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Articles are posted here individually soon after proof is returned from authors, before the corresponding journal issue is completed. All articles are in their final form, including issue number and pagination. For recently accepted articles, see Articles in press.

Vol. 60, no. 2 (2019)

 Linear Poisson structures and Hom-Lie algebroids. Esmaeil Peyghan, Amir Baghban, and Esa Sharahi Considering Hom-Lie algebroids in some special cases, we obtain some results of Lie algebroids for Hom-Lie algebroids. In particular, we introduce the local splitting theorem for Hom-Lie algebroids. Moreover, linear Hom-Poisson structure on the dual Hom-bundle will be introduced and a one-to-one correspondence between Hom-Poisson structures and Hom-Lie algebroids will be presented. Also, we introduce Hamiltonian vector fields by using linear Poisson structures and show that there exists a relation between these vector fields and the anchor map of a Hom-Lie algebroid. 299-313 On supersolvable groups whose maximal subgroups of the Sylow subgroups are subnormal. Pengfei Guo, Xingqiang Xiu, and Guangjun Xu A finite group $G$ is called an MSN$^{*}$-group if it is supersolvable, and all maximal subgroups of the Sylow subgroups of $G$ are subnormal in $G$. A group $G$ is called a minimal non-MSN$^{*}$-group if every proper subgroup of $G$ is an MSN$^{*}$-group but $G$ itself is not. In this paper, we obtain a complete classification of minimal non-MSN$^{*}$-groups. 315-322 Periodic solutions of Euler–Lagrange equations in an anisotropic Orlicz–Sobolev space setting. Fernando D. Mazzone and Sonia Acinas We consider the problem of finding periodic solutions of certain Euler–Lagrange equations which include, among others, equations involving the $p$-Laplace operator and, more generally, the $(p,q)$-Laplace operator. We employ the direct method of the calculus of variations in the framework of anisotropic Orlicz–Sobolev spaces. These spaces appear to be useful in formulating a unified theory of existence of solutions for such a problem. 323-341 Classification of left invariant Hermitian structures on 4-dimensional non-compact rank one symmetric spaces. Srdjan Vukmirović, Marijana Babić, and Andrijana Dekić The only 4-dimensional non-compact rank one symmetric spaces are $\mathbb{C}H^2$ and $\mathbb{R}H^4$. By the classical results of Heintze, one can model these spaces by real solvable Lie groups with left invariant metrics. In this paper we classify all possible left invariant Hermitian structures on these Lie groups, i.e., left invariant Riemannian metrics and the corresponding Hermitian complex structures. We show that each metric from the classification on $\mathbb{C}H^2$ admits at least four Hermitian complex structures. One class of metrics on $\mathbb{C}H^2$ and all the metrics on $\mathbb{R}H^4$ admit 2-spheres of Hermitian complex structures. The standard metric of $\mathbb{C}H^2$ is the only Einstein metric from the classification, and also the only metric that admits Kähler structure, while on $\mathbb{R}H^4$ all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy. 343-358 Almost additive-quadratic-cubic mappings in modular spaces. Mohammad Maghsoudi, Abasalt Bodaghi, Abolfazl Niazi Motlagh, and Majid Karami We introduce and obtain the general solution of a class of generalized mixed additive, quadratic and cubic functional equations. We investigate the stability of such modified functional equations in the modular space $X_\rho$ by applying the $\Delta_2$-condition and the Fatou property (in some results) on the modular function $\rho$. Furthermore, a counterexample for the even case (quadratic mapping) is presented. 359-379 SUSY $N$-supergroups and their real forms. Rita Fioresi and Stephen D. Kwok We study SUSY $N$-supergroups, $N=1,2$, their classification and explicit realization, together with their real forms. In the end, we give the supergroup of SUSY preserving automorphisms of $\mathbb{C}^{1|1}$ and we identify it with a subsupergroup of the SUSY preserving automorphisms of $\mathbb{P}^{1|1}$. 381-390 Eta-Ricci solitons on LP-Sasakian manifolds. Pradip Majhi and Debabrata Kar We consider $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds with Codazzi type of the Ricci tensor. Then we study $\eta$-Ricci solitons on $\varphi$-conformally semi-symmetric, $\varphi$-Ricci symmetric, and conformally Ricci semi-symmetric Lorentzian para-Sasakian manifolds. Finally, we construct an example of a three dimensional Lorentzian para-Sasakian manifold which admits $\eta$-Ricci solitons with non-constant scalar curvature. 391-405 Factorization of Frieze patterns. Moritz Weber and Mang Zhao In 2017, Michael Cuntz gave a definition of reducibility of a quiddity cycle of a frieze pattern: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the negative by giving counterexamples. We conclude that even under stronger assumptions, there is no canonical decomposition. 407-415 Conformal and Killing vector fields on real submanifolds of the canonical complex space form $\mathbb{C}^{m}$. Hanan Alohali, Haila Alodan, and Sharief Deshmukh In this paper, we find a conformal vector field as well as a Killing vector field on a compact real submanifold of the canonical complex space form $\left(\mathbb{C}^{m},J,\left\langle\,,\right\rangle\right)$. In particular, using immersion $\psi :M\rightarrow\mathbb{C}^{m}$ of a compact real submanifold $M$ and the complex structure $J$ of the canonical complex space form $\left(\mathbb{C}^{m},J,\left\langle\,,\right\rangle\right)$, we find conditions under which the tangential component of $J\psi$ is a conformal vector field as well as conditions under which it is a Killing vector field. Finally, we obtain a characterization of $n$-spheres in the canonical complex space form $\left(\mathbb{C}^{m},J,\left\langle\,,\right\rangle\right)$. 417-430 On $k$-circulant matrices involving the Jacobsthal numbers. Biljana Radičić Let $k$ be a nonzero complex number. We consider a $k$-circulant matrix whose first row is $(J_{1},J_{2},\dots,J_{n})$, where $J_{n}$ is the $n^\text{th}$ Jacobsthal number, and obtain the formulae for the eigenvalues of such matrix improving the formula which can be obtained from the result of Y. Yazlik and N. Taskara [J. Inequal. Appl. 2013, 2013:394, Theorem 7]. The obtained formulae for the eigenvalues of a $k$-circulant matrix involving the Jacobsthal numbers show that the result of Z. Jiang, J. Li, and N. Shen [WSEAS Trans. Math. 12 (2013), no. 3, 341–351, Theorem 10] is not always applicable. The Euclidean norm of such matrix is determined. We also consider a $k$-circulant matrix whose first row is $(J_{1}^{-1},J_{2}^{-1},\dots,J_{n}^{-1})$ and obtain the upper and lower bounds for its spectral norm. 431-442 Existence and uniqueness of solutions to distributional differential equations involving Henstock–Kurzweil–Stieltjes integrals. Guoju Ye, Mingxia Zhang, Wei Liu, and Dafang Zhao This paper is concerned with the existence and uniqueness of solutions to the second order distributional differential equation with Neumann boundary value problem via Henstock–Kurzweil–Stieltjes integrals.The existence of solutions is derived from Schauder's fixed point theorem, and the uniqueness of solutions is established by Banach's contraction principle. Finally, two examples are given to demonstrate the main results. 443-458 Conformal semi-invariant Riemannian maps to Kähler manifolds. Mehmet Akif Akyol and Bayram Şahin As a generalization of CR-submanifolds and semi-invariant Riemannian maps, we introduce conformal semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations, and obtain decomposition theorems by using the existence of conformal Riemannian maps. We also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic. 459-468