Volume 70 , issue 1 ( 2018 ) | back |

Local convergence of bilinear operator free methods under weak conditions | 1$-$11 |

**Abstract**

We study third-order Newton-type methods free of bilinear operators for solving nonlinear equations in Banach spaces. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

**Keywords:** ewton's method; bilinear operator; radius of convergence; local convergence.

**MSC:** 49M15, 74G20, 41A25

Time-like Hamiltonian dynamical systems in Minkowski space $\mathbb{r}^3_1$ and the nonlinear evolution equations | 12$-$25 |

**Abstract**

We show that all of the curve motions specified in the Frenet-Serret frame are described by the time evolution of an integral curve of a time-like Hamiltonian dynamical system in Minkowski space such that the integral curve under consideration is a geodesic curve on a leaf of the foliation determined by the Poisson structure. Accordingly, any nonlinear soliton equation related to curve dynamics is obtained as the time evolution of an integral curve of a Hamiltonian system. As an expository example, we define Hashimoto function in the Darboux frame which is reduced to the classical Hashimoto function provided that the Poisson vector corresponds to principal normal of an integral curve and show that the defocusing version of the nonlinear Schrödinger equation and the mKdV equation are obtained by the time evolution of this function.

**Keywords:** Hamiltonian systems; Poisson structure; motion of curves; Minkowski space; Darboux frame; geodesic line; Hashimoto function.

**MSC:** 37E35, 53B30, 53C44, 53D17, 53Z05

Initial--boundary value problems for Fuss-Winkler-Zimmermann and Swift--Hohenberg nonlinear equations of 4th order | 26$-$39 |

**Abstract**

We show that all of the curve motions specified in the Frenet-Serret frame are described by the time evolution of an integral curve of a time-like Hamiltonian dynamical system in Minkowski space such that the integral curve under consideration is a geodesic curve on a leaf of the foliation determined by the Poisson structure. Accordingly, any nonlinear soliton equation related to curve dynamics is obtained as the time evolution of an integral curve of a Hamiltonian system. As an expository example, we define Hashimoto function in the Darboux frame which is reduced to the classical Hashimoto function provided that the Poisson vector corresponds to principal normal of an integral curve and show that the defocusing version of the nonlinear Schrödinger equation and the mKdV equation are obtained by the time evolution of this function.

**Keywords:** This paper presents results of the investigation
of bifurcations of stationary solutions of the Swift--Hohenberg
equation and dynamic descent to the points of minimal values of the
functional of energy for this equation, obtained with the use of the
modification of the Lyapunov--Schmidt variation method and some
methods from the theory of singularities of smooth functions.
Nonstationary case is investigated by the construction of paths of
descent along the trajectories of the infinite-dimensional SH
dynamical system from arbitrary initial states to points of
the minimum energy.

**MSC:** 37M20, 35Q99, 34K18, 34C25

Blending type approximation by Bernstein-Durrmeyer type operators | 40$-$54 |

**Abstract**

In this note, we introduce the Durrmeyer variant of Stancu operators that preserve the constant functions depending on non-negative parameters. We give a global approximation theorem in terms of the Ditzian-Totik modulus of smoothness, a Voronovskaja type theorem and a local approximation theorem by means of second order modulus of continuity. Also, we obtain the rate of approximation for absolutely continuous functions having a derivative equivalent with a function of bounded variation. Lastly, we compare the rate of approximation of the Stancu-Durrmeyer operators and genuine Bernstein-Durrmeyer operators to certain function by illustrative graphics with the help of the Mathematica software.

**Keywords:** Stancu operators; global approximation; rate of convergence; \mbox{modulus} of continuity; Steklov mean.

**MSC:** 41A25, 26A15

New congruences for overcubic partition function | 55$-$63 |

**Abstract**

In 2010, Byungchan Kim introduced a new class of partition function $\overline{a}(n)$, the number of overcubic partitions of $n$ and established $\overline{a}(3n+2)\equiv 0\pmod{3}$. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan's congruences for the unrestricted partition function $p(n)$. We prove a number of results for $\overline{a}(n)$, for example, for $\alpha \ge 0$ and $n \ge 0$, $\overline{a}(8n+5)\equiv 0\pmod{16}$, $\overline{a}(8n+7)\equiv 0\pmod{32}$, $\overline{a}(8\cdot 3^{2\alpha+2}n+3^{2\alpha+2})\equiv 3^{\alpha} \overline{a}(8n+1)\pmod{8}$.

**Keywords:** Overcubic partitions; congruences; theta function.

**MSC:** 11P83, 05A15, 05A17

$\Theta \Gamma$ $N$-group | 64$-$78 |

**Abstract**

In this paper, we introduce the notion of $\Theta \Gamma$ $N$-group as a generalization of algebraic structures of $N$-group and gamma nearring. We present motivating examples of $\Theta \Gamma$ $N$-groups and prove classical isomorphism theorems.

**Keywords:** $N$-group, nearring, gamma nearring.

**MSC:** 16Y30

Generalized Raychaudhuri's equation for null hypersurfaces | 79$-$88 |

**Abstract**

Black hole kinematics and laws governing their event horizons in spacetimes are usually based on the expansion properties of families of null geodesics which generate such horizons. Raychaudhuri's equation is one of the most important tools in investigating the evolution of such geodesics. In this paper, we use the so-called Newton transformations to give a generalized vorticity-free Raychaudhuri's equation (Theorem $3.1$), with a corresponding null global splitting theorem (Theorem $3.5$) for null hypersurfaces in Lorentzian spacetimes. Two supporting physical models are also given.

**Keywords:** Null hypersurfaces; null horizons; Newton tranformations; mean curvature; black hole.

**MSC:** 53C42, 53C50, 53C80

Error locating codes and extended Hamming code | 89$-$94 |

**Abstract**

Error-locating codes, first proposed by J.~K. Wolf and B. Elspas, are used in fault diagnosis in computer systems and reduction of the retransmission cost in communication systems. This paper presents locating codes obtained from the famous Extended $(8, 4)$ Hamming code capable of identifying the sub-block that contains solid burst errors of length $2$ (or $3$) or less. We also make a comparison of information rate between the extended Hamming code and obtained codes. Further, comparisons in solid burst error detection and location probabilities of the codes over binary symmetrical channel are also provided.

**Keywords:** Parity check matrix, solid burst, syndrome, EL-codes, information rate, error probability.

**MSC:** 94B05, 94B20, 94B60