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"A normalized function $f$ on the open unit disc is starlike (or convex) univalent if the associated function $zf'(z)/f(z)$ (or $1+zf''(z)/f'(z)$) is a function with positive real part. The radius of starlikeness or convexity is usually obtained by using the estimates for functions with positive real part. Using subordination, we examine the radius of various starlikeness, in particular, radii of Janowski starlikeness and starlikeness of order $\beta$, for the function $f$ when the function $f$ is either convex or $(zf'(z)+\alpha z^2f''(z))/f(z)$ lies in the right-half plane. Radii of starlikeness associated with lemniscate of Bernoulli and exponential functions are also considered."

In the past few years, graph theory has emerged as one of the most powerful mathematical tools to model many types of relations and process dynamics in computer science, biological and social systems. Generally, a graph is depicted as a set of nodes which is called vertices connected by lines are called edges. A topological index is the numerical parameter of a graph that characterizes its topology and it is usually graph invariant. In this paper, we compute some important classes vertex degree-based graph invariants using the Zagreb index of some special graphs such as the co-normal product of graphs, concentric wheels graph and intersection graph.

The probabilistic neural networks (PNNs) are now being analysed to fix a variety of challenges in the diverse fields of science and technology. In chemical graph theory, there are several tools, such as polynomials, functions, etc. that can be used to characterize different network properties. The neighborhood M-polynomial (NM) is one of those that yields neighborhood degree sum based topological indices in a manner that is less time consuming than the usual approach. In this work, the NM-polynomial of 3-layered and 4-layered probabilistic neural networks are derived. Further, some neighborhood degree sum based topological indices are computed from those polynomials. Applications of the present work are interpreted by investigating the chemical importance of the indices. Some structure property models are derived. The graphical representations of the results are also reported.

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the rst derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.

An energy momentum-dependent potential is considered to study
nucleon-nucleon, nucleon-nucleus and nucleus-nucleus elastic scattering by
exploiting the variable phase method to potential scattering. The velocity dependent
potential, equivalent to Graz separable potential, has the ability to reproduce the
correct nature and numerical values of the NN phase parameters up to 400 MeV.
Also it works equally well for alpha-nucleon and alpha-alpha systems up to few
higher partial waves. Our results are in close conformity with the previous
theoretical and experimental data. For alpha-alpha scattering we have treated only
s-wave case.

"We study the distortion features of homeomorphisms of Sobolev class $W^{1,1}_{\rm loc}$ admitting integrability for $p$-outer dilatation. We show that such mappings belong to $W^{1,n-1}_{\rm loc},$ are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us to derive various distortion results like Lipschitz, H\""older, logarithmic H\""older continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms."

"The error function takes place in a wide range in the elds of mathe- matics, mathematical physics and natural sciences. The aim of the current paper is to investigate certain properties such as univalence and close-to-convexity of normalized imaginary error function, which its region is symmetric with respect to the real axis. Some other outcomes are also obtained."