Questions tagged [definitions]

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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 ...
Allawonder's user avatar
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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 $...
inoc's user avatar
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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}^...
inoc's user avatar
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6 votes
1 answer
508 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 ...
Jun Yang's user avatar
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2 answers
924 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 ...
lamlame's user avatar
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2 votes
2 answers
368 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 $$\...
Jannik Pitt's user avatar
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"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
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 ...
Student's user avatar
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3 votes
2 answers
373 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 ...
sriram's user avatar
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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 ...
Hans-Peter Stricker's user avatar
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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 ...
Qiaochu Yuan's user avatar
5 votes
0 answers
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
4 votes
0 answers
182 views

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
3 votes
1 answer
373 views

Is there a notion of projective dg category?

Since the paper Smooth and proper noncommutative schemes and gluing of DG categories by Orlov, dg categories are considered the non-commutative counterpart of algebraic geometry. More specifically, we ...
Federico Barbacovi's user avatar
12 votes
2 answers
1k views

Different definitions for integral de Rham cohomology classes

Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$. Now, one ...
G. Gallego's user avatar
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1 answer
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Definition of a system of recurrent events

[I asked a version of this question on MSE a few weeks ago and didn't get any useful feedback. Apologies if I am just being stupid.] I am reading the paper A note on the Borel-Cantelli lemma by Kochen ...
Rob Arthan's user avatar
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1 answer
265 views

Can we define cardinality that works under weaker grounds than Scott's cardinals?

Its known that within the perspective of $\sf ZF$ related theories, Scott's definition of cardinality can work under weaker grounds than Von Neumann's! Scott's cardinality works as long as the ...
Zuhair Al-Johar's user avatar
3 votes
0 answers
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
7 votes
1 answer
302 views

Constructive definition of noncommutative rational functions (aka free skew fields)

The question Let $F$ be a field. (I am fine with assuming $F=\mathbb{Q}$, but I suspect that a "right" answer will be independent of $F$.) Let $k$ be a nonnegative integer. Question. Is ...
darij grinberg's user avatar
1 vote
1 answer
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Extension of the definition of entropy to $\mathbb{Z}^d$ and $\mathbb{N}^d$

I read the paper Entropie d'un groupe abélien de transformation by Conze and the part of the book Dynamical systems of Algebraic Origin by Schmidt about the entropy for $\mathbb{Z}^d$ actions. I was ...
user502940'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 answer
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Is this integral transform related to the Laplace transform?

The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by $${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$ Let $\varphi: {\...
Into's user avatar
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Use of this space of very rapidly decreasing continuous functions

Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space $$ V_p := \left\{ f \in C([0,\infty)):\, \sum_{n=1}^{\infty} ...
ABIM's user avatar
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1 vote
1 answer
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What is the definition of Plancherel density?

I know about the Plancherel measure, but I don't know where the term "Plancherel density" is defined.
Not a grad student's user avatar
0 votes
1 answer
264 views

Terminology: "sufficiently large absolute constant"

I'm currently reading the paper "Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu" and have run into some terminology which I don't quite (rigorously) ...
ABIM'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
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0 answers
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Generalized compact open topology?

Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets $$ \left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\ U_{K,O}:=\...
ABIM's user avatar
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16 votes
2 answers
2k views

Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
Bumblebee's user avatar
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1 vote
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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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-2 votes
1 answer
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Undecidable definition of mathematical expressions?

I am arguing a bit on Facebook regarding the definition of a mathematical expression. Some argue that equations are not expressions (and there are a few possibly dubious online sources which states ...
Per Alexandersson's user avatar
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1 answer
89 views

Problematic definition of empirical distribution [closed]

Could you please me explain the definition of empirical distribution? In Wikipedia, the defining equality has a NUMBER on one side and a FUNCTION (the sum of functions) on the other, which seems a ...
Oro Rin's user avatar
6 votes
3 answers
725 views

Adjusting the definition of a well-powered category to category theory with universes: size issues

Wikipedia and Borceux (Handbook of Categorical Algebra, Part I) give the following definitions of subobjects and well-powered categories: A subobject of an object $X$ of a category $\mathsf{C}$ is an ...
Jxt921's user avatar
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1 vote
1 answer
138 views

Meaning of "quantitative result" [closed]

Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.
ABIM's user avatar
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2 votes
2 answers
238 views

Concise formulation of set of equation systems

I have the following set of equation systems, and I would like to find a short, formal way to write it down. My main difficulty is that I cannot find a good way to write the indices of the variables $\...
Mario Krenn's user avatar
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
28 votes
1 answer
2k views

How can I improve my formal definitions?

I am a Software Architect and not very familiarized with standard notation in mathematics. Nonetheless, I would like to write a paper explaining a normalization of a computing model for expert systems....
Izar Urdin's user avatar
5 votes
3 answers
260 views

Is there a name for this "stack" of graphs?

Let $G_1,\ldots,G_m$ be a sequence of graphs, all having the same number $n$ of vertices. For each pair $(G_i, G_{i+1})$ we add $n$ edges that connect the vertices of $G_i$ and $G_{i+1}$ bijectively. ...
Stephan Mertens's user avatar
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0 answers
23 views

A linear map satisfying the given property

Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying $T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$. If the algerba ...
Fermat's user avatar
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2 votes
2 answers
586 views

Explanation of definition of George Wilson's adelic Grassmannian

How is George Wilson's adelic Grassmannian from e.g. the paper https://link.springer.com/article/10.1007%2Fs002220050237 related to the adeles or (especially) the affine Grassmannian (a.k.a. the loop ...
Yellow Pig's user avatar
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3 votes
1 answer
1k views

Different definitions of a relatively compact operator

(Cross-post from Math Stackexchange, where some work has been done in the comments) Let $T,K$ be unbounded operators on a Hilbert space $H$. I've seen the following definition of a relatively compact ...
Jannik Pitt's user avatar
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3 votes
0 answers
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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1 vote
0 answers
669 views

Total complex of complexes

When we have a double complex of vector spaces $V^{p,q}$, we can produce a complex either taking direct sums or products along the anti-diagonals. Then, the differential in this new complex will be $$ ...
Federico Barbacovi's user avatar
3 votes
0 answers
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
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0 answers
77 views

What is meant by "roots in $Lie(N)$" in root space decomposition of Lie algebras?

Let $G = GL_n$ and $T$ the invertible diagonal matrices and $N$ the upper triangular matrices with only $1$'s on the diagonal. Then the Lie algebra $\mathcal{G}$ has the roots space decomposition $$ \...
Johnny T.'s user avatar
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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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26 votes
3 answers
3k views

Why is the standard definition of a $(p, q)$-tensor so bizarre?

At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows. Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
Arthur's user avatar
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0 answers
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Formal definition of episodic Markov Decision process?

David Silver, in his lecture 4 from his Youtube lectures, speaks about episodic Markov Decision Processes (MDPs) and Monte-Carlo Policy Evaluation. I could not find a formal definition of episodic ...
Vasileios Anagnostopoulos's user avatar
1 vote
0 answers
84 views

Basic notation question involving Lie Groups and Lie algebras

I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/...
Johnny T.'s user avatar
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4 votes
1 answer
331 views

Understanding the Hamilton's definition of $\ast$-operation

I'm studying by myself Mean Curvature Flow and I'm trying understand the definition of $\ast$-operation given by Richard Hamilton in the beginning of the section $13$ (page $40$) of his paper "Three-...
George's user avatar
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5 votes
0 answers
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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