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Questions tagged [definitions]

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46
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11answers
5k 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
135 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
232 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 ...
98
votes
1answer
28k 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
111 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 ...
34
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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
92 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
17 views

Admissible Quadruple of Type L for locally compact Hausdorff space

Hatori and Oi defined admissible quadruple of type L at definition $4,$ page $6.$ Let $X$ and $Y$ be compact Hausdorff spaces. Let $B$ and $\tilde{B}$􏰑 be unital point separating subalgebras of ...
0
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0answers
80 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
55 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 ...
2
votes
0answers
108 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
82 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
88 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
181 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
1answer
404 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
34 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
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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
247 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 ...
106
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59answers
13k 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 ...
30
votes
3answers
1k views

Why are there so many fractional derivatives?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator. I started with the book The Fractional Calculus ...
3
votes
1answer
66 views

Name of an algebraic structure that is an idempotent semiring but does not have right distributivity

As the title implies, I am looking for the right name for an algebraic structure, which is exactly as an idempotent semiring, apart from the fact that multiplication does not right-distribute over ...
13
votes
2answers
346 views

The $n$-th derivative has $n$ zeros. Can such a function be unbounded?

I asked this on Math.SE some days ago, but without any success. For some application I need a formal definition of bell-shaped function. So I had the following idea: Definition. A $C^\infty$-...
4
votes
1answer
240 views

Dualizing complex definition ubiquity

The following definition is the one, I found here http://stacks.math.columbia.edu/tag/0A7A. But let me recall it (out of consistency's sake): Definition For $A$ a Noetherian ring, a dualizing complex ...
6
votes
4answers
599 views

What is the essence of the constant factor in the standard definitions of the discriminant?

Let $f(x) = x^m+\sum_{j=0}^{m-1}f_{m-j}x^j\in P[x]$ be a monic polynomial over a field $P$ and let $f(x) = (x-\alpha_1)\cdot\ldots\cdot(x-\alpha_m)$ be a factorization of $f$ over an extension field $...
1
vote
0answers
73 views

Another characteristic of subsets of (finite) power sets

Consider a finite set $M$, its power set $\mathcal{P}(M)$ and a subset $S$ of the power set, i.e. $S \subset \mathcal{P}(M)$. Consider as a characteristic of $S$ the function $f_S: \mathbb{N} \...
3
votes
2answers
170 views

A notion of weak dependence

Let $(X_1,\ldots,X_n)$ be a collection of random variables. For $\alpha\ge1$, let us say that these are $\alpha$-weakly dependent if for all $1\le k\le n$ and all $1\le i_1<\ldots< i_k$, we have ...
11
votes
0answers
206 views

Is there a term for this graph subset?

Suppose $G$ is a (finite) graph which is $k$-vertex colourable (i.e. $\chi(G)\leqslant k$). Suppose $S$ is a set of vertices of $G$ with the following property: If $c:V(G)\rightarrow [k]$ is a vertex ...
5
votes
1answer
389 views

What would be the cotangent bundle of a Banach manifold?

Reference: Lang - Differential manifolds p.123 Quick question: Lang defines the cotangent bundle as the dual vector bundle of the tangent bundle, but shouldn't there be additionally a somewhat ...
2
votes
1answer
122 views

Graph algebras a la Lovasz

In the article (Lovasz, section 1.3) mentions graph algebra structures on the set of formal linear combinations (over a field?) of a collection of graphs. He also mentioned quantum graphs as an ...
3
votes
0answers
52 views

Is there a name for the general type of operation that sweeps a kernel over a function (e.g. like convolution, morph. dilation, registration, etc)

There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where: $f$ is a function $f : x \mapsto \mathbb{R}^N$ $k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$ $y$ ...
52
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12answers
4k views

Examples of advance via good definitions

In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...
5
votes
0answers
232 views

“Correct” definition of signed curvature in Minkowski plane

We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively. Looking at references such as ...
2
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1answer
126 views

$3$-Engel Group Definitions and Properties

This question is motivated by the definition of $2$-Engel group. The following two definitions of a $2$-Engel group are equivalent: A group $G$ such that $x$ commutes with $g^{-1}xg$ for all $x, g\in ...
23
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1answer
763 views

Why there is a Quot-scheme, not a Sub-scheme?

Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably ...
4
votes
1answer
309 views

Giving the same concept different names in the same paper

I found a seminal paper of renowned authors (Inference of Finite Automata Using Homing Sequences (1993) by Ron Rivest and Robert Schapire) in which the authors define the very same set-theoretic ...
1
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0answers
43 views

Is there optimal sparse and minimum energy solution?

I am interested to know if anyone worked on the the inverse solution of the problem defined below, which means I want to find $\tilde{y}$. $M_{\delta}^{\alpha}[\tilde{y}]=\|A\tilde{y}-\tilde{b}\|_{...
6
votes
1answer
506 views

Definition of “Lagrangian skeleton”

I'm looking for the precise definition of "Lagrangian skeleton", as I'm eventually going to give a talk on this topic. As I asked a professor in my university about references on Lagrangian skeleta, ...
3
votes
0answers
50 views

Compatible Finite Elements [closed]

I have a basic question as a matter of definition. I am wondering what is meant by compatible finite elements? Does it has to do with the spaces over which the trial functions are defined?
4
votes
0answers
121 views

When are descriptions of formal unramifiedness/smoothness via lifting properties equivalent to those via induced arrows to pullbacks?

Formal unramifiedness of an arrow $f:M\rightarrow N$ in algebraic geometry or synthetic differential geometry in defined by asking the lifting problem below to have at most one solution (existence is ...
6
votes
1answer
734 views

Short and elegant definition of the $C^1$ topology

A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map $$ C^\...
1
vote
0answers
134 views

Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$

The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...
6
votes
0answers
480 views

Competing notions of formal étaleness

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate. Here is a list of ...
7
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1answer
593 views

Definitions of Hilbert Bundles

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
2
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0answers
48 views

A proper class for smooth chaotic function

This might be a little, soft, but I'll try Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way: For every $...
5
votes
1answer
765 views

how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
9
votes
1answer
411 views

Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D @>...
4
votes
3answers
202 views

End points of continua

Whyburn (1942) defined an end point $x$ of a continuum $X$ to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines an end point ...
1
vote
1answer
174 views

If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space. Usually, we have a ...
15
votes
1answer
782 views

Definition of ind-schemes

What is the correct definition of an ind-scheme? I ask this because there are (at least) two definitions in the literature, and they really differ. Definition 1. An ind-scheme is a directed colimit ...
0
votes
1answer
95 views

What is the definition of size of an edge$?$

In page-$13$ of Graph minors. $X$. Obstructions to tree-decomposition, $\gamma(G)$ introduced as maximum size of an edge. What is the definition of size of an edge$?$ I think it may be number of edges ...