## PID expressed as finite union of subrings

There is a classical theorem that any field can’t be expressed as finite union of subfields.

In addition, there is a counterexample such that an integral domain can be expressed as finite union of subrings.

Therefore, I wonder whether there are any known results about the existence of Principal ideal domain (Or even Euclidean domain) that can be expressed as finite union of subrings?

## Is a family of Cohen-Macaulay modules again Cohen-Macaulay (non-noetherian case)

Let $$A$$ be a local non-noetherian $$\mathbb{C}$$-algebra, $$B$$ a finitely generated, regular $$\mathbb{C}$$-algebra and $$M$$ a finite $$B \otimes_{\mathbb{C}} A$$-module, flat over $$A$$. Suppose that $$M \otimes_A A/m$$ is a Cohen-Macaulay $$B$$-module (i.e., the restriction of $$M$$ to the special fiber is Cohen-Macaulay). Does this imply that $$M$$ is a Cohen-Macaulay module? If I understand correctly, this is true if $$A$$ is noetherian.

## Near-Legendre Conjecture

Ingham has shown that there is a prime between $$n^{3}$$ and $$(n+1)^{3}$$ for large enough $$n.$$

Legendre’s conjecture about the existence of primes between consecutive perfect squares is of course open.

What, if anything, is known about the existence of primes in the intervals $$[n^{2+\epsilon},(n+1)^{2+\epsilon}],$$ for $$n$$ large enough?

## smoothness of Hurwitz spaces with arbitrary ramification profiles

Fix integers $$n\ge 3,d\ge 2$$, and partitions $$\lambda_1,\ldots,\lambda_n$$ of $$d$$. Let $$\mathcal{H}$$ be the moduli space of degree $$d$$ covers $$f:C\to\mathbb{P}^1$$ that have ramification profiles $$\lambda_i$$ over pairwise distinct marked points $$x_i\in\mathbb{P}^1$$, where $$C$$ is smooth and projective and the $$x_i$$ form the set of all branch points of $$f$$. I believe it is true $$\mathcal{H}$$ is étale over $$M_{0,n}$$, and in particular smooth. Can someone point me to a reference for this fact? I know where to look in the case of simple ramification, $$\lambda_i=(2,1,1\ldots,1)$$, for instance https://perso.univ-rennes1.fr/matthieu.romagny/articles/hurwitz_spaces.pdf and the references therein, but not in general.

## Is every indecomposable homogeneous continuum unicoherent?

• Continuum = compact connected metrizable space

• Indecomposable = not the union of any two proper subcontinua.

• Homogeneous = for every two points $$x$$ and $$y$$ there is a homeomorphism of the space onto itself which maps $$x$$ to $$y$$.

• Unicoherent = the intersection of every two subcontinua is connected.

Examples of indecomposable homogeneous continua include solenoids, the pseudoarc, and solenoids of pseudoarcs (a solenoid with each point is blown up into a pseudoarc). I think it’s an open problem to determine whether this list is complete, but so far all of the examples are unicoherent. Is there a theorem stating that such continua must be unicoherent?

## How to solve optimization with matrix pseudo inverse

Thank everybody in advance. I’d like to solve an optimization problem for a matrix function $$f(C)$$. However, the matrix pseudo-inverse constraint gives me big troubles.

Vectors $$V_{n\times 1}$$, $$F_{m\times 1}$$ and matrix $$B_{m\times n}$$ are known. Matrix $$C_{n\times m}$$ is unknown. How to numerically solve: $$\min\limits_{E}f(C)=V^T CF$$ $${\rm subject\,\, to}: BC=I_{m\times m} \ m $$I_{m\times m}$$ is an $$m\times m$$ Identity matrix.

I know this problem can be (sort of, because non-square matrices) formulated into linear matrix inequality (LMI). Similar to: optimization of inverse matrix with constraint on matrix elements. But my processor does not allow such numerical method (LMI) to be used. Plus I do not know how to non-square LMI. So I am looking for a different approach to solve this problem.

I have an ugly gradient descent approach for this problem. But it is complicated and I do not like it. I will not post it for now to limit people’s thoughts. I will post it 4 days after the question.

In addition, I’m not sure if adding following constraint would make the problem easier: $${\rm subject\,\,to}: -I_{n\times n}<(CF)^T I_{n\times n}

Appendix – Formulate problem to (sort of) LMI: (May not be correct, since matrix is not sqaure) Define invertible matrix $$E_{n\times n}$$, $$BC = BEE^{-1}C=(BE)(E^{-1}C)=I \ E^{-1}C = pinv(BE) \ C=E\,\,pinv(BE)$$ Introducing $$E$$ is to represent all possible $$C$$ – parameterize $$C$$. Then the problem $$\min\limits_{E}f(C)=V^T CF$$ is equivalent of finding the $$\min$$ of scalar $$\alpha$$, that below inequality is feasible (i.e. some $$E$$ exists) $$V^T CF < \alpha \ V^T E\,\,pinv(BE) F < \alpha$$ With Schur complement, it is equivalent to following LMI $$\left[\begin{array}{cc} BE & F\ V^T E & \alpha \end{array} \right]<0$$

## The minimum of the reciprocals of some Poisson random variables

Let $$X_1,\dots,X_k$$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $$k$$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}},\dots,\frac{1}{1+X_{k}}\right\}$$ have a strictly positive limit as $$k$$ becomes large?

## Chern number of projection-Can one explain the magic?

I enclosed a computation where the Chern number of the first Landau level is computed (the result claimed is $$-1$$) and the full paper can be found here click me. I have difficulties understanding what happened here.

The projection is given by the integral kernel

$$\Pi(x,y) = \frac{qB}{h} e^{-(qB/4\hbar)(x-y)^2-i(qB/2\hbar)x\wedge y}.$$

The authors compute a “derivative” of this expression and get an integral expression for the Chern character and claim it is equal to $$-1$$. It does not seem like a very sophisticated computation, but it is just very hard to understand so I would like to ask here about this since it is very likely that I am missing something.

## The asymptotics of a vector sequence defined by a recursion relation

The sequence of vectors $$(\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2,\dots)$$ obeys the recursion relation that

$$A\mathbf{v}_j-\mathbf{v}_{j-1}=\sum_{k=0}^j diag(\mathbf{v}_k)B\mathbf{v}_{j-k}$$,

where A and B are given matrix. The first term $$\mathbf{v}_0$$ is also given.

How to compute the asymptotics of the elements in the vectors $$\mathbf{v}_j$$ or the norm of the vectors $$||\mathbf{v}_j||$$ or absolute average $$|u^T\mathbf{v}_j|$$. (From numerical results, I found these three has similar asymptotics.)

For number sequences, I know the method of generating function can solve similar problems like Motzkin numbers. Is there any methods for vector sequence or vector norm?

## Compact Kaehler submanifolds of projectivized Hilbert space

If we take a separable complex Hilbert space $$H$$, its projective space $$PH$$ is an infinite-dimensional Kähler manifold in a fairly obvious sense (see below). Suppose $$M \subset PH$$ is a finite-dimensional compact Kähler submanifold of $$PH$$. Must there be a finite-dimensional linear subspace $$V \subset H$$ such that $$M \subset PV$$?

(I don’t know a good reference for infinite-dimensional Kähler manifolds but the relevant concept here seems clear. $$PH$$ is a smooth Hilbert manifold that can be covered with charts modeled on a complex Hilbert space, with transition functions that are holomorphic (given in a neighborhood of each point by an absolutely convergent power series). This makes each tangent space of $$PH$$ into a complex vector space, and this complex vector space structure extends to a complex Hilbert space structure, which varies smoothly—in fact real-analytically—from point to point. The imaginary part of the inner product gives $$PH$$ a symplectic structure, and the real part gives $$PH$$ a Riemannian structure, both of which are strongly nondegenerate: i.e. they each give an isomorphism $$T_p PH \to T^*_p PH$$ of the underlying real Hilbert spaces.)