Serdica Mathematical Journal

Volume 48, Number 3, 2022

C O N T E N T S

A B S T R A C T S

ON THE DIAGONAL ACTIONS OF \(\mathbb{Z}\) ON \(\mathbb{Z}^n\)
Jordan Sahattchieve jantonov@umich.edu

2020 Mathematics Subject Classification: Primary 20F65; Secondary 20F16, 20F19.
Key words: Bounded packing, polycyclic groups, coset growth.

In this paper, I study the actions of \(\mathbb{Z}\) on \(\mathbb{Z}^n\) from a dynamical perspective. The motivation for this study comes from the notion of bounded packing introduced by Hruska and Wise in 2009. I shall also introduce the notion of coset growth for a finitely generated group. My analysis yields the following two results: bounded packing in certain semidirect products of \(\mathbb{Z}^n\) with \(\mathbb{Z}\) and a bound of the coset growth of the copy of \(\mathbb{Z}\) on the right in \(\mathbb{Z}^2 \rtimes \mathbb{Z}\) for the non-nilpotent groups of this type.

TOPICS ON REAL AND COMPLEX CONVEXITY
Jamel Abidi abidijamel1@gmail.com

2020 Mathematics Subject Classification: Primary 32A10, 32A60, 32U05, 32U15; Secondary 32W50.
Key words: Holomorphic, convex, plurisubharmonic functions, harmonic, holomorphic partial differential equation, complex structure, inequalities, strictly, maximum principle.

We investigate the study of convex, strictly plurisubharmonic and the special class consisted of convex and strictly plurisubharmonic functions in convex domains of \(\mathbb{C}^n\), \(n\geq1\).
Let \(h:\mathbb{C}^n\rightarrow\mathbb{C}\) be pluriharmonic. We prove that \(\{b\in\mathbb{C}\;/|h+b|\;\textrm{is a convex function on}\; \mathbb{C}^n\}=\emptyset\), or \(\{\alpha\}\), or \(\mathbb{C}\), where \(\alpha\in\mathbb{C}\).
Now let \(\varphi_1, \varphi_2, \varphi_3:D\rightarrow\mathbb{C}\) be three holomorphic functions, \(D\) is a domain of \(\mathbb{C}^n\). Put \(u(z,w)=| w-\overline{\varphi_1}(z)|| w-\overline{\varphi_2}(z)| | w-\overline{\varphi_3}(z)|\), for \((z,w)\in D\times\mathbb{C}\). We prove that \(u\) is psh on \(D\times\mathbb{C}\) if and only if \((\varphi_1+\varphi_2+\varphi_3)\) and \((\varphi_1\varphi_2+\varphi_1\varphi_3+ \varphi_2\varphi_3)\) are constant on \(D\), or \((\varphi_1+\varphi_2+\varphi_3)\) is non constant and \(\varphi_1=\varphi_2=\varphi_3\) on \(D\).