Authors:Mauricio Poletti Abstract: We prove that for Anosov maps of the 3-torus if the Lyapunov exponents of absolutely continuous measures in every direction are equal to the geometric growth of the invariant foliations then f is \(C^1\) conjugated to its linear part. PubDate: 2018-02-26 DOI: 10.1007/s00574-018-0079-7

Authors:Radu Miculescu; Alexandru Mihail Abstract: Taking as model the attractor of an iterated function system consisting of \(\varphi \) -contractions on a complete and bounded metric space, we introduce the set-theoretic concept of family of functions having attractor. We prove that, given such a family, there exist a metric on the set on which the functions are defined and take values and a comparison function \(\varphi \) such that all the family’s functions are \(\varphi \) -contractions. In this way we obtain a generalization for a finite family of functions of the converse of Browder’s fixed point theorem. As byproducts we get a particular case of Bessaga’s theorem concerning the converse of the contraction principle and a companion of Wong’s result which extends the above mentioned Bessaga’s result for a finite family of commuting functions with common fixed point. PubDate: 2018-02-26 DOI: 10.1007/s00574-018-0076-x

Authors:Tao Zhu Abstract: Some new Henry–Gronwall integral inequalities are established, which generalize some former famous inequalities and can be used as powerful tools in the study of differential and integral equations. PubDate: 2018-02-17 DOI: 10.1007/s00574-018-0074-z

Authors:Ale Jan Homburg Abstract: We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds. PubDate: 2018-01-31 DOI: 10.1007/s00574-018-0073-0

Authors:R. Tojeiro; S. Canevari Abstract: We develop a Ribaucour transformation for the class of conformally flat hypersurfaces \(f:M^{3} \rightarrow \mathbb {Q}_s^{4}(c)\) with three distinct principal curvatures of a pseudo-Riemannian space form of dimension 4, constant curvature c and index \(s\in \{0, 1\}\) , as well as for the class of hypersurfaces \(f:M^{3} \rightarrow \mathbb {Q}_s^{4}(c)\) with three distinct principal curvatures for which there exists another isometric immersion \(\tilde{f}:M^{3} \rightarrow \mathbb {Q}^{4}_{\tilde{s}}(\tilde{c})\) with \(\tilde{c}\ne c\) . It gives a process to produce a family of new elements of those classes starting from a given one and a solution of a linear system of PDE’s. This enables us to construct explicit new examples of hypersurfaces in both classes. PubDate: 2018-01-31 DOI: 10.1007/s00574-018-0072-1

Authors:Martino Garonzi; Igor Lima Abstract: Let G be a finite group and let c(G) be the number of cyclic subgroups of G. We study the function \(\alpha (G) = c(G)/ G \) . We explore its basic properties and we point out a connection with the probability of commutation. For many families \(\mathscr {F}\) of groups we characterize the groups \(G \in \mathscr {F}\) for which \(\alpha (G)\) is maximal and we classify the groups G for which \(\alpha (G) > 3/4\) . We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality \(\alpha (G) = \alpha (G/N)\) when G / N is a symmetric group. PubDate: 2018-01-30 DOI: 10.1007/s00574-018-0068-x

Authors:Gabriel Muñoz Abstract: We explore the relationship between limit linear series and fibers of Abel maps for compact type curves with three components. For compact type curves with two components, given an exact Osserman limit linear series \({\mathfrak {g}}\) , Esteves and Osserman associated a closed subscheme \({\mathbb {P}}({\mathfrak {g}})\) of the fiber of the corresponding Abel map. We generalize this definition to our case. Then, for \({\mathfrak {g}}\) the unique exact extension of an r-dimensional refined Eisenbud–Harris limit linear series, we find the irreducible components of \({\mathbb {P}}({\mathfrak {g}})\) and we show that \({\mathbb {P}}({\mathfrak {g}})\) is connected of pure dimension r, with the same Hilbert polynomial as the diagonal in \({\mathbb {P}}^{r}\times {\mathbb {P}}^{r}\times {\mathbb {P}}^{r}\) . PubDate: 2018-01-15 DOI: 10.1007/s00574-018-0071-2

Authors:Abhijit Das; Bikash Sahoo Abstract: The steady laminar flow and heat transfer of a non-Newtonian second grade fluid between two stretchable, co-axially rotating disks is considered. Using similarity transformations, partial differential equations governing the flow, are reduced to a set of highly coupled and nonlinear ordinary differential equations. These developed nonlinear equations are then integrated analytically using an effective analytical method called homotopy analysis method to obtain series solutions. The convergence of the obtained series solutions are also analyzed. Results obtained using 20th-order homotopy approximations, for different cases, such as the disks rotating in same (opposite) sense with same (different) angular velocities are shown graphically and discussed in detail for various parameters of interest, such as, stretching parameter, Reynolds number, non-Newtonian viscoelastic parameter. Of particular interest are the values of \(F''(0)\) and \(-G'(0)\) . And \(-G'(0)\) found to be decreasing function of the non-Newtonian viscoelastic parameter K whereas the values of \(F''(0)\) decreases with K, except when both the disk stretches and \(\Omega =0, 0.5\) . PubDate: 2018-01-12 DOI: 10.1007/s00574-018-0069-9

Authors:Grey Ercole Abstract: Let \(\left( X,\left\ \cdot \right\ _{X}\right) \) and \(\left( Y,\left\ \cdot \right\ _{Y}\right) \) be Banach spaces over \(\mathbb {R},\) with X uniformly convex and compactly embedded into Y. The inverse iteration method is applied to solve the abstract eigenvalue problem \(A(w)=\lambda \left\ w\right\ _{Y}^{p-q}B(w),\) where the maps \(A:X\rightarrow X^{\star }\) and \(B:Y\rightarrow Y^{\star }\) are homogeneous of degrees \(p-1\) and \(q-1,\) respectively. PubDate: 2018-01-10 DOI: 10.1007/s00574-018-0070-3

Authors:Xavier Carvajal; Amin Esfahani; Mahendra Panthee Pages: 505 - 550 Abstract: Considered in this work is an n-dimensional dissipative version of the Korteweg–de Vries (KdV) equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study well-posedness behavior of this equation when the dissipative effects are reduced. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0034-z Issue No:Vol. 48, No. 4 (2017)

Authors:H. Sano; Y. Kabata; J. L. Deolindo Silva; T. Ohmoto Pages: 623 - 639 Abstract: We present a local classification of smooth surfaces in \({\mathbb {P}}^3\) in terms of the singularity types (of codimension \(\le \) 4) of their central projections to a plane. Based on our classification result, we also give exact normal forms to surface germs at transition moments on bifurcations with respect to parabolic curves and flecnodal curves. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0036-x Issue No:Vol. 48, No. 4 (2017)

Authors:Angel Cano; Luis Loeza; Alejandro Ucan-Puc Pages: 641 - 647 Abstract: In this article we show that Bers’ simultaneous uniformization and Köebe’s retrosection theorem do not hold for discrete groups of projective transformations acting on complex projective space. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0035-y Issue No:Vol. 48, No. 4 (2017)

Authors:Masaru Hasegawa; Farid Tari Pages: 679 - 696 Abstract: The simplest way to have birth of surfaces is through transitions in the fibres of a function f with a Morse singularity of index 0 or 3. It is natural to seek to understand the geometry of newly born surfaces. We consider here the question of finding how many umbilics are on a newly born surface. We show that newly born surfaces in the Euclidean 3-space have exactly 4 umbilic points all of type lemon, provided that the Hessian of f at the singular point has pairwise distinct eigenvalues. This is true in both cases when f is an analytic or a smooth germ. When only two of such eigenvalues are equal, the number of umbilic points is either 2, 4, 6 or 8 when f is an analytic or a generic smooth germ. The same results holds for newly born surfaces in the Minkowski 3-space. In that case when the two eigenvalues associated to the two spacelike eigenvectors are distinct we get exactly 4 umbilic points all of type lemon. If they are equal, the number of umbilic points is either 2, 4, 6 or 8. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0037-9 Issue No:Vol. 48, No. 4 (2017)

Authors:Cícero Aquino; Halyson Baltazar Pages: 697 - 715 Abstract: The purpose of this article is to study the uniqueness of complete hypersurfaces satisfying some pinching curvature condition. Here, we use the generalized maximum principle of Omori–Yau to obtain uniqueness results for complete spacelike hypersurfaces immersed in a Lorentzian product space. In addition, we obtain the analogue results for complete hypersurfaces immersed in a Riemannian product space. PubDate: 2017-12-01 DOI: 10.1007/s00574-017-0041-0 Issue No:Vol. 48, No. 4 (2017)

Authors:Haimer A. Trejos Abstract: In Aledo et al. (Adv Math 224:2511–2530, 2010) and Espinar and Trejos (The Abresch–Rosenberg Shape Operator and Applications. arXiv:1512.02099, 2015) the authors showed the existence of a Codazzi pair defined on any constant mean curvature surface in the homogeneous spaces \(\mathbb {E}(\kappa , \tau )\) associated to the Abresch–Rosenberg differential. In this paper, we use the mentioned Codazzi pair to classify capillary disks in \(\mathbb {E}(\kappa , \tau )\) . As a consequence, the results presented in this paper generalize the previous classification of constant mean curvature disks in the product spaces \(\mathbb {S}^{2} \times \mathbb {R}\) and \(\mathbb {H}^{2} \times \mathbb {R}\) in do Carmo and Fernández (Forum Math 21:951–963, 2009) and Cavalcante and Lira (Mich Math J 55:163–181, 2007). PubDate: 2017-12-20 DOI: 10.1007/s00574-017-0063-7

Authors:Thiago F. da Silva; Nivaldo G. Grulha; Miriam S. Pereira Abstract: The Bi-Lipschitz geometry is one of the main subjects in the modern approach of singularity theory. However, it rises from works of important mathematicians of the last century, especially Zariski. In this work we investigate the Bi-Lipschitz equisingularity of families of essentially isolated determinantal singularities inspired by the approach of Mostowski and Gaffney. PubDate: 2017-12-16 DOI: 10.1007/s00574-017-0067-3

Authors:A. Aiolfi; G. Nunes; L. Sauer; R. Soares Abstract: We establish existence and uniqueness of compact graphs of cons tant mean curvature in \(M\times \mathbb {R}\) , over bounded domains contained in \(M\times \left\{ 0\right\} \) , with boundary lying in two horizontal slices of \(M\times \mathbb {R}\) . PubDate: 2017-12-11 DOI: 10.1007/s00574-017-0064-6

Authors:M. Martelo; B. Scárdua Abstract: We introduce a notion stability for subgroups of local complex analytic diffeomorphisms having a common fixed point, in several complex variables. This notion, called L-stability, is inspired in the notion of stability of Lyapunov for singular points and closed orbits of ordinary differential equations. It is also connected to the notion of stability for a proper leaf of a foliation in the classical sense of Reeb. We first classify in terms of the unitary group. Then we prove analytic linearization for a L-stable map and on the classification of L-stable linear groups. This is related to the study of subgroups of \({\mathbb {U}}(n)\) , the unitary matrix group. PubDate: 2017-12-08 DOI: 10.1007/s00574-017-0062-8

Authors:Qun Chen; Jianghai Shi Abstract: In this paper, we investigate the first non-zero eigenvalue problem of the following operator $$\begin{aligned} \left\{ \begin{array}{l} \mathrm {div} A\nabla {f}\mathrm =0 \quad \hbox {in}\quad \Omega ,\\ \frac{\partial f}{\partial v} =pf\ \quad \hbox {on}\quad \partial \Omega ,\\ \end{array} \right. \end{aligned}$$ where \(\Omega \) is a compact bounded domain in an m-dimensional complete Riemannian manifold \(M^{m}\) , v is the outward unit normal vector field of \(\partial \Omega \) and A is a positive definite symmetric (1,1)-tensor on \(M^{m}\) . By the Rayleigh-Ritz inequality and Hsiung–Minkowski formulas, we derive an upper bound for the first non-zero eigenvalue of these operators on bounded domain of complete manifolds isometrically immersed in a Euclidean space or a unit Sphere in terms of the r-th mean curvatures of its boundary \(\partial \Omega \) . PubDate: 2017-12-07 DOI: 10.1007/s00574-017-0066-4

Authors:Carlos R. Mamani; Alessandra A. Verri Abstract: Consider the Dirichlet Laplacian operator \(-\Delta ^D\) in a periodic waveguide \(\Omega \) . Under the condition that \(\Omega \) is sufficiently thin, we show that its spectrum \(\sigma (-\Delta ^D)\) is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in \(\sigma (-\Delta ^D)\) and locate it. PubDate: 2017-12-04 DOI: 10.1007/s00574-017-0065-5