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In this note, we study a discontinuous three-term recurrence relation which arises from seeking the steady states of a cellular neural network with step control function. Several collections of per...

The classical estimator of a quantile density function by orthogonal series depends on the empirical distribution function estimator Hn. The fact that Hn is a step function even when the underlying...

In this paper, the design procedure of a novel neuron is discussed. Starting from the activation function circuit which is based on the modification of regulated cascode structure, a new scheme is presented which can produce the logistic function with high percentage of accuracy. The main advantage of the proposed function generator circuit is its full programmability feature in which its output waveform slope can easily be adjusted by means of the control voltages applied to the bulks of input stage metal oxide semiconductor (MOS) transistors. Also, the output waveform can easily be converted to a step function with the help of bias changes for the input transistors. Following the concepts of the previous work by the authors, the designed activation function has a good compatibility with the employed synapse. Low power and small active-area consumption are the other features of the proposed circuit which qualify it for hardware implementation of neural networks. Post-layout simulation results bas...

In the current work, an advanced computational technique is formulated and performed to investigate the wave scattering problem from an infinitely transversally large anisotropic cylinder with arbitrary cross section. A meshless local Petrov–Galerkin method is used to discretize the anisotropic interior (inside the scatterer) and isotropic exterior (outside the scatterer) problems. The total electric fields inside and outside the scatterer are approximated by linear combinations of the generalized multiquadric radial basis functions. The local Heavisde step function is employed as the test function in weak formulation. Moreover, a simple collocation scheme is used to impose the interface and radiation boundary conditions directly. The proposed numerical technique is an efficient instrument to deal with practical complex scatterers and also has a good advantage in treating the material discontinuity at the interface between two different media. The performance and efficiency of the method are demonstrated ...

We study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our r...

In this paper we develop a method for constructing strong solutions of one-dimensional Stochastic Differential Equations where the drift may be discontinuous and unbounded. The driving noise is the Brownian Motion and we show that the solution is Sobolev-differentiable in the initial condition and Malliavin differentiable. This method is not based on a pathwise uniqueness argument. We will apply these results to the stochastic transport equation. More specifically, we obtain a continuously differentiable solution of the stochastic transport equation when the driving function is a step function.

We propose a semiparametric approach to estimate the existence and location of a statistical change-point to a nonlinear multivariate time series contaminated with an additive noise component. In particular, we consider a p-dimensional stochastic process of independent multivariate normal observations where the mean function varies smoothly except at a single change-point. Our approach involves conducting a Bayesian analysis on the empirical detail coefficients of the original time series after a wavelet transform. If the mean function of our time series can be expressed as a multivariate step function, we find our Bayesian-wavelet method performs comparably with classical parametric methods such as maximum likelihood estimation. The advantage of our multivariate change-point method is seen in how it applies to a much larger class of mean functions that require only general smoothness conditions.

Recently it was shown that in deformed helix ferroelectric (DHF) structure, the electrically controlled birefringence of chiral smectic C* phase with subwavelength helix pitch ( is known to be proportional to the square of the electric field E) depends on the cell thickness and on the frequency of applied voltage, given as a Step function of the time. The purpose of the present work is to investigate whether this quadratic effect depends also on the voltage function shape, comparing the electrically controlled birefringence and the Kerr constant in DHF cells with different thickness, driven by STAIR and Step voltage functions.

This paper addresses the scheduling problem of minimising the total weighted tardiness on a single machine with step-deteriorating jobs. With the assumption of deterioration, the job processing times are modelled by step functions of job starting times and pre-specified job deteriorating dates. The introduction of step-deteriorating jobs makes the single-machine total weighted tardiness problem more intractable. Until now, the computational complexity of this problem under consideration was not determined. In this study, it is proved to be strongly NP-hard. Then, a linear ordering mixed-integer programming model is derived for solving the problem instances optimally. In order to tackle large-sized problems, seven dispatching heuristic procedures are developed for near-optimal solutions. Meanwhile, the solutions delivered by the proposed heuristics are further improved by a pairwise swap movement. Computational results are presented to reveal the performance of all proposed approaches.

The applications of control chart have traditionally focused on the detection of step shifts in process mean. However, changes are usually gradual, not as perfect step shifts. The common consideration of a shift as a step function does not always adequately describe what actually happens in practice. Hence, there is a need for more realistic assumptions to be incorporated. This paper employs a Markov chain approach and provides a way to quantitatively measure the economic performance of control charts in the presence of a more general quality deterioration mechanism. The finite production run is considered in the model as it has become a very important production mode at present and the process failure mechanism is described by geometric distribution. The chart properties, particularly on the issues of the quality deterioration mechanism, are investigated. The findings provide critical insights on the use of step shift assumption when designing control charts.

This article considers the parallel machine scheduling problem with step-deteriorating jobs and sequence-dependent setup times. The objective is to minimize the total tardiness by determining the allocation and sequence of jobs on identical parallel machines. In this problem, the processing time of each job is a step function dependent upon its starting time. An individual extended time is penalized when the starting time of a job is later than a specific deterioration date. The possibility of deterioration of a job makes the parallel machine scheduling problem more challenging than ordinary ones. A mixed integer programming model for the optimal solution is derived. Due to its NP-hard nature, a hybrid discrete cuckoo search algorithm is proposed to solve this problem. In order to generate a good initial swarm, a modified Biskup–Hermann–Gupta (BHG) heuristic called MBHG is incorporated into the population initialization. Several discrete operators are proposed in the random walk of Levy flights and the cr...

Joint likelihood approaches have been widely used to handle survival data with time-dependent covariates. In construction of the joint likelihood function for the accelerated failure time (AFT) model, the unspecified baseline hazard function is assumed to be a piecewise constant function in the literature. However, there are usually no close form formulas for the regression parameters, which require numerical methods in the EM iterations. The nonsmooth step function assumption leads to very spiky likelihood function which is very hard to find the globe maximum. Besides, due to nonsmoothness of the likelihood function, direct search methods are conducted for the maximization which are very inefficient and time consuming. To overcome the two disadvantages, we propose a kernel smooth pseudo-likelihood function to replace the nonsmooth step function assumption. The performance of the proposed method is evaluated by simulation studies. A case study of reproductive egg-laying data is provided to demonstrate the...

An exact expression is derived relating the state average of the periodic solution to the average of the environmental carrying capacities for the periodic Beverton–Holt equation for arbitrary period. By studying numerically period 3 case, we show that the correlation coefficient of the intrinsic growth rates and , is not relevant in determining attenuation or resonance. By studying period 4 case, it is shown that if the intrinsic growth rate jumps upward along with steadily increasing carrying capacities, then resonance prevails. A period 7 example using out-of-step step functions is also seen to produce resonance.

Sufficient conditions are derived for simultaneous stabilisability of any finite number of linear, time-invariant, multi-input multi-output (MIMO) systems. Four general MIMO plant classes with arbitrary finite number of plants are shown to be simultaneously stabilisable using simple proportional + derivative controllers. For some of these plant classes it is also possible to achieve simultaneous stabilisation and asymptotic tracking of step-input references with zero steady-state error by using integral-action controllers. Three of these general MIMO plant classes are shown to be strongly simultaneously stabilisable.

Cooperative controllers are designed to force a group of N mobile agents with limited communication using only relative position between the agents to form a desired formation structure. The centre of the formation structure is the mean position of all the agents. The agents are stabilised at desired positions with respect to the centre of the structure. The proposed controllers also preserve initial communication connectivity and guarantees no collisions between the agents. The control design is based on smooth step functions and potential functions. The proposed control design is applied to solve gradient climbing and target capturing problems in both two- and three-dimensional spaces. The gradient average of a distributed field in nature or artificially generated by a target is first estimated over a bounded region using the field measurement by the agents. The gradient average is then used as the reference velocity to guide the agents.

In this paper, we study differential equations with piecewise constant argument of generalized (DEPCAGs) type, i.e., the argument is a general step function. They are hybrid equations combining properties of continuous and discrete equations. The play of the discrete part is always very important. The explicit solutions of the homogeneous and non-homogeneous linear DEPCAGs systems are obtained. Existence, uniqueness and stability of the solutions of the quasilinear DEPCAGs are under discussion. All previous results are improved. The importance of the advanced and delayed intervals will be clear. Cauchy and Green matrices type are deduced. The integral representation and Gronwall's inequality type obtained can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of DEPCAGs.

This article proposes an asymptotic expansion for the Studentized linear discriminant function using two-step monotone missing samples under multivariate normality. The asymptotic expansions related to discriminant function have been obtained for complete data under multivariate normality. The result derived by Anderson (1973) plays an important role in deciding the cut-off point that controls the probabilities of misclassification. This article provides an extension of the result derived by Anderson (1973) in the case of two-step monotone missing samples under multivariate normality. Finally, numerical evaluations by Monte Carlo simulations were also presented.

The empirical application of stochastic dominance in the selection of crop alternatives under risk has typically been based on a step function specification of the empirical cumulative distribution function (cdf). This specification has questionable consequences for the analysis, particularly with regards to the tails of the distribution. This article proposes an alternative nonparametric approach based on a nonparametric regression of the cdf. The effect of this alternative specification is then demonstrated using data from Northern Florida. Results are more realistic. The first-degree stochastic dominance results are less discriminating under the new specification while the second-degree results are more discriminating.

This paper shows that important insights can be lost when assessing the relative performance of balancing methods solely based on individual optima. This is demonstrated through a multi-objective assessment. A trade-off curve between RAS and sign-preserving absolute differences (SPAD) is obtained based on the 60×60 Norwegian 2001 input–output table. The trade-off curve takes on a form that is close to a step function. This demonstrates that the solution surface around the RAS and SPAD optimums are very flat. Solutions can be identified that improve on the other objective or measure with little or marginal cost to the original objective function. Motivation for the assessment is provided, the technique applied is presented and the implications of the findings are discussed in an input–output and industrial ecology context.

For a quantitative description of the measured dynamic mixing behavior of ethanol and glycerol, a computational fluid dynamics (CFD) simulation is carried out by using the ANSYS CFX-10 tool. An unstructured mesh with tetrahedral cells is generated by using ANSYS ICEM CFD 5.1. The newly modified algebraic slip model (MASM) which includes the ethanol droplets break-up as a function of time dp(t) by using a validated step function gives the real mixing behavior, i.e., viscosity as a function of time ηm(t) and mixing times (tm) in good agreement with the experimental results in semi-batch reactor (SBR) at different dosage times and in batch reactor (BR) at a total liquid height of 0.14 m. The modified model proposed in present paper which is based on the algebraic slip model (ASM) gives results closer to experimental results than its original model and the transport model (TRM). The prolongation of the mixing time by a factor of 1.5 caused by the pan cake effect is predicted by the MASM. The flow velo...

This article deals with the effects of a short-pulse laser of Gaussian temporal and different spatial profiles on the signals emanating from a two-dimensional rectangular participating medium containing localized inhomogeneities. An incident short-pulse laser of Gaussian temporal profile has spatial loading on the south boundary either as a Gaussian or as a step function. The absorbing and scattering participating medium consists of a localized circular or elliptical inhomogeneity. The extinction coefficient of the inhomogeneity is the same as the containing medium, while its scattering albedo is different. Transmittance and reflectance signals are analyzed for the effects of the extinction coefficient and the spatial profile of the incident short-pulse laser. Spatial distributions of the signals along the boundaries and temporal evolution of the heat flux distributions inside the medium are also studied. The problem is analyzed using the finite-volume method.

The accuracy of using step-function approximations to the Arrhenius exponential in computing the wavespeed in combustion wave propagation is investigated. Gaseous and gasless combustion, and first- and second-order reactions are included in the study. The theoretical analysis is based on Melnikov theory from dynamical systems. The error is shown to be small in most instances. The analytical results are supported with numerical simulations.

Rotating magnetic particle microrheometry has been a promising technique in measuring material properties in limited-sample high-viscosity fluids. Experimental limitations in the early motion require further theoretical exploration. In this work, the rotation of a ferromagnetic particle is considered under the influence of an external uniform magnetic field in an infinite highly viscous Newtonian fluid. The motion is restricted at the very low Reynolds number limit. Early-time analytical approximations are utilised to initiate numerical calculations in an attempt to describe the azimuthal velocity dependency on scaled time and radius. The equation of motion is solved by implementing a Crank-Nicholson finite-difference scheme, while the driving time-dependent boundary condition is discretised according to a Lax-Wendroff scheme. Stability and convergence criteria for the PDE are also discussed. It is demonstrated that the step function form of the applied magnetic field does not cause finite displacement other than that expected from Newtonian fluid flow for the typical magnetic field magnitude ranges encountered in micro-rheometric studies. The numerical solution is compared against analytical values available for particle 'zero-total-torque' condition and it was found to be second-order accurate in time and radial dimension.

The pressure–volume–temperature (pVT) relations for intermolecular interactions described by a repulsive step-function are obtained by molecular dynamics (MD) and Monte Carlo (MC) simulations. The system is modeled as a cubic unit cell with periodic boundary condition, where the unit cell contains 108 molecules. The pressure is obtained by the virial equation, and the phase diagram is estimated by Maxwell construction. The self-diffusion coefficients are calculated by MD simulation and used to assign phases as solid or fluid. High- and low-density solid and fluid phases are identified at low temperatures, and negative expansion is observed in the fluid phase near the low-density solid.

The linear system of difference equations , n = 0,1,2,… is considered. A non-trivial solution is small if . Conditions on the sequence of matrices are given guaranteeing that is necessary and sufficient for the existence of at least one small solution. The results are applied to the damped oscillator with step function coefficient a.

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function.

In this paper, we consider a reference input distribution problem in the decentralized systems. The problem is formulated in a two-level system with the relegator defined as the coordinator and the subsystems. A specific static reference input, composed by a series of step functions, is considered in this paper. The optimal local controllers for the subsystems are designed as a function of the distribution results. Principle of Optimality is applied to calculate the individual optimal cost after each step change of the reference input. The relegator is designed to further optimize the total cost among the closed-loop subsystems. The decentralized reference inputs, or the distribution results, are generated after this optimization. According to the communication capabilities between the relegator and the subsystems, we categorize the distribution methods into two types, Passive Reference Input Distribution and Active Reference Input Distribution. Both methods are defined in the paper. The results in the en...

The fixed-charge problem is a non-linear programming problem of practical interest in business and industry. The source-induced fixed-charge transportation problem (SIFCTP) is a variation of the regular fixed-charge transportation problem (FCTP) in which a fixed cost is incurred for every supply point that is used in the solution, along with a variable cost that is proportional to the amount shipped. This problem is significantly different from the widely studied FCTP, where a fixed cost is incurred upon activation of a route. The introduction of the fixed costs in addition to variable costs results in the objective function being a step function. Therefore, fixed-charge problems are usually solved using sophisticated analytical or computer software. This paper deviates from that approach. It presents a computationally simple algorithm for the solution of source-induced fixed-charge problems. The results of empirical tests of the effectiveness of the proposed algorithm are presented.

In order to perform the statistical mechanical calculations, we adopt the periodic cubic system with two molecules in the unit cell. Our model potential function consists of a step-function and the hard sphere wall. We assume the minimum image convention and get the canonical partition function. We find the fluid–fluid phase transition and the negative thermal expansion in the system. We discuss the thermodynamic properties vs. temperature plots under the constant volume. We show some results of the Monte Carlo (MC) simulations on the system with the same potential function under the periodic cubic boundary condition for a comparison. The theoretical results on pVT relations are in agreement with the MC simulations on the two-molecule system. The results of 108-molecule system with the MC simulations are expected to have the similar phase transition near the state in the case of the two-molecule system.

We examine a simple one-dimensional (1D) model of dislocation activity, including a stress-activated source and mutually interacting dislocations. We demonstrate, through numerical and analytical steps, that the dislocations emitted from a 1D stress-activated source evolve towards a distribution which is self-similar in time, and we derive the power-law forms and distribution function. We show that the asymptotic distribution is a step function, and the dislocation front moves out linearly in time. The spacing between dislocations in the asymptotic distribution is uniform and increases logarithmically in time. The number of dislocations increases as t/ln(t), and the strain increases as t 2/ln(t).

This paper considers a single-machine scheduling problem of minimizing the maximum completion time for a set of independent jobs. The processing time of a job is a non-linear step function of its starting time and due date. The problem is already known to be 𝒩𝒫-hard in the literature. In this paper, we first show this problem to be 𝒩𝒫-hard in the ordinary sense by proposing a pseudo-polynomial time dynamic programming algorithm. Then, we develop two dominance rules and a lower bound to design a branch-and-bound algorithm for deriving optimal solutions. Numerical results indicate that the proposed properties can effectively reduce the time required for exploring the solution space.

Recent results on discretization effects in unfolding intensity function of an indirectly observed nonhomogeneous Poisson process show that a critical feasibility condition for the strong L2 consistency of the histogram sieve maximum likelihood estimates requires that the approximation error of the function of interest with step functions tend to 0 faster than the squared singular values of the discretized folding operator. This condition may not be fulfilled in some standard inverse problems like, for example, the deconvolution problem and the Wicksell problem discretized in a standard way. The condition may be satisfied, however, with suitably modified discrete operator and suitably constructed parametric sets for the discrete problems. Motivated by these results, an additional smoothing step is added to the EMS (expectation-maximization-smoothing) algorithm and an automatic procedure for the choice of a smoothing parameter is proposed. An application to the Wicksell problem is presented. In this contex...

Inverse Gaussian (IG) distribution is used as a lifetime model to obtain optimal values for sample size (n) and action limit (d) for employing economic variable acceptance-sampling plans based on step-loss function. IG prior distribution is assumed for the lot mean. The cost function consists of three components: costs of acceptance, rejection, and inspection. Robustness of the proposed samling plan with respect to a choice of prior is studied. It indicates robustness as long as the tail specification of alternative prior distributions are close enough to that of IG distribution. Simulation studies indicate accuracy of Taylor series as an approximation method for intractable expected values. To facilitate employing the proposed sampling plan, Minitab macros are provided in Sec. 4.4 that search for optimal values of n, d and Expected Total Cost (ETC).

In this paper, a spatial-domain method for the reconstruction of the complex permittivity profile of two-dimensional large sizelow contrast objects is presented. The unknown dielectric scatterer is described as a superposition of step functions placed on a rectangular grid with unknown weighting coefficients. Furthermore, an improved Rytov approximation is developed for the description of the unknown internal field in the scattering integral equation. The inverse scattering problem is then solved by minimizing the squared error between measured and predicted values for the scattering amplitude. The efficiency of the direct and inverse scattering technique is validated by using synthetic scattering amplitude data, and a study of noise effects is undertaken. It is shown that significant improvement over conventional diffraction tomography methods based on the first-order Rytov approximation is possible with the proposed technique, and it can be efficiently used for soft tissue imaging in biomedical engineer...

We consider the discrete-time system x ( n )= g x ( n m 1)+ f ( y ( n m k )), y ( n )= g y ( n m 1)+ f ( x ( n m k )), n ] N describing the dynamic interaction of two identical neurons, where g ] (0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. We construct explicitly an attractive 2 k -periodic orbit in the case where f is a step function (McCulloch-Pitts Model). For the general nonlinear signal transmission functions, we use a perturbation argument and sharp estimates and apply the contractive map principle to obtain the existence and attractivity of a 2 k -periodic orbit. This is contrast to the continuous case (a delay differential system) where no stable periodic orbit can occur due to the monotonicity of the associated semiflow.

An important task in analysing time series of the current passed by a single ion channel is the restoration of the true signal from the noisy recording. Hodgson [1] devised a reversible jump Markov chain Monte Carlo (MCMC) algorithm to implement fully Bayesian single channel analysis. Noiseless ion channel signals are typically modelled as step functions with a discrete number of levels. As the number of steps is unknown, the joint posterior distribution under the Bayesian paradigm exhibits variable dimensionality, necessitating the use of reversible jump MCMC (Green [2]). This paper addresses the task of summarising posterior knowledge about the step function representing the noiseless channel signal, using Bayesian decision theory and the ideas of Rue [3]. We compute Bayes estimates of the true step function under a selection of loss functions from the same family.

We observe n events occurring in (0, T] taken from a Poisson process. The intensity function of the process is assumed to be a step function with multiple changepoints. This article proposes a Bayesian binary segmentation procedure for locating the changepoints and the associated heights of the intensity function. We conduct a sequence of nested hypothesis tests using the Bayes factor or the BIC approximation to the Bayes factor. At each comparison in the binary segmentation steps, we need only to compare a singlechangepoint model to a no-changepoint model. Therefore, this method circumvents the computational complexity we would normally face in problems with an unknown (large) number of dimensions. A simulation study and an analysis on a real dataset are given to illustrate our methods.

This paper presents evidence that returns to schooling in Libya are nonlinear. However, this evidence should not be taken as supporting the sheepskin effect. Although high-school graduation is associated with higher earnings over and above what is predicted by the linear model, other diploma years do not possess the same advantage when a spline function is used.When a step function is used, various years, some of which do not lead to a diploma, show high marginal returns.

It is shown that the simple Onsager second-virial approximation of density functional theory can successfully describe the orientational structure of a Gay-Berne film confined between aligning plates. The theory takes as input the density profile as determined by computer simulation and semi-quantitatively reproduces the variation in the nematic order parameter throughout the film, at different temperatures and for different surface potential strengths, without any adjustable parameters. In this context the validity of an earlier analytical approach is discussed where the density, order parameter and tilt angle profiles were assumed to be step functions.

There exist several interpolation methods used in conjunction with the order statistics for estimating a quantile function: the simple step function estimator, linear interpolation, quasi-quantile estimators, the Hermitian quantile function estimator, kernel estimators and Bernstein polynomial estimators, to name a few. In this note we explore using monotone rational splines for developing quantile function estimators. Delbourgo and Gregory [1] give a representation of a piecewise rational cubic function that is a monotone interpolant. The piecewise cubic interpolant is a generalization of their earlier rational quadratic monotone interpolant defined and analyzed in Gregory and Delbourgo [2] and Delbourgo and Gregory 3-4. We construct and examine the estimators based on these interpolants in terms of their exact bootstrap expressions for the finite sample size bias and variance estimates given sample profiles from a standard normal population. Recommendations for the choice of quantile estimator are given.

The non-equally-probable step-function (NEPSF) representation has been proposed as a new method for selecting scattering angles from a multigroup transfer matrix for a medium of a single nuclide or a mixture. This technique avoids the disadvantage of the equally-probable step-function representation, which cannot well present the group-to-group transfer lax for a mixture. Similar to the discrete angle representation, the NEPSF representation can preserve precisely the moments of the scattering law; moreover, it has the advantage of eliminating ray effects. We give semianalytical procedures for obtaining P3 and P5 NEPSF representations, which will facilitate greatly low-order transfer matrix treatment in multigroup Monte Carlo.

Time delays play an important role in many biological and ecological systems. However, they are usually incorporated into mathematical models in a way not explicitly dependent on space, even though the delay may be modelling some environmental aspect of the system. In this paper we study a scalar reaction-diffusion equation with a spatially inhomogeneous delay, taken for simplicity in the form of a step function of the spatial coordinate. We derive the dispersion relation from which analytical results on the stability or instability of the uniform steady states can be determined. We confirm and extended these results by numerical simulations which confirm the possibility of qualitatively different types of behaviour on different parts of the spatial domain.

Motivated by the successful application of mathematical programming techniques to difficult machine learning problems, we seek solutions of concave minimization problems over polyhedral sets with minimum number of nonzero components. We that if such problems have a solution, they have a vertex solution with a minimal number of zeros. This includes linear programs and general linear complementarity problems. A smooth concave exponential approximation to a step function solves the minimumsupport problem exactly for a finite value of the smoothing parameter. A fast finite linear-programming-based iterative method terminates at a stationary point, which for many important real world problems provides very useful answers. Utilizing the complementarity property of linear programs and linear complementarity problems, an upper bound on the number of nonzeros can be obtained by solving a single convex minimization problem on a polyhedral set

The coefficients of a truncated Legendre series are usually used in multigroup cross-section sets to treat the angular distribution for a group-to-group scattering event. Fine energy meshes and low-order Legendre expansions result in negative values in the corresponding multigroup Legendre expansions; therefore, special transfer matrix treatments for multigroup cross sections are needed. The difficulties of the truncated Legendre series representation in treating multigroup transfer are explained. In TRIMARAN-II, two existing standard methods, the equally probable step function (EPSF) representation and the discrete angle representation, which are based on preservation (at least approximately) of the first moments, are studied. The discrete angle representation has the advantage of accurately preserving the moments, but it may cause ray effects; the EPSF representation can eliminate ray effects, but it is not suitable for the treatment of the transfer matrix for material mixtures, because both forward- and backward-peaked scattering are present in this kind of cross section. A new method, the nonequally probable step function (NEPSF) representation, which combines the advantages of both the discrete angle and EPSF representations, is introduced. It can eliminate ray effects and accurately preserve the moments. The conjugate gradient method, powerful for solving multidimensional minimization problems is used to obtainmore » both the EPSF and NEPSF representations. A problem of neutron transmission in a hydrogenous material is used to compare the three representations. Comparisons of the TRIMARAN-II results with the three representations to those of the TRIPOLI-4 pointwise cross-section Monte Carlo code are given.« less

In this paper, we offer a natural way of acquainting the student with the notion of the distance function. Real‐life situations and graphs of ‘easy‐to‐see’ step functions should appeal to students’ intuition and make the topic of distance‐preserving functions easily accessible.

A family of kernel-based estimators parametrised by a bandwidth is used to estimate the location of the discontinuity in discretely observed data consisting of a discontinuous regression function with added noise. Asymptotic bias and variance calculations for the estimated location are given as a function of the bandwidth and the sample size. For functions Which are Smooth apart from a single discontinuity, bias is of order h 2, and is not in general negligible compared with variance, which is of order h/n. The asymptotic mean square error rate MSE = O(n −4/3), obtained by balancing squared bias and variance, can usually be improved to O(n −2+δ ) with δ > 0 arbitrarily small by choosing a smaller bandwidth h = O(n −1+δ ). Special cases are treated and bias and variance are calculated explicitly. Only bias depends on the type of discontinuous function considered: For a step function with added noise and for any function which is antisymmetric about the discontinuity point, squared bias becomes negligible c...

In L1 ordered alloys a dislocation can dissociate in the 111 plane into four 2 d Shockley partial dislocations connected by a complex stacking fault of width D and an antiphase boundary of width with specific energies and c A d b respectively. When 10 the classical treatment of dislocation interaction becomes inaccurate because treating the displacement as a step function is no longer adequate. A treatment in the Peierls model allows one to obtain more d D realistic values of and when and have been determined experic A mentally. Corrected values for Ni Al and Ni Fe are given. 3 3

We report the first demonstration of proton-exchanged optical planar waveguides in x-cut and z-cut LiNbO3 doped with 6 mol% ZnO using adipic acid as proton source. These waveguides exhibit a graded index profile which can be modeled by a linear step function with a surface index increase of 0.135 and 0.14 for x-cut and z-cut waveguides, respectively. The diffusion constant Do and the activation energy Q are characterized optically to be 1.64×109 μm2/h and 88.8 KJ/mol for x-cut waveguides, and 1.478×109 μm2/h and 91.25 KJ/mol for z-cut waveguides, respectively. The diffusion rate along the z-axis is slower than that along the x-axis, whereas the surface index increase on z-cut waveguides is larger than that on x-cut waveguides.

A review is given of some features of theories for inhomogeneous fluids of nonspherical molecules that take as input the direct correlation function of the corresponding homogeneous system. Two different methods are described for defining the structure of hard homonuclear molecules close to a hard planar wall. A spherical harmonics expanison (SHE) within the integral equation (IE) method is presented and, for comparison, a version of density functional theory for orientable hard bodies. In both cases the Pynn-Lado model is employed and a comparison is made with Monte Carlo data. The results indicate that for hard molecules the IE approach does not always capture the effects of orientation due to the characteristics of the SHE for the step function. This disadvantage is particularly true in the case of the orientationally averaged density profile.

A two-dimensional transient model is developed to simulate steady-state and transient operations of single-cell thermionic fuel elements (TFEs). Model predictions are in good agreement with published data to within 4.5 and 5.5% for fission and electrically heated TFEs of the TOPAZ-II type, respectively. In addition, the results of a transient analysis simulating the startup of an electrically heated TFE, following a step function increase in thermal power, are in presented and discussed.