Questions tagged [definitions]
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178
questions
1
vote
2answers
16 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 ...
30
votes
3answers
1k 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. ...
5
votes
2answers
226 views
Line graphs called “graph derivatives”: any intuition?
Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the ...
2
votes
1answer
113 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 ...
0
votes
1answer
79 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-...
0
votes
1answer
213 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 ...
0
votes
1answer
104 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 ...
0
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0answers
40 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 $...
1
vote
1answer
222 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:
$$ J(u)=\int_{\mathbb{R}^...
6
votes
1answer
285 views
Two definitions of automorphic forms on Lie groups
My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group.
The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
0
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2answers
229 views
Why are isotropic random vectors called isotropic if they aren't? [closed]
A random vector $X \in \mathbb{R}^n$ is isotropic if $\mathbb{E}XX^T = I_n$. However isotropic random vectors don't have the property of isotropy. See 1. So why are they called isotropic?
Similarly a ...
2
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2answers
261 views
What concept does covariance formalise?
So for me the definition the independence of two random variables $X,Y$ is intuitivly very clear.
But what I have never seen motivated is why the heck one would be interested in the covariance $$\...
0
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0answers
90 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
2answers
1k 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 ...
1
vote
1answer
177 views
What is the definition of brick product? Graph theory
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 ...
0
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0answers
108 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 ...
23
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0answers
740 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 ...
3
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0answers
26 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 $...
4
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0answers
157 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 &...
3
votes
1answer
201 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 ...
11
votes
2answers
643 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 ...
0
votes
1answer
55 views
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 ...
0
votes
1answer
241 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 ...
3
votes
0answers
60 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 ...
7
votes
1answer
219 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 ...
1
vote
1answer
96 views
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 ...
1
vote
0answers
118 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 ...
0
votes
1answer
101 views
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: {\...
0
votes
0answers
140 views
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
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
1
vote
1answer
103 views
What is the definition of Plancherel density?
I know about the Plancherel measure, but I don't know where the term "Plancherel density" is defined.
0
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1answer
185 views
Terminology: “sufficiently large absolute constant”
I'm currently reading the paper "Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu" and have run into some terminology which I don't quite (rigorously) ...
1
vote
0answers
57 views
Open/closed/constructible subsets of locally free sheaves
(Cross-posted from math.SE since I'm not sure what is a suitable platform. Link on https://math.stackexchange.com/questions/3597258/open-closed-constructible-subsets-of-locally-free-sheaves)
Is there ...
0
votes
0answers
49 views
Generalized compact open topology?
Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets
$$
\left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\
U_{K,O}:=\...
15
votes
2answers
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 ...
1
vote
0answers
51 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?
$$
\...
-2
votes
1answer
107 views
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 ...
0
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1answer
81 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 ...
6
votes
3answers
502 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 ...
1
vote
1answer
132 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.
2
votes
2answers
196 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 $\...
1
vote
0answers
90 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 ...
26
votes
1answer
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....
5
votes
3answers
243 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. ...
0
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0answers
21 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 ...
2
votes
2answers
441 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 ...
2
votes
1answer
405 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 ...
3
votes
0answers
162 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 $(...
0
votes
0answers
287 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
$$ ...
3
votes
0answers
152 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 ...
0
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0answers
70 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
$$
\...
