arXiv subjects

Ady Cambraia Jr

Publications and source records attributed to Ady Cambraia Jr.

4 recordsLinked to original sources

arXiv2021

On prime factors of Mersenne numbers

Let $(M_n)_{n\geq0}$ be the Mersenne sequence defined by $M_n=2^n-1$. Let $ω(n)$ be the number of distinct prime divisors of $n.$ In this short note, we present a description of the Mersenne numbers satisfying $ω(M_n)\leq3$. Moreover, we prove that the inequality, given $ε>0$, $ω(M_n)> 2^{(1-ε)\log\log n} -3 $ holds for almost all positive integers $n$. Besides, we present the integer solutions $(m,n,a)$ of the equation $M_m+M_n=2p^a$ with $m,n\geq2$, $p$ an odd prime number and $a$ a positive integer.

math.NT↗

arXiv2020

On Totally umbilical surfaces in the warped product $\mathbb{M}(κ)_f\times\mathbb{R}$

In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped product $\mathbb{M}(κ)_f\times I$ (here, $\mathbb{M}(κ)$ denotes the 2-dimensional space form, having constant curvature $κ$, $I$ an interval and $f$ the warping function) is invariant by an one-parameter group of isometries of the ambient space. We also find the first integral of the ordinary differential equation that the profile curve satisfies (we mean, the curve which generates a invariant totally umbilical surface). Moreover, we construct explicit examples of totally umbilical surfaces, invariant by one-parameter group of isometries of the ambient space, by considering certain non-trivial warping function.

math.DG↗

arXiv2019

Envelope of intermediate lines of a plane curve

For a pair of points in a smooth closed convex planar curve $γ$, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines ($EML$) is formed by the union of three affine invariants sets: Affine Envelope Symmetry Sets ($AESS$); Mid-Parallel Tangent Locus ($MPTL$) and Affine Evolute of $γ$. In this paper, we generalized these concepts by considering the envelope of the intermediate lines. For a pair of points of $γ$, its intermediate line is the line containing an intermediate point and the intersection point of the corresponding pair of tangent lines. Here, we present the envelope of intermediate lines ($EIL$) of the curve $γ$ and prove that this set is formed by three disconnected sets when the intermediate point is different from the mid-point: Affine Envelope of Intermediate Lines ($AEIL$); the curve $γ$ itself and the Intermediate-Parallel Tangent Locus ($IPTL$). When the intermediate point coincides with the mid-point, the $EIL$ coincides with the $EML$, and thus these sets are connected. Moreover, we introduce some standard techniques of singularity theory and use them to explain the local behavior of this set.

math.DG↗

arXiv2017

An additional structure over integer rings $\mathbb{Z}_{p^r}^n$

We present an algebraic structure in modules over integer rings with cardinality prime powers, which allows to define bases. With such structure, we prove a similar version for the basis extension theorem of linear algebra over fields. Moreover, we exhibit results involving the modules and their duals.

math.RA↗