Questions tagged [definitions]
For questions of the kind "What is the correct definition of property or object X?" or questions about how to define terms within specific theories.
235 questions
1 vote
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231 views
Definition of length structure in Burago, Burago, Ivanov's "A Course in Metric Geometry" does not imply Exercise 2.1.4?
In Burago, Burago, Ivanov's "A Course in Metric Geometry" (Definition 2.1.1, page 26 and 27) a length structure on a topological space $X$ is defined as a pair $(A,L)$ where $A$ is a set of ...
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6 votes
1 answer
358 views
A misunderstanding in Gorensteiness
I am reading the definition of a Gorenstein ring in Bruns-Herzog. A Noetherian local ring $R$ is called Gorenstein if $$ \operatorname{inj\,dim}_R R < \infty, $$ that is, if the injective ...
user569831
1 vote
0 answers
76 views
literature request: morphisms of plane trees as natural transformations in $[\omega^{op},\Delta_+]$?
This is a more open-ended followup to a related question: On Joyal's definition of a category of plane trees. That question recalled how rooted plane trees can be represented as contravariant ...
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12 votes
2 answers
746 views
On Joyal's definition of a category of plane trees
I'm trying to understand better the motivation for the definition of the category $Trees$ of finite plane trees in Joyal's unpublished manuscript "Disks, duality and Θ-categories" (1997, ...
- 3,623
8 votes
2 answers
876 views
Why is Galois cohomology usually defined only for discrete modules?
$\DeclareMathOperator{\Aut}{Aut}$If $K$ is a field, the Galois cohomology $H^n(K, M)$ can be defined using cochains (or by a derived construction, etc.). The absolute Galois group $G_K$ naturally ...
- 1,292
1 vote
0 answers
111 views
Epimorphisms with kernel pairs
I am a bit lost understanding some subtleties in various form of epimorphy. The nLab reports that an effective epimorphism is one that coequalizes its kernel pair. A regular epimorphism is simply one ...
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3 votes
1 answer
267 views
Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
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6 votes
0 answers
164 views
Is there a syntactic proof that first-order positive inductive definitions are conservative?
Every first-order positive inductive definition has a fixed point. It follows that, if the biconditional is thought of as an axiom in the language obtained from the background language by adding a new ...
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8 votes
0 answers
206 views
How to define Dedekind reals and Eudoxus reals such that they are equivalent to unmodulated Cauchy reals
In constructive mathematics without choice, we have three different versions of the real numbers (each embedding into the next). Regular Cauchy reals (functions $f : \mathbb N \to \mathbb Q$ such ...
- 7,128
12 votes
4 answers
1k views
Is there a name for finite unions of intervals?
Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
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1 vote
1 answer
155 views
Looking for definition of function spaces appearing in article of DiPerna & Lions
I am looking for the definition of various function spaces appearing in the following article, preferably with references to other sources where such spaces are discussed in greater detail: Article: ...
- 21
1 vote
0 answers
67 views
Definition of "interval of continuity" for function defined on sets
At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
- 13k
1 vote
0 answers
152 views
Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
- 5,004
1 vote
0 answers
246 views
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 ...
- 515
0 votes
1 answer
777 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://...
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1 vote
1 answer
302 views
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} \...
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1 vote
0 answers
319 views
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 ...
- 13k
-1 votes
1 answer
174 views
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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7 votes
0 answers
421 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 struck me that the definition of Grothendieck Topology bears some familiar ...
- 803
2 votes
1 answer
431 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 $...
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1 vote
0 answers
292 views
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 ...
5 votes
1 answer
536 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 $...
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2 votes
1 answer
264 views
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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2 votes
0 answers
99 views
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 ...
- 2,166
2 votes
1 answer
201 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....
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1 vote
1 answer
205 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 $\...
- 135
8 votes
1 answer
745 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 ...
- 3,644
6 votes
1 answer
953 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 ...
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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 ...
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1 vote
0 answers
246 views
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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5 votes
1 answer
682 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 ...
- 3,044
4 votes
0 answers
184 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 ...
- 1,548
3 votes
1 answer
127 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 &...
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3 votes
0 answers
99 views
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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5 votes
1 answer
603 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 ...
- 429
3 votes
0 answers
161 views
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, ...
- 1,548
4 votes
1 answer
617 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....
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3 votes
0 answers
87 views
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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3 votes
1 answer
306 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,...
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0 votes
0 answers
139 views
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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2 votes
0 answers
221 views
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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-4 votes
1 answer
187 views
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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2 votes
0 answers
315 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 ...
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4 votes
0 answers
338 views
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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0 votes
0 answers
135 views
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 "...
- 1
0 votes
0 answers
244 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 ...
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6 votes
2 answers
2k 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 $\partial S$....
user127837
1 vote
1 answer
127 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{...
1 vote
1 answer
198 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=...
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1 vote
0 answers
147 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}(...
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