Bulletin of the Brazilian Mathematical Society New Series

Miculescu, Radu; Mihail, Alexandru

doi: 10.1007/s00574-018-0076-xpmid: N/A

Taking as model the attractor of an iterated function system consisting of $$\varphi $$ φ -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 $$\varphi $$ φ such that all the family’s functions are $$\varphi $$ φ -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.

Poletti, Mauricio

doi: 10.1007/s00574-018-0079-7pmid: N/A

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 $$C^1$$ C 1 conjugated to its linear part.

Dela-Rosa, M.

doi: 10.1007/s00574-018-0075-ypmid: N/A

For an isolated hypersurface singularity $$f:(\mathbb {C}^{n+1},0)\rightarrow (\mathbb {C},0)$$ f : ( C n + 1 , 0 ) → ( C , 0 ) with Milnor number $$\mu $$ μ and good representative $$f:(X,0)\rightarrow (\Delta ,0)$$ f : ( X , 0 ) → ( Δ , 0 ) canonical $$\mu $$ μ -dimensional $$\mathbb {C}$$ C -bilinear vector spaces are associated: the Jacobian module, $$\Omega ^{f}$$ Ω f , which is isomorphic to the Milnor algebra $$A_f$$ A f up to a choice of coordinates; and the cohomology of the canonical Milnor fiber, H. Indeed, one has defined on $$\Omega ^f$$ Ω f , and hence in $$A_f$$ A f , the non-degenerate Grothendieck pairing $$res_{f,0}$$ r e s f , 0 which is a symmetric $$\mathbb {C}$$ C -bilinear form, and on the vanishing cohomology H it is defined a non-degenerate $$\mathbb {C}$$ C -bilinear form $$\mathbb {S}$$ S , induced by Poincaré duality, which is $$(-1)^{n+1}$$ ( - 1 ) n + 1 -symmetric on the generalized monodromy eigenspace $$H_{1}$$ H 1 and $$(-1)^{n}$$ ( - 1 ) n -symmetric on the direct sum of generalized monodromy eigenspaces $$H_{\ne 1}:=\oplus _{\lambda \ne 1}H_{\lambda }$$ H ≠ 1 : = ⊕ λ ≠ 1 H λ . On the other hand, there are two nilpotent $$\mathbb {C}$$ C -linear maps defined on $$\Omega ^f$$ Ω f and H, respectively; the first one is the map $$\{\mathbf {f}\}$$ { f } given by multiplication with f, which is $$res_{f,0}$$ r e s f , 0 -symmetric, and the other one is the $$\mathbb {S}$$ S -antisymmetric endomorphism N given by the logarithm of the unipotent part of the monodromy transformation. New bilinear forms can be constructed by composing on the left (or equivalently on the right) with powers of such nilpotent maps: $$res_{f,0}(\{\mathbf {f}\}^{\ell }\bullet ,\bullet )$$ r e s f , 0 ( { f } ℓ ∙ , ∙ ) and $$\mathbb {S}(N^{\ell }\bullet ,\bullet )$$ S ( N ℓ ∙ , ∙ ) for each integer $$\ell \ge 1$$ ℓ ≥ 1 . These new bilinear forms are called higher bilinear forms on $$\Omega ^f$$ Ω f resp. on H. In this paper, we show a formula which relates the powers $$\{\mathbf {f}\}^{\ell }$$ { f } ℓ , $$\ell \ge 1$$ ℓ ≥ 1 , to the powers $$N^{j}$$ N j , $$j\ge 1$$ j ≥ 1 . Our proof, which is inspired by a result of Varchenko obtained in 1981, uses the Laurent series (asymptotic) expansions of elements in the Jacobian module with respect to the Malgrange–Kashiwara’s $$\mathcal {V}$$ V -filtration. Finally, when the relation between Saito pairing and Grothendieck pairing is considered such a formula provides us with a result that gives an additive expansion for each higher bilinear form on $$\Omega ^f$$ Ω f expressed in terms of the higher bilinear forms on H and depending on the asymptotic expansions for the top forms on $$\Omega ^f$$ Ω f where these bilinear forms act.

Oleksik, Grzegorz; Różycki, Adam

doi: 10.1007/s00574-018-0078-8pmid: N/A

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.

Szirmai, Jenő

doi: 10.1007/s00574-018-0077-9pmid: N/A

In this paper we study the interior angle sums of geodesic triangles in $$\mathbf {Nil}$$ Nil geometry and prove that these can be larger, equal or less than $$\pi $$ π . We use for the computations the projective model of $$\mathbf {Nil}$$ Nil introduced by Molnár (Beitr. Algebra Geom. 38(2):261–288, 1997).

Kenmotsu, Katsuei

doi: 10.1007/s00574-018-0081-0pmid: N/A

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.

Hefez, A.; Rodrigues, J.; Salomão, R.

doi: 10.1007/s00574-018-0080-1pmid: N/A

The Milnor number of an isolated hypersurface singularity, defined as the codimension $$\mu (f)$$ μ ( 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 $$\mu (f)$$ μ ( 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.

Santos, J.; Matte, M.

doi: 10.1007/s00574-018-0082-zpmid: N/A

In this work we define a new set of integer partition, based on a lattice path in $${\mathbb {Z}}^2$$ Z 2 connecting the line $$x+y=n$$ 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 $$T_1^*(q)=\sum _{n=0}^{\infty }\dfrac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}}$$ T 1 ∗ ( q ) = ∑ n = 0 ∞ q n ( n + 1 ) ( - q 2 , q 2 ) n ( q , q 2 ) n + 1 .

Amri, Besma

doi: 10.1007/s00574-018-0083-ypmid: N/A

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 $$\mathfrak {T}_{\varphi ,\psi }(\sigma )$$ T φ , ψ ( σ ) associated with two wavelets $$\varphi ,\psi $$ φ , ψ and with symbol $$\sigma .$$ σ . We establish the boundedness and compactness of these operators. Last, we define the Schatten-von Neumann class $$S^p\ ;\ p\in \ [1,+\infty ],$$ S p ; p ∈ [ 1 , + ∞ ] , and we show that the Toeplitz operators belong to the class $$S^p$$ S p and we prove a formula of trace.

Bronzi, M.; Oler, J.

doi: 10.1007/s00574-018-0084-xpmid: N/A

Let $$L:[0,1]{\setminus }\{d\}\rightarrow [0,1]$$ L : [ 0 , 1 ] \ { d } → [ 0 , 1 ] be a one-dimensional Lorenz-like expanding map (d is the point of discontinuity), $$\mathcal {P}=\{ (0,d),(d,1) \}$$ P = { ( 0 , d ) , ( d , 1 ) } and $$C^{\alpha }([0,1],{\mathcal {P}})$$ C α ( [ 0 , 1 ] , P ) the set of piecewise Hölder-continuous potentials of [0, 1] with the usual $$\mathcal {C}^0$$ C 0 topology. In this context, applying a criteria by Buzzi and Sarig (Ergod Theory Dyn Syst 23(5):1383–1400, 2003, Th. 1.3), we prove that there exists an open and dense subset $$\mathcal {H}$$ H of $$C^{\alpha }([0,1],{\mathcal {P}})$$ C α ( [ 0 , 1 ] , P ) , such that each $$\phi \in \mathcal {H}$$ ϕ ∈ H admits exactly one equilibrium state.