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### Volume 60, number 1 (2019)

June 2019
 Primary decomposition and secondary representation of modules. Masoumeh Hasanzad, Jafar A'zami In this paper we study the notions of primary decomposition and secondary representation of modules over a commutative ring with identity. Also we review these concepts over injective and projective modules. 1–8 Lie $n$-multiplicative mappings on triangular $n$-matrix rings. Bruno L. M. Ferreira, Henrique Guzzo Jr. We extend to triangular $n$-matrix rings and Lie $n$-multiplicative maps a result about Lie multiplicative maps on triangular algebras due to Xiaofei Qi and Jinchuan Hou. 9–20 Higher order mean curvatures of SAC half-lightlike submanifolds of indefinite almost contact manifolds. Fortuné Massamba, Samuel Ssekajja We introduce higher order mean curvatures of screen almost conformal (SAC) half-lightlike submanifolds of indefinite almost contact manifolds, admitting a semi-symmetric non-metric connection. We use them to generalize some known results by Duggal and Sahin on totally umbilical half-lightlike submanifolds [Int. J. Math. Math. Sci. 2004, no. 68, 3737-3753]. Also, we derive a new integration formula via the divergence of some special vector fields tangent to these submanifolds, which we later use to characterize minimal and maximal submanifolds. Several examples, where possible, are also included to illustrate the main concepts. 21–44 Branching laws: some results and new examples. Oscar Márquez, Sebastián Simondi, Jorge A. Vargas For a connected, noncompact simple matrix Lie group $G$ so that a maximal compact subgroup $K$ has a three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contain the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that an arbitrary square integrable representation of the ambient group is admissible over the semisimple factor of $K$. 45–59 On the structure of split involutive Hom-Lie color algebras. Valiollah Khalili In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra $\mathcal{L}$ is of the form $\mathcal{L}=\mathcal{U}\oplus\sum_{[\alpha]\in\Pi/\sim} I_{[\alpha]}$, with $\mathcal{U}$ a subspace of the involutive abelian subalgebra $\mathcal{H}$ and any $I_{[\alpha]}$, a well-described involutive ideal of $\mathcal{L}$, satisfying $[I_{[\alpha]}, I_{[\beta]}]=0$ if $[\alpha]\neq[\beta]$. Under certain conditions, in the case of $\mathcal{L}$ being of maximal length, the simplicity of the algebra is characterized and it is shown that $\mathcal{L}$ is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras. 61–77 The multivariate bisection algorithm. Manuel López Galván The aim of this paper is to study the bisection method in $\mathbb{R}^n$. We propose a multivariate bisection method supported by the Poincaré–Miranda theorem in order to solve non-linear systems of equations. Given an initial cube satisfying the hypothesis of the Poincaré–Miranda theorem, the algorithm performs congruent refinements through its center by generating a root approximation. Through preconditioning we will prove the local convergence of this new root finder methodology and moreover we will perform a numerical implementation for the two dimensional case. 79–98 Some new $Z$-eigenvalue localization sets for tensors and their applications. Zhengge Huang, Ligong Wang, Zhong Xu , Jingjing Cui In this paper some new $Z$-eigenvalue localization sets for general tensors are established, which are proved to be tighter than those newly derived by Wang et al. [Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 187–198]. Also, some relationships between the $Z$-eigenvalue inclusion sets presented by Wang et al. and the new $Z$-eigenvalue localization sets for tensors are given. Besides, we discuss the effects of orthonormal transformations for the proposed sets. As applications of the proposed sets, some improved upper bounds for the $Z$-spectral radius of weakly symmetric nonnegative tensors are given. Numerical examples are also given to verify the advantages of our proposed results over some known ones. 99–119 Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces. Ramazan Akgün Jackson type direct theorems are considered in variable exponent Lebesgue spaces $L^{p(x)}$ with exponent $p(x)$ satisfying $1\leq \operatorname{ess\,inf}_{x\in [0,2\pi ]}p(x)$, $\operatorname{ess\,sup}_{x\in [0,2\pi]}p(x) < \infty$, and the Dini–Lipschitz condition. Jackson type direct inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages $\mathfrak{Z}_{v}f(\cdot) := \frac{1}{v}\int\nolimits_{0}^{v}f (\cdot+t) \,dt$ in these spaces. We give the main properties of the modulus of smoothness $\Omega_{r}(f,v)_{p(\cdot)} := \left\Vert (\mathbf{I}-\mathfrak{Z}_{v})^{r}f\right\Vert_{p(\cdot)}\quad (r\in \mathbb{N})$ in $L^{p(x)}$, where $\mathbf{I}$ is the identity operator. An equivalence of the modulus of smoothness and Peetre's $K$-functional is established. 121–135 Pricing American put options under stochastic volatility using the Malliavin derivative. Mohamed Kharrat The aim of this paper is to develop a methodology based on Malliavin calculus, in order to price American options under stochastic volatility. This leads to compute the conditional expectation $\mathbb{E}(P_{t}(X_{t}, V_{t})\mid(X_{l},V_{l}))$ for any $0\leq l < t$, where $V_{t}$ is generated by the Cox-Ingersoll-Ross (CIR) process. Some simulations and comparisons are given. 137–147 Trajectorial market models: arbitrage and pricing intervals. Sebastián E. Ferrando, Alfredo L. González, Iván L. Degano, Massoomeh Rahsepar The paper develops general, non-probabilistic market models based on trajectory sets and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of a martingale process as well as a generalization that allows a limited notion of arbitrage in the market while still providing coherent option prices. An illustrative example is described in detail. Several properties of the price bounds are obtained, in particular a connection with risk neutral pricing is established for trajectory markets associated to a continuous-time martingale model. 149–185 Relative modular uniform approximation by means of the power series method with applications. Kuldip Raj, Anu Choudhary We introduce the notion of relative convergence by means of a four dimensional matrix in the sense of the power series method, which includes Abel's as well as Borel's methods, to prove a Korovkin type approximation theorem by using the test functions $\{1,y,z,y^{2}+z^{2}\}$ and a double sequence of positive linear operators defined on modular spaces. We also endeavor to examine some applications related to this new type of approximation. 187–208 Top local cohomology modules over local rings and the weak going-up property. Asghar Farokhi, Alireza Nazari Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $\widehat{R}$ denote the $\mathfrak{m}$-adic completion of $R$. In this paper, we introduce the concept of the weak going-up property for the extension $R\subseteq \widehat{R}$ and we give some characterizations of this property. In particular, we show that this property is equivalent to the strong form of the Lichtenbaum–Hartshorne Vanishing Theorem. Also, when $R$ satisfies the weak going-up property, we show that for a finitely generated $R$-module $M$ of dimension $d$, and ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, we have $\operatorname{Att}_{R}(\operatorname{H}^{d}_{\mathfrak{a}}(M)) = \operatorname{Att}_{R}(\operatorname{H}^{d}_{\mathfrak{b}}(M))$ if and only if $\operatorname{H}^{d}_{\mathfrak{a}}(M)\cong \operatorname{H}^{d}_{\mathfrak{b}}(M)$, and we find a criterion for the cofiniteness of Artinian top local cohomology modules. 209–215 Formal torsors under reductive group schemes. Benedictus Margaux We consider the algebraization problem for torsors over a proper formal scheme under a reductive group scheme. Our results apply to the case of semisimple group schemes (which is addressed in detail). 217–224 Hörmander conditions for vector-valued kernels of singular integrals and their commutators. Andrea L. Gallo, Gonzalo H. Ibañez Firnkorn, María Silvina Riveros We study Coifman type estimates and weighted norm inequalities for singular integral operators and their commutators, given by the convolution with a vector-valued kernel. We define a weaker Hörmander type condition associated with Young functions for the vector-valued kernels. With this general framework we obtain as an example the result for the square operator and its commutator given in [M. Lorente, M. S. Riveros, and A. de la Torre, J. Math. Anal. Appl. 336 (2007), no. 1, 577–592]. 225–245 On the self-conjugateness of differential forms on bounded domains. Ricardo Abreu Blaya, Juan Bory Reyes, Efrén Morales Amaya, José María Sigarreta Almira Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n$ with boundary $\Gamma$ and let $\mathcal{W}$ be a non-homogeneous differential form harmonic in $\Omega$ and Hölder-continuous in $\Omega\cup\Gamma$. In this paper we study and obtain some necessary and sufficient conditions for the self-conjugateness of $\mathcal{W}$ in terms of its boundary value $\mathcal{W}|_\Gamma=\omega$. 247–256 A remark on trans-Sasakian 3-manifolds. Yaning Wang, Wenjie Wang Let $M$ be a trans-Sasakian $3$-manifold of type $(\alpha,\beta)$. In this paper, we give a negative answer to the question proposed by S. Deshmukh [Mediterr. J. Math. 13 (2016), no. 5, 2951–2958], namely we prove that the differential equation $\nabla\beta=\xi(\beta)\xi$ on $M$ does not necessarily imply that $M$ is homothetic to either a Sasakian or cosymplectic manifold even when $M$ is compact. Many examples are constructed to illustrate this result. 257–264 Finite dimensional Hopf algebras over the Kac-Paljutkin algebra $H_8$. Yuxing Shi Let $H_8$ be the Kac–Paljutkin algebra [Trudy Moskov. Mat. Obšč. 15 (1966), 224–261], which is the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra. All simple Yetter–Drinfel'd modules over $H_8$ are given, and finite-dimensional Nichols algebras over $H_8$ are determined completely. It turns out that they are all of diagonal type. In fact, they are of Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$, respectively. By the way, we calculate Gelfand–Kirillov dimensions for some Nichols algebras. As an application, we complete the classification of the finite-dimensional Hopf algebras over $H_8$ according to the lifting method. 265–298

### Volume 60, number 2 (2019)

December 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 A rigidity result for Kählerian manifolds endowed with closed conformal vector fields. Antonio Caminha We show that if a connected compact Kählerian surface $M$ with nonpositive Gaussian curvature is endowed with a closed conformal vector field $\xi$ whose singular points are isolated, then $M$ is isometric to a flat torus and $\xi$ is parallel. We also consider the case of a connected complete Kählerian manifod $M$ of complex dimension $n > 1$ and endowed with a nontrivial closed conformal vector field $\xi$. In this case, it is well known that the singularities of $\xi$ are automatically isolated and the nontrivial leaves of the distribution generated by $\xi$ and $J\xi$ are totally geodesic in $M$. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of $M$ along it is nonpositive, we show that $\xi$ is parallel and $M$ is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete Kählerian manifolds of complex dimension $n-1$. In particular, the universal covering of $M$ is isometric to a Riemannian product having $\mathbb{R}^2$ as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of $\xi$. 469–484 On the mixed integer randomized pattern search algorithm. Ebert Brea We analyze the convergence and performance of a novel direct search algorithm for identifying at least a local minimum of unconstrained mixed integer nonlinear optimization problems. The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA), so-called by the author, is based on a randomized pattern search, which is modified by two main operations for finding at least a local minimum of our problem, namely: moving operation and shrinking operation. The convergence properties of the MIRPSA are here analyzed from a Markov chain viewpoint, which is represented by an infinite countable set of states $\{d(q)\}_{q=0}^{\infty}$, where each state $d(q)$ is defined by a measure of the $q$th randomized pattern search $\mathcal{H}_{q}$, for all $q\in\mathbb{N}$. According to the algorithm, when a moving operation is carried out on a $q$th randomized pattern search $\mathcal{H}_{q}$, the MIRPSA Markov chain holds its state. Meanwhile, if the MIRPSA carries out a shrinking operation on a $q$th randomized pattern search $\mathcal{H}_{q}$, the algorithm will then visit the next $(q+1)$th state. Since the MIRPSA Markov chain never goes back to any visited state, we therefore say that the MIRPSA yields a birth and miscarriage Markov chain. 485–503 Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. Avik De, Tee-How Loo, and Changhwa Woo We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the non-existence of real hypersurfaces in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved. 505–515 Pseudoholomorphic curves in $\mathbb{S}^6$ and $\mathbb{S}^5$. Jost-Hinrich Eschenburg and Theodoros Vlachos The octonionic cross product on $\mathbb{R}^7$ induces a nearly Kähler structure on $\mathbb{S}^6$, the analogue of the Kähler structure of $\mathbb{S}^2$ given by the usual (quaternionic) cross product on $\mathbb{R}^3$. Pseudoholomorphic curves with respect to this structure are the analogue of meromorphic functions. They are (super-)conformal minimal immersions. We reprove a theorem of Hashimoto [Tokyo J. Math. 23 (2000), 137–159] giving an intrinsic characterization of pseudoholomorphic curves in $\mathbb{S}^6$ and (beyond Hashimoto's work) $\mathbb{S}^5$. Instead of the Maurer–Cartan equations we use an embedding theorem into homogeneous spaces (here: $\mathbb{S}^6 = G_2/SU_3$) involving the canonical connection. 517–537 Regularity of maximal functions associated to a critical radius function. Bruno Bongioanni, Adrián Cabral, and Eleonor Harboure In this work we obtain boundedness on BMO and Lipschitz type spaces in the context of a critical radius function. We deal with a local maximal operator and a maximal operator of a one parameter family of operators with certain conditions on its kernels that can be applied to the maximal of the semigroup in the context of a Schrödinger operator. 539–566 New extensions of Cline's formula for generalized Drazin–Riesz inverses. Abdelaziz Tajmouati, Mohammed Karmouni, and M. B. Mohamed Ahmed In this note, Cline's formula for generalized Drazin–Riesz inverses is proved. We prove that if $A, D \in \mathcal{B}(X, Y)$ and $B, C \in \mathcal{B} (Y, X)$ are such that $ACD =DBD$ and $DBA = ACA$, then $AC$ is generalized Drazin–Riesz invertible if and only if $BD$ is generalized Drazin–Riesz invertible, and that, in such a case, if $S$ is a generalized Drazin–Riesz inverse of $AC$ then $T := BS^2D$ is a generalized Drazin–Riesz inverse of $BD$. 567–572 Matrix moment perturbations and the inverse Szegő matrix transformation. Edinson Fuentes and Luis E. Garza Given a perturbation of a matrix measure supported on the unit circle, we analyze the perturbation obtained on the corresponding matrix measure on the real line, when both measures are related through the Szegő matrix transformation. Moreover, using the connection formulas for the corresponding sequences of matrix orthogonal polynomials, we deduce several properties such as relations between the corresponding norms. We illustrate the obtained results with an example. 573–593 Biconservative Lorentz hypersurfaces in $\mathbb{E}_{1}^{n+1}$ with complex eigenvalues. Ram Shankar Gupta and Ahmad Sharfuddin We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal. Also, we provide some examples of such hypersurfaces. 595–610 Functional analytic issues in $\mathbb{Z}_2^n$-geometry. Andrew James Bruce and Norbert Poncin We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fréchet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit. 611–636 Computing convex hulls of trajectories. Daniel Ciripoi, Nidhi Kaihnsa, Andreas Löhne, and Bernd Sturmfels We study the convex hulls of trajectories of polynomial dynamical systems. Such trajectories include real algebraic curves. The boundaries of the resulting convex bodies are stratified into families of faces. We present numerical algorithms for identifying these patches. An implementation based on the software Bensolve Tools is given. This furnishes a key step in computing attainable regions of chemical reaction networks. 637–662