The definitions tag has no wiki summary.

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### 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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### 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

191 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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67 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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70 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 ...

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### Name of Property $t=st \text{ and } s=ts$

What is the name of the property shared by a pair of functions $s,t$ with $$t=st \text{ and } s=ts$$
( Main example: relation-valued domain and range operations on relations, via ...

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### Non Borel Spaces: Gauge Integral

Question
Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology?
I'm wondering since it seems as the gauge crucially uses ...

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**1**answer

277 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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### 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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### On one class of Euclidean lattices

Let $\Lambda\subset \mathbb Z^3$ be 3D lattice with a basis
$$a_1=\left(\begin{smallmatrix} a_{11} \\ a_{21}\\
a_{31}
\end{smallmatrix}\right),a_2=\left(\begin{smallmatrix} a_{12} \\ a_{22}\\
a_{32}
...

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### 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 $(R,\mathfrak{m})$ ...

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### Does the notion of graphs with vertex multiplicity exist?

I need to use graphs where each vertex gets a natural number, $b(v)$, its multiplicity. These numbers indicate how many 'replications' of the vertex we have.
It is actually a way to write in a ...

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**1**answer

244 views

### Canonical differential on Tate curve

I am starting studying the theory of (algebraic) modular forms, and I have some trouble in understanding completely the construction of the Tate curve. My problem is the following: as far as I know ...

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281 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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295 views

### BRST cohomology definition

Is there written anywhere a full definition of BRST cohomology? All I have found so far is BRST cohomology in _______.
As far as I can see, BRST cohomology is the ...

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**1**answer

672 views

### Why do mathematicians prefer one definition over the other when they both define the same concept?

Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...

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### 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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### Understanding the left-separated spaces

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.
Could someone post some left-separated space to help me understand such ...

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### 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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### On similar concepts in mathematics whose similarity is a non-trivial fact.

Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this ...

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### Set theory definition of addition, negative numbers, and subtraction? [closed]

Using the definition of natural numbers $0 = \emptyset$ and $S(n) = n \cup \lbrace n \rbrace$ where S is the successor function, what is the definition of addition on natural numbers?
Concerning the ...

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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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### 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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### Grothendieck Topologies versus Pretopologies

The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...

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### What makes a distance?

In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popular and important one.
...

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### What is the analog of “monotonic” for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...

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### A scheme simple over Spec(A)?

What does it mean to say that a scheme $X$ is simple over $Spec(A)$ ?
I stumbled on this terminology in a paper of S. Lubkin entitled "On a conjecture of Andre Weil".

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### Simple Equivariant homology [no borel-Moore]

Hey. I'm working with Bredon's equivariant cohomology. At some point I need to compute the $4$th equivariant cohomology group of $S^1 \x D^3$ relatively to its boundary for the antipodal action of ...

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**1**answer

340 views

### equivalence of definitions of Carmichael numbers

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 $a$ $\iff a^{n-1} \equiv 1 \mod n$ ...

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### 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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### What is “augmented algebra”?

Really sorry for this question, but googling for some time did not help me. I was trying to understand the meaning of the following phrase:
Let B be an augmented algebra over a semi-simple algebra T.
...

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2k views

### Geometric picture of invariant differential of an elliptic curve

What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?
This question is an adjunct to MO Q1 on formal laws and L-series, which motivated Q2. Q1 (Silverman) and Darmon (pg. ...

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### Non-split groups

I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom

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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 ...

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608 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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### Locally constant functions with compact support = smooth ?

Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C_c^{\infty}(F)$ is the ...

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523 views

### What is the quantity 2(handles)+crosscaps called?

It is well-known that up to homeomorphism, the complete set of orientable surfaces is $\lbrace S_g : g=0,1,\dots \rbrace$, where $S_g$ is the sphere with $g$ handles. The complete set of ...

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623 views

### Unitary groups over number fields

When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...

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261 views

### Why is Pic^0(C) of a curve C a variety?

Let $C$ be an abstract non-singular curve.
I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety.
Any pointers towards a reference would be appreciated.

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### What does the adjective “natural” actually mean?

Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The ...

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796 views

### Lattices: why require bilinear form to be integral?

This is a quite localized question, but I hope it won't be closed as unfit to MO. Well, a lattice $\Lambda$ in $\mathbb{R}^n$ is a discrete subgroup generated by a basis. Such a lattice gets a ...

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### Transitive closure of multigraphs

The transitive closure of a directed graph, is another directed graph which encodes the reachability of nodes from other nodes. If $G$ is a graph, the edge $(v_1,v_2)$ is in it's transitive closure ...

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### Euclidean Function at 0

In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things ...

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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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### 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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### Is there a mathematical object called “ivy”?

As the title says, is there a mathematical object referred to as "ivy" or "ivy type" or similar?
I have a type of graph where this name fits perfectly, but I don't want it to clash with something ...

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**1**answer

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### 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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### What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...

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### Definition of forgetful functor

I was wondering if it is possible to make a formal definition of what it means for a functor to be forgetful. That is, using only the terminology of categories. I have seen so many examples of ...

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### Strings and “co-subsequences”

Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence ...