Authors:Federico Quallbrunn Pages: 335 - 345 Abstract: Abstract Following Suwa, we study unfoldings of algebraic foliations and their relationship with families of foliations, making focus on those unfoldings related to trivial families. The results obtained in the study of unfoldings are then applied to obtain information on the structure of foliations on projective spaces. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0024-6 Issue No:Vol. 48, No. 3 (2017)

Authors:V. Ramos; J. Siqueira Pages: 347 - 375 Abstract: Abstract We prove uniqueness of equilibrium states for a family of partially hyperbolic horseshoes associated to a class of Hölder continuous potentials with small variation and derive statistical properties for this unique equilibrium. We define a projection map associated to the horseshoe and prove a spectral gap for its transfer operator acting on some space of Hölder continuous observables. From this we deduce an exponential decay of correlations and a central limit theorem. We finally extend these results to the horseshoe via Rohlin’s disintegration of the equilibrium along the stable fibers. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0027-y Issue No:Vol. 48, No. 3 (2017)

Authors:Ruben Lizarbe Pages: 377 - 388 Abstract: Abstract We construct a family of irreducible components of space of holomorphic foliations of codimension one on \(\mathbb {P}^3\) associated to some affine Lie algebra. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0029-9 Issue No:Vol. 48, No. 3 (2017)

Authors:Nancy Guelman; Isabelle Liousse Pages: 389 - 397 Abstract: Abstract A group \(\Gamma \) is said to be periodic if for any g in \(\Gamma \) there is a positive integer n with \(g^n=id\) . We first prove that a finitely generated periodic group acting on the 2-sphere \({\mathbb S}^2\) by \(C^1\) -diffeomorphisms with a finite orbit, is finite and \(C^1\) -conjugate to a subgroup of \(\mathrm {O}(3,{\mathbb R})\) . This result is obtained by proving the more general statement: a finitely generated periodic group acting on any compact manifold by \(C^1\) -diffeomorphisms with a finite orbit, is finite. We use it for proving that a countable 2-group of spherical diffeomorphisms with bounded orders is finite. This gives a negative partial answer to a question posed by D. Fisher. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus is finite. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0028-x Issue No:Vol. 48, No. 3 (2017)

Authors:Adel Chala Pages: 399 - 411 Abstract: Abstract This paper studies the risk-sensitive optimal control problem for a backward stochastic system. More precisely, we set up a necessary stochastic maximum principle for a risk-sensitive optimal control of this kind of equations. The control domain is assumed to be convex and the generator coefficient of such system is allowed to be depend on the control variable. As a preliminary step, we study the risk-neutral problem for which an optimal solution exists. This is an extension of initial control system to this type of problem, where the set of admissible controls is convex. An example to carried out to illustrate our main result of risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0031-2 Issue No:Vol. 48, No. 3 (2017)

Authors:A. Bagatini; M. L. Matte; A. Wagner Pages: 413 - 437 Abstract: Abstract Considering the unsigned version of the mock theta functions \(\phi (q)\) , \(\psi (q)\) , \(f_0(q)\) , \(F_0(q)\) , \(f_1(q)\) and \(F_1(q),\) they represent partitions subject to some rules. Based on a two-line matrix representation, we can classify them according to the sum of second line and derive some results such as closed formulas and identities for other types of partitions. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0021-9 Issue No:Vol. 48, No. 3 (2017)

Authors:Zhengxin Zhou Pages: 439 - 447 Abstract: Abstract In this article, I have completely solved this problem: when one differential system is equivalent to a given differential system, what structure does this system and its reflecting integral have' At the same time, I have established the relationship between the reflecting integrals and the first integrals and integrating factors of the differential equations. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0026-4 Issue No:Vol. 48, No. 3 (2017)

Authors:Jinxin Gao; Xiuyun Guo Pages: 449 - 459 Abstract: Abstract Let H be a subgroup of a group G. Then we say that H is w-s-permutable in G if G has a normal subgroup K such that HK is s-permutable in G and \(H\cap K\) is nearly s-permutable in G. In this article, we analyse the structure of a group G by using the properties of w-s-permutable subgroups and obtain some new characterizations of p-nilpotent groups and supersoluble groups. Some known results are generalized. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0023-7 Issue No:Vol. 48, No. 3 (2017)

Authors:M. S. Alves; M. A. Jorge Silva; T. F. Ma; J. E. Muñoz Rivera Pages: 461 - 484 Abstract: Abstract The well-established Timoshenko system is characterized by a particular relation between shear stress and bending moment from its constitutive equations. Accordingly, a (thermal) dissipation added on the bending moment produces exponential stability if and only if the so called “equal wave speeds” condition is satisfied. This remarkable property extends to the case of non-homogeneous coefficients. In this paper, we consider a non-homogeneous thermoelastic system with dissipation restricted to the shear stress. To this new problem, by means of a delicate control observability analysis, we prove that a local version of the equal wave speeds condition is sufficient for the exponential stability of the system. Otherwise, we study the polynomial stability of the system with decay rate depending on the regularity of initial data. PubDate: 2017-09-01 DOI: 10.1007/s00574-017-0030-3 Issue No:Vol. 48, No. 3 (2017)

Authors:Feliz Minhós; Robert de Sousa Pages: 485 - 503 Abstract: Abstract This paper presents sufficient conditions for the solvability of the third order three point boundary value problem $$\begin{aligned} \left\{ \begin{array}{c} -u^{\prime \prime \prime }(t)=f(t,\,v(t),\,v^{\prime }(t)) \\ -v^{\prime \prime \prime }(t)=h(t,\,u(t),\,u^{\prime }(t)) \\ u(0)=u^{\prime }(0)=0,u^{\prime }(1)=\alpha u^{\prime }(\eta ) \\ v(0)=v^{\prime }(0)=0,v^{\prime }(1)=\alpha v^{\prime }(\eta ). \end{array} \right. \end{aligned}$$ The arguments apply Green’s function associated to the linear problem and the Guo–Krasnosel’skiĭ theorem of compression-expansion cones. The dependence on the first derivatives is overcome by the construction of an adequate cone and suitable conditions of superlinearity/sublinearity near 0 and \(+\infty \) . Last section contains an example to illustrate the applicability of the theorem. PubDate: 2017-09-01 DOI: 10.1007/s00574-016-0025-5 Issue No:Vol. 48, No. 3 (2017)

Authors:Pengcheng Niu; Leyun Wu Abstract: Abstract We consider the quasilinear degenerate elliptic equation with rough and singular coefficients of the form $$\begin{aligned} div\left( {A(x,u,\nabla u)} \right) = B(x,u,\nabla u) \end{aligned}$$ in an open set \(\Omega \subset {\mathbb {R}^n},\) where the quadratic form associated with the principle part of this equation allows vanishing and the coefficients in structural conditions on \(A(x,u,\nabla u)\) and \(B(x,u,\nabla u)\) require no smoothness but belong to some Stummel–Kato class. We prove a Fefferman–Phong type inequality related to the Stummel–Kato class and then an embedding inequality. Based on these inequalities, the local boundedness and Harnack’s inequality of the weak solutions are derived. As applications, the continuity and Hölder continuity for the nonnegative weak solutions are given. PubDate: 2017-10-09 DOI: 10.1007/s00574-017-0054-8

Authors:Waldemar Barrera; Adriana Gonzalez-Urquiza; Juan Pablo Navarrete Abstract: Abstract In this paper we give a generalization of the Conze–Guivarc’h limit set. With this definition the limit set has very similar properties to those of the limit set in hyperbolic spaces. Moreover, we prove a relation between this new limit set and the Kulkarni limit set. Additionally we show that some closed subsets can be approximated by the Conze–Guivarc’h limit set. This is a result in the theory of classic Kleinian groups. PubDate: 2017-09-02 DOI: 10.1007/s00574-017-0053-9

Authors:Dionicio Pastor Dallos Santos Abstract: Abstract In this paper we study the existence of solutions for a new class of nonlinear differential equations with three-point boundary conditions. Existence of solutions are obtained by using the Leray–Schauder degree. PubDate: 2017-08-24 DOI: 10.1007/s00574-017-0052-x

Authors:Yingbo Han Abstract: Abstract In this paper, we investigate a complete noncompact submanifold \(M^m\) in a sphere \(S^{m+t}\) with flat normal bundle. We prove that the dimension of the space of \(L^p\) p-harmonic l-forms (when \(m\ge 4\) , \(2\le l\le m-2\) and when \(m=3\) , \(l=2\) ) on M is finite if the total curvature of M is finite and \(m\ge 3\) . We also obtain that there are no nontrivial \(L^p\) p-harmonic l-forms on M if the total curvature is bounded from above by a constant depending only on m, p, l. PubDate: 2017-08-12 DOI: 10.1007/s00574-017-0051-y

Authors:Yin Chen Abstract: Abstract We study modular invariants of finite affine linear groups over a finite field \(\mathbb {F}_{q}\) under affine actions and linear actions. We generalize a result of Chuai (J Algebra 318:710–722, 2007, Theorem 4.2) to any m-folds affine actions. Suppose \(G\leqslant \mathrm{GL}(n,\mathbb {F}_{q})\) is a subgroup and W denotes the canonical module of \(\mathrm{GL}(n,\mathbb {F}_{q})\) . We denote by \(\mathbb {F}_{q}[W]^{G}\) the invariant ring of G acting linearly on W and denote by \(\mathbb {F}_{q}[W_{n+1}]^{AG(W^{*})}\) the invariant ring of the affine group \(AG(W^{*})\) of G acting canonically on \(W_{n+1}:=W\oplus \mathbb {F}_{q}\) . We show that if \(\mathbb {F}_{q}[W]^{G}=\mathbb {F}_{q}[f_{1},f_{2},\ldots ,f_{s}]\) , then \(\mathbb {F}_{q}[W_{n+1}]^{AG(W^{*})}=\mathbb {F}_{q}[f_{1},f_{2},\ldots ,f_{s},h_{n+1}]\) , where \(h_{n+1}\) denotes the \((n+1)\) -th Mui’s invariant of degree \(q^{n}\) . Let \(\mathrm{AGL}_{1}(\mathbb {F}_{p})\) be the 1-dimensional affine general linear groups over the prime field \(\mathbb {F}_{p}\) . We find a generating set for the ring of vector invariants \(\mathbb {F}_{p}[mW_{2}]^{\mathrm{AGL}_{1}(\mathbb {F}_{p})}\) and determine the Noether’s number \(\upbeta _{mW_{2}}(\mathrm{AGL}_{1}(\mathbb {F}_{p}))\) for any \(m\in \mathbb {N}^{+}\) . PubDate: 2017-07-24 DOI: 10.1007/s00574-017-0050-z

Authors:Talat Körpinar; Rıdvan Cem Demirkol Abstract: Abstract In this work, we firstly describe conditions for being elastica for a moving particle corresponding to different type of space curves in Minkowski space \(\mathsf{E}_2^4\) . Then, we investigate the energy on the elastic curves corresponding to a particular particle in the space and we also exploit its relationship with energy on the same particle in the Frenet vector fields. Finally, we characterize non-elastic curves in \(\mathsf{E}_2^4\) and we compute their energy to see the distinction between energies for the curves of elastic and non-elastic case in Minkowski space \(\mathsf{E}_2^4\) . PubDate: 2017-07-10 DOI: 10.1007/s00574-017-0047-7

Authors:Jeovanny de Jesus Muentes Acevedo Abstract: Abstract Let M be a compact Riemannian manifold. The set \(\text {F}^{r}(M)\) consisting of sequences \((f_{i})_{i\in {\mathbb {Z}}}\) of \(C^{r}\) -diffeomorphisms on M can be endowed with the compact topology or with the strong topology. A notion of topological entropy is given for these sequences. I will prove this entropy is discontinuous at each sequence if we consider the compact topology on \(\text {F}^{r}(M)\) . On the other hand, if \( r\ge 1\) and we consider the strong topology on \(\text {F}^{r}(M)\) , this entropy is a continuous map. PubDate: 2017-07-07 DOI: 10.1007/s00574-017-0049-5

Authors:Mário Bessa; Jairo Bochi; Michel Cambrainha; Carlos Matheus; Paulo Varandas; Disheng Xu Abstract: Abstract A theorem of Viana says that almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. In this note we extend this result to cocycles on any noncompact classical semisimple Lie group. PubDate: 2017-07-04 DOI: 10.1007/s00574-017-0048-6

Authors:Tiago Macedo; Thiago Castilho de Mello Abstract: Abstract Let R be a commutative integral domain with unit, f be a nonconstant monic polynomial in R[t], and \(I_f \subset R[t]\) be the ideal generated by f. In this paper we study the group of R-algebra automorphisms of the R-algebra without unit \(I_f\) . We show that, if f has only one root (possibly with multiplicity), then \({{\mathrm{Aut}}}(I_f) \cong R^\times \) . We also show that, under certain mild hypothesis, if f has at least two different roots in the algebraic closure of the quotient field of R, then \({{\mathrm{Aut}}}(I_f)\) is a cyclic group and its order can be completely determined by analyzing the roots of f. PubDate: 2017-07-04 DOI: 10.1007/s00574-017-0046-8

Authors:Thaís Maria Dalbelo; Marcelo Messias; Alisson C. Reinol Abstract: Abstract In this paper we give the normal form of all polynomial differential systems in \(\mathbb {R}^3\) having a weighted homogeneous surface \(f=0\) as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. Using the obtained results we give some examples of stratified vector fields, when \(f=0\) is a singular surface. We also apply the obtained results to study the Vallis system, which is related to the so-called El Niño atmospheric phenomenon, when it has a cone as an invariant algebraic surface, performing a dynamical analysis of the flow of this system restricted to the invariant cone and providing a stratification for this singular surface. PubDate: 2017-06-27 DOI: 10.1007/s00574-017-0045-9