Generalized Close-to-Convexity Related with Bounded Boundary Rotation

The class Pα,m[A, B] consists of functions p, analytic in the open unit disc E with p(0) = 1 and satisfy p(z) = (m/4 + ½) p1(z) – (m/4 – 1/2) p2(z), m ≥ 2, and p1, p2 are subordinate to strongly Janowski function (1+Az/1+Bz)α, α ∈ (0, 1] and −1 ≤ B < A ≤ 1. The class Pα,m[A, B] is used to define Vα,m[A, B] and Tα,m[A, B; 0; B1], B1 ∈ [−1, 0). These classes generalize the concept of bounded boundary rotation and strongly close-to-convexity, respectively. In this paper, we study coefficient bounds, radius problem and several other interesting properties of these functions. Special cases and consequences of main results are also deduced.

The evolution of triple junctions: from failure to success

Divergent triple junctions are stable plate margins where three spreading ridges meet. Although it is accepted that this configuration is inherited from an earlier phase of continental rifting, how post-breakup triple junctions emerge from the separation of two plates remains unclear. By documenting the strain rate history recorded in the three rift-arms of several modern and ancient triple junctions, we show that deformation is episodic and localized in only one or two rifts at any given time. We further investigate this behavior in three-dimensional (3D) analog experiments of rifting, under a range of kinematic boundary conditions and containing a variety of pre-existing lithospheric heterogeneities. Deformation in the experiments is characterized by strain jumps and rift abandonment, comparable to natural observations. Boundary rotation during extension induces oblique stretching directions, along-strike strain gradients and forces significant strain jump to reduce the number of rifts segments active. Models that comprise lithospheres ranging from homogenous to containing a triple junction-like pre-existing heterogeneities, never developed a three-armed rift, where all rift segments are active at same time, at any stage. Our experimental results indicate that, unlike mature, successful, and stable oceanic triple junctions, early-stage continental rifting progresses through unstable “double-junctions” characterized by repeated strain jumps and rift failures and reactivations.

Some properties for certain class of bi-univalent functions defined by $ q $-Cătaş operator with bounded boundary rotation

<abstract><p>Throughout the paper, we introduce a new subclass $ \mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}\ f(z)$ by using the Bazilevič functions with the idea of bounded boundary rotation and $ q $-analogue Cătaş operator. Also we find the estimate of the coefficients for functions in this class. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward $ (p, q) $-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter $ p $ is obviously redundant.</p></abstract>

Molecular dynamics simulation of grain size effect on friction and wear of nanocrystalline zirconium

In this study, molecular dynamics simulation was conducted to investigate the frictional behaviors between diamond tool and zirconium (Zr) substrates at the nanoscale. The effects of grain size on friction and wear were discussed under different sliding velocities. The simulation results showed that the friction forces had similar variation tendencies under different sliding velocities. Besides, the friction responses were stronger at high sliding velocities because of the atomic adhesion while the ploughing effect was more obvious at slower sliding velocity. Moreover, both the friction forces and the wear amounts increased with the decrease in the average grain sizes of the substrates. To explain this phenomenon, the internal mechanism was investigated by using the dislocation extract algorithm and the atomic displacement analyses. The results showed that the [0001]-oriented single crystalline substrate was prone to form continuous dislocation structures moving tangentially along the sliding direction due to the characteristic of Zr's slip systems, whereas grain boundaries conducted the deformation further into the polycrystalline substrates, increasing the contact areas and causing atomic accumulation in front, both resulted in stronger friction responses and wear. Accordingly, with the decrease in average grain sizes, the substrates experienced more severe subsurface damage and the deformation mechanism of nanocrystalline Zr had evolved from dislocation emission to grain boundary rotation and sliding.