Questions tagged [definitions]
The definitions tag has no usage guidance.
210
questions
8
votes
1
answer
579
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 ...
- 3,366
6
votes
1
answer
347
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 ...
- 347
12
votes
2
answers
1k
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 ...
- 221
1
vote
0
answers
189
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)=\...
5
votes
1
answer
270
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 ...
- 2,509
4
votes
0
answers
144
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 ...
- 1,458
3
votes
1
answer
98
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 &...
- 35
2
votes
0
answers
53
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 &...
- 387
5
votes
1
answer
191
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 ...
- 387
2
votes
0
answers
84
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, ...
- 1,458
4
votes
1
answer
381
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....
- 113
3
votes
0
answers
56
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_{...
- 181
3
votes
0
answers
110
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,...
- 191
0
votes
0
answers
32
views
Orientation-compatible intersections
(Note: Cross-posted from MSE in https://math.stackexchange.com/questions/4490846/orientation-compatible-intersections since it isn't clear if this is well-known and couldn't find MO questions on ...
- 501
0
votes
0
answers
85
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 ...
- 119
2
votes
0
answers
146
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}^{+} \...
- 87
-4
votes
1
answer
125
views
Hilbert’s third problem and what a polyhedron is [closed]
What is the definition of a polyhedron used by Hilbert’s third problem?
- 2,415
2
votes
0
answers
217
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 ...
- 2,261
4
votes
0
answers
150
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'...
- 243
0
votes
0
answers
82
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
104
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 ...
- 501
5
votes
2
answers
979
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
94
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
119
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=...
- 123
1
vote
0
answers
91
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}(...
- 119
2
votes
0
answers
134
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 ...
- 121
4
votes
1
answer
318
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 ...
- 253
2
votes
0
answers
95
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 ...
- 593
8
votes
2
answers
767
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 ...
- 72.9k
1
vote
0
answers
35
views
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.
...
- 759
1
vote
1
answer
59
views
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 ...
- 291
1
vote
0
answers
86
views
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 ...
- 3,373
2
votes
0
answers
120
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\...
- 851
1
vote
2
answers
130
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 ...
- 125
33
votes
5
answers
2k
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. ...
- 58.7k
2
votes
1
answer
191
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 ...
- 1,345
0
votes
1
answer
95
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-...
- 409
0
votes
1
answer
259
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 ...
- 155
0
votes
1
answer
122
views
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 ...
- 335
0
votes
0
answers
44
views
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 $...
- 339
0
votes
1
answer
737
views
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:
$$ J(u)=\int_{\mathbb{R}^...
- 339
6
votes
1
answer
453
views
Two definitions of automorphic forms on Lie groups
My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group.
The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
- 347
0
votes
2
answers
668
views
Why are isotropic random vectors called isotropic if they aren't? [closed]
A random vector $X \in \mathbb{R}^n$ is isotropic if $\mathbb{E}XX^T = I_n$. However isotropic random vectors don't have the property of isotropy. See 1. So why are they called isotropic?
Similarly a ...
- 153
2
votes
2
answers
333
views
What concept does covariance formalise?
So for me the definition the independence of two random variables $X,Y$ is intuitivly very clear.
But what I have never seen motivated is why the heck one would be interested in the covariance $$\...
- 944
0
votes
0
answers
272
views
"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 ...
15
votes
2
answers
2k
views
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 ...
- 4,560
3
votes
2
answers
341
views
What is the definition of brick product of graphs?
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 ...
- 101
0
votes
0
answers
129
views
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 ...
- 9,452
23
votes
0
answers
1k
views
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 ...
- 112k
4
votes
0
answers
137
views
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 $...
- 163