Transitioning from Discrete to Continuous Distribution Mathematica vs. Excel —An Example

Frequencies of the repeated integers of the first n digits of e.g. π utilizing commercial software are listed. The discrete distribution is utilized to evaluate its statistical moments. The distribution is fitted with a polynomial generating a continuous replica of the former. Its statistical moments are evaluated and compared to the former. The procedure clarifies the mechanism transiting from discrete to a continuous domain. Applying Mathematica the fitted polynomial is replaced with an interpolated function with controlled smoothing factor refining the quality of the fit and its corresponding moments. Knowledge learned assists in the understanding of the standard procedure calculating moments of e.g. Maxwell-Boltzmann continuous distribution in kinetic Theory of gases.

Application of a Discrete Phase-Randomized Coherent State Source in Round-Robin Differential Phase-Shift Quantum Key Distribution

Recently, a novel kind of quantum key distribution called the round-robin differential phase-shift （RRDPS） protocol was proposed, which bounds the amount of leakage without monitoring signal disturbance. The protocol can be implemented by a weak coherent source. The security of this protocol with a simply characterized source has been proved. The application of a common phase shift can improve the secret key rate of the protocol. In practice, the randomized phase is discrete and the secret key rate is deviated from the continuous case. In this study, we analyze security of the RRDPS protocol with discrete-phase-randomized coherent state source and bound the secret key rate. We fix the length of each packet at 32 and 64, then simulate the secret key rates of the RRDPS protocol with discrete-phase randomization and continuous-phase randomization. Our simulation results show that the performance of the discrete-phase randomization case is close to the continuous counterpart with only a small number of discrete phases. The research is practically valuable for experimental implementation.

Improving the utility of locally differentially private protocols for longitudinal and multidimensional frequency estimates

This paper investigates the problem of collecting multidimensional data throughout time(i.e.,longitudinal studies)for the fundamental task of frequency estimation under Local Differential Privacy(LDP)guarantees.Contrary to frequency estimation of a single attribute,the multidimensional aspect demands particular attention to the privacy budget.Besides,when collecting user statistics longitudinally,privacy progressively degrades.Indeed,the“multiple”settings in combination(i.e.,many attributes and several collections throughout time)impose several challenges,for which this paper proposes the first solution for frequency estimates under LDP.To tackle these issues,we extend the analysis of three state-of-the-art LDP protocols(Generalized Randomized Response–GRR,Optimized Unary Encoding–OUE,and Symmetric Unary Encoding–SUE)for both longitudinal and multidimensional data collections.While the known literature uses OUE and SUE for two rounds of sanitization(a.k.a.memoization),i.e.,L-OUE and L-SUE,respectively,we analytically and experimentally show that starting with OUE and then with SUE provides higher data utility(i.e.,L-OSUE).Also,for attributes with small domain sizes,we propose Longitudinal GRR(L-GRR),which provides higher utility than the other protocols based on unary encoding.Last,we also propose a new solution named Adaptive LDP for LOngitudinal and Multidimensional FREquency Estimates(ALLOMFREE),which randomly samples a single attribute to be sent with the whole privacy budget and adaptively selects the optimal protocol,i.e.,either L-GRR or L-OSUE.As shown in the results,ALLOMFREE consistently and considerably outperforms the state-of-the-art L-SUE and L-OUE protocols in the quality of the frequency estimates.

Global exponential stability of mixed discrete and distributively delayed cellular neural network

This paper concernes analysis for the global exponential stability of a class of recurrent neural networks with mixed discrete and distributed delays. It first proves the existence and uniqueness of the balance point, then by employing the Lyapunov-Krasovskii functional and Young inequality, it gives the sufficient condition of global exponential stability of cellular neural network with mixed discrete and distributed delays, in addition, the example is provided to illustrate the applicability of the result.

Synchronization of stochastically hybrid coupled neural networks with coupling discrete and distributed time-varying delays

A general model of linearly stochastically coupled identical connected neural networks with hybrid coupling is proposed, which is composed of constant coupling, coupling discrete time-varying delay and coupling distributed timevarying delay. All the coupling terms are subjected to stochastic disturbances described in terms of Brownian motion, which reflects a more realistic dynamical behaviour of coupled systems in practice. Based on a simple adaptive feedback controller and stochastic stability theory, several sufficient criteria are presented to ensure the synchronization of linearly stochastically coupled complex networks with coupling mixed time-varying delays. Finally, numerical simulations illustrated by scale-free complex networks verify the effectiveness of the proposed controllers.

The Lotka-Volterra Competition System with Discrete and Distributed Time Delays

In this paper, the Lotka-Volterra competition system with discrete and distributed time delays is considered. By analyzing the characteristic equation of the linearized system, the local asymptotic stability of the positive equilibrium is investigated. Moreover, we discover the delays don＇t effect the stability of the equilibrium in the delay system. Finally, we can conclude that the positive equilibrium is global asymptotically stable in the delay system.

TESTING FOR VARYING DISPERSION IN DISCRETE EXPONENTIAL FAMILY NONLINEAR MODELS

It is necessary to test for varying dispersion in generalized nonlinear models.Wei,et al(1998) developed a likelihood ratio test,a score test and their adjustments to test for varying dispersion in continuous exponential family nonlinear models.This type of problem in the framework of general discrete exponential family nonlinear models is discussed.Two types of varying dispersion,which are random coefficients model and random effects model,are proposed,and corresponding score test statistics are constructed and expressed in simple,easy to use,matrix formulas.

On the MLE of the Waring distribution

The two-parameter Waring is an important heavy-tailed discrete distribution,which extends the famous Yule Simon distribution and provides more flexibility when modelling the data.The commonly used EFF(Expectation-First Frequency)for parameter estimation can only be applied when the first moment exists,and it only uses the information of the expectation and the first frequency,which is not as efficient as the maximum likelihood estimator(MLE).However,the MLE may not exist for some sample data.We apply the profle method to the log-likelihood function and derive the necessary and sufficient Conditions for the existence of the MLE of the Waring parameters.We use extensive simulation studies to compare the MLE and EFF methods,and the goodness-of-fit comparison with the Yule Simon distribution.We also apply the Waring distribution to fit an insurance data.

Discrete element modeling of the effect of particle size distribution on the small strain stiffness of granular soils

Discrete element modeling was used to investigate the effect of particle size distribution on the small strain shear stiffness of granular soils and explore the fundamental mechanism controlling this small strain shear stiffness at the particle level. The results indicate that the mean particle size has a negligible effect on the small strain shear modulus. The observed increase of the shear modulus with increasing particle size is caused by a scale effect. It is suggested that the ratio of sample size to the mean particle size should be larger than 11.5 to avoid this possible scale effect. At the same confining pressure and void ratio, the small strain shear modulus decreases as the coefficient of uniformity of the soil increases. The Poisson＇s ratio decreases with decreasing void ratio and increasing confining pressure instead of being constant as is commonly assumed. Microscopic analyses indicate that the small strain shear stiffness and Poisson＇s ratio depend uniquely on the soil＇s coordination number.

Modeling the influence of forced ventilation on the dispersion of droplets ejected from roadheader-mounted external sprayer

In order to reveal the influence of forced ventilation on the dispersion of droplets ejected from roadheader-mounted external sprayer,the paper studies the air-flowing field and the droplet distribution under the condition of gentle breeze and normal forced ventilation in heading face using the particle tracking technology of computational fluid dynamics(CFD).The results show that air-flowing tendency in the same section presents great comparability in the period of gentle breeze and forced ventilation,and the difference mainly embodies in the different wind velocity.The influence of ventilation on the dispersion of droplets is faint under the gentle breeze condition.The droplet can be evenly distributed around the cutting head.However,under the normal forced ventilation,a large number of droplets will drift to the return air side.At the same time,droplet clusters are predominantly presented in the lower part of windward side and the middle of the leeward side around the cutting head.In contrast,the droplet concentration in other parts around cutting head decreases a lot and the droplets are unable to form close-grained mist curtain.So the dust escape channel is formed.In addition,the simulation results also reveal that the disturbance of air flow on the droplet distribution can be effectively relieved when using ventilation duct with Coanda effect(VDCE).Field experiment results show that the dust suppression efficiency of total dust and respirable dust increases respectively by 10.5%and 9.3%when using VDCE,which proves that it can weaken the influence of airflow on droplet dispersion.

A Stochastic Inventory System with Postponed Demands and Infinite Pool in Discrete-Time Setup

In this article,we consider a discrete-time inventory model in which demands arrive according to a discrete Markovian arrival process.The inventory is replenished according to an es;ST policy,and the lead time is assumed to follow a discrete phase-type distribution.The demands that occur during stock-out periods either enter a pool which has an infinite capacity or leave the system with a predefined probability.The demands in the pool are selected one by one,if the on-hand inventory level is above s t 1;and the interval time between any two successive selections is assumed to have a discrete phase-type distribution.The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady-state case.We derive the system performance measures under steady state and using these measures,the total expected cost rate of the system is calculated.The impacts of arrival rate on the performance measures are graphically illustrated.Finally,we study the impact of cost on the optimal values of the total expected cost rate,inventory level and the reorder point.

Adaptive Fuzzy Dynamic Surface Control for a Class of Perturbed Nonlinear Time-varying Delay Systems with Unknown Dead-zone

In this paper,adaptive dynamic surface control(DSC) is developed for a class of nonlinear systems with unknown discrete and distributed time-varying delays and unknown dead-zone.Fuzzy logic systems are used to approximate the unknown nonlinear functions.Then,by combining the backstepping technique and the appropriate Lyapunov-Krasovskii functionals with the dynamic surface control approach,the adaptive fuzzy tracking controller is designed.Our development is able to eliminate the problem of 'explosion of complexity' inherent in the existing backstepping-based methods.The main advantages of our approach include:1) for the n-th-order nonlinear systems,only one parameter needs to be adjusted online in the controller design procedure,which reduces the computation burden greatly.Moreover,the input of the dead-zone with only one adjusted parameter is much simpler than the ones in the existing results;2) the proposed control scheme does not need to know the time delays and their upper bounds.It is proven that the proposed design method is able to guarantee that all the signals in the closed-loop system are bounded and the tracking error is smaller than a prescribed error bound,Finally,simulation results demonstrate the effectiveness of the proposed approach.

Characteristic model-based consensus of networked heterogeneous robotic manipulators with dynamic uncertainties

In this paper, we address the characteristic model-based discrete-time consensus problem of networked robotic manipulators with dynamic uncertainties. The research objective is to achieve joint-position consensus of multiple robotic agents interconnected on directed graphs containing a spanning tree. A novel characteristic model-based distributed adaptive control scenario is proposed with a state-relied projection estimation law and a characteristic model-based distributed controller. The performance analysis is also unfolded where the uniform ultimate boundedness(UUB) of consensus errors is derived by resorting to the discrete-time-domain stability analysis tool and the graph theory. Finally, numerical simulations illustrate the effectiveness of the proposed theoretical strategy.