Mathematics, Volume 6, Issue 8 (August 2018) – 14 articles

In this article, we study the chemical graph of a cyclic octahedron structure of dimension n and compute the eccentric connectivity polynomial, the eccentric connectivity index, the total eccentricity, the average eccentricity, the first Zagreb index, the second Zagreb index, the third Zagreb index, the atom bond connectivity index and the geometric arithmetic index of the cyclic octahedron structure. Furthermore, we give the analytically closed formulas of these indices which are helpful for studying the underlying topologies. Full article

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. We study when a finite set $S\subset X$ evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding $X_{n,d}$ of ${\mathbb{P}}^n$ and $\left|S\right|\le (\genfrac{}{}{0pt}{}{n+\lfloor d/2\rfloor }{n})$ . For the tensor rank, we describe the cases with $\left|S\right|\le 3$ . For $X_{n,d},$ we raise some questions of the maximum rank for $d\gg 0$ (for a fixed n) and for $n\gg 0$ (for a fixed d). Full article

In this paper, we demonstrate that interventions and stressors do not necessarily cause the same distractions in all people; therefore, it is impossible to evaluate the impacts of interventions and stressors on traffic accidents. We analyzed publicly available multimodal data that was collected through one of the largest controlled experiments on distracted driving. A crossover design was used to examine the effects of actual and perceived interventions and stressors in driving behaviors and parallel designs on reactivity to a startling event. To analyze this data and make recommendations, we developed and compared a wide variety of mixed effects statistical models and machine learning methods to evaluate the effects of interventions and stressors on driving behaviors. Full article

Intuitionistic falling shadow is introduced, and applied to $BCK/BCI$ -algebras. Falling intuitionistic subalgebra and falling intuitionistic ideal of $BCK/BCI$ -algebras are introduced, and related properties are investigated. Relations between falling intuitionistic subalgebra and falling intuitionistic ideal are discussed. A characterization of falling intuitionistic ideal is established. Full article

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision. Full article

Pythagorean fuzzy sets (PFSs), an extension of intuitionistic fuzzy sets (IFSs), inherit the duality property of IFSs and have a more powerful ability than IFSs to model the obscurity in practical decision-making problems. In this research study, we compute the energy and Laplacian energy of Pythagorean fuzzy graphs (PFGs) and Pythagorean fuzzy digraphs (PFDGs). Moreover, we derive the lower and upper bounds for the energy and Laplacian energy of PFGs. Finally, we present numerical examples, including the design of a satellite communication system and the evaluation of the schemes of reservoir operation to illustrate the applications of our proposed concepts in decision making. Full article

The Baker–Campbell–Hausdorff (BCH) expansion is a general purpose tool of use in many branches of mathematics and theoretical physics. Only in some special cases can the expansion be evaluated in closed form. In an earlier article we demonstrated that whenever $[X,Y]=uX+vY+cI$, BCH expansion reduces to the tractable closed-form expression $$Z(X,Y)=\ln \left(e^X{e}^Y\right)=X+Y+f(u,v)[X,Y],$$ where $f(u,v)=f(v,u)$ is explicitly given by the the function $${\displaystyle f(u,v)=\frac{(u-v)e^{u+v}-(u{e}^u-v{e}^v)}{uv(e^u-e^v)}=\frac{(u-v)-(u{e}^{-v}-v{e}^{-u})}{uv(e^{-v}-e^{-u})}.}$$ This result is much more general than those usually presented for either the Heisenberg commutator, $[P,Q]=-i\hslash I$, or the creation-destruction commutator, $[a,a^{†}]=I$. In the current article, we provide an explicit and pedagogical exposition and further generalize and extend this result, primarily by relaxing the input assumptions. Under suitable conditions, to be discussed more fully in the text, and taking $L_{A}B=[A,B]$ as usual, we obtain the explicit result $${\displaystyle \ln \left(e^X{e}^Y\right)=X+Y+\frac{I}{e^{-L_X}-e^{+L_Y}}\left(\frac{I-e^{-L_X}}{L_X}+\frac{I-e^{+L_Y}}{L_Y}\right)[X,Y].}$$ We then indicate some potential applications. Full article

Let $T(X,Y)$ be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation $\rho$ on X, let $\widehat{\rho }$ be the restriction of $\rho$ on Y, R a cross-section of $Y/\widehat{\rho }$ and define $T(X,Y,\rho ,R)$ to be the set of all total transformations $\alpha$ from X into Y such that $\alpha$ preserves both $\rho$ (if $(a,b)\in \rho$, then $(a\alpha ,b\alpha )\in \rho$) and R (if $r\in R$, then $r\alpha \in R$). $T(X,Y,\rho ,R)$ is then a subsemigroup of $T(X,Y)$. In this paper, we give descriptions of Green’s relations on $T(X,Y,\rho ,R)$, and these results extend the results on $T(X,Y)$ and $T(X,\rho ,R)$ when taking $\rho$ to be the identity relation and $Y=X$, respectively. Full article

In this paper, several portfolio choice models are studied: a purely possibilistic model in which the return of the risky is a fuzzy number, and four models in which the background risk appears in addition to the investment risk. In these four models, risk is a bidimensional vector whose components are random variables or fuzzy numbers. Approximate formulas of the optimal allocation are obtained for all models, expressed in terms of some probabilistic or possibilistic moments, depending on the indicators of the investor preferences (risk aversion, prudence). Full article

Hydraulic fracturing has played a crucial role in enhancing the extraction of oil and gas from deep underground sources. The two main objectives of hydraulic fracturing are to produce fractures with a desired fracture geometry and to achieve the target proppant concentration inside the fracture. Recently, some efforts have been made to accomplish these objectives by the model predictive control (MPC) theory based on the assumption that the rock mechanical properties such as the Young’s modulus are known and spatially homogenous. However, this approach may not be optimal if there is an uncertainty in the rock mechanical properties. Furthermore, the computational requirements associated with the MPC approach to calculate the control moves at each sampling time can be significantly high when the underlying process dynamics is described by a nonlinear large-scale system. To address these issues, the current work proposes an approximate dynamic programming (ADP) based approach for the closed-loop control of hydraulic fracturing to achieve the target proppant concentration at the end of pumping. ADP is a model-based control technique which combines a high-fidelity simulation and function approximator to alleviate the “curse-of-dimensionality” associated with the traditional dynamic programming (DP) approach. A series of simulations results is provided to demonstrate the performance of the ADP-based controller in achieving the target proppant concentration at the end of pumping at a fraction of the computational cost required by MPC while handling the uncertainty in the Young’s modulus of the rock formation. Full article

This paper considers resource allocation among producers (agents) in the case where the Principal knows nothing about their cost functions while the agents have Markovian awareness about his/her strategies. We use a dynamic setup of the stochastic inverse Stackelberg game as the model. We suggest an algorithm for solving this game based on Q-learning. The associated Bellman equations contain functions of one variable for the Principal and also for the agents. The new results are illustrated by numerical examples. Full article

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, ${\mathbb{E}}^n$, is said to be of generalized 1-type if, for the Laplace operator, $\Delta$, on the submanifold, it satisfies $\Delta G=fG+gC$, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in ${\mathbb{E}}^3$. Second, we show that the Gauss map of any cylindrical surface in ${\mathbb{E}}^3$ is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in ${\mathbb{E}}^3$, except planes. Finally, we show that cylindrical hypersurfaces in ${\mathbb{E}}^{n+2}$ always have generalized 1-type Gauss maps. Full article

It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method. Full article

This paper deals with an infective process of type SIS, taking place in a closed population of moderate size that is inspected periodically. Our aim is to study the number of inspections that find the epidemic process still in progress. As the underlying mathematical model involves a discrete time Markov chain (DTMC) with a single absorbing state, the number of inspections in an outbreak is a first-passage time into this absorbing state. Cumulative probabilities are numerically determined from a recursive algorithm and expected values came from explicit expressions. Full article