Narrow Results

The generalized Hamiltonian function is proposed for the brushless DC motor (BLDCM) chaotic system. The Hamiltonian and Casimir functions are derived from the generalized Hamiltonian function. In this way the Casimir energy is proven to be a special type of the generalized Hamiltonian function. The derivative of the Hamiltonian function is used for analyzing the various dynamical behaviors under different combination of energy components. An analytical optimal bound of the BLDCM is simply proposed from the Hamiltonian power. Along the study, the comparison between the Hamiltonian and Casimir powers is conducted, and physical interpretations and mechanism revealing the onset of chaos are provided for the BLDCM chaotic system. Bifurcation analysis through the Hamiltonian power and Casimir power identifies the different dynamic patterns.

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

A graph invariant is any function on a graph that does not depend on a labeling of its vertices. One of the best known graph invariants successfully applied in chemical graph theory over the last decade is the harmonic index. It is defined for a graph $G$ as the sum of the terms $\frac{2}{d_G(u)+d_G(v)}$ over all edges $uv$ of $G$, where $d_G(u)$ and $d_G(v)$ denote the degrees of the vertices $u$ and $v$ in $G$, respectively. The eccentric version of harmonic index has recently been proposed in an analogous way by replacing the vertex degrees with the vertex eccentricities. One of the main topics in chemical graph theory is to study how certain invariants of product graphs are related to the corresponding invariants of their components. Due to this, we investigate here the behavior of the eccentric version of harmonic index under various families of graph products and apply the derived results on some graphs of chemical and general interest.

The disadvantage of a nondimensionalized model of a permanent-magnet synchronous Motor (PMSM) is identified. The original PMSM model is transformed into a Kolmogorov system to aid dynamic force analysis. The vector field of the PMSM is analogous to the force field including four types of torque — inertial, internal, dissipative, and generalized external. Using the feedback thought, the error torque between external torque and dissipative torque is identified. The pitchfork bifurcation of the PMSM is performed. Four forms of energy are identified for the system — kinetic, potential, dissipative, and supplied. The physical interpretations of the decomposition of force and energy exchange are given. Casimir energy is stored energy, and its rate of change is the error power between the dissipative energy and the energy supplied to the motor. Error torque and error power influence the different types of dynamic modes. The Hamiltonian energy and Casimir energy are compared to find the function of each in producing the dynamic modes. A supremum bound for the chaotic attractor is proposed using the error power and Lagrange multiplier.

Subspace code is very useful in the error correction for random network coding. In this paper, subspace code based on flats in affine space over finite fields is constructed. Sphere-packing bound, Wang–Xing–Safavi-Naini bound, anticode bound, Ahlswede–Aydinian bound and Gilbert–Varshamov bound for the size of code based on flats in affine space over finite fields are provided. Finally, we introduce another method of obtaining the anticode bound for the size of code based on flats in affine space from Erdős–Ko–Rado theorem.

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.