Autoregressive representations of multivariate stationary stochastic processes

Consider a q-variate weakly stationary stochastic process {X _n } with the spectral density W. The problem of autoregressive representation of {X _n } or equivalently the autoregressive representation of the linear least squares predictor of X _n , based on the infinite past is studied. It is shown that for every W in a large class of densities, the corresponding process has a mean convergent autoregressive representation. This class includes as special subclasses, the densities studied by Masani (1960) and Pourahmadi (1985). As a consequence it is shown that the condition W ^-1∈L _qxq ¹ or minimality of {X _n } is dispensable for this problem. When W is not in this class or when W has zeros of order 2 or more, it is shown that {X _n } has a mean Abel summable or mean compounded Cesáro summable autoregressive representation. Research supported by the NSF Grant MCS-8301240 and the AFOSR, Grant F49620 82 C0009. This work was done while the author was visiting Center for Stochastic Processes, University of North Carolina, Chapel Hill     

Legendre wavelets method for constrained optimal control problems

A numerical method for solving non‐linear optimal control problems with inequality constraints is presented in this paper. The method is based upon Legendre wavelet approximations. The properties of Legendre wavelets are first presented. The operational matrix of integration and the Gauss method are then utilized to reduce the optimal control problem to the solution of algebraic equations. The inequality constraints are converted to a system of algebraic equalities; these equalities are then collocated at the Gauss nodes. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2002 John Wiley & Sons, Ltd.     

An analytical solution of the diffusion- convection equation over a finite domain

Numerical solutions to the diffusion-convection equation are usually evaluated through comparison with analytical solutions in one dimension. Literature survey indicates the most frequently used analytical solution is one derived for asemi-infinite domain. This paper presents an analytical solution to this problem over a finite domain. Comparison is made with a solution to a mathematically similar problem in heat conduction with radiation.     

Antioxidant Activity and Molecular Docking Study of Volatile Constituents from Different Aromatic Lamiaceous Plants Cultivated in Madinah Monawara,Saudi Arabia

A comparative study of volatile constituents, antioxidant activity, and molecular docking was conducted between essential oils from Mentha longifolia L., Mentha spicata L., and Origanum majorana L., widely cultivated in Madinah. The investigation of volatile oils extracted by hydrodistillation was performed using Gas Chromatography-Mass Spectrometry (GC-MS). A total number of 29, 42, and 29 components were identified in M. longifolia, M. spicata, and O. majorana representing, respectively, 95.91, 94.62, and 98.42, of the total oils. Pulegone (38.42%), 1,8-cineole (15.60%), menthone (13.20%), and isopulegone (9.81%) were the dominant compounds in M. longifolia oil; carvone (35.14%), limonene (27.11%), germacrene D (4.73%), and β-caryophyllene (3.02%) were dominant in M. spicata oil; terpin-4-ol (42.47%), trans-sabinene hydrate (8.52%), γ-terpinene (7.90%), α-terpineol (7.38%), linalool (6.35%), α-terpinene (5.42%), and cis-sabinene hydrate (3.14%) were dominant in O. majorana oil. The antioxidant activity, assessed using DPPH free radical–scavenging and ABTS assays, was found to be the highest in O. majorana volatile oil, followed by M. spicata and M. longifolia, which is consistent with the differences in total phenolic content and volatile constituents identified in investigated oils. In the same context, molecular docking of the main identified volatiles on NADPH oxidase showed a higher binding affinity for cis-verbenyl acetate, followed by β-elemene and linalool, compared to the control (dextromethorphan). These results prove significant antioxidant abilities of the investigated oils, which may be considered for further analyses concerning the control of oxidative stress, as well as for their use as possible antioxidant agents in the pharmaceutical industry.     

An efficient heuristic algorithm for the capacitated p-\! median problem

The capacitated $p$ -median problem (CPMP) is one of the well-known facility-location problems. The objective of the problem is to minimize total cost of locating a set of capacitated service points and allocating a set of demand points to the located service points, while the total allocated demand for each service point is not be greater than its capacity limit. This paper presents an efficient heuristic algorithm based on the local branching and relaxation induced neighborhood search methods for the CPMP. The proposed algorithm is a heuristic technique that utilizes a general mixed integer programming solver to explore neighborhoods. The parameters of the proposed algorithm are tuned by design of experiments. The proposed method is tested on a large set of benchmark instances. The results show that the method outperforms the best method found in the literature.     

The generalized cubic functional equation and the stability of cubic Jordan *-derivations

In the current work, we obtain the general solution of the following generalized cubic functional equation $$\begin{aligned}&f(x+my)+f(x-my)\\&\quad =2\left( 2\cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(x)-\frac{1}{2}\left( \cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(2x)\\&\qquad +m^2\{f(x+y)+f(x-y)\} \end{aligned}$$ for an integer $m \ge 1$ . We prove the Hyers–Ulam stability and the superstability for this cubic functional equation by the directed method and a fixed point approach. We also employ the mentioned functional equation to establish the stability of cubic Jordan $*$ -derivations on $C^*$ -algebras and $JC^*$ -algebras.