Questions tagged [definitions]

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2 votes
1 answer
610 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 ...
-4 votes
0 answers
52 views

pixels on a graphing calculator [closed]

Some time ago I asked a question that remained unanswered. Now I changed the equation a bit. https://www.desmos.com/3d/018fa463e0 A little clarification, t is the base of the square and it is variable,...
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 ...
0 votes
1 answer
391 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://...
0 votes
0 answers
121 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} \...
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 ...
2 votes
1 answer
78 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....
106 votes
2 answers
32k views

What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
-1 votes
1 answer
122 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 ...
6 votes
0 answers
216 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 ...
147 votes
71 answers
20k views

Nonequivalent definitions in Mathematics

I would like to ask if anyone could share any specific experiences of discovering nonequivalent definitions in their field of mathematical research. By that I mean discovering that in different ...
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,...
2 votes
1 answer
373 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 $...
1 vote
0 answers
186 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 ...
1 vote
1 answer
110 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 ...
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 $...
2 votes
1 answer
151 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 ...
2 votes
0 answers
68 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 ...
1 vote
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 $\...
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. ...
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 ...
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 ...
6 votes
1 answer
464 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 ...
1 vote
0 answers
202 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
394 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 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 &...
14 votes
4 answers
464 views

Axiomatic approach to means

Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice ...
4 votes
0 answers
156 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 ...
3 votes
2 answers
376 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 ...
27 votes
4 answers
3k views

What are examples of theorems which were once "valid", then became "invalid" as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
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 &...
5 votes
1 answer
299 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 ...
2 votes
0 answers
111 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, ...
4 votes
1 answer
446 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....
3 votes
0 answers
67 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_{...
7 votes
2 answers
2k views

What is a proper stack?

I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism ...
30 votes
3 answers
4k views

When is a classification problem "wild"?

I hope someone can point me to a quick definition of the following terminology. I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
0 votes
0 answers
98 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 ...
2 votes
0 answers
155 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}^{+} \...
-4 votes
1 answer
134 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 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 ...
4 votes
0 answers
205 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'...
0 votes
0 answers
93 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 "...
0 votes
0 answers
156 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 ...
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 $\partial S$....
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{...
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=...
1 vote
0 answers
109 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}(...
0 votes
1 answer
997 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}^...
2 votes
0 answers
161 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 ...

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