The definitions tag has no wiki summary.
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122 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 ...
6
votes
2answers
516 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 ...
2
votes
1answer
166 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 ...
5
votes
1answer
211 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? ...
3
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0answers
77 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 : ...
1
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1answer
247 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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0answers
75 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 ...
1
vote
1answer
71 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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0answers
113 views
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 ...
3
votes
1answer
287 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 ...
2
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0answers
149 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 ...
3
votes
0answers
90 views
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}
...
2
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0answers
244 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})$ ...
10
votes
7answers
446 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
votes
1answer
249 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 ...
4
votes
5answers
288 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
304 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
votes
1answer
689 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
586 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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vote
2answers
161 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 ...
2
votes
2answers
191 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
1k views
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
2k 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 ...
8
votes
3answers
823 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
419 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 ...
8
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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 ...
9
votes
3answers
667 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.
...
2
votes
0answers
243 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 ...
4
votes
2answers
609 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".
1
vote
1answer
278 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
vote
1answer
385 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 $a$ $\iff a^{n-1} \equiv 1 \mod n$ ...
16
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1answer
834 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 ...
3
votes
1answer
1k 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
votes
1answer
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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2answers
635 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
622 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 ...
5
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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 ...
0
votes
4answers
531 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
votes
2answers
643 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 $U(V)$ ...
5
votes
0answers
264 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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votes
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
votes
3answers
851 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
votes
3answers
666 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
452 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 ...
63
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18answers
8k 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 ...
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2answers
757 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
votes
1answer
791 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 ...
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votes
2answers
269 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 ...
26
votes
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 ...
