of fractions of a domain as The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded. What does ...
0
votes
2answers
66 views

How to make precise the notion of “the multiset of roots of a polynomial function”?

A (real) polynomial function can be defined as a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a sequence $a : \mathbb{N} \rightarrow \mathbb{R}$ such that the terms of $a$ ...
0
votes
2answers
530 views

What is a bijective linear mapping called?

Friedberg - Linear Algebra p.102 This book states that "a bijective linear map from a vector space to another vector space is called an isomorphism". As far as know, generally isomorphism means ...
11
votes
5answers
417 views

what is the definition of $=$?

what is the definition of $=$? Above is the question that I would like to be answered, below are some of my thoughts. I've been thinking about what it means to say $A = B$ I came to this from ...
1
vote
2answers
376 views

Difference between closure and the boundary

I'm having a hard time distinguising the difference between the boundary and the closure of sets. They seem so similar, but that almost sounds too good to be true. So if the boundary is just the ...
1
vote
0answers
38 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
2
votes
1answer
96 views

Affine algebra of an algebraic group

From what I understand there are two approaches to defining an algebraic group. One can start talking about varieties and the Zariski topology and such and get to a definition of an algebraic group. ...
6
votes
1answer
294 views

Axioms vs. Universal Constructions/Properties

What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more ...
2
votes
1answer
830 views

Definition of the complement of a set

My book defines the complement of a set as, "Let $U$ be the universal set. The complement of the set $A$, denoted by $\bar{A}$, is the complement of $A$ with respect to $U$. Therefore, the complement ...
5
votes
1answer
63 views

Can you define a vector space in terms of a pre-existing projective space?

Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less ...
2
votes
4answers
201 views

Fourier transform

Could anyone explain to me how do we change Fourier transform equation from this [Wiki - look at the top of the page]: $$ \mathcal{F}(x) = \int\limits_{-\infty}^{\infty} \mathcal{G}(k)\, e^{-2\pi i ...
1
vote
1answer
81 views

How do we distinguish “walks” or “paths”?

For example, let $G(V,E)$ be a graph such that $V=\{v_1,v_2\}$ and $E=\{(v_1,v_2)\}$. And let $s_1:\{1,2\}\rightarrow V$ be a walk such that $s_1(1)=v_1$ and $s_1(2)=v_2$. And let ...
0
votes
1answer
48 views

What is this matrix called?

Let $G=(V,E)$ be a finite graph where $V$ has $n$ elements so that $V=\{v_1,...,v_n\}$. Now, define $a_{ij}$ to be 1 if $(v_i,v_j)\in E$ and 0 otherwise. What is this $n\times n$ matrix $(a_{ij})$ ...
0
votes
2answers
58 views

Question about power of sets

If two sets are finite and they have the same power, can we say that the two sets are equivalent? Is every finite set countable?
0
votes
3answers
179 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
1
vote
1answer
205 views

Does *pair* always mean a pair of distinct elements in graph theory?

Definition of edge in wikipedia: An edge of a graph is a set of 2-elements in a set of vertices. Definition of tournament in my text: A tournament is a directed graph such that each pair of vertices ...
3
votes
1answer
118 views

questions on the completely accumulation

Could somebody help me to understand the definition of completely accumulation? And help me show that this claim: A space $X$ is compact iff every infinite set in $X$ has a point of complete ...
8
votes
3answers
81 views

$(\Bbb R \to \Bbb R : x\mapsto x^2)\equiv(\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2) \not\equiv (\Bbb C \to \Bbb C:x\mapsto x^2)$

Consider the following functions: $f:\Bbb R \to \Bbb R : x\mapsto x^2$ $g:\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2$ $h:\Bbb C \to \Bbb C:x\mapsto x^2$ I'm quite sure that $h$ is not equal to $f$ ...
3
votes
2answers
106 views

Associativity with one operation or two (or more) operations

It seems to me there are different 'types' of associative law that are all said to simply have the property of associativity. For example this term is applied if we are only considering one operator ...
1
vote
1answer
142 views

Metrics with infinite distances.

I've been wondering about the spaces $\Bbb R\cup\{-\infty,+\infty\}$ and $\Bbb C\cup\{\infty\}.$ Is there a useful generalization of the definition of a metric they satisfy? I thought it would be ...
4
votes
1answer
101 views

Are bicategories and lax 2-categories the same?

My question is that whether the definition of bicategories is the same as the definition of lax 2-categories. I heard that they are both week versions of 2-categories. Are they the same? If not, how ...
1
vote
2answers
98 views

Understanding the definition of monotonically monolithic

A collection $\mathcal{N}$ of subsets of $X$ is called an external network of $A \subset X$, when for every $x \in A$ and every neighbourhood $U$ of $x$, there exists some $N \in \mathcal{N}$ such ...
2
votes
1answer
75 views

“[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds. ...
4
votes
2answers
1k views

Definition of an “Experiment” in Probability

One can define the fundamental concepts of probability theory (such as a probability measure, random variable, etc) in a purely axiomatic manner. However, when we teach probability, we start off with ...
0
votes
0answers
66 views

Improper or Undefined

Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral $\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined? If we take it as a legitimate function for improper Riemann ...
12
votes
8answers
2k views

Why do we accept Kuratowski's definition of ordered pairs?

I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly ...
0
votes
2answers
451 views

About the definition of n-tuple

I've read from the theory of sets that the definition of ordered pair is foundational. To define formally the term of $n$-tuple I suppose we need to use the concept of ordered pair as well as the ...
2
votes
3answers
753 views

Expected Value Function

My text-book defines expected value as $$E(X) = \mu_x = \sum_{x \in D} ~x \cdot p(x)$$ And so, if I was to find the expected value of a random variable $X$, where $X = 1,2,3$, then it would resemble ...
12
votes
1answer
1k views

Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
16
votes
6answers
1k views

How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity?

I'm told that a function defined on an interval $[a,b]$ or $(a,b)$ is uniformly continuous if for each $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that $|x-t|\lt \delta$ implies that ...
2
votes
3answers
331 views

How can the real numbers be a field if $0$ has no inverse?

I'm reading a linear algebra book (Linear Algebra by Georgi E. Shilov, Dover Books) and the very start of the book discusses fields. 9 field axioms discussing addition and multiplication are given ...
2
votes
3answers
1k views

Linear independence and dependence of vectors

I am really stuck in this problem, I have only 2 days to learn matrix's base, and its generator. My problem is that I know definitions but I don't understand intuitively what they mean. What I know: ...
10
votes
1answer
458 views

Is there a way to relate convexity to Gaussian curvature?

This is a vague question because I'm not sure what I want to ask. An ellipsoid has positive curvature everywhere, and bounds a convex subset of $\mathbb R^3$. What I want to say now is "It seems as ...
1
vote
1answer
727 views

Definition of induced cycle

According to Diestel (page 4): "If $G' \subseteq G$ and $G'$ contains all the edges $xy \in E$ with $x, y \in V'$, then $G'$ is an induced subgraph of $G$" According to Wikipedia "induced cycle is a ...
1
vote
1answer
231 views

Definition of the Support of a Real Valued Random Variable

This is an embarrassingly simple question: If $X$ is a normally distributed random variable, what is the support of $Y=X^2$? It clearly should be the positive real line. However, I cannot find a clear ...
1
vote
2answers
97 views

What is the relation between $ \kappa$-monolithic and monotonically monolithic?

For an infinite cardinal $\kappa$, a space $X$ is called $\kappa$-monolithic if $nw(\overline{A}) \le \kappa $ for any set $A \subset X$ with $|A| \le \kappa$. And you can see this definition of ...
4
votes
4answers
144 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
4
votes
2answers
3k views

What are relative open sets?

I came across the following: Definition 15. Let $X$ be a subset of $\mathbb{R}$. A subset $O \subset X$ is said to be open in $X$ (or relatively open in $X$) if for each $x \in O$, there exists ...
1
vote
1answer
87 views

Definition of purely oscillatory

This is question about a term whose definition I can find anywhere. I am given to solve a differential equation and one of the questions asks to show that the solution (we are given initial data) is ...
1
vote
1answer
763 views

complement of a function $f: \{2n | n\in \mathbb{N}_0 \}: n \rightarrow n+1$

i am reading a textbook here and i saw, there is notion of Complement of a function. or Negation of a function definiton, this is whow i understood but it is definitely wrong how i do it, i know. in ...
1
vote
1answer
112 views

Question on definition: primitives in the enveloping algebra of a Lie algebra.

Let $C$ be a coalgebra, and take $c\in C$. Then $c$ is group-like if $\Delta c=c\otimes c$ and $\epsilon(c)=1_k$, and the set of group-like elements is denoted $G(C)$. For $g,h\in G(C)$, $c$ ...
4
votes
2answers
276 views

Trying to understand Hilbert Spaces…

I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert ...
5
votes
2answers
254 views

Motivation for a new definition of the derivative using the concept of average velocity

As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that the derivative of $ f $ at $ a $ equals $ L $ if and only if $$ \lim_{x \to a} ...