Bulletin of the Brazilian Mathematical Society, New Series

Viana, Marcelo

doi: 10.1007/s00574-018-00125-wpmid: N/A

Abadie, Fernando; Buss, Alcides; Ferraro, Damián

doi: 10.1007/s00574-018-0088-6pmid: N/A

We introduce notions of weak and strong equivalence for non-saturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional $${\mathrm {C}}^*$$ C ∗ -algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.

Fukunaga, Tomonori; Takahashi, Masatomo

doi: 10.1007/s00574-018-0090-zpmid: N/A

A framed surface is a smooth surface in the Euclidean space with a moving frame. The framed surfaces may have singularities. We treat smooth surfaces with singular points, that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves.

Chirre, Andrés

doi: 10.1007/s00574-018-0092-xpmid: N/A

In this paper, we exhibit upper and lower bounds with explicit constants for some objects related to entire L-functions in the critical strip, under the generalized Riemann hypothesis. The examples include the entire Dirichlet L-functions $$L(s,\chi )$$ L ( s , χ ) for primitive characters $$\chi $$ χ .

Michalska, Maria; Walewska, Justyna

doi: 10.1007/s00574-018-0094-8pmid: N/A

The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularity f. Assuming that f is irreducible, one can write $$ f = \sum _{q\alpha + p\beta \ \ge \ pq}c_{\alpha \beta }\ x^{\alpha }y^{\beta } $$ f = ∑ q α + p β ≥ p q c α β x α y β where $$c_{p0}c_{0q}\ne 0$$ c p 0 c 0 q ≠ 0 , $$2\le p<q$$ 2 ≤ p < q and p, q are coprime. We show that as Milnor numbers of deformations of f one can attain all numbers from $$\mu (f)$$ μ ( f ) to $$\mu (f)-r(p-r)$$ μ ( f ) - r ( p - r ) , where $$q\equiv r (\mathrm{mod\ }p)$$ q ≡ r ( mod p ) . Moreover, we provide an algorithm which produces the desired deformations.

Lima, Ronaldo; Andrade, Rubens

doi: 10.1007/s00574-018-0095-7pmid: N/A

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by do Carmo and Lima, as well as by do Carmo and Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem for hypersurfaces with prescribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one.

Anciaux, Henri; Bayard, Pierre

doi: 10.1007/s00574-018-0096-6pmid: N/A

It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface $$\mathcal{S}$$ S the set of oriented lines that cross $$\mathcal{S}$$ S orthogonally is a Lagrangian submanifold. Conversely, if $$\overline{\mathcal{S}}$$ S ¯ an n-dimensional family of oriented lines is Lagrangian, there exists, locally, a 1-parameter family of immersed, oriented, parallel hypersurfaces $$\mathcal{S}_t$$ S t whose tangent spaces cross orthogonally the lines of $$\overline{\mathcal{S}}.$$ S ¯ . The purpose of this paper is to generalize these facts to higher dimension: to any point x of a submanifold $$\mathcal{S}$$ S of $${\mathbb {R}^{}} ^m$$ R m of dimension n and co-dimension $$k=m-n,$$ k = m - n , we may associate the affine k-space normal to $$\mathcal{S}$$ S at x. Conversely, given an n-dimensional family $$\overline{\mathcal{S}}$$ S ¯ of affine k-spaces of $${\mathbb {R}^{}} ^m$$ R m , we provide certain conditions granting the local existence of a family of n-dimensional submanifolds $$\mathcal{S}$$ S which cross orthogonally the affine k-spaces of $$\overline{\mathcal{S}}$$ S ¯ . We also define a curvature tensor for a general family of affine spaces of $${\mathbb {R}^{}} ^m$$ R m which generalizes the curvature of a submanifold, and, in the case of a 2-dimensional family of 2-planes in $${\mathbb {R}^{}} ^4$$ R 4 , show that it satisfies a generalized Gauss–Bonnet formula.

Zaeim, Amirhesam; Karami, Ramisa

doi: 10.1007/s00574-018-0097-5pmid: N/A

Geometry of four dimensional pseudo-Riemannian Lie groups of signature (2, 2) studied. A rich family of Einstein, locally symmetric and conformally flat examples is presented.

Ghosh, Sekhar; Choudhuri, Debajyoti; Giri, Ratan

doi: 10.1007/s00574-018-0100-1pmid: N/A

In this paper we prove existence of solutions for a partial differential equation involving a singularity with a general nonnegative, Radon measure as source term which is given as $$\begin{aligned} (-\Delta )^s u&= f(x)h(u)+\mu ~\text {in}~\Omega , \\ u&=0~\text {in}~\mathbb {R}^N{\setminus }\Omega , \\ u&> 0~\text {in}~\Omega , \end{aligned}$$ ( - Δ ) s u = f ( x ) h ( u ) + μ in Ω , u = 0 in R N \ Ω , u > 0 in Ω , where $$\Omega $$ Ω is a bounded domain of $$\mathbb {R}^N$$ R N , f is a nonnegative function over $$\Omega $$ Ω .

Sun, Qinxiu; Akinocho, Kamalou

doi: 10.1007/s00574-018-0105-9pmid: N/A

The goal of this paper is to study cohomological theory of Hom-associative H-pseudoalgebras and Hom–Lie H-pseudoalgebras. We define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative H-pseudoalgebra. Furthermore, the symmetric Schouten product and alternating Schouten product are studied. Using the Gerstenhaber bracket and alternating Schouten product, differential graded Lie algebra are constructed on the space of multilinear mappings of Hom-associative H-pseudoalgebra and Hom-Lie H-pseudoalgebras.