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 ├ętale 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 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<n $ $ $ 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}<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 $ $

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.)