# All Questions

**12**

votes

**1**answer

906 views

### L-functions and higher-dimensional Eichler-Shimura relation

From what I have been reading I understand that it is a part of the Langlands program to express Zeta-function of a Shimura variety associated to the group G in terms of L-functions attached to G. I ...

**6**

votes

**2**answers

553 views

### Does anyone recognize this quiver-with-relations?

Below I describe an infinite (but locally finite) quiver with relations. My question is whether anyone recognizes it and can provide appropriate pointers to the literature. I'm mainly interested in ...

**3**

votes

**0**answers

152 views

### coordinate free Euler-Lagrange

The variational approach is to seek critical points in terms
the Euler-Lagrange variational derivatives
$E_a(S_0)$ of a local function $S_0.$ The zero locus does not depend
on coordinates. Where is ...

**2**

votes

**1**answer

189 views

### A question on Lusztig's `graph with automorphism' construction?

Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix ...

**8**

votes

**2**answers

672 views

### Asymptotics for Ramsey Theory

Ramsey Theory says that every sufficently large (but finite) complete graph having $d-$coloured edges contains a monochromatic complete subgraph with $k$ vertices.
One could ask for asymptotics: Let ...

**5**

votes

**0**answers

440 views

### Parabolic cylinder functions - explicit estimates?

I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker).
Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in ...

**4**

votes

**1**answer

174 views

### Generator of a $C_0$-semigroup restricted to a subspace

Suppose we have a decreasing filtration of Banach spaces $(E_t)\_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...

**1**

vote

**2**answers

218 views

### Minimum 1st-neghbors distance between N random points on a ring

We have $N$ points randomly and uniformly distributed on a ring of length 1.
Let $d_i$ be the distance between point $i$ and its first neighbor.
We want to know the expected value of the smallest ...

**2**

votes

**2**answers

348 views

### Principal G-covers with G finite abelian

Let $X$ be a smooth, complex projective variety, and $G$ a finite abelian group. We want to study $G$-principal bundles over $X$, or, in other words, étale $G$-covers over $X$.
Topologically, these ...

**3**

votes

**2**answers

720 views

### Maximum area of intersection between annulus and circle? [closed]

Given two concentric circles $[C_1,C_2 ]$ with radii $(R_1 < R_2) $ creating an annulus; where should a third circle ($C_3$, radius $R_3$) be located such that the area of intersection between ...

**10**

votes

**1**answer

1k views

### When are complex polynomial maps almost surjective?

Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...

**6**

votes

**2**answers

485 views

### Bolyai's construction

Here is Bolyai's construction (in Klein model), which I learned recently from answer of Will Jagy to this question.
It is a Compass-and-straightedge construction of asymptotically parallel line in ...

**5**

votes

**2**answers

755 views

### What is the relationship between the finiteness of the Tate-Shafarevich group and the Tate conjectures?

(I asked this on math-stackexchange, but it seems more appropriate to this forum, so I took it off from there and am posting it here)
After the great answer I got for my previous question about the ...

**2**

votes

**3**answers

399 views

### Why is this graph not generically globally rigid?

Let us assume that we are given a connected, undirected graph. Under the assumption that no three points are collinear, such a graph is uniquely realizable in the plane iff we can certify that it is ...

**1**

vote

**1**answer

165 views

### What can we infer about the size of a complete Boolen algebra, given it is $\kappa$-c.c.?

More specifically, if we only know that a complete Boolean algebra, $\mathbf{B}$, is $\kappa$-c.c., can we give a (reasonably tight) upper bound to the size of $\mathbf{B}$ in terms of $\kappa$?
...

**6**

votes

**2**answers

835 views

### Is there a ruler and compass construction of the common perpendicular of two geodesics in H^3?

Assume we have two geodesics in the Poincaré ball model of $\mathbb{H}^3$, viewed as arcs intersecting the boundary of and contained in the Euclidean unit sphere in $\mathbb{R}^3$. Is there a ruler ...

**1**

vote

**1**answer

377 views

### Integrally closed

Is there any idea to prove that $F + xk[[x]]$ is not integrally closed when the field $k$ is a proper extension of the field $F.$

**3**

votes

**1**answer

194 views

### Decomposability of exterior two-forms

Hello,
The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is ...

**2**

votes

**2**answers

492 views

### Consecutive integers with no large prime factors

I need answer of following Question for my study of an irrational number.
(The raw problem is slightly different.)
Let $k$ be an arbitrary large positive integer, and let $A$ is a positive integer ...

**7**

votes

**1**answer

284 views

### Clarifying the link between deformation/rigidity and dual cocycles

Suppose that a type $II_{1}$ factor $M$ decomposes in two ways as a group von Neumann algebra, e.g. as $L\Gamma$ and as $L\Lambda$. The decomposition $L\Gamma$ gives rise to a comultiplication ...

**4**

votes

**2**answers

2k views

### Is there some algorithms for solving non-linear matrix equations?

Is there some algorithms for solving non-linear matrix equations on field $\mathbb{C}$?
Especially, solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix satisfying
...

**7**

votes

**1**answer

337 views

### Labeling a Square Array

Suppose that the $n^2$ cells of an $n\times n$ array are labeled with the integers $1, \dots, n^2$. Under the traditional left-to-right and top-to-bottom labeling, the labels of horizontally adjacent ...

**1**

vote

**1**answer

221 views

### On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...

**7**

votes

**2**answers

566 views

### Intuition behind the age grading in quantum cohomology of orbifolds

Let $\mathscr{X}$ be a smooth DM-stack with projective coarse moduli space. I am interested in the orbifold cohomology ring $H^\mathrm{orb}(\mathscr{X})$, as defined by Chen-Ruan (for orbifolds) and ...

**10**

votes

**2**answers

926 views

### Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...

**2**

votes

**0**answers

104 views

### classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...

**0**

votes

**1**answer

681 views

### Interplay between Riemann and Swinnerton-Dyer

Hello everyone,
After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes ...

**0**

votes

**0**answers

142 views

### Affine varieties as Stein surfaces

I have a somewhat general and vague question in mind. Is there anything in literature related to Affine varieties as examples of Stein manifolds? I know that there is a topological approach to Stein ...

**1**

vote

**1**answer

125 views

### Union of Associated Primes.

Let $R$ be a Noetherian ring. Let $I=(x_1,x_2,...,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n,x_2^n,...,x_t^n)$. Are there any results about finiteness of $\cup_n Ass_R(I^n/I_n)$?
More ...

**12**

votes

**3**answers

1k views

### What are the endofunctors on the simplex category?

Is there a 'classification' of the endofunctors F: Δ --> Δ where Δ denotes the simplex category with objects [n] and the weakly monotone maps [m] --> [n] as morphisms (Actually, I ...

**9**

votes

**0**answers

303 views

### Geometrizing the Third Cohomology of a Complex Lie Group

If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural ...

**0**

votes

**0**answers

205 views

### functional equation, how to solve

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
$$F(x, y) = \frac{x\circ Ay}{x^TAy}$$
$$G(x, y) = \frac{x\circ By}{x^TBy}$$
where $A$ and $B$ are ...

**2**

votes

**2**answers

267 views

### springer resolution over $\wedge^3 \mathbb{C}^6$

The action of $GL_6$ on $P(\wedge^3 \mathbb{C}^6)=P^{19}$ has 4 orbits (of dim 19, 18, 14, 9). Can you describe how the springer resolution applies to each of these orbits? It should have positive ...

**4**

votes

**1**answer

470 views

### Infinite monkeys computing … triangle area?

I wonder if it is possible to specialize the question:
(a) What is the probability that a random Turing Machine program
will halt?, to: (b) What is the probability that a random Turing Machine
...

**2**

votes

**0**answers

160 views

### Enumerating certain solutions to the equation XAX=B in the Symmetric Group

I'm interested in understanding how to enumerate a certain subset of the solutions to the equation $XAX=B$ in the symmetric group $\Sigma_{n}$. This is related to a topological problem- counting a ...

**1**

vote

**1**answer

490 views

### Consistency and inaccessible cardinals [closed]

I just want to make sure I understand certain things well. I believe my questions are quite simple. Are the following statments true ?
1/ One cannot prove Con(ZFC) in ZFC. However ZFC might not be ...

**4**

votes

**4**answers

668 views

### cuspidal types and Iwahori subgroup for $SL(2,F)$

Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.
Is there any possibility that $J\subset I$ or even a subgroup?

**6**

votes

**3**answers

589 views

### Does every compact Hausdorff ring admit a decomposition into primitive idempotents?

Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in ...

**5**

votes

**3**answers

680 views

### Are there primes p, q such that p^4+1 = 2q^2 ?

$\exists p, q \in \mathbb{P}: p^4+1 = 2q^2$? I suspect there is some simple proof that no such p, q can exist, but I haven't been able to find one.
Solving the Pell equation gives candidates for ...

**25**

votes

**1**answer

3k views

### Torsors for finite group schemes

Let $k$ be a field of characteristic $p > 0$ (assume $k$ is perfect if it helps). Let $G$ be a connected finite group scheme of height one over $k$. Then $G$ is determined by its Lie algebra ...

**9**

votes

**1**answer

411 views

### Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively

I'm interested in a complexity question related to problems like the slice-ribbon problem.
To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...

**7**

votes

**3**answers

370 views

### Group Extensions and Line Bundles on $BG$

I am sure the answer to this question is well-known, but
It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a ...

**7**

votes

**2**answers

757 views

### algebraic proof of Atiyah-Bott fixed point formula?

Hi,
Atiyah and Bott apparently proved the following theorem:
Let $X$ be a smooth projective complex variety and $L$ a line bundle on $X$.
Let $f:X\to X$ be an automorphism of $(X,L)$ with finitely ...

**6**

votes

**2**answers

300 views

### Are Vitali-type nonmeasurable sets determinate?

Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from ...

**17**

votes

**3**answers

2k views

### Origin of the term “localization” for the localization of a ring

I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source ...

**1**

vote

**1**answer

163 views

### Algebraicity of the canonical projection $X(\Gamma)\to X(1)$ and of $X(\Gamma)$

Let $\Gamma\subset {SL}_2(\mathbb Z)$ be a congruence subgroup, and let $X(\Gamma)$ be the associated compact modular curve. The inclusion $\Gamma \subset SL_2(\mathbb Z)$ induces a canonical ...

**14**

votes

**2**answers

1k views

### Does “Algebraic numbers coloured by degree” form a fractal?

This picture from Wikipedia's article on Algebraic numbers shows a visualization of Algebraic numbers coloured by degree.
I'm wondering if this is a fractal?

**18**

votes

**0**answers

758 views

### Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...

**2**

votes

**1**answer

188 views

### cohomology of restrictions of vector bundles to deformations

Suppose $X \subset Y$ is a pair of varieties, and $s \in H^0(N_{X/Y})$ is a section. This corresponds to a first-order deformation $X' \subset Y \times \text{Spec}(\mathbb{C}[\epsilon]/\epsilon^2)$ of ...

**5**

votes

**3**answers

522 views

### Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields

Let $k$ be a finite field, and $A$, $B$ abelian varieties over $k$. Let $T_p(A)$, $T_p(B)$ be
the (contravariant) Dieudonn\'e modules associated to the p-divisible groups attached to $A$ and $B$, ...