Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...
3
votes
2answers
62 views
two notation: semi-metric and pesudometric
There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different?
Thanks ahead.
1
vote
3answers
46 views
Precise differences in meaning of Power Series, Taylor Series
Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...
1
vote
3answers
58 views
Formula for Product of Subgroups of $\mathbb Z$, Problem
What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$?
Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
1
vote
2answers
93 views
What does it mean to have no proper non-trivial subgroup
I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper ...
3
votes
2answers
48 views
Definition of complete in the context of Lebesgue measurable sets
I came across this statement on Lebesgue measurable sets.
The Lebesgue measurable sets are said to be complete because every subset of a null set is again measurable and the lebesgue measurable ...
2
votes
3answers
109 views
Example of a (dis)continuous function
The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous?
I ...
5
votes
1answer
89 views
Is uncountably summation defined?
We know that finite and countably summation is defined. But How about uncountably summation, say $$\sum_{i\in \mathbb{R}}0$$
Is it defined?
1
vote
1answer
50 views
Limit point definition
I have read the definition of a limit point of a set in Real Analysis.
The definition goes like:
A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
-3
votes
3answers
108 views
Do dihedral groups $D_n$ for $n\geq 5$ exist?
I know we can generate dihedral group of order three ($D_3$) and four ($D_4$) but my question is whether we can generate dihedral group of order five?
7
votes
4answers
334 views
Congruent Modulo $n$: definition
In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$, and $9 \equiv -1 \pmod 5$. But when I first learned about ...
1
vote
2answers
45 views
What does relax mean in the mathematical context
Here is a direct citation from wikipedia:
The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Friedrich Götze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu ...
3
votes
0answers
51 views
Define composition of small cyles and making a big graph
I am having following sub graphs and wish to compose all and make a
one bigger graph (say G). After that, I want to select the closed path
where it is passing along the outer vertices of that ...
5
votes
2answers
62 views
Standard definition of group isomorphism
ProofWiki defines a group isomorphism as a bijective homomorphism. In Topics in Algebra 2$\varepsilon$, Herstein defines a group isomorphism as an injective homomorphism:
Definition. A ...
3
votes
1answer
78 views
Basic question about analyticity vs. differentiability in complex analysis.
In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula,"
3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
3
votes
1answer
90 views
What is Absolute convergence?
Take:
$$
(u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i).
$$
$k$ is there, it's because you want to define
$$
\ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), (u*v)(3), ...
1
vote
1answer
23 views
Change Along A Tangent Line
I am taking some time to review differentials. What I don't quite get is why the change along the tangent line is $f'(x) \Delta x$, and how it leads to $f'(x)dx$
0
votes
1answer
62 views
Different formulation of a Traveling Salesman Problem
Given a undirected, weighted, complete graph $(V,E,c)$ with $c \to \mathbb{N}$ and $v_0 \in V$ we are looking for a set $E' \subset E$ minimal with respect to $c$ with the following conditions:
for ...
1
vote
2answers
71 views
What is the meaning of the expression $\liminf f_n$?
I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
6
votes
2answers
77 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 ...
-3
votes
2answers
49 views
correct order of lemma, proof and so on? [closed]
What is the right order of the following?
lemma, theorem, definition, corollary, proof
I do not want to know the difference, what they mean and so. Just (if it is possible) something like a ...
8
votes
3answers
208 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
37 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
108 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
42 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
95 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
37 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 ...
6
votes
4answers
169 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
41 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
88 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 ...
10
votes
5answers
314 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
3answers
72 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
32 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
46 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. ...
5
votes
1answer
113 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
86 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 ...
4
votes
1answer
41 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 ...
1
vote
3answers
46 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
48 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
41 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
45 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
2answers
76 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 $$P=\{\bar x\in ...
1
vote
1answer
57 views
Does *pair* always mean a pair of distinct elements in graph theory?
Definition of edge in wikipedia: An edge of a graph is a set of 2-elements in a set of vertices.
Definition of tournament in my text: A tournament is a directed graph such that each pair of vertices ...
4
votes
1answer
53 views
questions on the completely accumulation
Could somebody help me to understand the definition of completely accumulation? And help me show that this claim: A space $X$ is compact iff every infinite set in $X$ has a point of complete ...
8
votes
3answers
73 views
$(\Bbb R \to \Bbb R : x\mapsto x^2)\equiv(\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2) \not\equiv (\Bbb C \to \Bbb C:x\mapsto x^2)$
Consider the following functions:
$f:\Bbb R \to \Bbb R : x\mapsto x^2$
$g:\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2$
$h:\Bbb C \to \Bbb C:x\mapsto x^2$
I'm quite sure that $h$ is not equal to $f$ ...
2
votes
2answers
89 views
Associativity with one operation or two (or more) operations
It seems to me there are different 'types' of associative law that are all said to simply have the property of associativity.
For example this term is applied if we are only considering one operator ...
1
vote
1answer
69 views
Metrics with infinite distances.
I've been wondering about the spaces $\Bbb R\cup\{-\infty,+\infty\}$ and $\Bbb C\cup\{\infty\}.$ Is there a useful generalization of the definition of a metric they satisfy? I thought it would be ...
3
votes
1answer
52 views
Are bicategories and lax 2-categories the same?
My question is that whether the definition of bicategories is the same as the definition of lax 2-categories.
I heard that they are both week versions of 2-categories.
Are they the same? If not, how ...
0
votes
0answers
38 views
Definition: Plotting a 2D surface
I would very much appreciate a definition clarification.
Suppose we have 2 functions of a variable $x$: $f(x)$ and $g(x)$
And we have that $$f'^2+g=c$$ where $c$ is a constant
What does it mean to ...
2
votes
2answers
40 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
54 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.
...
