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

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2
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1answer
252 views

The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
1
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0answers
112 views

How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then ...
1
vote
1answer
2k views

Definition of correspondence

A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for ...
2
votes
1answer
109 views

Defining Test-Objects

In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
1
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1answer
118 views

geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
6
votes
1answer
137 views

Defining $\Bbb{Q}$ without the axiom of infinity

(TL;DR version: I want a meaningful definition of $\Bbb{Q}$ without $\sf{Inf}$.) In the "conventional" construction of the rationals, we define $\Bbb{Q}$ as follows: $\omega$ is the first limit ...
2
votes
2answers
72 views

Name of the order with irreflexivity, antisymmetry and transitivity?

I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
1
vote
2answers
65 views

Equivalence relation proofs: general or specific?

I'm confused about whether a specific example must exist to prove an aspect of an equivalence relation. For example: if a set, $A$, only contains one element, $A = \{1\}$, and a relation, $R$, on ...
2
votes
2answers
301 views

Definition of logarithm in complex domain

My first question is: What is the proper definition of logarithmic function $f(z)=\ln{z}$. where $z\in \mathbb{C}$. quoting Wikipedia. a complex logarithm function is an "inverse" of the ...
0
votes
1answer
67 views

Difference in reference of L space in Fubini Tonelli

I think my understanding of how $L^+$ and $L^1$ spaces are defined (I'm using Folland) is a little hazy. For example, in the Fubini-Tonelli theorem: $\textbf{For the Tonelli part:}$ We start with if ...
1
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3answers
337 views

Cantor Set and Fractals

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but ...
3
votes
1answer
59 views

What does it mean for the determnant of a matrix to be independent of the vector space

Explain what it means for the determinant of the matrix, representing an operator $F$, to be independent of the basis of the vector space. Prove this property of the determinant. I'm not exactly ...
5
votes
1answer
143 views

Allegories in easy words?

1) What is, in easy words, the definiton of an allegory? 2) And when are allegories useful? What does it have to do with the category theory and categories? With the definiton of category, ...
0
votes
0answers
50 views

Are my definitions of cotangent space, differential and differential forms and coboundary operator correct?

Define the cotangent space $T_a^*\mathbb{R}^n$. Define the differential of a dunction $f$ at the point $a, df \in T_a^*\mathbb{R}^n$. Write down the explicit formula for the deffertial $df$ in ...
3
votes
0answers
127 views

Infinite set ordered like a “infinite tree”

Obviously I don't need to say that I'm still a newbie, so I'll try to explain my question with some pics too. I have a infinite set $T$ I have a visual idea of how this set is ordered, but I don't ...
7
votes
1answer
697 views

Can a function be increasing *at a point*?

From what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might call ...
4
votes
1answer
604 views

What does it mean for a set to be bounded?

I've been trying to prove a slightly different statement to the definition of the well ordering principle given by Lang in Undergraduate Algebra and given here for reference: Every non-empty set of ...
1
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1answer
36 views

A follow up question on completeness of filter (generalised to p.o.s)

Now I'm looking at the following generalisation of $\kappa$-closedness of a filter on a set to a filter in a partial order: Let $\kappa$ be a regular uncountable cardinal. A partial order $P$ is ...
4
votes
1answer
83 views

Question about definition of $\kappa$-completeness of filter

I am looking at the following definition: Let $\kappa$ be a regular uncountable cardinal and let $\mathcal F$ be a filter on a non-empty set $X$. We say that $\mathcal F$ is $\kappa$-complete if ...
1
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1answer
35 views

$\tau_1$ coarser topology than $\tau_2$

I know the following definition of coarser topologies: If $\tau_1$ and $\tau_2$ are two topologies on $X$, we say $\tau_1$ is coarser than $\tau_2$ if $\tau_1\subseteq\tau_2$. In my book about ...
3
votes
3answers
470 views

What does it mean for elements to be algebraically independent?

Wikipedia's definition: "In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation ...
0
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1answer
56 views

Proposition from the book “Convex analysis and measurable multifunction ”

Please , what is $F_{\sigma}$? And all $U$ can be writen as $\cup F_n$, or only open set ? Thank you
0
votes
1answer
55 views

Definition of multiple sum

Suppose we have an abelian group $(G,+)$. What is the formal definition of multiple sums such as $\sum_{i_1 \in A} \sum_{i_2 \in A_{i_1}} \cdots \sum_{i_n \in A_{i_{n-1}}}f(i_1,\ldots,i_n)$? Thanks ...
5
votes
2answers
202 views

Intuitive explanation of ball-based definition for continuity of functions in metric spaces

First of all, hat tip to @Fayz for providing this definition. Backstory: I broke my glasses several days ago and, in the meantime, this important definition was written on a board I could not see. ...
3
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0answers
185 views

Exercise on measure theory

I have this exercise: Let $\mathscr {S}$ be a collection of subsets of the set $X$. Show that for each $A$ in $\sigma(\mathscr S)$ there is a countable subfamily $\mathscr C_0$ of $\mathscr S$ ...
1
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1answer
133 views

Prüfer sequence for an order-2 tree?

All the algorithms for constructing a Prüfer sequence state that the input is a tree, but none give any output corresponding to an order-2 tree. And Wikipedia gives this definition:" A Prüfer ...
14
votes
2answers
197 views

Is this an equivalent definition of a normal subgroup?

Let $G$ be a group and $N$ a subgroup. Consider the condition $$(\forall g\in G)(\exists x,y\in G)\ NgN=xNy.\tag1$$ If $N\lhd G$, then for each $g\in G$ we have $NgN=gNN=gN=gN\cdot1$, so the ...
1
vote
3answers
88 views

A question about root of a polynomial

If we plug in $x$ into a polynomial and we get the value of $0$ as a result, can we be certain that $x$ is the root of the polynomial? If that is the case, why in this Wikipedia article, it says ...
6
votes
2answers
314 views

What are central automorphisms used for?

A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$. It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
4
votes
1answer
2k views

What is a residue class?

My number theory book has hopelessly lost me on the topic of residue classes. I understand the very basics of congruence and modular arithmetic, but if someone could give not only a formal, but ...
1
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4answers
2k views

Why aren't logarithms defined for negative $x$?

Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...
2
votes
1answer
1k views

Definition of local maxima, local minima

Wikipedia says that: A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when |x − ...
1
vote
1answer
252 views

Any collection of subsets of $X$ can serve as a sub-base for a topology

In my lecture notes there is this observation: "Any collection $S$ whatsoever of subsets of the non-empty set $X$ can serve as sub-basis for a topology on $X$." So, taking the set $X:=\{1,2,3,4,5\}$ ...
2
votes
1answer
81 views

Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...
3
votes
1answer
25 views

What is the name for the unique “simplification” of a graph?

Is there a conventional name for the resultant graph (H) obtained by deleting all loops and multiple edges from the original graph (G)? Something along the lines of "Let H be the simple graph of G.." ...
1
vote
1answer
88 views

Elementary definition: what's a parallel volume-form?

This is a very elementary question, What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric? To find out more about the concept, what kind of topic do I need ...
5
votes
1answer
574 views

Topology of uniform convergence?

Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergence I am having a hard ...
2
votes
1answer
97 views

Property of Division by vector for a field

Serge Lang in "Linear Algebra" on page 2 says that The essential thing about a field is that it is a set of elements which can be added and multiplied, in such a way that additon and ...
2
votes
1answer
145 views

how to write the process of decomposition of a graph into shortest closed sub graphs

If I want to decompose a graph in to possible shortest closed cycles (as shown in right side). then how can i describe this process with mathematical notations. to understand please refer below ...
0
votes
1answer
296 views

Mathematical notation of graph subdivision

If anyone can define a directed graph subdivision with mathematical notation, please post a response. My second question is: Irrespective from the planar embedded graph or not, is this definition ...
2
votes
2answers
131 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
0
votes
1answer
107 views

Meaning of $L_A$?

Let $A$ be an m*n matrix with entries from a field $F$. $L_A: F^n \rightarrow F^m$ defined by $L_A=Ax$. I'm a bit confused about this definition. $L_A$ is a matrix representation of linear ...
5
votes
2answers
189 views

Definition of Unipotent in Positive Characteristic

Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$. Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all ...
1
vote
1answer
54 views

What is the graph $K^c_m$?

The book "Graph Theory with applications" by J.A. Bondy and U.S.R. Murty, which is available here. The Theorem $4.6$ of this book says that: If $G$ is a non-Hamiltonian simple graph with $n≥3$ ...
5
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0answers
198 views

I think a definition is wrong in “Model Categories” by Hovey.

I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read: Define a map $f:X\rightarrow Y$ to be a ...
5
votes
2answers
111 views

Is an “open system” just a topological space?

I think of an "open set" as being "roomy" or "spacious," in the sense that around every point, there is a little bit of room. This motivates the following definition. Definition. An "open system" ...
3
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2answers
219 views

Definition of “Up to” (homeomorphism,isotopy, etc), and Examples?

I've tried googling this usage and understanding the results but I'm struggling to make intuitive sense of it. So my question is, what is the phrase "up to" understood to mean, and what are some ...
2
votes
3answers
480 views

Positive constant scalar definition

In French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one explain ...
0
votes
2answers
39 views

Defined amount and value amount of a function

What is the defined amount and value amount for this function: $$f(x)=\sqrt{(x+7)(1-x)}?$$ The defined amount is all the x-values the function can be and the value amount is all the y-values the ...
1
vote
1answer
141 views

Differentiable manifold in dimension 1 and its critical point

Please, I want to know how to define a differentiable manifold in dimension 1, and if the circle is a differentiable manifold in dimension $1$, and what is its critical point. Thank you.