The definitions tag has no wiki summary.
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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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0answers
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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}
...
3
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0answers
222 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})$ ...
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7answers
420 views
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 ...
5
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1answer
227 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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4answers
236 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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1answer
277 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 ...
5
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1answer
629 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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2answers
474 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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2answers
139 views
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 ...
3
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2answers
173 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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4answers
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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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2answers
1k views
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 ...
7
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3answers
675 views
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 ...
2
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3answers
344 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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4answers
1k views
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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3answers
658 views
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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0answers
218 views
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 ...
3
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2answers
592 views
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".
0
votes
1answer
259 views
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 ...
1
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1answer
311 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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1answer
748 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 ...
1
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1answer
829 views
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.
...
5
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1answer
1k 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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2answers
565 views
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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votes
2answers
2k views
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 ...
2
votes
2answers
585 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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4answers
1k views
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 ...
1
vote
4answers
510 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 ...
5
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2answers
604 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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0answers
258 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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11answers
5k views
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 ...
4
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3answers
744 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 ...
2
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3answers
568 views
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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2answers
441 views
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 ...
49
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16answers
7k views
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 ...
7
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2answers
737 views
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 ...
4
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1answer
780 views
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 ...
2
votes
1answer
208 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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1answer
2k views
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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4answers
1k views
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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5answers
341 views
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 ...
23
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1answer
1k views
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 ...
4
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0answers
268 views
Q-construction and Gabriel-Zisman Localization
It might be a stupid question.
When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...
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What are the relationship between various definitions for quasi coherent sheaves?
It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves
My questions are:
Are there any other definitions of quasi coherent ...
4
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1answer
836 views
What is a proper stack?
I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism ...
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3answers
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When is a classification problem “wild”?
I hope someone can point me to a quick definition of the following terminology.
I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
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23answers
20k views
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 ...
3
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2answers
297 views
Definition modifications without choice
What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? ...
