Recently Active Questions
159,037 questions
9
votes
2
answers
359
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When do PROP-morphisms induce adjunctions?
If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where B-Alg(C)=tensor-...
- 1,872
18
votes
1
answer
795
views
Is there a Murnaghan-Nakayama Rule for GL(n,q)?
The Murnaghan-Nakayama rule for S_n is a combinatorial rule to compute the irreducible characters of the symmetric group. Is there a q-analogue of this rule for GL(n,q) to compute the irreducible ...
- 11.2k
4
votes
1
answer
321
views
Reverse Langlands transform
What os the meaning of a reverse Langlands transform to which Drinfeld seems to refer?
CommunityBot
- 1
2
votes
2
answers
550
views
Algebraic Geometry in an applied setting?
I just saw this paper recently which mentioned that the optimization on a Grassmanian Manifold can be used to get an achieve an best approximation of a multilinear rank of a tensor (in the sense of a ...
- 8,681
6
votes
1
answer
1k
views
What is Drinfeld's manuscript "Best Dream" (in Russian!) about?
I would like to know what Drinfeld's scanned manuscript "Best Dream" is about: the title makes me curious.
It's in Russian.
- 15.1k
1
vote
3
answers
1k
views
Important (interesting) unsolved problems [closed]
I think it would be interesting to have a list of important unsolved problems in mathematics.
What are the important (interesting) problems in your field of work? It would be especially nice, to have ...
- 28.1k
11
votes
3
answers
932
views
Line bundles on moduli spaces
This is perhaps too broad or vague (or silly) a question, but here it is anyway: why should I care about constructing line bundles on a moduli space? This comes up all of the time, but I seem to be ...
- 4,394
1
vote
1
answer
162
views
Does automatic decomposition of varieties into irreducibles exist?
Varieties decompose uniquely into finitely many irreducibles, and each variety is generated by only finitely polynomials. These two finiteness properties make varieties seemingly "manageable" objects, ...
- 5,243
12
votes
3
answers
530
views
Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 5,992
6
votes
2
answers
540
views
How do quantum knot invariants change when I pick a funny ribbon element?
So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ...
- 28.1k
1
vote
2
answers
264
views
Antipode for quantum matrices.
Am I right in assuming that one cannot define an antipode for $M_q(n)$ the bi-algebra of $nXn$ quantum matrices? If so, does anyone know a proof?
- 6,131
8
votes
2
answers
626
views
Which commutative rigs arise from a distributive category?
A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...
- 54.6k
14
votes
2
answers
882
views
A complex manifold which is quasiprojective in two different ways
Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two ...
- 5,062
8
votes
1
answer
1k
views
Learning about Galois representations
My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.
Thus take pt = Spec F_q, ...
- 15.1k
6
votes
2
answers
509
views
Can I finitely color Z^2 such that (x,a) and (a,y) are different for every x,y,a?
I ran into this obstacle in a harmonic analysis problem; I know epsilon about coloring problems.
Is it possible to finitely color Z^2 so that the points (x,a) and (a,y) are differently colored for ...
- 1,975
4
votes
6
answers
689
views
Placing checkers with some restrictions
We are going to put n checkers on an (n x n) checkers board, with the following restrictions:
1) In each column there is EXACTLY one checker.
2) For i=1,2,...,(n-1), the first i rows cannot have ...
- 1,975
17
votes
2
answers
5k
views
Exactness of filtered colimits
Are filtered colimits exact in all abelian categories?
In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase ...
- 8,681
5
votes
1
answer
1k
views
Equivalence of boundedness and total boundedness
Compact subspaces of metric spaces are totally bounded. In some spaces, however, this is equivalent to just being bounded. This (supposedly) holds in finite dimensional Banach spaces.
Can we ...
- 1,069
2
votes
3
answers
1k
views
Computing a Factor Group
I have a problem in computing (i.e. classify) a factor group.
For example The group Z*Z*Z/<(3,6,9)> is isomorphic to Z_3*Z*Z. I can show this by contructing a homomorphism f
f(a,b,c) = ( a mod 3 ,...
- 1,756
3
votes
1
answer
2k
views
Hilbert Space as direct sum of subspaces with cyclic vectors
Ok,so this should be easy, however I havent taken functional analysis for a while. But given a compact self-adjoint operator on a hilbert space H(over the complex numbers), we define v to be a cyclic ...
- 31.2k
7
votes
2
answers
474
views
Density of a subset of the reals
The rationals are clearly dense in the real number system, i.e. for every pair a < b of real numbers there exists a rational number p/q s.t. a < p/q < b. I conjecture the same to be true with ...
- 2,479
2
votes
2
answers
454
views
Is the center of a free (as a module) algebra free?
A submodule of a free module need not be free (for instance, in the free Z[X]-module Z[X] the submodule generated by 2 and X is not free). But over a principal ideal domain, submodules of free modules ...
- 53.3k
8
votes
3
answers
832
views
Why is the Hodge class of \bar{M_g} big and nef?
Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...
- 10.5k
11
votes
2
answers
2k
views
Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 13.6k
2
votes
1
answer
493
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Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 389
15
votes
2
answers
1k
views
Categorifying the Reals via von Neumann Algebras?
So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...
CommunityBot
- 1
2
votes
1
answer
226
views
Are non-maximal orders in number fields Grothendieck rings?
Recall that a ring homomorphism A->B is geometrically regular if for all primes p of A, the fiber of B over p is geometrically regular over k(p). A Grothendieck ring (or, G-ring) is one for which A_p->...
- 1,806
5
votes
1
answer
322
views
is amalgamation of groups associative
Given groups $G_1, G_2, G_3$ and injections $A_1 \to G_1$ and $A_1
\to G_2$ , from $A_2 \to G_2$ and $A_2 \to G_3$, let $G_1 *_{A_1} *G_2 *_{A_2} G_3$ be the amalgam formed these groups and maps.
...
- 10.7k
9
votes
1
answer
643
views
Determinant of a pullback diagram
Suppose that X and Y are finite sets and that f : X → Y is an arbitrary map. Let PB denote the pullback of f with itself (in the category of sets) as displayed by the commutative diagram
PB &...
- 25.2k
9
votes
2
answers
775
views
How can I prove that a sequence of squares of graph norms is never cyclotomic?
The norm of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||G|| for the norm of G.
Now, fix some graph <...
- 156k
5
votes
1
answer
332
views
Extending Functions on Closed Submanifolds of C^n
Functions on an algebraic subvariety X of A^n are the same as functions on A^n restricted to X. So the statement that functions on X extend to all of A^n follows by the definition. My question is: ...
- 10.5k
3
votes
2
answers
707
views
Is an nth root of unity a square?
Suppose w^(2n)=1 (w is a complex number).
For which n (if any) \sqrt(w) \in Q(w) ?
- 156k
6
votes
1
answer
494
views
Non-free projective modules for a Universal Enveloping Algebra?
Let g be a finite dimensional Lie algebra over k, and let U be its universal enveloping Lie algebra. Is there a left module M of U which is projective but not free? That is, is the Quillen-Suslin ...
- 3,545
8
votes
1
answer
572
views
Lifting bases for (Z/pZ)^n to Z^n
The following question came up in my research. I suspect that it has a slick answer,
but I can't seem to find it.
Fix an integer n>=2 and a prime p. Define X(n) to be the set of primitive
vectors ...
- 6,589
13
votes
6
answers
3k
views
Gromov-Witten theory and compactifications of the moduli of curves
Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather ...
- 2,706
6
votes
1
answer
187
views
Homotopy type of stabilizers
Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y).
My question is the following: is it ...
- 52.7k
-1
votes
1
answer
338
views
about Function of Random variables [closed]
Hello,
I am studying random variables.
Question is this:
if rv X & a function g is known, what is the pdf of random variable Y = g(x)?
in the textbook answer is explained as follows.
P[y ≤ Y ≤...
- 7,800
9
votes
1
answer
10k
views
What is the difference between a homogeneous stochastic process and a stationary one?
Hello.
I am studying stochastic process.
here,
I don't know what is difference of
"the process is homogeneous"
and
"the process is stationary"
I feel confusing. It seems to similar to me.
- 7,800
3
votes
1
answer
914
views
Range of a Certain Linear Operator
Consider the following hermitian form on the sobolev space H^1(I), of an interval I:
g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I.
Riesz representation ...
- 25.8k
9
votes
2
answers
1k
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Is there a free digraph associated to a graph?
A little bit of background: A graph G is, of course, a set of vertices V(G) and a multiset of edges, which are unordered pairs of (not necessarily distinct) vertices. We say that two vertices v_1, v_2 ...
- 25.2k
7
votes
2
answers
559
views
Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?
Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
- 26.8k
7
votes
3
answers
3k
views
Beilinson-Bernstein and Koszul duality
For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed ...
- 24.1k
9
votes
6
answers
3k
views
Primes are pseudorandom?
I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
- 41.1k
1
vote
1
answer
261
views
Correlation measure between signals of different dimensions?
I have several temporal signals of different dimensions, for example the motion of a point throughout time which would be of dimension 3, and the value of a temperature sensor, of dimension 1.
I ...
- 592
3
votes
3
answers
2k
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singular cohomology of SO(4)
I'm trying to compute the singular cohomology of SO(4), just as practice for using spectral sequences. I got H0=Z, H1=0, H2=Z/2Z, H3=Z⊕Z, H4=0, H5=Z/2Z, and H6=Z. Are these correct? I'm not ...
- 8,167
11
votes
2
answers
1k
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What is the size of the category of finite dimensional F_q vector spaces?
The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j ...
- 2,600
1
vote
2
answers
132
views
Multiplicative stability for integration [closed]
I remember back in undergraduate to ask myself this question :
In the general case, I is an interval,
\int_I fg =! \int_I f \int_I g (*)
But how to describe the egality case, i.e find all couples (f,g)...
- 21
4
votes
1
answer
714
views
How do you rotate a matrix to maximum sparsity?
Given a matrix M, I want to find an orthogonal matrix U that maximizes the number of entries that are zero in the product MU. How do I go about doing this?
- 5,548
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 8,828
11
votes
1
answer
336
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cardinality of final coalgebras in Top
Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. ...
- 27.7k