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

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Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?
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0answers
18 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
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1answer
32 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
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3answers
363 views

Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
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1answer
42 views

A question about combinatorial commutative diagrams.

In the comments of this question the directed graphs that can appear as commutative diagrams are axiomatized: A graph is a combinatorial commutative diagram iff it's nonempty and such that any ...
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1answer
30 views

A subset on which a measure is strictly positive

There is a "cake" which is represented by the interval $[0,1]$. There is a non-atomic value measure $V$ defined on the cake. I would like to define an algorithm for dividing the cake between two ...
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1answer
15 views

Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Let $P\to $ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?
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1answer
53 views

Is there a name for this simple structure?

Is there a name for $(X,S)$ where $X$ is a set and $S\subseteq X$ and a morphism $(X,S)\overset{\alpha}{\longrightarrow}(X^\prime,S^\prime)$ is a function $\alpha:X\rightarrow X^\prime$ such that ...
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1answer
37 views

Is 1 coprime to itself? [on hold]

Is $\{1,1\}$ a pair of co-prime numbers? According to the definition, two numbers are coprime if $\gcd(a,b)=1$, and for $\{1,1\}$ it is true that $\gcd(1,1)=1$.
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2answers
77 views

Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
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0answers
31 views

Are permutation group block only defined in the context of finite sets?

From Dummit and Foote (emphasis mine): Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ s.t. $\forall \sigma\in G:$ either $\sigma(B)\cap B ...
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2answers
69 views

Why do we care if a function is uniformly continuous? [duplicate]

There are a lot of question regarding whether a function is or is not uniformly continuous or just continuous and there are a lot of $\epsilon_s$ and $\delta_s$ trying to show whether a function is ...
3
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1answer
49 views

What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
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1answer
24 views

How do we define discriminant over a commutative ring?

Let $f$ be a nonconstant polynomial over a field $F$. Since there exists a splitting field of $f$ over $F$, $f$ can be decomposed as $f=c\prod_{i=1}^n (X-\alpha_i)$ Hence, it is possible to define ...
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1answer
43 views

Is there a specific mathematical term for a shape whose dimensions are defined?

When I say the word "circle", I know that I have described a "shape". Specifically, a "circle" is the shape formed by the set of all points in a plane that are at a given distance from a given point. ...
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0answers
55 views

Operator form $L^2$ space to$L^1$

Can we have an operator such that it transforms an element of $L^2$ to $L^1$? Is this a valid question or this is incorrect? We can consider the measure space as finite.
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1answer
15 views

Simplicial homology: chain group with basis open n-simplices vs. chain group with basis closed n-simplices

In his Algebraic Topology book, Hatcher defines the chain groups for simplicial homology as free abelian groups with basis the open $n$-simplices of some simplicial complex X. Is there any ...
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3answers
112 views

What is the precise definition of 'between'?

I'm wondering what the precise definition of 'between' is in a mathematical context. For example, many statements of the intermediate value theorem state that a value $k$ is 'between' $f(a)$ and ...
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0answers
21 views

Is there one to one relation Positive definite(PD) matrix and PD function?

Is it correct to say that a PD matrix can be built from a PD function? For example circulant matrix or Toeplitz seems to be built from a positive definite function. Positive definite function is ...
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1answer
34 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
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4answers
75 views

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different ...
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2answers
28 views

Why does the Big Oh (and similar) notations needs $n_0$?

The generally agreed definition of the Big Oh notation (afaik) is as follows: The function $f(n)$ is $O(g(n))$ if there exists constants $c$ and $n_0$ such that for all $n \ge n_0$, $f(n) \le c ...
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1answer
15 views

Why should the matrix $A$ in an ILP be integer?

Almost everywhere I read about integer linear programming (ILP), I found that the matrix has to be integer (by definition). More precisely, an ILP is defined as follows: An ILP in canonical form is ...
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1answer
32 views

General meaning of the term “Functional”

In the most general is a "functional" simply a function which can accept a function as input? So, is it natural to describe: $f: \mathbb{N} \rightarrow \mathbb{N}$ as a function. Whereas it is more ...
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1answer
45 views

Priority of the 3 axioms of groups [duplicate]

In my book about Abstract Algebra, it is stated that A group $\langle G,*\rangle$ is a set $G$, closed under a binary operation $*$, such that these 3 axioms are satisfied: $g_1$: For all ...
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1answer
13 views

Name for a collection of subsets of a set $E$ such that every element of $E$ is in some member?

Let $X$ be a set. A partition of $X$ is a collection of subsets $\{E_\alpha\}\subset\mathcal{P}(X)$ such that the following holds. For every $x\in X$, there is some $\alpha$ such that $x\in ...
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1answer
42 views

definitions of order-dense Riesz spaces

I would like to get an intuitive idea as to why the definitions of order-dense and locally order-dense Riesz subspaces were chosen in the following way: A Riesz subspace $F$ of $E$ is order-dense if ...
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1answer
28 views

Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
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2answers
82 views

Notation for Tautologies

I've been stuck for a while in this question and so far I don't understand the flaw of my reasoning please if you guys could help me out. See, this is my context. From the definition of argument we ...
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0answers
31 views

$\mathbf{A}$ is unimodular $\Rightarrow$ $\mathbf{A}$ has entry in $\{-1, 0, 1\}$?

Is it true that $$\mathbf{A}\;\text{is unimodular}\;\Rightarrow\mathbf{A}\;\text{has entry in}\; \{-1, 0,1\}?$$ Also can an unimodular matrix $\mathbf{A}$ has entry in $\mathbb{R}$?
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2answers
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Rudin's Chain Rule

Rudin's chain rule theorem goes like this: Suppose $f$ is continuous on ${[a,b]}$, $f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which contains the range of $f$, and ...
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2answers
31 views

Possible definition of the matrix representation of a linear transformation with respect to given bases

Let $E$, $F$ be vector spaces with basis $\{e_1,\dots,e_m\}$, $\{f_1,\dots,f_n\}$. Let $T:E\to F$ be a linear transformation. We say that the matrix $A\in\mathbb{R}^{m\times n}$ represents $T$ with ...
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4answers
362 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
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1answer
48 views

Help in understanding Bochner's theorem

This relates to the Bochner's theorem stated in the link: https://en.wikipedia.org/wiki/Bochner%27s_theorem My question is related to the unique probability measure μ on G. I want to express the ...
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1answer
29 views

Ambient Isotopy of Knots

Statement 1: Knots of opposite chirality have ambient isotopy, but not regular isotopy. Statement 2: We can then define two such knots to be equivalent if they are ambient isotopic, meaning that ...
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0answers
31 views

Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...
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1answer
23 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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1answer
34 views

a real number as a point

An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) $p$ in the domain of a real-valued function $f$ of a real variable where ...
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1answer
22 views

Terminology help for a set relation: for sets $X, Y$, not necessarily disjoint, such that neither is a subset of the other.

Is there an existing term for pairs of sets $X, Y$, not necessarily disjoint, such that neither $X \subseteq Y$ nor $Y \subseteq X$? Would it be incorrect (or misleading) to call them something like ...
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0answers
32 views

Are there non-holomorphic or non-analytic polynomials of several complex variables?

Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply ...
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1answer
17 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
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1answer
42 views

Unknowledgable of single variabled integral

Browsing Stack-Exchange and other sites, I have noticed this come up quite a few times, an integral with only one letter or number! This would be a great example: $$\int_{a}$$ What in the world does ...
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0answers
43 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
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5answers
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Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
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3answers
119 views

If $AB\not=BA,$ then do $A$ and $B$ have to be matrices

Is this solvable? Or are there other things that fit the bill for $A$ and $B$?
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2answers
23 views

What is “an increasing sequence of step functions”?

I'm reading Alan Weir's "Lebesgue Integration and Measure". In exercise 8 on page 30 he talks about "...an increasing sequence of step functions $\{\phi_n\}$..." and "...an increasing sequence of ...
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1answer
32 views

how to define a “directed spanning tree”?

In all my books and articles about "graph theory", I didn't find the definition of "directed spanning tree". Could you please give this definition and the reference? How to judge if a directed graph ...
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1answer
23 views

Filters and filter bases in order theory

Hi I would like to confirm the following ideas regarding filters in order theory: By definition I have that a filter is a subset of a poset $(P, \leq)$ which satisfies: $\mathcal{F}$ is non-empty. ...
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11answers
2k views

Why every definition is an “iff”-type statement? [duplicate]

Suppose that we are trying to define a mathematical object $M$. The statement of the definition generally takes the form (or some of its equivalent variant), A mathematical object is said to be ...
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1answer
73 views

Explanation of a joke on abelian groups (grapes).

Q: What's purple and commutes? A: An Abelian grape. Q: What is lavender and commutes? A: An Abelian semigrape. Q: What's purple, commutes, and is worshipped by a limited number of people? A: A ...