Questions tagged [definitions]
The definitions tag has no usage guidance.
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What is an "open Baire set"?
In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
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Generalizing uniform structures as Grothendieck topologies
Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
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Impredicativity, definition, recursion and conservatism
Suppose we in an impredicative framework isolate the fixed point
$$Gx\leftrightarrow A(G,x)$$
from a $Gx$ obtained by $\Pi^1_1$-comprehension as equivalent to $\forall K((A(K,x)\to Kx)\to Kx)$, where $...
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Definition of “morphism of schemes that induces a bijection between irreducible components ”
$\def\sO{\mathcal{O}}\def\sF{\mathcal{F}}$On the Stacks Project there are several instances where the seemingly undefined notion of a “morphism of schemes that induces a bijection between irreducible ...
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Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane?
Suppose $\lambda^{*}$ is the Lebesgue outer measure.
Question:
Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where:
The range of $f$ is $\mathbb{R}$
For all real $x_1,x_2,y_1,y_2$, where $...
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Finding an explicit & bijective function that satisfies the following properties?
Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$.
Question:
Does there exist an explicit ...
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Justification of modular law in allegories
The modular law in modular lattices can be described as an isomorphism between opposite edges of the square $(a \land b), a, b, (a\lor b)$. A fancier way of saying this is an adjoint equivalence with ...
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"Balanced" separator which is independent set
I am looking for existence results on separators of $r$-regular graphs $G=(V,E)$, which have the property that
$S\subset V$ is a separator
for all $v,w\in S$ the edge $\langle v,w\rangle\not\in E$ (i....
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Convergence rate of a sequence of sets to a set-theoretic limit?
Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$.
If we define a sequence of sets $\left(F_r\right)_{r\in\mathbb{N}}$ with a set theoretic limit of $A$; how do we define the rate at which $\...
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Measure without measurable sets
This question is a little on the softer and speculative side, so bear with me.
Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
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Definition of locally symmetric space of reductive groups
This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless.
In my attempt to study Shimura varieties, I came across ...
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Would it be possible to propose a satisfying categorical definition for the notion of basis?
I once came across a definition for the notion of basis that was independent of the type of the algebraic structure considered (although I cannot find where). Translated into category terminology, it ...
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What is a definition of $A(P_v)$ in the definition of Brauer-Manin obstruction?
This is a question related to the definition of Brauer-Manin obstruction.
Let $K$ be a number field. $X/K$ be an algebraic variety over $K$.
Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\...
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What is the correct definition of semisimple linear category?
I have been using the notion of "semisimple linear category" for a while now, but I never bothered to write down a definition. I've always just waved my hands and said "Schur's lemma ...
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Definitions of torch ring
Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions.
An FGC ring is a commutative ring whose ...
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References on coefficient quivers
I would like to study about coefficient quivers, but I cannot find a good reference, as book for example. I could find many papers working with coefficient quivers, but none of them give a book or a &...
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The most general (but useful) definition of "attractor" for dynamical systems
Consider J. Milnor's paper: On the concept of attractor.
There he writes: "A less restrictive definition" [than some of the previous ones he had considered] of the concept of an attract is &...
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On the correct definition of attractors
It is well-known in dynamical systems that the concept of "attractor" differs in the literature.
My question is whether attractors need to be defined as subsets of $\omega$-limit sets of ...
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Which definitions of "local module" have gotten traction?
It seems like "local module" has been defined a lot of ways:
if 𝑀 has a largest proper submodule. (This math.se post)
if it is hollow and has a unique maximal submodule (Singh, Surjeet, ...
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On the definition of a continuous function
I remember once reading that "a continuous function can be loosely described as a function whose graph can be drawn without lifting the pen from the paper". We all know that this is not true....
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Characterizing image of integral transform applied to sections of a fiber bundle
Geometry is not my area, and so, I am not sure the title accurately captures what I am interested in exactly... I hope the tags are appropriate.
For any vector $v$, denote it's $i$-th component by $v_{...
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Extending the class of primitive recursive functions with higher order recursion schema
I'm trying to extend the class of primitive recursive functions by extending the recursion schema over higher types.
We usually define the class of primitive recursive functions by using zero function,...
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definition of level-preserving diffeomorphism
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) page 10 we have :
Up to level-preserving ...
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A question on terminology for sequences satisfying $\gcd(a_m,a_n)=a_{\gcd(m,n)}$
How do you refer to those sequences $\{a_{n}\}_{n \in \mathbb{Z}^{+}}$ of integers that satisfy the condition $\text{gcd}(a_{m}, a_{n}) = a_{\text{gcd}(m,n)}$ for every $(m,n) \in \mathbb{Z}^{+} \...
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Hilbert’s third problem and what a polyhedron is [closed]
What is the definition of a polyhedron used by Hilbert’s third problem?
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A formal definition of a useful theorem?
Sorry if this feels a bit squishy, but I'm wondering if there is any published work trying to give a fully formal definition of the notion of a useful theorem. I mean, in mathematics we all know that ...
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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 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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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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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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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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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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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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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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