We characterize clean elements of $ℛ\left(L\right)$ and show that $\alpha \in ℛ\left(L\right)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that ${𝔠}_L\left(\mathrm{coz}(\alpha -\mathbf{1})\right)\subseteq U\subseteq {𝔬}_L\left(\mathrm{coz}\left(\alpha \right)\right)$. Also, we prove that $ℛ\left(L\right)$ is clean if and only if $ℛ\left(L\right)$ has a clean prime ideal. Then, according to the results about $ℛ\left(L\right),$ we immediately get results about ${𝒞}_c\left(L\right).$

We will consider the following problem $$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u\>0\text{in}\Omega ,$$ where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c\>0$ and $\lambda \>0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is novery weak supersolutionif $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p\<p_{+}\left(\lambda \right)$, for $c\>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

Let ${\left(F_n\right)}_{n\in \mathbb{N}}$ be a sequence of non-decreasing functions from $[0,+\infty )$ into $[0,+\infty )$. Under some suitable hypotheses of ${\left(F_n\right)}_{n\in \mathbb{N}}$, we will prove that if $g\in L^p\left({ℝ}^N\right)$, $1<p<+\infty$, satisfies ${lim\; inf}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy<+\infty$, then $g\in W^{1,p}\left({ℝ}^N\right)$ and moreover ${lim}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy=K_{N,p}{\int }_{{ℝ}^N}{\left|\nabla g\left(x\right)\right|}^{p}dx$, where $K_{N,p}$ is a positive constant depending only on $N$ and $p$. This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some...

Let $\varphi$ be a substitution of Pisot type on the alphabet $𝒜=\{1,2,...,d\}$; $\varphi$ satisfies theif for every $i,j\in 𝒜$, there are integers $k,n$ such that ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same $k$-th letter, and the prefixes of length $k-1$ of ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d=2$ and provide a partial result for $d\ge 2$.

We prove that, assuming , if $X$ is a space with ${\aleph }_1$-calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_{\delta }$-diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with ${\aleph }_1$-calibre.

We introduce two entire functions $f_{{\mathrm{A}}_{\frac{1}{2}\infty }}$ and $f_{{\mathrm{D}}_{\frac{1}{2}\infty }}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps ${\mathbf{C}}^2\to \mathbf{C}$ define topologically locally trivial fibrations over $\mathbf{C}\setminus \{0,1\}$. All critical points in the singular fibers over $0$ and $1$ are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed infinite quivers of type ${\mathrm{A}}_{\frac{1}{2}\infty }$ and ${\mathrm{D}}_{\frac{1}{2}\infty }$, respectively. Coxeter elements...

Let $𝒜$ be a commutative algebraic group defined over a number field $k$. We consider the following question:A complete answer for the case of the multiplicative group ${𝔾}_m$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $𝒜$ is either an elliptic curve or a torus of small dimension with respect to $r$. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime. ...

A vertex $k$-coloring of a graph $G$ is a if $M\left(u\right)\ne M\left(v\right)$ for every edge $uv\in E\left(G\right)$, where $M\left(u\right)$ and $M\left(v\right)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the ${\chi }_m\left(G\right)$ of $G$. For an integer $\ell \ge 0$, the $\ell$- of a graph $G$, ${\mathrm{cor}}^{\ell }\left(G\right)$, is the graph obtained from $G$ by adding, for each vertex $v$ in $G$, $\ell$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for $\ell$- of all complete graphs, the regular complete...

Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega \left(𝒟\right)$ be the Burnside ring of $𝒟$, $\Delta \left(𝒟\right)$ be the augmentation ideal of $\Omega \left(𝒟\right)$. Denote by ${\Delta }^n\left(𝒟\right)$ and $Q_n\left(𝒟\right)$ the $n$th power of $\Delta \left(𝒟\right)$ and the $n$th consecutive quotient group ${\Delta }^n\left(𝒟\right)/{\Delta }^{n+1}\left(𝒟\right)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for ${\Delta }^n\left(𝒟\right)$ and determines the isomorphism class of $Q_n\left(𝒟\right)$ for each positive integer $n$.

Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain...

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences...

In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R\left[x\right]$ is $P$-semicommutative if and only if $R[x,x^{-1}]$ is $P$-semicommutative. Also, if $R\left[\right[x\left]\right]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P\left(R\right)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative...

Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\ge 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D:R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D\left(xy\right)=D\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. An additive mapping $F:R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F\left(xy\right)=F\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. We prove...

Random walks in random scenery are processes defined by $Z_n:={\sum }_{k=1}^n{\xi }_{X_1+\cdots +X_k}$, where $(X_k,k\ge 1)$ and $({\xi }_y,y\in {\mathbb{Z}}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb{Z}}^d$ and $ℝ$ respectively. We suppose that the distributions of $X_1$ and ${\xi }_0$ belong to the normal basin of attraction of stable distribution of index $\alpha \in (0,2]$ and $\beta \in (0,2]$. When $d=1$ and $\alpha \ne 1$, a functional limit theorem has been established in ( (1979) 5–25) and a local limit theorem in (To appear). In this paper, we establish the convergence in distribution...