The definitions tag has no usage guidance.

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### What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...

**6**

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**0**answers

311 views

### Competing notions of 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**

votes

**1**answer

300 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 ...

**19**

votes

**1**answer

1k views

### Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...

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**3**answers

154 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 ...

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**1**answer

175 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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**0**answers

25 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 ...

**4**

votes

**1**answer

127 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 ...

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**26**answers

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### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

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votes

**4**answers

814 views

### What is a Kelley ring?

I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...

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vote

**1**answer

90 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 ...

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votes

**1**answer

374 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 ...

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**1**answer

58 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 ...

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**2**answers

577 views

### equivalence of definitions of Carmichael numbers [closed]

I would like to prove the equivalence of the two most common definitions of a composite integer $n > 1$ being a Carmichael number:
$a^n \equiv a \mod n $ for all $a$ $\iff a^{n-1} \equiv 1 \mod n$ ...

**5**

votes

**1**answer

210 views

### Definition of a normed ring

A normed ring "should" be a monoid object in the monoidal category of normed abelian groups. There are (at least) two choices of morphisms of normed groups, namely bounded or short homomorphisms, ...

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**5**answers

1k views

### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...

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vote

**1**answer

84 views

### Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:
"A finite graph whose edges are smoothly embedded arcs in a surface, ...

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votes

**1**answer

183 views

### Profinite groups, directed sets and $H^1$

Usually whenever one reads the definition of profinite group, one starts with an ordered set $I$ which is directed, meaning that for every $i,j\in I$ there is some $k\in I$ such that $i\leq k$ and ...

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**1**answer

181 views

### Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly.
For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...

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votes

**1**answer

147 views

### Definition of Strongly Stable 0-cycle

I am not sure whether this question deserves to be asked in this forum, but I have no other choice as I can't find the definition anywhere. So here is the question:
When is a 0-cycle on $\mathbb P^n$ ...

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votes

**1**answer

122 views

### Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: ...

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votes

**2**answers

711 views

### On figurate numbers

Do you know a text where I can find a definition of polygonal number that is both geometrically and operationally sound?
I've basically seen two ways in which this topic is approached in the ...

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**0**answers

85 views

### Can we have extension of Mercer theorem to interpolation? [closed]

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...

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**0**answers

90 views

### One-sided local $L^p$ spaces

Consider the vector space $L^p_{\text{left-loc}}$ of measurable functions $f:[0,1]\to\mathbb R$ so that for all $x\in(0,1]$ there exists $\delta>0$ so that $f|_{[x-\delta,x]}\in L^p$.
Does this ...

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**0**answers

177 views

### What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...

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**3**answers

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### Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...

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**2**answers

1k views

### Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...

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**2**answers

177 views

### Axiomatic approach to means

Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice ...

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**1**answer

277 views

### Gerstenhaber versus Schouten

In terms of formal definitions, is there any distinction between Schouten and Gerstenhaber algebras?

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**1**answer

777 views

### What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?

Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative ...

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### Riemannian simplicial complex and quasi-conformal complex

In this paper by Robert Young, the author defines
We define a riemannian simplicial complex to be a simplicial
complex with a metric which gives each simplex the structure of a riemannian
...

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131 views

### Fiber products of adic spaces

In the notes from Peter Scholze's class at Berkeley he makes the following remark: "Let us call a Huber pair $(A, A^+)$ admissible if $A$ is finitely generated over a ring of definition $A_0 \subset ...

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**2**answers

299 views

### Do you know of any asymmetric, nonparametric measure of dependence?

A measure of dependence is a way to assign a number (usually normalized between 0 and 1) to a couple of random variable, such that $\delta(X,Y)=0$ if and only of $X$ and $Y$ are independent, and ...

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632 views

### Is this a vertex algebroid?… What is vertex algebroid?

A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are.
During our discussion, I came up with a guess of what a vertex algebroid might be.
I'm ...

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237 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by ...

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**1**answer

248 views

### Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function ...

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### Category Theoretic Interpretation of Matroids?

Hello everyone! First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category ...

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134 views

### How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...

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**1**answer

231 views

### Meaning of $g_d^r$ in algebraic geometry

As an editor I often encounter the symbol $g_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? ...

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**1**answer

348 views

### Concise definition of subobjects

Higher category theory tells us that it is a bad idea to identify isomorphic things. Rather, the isomorphism should belong to some additional data. Also, categorification tells us that one should, ...

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### Can a mathematical definition be wrong?

This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...

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97 views

### A certain Acyclic Partition of a digraph

Has the following object been defined in the literature? What is it called? And what literature studies it? Are there other characterizations of this? What properties are known?
Let $G$ be a directed ...

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vote

**1**answer

76 views

### On Severi's definition of the complementary correspondence

In Weil's short note entitled "On the Riemann hypothesis in function-fields"
he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where ...

**3**

votes

**1**answer

297 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

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300 views

### procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...

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576 views

### Defining the integral of a function using the product measure

Imagine that we're trying to define the expression
$$\int_U f(x)dx$$
in a rigorous way.
Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...

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**0**answers

173 views

### Alternate definition for the torsion tensor

I would be pleased to have some information about an alternate definition for the torsion tensor.
Let us consider a smooth manifold $\mathcal{M}$ together with an arbitrary connection $\nabla$. The ...

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**3**answers

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### What are Penrose Tilings, and how do they relate to Quasicrystals?

The question is in the title, but let me elaborate a little.
Background
Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...

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**2**answers

208 views

### On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $ \mathbb{R}^{d} $, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet ...

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261 views

### PAC field : Algebraically closed field :: ? : Henselian local ring

I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity.
I'd want to call a DVR $(R,\mathfrak{m})$ ...