Abstract: We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy. PubDate: 2019-03-02

Abstract: In the present work, we study the decompositions of codimension-one transitions that alter the singular set the of stable maps from \(S^3\) to \(\mathbb {R}^3,\) the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets. PubDate: 2019-02-25

Abstract: Let \(r_d\) be the maximum number of skew lines that a smooth projective surface of degree d (over the complex numbers) can have. It is known that \(r_3=6\) , \(r_4=16\) (Schläfli in Q J Math Soc 2:55–65, 110–121, 1858; Nikulin in Math USSR Izv 9:261–275, 1975) and was proven by Miyaoka in 1975 that \(r_d\le 2d(d -2)\) if \(d\ge 4\) (Miyaoka in Math Ann 268:159–172, 1984). Up to now \(r_d\) remains unknown for \(d\ge 5\) . However, the lower bound \(d(d-2)+2\) was found by Rams (Proc Am Math Soc 133(1):11–13, 2005) which was improved by Boissière and Sarti (Ann Scuola Norm Sup Pisa Cl Sci 5:39–52, 2007), who showed that \(d(d-2)+4\le r_d\) for \(d \ge 5\) and odd. In this work, we take the family of degree d smooth surfaces \(\mathcal{R}_d\) in \({\mathbb {P}}^3\) (cf. (1)), considered by Boissière and Sarti (2007) and study \(r(\mathcal{R}_d)\) , the maximum number of skew lines that \(\mathcal{R}_d\) can have. In fact, we prove that \(r(\mathcal{R}_d)\! =\! d(d-2)+4\) if \(d \ge 5\) and odd. Otherwise, we prove that \(r(\mathcal{R}_6)\ge 48\) , which implies that Miyaoka’s bound is sharp for \(d=6\) , i.e. \(r_6=48\) . Still in the even case, we show that \(\mathcal{R}_d\) contains \(d(d-2)+4\) skew lines and we improve the Miyaoka’s bound for the family \(\mathcal{R}_d\) if d is even (Theorem 4.11). PubDate: 2019-02-23

Abstract: The objective of this paper is to study the existence, multiplicity and non existence of solutions for semilinear elliptic problems under a local Landesman–Lazer condition. There is no growth restriction at infinity on the nonlinear term and it may change sign. In order to establish the existence of solution we combine the Lyapunov–Schmidt reduction method with truncation and approximation arguments via bootstrap methods. In our applications we also consider the existence of a bifurcation point which may have multiple positive solutions for a fixed value of the parameters. PubDate: 2019-02-22

Abstract: If \(\gamma \) is a knot in \( S^3 \cong \mathbb {H}^2 _{\mathbb {C}} \subset \mathbb {P}_{\mathbb {C}}^2\) , then the set \(\Lambda (\gamma )\subset \mathbb {P}_{\mathbb {C}}^2\) is defined as the union of all the complex lines tangent to \(\partial \mathbb {H}^2 _{\mathbb {C}}\) at points in the image of \(\gamma \) . The following result is obtained: the number of components of \(\Omega (\gamma )=\mathbb {P}_{\mathbb {C}}^2 {\setminus } \Lambda (\gamma )\) is greater or equal to the number of distinct integers in the set \(\{\ell (\gamma , C): C \text { is a positively oriented chain disjoint to } \gamma \}\) , where \(\ell (\gamma , C)\) denotes the linking number between \(\gamma \) and C. PubDate: 2019-02-21

Abstract: An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied. PubDate: 2019-02-11

Abstract: In this paper we propose a new class of probability distributions, so called multivariate alpha skew normal distribution. It can accommodate up to two modes and generalizes the distribution proposed by Elal-Olivero [Proyecciones (Antofagasta) 29(3):224–240, 2010] in its marginal components. Its properties are studied. In particular, we derive its standard and non-standard densities, moment generating functions, expectations, variance-covariance matrixes, marginal and conditional distributions. Estimation is based on maximum likelihood. The asymptotic properties of the inferential procedure are verified in the light of a simulation study. The usefulness of the new distribution is illustrated in a real benchmark data. PubDate: 2019-01-05

Abstract: This paper presents an operator theory approach for the abstract structure of Banach function modules over coset spaces of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Let \(\mu \) be the normalized G-invariant measure over the homogeneous space G / H associated to the Weil’s formula and \(1\le p<\infty \) . We then introduce the notion of convolution left-module action of \(L^1(G/H,\mu )\) on the Banach function spaces \(L^p(G/H,\mu )\) . PubDate: 2019-01-01

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: 2018-12-01

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: 2018-12-01

Authors:Daniyal M. Israfilov; Elife Gursel; Esra Aydin Abstract: The maximal convergence properties of the partial sums of the Faber series in the variable exponent Smirnov classes are investigated. PubDate: 2018-05-02 DOI: 10.1007/s00574-018-0086-8

Authors:Ahmed Mostafa Khalil; Sheng-Gang Li; Fei You; Sheng-Quan Ma Abstract: In this paper, we present and study the concepts of fuzzifying pre- \(\theta \) -neighborhood system of a point, fuzzifying pre- \(\theta \) -closure of a set, fuzzifying pre- \(\theta \) -interior of a set, fuzzifying pre- \(\theta \) -open sets and fuzzifying pre- \(\theta \) -closed sets in fuzzifying topological spaces. The basic properties of these concepts are investigated. Two types of functions in a fuzzifying topological spaces called fuzzifying strongly pre-irresolute and fuzzifying weakly pre-irresolute functions are introduced. Then the interrelations of these functions with the parallel existing allied concepts are established. Finally, several characterizations of these functions along with different conditions for their existence are obtained. PubDate: 2018-04-28 DOI: 10.1007/s00574-018-0089-5

Authors:Sorina Barza; Anca N. Marcoci; Liviu G. Marcoci Abstract: We present factorizations of weighted Lebesgue, Cesàro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy’s integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities. PubDate: 2018-04-28 DOI: 10.1007/s00574-018-0087-7

Authors:Jihua Yang Abstract: The sixteen generators of Abelian integral \(I(h)=\oint _{\Gamma _h}g(x,y)dx-f(x,y)dy\) , which satisfy eight different Picard–Fuchs equations respectively, are obtained, where \(\Gamma _h\) is a family of closed orbits defined by \(H(x,y)=ax^4+by^4+cx^8=h\) , \(h\in \Sigma \) , \(\Sigma \) is the open intervals on which \(\Gamma _h\) is defined, and f(x, y) and g(x, y) are real polynomials in x and y of degree n. Moreover, an upper bound of the number of zeros of I(h) is obtained for a special case $$\begin{aligned} f(x,y)=\sum \limits _{0\le i\le 4k+1=n}a_ix^{4k+1-i}y^i,\ \ \ g(x,y)=\sum \limits _{0\le i\le 4k+1=n}b_ix^{4k+1-i}y^i. \end{aligned}$$ PubDate: 2018-04-23 DOI: 10.1007/s00574-018-0085-9

Authors:Besma Amri Abstract: We define wavelets and wavelet transforms associated with spherical mean operator. We establish a Plancherel theorem, orthogonality property and inversion formula for the wavelet transform. Next, we define the Toeplitz operators \(\mathfrak {T}_{\varphi ,\psi }(\sigma )\) associated with two wavelets \(\varphi ,\psi \) and with symbol \(\sigma .\) We establish the boundedness and compactness of these operators. Last, we define the Schatten-von Neumann class \(S^p\ ;\ p\in \ [1,+\infty ],\) and we show that the Toeplitz operators belong to the class \(S^p\) and we prove a formula of trace. PubDate: 2018-03-23 DOI: 10.1007/s00574-018-0083-y

Authors:J. P. O. Santos; M. L. Matte Abstract: In this work we define a new set of integer partition, based on a lattice path in \({\mathbb {Z}}^2\) connecting the line \(x+y=n\) to the origin, which is determined by the two-line matrix representation given for different sets of partitions of n. The new partitions have only distinct odd parts with some particular restrictions. This process of getting new partitions, which has been called the Path Procedure, is applied to unrestricted partitions, partitions counted by the 1st and 2nd Rogers–Ramanujan Identities, and those generated by the Mock Theta Function \(T_1^*(q)=\sum _{n=0}^{\infty }\dfrac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}}\) . PubDate: 2018-03-16 DOI: 10.1007/s00574-018-0082-z

Authors:A. Hefez; J. H. O. Rodrigues; R. Salomão Abstract: The Milnor number of an isolated hypersurface singularity, defined as the codimension \(\mu (f)\) of the ideal generated by the partial derivatives of a power series f that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation f representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series f in two indeterminates over the complex numbers, it was shown by Milnor that \(\mu (f)\) always coincides with the conductor c(f) of the semigroup of values S(f) of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup S(f) is tame, that is, the characteristic of the field does not divide any of its minimal generators. PubDate: 2018-03-14 DOI: 10.1007/s00574-018-0080-1

Authors:Katsuei Kenmotsu Abstract: The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: when the curvature of the ambient space is positive, there exists a unique such surface up to rigid motions of the target space. On the other hand, when the curvature of the ambient space is negative, there are ‘non-trivial’ complete parallel mean curvature surfaces generated by Jacobi elliptic functions and they exhaust such surfaces. PubDate: 2018-03-13 DOI: 10.1007/s00574-018-0081-0

Authors:Jenő Szirmai Abstract: In this paper we study the interior angle sums of geodesic triangles in \(\mathbf {Nil}\) geometry and prove that these can be larger, equal or less than \(\pi \) . We use for the computations the projective model of \(\mathbf {Nil}\) introduced by Molnár (Beitr. Algebra Geom. 38(2):261–288, 1997). PubDate: 2018-03-03 DOI: 10.1007/s00574-018-0077-9

Authors:Grzegorz Oleksik; Adam Różycki Abstract: Let f be a real polynomial, non-negative at infinity with non-compact zero-set. Suppose that f is non-degenerate in the Kushnirenko sense at infinity. In this paper we give a formula for the Łojasiewicz exponent at infinity of f and a formula for the exponent of growth of f in terms of its Newton polyhedron. PubDate: 2018-03-02 DOI: 10.1007/s00574-018-0078-8