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

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8
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4answers
812 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
109 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
71 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 ...
4
votes
2answers
119 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
2answers
221 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
106 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
32 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
199 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 ...
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2answers
174 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
97 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
237 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
365 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, ...
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vote
5answers
873 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
67 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
192 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
873 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 ...
8
votes
4answers
565 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
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2answers
62 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
424 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
407 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
309 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
37 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
92 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
267 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
795 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
62 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
167 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
80 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
56 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
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2answers
165 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
177 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
113 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
105 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
118 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
91 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
89 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
838 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
369 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 ...
1
vote
3answers
620 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 ...
10
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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 ...
15
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
324 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
399 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
618 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 ...