# Questions tagged [definitions]

The definitions tag has no usage guidance.

178
questions

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### 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

**3**answers

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. ...

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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 ...

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votes

**1**answer

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 ...

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**1**answer

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-...

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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 ...

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votes

**1**answer

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 ...

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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 $...

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**1**answer

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}^...

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votes

**1**answer

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 ...

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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 ...

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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 $$\...

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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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 ...

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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 $...

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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 &...

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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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: {\...

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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} ...

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**1**answer

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.

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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) ...

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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 ...

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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}:=\...

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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 ...

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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?
$$
\...

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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 ...

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**1**answer

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 ...

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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 ...

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**1**answer

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.

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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 $\...

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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 ...

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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....

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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. ...

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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 ...

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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 ...

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**1**answer

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 ...

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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 $(...

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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
$$ ...

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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 ...

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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
$$
\...