# César Adolfo Hernández MeloUniversidade Estadual de Maringá | UEM · Departamento de Matemática

César Adolfo Hernández Melo

Ph. D.

## About

20

Publications

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33

Citations

Introduction

**Skills and Expertise**

## Publications

Publications (20)

Exploring the epsilon-delta definition of continuity, we purpose a method to analyze the uniform continuity of real-valuad functions of one real variable.

This paper considers a one-dimensional generalized Allen-Cahn equation of the form \[ u_t = \varepsilon^2 (D(u)u_x)_x - f(u), \] where $\varepsilon>0$ is constant, $D=D(u)$ is a positive, uniformly bounded below diffusivity coefficient that depends on the phase field $u$ and $f(u)$ is a reaction function that can be derived from a double-well poten...

In this paper we provide detailed information about the instability of equilibrium solutions of a nonlinear family of localized reaction-difussion equations in dimensione one. Beyond we provide explicit formulas to the equilibrium solutions, via perturbation method and we calculate the exact number of positive eigenvalues of the linear operator ass...

In the present work, a formula is provided for determining the idempotent elements of a commutative ring R from those of the quotient ring R/N, where N is in most cases a nilpotent ideal of R. As an application of this formula, idempotent elements of certain commutative rings are described. Several examples are included illustrating the main result...

In this paper, a nonlinear Schrödinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that...

Let R be a commutative ring with a collection of ideals \(\{ N_1, N_2, \dots , N_{k-1}\}\) satisfying certain conditions, properties of the set of invertible quadratic residues of the ring R are described in terms of properties of the set of invertible quadratic residues of the quotient ring \(R/N_1\)

Let $R$ be a commutative ring with a collection of ideals $\{ N_1, N_2, \dots, N_{k-1}\}$ satisfying certain conditions, properties of the set of invertible quadratic residues of the ring $R$ are described in terms of properties of the set of invertible quadratic residues of the quotient ring $R/N_1$.

In this work a linear complementary dual code associated to the Haar wavelet transform over the finite field Zp for certain values for p is given. Examples are presented illustrating the results.

Considering Zn the ring of integers modulo n, the classical Fermat-Euler theorem establishes the existence of a specific natural number φ(n) satisfying the following property: xφ(n)=1
for all x belonging to the group of units of Zn. In this manuscript, this result is extended to a class of rings that satisfies some mild conditions.

Considering $\mathbb{Z}_n$ the ring of integers modulo $n$, the classical Fermat-Euler theorem establishes the existence of a specific natural number $\varphi(n)$ satisfying the following property: $ x^{\varphi(n)}=1%\hspace{1.0cm}\text{for all}\hspace{0.2cm}x\in \mathbb{Z}_n^*, $ for all $x$ belonging to the group of units of $\mathbb{Z}_n$. In th...

This paper considers a one-dimensional generalized Allen–Cahn equation of the form $$\begin{aligned} u_t = \varepsilon ^2 (D(u)u_x)_x - f(u), \end{aligned}$$where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field u, and f(u) is a reaction function that can b...

Let $R$ be an associative ring with identity and let $N$ be a nil ideal of $R$. It is shown that units of $R/N$ can be lifted to units in $R$. Under some mild conditions on the ring, a procedure is given to determine those lifted units in a recursive way. As an application, the units of several classes of rings are determined, including: matrix rin...

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrödinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literatur...

In this paper, a nonlinear Schr\"odinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved tha...

This problem deals with the convergence and divergence of a serie whose nth term is defined by a convex differentiable function.

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrodinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literatur...

Under certain general conditions, an explicit formula to compute the greatest delta-epsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given. Several examples illustrating the theory are discussed.

We present two heuristic methods to get epsilon-delta
proofs. From these methods, a new approach to study uniform continuity
of real functions comes up. In addition, some results on uniform
continuity of homeomorphisms in the real line are established.

The main purpose of this paper is to investigate the existence and stability of periodic and non-periodic equilibrium solutions related to the nonlinear heat equation: u(t) = U-xx, + wu + u(3) + u(5). (1) The existence of periodic equilibriums with a fixed period L is deduced from the Theory of Jacobian Elliptical Functions and the Implicit Functio...

For bounded and unbounded domains in R3, we establish the localization and the structure of the spectrum of normal vibrations described by systems of partial differential equations modelling small displacements of compressible stratified fluid in the homogeneous gravity field. We also compare the spectral properties of gravitational and rotational...