Explicit Mathematical Expressions Research Articles

Teleportation of displaced Fock states: Fidelity and their teleported photon number distributions

We consider the teleportation of displaced Fock states which are highly non-classical states of the quantized electromagnetic field which have a set of remarkable quantum properties that include the peculiar oscillations of their photon number distributions. We use the transfer operator formalism to show that the quantum teleportation of a DFS renders a finite superposition of orthonormal DFS's and find its explicit mathematical expression in terms of the compression parameter of the correlated EPR states of the quantum channel. The expression for a teleported Fock state is also derived as a particular case of DFS's teleportation. We finally apply these results to study the fidelity of the teleportation of DFS's and the teleportation of their photon number statistics.

Modelling and simulation of Rayleigh fading, path loss, and shadowing fading for wireless mobile networks

In this paper, we first derive two explicit mathematical expressions of bit error rates (BERs) with Rayleigh fading for differential phase shift keying (DPSK) and quadrature amplitude modulation (QAM), and establish corresponding simulation models to realize the effect of Rayleigh fading. Next, we propose a mathematical expression to define the effect of shadowing fading and an approach to computing it. Then, we optimize the effect of path loss through adapting the value of the path loss exponent to different environments. Finally, using a packet-oriented simulation tool, OPtimized Network Engineering Tool (OPNET), we realize the effects of Rayleigh fading and shadowing fading, optimize the effect of path loss, and investigate the integrated impact of these effects on the performance of wireless mobile networks including wireless local area networks (WLANs) and mobile ad hoc networks (MANETs). Simulation results not only coincide with the experimental results in present literature, but also show that these effects exert a non-negligible impact on the performance of wireless mobile networks. Accordingly, our new models and approaches make the present over-optimistic simulation results more accurate and reasonable.

Analytical approach to the strain rate effect on the dynamic tensile strength of brittle materials

An explicit mathematical expression for the dynamic load-carrying capacity of brittle materials under dynamic tensile loads is derived based on a kind of structural-temporal failure criterion [1] and the one-dimensional longitudinal plane wave propagation model. It is shown that the dependence of the dynamic load-carrying capacity on the strain rate can be determined only by the static material parameters such as tensile strength, density, incubation time, critical failure length and constitutive constants, which verifies that the well known strain rate effect on material strength can be considered as an structural rather than material behavior, as pointed out by Cotsovos and Pavlović [2] recently. Moreover, it is found that, under constant strain rate, the dynamic load-carrying capacity depends also on the amplitudes of imposed boundary loads, which explains, to a significant extent, the scatter that characterizes the available experimental data. Furthermore, the derived expression can also be used as a foundation of theoretical analyses on other problems involving the strain rate effect such as dynamic size effect, dynamic failure of quasi-brittle materials and composites.

AN ENTIRELY ANALYTICAL COSMOLOGICAL MODEL

The purpose of the present study is to show that in a particular cosmological model, with an affine equation of state, one can obtain, besides the background given by the scale factor, Hubble and deceleration parameters, a representation in terms of scalar fields and, more important, explicit mathematical expressions for the density contrast and the power spectrum. Although the model so obtained is not realistic, it reproduces features observed in some previous numerical studies, and as such it indicates that the Chaplygin gas by itself is not competitive when compared to the usual Λ-CDM model.

System optimization for HVAC energy management using the robust evolutionary algorithm

For an installed centralized heating, ventilating and air conditioning (HVAC) system, appropriate energy management measures would achieve energy conservation targets through the optimal control and operation. The performance optimization of conventional HVAC systems may be handled by operation experience, but it may not cover different optimization scenarios and parameters in response to a variety of load and weather conditions. In this regard, it is common to apply the suitable simulation–optimization technique to model the system then determine the required operation parameters. The particular plant simulation models can be built up by either using the available simulation programs or a system of mathematical expressions. To handle the simulation models, iterations would be involved in the numerical solution methods. Since the gradient information is not easily available due to the complex nature of equations, the traditional gradient-based optimization methods are not applicable for this kind of system models. For the heuristic optimization methods, the continual search is commonly necessary, and the system function call is required for each search. The frequency of simulation function calls would then be a time-determining step, and an efficient optimization method is crucial, in order to find the solution through a number of function calls in a reasonable computational period. In this paper, the robust evolutionary algorithm (REA) is presented to tackle this nature of the HVAC simulation models. REA is based on one of the paradigms of evolutionary algorithm, evolution strategy, which is a stochastic population-based searching technique emphasized on mutation. The REA, which incorporates the Cauchy deterministic mutation, tournament selection and arithmetic recombination, would provide a synergetic effect for optimal search. The REA is effective to cope with the complex simulation models, as well as those represented by explicit mathematical expressions of HVAC engineering optimization problems.

General stochastic user equilibrium traffic assignment problem with link capacity constraints

AbstractThis paper addresses a general stochastic user equilibrium (SUE) traffic assignment problem with link capacity constraints. It first proposes a novel linearly constrained minimization model in terms of path flows and then shows that any of its local minimums satisfies the generalized SUE conditions. As the objective function of the proposed model involves path‐specific delay functions without explicit mathematical expressions, its Lagrangian dual formulation is analyzed. On the basis of the Lagrangian dual model, a convergent Lagrangian dual method with a predetermined step size sequence is developed. This solution method merely invokes a subroutine at each iteration to perform a conventional SUE traffic assignment excluding link capacity constraints. Finally, two numerical examples are used to illustrate the proposed model and solution method.

Speculation in the feasibility of evaporative cooling

The use of evaporative air cooling, for residential air conditioning, cannot be taken for granted in all situations. It depends on the climatic conditions and the specific nature of application. This work establishes a general foundation for judging the feasibility of evaporative cooling with different evaporative-system configurations, under different climatic conditions and for different applications. Two feasibility criteria were stipulated; the rate of air supply to space and the indoor relative humidity. Systematic procedures are presented for evaluating the required air-flow rate and predicting the achievable indoor condition. Explicit mathematical expressions are derived to define the limitations on outdoor conditions for any allowable specific air flow. The impacts of various pertinent factors are investigated. These include the required indoor temperature, the quality of space load represented by its SHF and the performance index of the system. Computer programs were devised to automate, hence facilitate, the repetitive computations and to evade the graphical work on the psychrometric chart. Samples of program results are graphically displayed.

Efficiency and Goodput Analysis of Dly-ACK in IEEE 802.15.3

An analytical model for studying the performance of the delayed acknowledgment (Dly-ACK) mechanism in the IEEE 802.15.3 standard over wireless Rayleigh-fading channel is developed. A three-state Markov chain is applied to approximate the correlated transmission errors. Explicit mathematical expressions for the goodput and efficiency of Dly-ACK are derived. It is found that the correlation between consecutive transmission errors has a significant impact on the goodput and efficiency of the Dly-ACK mechanism. The goodput tends to increase as the size of the Dly-ACK burst increases; however, the amount of increase depends on the underlying delay. Simulation results are given to validate the analytical results.

Utility of the Coefficient of Determination (r2) in Assessing the Accuracy of Interspecies Allometric Predictions: Illumination or Illusion?

The appropriateness of relying on the coefficient of determination (r(2)) as a statistical metric for judging the predictability of human clearance (CL) based on interspecies animal data was assessed. An explicit mathematical expression was derived for r(2) as a function of species body weight and the corresponding measured value of CL. The derived mathematical function demonstrated that r(2) is numerically large in most instances. Simulations using random CL generated from a common combination of species of mouse, rat, and monkey resulted in an r(2) of 0.75 as the minimum, and 0.95 and 0.98 at 50th and 75th percentiles, respectively, given that total CL values increase with increasing species body weight. Analysis of literature data also indicated that the prediction accuracy of human CL was not correlated with values of r(2). Therefore, it is concluded that r(2) is a limited statistical measure when assessing allometric scaling for the purpose of predicting human CL.

Prediction model of tunnel boring machine performance by ensemble neural networks

The penetration rate of a tunnel boring machine (TBM) depends on many factors ranging from the machine design to the geological properties. Therefore it may not be possible to capture this complex relationship in an explicit mathematical expression. In this paper, we propose an ensemble neural network (ENN) to predict TBM performance. Based on site data, a four-parameter ENN model for the prediction of the specific rock mass boreability index is constructed. Such a neural-network-based model has the advantages of taking into account the uncertainties embedded in the site data and making appropriate inferences using very limited data via the re-sampling technique. The ENN-based prediction model is compared with a non-linear regression model derived from the same four parameters. The ENN model outperforms the non-linear regression model.

Transfer Delay Analysis of WAP 2.0 for Short-Lived Flows

In this paper, an analytical framework for studying the transfer delay of wireless application protocol (WAP) 2.0 for short-lived flows is developed based on a two-state Markov chain that approximates both correlated and independent packet losses. For a given wireless link and protocol parameters, an explicit mathematical expression which yields a good estimate of the WAP 2.0 transfer delay is derived. The analytical results are validated by simulation. It is shown that for large file sizes (> 30 kB), WAP 2.0 is more sensitive to bursty packet losses than random packet losses. It is also shown that the transfer delay of WAP 2.0 can be improved by increasing the size of the initial window in a low-rate bursty error environment but degrades in a high-rate bursty error environment

Methodology of Concept Control Synthesis to Avoid Unmoving and Moving Obstacles (II)

The dynamic path generation problem of robots in environments with other unmoving and moving objects is considered. Generally, the problem is known in the literature as find path or robot motion planning. In this paper, we apply the behavioral cloning approach to design the robot controller. In behavioral cloning, the system learns from control traces of a human operator. The task for the given problem is to find a controller in the form of an explicit mathematical expression. Thus, machine learning programs to induce the operator's trajectories as a set of symbolic constraints are used. Then, mathematical induction to generalize the obtained equations in order to apply them in situ with an infinite number of obstacles is also used. A method to evaluate cloning success is proposed. The typical kind of noise is included.

STOKES' DRIFT AND HYPERSENSITIVE RESPONSE WITH DICHOTOMOUS MARKOV NOISE

Stochastic Stokes' drift and hypersensitive transport driven by dichotomous noise are theoretically investigated. Explicit mathematical expressions for the asymptotic probability density and drift velocity are derived including the situation in which particles cross unstable fixed points. The results are confirmed by numerical simulations.

Explicit expressions for the plastic normality-flow rule associated to the Tresca yield criterion

The Tresca yield criterion is classical and important to the theory of plasticity. It is usually formulated in terms of the difference between the maximum and minimum principal stresses. Difficulties have been encountered in attempts to explicitly express the plastic normality-flow rule associated to it, because the difference between the maximum and minimum principal stresses is generally not differentiable but subdifferentiable with respect to the stress tensor. In this work, the corresponding subdifferential is determined and specified for all possible cases; the explicit mathematical expressions are obtained for the plastic normality-flow rule relative to the Tresca yield criterion.

Multi-Speed Gearbox Design Using Multi-Objective Evolutionary Algorithms

Optimal design of a multi-speed gearbox involves different types of decision variables and objectives. Due to lack of efficient classical optimization techniques, such problems are usually decomposed into tractable subproblems and solved. Moreover, in most cases the explicit mathematical expressions of the problem formulation is exploited to arrive at the optimal solutions. In this paper, we demonstrate the use of a multi-objective evolutionary algorithm, which is capable of solving the original problem involving mixed discrete and real-valued parameters and more than one objectives, and is capable of finding multiple nondominated solutions in a single simulation run. On a number of instantiations of the gearbox design problem having different complexities, the efficacy of NSGA-II in handling different types of decision variables, constraints, and multiple objectives are demonstrated. A highlight of the suggested procedure is that a post-optimal investigation of the obtained solutions allows a designer to discover important design principles which are otherwise difficult to obtain using other means.

Off-State Breakdown Modeling for High-Schottky-Barrier δ-Doped In0.49Ga0.51P/In0.25Ga0.75As/InP High Electron Mobility Transistor

An off-state two-terminal gate-drain current model has been proposed to investigate the breakdown phenomenon for δ-doped high electron mobility transistors (δ-HEMTs) with a high Schottky barrier. The proposed model is based on the integrated mechanisms of thermionic-field emission theory for gate-barrier tunneling and impact ionization multiplication within the channel regime by self-consistently solving the 2-D Poisson equation. Explicit mathematical expressions have been derived to provide an efficient estimation of breakdown criteria and illustrations of sophisticated breakdown characteristics. Calculations of breakdown criteria for δ-HEMTs with a high Schottky barrier, including Au/In0.49Ga0.51P and Au/In0.34Al0.66As0.85Sb0.15, have also been verified and discussed. This model is suitable for the implementation of computer-aided design (CAD) design tools.

Aeroelastic Divergence of Stiffened Composite Multicell Wing Structures

When the wing is treated as a composite sandwich plate and it is assumed that the wing chordwise section is rigid, mathematical formulations for the stiffened composite multicell wing structures are provided, and associated governing equations for the aeroelastic divergence are also derived by a direct approach. When the matrix notation is used, the system of equations is written into an explicit and simple mathematical expression that can then be solved explicitly by using the technique of Laplace integral transform. In this modeling, the composite wing skins and stringers (including the spar flanges) are simulated as the faces, whereas the spar webs and ribs are simulated as the core of the sandwiches. Because the wing cross section must have a streamline shape, unlike the usual uniform thickness modeling, it is more appropriate to simulate the wings as variable thickness sandwiches where the thickness is a function of the airfoil. Moreover, as is usual for assumptions for the sandwich plates, the effects of the transverse shear deformation are also considered. Because several special conditions have been studied in the literature, we first compare our solutions with some existing solutions. To show the generality, several illustrative examples are given to consider the effects of spars, skins (including the ply orientation and stacking sequence), stringers, swept angles, aspect ratio, shape of airfoil, and the warping restraints on the divergence dynamic pressures and the lift loads redistribution.

The Formal Underpinnings of The Response Functions Used in X‐Ray Spectral Analysis

This work provides an in-depth mathematical description of the response functions that are used for spatial and spectral analysis of X-ray data. The use of such functions is well known to anyone familiar with the analysis of X-ray data where they may be identified with the quantities contained in the ancillary response file (ARF), the redistribution matrix file (RMF), and the exposure map. Starting from first principles, explicit mathematical expressions for these functions, for both imaging and dispersive modes, are arrived at in terms of the underlying instrumental characteristics of the telescope including the effects of pointing motion. The response functions are presented in the context of integral equations relating the expected detector count rate to the source spectrum incident upon the telescope. Their application to the analysis of several source distributions is considered. These include multiple, possibly overlapping, spectrally distinct point sources, as well as extended sources. Assumptions and limitations behind the usage of these functions, as well as their practical computation, are addressed.

Free vibration analysis of a Timoshenko beam carrying multiple spring–mass systems by using the numerical assembly technique

From the equation of motion of the ‘bare’ Timoshenko beam (without any spring–mass systems attached), an eigenfunction in terms of four unknown integration constants is obtained. The substitution of the eigenfunction into the three compatible equations, one force–equilibrium equation and one governing equation for the sprung mass (ν=1,…,n) yields a matrix equation of the form [Bν] {Cν}=0. Similarly, when the eigenfunction is substituted into the two boundary-condition equations at the ‘left’ end and those at the ‘right’ end of the beam, one obtained [BL] {CL}=0 and [BR] {CR}=0, respectively. Assembly of the coefficient matrices [BL], [Bν] and [BR] will arrive at the eigen equation [B̄] {C̄}=0, where the elements of {C̄} are composed of the integration constants Cνi (ν=1,…,n and i=1,…,4) and the modal displacements of the sprung mass, Zν. For a Timoshenko beam carrying n spring–mass systems, the order of the overall coefficient matrix [B̄] is 5n+4. The solutions of ∣B̄∣=0 (where ∣·∣ denotes a determinant) give the natural frequencies of the ‘constrained’ beam (carrying multiple spring–mass systems) and the substitution of each corresponding values of Cνi into the associated eigenfunction at the attaching points will define the corresponding mode shapes. In the existing literature the eigen equation [B̄] {C̄}=0 was denoted in explicit form and then solved analytically or numerically. Because of the lengthy explicit mathematical expressions, the existing approach becomes impractical if the total number of the spring–mass systems is larger than ‘two’. But any number of spring–mass systems will not make trouble for the numerical assembly technique presented in this paper. Copyright © 2001 John Wiley & Sons, Ltd.

Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique

From the equation of motion of the ‘bare’ Timoshenko beam (without any spring–mass systems attached), an eigenfunction in terms of four unknown integration constants is obtained. The substitution of the eigenfunction into the three compatible equations, one force–equilibrium equation and one governing equation for the sprung mass (ν=1,…,n) yields a matrix equation of the form [Bν] {Cν}=0. Similarly, when the eigenfunction is substituted into the two boundary-condition equations at the ‘left’ end and those at the ‘right’ end of the beam, one obtained [BL] {CL}=0 and [BR] {CR}=0, respectively. Assembly of the coefficient matrices [BL], [Bν] and [BR] will arrive at the eigen equation [B̄] {C̄}=0, where the elements of {C̄} are composed of the integration constants Cνi (ν=1,…,n and i=1,…,4) and the modal displacements of the sprung mass, Zν. For a Timoshenko beam carrying n spring–mass systems, the order of the overall coefficient matrix [B̄] is 5n+4. The solutions of ∣B̄∣=0 (where ∣·∣ denotes a determinant) give the natural frequencies of the ‘constrained’ beam (carrying multiple spring–mass systems) and the substitution of each corresponding values of Cνi into the associated eigenfunction at the attaching points will define the corresponding mode shapes. In the existing literature the eigen equation [B̄] {C̄}=0 was denoted in explicit form and then solved analytically or numerically. Because of the lengthy explicit mathematical expressions, the existing approach becomes impractical if the total number of the spring–mass systems is larger than ‘two’. But any number of spring–mass systems will not make trouble for the numerical assembly technique presented in this paper. Copyright © 2001 John Wiley & Sons, Ltd.