Questions tagged [definitions]

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What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
HASouza's user avatar
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1 answer
406 views

What is the mathematician's definition of the determinant? [closed]

I am trying really hard to find a good definition of the determinant. I have looked virtually every single resource online and everybody gives a different answer: sum of cofactors or minors https://...
Olórin's user avatar
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Defining the number of rightmost frozen digits of Graham's number

It is well-known that (in radix-$10$) Graham's number, $G$, can be expressed as a tetration with base $3$ and a very large hyperexponent $\tilde{b}$. Thus, we can write that $\exists! \hspace{1mm} \...
Marco Ripà's user avatar
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Can the following definition of choice principle salvage the prior attempts?

In prior postings 1 , 2, I've presented a definition of choice principle as what is equivalent to a selection principle. However, it was proved that it is inadequate in the sense that it admits ...
Zuhair Al-Johar's user avatar
-1 votes
1 answer
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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 ...
i like math's user avatar
7 votes
0 answers
218 views

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 ...
Nik Bren's user avatar
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2 votes
1 answer
374 views

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 $...
Frode Alfson Bjørdal's user avatar
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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 ...
Elías Guisado Villalgordo's user avatar
5 votes
1 answer
358 views

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 $...
Arbuja's user avatar
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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 ...
Arbuja's user avatar
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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 ...
Trebor's user avatar
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2 votes
1 answer
95 views

"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....
Jens Fischer's user avatar
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1 answer
140 views

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 $\...
Arbuja's user avatar
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8 votes
1 answer
664 views

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 ...
Amir Sagiv's user avatar
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6 votes
1 answer
481 views

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 ...
Coherent Sheaf's user avatar
12 votes
2 answers
2k views

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 ...
Contactomorph's user avatar
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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)=\...
Duality's user avatar
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5 votes
1 answer
407 views

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 ...
Milo Moses's user avatar
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4 votes
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158 views

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 ...
rschwieb's user avatar
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3 votes
1 answer
102 views

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 &...
IMP's user avatar
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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 &...
alhal's user avatar
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1 answer
315 views

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 ...
alhal's user avatar
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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, ...
rschwieb's user avatar
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4 votes
1 answer
450 views

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....
mamediz's user avatar
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3 votes
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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_{...
Atom Vayalinkal's user avatar
4 votes
1 answer
175 views

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,...
Jii's user avatar
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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 ...
Usa's user avatar
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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}^{+} \...
Jamai-Con's user avatar
-4 votes
1 answer
135 views

Hilbert’s third problem and what a polyhedron is [closed]

What is the definition of a polyhedron used by Hilbert’s third problem?
Daniel Sebald's user avatar
2 votes
0 answers
231 views

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 ...
Peter Gerdes's user avatar
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4 votes
0 answers
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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'...
Keba's user avatar
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0 answers
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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 "...
Lina's user avatar
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0 answers
159 views

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 ...
modnar's user avatar
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5 votes
2 answers
1k views

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 vote
1 answer
98 views

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{...
Devashish Sonowal's user avatar
1 vote
1 answer
154 views

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=...
Luis Alexandher's user avatar
1 vote
0 answers
110 views

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}(...
Usa's user avatar
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2 votes
0 answers
163 views

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 ...
Eduardo Reis's user avatar
4 votes
1 answer
347 views

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 ...
hivert's user avatar
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2 votes
0 answers
113 views

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 ...
Ryze's user avatar
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9 votes
2 answers
1k views

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 ...
Timothy Chow's user avatar
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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. ...
Joe's user avatar
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1 vote
1 answer
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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 ...
HDB's user avatar
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1 vote
0 answers
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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 ...
Johnny T.'s user avatar
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2 votes
0 answers
127 views

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\...
G. Blaickner's user avatar
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1 vote
2 answers
244 views

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 ...
Norman's user avatar
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33 votes
5 answers
3k views

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. ...
3 votes
1 answer
231 views

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 ...
Matthieu Latapy's user avatar
0 votes
1 answer
101 views

(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-...
John Samples's user avatar
0 votes
1 answer
276 views

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 ...
user2925716's user avatar

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