Questions tagged [definitions]

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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 ...
Qiaochu Yuan's user avatar
11 votes
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225 views

Is there a term for this graph subset?

Suppose $G$ is a (finite) graph which is $k$-vertex colourable (i.e. $\chi(G)\leqslant k$). Suppose $S$ is a set of vertices of $G$ with the following property: If $c:V(G)\rightarrow [k]$ is a vertex ...
JonCC's user avatar
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What are the relationship between various definitions for quasi coherent sheaves?

It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves My questions are: Are there any other definitions of quasi coherent ...
Shizhuo Zhang's user avatar
7 votes
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504 views

Competing notions of formal étaleness

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate. Here is a list of ...
Arrow's user avatar
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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 ...
Nik Bren's user avatar
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287 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 $...
user449595's user avatar
5 votes
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450 views

Basic questions about crystals and Grothendieck connections

I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...
Arrow's user avatar
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"Correct" definition of signed curvature in Minkowski plane

We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively. Looking at references such as ...
Ivo Terek's user avatar
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340 views

Why is Pic^0(C) of a curve C a variety?

Let $C$ be an abstract non-singular curve. I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety. Any pointers towards a reference would be appreciated.
andre's user avatar
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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 ...
rschwieb's user avatar
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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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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 &...
user221330's user avatar
4 votes
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236 views

Exterior tensor product of $D$ Modules

The exterior tensor product of sheaves of modules is defined as: $M \boxtimes N = p_1^{*}M \otimes_{\mathcal{O}_{X \times Y}} p_2^{*}N \cong \mathcal{O}_{X \times Y} \otimes_{p_1^{-1}\mathcal{O}_X \...
Federico Barbacovi's user avatar
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129 views

When are descriptions of formal unramifiedness/smoothness via lifting properties equivalent to those via induced arrows to pullbacks?

Formal unramifiedness of an arrow $f:M\rightarrow N$ in algebraic geometry or synthetic differential geometry in defined by asking the lifting problem below to have at most one solution (existence is ...
Arrow's user avatar
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4 votes
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What does "control of a deformation problem" mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
Jim Stasheff's user avatar
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Q-construction and Gabriel-Zisman Localization

It might be a stupid question. When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...
Shizhuo Zhang's user avatar
3 votes
0 answers
66 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 &...
alhal's user avatar
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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
3 votes
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76 views

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 ...
JacobsonRadical's user avatar
3 votes
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197 views

Generalization of normal subgroup

I am wondering whether the following concept appears in the group theory literature under some (perhaps different) name. Let $G$ be a group and let $A,B$ be subgroups of $G$. Definition. Say that $(...
pre-kidney's user avatar
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3 votes
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284 views

Equivariant sheafs and $G$ actions on modules

I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf. The set up ...
Louis Jaburi's user avatar
3 votes
0 answers
60 views

What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)

Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...
Johnny T.'s user avatar
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3 votes
0 answers
71 views

Is there a name for the general type of operation that sweeps a kernel over a function (e.g. like convolution, morph. dilation, registration, etc)

There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where: $f$ is a function $f : x \mapsto \mathbb{R}^N$ $k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$ $y$ ...
Tasos Papastylianou's user avatar
3 votes
0 answers
434 views

How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below. If $(X,\Sigma)$ is a measurable space, then the function $\mu : \Sigma\...
zeh's user avatar
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334 views

What is the analog of "monotonic" for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...
Alec Jacobson's user avatar
2 votes
0 answers
67 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 ...
Trebor's user avatar
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1 answer
76 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
2 votes
0 answers
110 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, ...
rschwieb's user avatar
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2 votes
0 answers
154 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}^{+} \...
Jamai-Con's user avatar
2 votes
0 answers
230 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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2 votes
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159 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
2 votes
0 answers
112 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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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
  • 1,137
2 votes
1 answer
587 views

A "boundary map" for the algebraic equivalence relation of cycles

In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want. Let $X$ be ...
Vincenzo Zaccaro's user avatar
2 votes
0 answers
126 views

The premises of Aczel's inductive definitions

This is a follow-up to https://stackoverflow.com/questions/49650053/are-inductive-definitions-finitely-generated-in-isabelle As I said there, Aczel writes in his paper An Introduction to Inductive ...
Gergely's user avatar
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2 votes
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A proper class for smooth chaotic function

This might be a little, soft, but I'll try Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way: For every $...
Amir Sagiv's user avatar
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2 votes
0 answers
409 views

Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given by $$H(\mathcal{A}...
Turbo's user avatar
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2 votes
0 answers
372 views

Alternate definition for the torsion tensor

I would be pleased to have some information about an alternate definition for the torsion tensor. Let us consider a smooth manifold $\mathcal{M}$ together with an arbitrary connection $\nabla$. The ...
user avatar
2 votes
0 answers
318 views

PAC field : Algebraically closed field :: ? : Henselian local ring

I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity. I'd want to call a DVR $(R,\mathfrak{m})$ pseudo-...
Bobby Grizzard's user avatar
1 vote
0 answers
108 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 ...
HASouza's user avatar
  • 293
1 vote
0 answers
272 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 ...
Zuhair Al-Johar's user avatar
1 vote
0 answers
185 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 ...
Elías Guisado Villalgordo's user avatar
1 vote
0 answers
201 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)=\...
Duality's user avatar
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1 vote
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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}(...
Usa's user avatar
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1 vote
0 answers
36 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. ...
Joe's user avatar
  • 759
1 vote
0 answers
379 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 ...
Christine McMeekin's user avatar
1 vote
0 answers
317 views

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 ...
VS.'s user avatar
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1 vote
0 answers
61 views

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 ...
modnar's user avatar
  • 501
1 vote
0 answers
53 views

Standard definition: vector-valued essential support

Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used? $$ \...
ABIM's user avatar
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1 vote
0 answers
96 views

What's the name of functions that produces a non deterministic solution without losing the exact solution?

I know that Turing reductions, function reductions and aproximation algorithms can produce good results and aproaches to the solution of a problem, but sometimes they lost the exact solution. Is there ...
Izar Urdin's user avatar