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      Volume 91, Issue 2

      August 2018

    • On non-consensus motions of dynamical linear multiagent systems


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      The non-consensus problems of high-order linear time-invariant dynamical homogeneous multiagentsystems are studied. Based on the conditions of consensus achievement, the mechanisms that lead to non-consensus motions are analysed. Besides, a comprehensive classification of diverse types of non-consensus phases corresponding to different conditions is conducted, which is jointly depending on the self-dynamics of the agents, the interactive protocol and the graph topology. A series of numerical examples are explained to illustrate thetheoretical analysis.

    • Cross-like terahertz metamaterial absorber for sensing applications


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      In this work, a new multiband terahertz metamaterial absorber is designed and characterised by numerical simulation method. In addition, the utilisation of the proposed absorber as a sensor is also investigated. The dielectric and thickness sensing characteristics are analysed. The proposed multiband metamaterial absorber has the ability for utilising the terahertz region up to 2THz. According to the results, it is found that the proposed absorber is capable of sensing unknown materials and material thickness with any of its five absorption bands. The sensitivity of the proposed sensor is 6.57GHz/unit sensitivity for dielectric sensing and 7.66 GHz/$\mu$m for thickness sensing.

    • Ab-initio calculations of structural, phonon and thermal properties of $\rm{Al_{x}Ga_{1−x}As}$ alloy


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      Structural, phonon and thermal properties of $\rm{Al_{x}Ga_{1−x}As}$ alloy for different values of x (x = 0.0, 0.25, 0.5, 0.75 and 1.0) have been investigated by quasiharmonic Debye–Einstein model and Quantum Espresso package. The correction of Vegard’s law for lattice constant has been examined and has a good agreement with other experimental and theoretical works. TO–LO splitting at gamma point, which is related to breaking of cubic symmetry, has been calculated by optical phonon mode calculation. It is found that by increasing the amount of Al in $\rm{Al_{x}Ga_{1−x}As}$ alloy, specific heat at constant pressure and Debye temperature will be increased.

    • Analytical results for periodically-driven two-level models in relation to Heun functions


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      We introduce three different types of periodically-driven multiparametric two-level models whoseanalytical solutions are given in terms of Heun functions. These results are applied to obtain exact analytical results for certain types of periodic potentials and asymmetric double-well potentials. In particular, it is shown that underspecial parameter conditions, an experimentally realised periodic potential supports the exact in-gap solutions. In the asymmetric double-well potentials, some exact results of the bound-state wave functions and associated energiesare found in explicit form.

    • Combination–combination synchronisation of time-delay chaotic systems for unknown parameters with uncertainties and external disturbances


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      In this article, an adaptive control method is proposed to study the combination–combination synchronisation phenomenon of four non-identical time-delayed chaotic systems for fully unknown parameters with parametric uncertainties and external disturbances. Based on the Lyapunov–Krasovskii functional theory, an appropriate adaptive controller is constructed so that a globally and asymptotically stable synchronisation state can be established between the master and the slave systems. Unknown parameters are identified by designing suitable parameter update laws. To elaborate the presented scheme, double-delay Rossler and time-delay Chen systems are considered as the master systems and time-delay Shimizu–Morioka and time-delay modified Lorenz systems are considered as the slave systems. Numerical simulations are presented to justify the theoretical analysis.

    • Theoretical lower limit of mass of phonon and critical mass for matter–dark matter conversion


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      From Planck’s equation for black body radiation and de Broglie’s wave–particle duality relation, we can get a relation between the mass of a phonon and frequency of the emitted radiation. From this relation, we get the theoretical lower limit of the mass of a phonon and critical mass for matter–dark matter conversion. The maximum matter density and limit of the string length are also discussed in this respect. It is observed that there is a critical mass of the smallest particle, which is $7.367 \times 10^{−51}$ kg, above which we get normal matter and below, the dark matter. It is also observed that if phonon obeys the de Broglie’s equation, generation of an electromagnetic radiation of frequency less than 56638721410 Hz is not possible by thermal heating.

    • Characterisation of electric discharge in hollow electrode Z-pinch device by means of Rogowski coils


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      The hollow electrode Z-pinch (HEZP) is expressed as a new shape of Z-pinch devices in which one of the electrodes is ring-shaped. The periodic time of the discharge current is 35 $\mu$s with a total system inductance of 288 nH, total system resistance of 14 m$\Omega$, and 34% deposited energy for a charging voltage of 8 kV. The pinch effect appears in the shape of a sharp spike in the signal of the discharge voltage and dip in the signal of discharge current, which leads to an increase in the plasma inductance at the pinch time. The plasma current density, which is measured using miniature Rogowski coil for 8 kV charging voltage and 1 torr pressure, has a maximum value of 12.1 $\rm{kA/cm^{2}}$ near the axis of the discharge tube and decreases toward the wall. The helium gas pressure in the range of 1–2 torr expresses the situation of the maximum current density. The pinch time increases by increasing the gas pressure and also by decreasing the charging voltage leading to a decrement of the peak discharge current and hence the magnetic field is also decreased. A delay time of at least 4.1 $\mu$s is found to be required to form the pinch for the implemented set-up of anode–cathode dimensions and interdistance. The calculated sheath velocity is in the range of 1.2–6 cm/$\mu$s which is directly proportional to the charging voltage and inversely proportional to the gas pressure.

    • Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation


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      In this paper, we propose a generalised perturbation ($n, N − n$)-fold Darboux transformation (DT) of the modified Korteweg–de Vries (mKdV) equation using the Taylor expansion and a parameter limit procedure. We apply the generalised perturbation ($1, N − 1$)-fold DT to find the new explicit higher-order rational soliton (RS) solutions in terms of determinants of the mKdV equation. These higher-order RS solutions are different from those known soliton results in terms of hyperbolic functions which are obtained from the classical iterated DT. The dynamics behaviours of the first-, second-, third-, and fourth-order RS solutions are shown graphically. The wave propagation characteristics and stability are also discussed using numerical simulations. We find that the initial constant seed solution plays an important role on the wave propagation stability of RS. Through Miura transformation, we give some complex higher-order rational solutions of the Korteweg–de Vries (KdV) equation which are different from the known results. The relevant structures also are discussed using some figures. The method used can also be extended to seek explicit rational solutions of other nonlinear integrable equations.

    • Trivariate analysis of two qubit symmetric separable state


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      One of the main problems of quantum information theory is developing the separability criterion which is both necessary and sufficient in nature. Positive partial transposition test (PPT) is one such criterion which is both necessary and sufficient for $2 \times 2$ and $2 \times 3$ systems but not otherwise. We express this strong PPT criterion for a system of 2-qubit symmetric states in terms of the well-known Fano statistical tensor parameters and prove that a large set of separable symmetric states are characterised by real statistical tensor parameters only. The physical importance of these states are brought out by employing trivariate representation of density matrix wherein the components of J, namely $J_{x} , J_{y} , J_{z}$ are considered to be the three variates.We prove that this set of separable states is characterised by the vanishing average expectation value of $J_{y}$ and its covariance with $J_{x}$ and $J_{z}$ . This allows us to identify a symmetric separable state easily. We illustrate our criterion using 2-qubit system as an example.

    • Meyer–Neldel energy in Se-based binary and ternary chalcogenide glasses


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      The integral equations for DC conductivity and external conductance for the network of localised states in amorphous solids are solved by iteration method. The random free energy barriers and single polaron hoppingmodel are used to obtain the DC conductivity $\sigma_\rm{DC}$ and Meyer–Neldel energy $E_\rm{MN}$. The experimental estimates of optical band gap $E_\rm{g}$, dielectric function $\epsilon$, glass transition temperature $T_\rm{g}$ and $\sigma_\rm{DC}$ are used to calculate $E_\rm{MN}$ for Se-based binary and ternary chalcogenide glasses. The calculated values are found to be in agreement with the available experimental data. $E_\rm{MN}$ increases with increase of attempt frequency. The true pre-exponential factor $\sigma_\rm{00}$ is related to $E_\rm{MN}$ as ln $\sigma_\rm{00} = p − q E_\rm{MN}$, where $p$ is nearly 7.3 and $q$ is material-dependent. The calculated values of $E_\rm{MN}$ and $\sigma_\rm{00}$ suggest that DC conduction in these chalcogenides is due to acoustic and optical phonon-assisted polaron hopping.

    • Modified KdV–Zakharov–Kuznetsov dynamical equation in a homogeneous magnetised electron–positron–ion plasma and its dispersive solitary wave solutions


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      Propagation of three-dimensional nonlinear ion-acoustic solitary waves and shocks in a homogeneous magnetised electron–positron–ion plasma is analysed. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magneticfields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wave solutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, bright and dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. Theseresults demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical problems.

    • Strange quark star with Tolman IV background


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      In the present article, we have studied the behaviour of static, spherically symmetric, anisotropic stellar models within the framework of MIT Bag model in the Tolman IV background. Different physical properties like energy conditions, stability, compactness factor and surface red-shift are investigated through graphical plots and mathematical calculations. The interior solutions found are non-singular in nature, i.e., regular at the centre. The interior spherically symmetric solutions have been matched to an exterior Schwarzschild geometry.

    • Modified extended exp-function method for a system of nonlinear partial differential equations defined by seismic sea waves


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      Nonlinear partial differential equations are the main area of focus for researchers and scientists doing research in applied mathematics. Finding solutions of these nonlinear partial differential equations had gained considerable importance over the last few decades. In this work, an analytical technique named extended exp-function method is introduced for finding archetype exact solutions of innovative nonlinear coupled Konno–Oono equation. Different types of travelling wave solutions, i.e. complex hyperbolic function and complex trigonometric function solutions, with numerous capricious parameters are revealed. Subsequently, by using Maple 16, we plot2D and 3D surfaces of analytical solutions obtained in this article. The depiction of the technique is straight, useful and can be applied to other nonlinear systems of partial differential equations.

    • Exact solutions to complex Ginzburg–Landau equation


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      In this paper, the trial equation method and the complete discrimination system for polynomial method are applied to retrieve the exact travelling wave solutions of complex Ginzburg–Landau equation. Both the Kerr and power laws of nonlinearity are considered. All the possible exact travelling wave solutions consisting of the rational function-type solutions, solitary wave solutions, triangle function-type periodic solutions and Jacobianelliptic functions solutions are obtained, and some of them are new solutions. In addition, concrete examples are presented to ensure the existence of obtained solutions. Moreover, four types of representative solutions are depicted to present the nature of the obtained solutions.

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