Questions tagged [definitions]

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In mathematics, what are some complicated objects that can be intuitively described using plain English? [closed]

I was thinking of the following problem: Let $X$ be a random variable that realizes a piece of observable (data) $x_n \in \mathbb{R}^n$ (say, vectorized images of cats), what is the meaning of the ...
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What is a finitely connected domain?

(Cross-posted from MSE.) The paper Chang, S.-Y. A. and Yang, P. C.: Conformal deformation of metrics on S 2 . J. Differential Geom., 27(2), 1988 (DOI, MathSciNet) states in Proposition 2.3 that Moser'...
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What is "inn" in this paper?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030.,in the page 18 we have: $$ \begin{aligned} ...
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What is the average degree of a d-simplex?

I am a beginner in network topology topics and while I was reading an article about simplicial complexes where the authors had used random simplicial complexes, I came across a formula using "...
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Definition of union of simplicial complex and a subset

(Cross-posted from MSE: https://math.stackexchange.com/questions/4425225/definition-of-union-of-simplicial-complex-and-a-subset) Consider a simplicial complex $\Delta$ with vertex set equal to some ...
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Mapping class group and pure mapping class group

"A Primer on Mapping Class Groups" wrote Let $\mathrm{Homeo}_+(S, \partial S)$ denote the group of orientation-preserving homeomorphisms of $S$ that restrict to the identity on $\partial S$....
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1 answer
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What are semipositone functions? [closed]

I am reading a paper on multiple solutions for boundary value problems of fourth-order differential systems. In the paper, there is a nonlinear term $f\in C\left[(0,1)\times \mathbb{R}^+\times \mathbb{...
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Confusion in definition of class of structures and combinatorial class [closed]

I understand that a combinatorial class $\mathcal{A}$ is a set of objects, with a function of size $\lvert\cdot\rvert_{\mathcal{A}}:\mathcal{A}\to \mathbb{N}$. With objects of size $n$: $\mathcal{A}_n=...
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what are definitions of born or die (birth-death point) and crossing point?

in this paper we have : A presentation for the mapping class group of a closed orientable surface.by Hatcher.W.Thurston ...(a) $f_{t_{0}}$ has exactly one degenerate critical point, of the form $f_{t}(...
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Involutory vs Involutary: Are both terms correct?

I have seen references for both terms, apparently referring to the same notion of a "self-inverse function". Do both of these terms really mean the same thing? Is one a misspelling of the ...
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4 votes
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Group presentation in the category of finite group

Context: I'm trying to deal with presentations in the framework of Gonthier et al. formalization of the group theory in the proof assistant Coq. It was used to machine check the Feit-Thompson odd ...
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Potential on a quiver

I found two definitions of potential on a quiver. Selfinjective quivers with potential and 2-representation-finite algebras, Martin Herschend and Osamu Iyama 2.1 Quivers with potential. Let $Q$ be a ...
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Analytic/synthetic distinction in mathematics besides geometry?

In a recent answer to an old MO question, I made a distinction between a "definition" of a mathematical object in the sense of axioms that characterize it, and a "definition" that ...
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Precise definition of locally closed complex curve

In Stein Manifold and Holomorphic Mappings, by Forstnerič, I refer to Definition 8.9.9: An exposed point is a point belonging to a certain subset $\Sigma$ of $\Bbb C^2$, enjoying certain properties. ...
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Conditions such that split coequalizers are a symmetric notion

Consider the notion of a split coequalizer (see the nLab for the definition). Note that the definition seems to be non-symmetric. Are there any conditions on the ambient category such that it becomes ...
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Definition of a unit ball in an Euclidean subspace? [closed]

Suppose $\Lambda$ is a $3$ dimensional lattice inside $\mathbb{R}^4$ and let $E$ be the subspace $\mathbb{R}$-spanned by $\Lambda$. What exactly is meant by the unit ball in $E$? This is something ...
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Definition of trace in topological BF-theories

I very important example of topological field theories are "BF-theories", which are usually defined as follows: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\...
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Should mixed strategies in normal form games be interpreted as measurable functions or probability vectors?

I have recently been stuck trying to understand how game theorists extend a normal form game (matrix game) into a game with mixed strategies (so called mixed extension). I feel like I am missing ...
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4 answers
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Shapes for category theory

Most texts on category theory define a (small) diagram in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the shape of the diagram. ...
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1 answer
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Temporal generalization of graphs: density vs $n$ and $m$?

In short: we generalize graphs to the temporal case, but fail to fully preserve the usual relation between density, number of vertices, and number of edges; how to make better? Context. We propose a ...
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(Seeking Definition) What Does it Mean for a Space to have Rim-Type $\alpha$? Or the 'derivative' of a Countable Set?

I've encountered a definition in several papers, but literally none of them define the term. They all instead reference a book by Menger that has never been printed in English. The term is "rim-...
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How do "Galois-type" and "saturation" for AECs generalize "type" and "saturation" in first-order model theory?

As I'm not allowed to ask a new question due to limit reached matter, I still want to EDIT this one as communicated with @Alex Kruckman in the comments below. I would like to understand the ...
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1 answer
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Does the definition of limit correspond to the intuitive notion? [closed]

I have been pondering the question of whether the formal definition of limit captures well our intuitive notion of it now for the past few days, with no headway at all. Perhaps I could find some ...
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definition of functions that "weakly vanishes as $y\to\infty$" and find a proof of Theorem 9 in the article

I'm reading "Extension problem and fractional operators: semigroups and wave equations" by P.R. Stinga, i have two questions: in Theorem 7 the author use the state "weakly vanishes as $...
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Euler-Lagrange equation for a functional

What does it mean that the equation: $$ \text{div}_{x,y}(y^a\nabla_{x,y}u)=0,\quad \text{in }\mathbb{R}^n\times(0,\infty),$$ is the Euler-Lagrange equation for the functional: $$ J(u)=\int_{\mathbb{R}^...
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6 votes
1 answer
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Two definitions of automorphic forms on Lie groups

My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group. The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
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2 answers
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Why are isotropic random vectors called isotropic if they aren't? [closed]

A random vector $X \in \mathbb{R}^n$ is isotropic if $\mathbb{E}XX^T = I_n$. However isotropic random vectors don't have the property of isotropy. See 1. So why are they called isotropic? Similarly a ...
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2 votes
2 answers
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What concept does covariance formalise?

So for me the definition the independence of two random variables $X,Y$ is intuitivly very clear. But what I have never seen motivated is why the heck one would be interested in the covariance $$\...
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"totally positive" elements in a field that is not totally real

In a number field that is not necessarily totally real, could it make sense to consider "totally positive" elements as elements that are positive in all real embeddings? (So in a totally ...
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15 votes
2 answers
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Formal definition of homotopy type theory

The HoTT community is quite friendly, and produces many motivational introductions to HoTT. The blog and the HoTT book are quite helpful. However, I want to get my hands directly onto that, and am ...
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1 vote
1 answer
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What is the definition of brick product? Graph theory

Can anyone help me with the exact definition of brick product of graphs, say path, cycle. I am not able to find a paper with a clear definition on the internet. Can anyone give me a URL to such a ...
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Graph theory: Closed neighourhoods and generalized clustering coefficients

The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$. The number of edges between neighbours divided by the number of pairs of neighbours is ...
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23 votes
0 answers
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What, precisely, do we mean when we say that a f.d. vector space is canonically isomorphic to its double dual?

I've been reading the Xena Project blog, which has been loads of fun. In the linked post Kevin gives the natural isomorphism $V \to V^{\ast \ast}$ from a f.d. vector space to its dual as an example of ...
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4 votes
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Equivalent definitions of unramified characters

Let $G$ be a connected reductive group over a local field $F$. An unramified character of $G(F)$ is a continuous character $\chi: G(F)\to\mathbb{C}^\times$ that is trivial on all compact subgroups of $...
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4 votes
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Issue with "definition" of pseudo algebraically closed fields

I'm having an issue with a sentence in Chapter 11 of Fried & Jarden's Field Arithmetic. As a "motto" for pseudo algebraically closed (PAC) fields, they say they are fields $K$ such that &...
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3 votes
1 answer
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Is there a notion of projective dg category?

Since the paper Smooth and proper noncommutative schemes and gluing of DG categories by Orlov, dg categories are considered the non-commutative counterpart of algebraic geometry. More specifically, we ...
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12 votes
2 answers
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Different definitions for integral de Rham cohomology classes

Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$. Now, one ...
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0 votes
1 answer
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Definition of a system of recurrent events

[I asked a version of this question on MSE a few weeks ago and didn't get any useful feedback. Apologies if I am just being stupid.] I am reading the paper A note on the Borel-Cantelli lemma by Kochen ...
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1 vote
1 answer
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Can we define cardinality that works under weaker grounds than Scott's cardinals?

Its known that within the perspective of $\sf ZF$ related theories, Scott's definition of cardinality can work under weaker grounds than Von Neumann's! Scott's cardinality works as long as the ...
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3 votes
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Confusion on the assumption when discussing the kneading invariants for unimodal maps

A unimodal map is a continuous map $f:[0,1]\longrightarrow [0,1]$ such that there is only one turning point (critical point), denoted by $c$, and $f(0)=f(1)=0$. Unimodal map is related to kneading ...
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7 votes
1 answer
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Constructive definition of noncommutative rational functions (aka free skew fields)

The question Let $F$ be a field. (I am fine with assuming $F=\mathbb{Q}$, but I suspect that a "right" answer will be independent of $F$.) Let $k$ be a nonnegative integer. Question. Is ...
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1 vote
1 answer
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Extension of the definition of entropy to $\mathbb{Z}^d$ and $\mathbb{N}^d$

I read the paper Entropie d'un groupe abélien de transformation by Conze and the part of the book Dynamical systems of Algebraic Origin by Schmidt about the entropy for $\mathbb{Z}^d$ actions. I was ...
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1 vote
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What is the standard definition of dual of disconnected planar graph when underlying graph derives 'product structure' over connected graphs?

Dual graph of a plane graph has a standard definition https://en.wikipedia.org/wiki/Dual_graph and an edgeless graph on $n$ vertices is planar. What is the standard dual graph of such a graph? Update ...
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Is this integral transform related to the Laplace transform?

The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by $${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$ Let $\varphi: {\...
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Use of this space of very rapidly decreasing continuous functions

Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space $$ V_p := \left\{ f \in C([0,\infty)):\, \sum_{n=1}^{\infty} ...
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1 answer
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What is the definition of Plancherel density?

I know about the Plancherel measure, but I don't know where the term "Plancherel density" is defined.
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Terminology: "sufficiently large absolute constant"

I'm currently reading the paper "Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu" and have run into some terminology which I don't quite (rigorously) ...
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1 vote
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Open/closed/constructible subsets of locally free sheaves

(Cross-posted from math.SE since I'm not sure what is a suitable platform. Link on https://math.stackexchange.com/questions/3597258/open-closed-constructible-subsets-of-locally-free-sheaves) Is there ...
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Generalized compact open topology?

Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets $$ \left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\ U_{K,O}:=\...
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16 votes
2 answers
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Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
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