We construct a spectral, non-tiling set of size $2p$ in $\mathbb Z/p\mathbb Z{4}$, $p$ odd prime. This example complements a previous counterexample in [C. Aten et al., \textit{Tiling sets and spectral sets over finite fields}, arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$. On the contrary we show that the conjecture does hold in $(\mathbb Z/2\mathbb Z)^4$.

In this paper, we prove the nonexistence of $L^{2p}_{f}$-harmonic $1$-forms on a complete noncompact $f$-minimal hypersurface in a weighted manifold with nonpositive sectional curvature. This results can be viewed as an extension of Yun and Seo's results on $L^{2}_{f}$-harmonic $1$-forms.

Let $R$ be an integral domain. Then $R$ is said to be a $\lambda$-domain if the set of all overrings of $R$ is linearly ordered by inclusion. If $R_1 + R_2$ is an overring of $R$ for each pair of overrings $R_1, R_2$ of $R$, then $R$ is said to be a $\Delta$-domain. We show that if $R\subset T$ is an extension of integral domains such that each proper subring of $T$ containing $R$ is a $\lambda$-domain (resp., $\Delta$-domain), then $T$ is a $\lambda$-domain (resp., $\Delta$-domain under some conditions). Moreover, the pair $(R, T)$ is a residually algebraic pair. Two new ring theoretic properties, namely $\lambda$-property of domains and $\Delta$-property of domains are introduced and studied.

In the class of torsion-free abelian groups of finite rank, Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups is reduced to indecomposable groups. Torsion-free abelian groups $A$ of rank two such that every associative ring on $A$ is commutative but there exists a~non-commutative ring with the additive group $A$, are classified. Moreover, several valuable results concerning rings on torsion-free abelian groups of rank two achieved by Beaumont, Wisner, Jackett, Aghdam and Najafizadeh are complemented and their proof are greatly simplified.$

Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. Let $t\in\Bbb{N}_0$ be an integer, $M$ a finitely generated $R$-module and $X$ be an $R$-module such that $\Ext^i_R(R/I,X)$ is finitely generated (resp. minimax, weakly Laskerian) for all $i\leq t+1$. We prove that if $H^{i}_{I}(M, X)$ is ${\rm FD_{\leq 1}}$ for all $i<t$, then the $R$-modules $H^{i}_{I}(M,X)$ are $I$-cofinite (resp. $I$-cominimax, $I$-weakly cofinite) for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $H^{t}_{I}(M,X)$, the $R$-module $\Ext^i_R(R/I,H^{t}_{I}(M,X)/N)$ is finitely generated (resp. minimax, weakly Laskerian) for $i=0,1$. In particular the set $Ass_R(H^{t}_{I}(M,X)/N)$ is a finite set.

We give an analytic characterization of nef and good line bundles on smooth projective varieties. This supplements F. Russo's paper ``A characterization of nef and good divisors by asymptotic multiplier ideals'' (Bull. Belg. Math. Soc. Simon Stevin 2009) and extends it in the analytic direction of singular hermitian metrics.

In this paper, we consider a generalization of well-posedness for a class of split multi-valued variational inequalities and establish some metric characterizations for them. Moreover, we show that the existence and uniqueness of solution for a split multi-valued variational inequality is equivalent to the strong well-posedness.

Let $R$ be a commutative Noetherian ring with non-zero identity and $I$ be an ideal of $R$. Let $M$ and $N$ be two finitely generated $R$-modules. In this paper it is shown that, if ${\rm cd}(I,R)\leq 1$ and ${\rm pd_{R}}(M)<\infty$, then $H_{I}^{i}(M,N)$ are $I$-cofinite for each $i\geq 0$. Moreover, it is shown that if $\mbox{q}(I,R) \leq 1$ and ${\rm pd_{R}}(M)<\infty$, then the Bass numbers of generalized local cohomology modules $H_{I}^{i}(M,N)$ are finite for each $i\geq 0$. We also characterize the greatest integer $i$ such that $H_{I}^{i}(M,N)$ is not Artinian and $I$-cofinite.

We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their duals are studied in detail and compared to other approaches. As application we discuss the convergence of the complete growth function introduced by Grigorchuk and Nagnibeda.

We investigate the complex symmetric structure of the weighted composition operator $W_{\psi,\phi}$ on the subspace $\mathcal{H}_{\alpha, \beta}^2(\mathbb{D})$ of the Hardy space $\mathcal{H}^2(\mathbb{D})$ of codimension one. We provide characterizations of symbols $\phi$ and $\psi$ such that $W_{\psi,\phi}$ is complex symmetric with respect to a special conjugation. In addition, we discuss isometric properties of the complex symmetric operator $W_{\psi,\phi}$ on $\mathcal{H}_{\alpha, \beta}^2(\mathbb{D})$.

We study multiorthomorphisms on Riesz spaces. As an application, we introduce and we describe a new class of order bounded orthogonally additive homogeneous polynomials.

In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces, which generalizes the previous works done by Sharma et al.