It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the nonexistence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to $14$, the above condition is also sufficient. Lastly, we present an infinite new class of prime polyomino ideals.

In this work, we obtain existence of nontrivial solutions to a minimization problem involving a fractional Hardy–Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda >0$, we analyze the attainability of the optimal constant $${\mu }_{\alpha ,\lambda }(\Omega ):=inf\hspace{0.17em}\left\{\right[u{]}_{s,\Omega }^2+\lambda {\int }_{\Omega }|u{|}^{2}dx:u\in H^s(\Omega ),{\int }_{\Omega }\frac{\left|u\right(x){|}^{2_{s,\alpha }}}{|x{|}^{\alpha }}dx=1\},$$ where $0<s<1$, $n>4s$, $0\le \alpha <2s$, $2_{s,\alpha }=\frac{2(n-\alpha )}{n-2s}$, and $\Omega \subset {\mathbb{R}}^n$ is a bounded domain such that $0\in \Omega$.

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let $\mathrm{pr}\left(n\right)$ denote the largest prime $p$ with $p\#\mid \varphi \left(n\right)$, where $\varphi$ is Euler’s totient function. We show that the normal order of $\mathrm{pr}\left(n\right)$ is $loglogn/logloglogn$; that is, $\mathrm{pr}\left(n\right)\sim loglogn/logloglogn$ as $n\to \infty$ on a set of integers of asymptotic density 1. In fact, we show there is an asymptotic secondary term and, on a tertiary level, there is an asymptotic Poisson distribution. We also show an analogous result for the largest integer $k$ with $k!\mid \varphi \left(n\right)$.

In the first part, we show that a Banach space-valued function $f$ is holomorphic (harmonic) if and only if it is dominated by an $L_{\mathrm{loc}}^1$ function and there exists a separating set $W\subset X\text{'}$ such that $\langle f,x\text{'}\rangle$ is holomorphic (harmonic) for all $x\text{'}\in W$. This improves a known result which requires $f$ to be locally bounded. In the second part, we consider classical results in the $L^p$ theory for elliptic differential operators of second order. In the vector-valued setting, these results are shown to be equivalent to the UMD property.

Let $R$ be a Cohen–Macaulay local ring possessing a canonical module. In this paper, we consider when the maximal ideal of $R$ is self-dual—i.e., it is isomorphic to its canonical dual as an $R$-module. Local rings satisfying this condition are called Teter rings, studied by Teter, Huneke–Vraciu, Ananthnarayan–Avramov–Moore, and others. In the one-dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness.

A portrait $\mathcal{P}$ on ${\mathbb{P}}^N$ is a pair of finite point sets $Y\subseteq X\subset {\mathbb{P}}^N$, a map $Y\to X$, and an assignment of weights to the points in $Y$. We construct a parameter space ${End}_d^N\left[\mathcal{P}\right]$ whose points correspond to degree $d$ endomorphisms $f:{\mathbb{P}}^N\to {\mathbb{P}}^N$ such that $f:Y\to X$ is as specified by a portrait $\mathcal{P}$, and prove the existence of the GIT quotient moduli space ${\mathcal{M}}_d^N\left[\mathcal{P}\right]:={End}_d^N//{SL}_{N+1}$ under the ${SL}_{N+1}$-action $(f,Y,X{)}^{\varphi }=({\varphi }^{-1}\circ f\circ \varphi ,{\varphi }^{-1}\left(Y\right),{\varphi }^{-1}\left(X\right))$ relative to an appropriately chosen line bundle. We also investigate the geometry of ${\mathcal{M}}_d^N\left[\mathcal{P}\right]$ and give two arithmetic applications.