Recently Active Questions
159,064 questions
14
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2
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900
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Do torsion-free groups give projectionless group ($C^\ast$) algebras?
One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
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4
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1
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334
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Non-commutative versions of X/G
Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on ...
- 8,559
15
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2
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987
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Can an infinite sequence of integers generate integer-area triangles?
(asked by Shanzhen Gao, shanzhengao at yahoo.com, on the Q&A board at JMM)
Does there exist an infinite, monotonically increasing sequence of integers $\{ a_n \}_{n \geq 0}$ such that for any $n$,...
- 156k
8
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2
answers
2k
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(nontrivial) isotrivial family of elliptic curves
I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
- 6,247
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3
answers
3k
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Seeking for a formula or an expression to generate non-repeatative random number .. [closed]
With my personal interest and hobby I started this ..
Given a sequence of numbers 1,2,3 .... N
where N is the highest among the sequence and length of the sequence as well ..I tried my best to bring ...
3
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1
answer
899
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Eisenstein series and the Kronecker limit theorem
It is well known that the first Kronecker limit theorem gives the Laurent expansion of the Eisenstein series $E(z,s)$ over $SL(2,Z)$ at $s=1$; see, for example, Serge Lang's book Elliptic Curves, ...
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1
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Closed form of a nonlinear recurrence sequence.
The master theorem seems to fail on nonlinear recursive functions. Is there a standard tool for finding the closed forms of recursive functions of this form?
The question comes from trying to find ...
- 1,981
1
vote
2
answers
2k
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A random variable: is it a function or an equivalence class of functions? [closed]
A random variable: is it a function or an equivalence class of functions?
- 29k
5
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1
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309
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(Z/n)^(I) is a direct summand of (Z/n)^I
Dear group theorists,
Let $n \geq 1$ and $I$ be an infinite set (you may assume $I$ to be countable). Is the abelian group $(\mathbb{Z}/n)^{(I)}$ (direct sum of copies $\mathbb{Z}/n$) a direct ...
CommunityBot
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9
votes
1
answer
604
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Which changes of metric fix all open balls of a metric space?
In an earlier question, I was interested in counting the number of metric spaces on N points, where I considered two metric spaces to be the same if they had the same collection of open balls. Two ...
CommunityBot
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14
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1
answer
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What is the meaning of symplectic structure? [closed]
Answers can come in mathematical, physical, and philosophical flavors.
Edit: There seems to be a consensus that this question is not formulated well. I must respectfully disagree. My interest in the ...
- 3,203
4
votes
1
answer
1k
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references for models of stable infinity categories
There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...
- 27.5k
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votes
2
answers
1k
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Maximal Ellipsoid
John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
- 31.5k
0
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3
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171
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Does there exist an $A$ and $\mathfrak{su}(2) \subset B$ such that $\mathfrak{su}(3) \simeq A \otimes B$
I'm in the middle of trying to prove something at the moment and am looking for a decomposition of the Lie algebra $\mathfrak{su}(3)$ into a tensor product of some algebra $A$, and another $B$ ...
- 15.4k
5
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3
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841
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Integer subset that only occupies (p-1)/2 equivalence classes mod p?
I'm not quite sure the best way to ask this, so bear with me: Does anyone know of a subset of integers such that, for any odd prime p, the subset only occupies (p-1)/2 equivalence classes mod p (and ...
- 25.7k
1
vote
1
answer
248
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Unusual ray tracing
Background
Ray tracing is very common in computational geometry and the problem is then to find the point of intersection between the equation of a line and the equation of a plane in 3D.
The ...
- 3,059
8
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1
answer
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How does one get the short exact sequence in a two-column spectral sequence?
In a two-column double complex, one gets from the associated spectral sequence short exact sequences $0\to E_2^{1,n-1}\to H^n\to E_2^{0,n}\to 0$, where $H^n$ is the cohomology of the total complex, ...
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20
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3
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What is the origin of the term "spectrum" in mathematics?
The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; ...
- 6,523
10
votes
3
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3k
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The localisation long exact sequence in K-theory over an arbitrary base
If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory
... Kn+1(Dx) --> Kn(k) --...
CommunityBot
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8
votes
1
answer
969
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Moshe Rosenfeld's Salmon Problem
As an amusement at the start of this talk, Moshe Rosenfeld poses the following question.
Suppose that there are n salmon which
begin at distinct points on a unit
circle, each facing either ...
- 723
0
votes
1
answer
207
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Correlation of Statistical Tests
Suppose I have a sequence $\{x_i\}_{i=1}^\infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are ...
- 498
4
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2
answers
592
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Five lemma in HoTop* and arbitrary pointed model categories
Let $\textbf{HoTop}^*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $\textbf{HoTop}^*$, i.e. pointed homotopy ...
CommunityBot
- 1
5
votes
2
answers
675
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Terminology: Is there a name for a category with biproducts?
Many people are familiar with the notion of an additive category. This is a category with the following properties:
(1) It contains a zero object (an object which is both initial and terminal).
...
- 7,329
11
votes
2
answers
2k
views
Meaning of orientation/orientability over rings other than the integers
This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately.
We consider the homology definition of an orientation for a manifold, as ...
CommunityBot
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4
votes
1
answer
412
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F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 65.4k
2
votes
1
answer
337
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Switching function for Bang-Bang nagivation
I'm attempting to develop an equation to determine the "switching time" for a control system. I've managed to work out a specific solution for when starting and ending velocities are are the same, ...
- 1,981
6
votes
1
answer
2k
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moduli space and modularity
I recently realized some kind of analogy when considering modularity results (such as the modularity of elliptic curves over Q). The analogy comes from algebraic groups. Take one point (say, the ...
- 57.6k
5
votes
2
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1k
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Relating Euler characteristic, intersection product, Morse theory (plus SU(2) and 3-manifolds)
Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F* denote F with a puncture. Then the space H of representations of pi_1(F*) on SU(2) is just SU(2)^2g, and ...
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1
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807
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A question about fibrations of simplicial sets and their fibers
I couldn't think of a title for this, but here we go:
Fix $p:S\rightarrow T$, a left fibration of simplicial sets, and an edge $f:\Delta^1 \rightarrow T$. Let $t$ be the first vertex of $f$, and $t'$...
- 19.6k
4
votes
0
answers
287
views
Rozansky-Witten class associated to the Theta Graph
Suppose I have a holomorphic symplectic manifold, and a smooth $(1,0)$ connection on the tangent bundle which is compatible with both the complex and the symplectic structures. Say that the ...
1
vote
1
answer
333
views
Extension of some feature of SDE Ornstein-Uhlenbeck type
Hi everyone,
I am looking for some ideas (or references) in order to get an explicit SDE (if it exists) which would have a stylised property extending in some sense the mean-reversion property of SDE ...
- 28k
9
votes
2
answers
2k
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How natural is the reciprocity map?
For local field, the reciprocity map establishes almost an isomorphism from the multiplicative group to the Abelian Absolute Galois group. (In global case the relationship is almost as nice). It is ...
- 57.6k
15
votes
27
answers
3k
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Justifying a theory by a seemingly unrelated example
Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...
CommunityBot
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12
votes
0
answers
851
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Compact Symplectic Fano (strongly monotone) manfiolds
What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry?
We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that
$[c_1(...
- 28.9k
3
votes
1
answer
895
views
Bernstein inequality for multivariate polynomial
Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max_{|z|=1} |Q(z)|$.
So, are there ...
- 3
4
votes
1
answer
281
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FTR Quantization for any Subalgebra of $GL(n)$?
As is well known, the quantum groups $SU_q(n)$, amongst others, arise from $R$-matrix solutions of the Yang-Baxter Equation. My question is: For any subalgebra of $GL(n)$, does there exist an $R$-...
- 6,131
2
votes
3
answers
1k
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Baire category theorem
Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$.
Let's call the following statement (2): ...
- 498
10
votes
5
answers
632
views
is there a good computer package for working with bicomplexes?
I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
- 4,394
42
votes
4
answers
8k
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Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
- 4,909
10
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0
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544
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What is the "category of bifurcations"?
While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added):
In this paper we show that every bifurcation set contains a copy of the boundary of the ...
- 5,992
10
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1
answer
2k
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When is the canonical divisor of an algebraic surface smooth?
Is there some condition on a complex algebraic surface that implies it has a smooth canonical divisor? I am searching for the sharpest possible condition, but sufficient criteria would be nice as ...
- 13.1k
9
votes
4
answers
820
views
How much of the current logic is about syntax?
The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current ...
- 3,267
12
votes
3
answers
7k
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Is functional programming a branch of mathematics?
In Theory mainly concerned with lambda-calculus?, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:
That would never stick unless there's another good reason. ...
CommunityBot
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3
votes
3
answers
212
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Rank(A) and other algorithms as a polynomial
If $A = (\alpha_{ij}) \in \mathbb{C}^{nxm}$ we have simple algorithms by which to determine $\mathrm{rank}(A)$. However, is there a polynomial $f \in \mathbb{C}[\alpha_{ij}]$ where $f \colon \mathbb{C}...
- 44.4k
8
votes
3
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698
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L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...
- 31.5k
7
votes
2
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232
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Local view of setting p*n out of n bits to 1
For p a constant in (0,1) and n going to infinity such that pn is an integer,
consider the distribution on n bits that selects a random subset of pn bits, sets those to 1, and sets the others to 0.
...
- 6,694
2
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2
answers
3k
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Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 33.8k
3
votes
1
answer
2k
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About the Shannon Switching Game
I was playing around with the Shannon Switching Game for some planar graphs, trying to get some intuition for the strategy, when I noticed a pattern. Since I only played on planar graphs, I'll ...
- 19.1k
11
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2
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862
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Monotone Lipschitz embedding ?
In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the ...
- 61.9k
9
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2
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744
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Cobordisms of bundles?
Is there a notion of a cobordism which is compatible with bundle structure?
That is, if I have bundles $E$ and $F$, is it the case that the manifold $W$ with $E$ and $F$ as boundary components, can ...
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