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### Volume 61, number 1 (2020)

June 2020
 Local solvability of elliptic equations of even order with Hölder coefficients. María Amelia Muschietti and Federico Tournier We consider elliptic equations of order $2m$ with Hölder coefficients. We show local solvability of the Dirichlet problem with $m$ conditions on the boundary of the upper half space. First we consider local solvability in free space and then we treat the boundary case. Our method is based on applying the operator to an approximate solution and iterating in the Hölder spaces. A priori estimates for the approximate solution is the essential part of the paper. 1–47 On convergence of subspaces generated by dilations of polynomials. An application to best local approximation. Fabián E. Levis and Claudia V. Ridolfi We study the convergence of a net of subspaces generated by dilations of polynomials in a finite dimensional subspace. As a consequence, we extend the results given by Zó and Cuenya [Advanced Courses of Mathematical Analysis II (Granada, 2004), 193–213, World Scientific, 2007] on a general approach to the problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials. 49–62 Perturbation of Ruelle resonances and Faure–Sjöstrand anisotropic space. Yannick Guedes Bonthonneau Given an Anosov vector field $X_0$, all sufficiently close vector fields are also of Anosov type. In this note, we check that the anisotropic spaces described by Faure and Sjöstrand and by Dyatlov and Zworski can be chosen adapted to any smooth vector field sufficiently close to $X_0$ in $C^1$ norm. 63–72 Generalized metallic structures. Adara M. Blaga and Antonella Nannicini We study the properties of a generalized metallic, a generalized product and a generalized complex structure induced on the generalized tangent bundle of a smooth manifold $M$ by a metallic Riemannian structure $(J,g)$ on $M$, providing conditions for their integrability with respect to a suitable connection. Moreover, by using methods of generalized geometry, we lift $(J,g)$ to metallic Riemannian structures on the tangent and cotangent bundles of $M$, underlying the relations between them. 73–86 A heat conduction problem with sources depending on the average of the heat flux on the boundary. Mahdi Boukrouche and Domingo A. Tarzia Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\mathbb{R}^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0, y, s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of the second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method. 87–101 On classes of finite rings. Aleksandr Tsarev A class of rings is a formation whenever it contains all homomorphic images of its members and if it is subdirect product closed. In the present paper, it is shown that the lattice of all formations of finite rings is algebraic and modular. Let $R$ be a finite commutative ring with an identity element. It is established that there is a one-to-one correspondence between the set of all invariant fuzzy prime ideals of $R$ and the set of all fuzzy prime ideals of each ring of the formation generated by $R$. 103–111 A canonical distribution on isoparametric submanifolds I. Cristián U. Sánchez We show that on every compact, connected homogeneous isoparametric submanifold $M$ of codimension $h\geq 2$ in a Euclidean space, there exists a canonical distribution which is bracket generating of step 2. An interesting consequence of this fact is also indicated. In this first part we consider only the case in which the system of restricted roots is reduced, reserving for a second part the case of non-reduced restricted roots. 113–130 Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential operators. Imed Basdouri, Sarra Hammami, and Olfa Messaoud We consider the $\mathfrak{sl}(2)$-module structure on the spaces of bilinear bidifferential operators acting on the spaces of weighted densities. We compute the second cohomology group of the Lie algebra $\mathfrak{sl}(2)$ with coefficients in the space of bilinear bidifferential operators that act on tensor densities $\mathcal{D}_{\lambda,\nu,\mu}$. 131–143 Kreĭn space unitary dilations of Hilbert space holomorphic semigroups. Stefania A. M. Marcantognini The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that $B_\beta:=A-\beta$ is a sectorial operator of angle less than $\frac{\pi}{2}$ for some $\beta \geq 0$. If $B_\beta$ is dissipative in some equivalent scalar product then the Naimark–Arocena representation theorem is applied to obtain a Kreĭn space unitary dilation of the semigroup. 145–160 Lifting vector fields from manifolds to $r$-jet prolongation of the tangent bundle. Jan Kurek and Włodzimierz M. Mikulski If $m\geq 3$ and $r\geq 0$, we deduce that any natural linear operator lifting vector fields from an $m$-manifold $M$ to the $r$-jet prolongation $J^{r}TM$ of the tangent bundle $TM$ is the composition of the flow lifting $\mathcal{J}^r$ corresponding to the $r$-jet prolongation functor $J^r$ with a natural linear operator lifting vector fields from $M$ to $TM$. If $0\leq s\leq r$ and $m\geq 3$, we find all natural linear operators transforming vector fields on $M$ into base-preserving fibred maps $J^{r}TM\to J^{s}TM$. 161–168 Localization operators and scalogram associated with the generalized continuous wavelet transform on $\mathbb{R}^d$ for the Heckman–Opdam theory. Hatem Mejjaoli and Khalifa Trimèche We consider the generalized wavelet transform $\Phi_{h}^{W}$ on $\mathbb{R}^{d}$ for the Heckman–Opdam theory. We study the localization operators associated with $\Phi_{h}^{W}$; in particular, we prove that they are in the Schatten–von Neumann class. Next we introduce some results on the scalogram for this transform. 169–196

### Volume 61, number 2 (2020)

December 2020
 $\mathfrak{D}^\perp$-invariant real hypersurfaces in complex Grassmannians of rank two. Ruenn-Huah Lee and Tee-How Loo Let $M$ be a real hypersurface in a complex Grassmannian of rank two. Denote by $\mathfrak{J}$ the quaternionic Kähler structure of the ambient space, $TM^\perp$ the normal bundle over $M$, and $\mathfrak{D}^\perp=\mathfrak{J}TM^\perp$. The real hypersurface $M$ is said to be $\mathfrak{D}^\perp$-invariant if $\mathfrak{D}^\perp$ is invariant under the shape operator of $M$. We show that if $M$ is $\mathfrak{D}^\perp$-invariant, then $M$ is Hopf. This improves the results of Berndt and Suh [Int. J. Math. 23 (2012) 1250103] and [Monatsh. Math. 127 (1999), 1–14]. We also classify $\mathfrak{D}^\perp$ real hypersurfaces in complex Grassmannians of rank two of noncompact type with constant principal curvatures. 197–207 Drazin inverse of singular adjacency matrices of directed weighted cycles. Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Adrián Pastine We present a necessary and sufficient condition for the singularity of circulant matrices associated with directed weighted cycles. This condition is simple and independent of the order of matrices from a complexity point of view. We give explicit and simple formulas for the Drazin inverse of these circulant matrices. We also provide a Bjerhammar-type condition for the Drazin inverse. 209–227 On the atomic and molecular decomposition of weighted Hardy spaces. Pablo Rocha The purpose of this article is to give another molecular decomposition for members of weighted Hardy spaces, different from that given by Lee and Lin [J. Funct. Anal. 188 (2002), no. 2, 442–460], and to review some overlooked details. As an application of this decomposition, we obtain the boundedness on $H^{p}_{w}(\mathbb{R}^{n})$ of every bounded linear operator on some $L^{p_0}(\mathbb{R}^n)$ with $1 < p_0 < +\infty$, for all weights $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ if $1< \frac{r_w -1}{r_w} p_0$, or all $0 < p < \frac{r_w -1}{r_w} p_0$ if $\frac{r_w -1}{r_w} p_0 \leq 1$, where $r_w$ is the critical index of $w$ for the reverse Hölder condition. In particular, the well-known results about boundedness of singular integrals from $H^{p}_w(\mathbb{R}^{n})$ into $L^{p}_{w}(\mathbb{R}^{n})$ and on $H^{p}_{w}(\mathbb{R}^{n})$ for all $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ are established. We also obtain the $H^{p}_{w^{p}}(\mathbb{R}^{n})$-$H^{q}_{w^{q}}(\mathbb{R}^{n})$ boundedness of the Riesz potential $I_{\alpha}$ for $0 < p \leq 1$, $\frac{1}{q}=\frac{1}{p} - \frac{\alpha}{n}$, and certain weights $w$. 229–247 On some fixed point theorems in abstract duality pairs. Kinga Cichoń, Mieczysław Cichoń, and Mohamed M. A. Metwali We study the existence of fixed points for some functional problems. We focus on bilinear operators which are extensions of the product of operators (Banaś–Dhage-type results) and obtain some results on abstract duality pairs (abstract triples). However, we do not restrict ourselves to these cases; in fact, some special cases which do no fit in the above description are indicated. In particular, our approach allows us to study some differential or integral problems, for which the operators are not acting on a fixed Banach space. 249–266 On Jacobson's lemma and Cline's formula for Drazin inverses. Dijana Mosić Under new conditions $bac=bdb$ and $cdb=cac$, we present extensions of Jacobson's lemma and Cline's formula for the generalized Drazin inverse and pseudo Drazin inverse in a ring. Applying these results, we give Jacobson's lemma for the Drazin inverse, group inverse, and ordinary inverse, and Cline's formula for the Drazin inverse. 267–276 Reflexivity of rings via nilpotent elements. Abdullah Harmanci, Handan Kose, Yosum Kurtulmaz, and Burcu Ungor An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in \operatorname{nil}(R)$ and $b\in R$, $aRb \subseteq I$ implies $bRa \subseteq I$, where $\operatorname{nil}(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive rings, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal $I$ of a ring $R$, $R/I$ is left N-reflexive. If an ideal $I$ of a ring $R$ is reduced as a ring without identity and $R/I$ is left N-reflexive, then $R$ is left N-reflexive. If $R$ is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in $R[x]$ are nilpotent in $R$, it is proved that $R$ is left N-reflexive if and only if $R[x]$ is left N-reflexive. We show that the concept of left N-reflexivity is weaker than that of reflexivity and stronger than that of right idempotent reflexivity. 277–290 $p$-Dual frames and $p$-Riesz sequences in quasinormed spaces. José Alfonso López Nicolás The present contribution is aimed at obtaining new results in duality between $p$-dual frames and $p$-Riesz sequences in quasinormed spaces with a normalized Schauder basis. We obtain two results which show the relationship of duality between these concepts. We split a quasinormed space into a topological sum of two subspaces and use the Schauder basis to establish a relationship between the $p$-dual frames of one of these subspaces and the $p$-Riesz sequences of the dual of the other one. In fact, the main results are stated in a lightly more general context. 291–301 Hyponormality of Toeplitz operators on the Bergman space of an annulus. Houcine Sadraoui and Mohammed Guediri A bounded operator $S$ on a Hilbert space is hyponormal if $S^{\ast}S-SS^{\ast}$ is positive. In this work we find necessary conditions for the hyponormality of the Toeplitz operator $T_{f+\overline{g}}$ on the Bergman space of the annulus $\{1/2<|z|<1\}$, where $f$ and $g$ are analytic and $f$ satisfies a smoothness condition. 303–313 The geodesic flow on nilmanifolds associated to graphs. Gabriela P. Ovando We study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. We start on the corresponding Lie group equipped with a left-invariant metric, which is induced to the quotients. Also examples of integrable geodesic flows and of non-integrable ones are shown. 315–338 Quasi-modal operators on distributive nearlattices. Ismael Calomino, Sergio A. Celani, and Luciano J. González We introduce the notion of quasi-modal operator in the variety of distributive nearlattices, which turns out to be a generalization of the necessity modal operator studied in [S. Celani and I. Calomino, Math. Slovaca 69 (2019), no. 1, 35–52]. We show that there is a one to one correspondence between a particular class of quasi-modal operators on a distributive nearlattice and the class of possibility modal operators on the distributive lattice of its finitely generated filters. Finally, we consider the concept of quasi-modal congruence, and we show that the lattice of quasi-modal congruences of a quasi-modal distributive nearlattice is isomorphic to the lattice of congruences of the lattice of finitely generated filters with a possibility modal operator. 339–352 Geometry of pointwise CR-slant warped products in Kaehler manifolds. Bang-Yen Chen, Siraj Uddin, and Falleh R. Al-Solamy We call a submanifold $M$ of a Kaehler manifold $\tilde{M}$ a pointwise CR-slant warped product if it is a warped product, $B\times_{\!f} N_\theta$, of a CR-product $B=N_T\times N_\perp$ and a proper pointwise slant submanifold $N_\theta$ with slant function $\theta$, where $N_T$ and $N_\perp$ are complex and totally real submanifolds of $\tilde{M}$. We prove that if a pointwise CR-slant warped product $B\times_{\!f} N_\theta$ with $B=N_T\times N_\perp$ in a Kaehler manifold is weakly ${\mathfrak{D}^\theta}$-totally geodesic, then it satisfies $\|\sigma\|^2\geq 4s\left\{ (\csc^2\theta+\cot^2\theta)\|\nabla^T(\ln f)\|^2 + (\cot^2\theta)\|\nabla^\bot(\ln f)\|^2 \right\},$ where $N_T$, $N_\perp$, and $N_\theta$ are complex, totally real and proper pointwise slant submanifolds of $\tilde{M}$, respectively, and $s=\frac{1}{2}\dim N_\theta$. In this paper we also investigate the equality case of the inequality. Moreover, we give a non-trivial example and provide some applications of this inequality. 353–365 Triangular spherical dihedral f-tilings: the $(\pi/2, \pi/3, \pi/4)$ and $(2\pi/3, \pi/4, \pi/4)$ family. Catarina P. Avelino and Altino F. Santos We classify all the dihedral f-tilings with spherical triangles $\big(\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}\big)$ and $\big(\frac{2\pi}{3}, \frac{\pi}{4}, \frac{\pi}{4}\big)$, and give the combinatorial structure, including the symmetry group of each tiling. 367–387 Viscosity approximation method for modified split generalized equilibrium and fixed point problems. Hammed Anuoluwapo Abass, Chinedu Izuchukwu, and Oluwatosin Temitope Mewomo We introduce a viscosity iterative algorithm for approximating a common solution of a modified split generalized equilibrium problem and a fixed point problem for a quasi-pseudocontractive mapping which also solves some variational inequality problems in real Hilbert spaces. The proposed iterative algorithm is constructed in such a way that it does not require the prior knowledge of the operator norm. Furthermore, we prove a strong convergence theorem for approximating the common solution of the aforementioned problems. Finally, we give a numerical example of our main theorem. Our result complements and extends some related works in the literature. 389–411 Minimal solutions of the rational interpolation problem. Teresa Cortadellas Benítez, Carlos D'Andrea, and Eulàlia Montoro We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies, and the standard method provided by the Extended Euclidean Algorithm (EEA). As a consequence, we obtain explicit descriptions for solutions of minimal degrees in terms of the degrees of elements appearing in the EEA. This result allows us to describe the minimal degree in a $\mu$-basis of a polynomial planar parametrization in terms of a critical degree arising in the EEA. 413–429 On the restricted partition function via determinants with Bernoulli polynomials. II. Mircea Cimpoeaş Let $r\geq 1$ be an integer, $\mathbf{a}=(a_1,\dots,a_r)$ a vector of positive integers, and let $D\geq 1$ be a common multiple of $a_1,\dots,a_r$. We prove that if $D=1$ or $D$ is a prime number then the restricted partition function $p_{\mathbf{a}}(n) :=$ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_j x_j=n$, with $x_1\geq 0, \dots, x_r\geq 0$, can be computed by solving a system of linear equations with coefficients that are values of Bernoulli polynomials and Bernoulli–Barnes numbers. 431–440 Weakly convex and convex domination numbers for generalized Petersen and flower snark graphs. Jozef Kratica, Dragan Matić, and Vladimir Filipović We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph $GP(n,k)$, we prove that if $k=1$ and $n\geq 4$ then both the weakly convex domination number $\gamma_\mathit{wcon}(GP(n,k))$ and the convex domination number $\gamma_\mathit{con}(GP(n,k))$ are equal to $n$. For $k\geq 2$ and $n\geq 13$, $\gamma_\mathit{wcon}(GP(n,k))=\gamma_\mathit{con}(GP(n,k))=2n$, which is the order of $GP(n,k)$. Special cases for smaller graphs are solved by the exact method. For a flower snark graph $J_n$, where $n$ is odd and $n\geq 5$, we prove that $\gamma_\mathit{wcon}(J_n)=2n$ and $\gamma_\mathit{con}(J_n)=4n$. 441–455 Characterization of hypersurface singularities in positive characteristic. Amir Shehzad, Muhammad Ahsan Binyamin, and Hasan Mahmood The classification of right unimodal and bimodal hypersurface singularities over a field of positive characteristic was given by H. D. Nguyen. The classification is described in the style of Arnold and not in an algorithmic way. This classification was characterized by M. A. Binyamin et al. [Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 61(109) (2018), no. 3, 333–343] for the case when the corank of hypersurface singularities is $\leq 2$. The aim of this article is to characterize the right unimodal and bimodal hypersurface singularities of corank $3$ in an algorithmic way by means of easily computable invariants such as the multiplicity, the Milnor number of the given equation, and its blowing-up. On the basis of this characterization we implement an algorithm to compute the type of the right unimodal and bimodal hypersurface singularities without computing the normal form in the computer algebra system Singular. 457–468