ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition \(c_2=1\) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with \(c_2=1\) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine \(2\)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with \(c_2=1\) that admit an automorphism group acting \(2\)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with \(c_2=1\) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.

Tuning Hybrid Nano-semiconductor-glass Via High Intensity Laser

Conceptually, the high intensity laser represents the simplest thin film deposition techniques that consists of both a target and a substrate holders housed in a vacuum chamber with a high powered pulsed laser as the external energy source for evaporation of target material (Semiconductor Glass). Using deposit thin laser films three ranges of frequencies were produced: (0-15,000 mJ/cm2) as a result tuning of semiconductor was satisfying condition, while the second, 0-33,000 mJ/cm2 as a result tuning of semiconductor had a stable condition and the last 0-100,000 mJ/cm2 as a result tuning of semiconductor was unstable condition. The results demonstrate a decrease in resistance due to charging the semiconductor glass by high intensity laser as well as a superior charge efficiency and lifetime of semiconductor glass coated cells compared to high intensity laser. The current increase in the charge appeared proportional with extra energy stored of the semiconductor glass coated electrodes at 2Co (>23%) in comparison with control. It is concluded that an increase in the capacity of semiconductor glass may address the main difficulty for utilizing the high intensity laser chemistry for future demands.

Fixed Point Approximation for a Class of Generalized Nonexpansive Mappings in Hadamard Spaces

In this paper, we establish strong and Δ convergence results for mappings satisfying condition B γ , μ through a newly introduced iterative process called JA iteration process. A nonlinear Hadamard space is used the ground space for establishing our main results. A novel example is provided for the support of our main results and claims. The presented results are the good extension of the corresponding results present in the literature.

AN ITERATIVE SCHEME FOR FIXED POINT PROBLEMS

In this paper, we introduce a new three steps iteration process, prove that our newly proposed iterative scheme can be used to approximate the fixed point of a contractive-like mapping and establish some convergence results for our newly proposed iterative scheme generated by a mapping satisfying condition (E) in the framework of uniformly convex Banach space. In addition, with the aid of numerical examples, we established that our newly proposed iterative scheme is faster than the iterative process introduced by Ullah et al., [26], Karakaya et al., [16], Abass et. al. [1] and some existing iterative scheme in literature. More so, the stability of our newly proposed iterative process is presented and we also gave some numerical examples to display the efficiency of our proposed algorithm.

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$ -groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$ , the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$ , which implies hyperbolicity.

On variational nonlinear equations with monotone operators

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

Iterative Approximations for a Class of Generalized Nonexpansive Operators in Banach Spaces

In this article, we prove some weak and strong convergence theorems for mappings satisfying condition (E) using the AK iterative scheme in the setting of Banach spaces. We offer a new example of mapping with condition (E) in support of our main result. Our results extend and improve many well-known corresponding results of the current literature.

On groups with action on itself

We introduce the notions of center, singularity and nilpotency (class) of a group with action on itself. Also, we describe a new package GwA for GAP4, including functions checking some properties of groups with action on itself with finite underlying group. As applications of the implemented functions, we give examples of groups with action satisfying Condition 1 stated in [3]. In other words, we get concrete examples of “coquecigrue” in the terminology of Loday [8, 9].

On coefficients of some p-valent starlike functions

We consider the class Ap of functions f analytic in the unit disk |z| < 1 in the complex plane, of the form f(z) = zp + ... such that Rezf(p)(z)/f (p-1)(z) > 0 in the unit disc. The object of the present paper is to derive some bounds for coefficients in this class and relation with the functions satisfying condition Ref(k)(z)/f(p-k)(z) > 0 in the unit disc.