All Questions
153,396
questions
8
votes
1
answer
227
views
commutative "weakly" Frobenius algebras and 2d TQFT
Fix a field $k$. A classic result written up carefully by Abrams in the article "Two-Dimensional Topological Quantum Field Theories and Frobenius Algebras"
says that there is a bijective ...
- 27.8k
3
votes
1
answer
294
views
Local to global deformation of invertible sheaves
Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an ...
- 15.3k
3
votes
1
answer
309
views
Solution to an exponential Diophantine equation
I am trying to solve the following exponential Diophantine equation:
$$ 9^{k_1} -2^{j_1} = 9^{k_2}-2^{j_2}$$
My conjecture is that this implies $k_1=k_2$ and $j_1=j_2$, apart from eventually some ...
- 99.2k
4
votes
0
answers
101
views
Extending a holomorphic map on diffeomorphic affine varieties
Suppose I have two smooth complex affine varieties $X$ and $Y$. Assume that they are each diffeomorphic to $\mathbb{R}^{2n}$ (where $n\geq 3$).
Question: If there exists open dense subsets $U\...
- 8,394
8
votes
1
answer
559
views
About fibrations with fibre Eilenberg-MacLane spaces
Let $f: E\rightarrow B$ be a Kan fibration between pointed connected Kan complexes with fibre the Eilenberg-MacLane space $\mathrm{K}(M, n), n\geq 2, M$ an abelian group. Assume $f$ induces an ...
- 9,246
2
votes
0
answers
63
views
Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
- 603
1
vote
0
answers
812
views
Is dimension invariant under blow-ups?
Let $X'\rightarrow X$ be a blow-up of a finitely dimensional scheme $X$ in a center $D$.
Under which assumptions one has $\dim X'=\dim X$? Do you know a proof or a reference for a proof? Do you know ...
CommunityBot
- 1
1
vote
1
answer
128
views
Number of distinct points in an n-dimensional tetrahedron
Consider an n-dimensional tetrahedron with $n+1$ vertices $\langle v_0, v_1, \dots,v_n\rangle$. $v_0$ is the origin while $v_i$ lies on $e_i$ (the $i^{th}$ coordinate axis) at a distance $D$ from the ...
- 15.6k
1
vote
0
answers
52
views
metric density of a set in the plane with respect to distinct metrics
Let $A \subseteq \mathbb{R}^2$ be Borel (or even open or closed), let $\mathbf{0} = ( 0 , 0)$, and let $\lambda$ be the Lebesgue measure in $\mathbb{R}^2$. Let $
\mathcal{D}^2_A ( \mathbf{0} ) = \lim_{...
-1
votes
1
answer
115
views
probability of m to be a primality radius of n
Disclaimer : this is a crosspost from MSE, as the question got one upvote but no comment or answer whatsoever.
Assuming Goldbach's conjecture, let's denote by $r_{ 0}(n):=\inf\{r\geq 0,(n−r,n+r)\in\...
CommunityBot
- 1
1
vote
1
answer
169
views
On projection theory for inseparable Hilbert spaces
How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
- 3,554
2
votes
1
answer
574
views
Asymptotic formula for absolute difference of number of prime factors between consecutive integers
For $ n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ define $\Omega(n)= \alpha_1+\cdots+\alpha_k$.
What is known about the asymptotic behavior as $N\rightarrow\infty$ for sums of the form
$$\sum_{n=1}^N |\...
- 2,141
3
votes
1
answer
181
views
Vanishing question for self-products in Galois cohomology
Suppose that $k$ is a number field. Let $G$ be the absolute Galois group of $k$ , let $M$ be a torsion $G$-module and $\alpha \in H^{1} (G, M)$. Is it true that
$$\alpha \cup \alpha \cup \ldots \cup \...
- 2,170
1
vote
1
answer
176
views
Can we have a nearily unrestricted class comprehension over predicates that do not mention the class membership symbol
Suppose that $T$ is a consistent first order theory. Now let the language of $T$ be $L_T$.
Question: is it always consistent to add a new primitive constant $D$, and a new primitive binary relation $...
- 9,266
5
votes
1
answer
123
views
Left split subobject in a $2$-category
Let $\mathcal{K}$ be a $2$-category. Keeping in mind the Cat-intuition on $\mathcal{K}$ I say that:
Def : A $1$-cell $A \stackrel{f}{\to} B$ is a left split subobject if $f$ is a right adjoint and ...
- 65.1k
3
votes
1
answer
647
views
Use of weights in the GPY's and Tao-Maynard's work on the twin prime conjecture
I am going through James Maynard's paper, Small Gaps between Primes, and have a number of questions regarding his approach. First, I am wondering why uses weights in his approach. While I generally ...
- 99.2k
2
votes
0
answers
130
views
An isocline geodesic characterization of $2$ dimensional flat metrics
Lets we have a Riemannian metric on an open subset of the plane which satisfies the following local property.
Local Property: For every point $x$ and every foliation by geodesics around $x$, ...
- 312
3
votes
1
answer
861
views
Is Quantum Mechanics (norm)-consistent?
I edited a few small comments to the question in order to make it perhaps more comprehensible.
Today I came across the following question in quantum mechanics.
In Quantum mechanics it is common to ...
1
vote
0
answers
113
views
density of fractal measures
Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
- 11
2
votes
1
answer
85
views
Maximal Vertex Degree of MSTs in Euclidean Spaces
Are there any Euclidean spaces, in which the maximal vertex degree of MSTs (Minimum Spanning Trees) of a finite set of points and edge weights equal to Euclidean distance, isn't equal to the kissing ...
- 1,991
1
vote
0
answers
141
views
The specific elliptic fibration on the Kummer surface of the superspecial abelian surface
Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve
$$
y^2 = x^3 - 1\qquad (y^2 = x^4 - 1)
$$
over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that
$$p
\...
- 1,801
2
votes
2
answers
214
views
Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?
Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $...
- 8,657
6
votes
1
answer
420
views
Cocycle superrigidity
Let $\Gamma$ be a group with a probability measure preserving action on $(X,\mu)$, and $H$ another group. Recall that a cocycle is a map $c:\Gamma\times X\to H$ such that $c(gg',x)=c(g,g'x)c(g',x)$. ...
- 11.4k
7
votes
2
answers
738
views
Which large cardinals have a Matryoshka characterization?
What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how:
As illustrated in the pictures, a Matryoshka set is a self-...
CommunityBot
- 1
9
votes
3
answers
961
views
Number of Geodesic Paths Passing Through a Vertex in an Expander Graph
Let $\{G_{n}\}$ be a sequence of $k$-regular expander graphs. For each $n$ assume that each pair of nodes in the graph is transmitting a unit load of traffic and the traffic goes through the minimum ...
- 1,042
0
votes
0
answers
106
views
Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme
I'm seeking to understand the de Rham cohomology of a Hilbert scheme $K3^{[4]}$ of the K3 surface. By Beauville, this 8-dimensional compact manifold is Kaehler, irreducible, holomorphically symplectic ...
- 35
8
votes
1
answer
289
views
Subspaces isomorphic with quotients
Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?
- 11.3k
6
votes
1
answer
283
views
How are simplicial sets with Quillen model structure a simplicial model category?
I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category.
Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\...
- 19.4k
0
votes
1
answer
156
views
Convergence of an integral with respect to the Wiener measure
Most probably this question should be well studied in the theory of stochastic processes, but I am not educated in that area. Sorry if this question is too elementary.
Let $V\colon \mathbb{R}\to \...
- 5,282
1
vote
0
answers
187
views
Type III factor examples?
How to prove the crossed product of $G$ and von Neumann algebra $M$, where $G$ is locally compact group acting on $M$ via free ergodic action and $M$ is type $II_{\infty}$ factor, is type $III$ factor,...
- 217
12
votes
5
answers
7k
views
Applied mathematics Books (graduate level)
What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?
There is a lot of books on ...
- 4,618
1
vote
2
answers
213
views
vector valued BVP for ODE's
I am dealing with a vector valued second order homogeneous BVP:
$\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$
where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and $...
- 51.6k
1
vote
0
answers
196
views
Growth of Selmer Groups
If $E$ is an elliptic curve over $K$, is there any effective estimate for the discriminant of the extension $L/K$ for which the $p$-part of the
Selmer or Tate-Shafarevich groups become large?
I will ...
- 1,191
3
votes
1
answer
183
views
Reference request: denominators of lonely runner numbers
The lonely runner conjecture states that for $M=\{m_1,...,m_n\}$ a set of distinct positive integers the quantity
$$
\kappa(M):=\sup_{t \in \mathbb{R}} \min_i ||tm_i||
$$
satisfies $\kappa(M) \geq \...
- 1,991
6
votes
3
answers
389
views
Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom
This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze ...
- 65.1k
1
vote
1
answer
214
views
Torus actions on $Sp(n)$-spheres
In this old question of mine
https://math.stackexchange.com/questions/1651906/spheres-as-symplectic-homogeneous-spaces
the presentation of spheres as symplectic group homogeneous spaces was ...
- 30.2k
1
vote
0
answers
184
views
A nested sequence of closed subspaces of $\ell^2$
Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to u$.
Is it possible extract a ...
- 4,618
2
votes
1
answer
97
views
In search of disconnected indecomposable self-injective finite-dimensional algebras
I wanted to know if it is possible to construct an indecomposable self-injective finite-dimensional algebra $\Lambda$ whose Auslander-Reiten quiver $\Gamma_\Lambda$ is not connected. I'd love to see ...
- 33.9k
3
votes
1
answer
236
views
Question on Eilenberg-Watts theorem
I'm not sure if this is a research level question, but:
Let $F:Rep_A \to Rep_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has ...
- 1,661
0
votes
0
answers
139
views
Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.2k
0
votes
1
answer
393
views
Regularity of Laplace equation with Dirichlet data on a part of the boundary
From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system
\begin{align}
-\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\
u &= g &\text{on}\ \...
- 363
4
votes
0
answers
193
views
Canonical differential on K3 surface
On an elliptic curve over $\mathbb{Q}$, we can associate a canonical Neron model and with it a Neron differential, whose coefficients in some natural coordinates yield the Dirichlet coefficients of ...
- 1,964
8
votes
1
answer
441
views
Singularity structure of integrals of rational functions
Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the ...
- 59.8k
5
votes
1
answer
334
views
When is a linear combination of the elementary symmetric polynomials reducible?
Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...
- 49.5k
4
votes
0
answers
133
views
Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
2
votes
0
answers
335
views
Hölder-Zygmund spaces of negative order
In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the ...
- 56.6k
6
votes
2
answers
308
views
Computing the relative class group (with Galois action) of relatively large cyclotomic groups
For a cyclotomic field $K = \mathbb Q(\zeta_n)$, let $K^+$ be its maximal totally real subfield. We know that $H^+ = Cl(K^+)$ injects into $H = Cl(K)$. I am interested in computing the group $H/H^+$ ...
- 5,630
2
votes
0
answers
83
views
p-group of maximal class
I am trying to prove that if $G$ is a $p$-group of maximal class and order $p^4$ ($p$ odd), then its unique two-step centraliser $G_1=C_G([G,G])$ is of the form $C_{p^2}\times C_p$. It is clear from ...
- 405
2
votes
0
answers
63
views
Transfer map between simplicial manifolds
Let $M^m$ and $N^n$ be two triangulated oriented and closed manifolds and $f:M\to N$ a simplicial map. For each $a\in H_p(N)$ we may consider its homological transfer $f_!a\in H_{m-n+p}(M)$.
I want ...
- 1,623
8
votes
0
answers
181
views
Geodesics between boundary points of a hyperbolic space
Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not ...
- 2,508