OMS-Vol. 3 (2019), Issue 1, pp. 398 – 403 Open Access Full-Text PDF

Abstract: In this paper, Petrović’s inequality is generalized for \(h-\)convex functions on coordinates with the condition that \(h\) is supermultiplicative. In the case, when \(h\) is submultiplicative, Petrović’s inequality is generalized for \(h-\)concave functions. Also particular cases for \(P-\)function, Godunova-Levin functions, \(s-\)Godunova-Levin functions and \(s-\)convex functions has been discussed.

OMS-Vol. 3 (2019), Issue 1, pp. 390 – 397 Open Access Full-Text PDF

Abstract: Inspired by the observation that adjacent vertices need possess their own characteristics in terms of total coloring, we study the smarandachely adjacent vertex total coloring (abbreviated as SAVTC) of a graph \(G\), which is a proper total coloring of \(G\) such that for every vertex \(u\) and its every neighbor \(v\), the color-set of \(u\) contains a color not in the color-set of \(v\), where the color-set of a vertex is the set of colors appearing at the vertex or its incident edges. The minimum number of colors required for an SAVTC is denoted by \(\chi_{sat}(G)\). Compared with total coloring, SAVTC would be more likely to be developed for potential applications in practice. For any graph \(G\), it is clear that \(\chi_{sat}(G)\geq \Delta(G)+2\), where \(\Delta(G)\) is the maximum degree of \(G\). We, in this work, analyze this parameter for general subcubic graphs. We prove that \(\chi_{sat}(G)\leq 6\) for every subcubic graph \(G\). Especially, if \(G\) is an outerplanar or claw-free subcubic graph, then \(\chi_{sat}(G)=5\).

OMS-Vol. 3 (2019), Issue 1, pp. 382 – 389 Open Access Full-Text PDF

Abstract: In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form \(\frac{d}{dt} \Biggl[r(t) \frac{d}{dt} \biggl [x(t)+p(t)x(t-\tau)\biggr]\Biggr]+q(t)G\bigl(x(t-\sigma_1)\bigr)+v(t)H\bigl(x(t-\sigma_2)\bigr)=0, \;\; t \geq t_0,\) under the assumptions \(\int_{0}^{\infty}\frac{d\eta}{r(\eta)}=\infty\) and \(\int_{0}^{\infty}\frac{d\eta}{r(\eta)}<\infty\) for \(|p(t)|<+\infty\). Two illustrative examples are included.

OMS-Vol. 3 (2019), Issue 1, pp. 358 – 381 Open Access Full-Text PDF

Abstract: This study investigate movements of molecule on the biological cell via the cell walls at any given time. Specifically, we examined the movement of a particle in tiling, i.e. in hexagonal and square tiling. The specific questions we posed includes (i) whether particles moves faster in hexagonal tiling or in square tiling (ii) whether the starting point of particles affect the movement toward attainment of stationary distribution. We employed the transitional probabilities and stationary distribution to derive expected passage time to state \(j\) from state \(i\), and the expected recurrence time to state \(i\) in both hexagonal and square tiling. We also employed aggregation of state symmetries to reduce the number of state spaces to overcome the problems (i.e. the difficulty to perform algebraic computation) associated with large transition matrix. This approach leads to formation of a new Markov chain \(X_t\) that retains the original Markov chains properties, i.e. by aggregation of states with the same stochastic behavior to the process. Graphical visualization for how fast the equilibrium is attained with different values of the probability parameter \(p\) in both tilings is also provided. Due to difficulties in obtaining some analytical results, numerical simulation were performed to obtains useful results like expected passage time and recurrence time.

OMS-Vol. 3 (2019), Issue 1, pp. 349 – 357 Open Access Full-Text PDF

Abstract: 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.