Narrow Results

In this paper, we investigate the fractal nature of the local fractional Landau–Ginzburg–Higgs Equation (LFLGHE) describing nonlinear waves with weak scattering in a fractal medium. The main goal of the paper is to introduce and apply the Local Fractional Elzaki Variational Iteration Method (LFEVIM) for solution of LFLGHE. Convergence analysis of LFEVIM solution for general nonlinear local fractional partial differential equation is also provided. Two examples of the local fractional LFLGHE are considered to demonstrate the applicability of the proposed technique with numerical simulations on Cantor set.

For $n\in \mathbb{N}$ the $n$th alternating harmonic number

Approximations of nonhomogeneous discrete Markov chains (NDMC) play an essential role in both probability and statistics. In all these settings, it is crucial to consider random variables in appropriate spaces. Therefore, the abstract considerations of such spaces lead to investigating the approximations in ordered Banach space scheme. In this paper, we consider two topologies on the set of NDMC of abstract state spaces. We establish that the set of all uniformly $P$-ergodic NDMC is norm residual in NDMC. The set of point-wise weak $P$-ergodic NDMC is also considered and such sets are shown to be a $G_{\delta }$-subset (in strong topology) of NDMC. We point out that all the deduced results are new in the classical and non-commutative probabilities, respectively, since in most of earlier results the limiting projection is taken as a rank one projection. Indeed, the obtained results give new insight into data-analysis and statistics.

In this survey, we present several results on the regularizing effect, rigidity and approximation of 2D unit-length divergence-free vector fields. We develop the concept of entropy (coming from scalar conservation laws) in order to analyze singularities of such vector fields. In particular, based on entropies, we characterize lower semicontinuous line-energies in 2D and we study by Γ-convergence method the associated regularizing models (like the 2D Aviles–Giga and the 3D Bloch wall models). We also present some applications to the analysis of pattern formation in micromagnetics. In particular, we describe domain walls in the thin ferromagnetic films (e.g. symmetric Néel walls, asymmetric Néel walls, asymmetric Bloch walls) together with interior and boundary vortices.

The Poset Cover Problem is the problem where the input is a set of linear orders and the goal is to find a minimum set of posets that generates exactly all the given linear orders. The computational complexity of the decision version of the problem is already known; it is NP-Complete. However, the approximation complexity or approximability of the problem is not yet known.

In this paper, we show the approximability of two simple variations of the problem where the poset being considered are Hammock posets having hammocks of size 2. The first is Hammock(2, 2, 2)-Poset Cover Problem where the solution is a set of Hammock posets with 3 hammocks of size 2. This problem has been shown to be NP-Complete and in this paper we show that it is 2.7-approximable. The second variation which is more general than the first one considers Hammock posets with any number of hammocks of size 2 and we show that it is $H(n)-\frac{196}{300}-\text{approximable}$.