# Tagged Questions

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 ...

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.
3answers
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### 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 ...
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, ...
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 ...
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 ...
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 ...
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?
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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), ...
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 $f'(x)dx$
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 ...
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.
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $a$ ...
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 ...
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 ...
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 ...
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.
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $(a_{ij})$ ...
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?
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 ...
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 ...
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 ...
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 $f$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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. ...
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