A graceful difference labeling (gdl for short) of a directed graph \(G\) with vertex set \(V\) is a bijection \(f:V\rightarrow\{1,\ldots,\vert V\vert\}\) such that, when each arc $uv$ is assigned the difference label \(f(v)-f(u)\), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.

The author considers a mathematical model of immunotherapy and anti-angiogenesis inhibitor therapy for cancer patients over a fixed time horizon. Disease dynamics are captured by a system of ODEs developed in [1], describing dynamics among host cells, cancer cells, endothelial cells, effector cells, and anti-angiogenesis. Existence, uniqueness, and characterization of optimal treatment profiles that minimize the tumor and drug usage, while maintaining healthy levels of effector and host cells are determined. A theoretical analysis is performed to characterize the optimal control. Numerical simulations are performed to illustrate optimal control profiles for a variety of different patients, each leading to different treatment protocols.

The objective of this paper is to investigate the existence and uniqueness theorem for stochastic partial differential equations with poisson jumps and delays. The existence of mild solutions of the problem is studied by using a different resolvent operator defined in [1] and fixed point theorem.

In this work, we study the small oscillations of a system formed by an elastic container with negligible density and a heavy barotropic gas (or a compressible fluid) filling the container. By means of an auxiliary problem, that requires a careful mathematical study, we deduce the problem to a problem for a gas only. From its variational formulation, we prove that is a classical vibration problem.

In this paper we define a new class of hyperholomorphic functions, which is known as \(F^{\alpha}_{G}(p,q,s)\) spaces. We characterize hyperholomorphic functions in \(F^{\alpha}_{G}(p,q,s)\) space in terms of the Hadamard gap in Quaternion analysis.

The effects of shear deformation and rotary inertia on the dynamics of anisotropic plates traversed by varying moving load resting on Vlasov foundation is investigated in this work. The problem is solved for concentrated loads with simply supported boundary conditions. An analytic solution based on the Galerkin’s method is used to reduce the fourth order partial differential equation into a system of coupled fourth order differential equation and a modification of the Struble’s technique and Laplace transforms are used to solve the resulting fourth order differential equation. Results obtained indicate that shear deformation and rotary inertia have significant effect on the dynamics of the anisotropic plate on the Vlasov foundation. Solutions are obtained for both the moving force and the moving mass problems. From the graphical results obtained, the amplitude of vibrations of the plate under moving mass is greater than that of the moving force and increasing the value of rotary inertia \({R_0}\) reduces the amplitude of vibration of the plate. increasing the mass ratio increases the amplitude of vibration of the plate.

We explore the possibility of using an iron-based anodic material (\(\alpha\)-hematite) synthesized with a hierarchical 3D urchin-like morphology, as an OER catalyst. The electrodes are prepared by pulsed laser deposition followed by thermal annealing leading to the hierarchical 3D urchin-like morphology. The effect of the deposition parameter on the catalyst phase and morphology are investigated by microRaman spectroscopy and scanning electron microscopy, while the electrode metrics are determined by voltammetric methods and Tafel analysis. We observe that the material is highly electroactive towards the OER, with performance in-line with that of noble-metal based state-of-the-art catalysts.

In literature, there are many methods for solving nonlinear partial differential equations. In this paper, we develop a new method by combining Adomian decomposition method and Shehu transform method for solving nonlinear partial differential equations. This method can be named as Shehu transform decomposition method (STDM). Some examples are solved to show that the STDM is easy to apply.

Dynamic analysis of isotropic thin rectangular plate resting on two-parameter elastic foundations is investigated. The governing system is converted to system of nonlinear ordinary differential equation using Galerkin method of separation. The Ordinary differential equation is analyzed using hybrid method of Laplace transform and Variation of iteration Method. The accuracies of the analytical solutions obtained are verified with existing literature and confirmed in good agreement. Thereafter, the analytical solutions are used for parametric studies. From the results, it is observed that, increase in elastic foundation parameters increases the natural frequency. Increase in aspect ratios increases the natural frequency. It is expected that the present study will add value to the existing knowledge in the field of vibration.

Given a set of locations or cities and the cost of travel between each location, the task is to find the optimal tour that will visit each locations exactly once and return to the starting location. We solved a routing problem with focus on Traveling Salesman Problem using two algorithms. The task of choosing the algorithm that gives optimal result is difficult to accomplish in practice. However, most of the traditional methods are computationally bulky and with the rise of machine learning algorithms, which gives a near optimal solution. This paper studied two methods: branch-and-cut and machine learning methods. In the machine learning method, we used neural networks and reinforcement learning with 2-opt to train a recurrent network that predict a distribution of different location permutations using the negative tour-length as the reward signal and policy gradient to optimize the parameters of recurrent network. The improved machine learning with 2-opt give near-optimal results on 2D Euclidean with upto 200 nodes.