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 Bulletin of the Brazilian Mathematical Society, New Series   [SJR: 0.436]   [H-I: 19]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1678-7544 - ISSN (Online) 1678-7714    Published by Springer-Verlag  [2350 journals]
• Automorphisms of Ideals of Polynomial Rings
• Authors: Tiago Macedo; Thiago Castilho de Mello
Pages: 1 - 15
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: 2018-03-01
DOI: 10.1007/s00574-017-0046-8
Issue No: Vol. 49, No. 1 (2018)

• Large Deviations for Equilibrium Measures and Selection of Subaction
• Authors: Jairo K. Mengue
Pages: 17 - 42
Abstract: Given a Lipschitz function $$f:\{1,\ldots ,d\}^\mathbb {N} \rightarrow \mathbb {R}$$ , for each $$\beta >0$$ we denote by $$\mu _\beta$$ the equilibrium measure of $$\beta f$$ and by $$h_\beta$$ the main eigenfunction of the Ruelle Operator $$L_{\beta f}$$ . Assuming that $$\{\mu _{\beta }\}_{\beta >0}$$ satisfy a large deviation principle, we prove the existence of the uniform limit $$V= \lim _{\beta \rightarrow +\infty }\frac{1}{\beta }\log (h_{\beta })$$ . Furthermore, the expression of the deviation function is determined by its values at the points of the union of the supports of maximizing measures. We study a class of potentials having two ergodic maximizing measures and prove that a L.D.P. is satisfied. The deviation function is explicitly exhibited and does not coincide with the one that appears in the paper by Baraviera-Lopes-Thieullen which considers the case of potentials having a unique maximizing measure.
PubDate: 2018-03-01
DOI: 10.1007/s00574-017-0044-x
Issue No: Vol. 49, No. 1 (2018)

• Liouville Type Results for Two-Sided Hypersurfaces in Weighted Killing
Warped Products
• Authors: Henrique F. de Lima; Eraldo Lima; Adriano Medeiros; Márcio S. Santos
Pages: 43 - 55
Abstract: We establish Liouville type results concerning two-sided hypersurfaces immersed in a weighted Killing warped product, under suitable constraints either on the Bakry-Émery-Ricci tensor of the base of the ambient space or on the height function of the hypersurface.
PubDate: 2018-03-01
DOI: 10.1007/s00574-017-0043-y
Issue No: Vol. 49, No. 1 (2018)

• Modular Invariants of Finite Affine Linear Groups
• Authors: Yin Chen
Pages: 57 - 72
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: 2018-03-01
DOI: 10.1007/s00574-017-0050-z
Issue No: Vol. 49, No. 1 (2018)

• Positivity of the Top Lyapunov Exponent for Cocycles on Semisimple Lie
Groups over Hyperbolic Bases
• Authors: Mário Bessa; Jairo Bochi; Michel Cambrainha; Carlos Matheus; Paulo Varandas; Disheng Xu
Pages: 73 - 87
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: 2018-03-01
DOI: 10.1007/s00574-017-0048-6
Issue No: Vol. 49, No. 1 (2018)

• On the Continuity of the Topological Entropy of Non-autonomous Dynamical
Systems
• Authors: Jeovanny de Jesus Muentes Acevedo
Pages: 89 - 106
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: 2018-03-01
DOI: 10.1007/s00574-017-0049-5
Issue No: Vol. 49, No. 1 (2018)

• p -Harmonic l -forms on Complete Noncompact Submanifolds in Sphere with
Flat Normal Bundle
• Authors: Yingbo Han
Pages: 107 - 122
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: 2018-03-01
DOI: 10.1007/s00574-017-0051-y
Issue No: Vol. 49, No. 1 (2018)

• Problems with Mean Curvature-Like Operators and Three-Point Boundary
Conditions
• Authors: Dionicio Pastor Dallos Santos
Pages: 123 - 136
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: 2018-03-01
DOI: 10.1007/s00574-017-0052-x
Issue No: Vol. 49, No. 1 (2018)

• Polynomial Differential Systems in $$\mathbb {R}^3$$ R 3 Having Invariant
Weighted Homogeneous Surfaces
• Authors: Thaís Maria Dalbelo; Marcelo Messias; Alisson C. Reinol
Pages: 137 - 157
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: 2018-03-01
DOI: 10.1007/s00574-017-0045-9
Issue No: Vol. 49, No. 1 (2018)

• A New Approach on the Energy of Elastica and Non-Elastica in Minkowski
Space E $$_{2}^{4}$$ 2 4
• Authors: Talat Körpinar; Rıdvan Cem Demirkol
Pages: 159 - 177
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: 2018-03-01
DOI: 10.1007/s00574-017-0047-7
Issue No: Vol. 49, No. 1 (2018)

• Borsuk-Ulam Theorems and Their Parametrized Versions for $$\mathbb {F}P^m\times \mathbb {S}^3$$ F P m × S 3
• Authors: Somorjit Konthoujam Singh; Hemant Kumar Singh; Tej Bahadur Singh
Pages: 179 - 197
Abstract: Let $$G=\mathbb {Z}_p,$$ $$p>2$$ a prime, act freely on a finitistic space X with mod p cohomology ring isomorphic to that of $$\mathbb {F}P^m\times \mathbb {S}^3$$ , where $$m+1\not \equiv 0$$ mod p and $$\mathbb {F}=\mathbb {C}$$ or $$\mathbb {H}$$ . We wish to discuss the nonexistence of G-equivariant maps $$\mathbb {S}^{2q-1}\rightarrow X$$ and $$X\rightarrow \mathbb {S}^{2q-1}$$ , where $$\mathbb {S}^{2q-1}$$ is equipped with a free G-action. These results are analogues of the celebrated Borsuk-Ulam theorem. To establish these results first we find the cohomology algebra of orbit spaces of free G-actions on X. For a continuous map $$f\!:\! X\rightarrow \mathbb {R}^n$$ , a lower bound of the cohomological dimension of the partial coincidence set of f is determined. Furthermore, we approximate the size of the zero set of a fibre preserving G-equivariant map between a fibre bundle with fibre X and a vector bundle. An estimate of the size of the G-coincidence set of a fibre preserving map is also obtained. These results are parametrized versions of the Borsuk-Ulam theorem.
PubDate: 2018-03-01
DOI: 10.1007/s00574-017-0040-1
Issue No: Vol. 49, No. 1 (2018)

• On the Algebraic Structure and the Number of Zeros of Abelian Integral for
a Class of Hamiltonians with Degenerate Singularities
• 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

• Equilibrium State for One-Dimensional Lorenz-Like Expanding Maps
• Authors: M. A. Bronzi; J. G. Oler
Abstract: Let $$L:[0,1]{\setminus }\{d\}\rightarrow [0,1]$$ be a one-dimensional Lorenz-like expanding map (d is the point of discontinuity), $$\mathcal {P}=\{ (0,d),(d,1) \}$$ and $$C^{\alpha }([0,1],{\mathcal {P}})$$ the set of piecewise Hölder-continuous potentials of [0, 1] with the usual $$\mathcal {C}^0$$ 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 $$\mathcal {H}$$ of $$C^{\alpha }([0,1],{\mathcal {P}})$$ , such that each $$\phi \in \mathcal {H}$$ admits exactly one equilibrium state.
PubDate: 2018-04-06
DOI: 10.1007/s00574-018-0084-x

• Toeplitz Operators for Wavelet Transform Related to the Spherical Mean
Operator
• 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

• A New Approach to Integer Partitions
• 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

• The Milnor Number of Plane Branches with Tame Semigroups of Values
• 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

• Complete Parallel Mean Curvature Surfaces in Two-Dimensional Complex
Space-Forms
• 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

• $$\mathbf {Nil}$$ Nil Geodesic Triangles and Their Interior Angle Sums
• 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

• The Łojasiewicz Exponent at Infinity of Non-negative and
Non-degenerate Polynomials
• 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

• On a Lemma of Varchenko and Higher Bilinear Forms Induced by Grothendieck
Duality on the Milnor Algebra of an Isolated Hypersurface Singularity
• Authors: M. A. Dela-Rosa
Abstract: For an isolated hypersurface singularity $$f:(\mathbb {C}^{n+1},0)\rightarrow (\mathbb {C},0)$$ with Milnor number $$\mu$$ and good representative $$f:(X,0)\rightarrow (\Delta ,0)$$ canonical $$\mu$$ -dimensional $$\mathbb {C}$$ -bilinear vector spaces are associated: the Jacobian module, $$\Omega ^{f}$$ , which is isomorphic to the Milnor algebra $$A_f$$ up to a choice of coordinates; and the cohomology of the canonical Milnor fiber, H. Indeed, one has defined on $$\Omega ^f$$ , and hence in $$A_f$$ , the non-degenerate Grothendieck pairing $$res_{f,0}$$ which is a symmetric $$\mathbb {C}$$ -bilinear form, and on the vanishing cohomology H it is defined a non-degenerate $$\mathbb {C}$$ -bilinear form $$\mathbb {S}$$ , induced by Poincaré duality, which is $$(-1)^{n+1}$$ -symmetric on the generalized monodromy eigenspace $$H_{1}$$ and $$(-1)^{n}$$ -symmetric on the direct sum of generalized monodromy eigenspaces $$H_{\ne 1}:=\oplus _{\lambda \ne 1}H_{\lambda }$$ . On the other hand, there are two nilpotent $$\mathbb {C}$$ -linear maps defined on $$\Omega ^f$$ and H, respectively; the first one is the map $$\{\mathbf {f}\}$$ given by multiplication with f, which is $$res_{f,0}$$ -symmetric, and the other one is the $$\mathbb {S}$$ -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: $$res_{f,0}(\{\mathbf {f}\}^{\ell }\bullet ,\bullet )$$ and $$\mathbb {S}(N^{\ell }\bullet ,\bullet )$$ for each integer $$\ell \ge 1$$ . These new bilinear forms are called higher bilinear forms on $$\Omega ^f$$ resp. on H. In this paper, we show a formula which relates the powers $$\{\mathbf {f}\}^{\ell }$$ ,
PubDate: 2018-03-01
DOI: 10.1007/s00574-018-0075-y

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