Questions tagged [definitions]
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153
questions
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0answers
54 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 ...
-5
votes
2answers
418 views
Coming up with a equivalent (or close) definition for an average which is easier to compute? [closed]
Continuing from my last question, I understand that my definition is unclear so I have modified it.
Since no one has answered my question on math stack exchange, I decided to ask here.
Definition
...
0
votes
1answer
91 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: {\...
1
vote
0answers
137 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
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
0
votes
0answers
27 views
Extended-value subgradients
(I am not an expert in convex analysis, as may become clear.)
Let $R$ be the extended reals, $R = \mathbb{R} \cup \{\pm \infty\}$.
Standard texts define the subgradient of a convex function $f: \...
1
vote
1answer
94 views
What is the definition of Plancherel density?
I know about the Plancherel measure, but I don't know where the term "Plancherel density" is defined.
1
vote
1answer
157 views
Terminology: “sufficiently large absolute constant”
I'm currently reading the paper "Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu" and have run into some terminology which I don't quite (rigorously) ...
1
vote
0answers
53 views
Open/closed/constructible subsets of locally free sheaves
(Cross-posted from math.SE since I'm not sure what is a suitable platform. Link on https://math.stackexchange.com/questions/3597258/open-closed-constructible-subsets-of-locally-free-sheaves)
Is there ...
2
votes
0answers
44 views
Generalized compact open topology?
Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets
$$
\left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\
U_{K,O}:=\...
14
votes
2answers
1k 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 ...
3
votes
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
104 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
votes
1answer
77 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
431 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 ...
1
vote
1answer
127 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
183 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
87 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
242 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
votes
0answers
20 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
363 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 ...
0
votes
1answer
217 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
151 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
175 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
107 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
votes
0answers
67 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
$$
\...
0
votes
0answers
81 views
How is the following graph operation defined in the given research paper?
enter image description hereI am unable to understand the graph operation defined in the following paper
https://www.sciencedirect.com/science/article/pii/S0972860017301706.
Can someone kindly ...
3
votes
0answers
48 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 ...
0
votes
0answers
30 views
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 ...
1
vote
0answers
79 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/...
2
votes
1answer
291 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-...
5
votes
0answers
263 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. ...
51
votes
11answers
6k views
What definitions were crucial to further understanding?
Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
2
votes
1answer
296 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 ...
6
votes
1answer
250 views
“Cyclic” continuum
On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:
"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients ...
105
votes
1answer
30k 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
vote
0answers
253 views
Different types of a test functions in weak solutions of a PDEs
In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the most usual definition is given bellow.
Let's ...
33
votes
7answers
3k views
Paradoxical Mathematical Objects Pending for Construction [duplicate]
The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
1
vote
2answers
118 views
Definition and examples of operator-stable distributions
I was trying to understand the basic ideas of the operator-stable distributions. I found the papers by Hudson and Sato. However, unfortunately, I am being unable to understand the mathematical ...
0
votes
0answers
89 views
Amalgamated free-product of semigroups (definition)
I am self-studying some concepts including the title one. I reached the definition of an amalgamated free-product ${S_1}{*_U}S_2$ where $[S_1, S_2; U, w_1,w_2]$ is an amalgam of semigroups. Let $S_1=\...
2
votes
1answer
65 views
Name for “partially complete” invariants in classification problems?
For any equivalence $\sim$ on some collection of objects $C$ consider the problem of trying to determine if two arbitrary objects $x$ and $y$ in $C$ are equivalent i.e. if $x\sim y$ now by definition ...
3
votes
0answers
156 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 \...
2
votes
0answers
88 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 ...
3
votes
2answers
101 views
In the context of directed graphs is it standard notation to allow an element of an independent vertex set to be contained in a loop?
Given any relation $R$, that is, any set of ordered pairs, we can associate a unique digraph $D$ to our relation $R$ by setting $D=(\text{fld}(R),R)$ where $\text{fld}(R)=\text{dom}(R)\cup\text{rng}(R)...
3
votes
3answers
276 views
On 2-actions of strict 2-groupoids?
I'm looking for an opinion if the following makes sense.
A linear representation of a groupoid $\mathcal{G}$ is a functor $$\nabla: \mathcal{G}\longrightarrow \mathsf{Vect}_{\mathbb K},$$ where $\...
8
votes
2answers
533 views
Reference Request: A definition of topology using monads (a.k.a. halos)
In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all ...
1
vote
0answers
48 views
How is the minimax oracle used to find the oracle complexity of projected subgradient?
I am going through a set of blog posts on the complexity of projected gradient method.
https://blogs.princeton.edu/imabandit/2013/03/15/orf523-oracle-complexity-large-scale-optimization/ defines the ...
26
votes
5answers
2k views
Main statement as theorem or corollary
In a text there will often be a few important results that are usually called Theorems. Intermediate statements are called Lemmas, and statements that follow immediately from previous results are ...
1
vote
0answers
250 views
Characterization of non-Zeno functions $f:\mathbb{R}\rightarrow \{0,1\}$
[Edit: I tried to integrate Nate's comments (see below).]
In the context of automata over continuous time, consider Boolean-valued functions $f:\mathbb{R}\rightarrow \{0,1\}$. There are uncountably ...
129
votes
64answers
16k 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 ...
