Authors:Jongchon Kim Pages: 773 - 789 Abstract: We consider quasiradial Fourier multipliers, i.e., multipliers of the form \(m(a(\xi ))\) for a class of distance functions a. We give a necessary and sufficient condition for the multiplier transformations to be bounded on \(L^p\) for a certain range of p. In addition, when m is compactly supported in \((0,\infty )\) , we give a similar result for associated maximal operators. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0595-2 Issue No:Vol. 196, No. 3 (2017)

Authors:Laura Gioia Andrea Keller Pages: 791 - 818 Abstract: In the present article we investigate how geometric microstructures of a domain can affect the diffusion on the macroscopic level. More precisely, we look at a domain with additional microstructures of two kinds, the first one are periodically arranged “horizontal barriers” and the second one are “vertical barriers” which are not periodically arranged, but uniform on certain intervals. Both structures are parametrized in size by a small parameter \(\varepsilon \) . Starting from such a geometry combined with a diffusion(-reaction) model, we derive the homogenized limit and discuss the differences of the resulting limit problems for various particular arrangements of the microstructures. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0596-1 Issue No:Vol. 196, No. 3 (2017)

Authors:Luca Rizzi; Francesco Zucconi Pages: 819 - 836 Abstract: We prove a full generalization of the Castelnuovo’s free pencil trick. We show its analogies with Rizzi and Zucconi (Differential forms and quadrics of the canonical image. arXiv:1409.1826, Theorem 2.1.7); see also Pirola and Zucconi (J Algebraic Geom 12(3):535–572, Theorem 1.5.1). Moreover we find a new formulation of the Griffiths’s infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic 1-forms. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0597-0 Issue No:Vol. 196, No. 3 (2017)

Authors:Lander Cnudde; Hendrik De Bie Pages: 837 - 862 Abstract: Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel function of this integral transform is constructed using the Mehler formula. An explicit expression for the integral transform is obtained and allows for the study of its properties. In the second part, two kinds of corresponding convolutions are examined: Mustard convolutions and convolutions based on generalised translation operators. The paper finishes by demonstrating the connection between both. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0598-z Issue No:Vol. 196, No. 3 (2017)

Authors:Ana Irina Nistor Pages: 863 - 875 Abstract: In the present paper, we classify curves and surfaces in \({\mathbb {S}}^2\times \mathbb {R}\) , which make constant angle with a rotational Killing vector field. We obtain the explicit parametrizations of such curves and surfaces, and we find examples in some particular cases. Finally, we give the complete classification of minimal constant angle surfaces in \({\mathbb {S}}^2\times \mathbb {R}\) . PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0599-y Issue No:Vol. 196, No. 3 (2017)

Authors:Tommaso Leonori; Alessio Porretta; Giuseppe Riey Pages: 877 - 903 Abstract: We prove comparison principles for quasilinear elliptic equations whose simplest model is $$\begin{aligned} \lambda u -\Delta _p u + H(x,Du)=0 \quad x\in \Omega , \end{aligned}$$ where \(\Delta _p u = \text { div }( Du ^{p-2} Du)\) is the p-Laplace operator with \(p> 2\) , \(\lambda \ge 0\) , \(H(x,\xi ):\Omega \times \mathbb {R}^{N}\rightarrow \mathbb {R}\) is a Carathéodory function and \(\Omega \subset \mathbb {R}^{N}\) is a bounded domain, \(N\ge 2\) . We collect several comparison results for weak sub- and super-solutions under different setting of assumptions and with possibly different methods. A strong comparison result is also proved for more regular solutions. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0600-9 Issue No:Vol. 196, No. 3 (2017)

Authors:David Brander Pages: 905 - 928 Abstract: We study a generalization of constant Gauss curvature \(-1\) surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyse the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solution is unique for most curves, but for some curves there are infinitely many solutions, and this is encoded in the curvature and torsion of the curve. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0601-8 Issue No:Vol. 196, No. 3 (2017)

Authors:Yann Bugeaud; Jia-Yan Yao Pages: 929 - 946 Abstract: The irrationality exponent of an irrational p-adic number \(\xi \) , which measures the approximation rate of \(\xi \) by rational numbers, is in general very difficult to compute explicitly. In this work, we shall show that the irrationality exponents of large classes of automatic p-adic numbers and Mahler p-adic numbers (which are transcendental) are exactly equal to 2. Our classes contain the Thue–Morse–Mahler p-adic numbers, the regular paperfolding p-adic numbers, the Stern p-adic numbers, among others. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0602-7 Issue No:Vol. 196, No. 3 (2017)

Authors:Stefano Meda; Sara Volpi Pages: 947 - 981 Abstract: In this paper, we define a space \({\mathfrak h}^1(M)\) of Hardy–Goldberg type on a measured metric space satisfying some mild conditions. We prove that the dual of \({\mathfrak h}^1(M)\) may be identified with \(\mathfrak {b}\mathfrak {m}\mathfrak {o}(M)\) , a space of functions with “local” bounded mean oscillation, and that if p is in (1, 2), then \(L^p(M)\) is a complex interpolation space between \({\mathfrak h}^1(M)\) and \(L^2(M)\) . This extends previous results of Strichartz, Carbonaro, Mauceri and Meda, and Taylor. Applications to singular integral operators on Riemannian manifolds are given. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0603-6 Issue No:Vol. 196, No. 3 (2017)

Authors:Angelo R. F. de Holanda; Olivaine S. de Queiroz; Cesar K. S. dos Santos Pages: 983 - 1000 Abstract: We obtained existence and pointwise regularity results for the following parabolic free boundary problem: $$\begin{aligned} u_t-\Delta u = \chi _{\{u>0\}} \log u \ \ \hbox {in} \ \ \Omega \times (0,T], \end{aligned}$$ with initial and boundary conditions in some appropriate spaces. The equation is singular along the set \(\partial \{u>0\}\) , and the logarithmic nonlinearity does not have scaling properties. Thus, the machinery from regularity theory for free boundary problems, which strongly relies on the homogeneity of the problem, can not be applied directly. We prove that, near the free boundary, an approximate solution grows at most like \(r^2\log r.\) This is the so-called supercharacteristic growth, and its study has intriguing open questions. Our estimates are crucial to understand further analytic and geometric properties of the free boundary. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0604-5 Issue No:Vol. 196, No. 3 (2017)

Authors:Xiaoxiang Jiao; Mingyan Li Pages: 1001 - 1023 Abstract: In this paper, we investigate geometry of conformal minimal two-spheres immersed in \(G(2,n;\mathbb {R})\) and give a classification theorem of linearly full conformal minimal immersions of constant curvature from \(S^2\) to \(G(2,n;\mathbb {R})\) , or equivalently, a complex hyperquadric \(Q_{n-2}\) under some conditions. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0605-4 Issue No:Vol. 196, No. 3 (2017)

Authors:Mateusz Michałek; Hyunsuk Moon; Bernd Sturmfels; Emanuele Ventura Pages: 1025 - 1054 Abstract: We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics, we determine the real rank boundary: It is a hypersurface of degree 168. For quartics, sextics and septics, we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0606-3 Issue No:Vol. 196, No. 3 (2017)

Authors:Paweł Zapałowski Pages: 1055 - 1071 Abstract: Answering all questions—concerning proper holomorphic mappings between generalized Hartogs triangles—posed by Jarnicki and Plfug (First steps in several complex variables: Reinhardt domains, EMS Textbooks in Mathematics, European Mathematical Society Publishing House, 2008), we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, we completely describe the group of holomorphic automorphisms of such domains and establish rigidity of proper holomorphic self-mappings on them. Moreover, we also complete the classification of proper holomorphic mappings in the class of complex ellipsoids which was initiated by Landucci and continued by Dini and Selvaggi Primicerio. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0607-2 Issue No:Vol. 196, No. 3 (2017)

Authors:Giampiero Chiaselotti; Tommaso Gentile; Federico G. Infusino; Paolo A. Oliverio Pages: 1073 - 1112 Abstract: In this paper we continue a research project concerning the study of a graph from the perspective of granular computation. To be more specific, we interpret the adjacency matrix of any simple undirected graph G in terms of data information table, which is one of the most studied structures in database theory. Granular computing (abbreviated GrC) is a well-developed research field in applied and theoretical information sciences; nevertheless, in this paper we address our efforts toward a purely mathematical development of the link between GrC and graph theory. From this perspective, the well-studied notion of indiscernibility relation in GrC becomes a symmetry relation with respect to a given vertex subset in graph theory; therefore, the investigation of this symmetry relation turns out to be the main object of study in this paper. In detail, we study a simple undirected graph G by assuming a generic vertex subset W as reference system with respect to which examine the symmetry of all vertex subsets of G. The change of perspective from G without reference system to the pair (G, W) is similar to what occurs in the transition from an affine space to a vector space. We interpret the symmetry blocks in the reference system (G, W) as particular equivalence classes of vertices in G, and we study the geometric properties of all reference systems (G, W), when W runs over all vertex subsets of G. We also introduce three hypergraph models and a vertex set partition lattice associated to G, by taking as general models of reference several classical notions of GrC. For all these constructions, we provide a geometric characterization and we determine their structure for basic graph families. Finally, we apply a wide part of our work to study the important case of the Petersen graph. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0608-1 Issue No:Vol. 196, No. 3 (2017)

Authors:Hongjie Li; Hongyu Liu Pages: 1113 - 1135 Abstract: We consider plasmon resonances for the elastostatic system in \({\mathbb {R}}^3\) associated with a very broad class of sources. The plasmonic device takes a general core–shell–matrix form with the metamaterial located in the shell. It is shown that the plasmonic device in the literature which induces resonance in \({\mathbb {R}}^2\) does not induce resonance in \({\mathbb {R}}^3\) . We then construct two novel plasmonic devices with suitable plasmon constants, varying according to the source term or the loss parameter, that can induce resonances. If there is no core, we show that resonance always occurs. If there is a core of an arbitrary shape, we show that the resonance strongly depends on the location of the source. In fact, there exists a critical radius such that resonance occurs for sources lying within the critical radius, whereas resonance does not occur for sources lying outside the critical radius. Our argument is based on the variational technique by making use of the primal and dual variational principles for the elastostatic system, along with a highly technical construction of the associated perfect plasmon elastic waves. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0609-0 Issue No:Vol. 196, No. 3 (2017)

Authors:Dmitri Alekseevsky; Liana David Pages: 1137 - 1164 Abstract: The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated with second-order differential equations, and structures defined by gradings of Lie algebras (in the framework of parabolic geometries). Tanaka’s prolongation procedure associates with a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description of local invariants, investigation of the automorphism group, and equivalence problem. In this paper, we develop an alternative constructive approach for Tanaka’s prolongation procedure, based on the theory of quasi-gradations of filtered vector spaces, G-structures, and their torsion functions. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0610-7 Issue No:Vol. 196, No. 3 (2017)

Authors:Iva Dřímalová; Werner Kratz; Roman Šimon Hilscher Pages: 1165 - 1183 Abstract: In this paper we study a general even-order symmetric Sturm–Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm–Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh’s principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even-order symmetric Sturm–Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0611-6 Issue No:Vol. 196, No. 3 (2017)

Authors:Miroslav Bulíček; Martin Kalousek; Petr Kaplický Pages: 1185 - 1202 Abstract: We combine two-scale convergence, theory of monotone operators and results on approximation of Sobolev functions by Lipschitz functions to prove a homogenization process for an incompressible flow of a generalized Newtonian fluid. We avoid the necessity of testing the weak formulation of the initial and homogenized systems by corresponding weak solutions, which allows optimal assumptions on lower bound for a growth of the elliptic term. We show that the stress tensor for homogenized problem depends on the symmetric part of the velocity gradient involving the limit of a sequence selected from a family of solutions of initial problems. PubDate: 2017-06-01 DOI: 10.1007/s10231-016-0612-5 Issue No:Vol. 196, No. 3 (2017)

Authors:Nicola Garofalo; Isidro H. Munive Abstract: In a cylinder \(D_T = \Omega \times (0,T)\) , where \(\Omega \subset \mathbb {R}^{n}\) , we examine the relation between the L-caloric measure, \(\mathrm{d}\omega ^{(x,t)}\) , where L is the heat operator associated with a system of vector fields of Hörmander type, and the measure \(\mathrm{d}\sigma _X\times \mathrm{d}t\) , where \(\mathrm{d}\sigma _X\) is the intrinsic X-perimeter measure. The latter constitutes the appropriate replacement for the standard surface measure on the boundary and plays a central role in sub-Riemannian geometric measure theory. Under suitable assumptions on the domain \(\Omega \) we establish the mutual absolute continuity of \(\mathrm{d}\omega ^{(x,t)}\) and \(\mathrm{d}\sigma _X\times \mathrm{d}t\) . We also derive the solvability of the initial-Dirichlet problem for L with boundary data in appropriate \( L^p\) spaces, for every \(p>1\) . PubDate: 2017-05-25 DOI: 10.1007/s10231-017-0669-9

Authors:Adriana A. Cintra; Irene I. Onnis Abstract: In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz–Minkowski space \({\mathbb {L}}^{3}\) , using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in \({\mathbb {L}}^{3}\) constructed via the Enneper representation formula that it is equivalent to the Weierstrass representation obtained by Kobayashi (for spacelike immersions) and by Konderak (for the timelike ones). PubDate: 2017-05-25 DOI: 10.1007/s10231-017-0666-z