# Questions tagged [definitions]

The definitions tag has no usage guidance.

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### 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**

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**1**answer

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### 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 ...

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**1**answer

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### “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**

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**1**answer

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### 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 ...

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### 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 ...

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**7**answers

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### 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 ...

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**2**answers

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### 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 ...

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### 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 ...

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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=\...

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**1**answer

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 ...

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### 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 \...

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### 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 ...

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**2**answers

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)...

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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 $\...

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**1**answer

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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**59**answers

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### 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 ...

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**3**answers

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### 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 ...

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**1**answer

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### 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 ...

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### 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 $C^\infty$-...

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**1**answer

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### 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 ...

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### 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 $...

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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} \...

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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
...

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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 ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 $k : x \mapsto \mathbb{R}^N$
$y$ ...

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### 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 ...

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### “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 ...

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**1**answer

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 ...

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### 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 ...

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### 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 ...

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### 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}\|_{...

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### 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, ...

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### 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?

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### 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 ...

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### 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^\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 @>...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...