General Discrete Model Research Articles

Monotone traveling waves in a general discrete model for populations with long term memory

In this paper we consider the existence of monotone traveling waves for a class of general integral difference models for populations that are dependent on the previous state term and also on long term memory. This allows us to consider multiple past states. For this model we will have to deal with the non-compactness of the evolution operator when we prove the existence of a fixed point. This difficulty will be overcome by using the Monotone Iteration Method and Dini’s Theorem to show uniform convergence of an iterative evolution operator to a continuous wave function.

Kinetic models for semiflexible polymers in a half-plane

Based on a general discrete model for a semiflexible polymer chain, we introduce a formal derivation of a kinetic equation for semiflexible polymers in the half-plane via a continuum limit. It turns out that the resulting equation is the kinetic Fokker-Planck-type equation with the Laplace-Beltrami operator under a non-local trapping boundary condition. We then study the well-posedness and the long-chain asymptotics of the solutions of the resulting equation. In particular, we prove that there exists a unique measure-valued solution for the corresponding boundary value problem. In addition, we prove that the equation is hypoelliptic and the solutions are locally H\"older continuous near the singular boundary. Finally, we provide the asymptotic behaviors of the solutions for large polymer chains.

One-shot detection limits of time-alignment two-photon illumination radar

Quantum radar has recently gained increasing importance in a number of military applications. The estimation accuracy of one-shot quantum illumination events is significant in target detection. However, the accuracy is inevitably deteriorated by measurement noises. The traditional one-shot illumination emits a single photon towards a certain area which thermal noise exists in the path to, and the states of the received photons are hard to distinguish in the following processing. Therefore, a new optical probe source is proposed in this work. The independent detecting unit in the enhanced illumination is comprised of two photons aligned in time by using Hong–Ou–Mandel (HOM) interferometer. Further, one-shot detection in a general discrete model is realized and it proves a significant promotion in accuracy. The expansion of useful parts in parameter space and the lower minimal error probability for hypothesis testing have been mathematically demonstrated. The accuracy of one-shot detection can be effectively improved by the proposed scheme implying that it possesses great potential applications in quantum illumination and imaging.

On the network of synchronized switch damping for blisks

This paper presents a new semi-active piezoelectric damping strategy for the vibration suppression of integrally bladed disk (blisk). The main novelty is the networking of the synchronized switch damping (SSD). Different from the independently shunted SSD broadly studied in the literature, in our approach the piezoelectric materials distributed to different blade sectors are first connected in few electric networks. The blades in the same network share a single SSD circuit. In this way, additional electrical energy channels are established, and the vibration energy can be thereby dissipated and redistributed globally. In addition, the overall number of the SSD circuits will be drastically reduced, decreasing the complexity of the system. To illustrate and demonstrate such an idea, numerical and experimental studies are conducted. First, We presents the governing equations of a general discrete blisk model with the SSD network and derive a linearization method to solve the steady-state solutions. Next, the optimal topologies of the SSD network for different vibrational modes are summarized through numerical simulation. The proposed network exhibits better performance than other candidates for the mistuned case. Then, through the Mont–Carlo simulation, statistical results prove that the network is also effective for the suppression of the amplitude magnification due to mistuning. Eventually, these findings are validated experimentally through a dummy blisk.

Inlet particle-sorting cyclones configured along a spiral channel for the enhancement of PM2.5 separation

In order to control fine particulate matter (PM2.5) emissions from industrial sources, efficient separation technologies are absolutely crucial, in this work, a novel method was developed for enhancing PM2.5 separation by regulating particle size distribution of the cyclone’s inlet using a spiral distributor. A general discrete model was further created for evaluating flow and pressure drop uniformity in parallel mini-cyclones configured along a spiral channel to obtain high separation efficiency and large handling capacity simultaneously. The results showed that the spiral distributor provided a “particle-sorting” effect, which made the small and large particles enter cyclone radially from the outer side and to the inner side at the rectangular inlet, respectively. The large particles on the inside of the cyclones could form a radially mobile granular membrane that could further drive the relatively small particles to be separated. The mean diameter of the inlet particles was 8.237 μm, the overall separation efficiency with the spiral distributor reached 90.5%, which was13.4 % higher than the initial cyclone. The maldistribution of the flow and pressure drop decreased significantly when the flow rate and cyclone number increased, and the maximum maldistribution of the flow was less than 10%. It is the first time for this novel method successfully applied to the deep purification of tail gas in the decoking process of an ethylene cracking furnace, in which the separation efficiency of coke particles was considerably improved.

One-shot detection limits of quantum illumination with discrete signals

To detect a stealth target, one may directly probe it with a single photon and analyze the reflected signals. The efficiency of such conventional detection scheme can potentially be enhanced by quantum illumination, where entanglement is exploited to break the classical limits. The question is what is the optimal signal state for achieving the detection limit? Here, we address this question in a general discrete model, and derive a complete set of analytic solutions. For one-shot detection, the parameter space can be classified into three distinct regions, in the form of a “phase diagram” for both conventional and quantum illumination. Interestingly, whenever the reflectivity of the target is less than some critical value, all received signals become useless, which is true even if entangled resources are employed. However, there does exist a region where quantum illumination can provide advantages over conventional illumination; there, the optimal signal state is an entangled state with an entanglement spectrum inversely proportional to the spectrum of the environmental noise state and is, surprisingly, independent of the occurrence probability and the reflectivity of the object. The entanglement of the ideal probe state increases with the entropy of the environment; it becomes more entangled as the temperature of the environment increases. Finally, we show that the performance advantage cannot be fully characterized by any measure of quantum correlation, unless the environment is a complete mixed state.

Interannual population dynamics of the green spruce aphid Elatobium abietinum (Walker) in France

AbstractThe hypothesis that similar processes govern interannual dynamics of green spruce aphid in the UK and France, is generally supported by the application of a general discrete model. A simple model based on relatively few parameters was able to closely characterise interannual population dynamics from completely independent aerial and arboreal samples of aphids. Long‐term field population estimates of the green spruce aphid Elatobium abietinum (Walker) in France have provided the opportunity to select and evaluate the generality of a model, which was developed in the UK to explain the year‐to‐year variations in peak abundance of the aphid. The objective was to observe the influence of the local climates and disturbing climate factors on the population densities of the insect in two regions of France. The model uses climate variables and aphid population data from regular samples in the two regions that were investigated. A general discrete model was used to predict aphid population densities. The model performed well in tracking the interannual patterns of population but was less likely to predict absolute population density. To improve predictions, further account would need to be taken of additional site‐specific climate variables and the strength of overcompensating density dependence. Nevertheless, it is clear that broadly similar processes are at work in the population dynamics of this insect across its biogeographical range.

A general discrete degradation model with fatal shocks and age- and state-dependent nonfatal shocks

A general discrete competing risks model with age- and state-dependent random shocks is developed in this research. This model allows for a more generalized structure of dependence by considering the current degradation state. Random shocks are classified into fatal shocks and nonfatal shocks with corresponding probability. Fatal shocks can bring immediate failure, and nonfatal shocks will affect the degradation process. In this sense, the dependent relationships have two folds: a) the dependence between nonfatal shocks and the natural degradation process, reflected by a time-scaled accelerated factor, and b) the dependence between the current state and nonfatal shocks, which is modulated by three functions of the current state representing the effects on degradation increments and degradation rate acceleration of nonfatal shocks. The extended Kalman filter (EKF) is used together with measured data to estimate the distribution of the current state. Based on the degradation state equation and measurement function, a closed-form reliability function is derived, and the reliability can be updated given new measured data. An illustrative example is provided to demonstrate the advantages of the proposed model. The result shows that the reliability evaluation is more realistic when the impact of the current state on random shocks is considered.

A general mathematical model for two-parameter generating machining of involute cylindrical gears

• A general mathematical model for two-parameter generating machining of involute gears is developed. • A general discrete assisted model for two-parameter generating machining is proposed. • The instant contact line is derived by implicated information during discrete enveloping. • The engaged cutting edge segments can be distinguished when multiple-cutting edge are adopted. Instead of the common analytical model for involute gears machining, a general mathematical model for two-parameter generating machining is developed in this study with adopting discrete enveloping to increase its robustness and generality for machined profile calculation and instant contact analysis. The geometric parameters and motion relationships are integrated according to the cross-axis gear meshing at first, and the enveloping theory for two-parameter generating process is analyzed briefly with involving middle gear rack. By transforming the multiple cutting edges relative to the gear blank according to the generating motions, a numerical algorithm is proposed to distinguish the external enveloping profile on the shaft section, then the derivation of instant contact points via the implied time sequences of profile points is introduced. Finally, the coordinate transformations for the spatial trajectory of instant contact points identifying and the engaged cutting edge segments distinguishing are investigated. At last, a non- involute profile skiving is performed to verify the generality and the robustness of proposed method, and several typically generating machining operations for cylindrical involute gear are simulated numerically to proof the accuracy of the proposed model by finding the deviations from the standard involute curves and the geometries of the instant contact points.

Performance Analyses of Four-Instant Discretization Formulas With Application to Generalized-Sylvester-Type Future Matrix Equation

For solving generalized-Sylvester-type future matrix equation (GS-type FME) effectively, the continuous zeroing neural network (ZNN) model should be discretized by appropriate discretization formula. In this paper, theoretical upper bound of truncation errors of four-instant discretization formulas is presented, of which the upper bound is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(g^{2})$ </tex-math></inline-formula> . In addition, an inspirational method to decrease the truncation error of general discretization formula is proposed and analysed. Specifically, first of all, on the basis of Taylor expansion, different four-instant discretization formulas are presented by exploiting the higher-order derivative elimination (HODE) methods, which include second-order derivative elimination (SODE) method, third-order derivative elimination (TODE) method and fourth-order derivative elimination (FODE) method. Then, the theoretical upper bound of truncation errors of these discretization formulas is presented, investigated and analysed with the theoretical analyses provided. Secondly, by analyzing the parameter of general discretization formula, we propose and analyse an inspirational method to decrease the truncation error of general discretization formula. Thirdly, by exploiting the general discretization formula to discretize continuous ZNN model for solving GS-type FME, general discrete ZNN model is presented. Finally, an illustrative numerical experiment is presented to substantiate the effectiveness of theoretical results.

Inference of multiple-wave admixtures by length distribution of ancestral tracks.

The ancestral tracks in admixed genomes are valuable for population history inference. While a few methods have been developed to infer admixture history based on ancestral tracks, these methods suffer the same flaw: only population admixture history under some specific models can be inferred. In addition, the inference of history might be biased or even unreliable if the specific model deviates from the real situation. To address this problem, we firstly proposed a general discrete admixture model to describe the admixture history with multiple ancestral populations and multiple-wave admixtures. We next deduced the length distribution of ancestral tracks under the general discrete admixture model. We further developed a new method, MultiWaver, to explore multiple-wave admixture histories. Our method could automatically determine an optimal admixture model based on the length distribution of ancestral tracks, and estimate the corresponding parameters under this optimal model. Specifically, we used a likelihood ratio test (LRT) to determine the number of admixture waves, and implemented an expectation-maximization (EM) algorithm to estimate parameters. We used simulation studies to validate the reliability and effectiveness of our method. Finally, good performance was observed when our method was applied to real data sets of African Americans and Mexicans, and new insights were gained into the admixture history of Uyghurs and Hazaras.

Nonuniform behavior and stability of Hopfield neural networks with delay

Based on a new abstract result on the behavior of nonautonomous delayed equations, we obtain a stability result for the solutions of a general discrete nonautonomous Hopfield neural network model with delay. As an application we improve some existing results on the stability of Hopfield models.

Patches Approach to Investigate the Populational Dynamics in Dengue

In areas where resources are located in patches or discrete locations, human dispersal is more conveniently modeled, in which the population is divided into discrete patches. In this work we develop a general discrete model to analyze the spread of Dengue disease. In the process of mathematical modeling we take into account the human populations and the circulation of a single serotype of dengue mosquitoes. The movements of susceptible, infected and recovered humans among all patches are considered. Aquatic phases with different carrying capacities are considered within the patches. Also an arbitrary number of patches can be used to simulate the spread of dengue disease. In this paper we performed numerical experiments to show the applicability of this methodology to investigate the dengue disease problem. The general discrete space model was developed for solving epidemiological problems whereas the human-vector interactions and human mobilities play an important role. Based on our numerical results, we may recommend the general patches model for solving epidemiological problems in Population Dynamics.

Monotone traveling waves in a general discrete model for populations

In this paper we consider the existence of monotone traveling waves for a class of general integral difference model for populations that allows the dispersal probability to have no continuous density functions but the fecundity functions to generate a monotone dynamical systems. In this setting we deal with the non-compactness of the evolution operator by using the monotone iteration method.

Boundary Layers and Shock Profiles for the Broadwell Model

We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.

Asymptotic Behaviour of Ruin Probabilities in a General Discrete Risk Model Using Moment Indices

We study the rough asymptotic behaviour of a general economic risk model in a discrete setting. Both financial and insurance risks are taken into account. Loss during the first \(n\) years is modelled as a random variable \(B_1+A_1B_2+\cdots +A_1\ldots A_{n-1}B_n\), where \(A_i\) corresponds to the financial risk of the year \(i\) and \(B_i\) represents the insurance risk, respectively. Risks of the same year \(i\) are not assumed to be independent. The main result shows that ruin probabilities exhibit power law decay under general assumptions. Our objective is to give a complete characterisation of the relevant quantities that describe the speed at which the ruin probability vanishes as the amount of initial capital grows. These quantities can be expressed as maximal moments, called moment indices, of suitable random variables. In addition to the study of ultimate ruin, the case of finite time interval ruin is considered. Both of these investigations make extensive use of the new properties of moment indices developed during the first half of the paper.

Degraded Broadcast Diamond Channels With Noncausal State Information at the Source

A state-dependent degraded broadcast diamond channel is studied where the source-to-relays cut is modeled with two noiseless, finite-capacity digital links with a degraded broadcasting structure, while the relays-to-destination cut is a general multiple access channel controlled by a random state. It is assumed that the source has noncausal channel state information and the relays have no state information. Under this model, first, the capacity is characterized for the case where the destination has state information, i.e., has access to the state sequence. It is demonstrated that in this case, a joint message and state transmission scheme via binning is optimal. Next, the case where the destination does not have state information, i.e., the case with state information at the source only, is considered. For this scenario, lower and upper bounds on the capacity are derived for the general discrete memoryless model. Achievable rates are then computed for the case in which the relays-to-destination cut is affected by an additive Gaussian state. Numerical results are provided that illuminate the performance advantages that can be accrued by leveraging noncausal state information at the source.

Continuous-time evolutionary stock and bond markets with time-dependent strategies

This paper develops a general continuous-time evolutionary finance model with time-dependent strategies based on evolutionary game theory. We show that the continuous model, which is a limit of a general discrete model, is well-defined and if there exists one completely diversified strategy in the market, then there is no sudden bankruptcy. We study in detail, a deterministic evolutionary bond market and certify that a bond market is evolutionary stable if and only if the total returns across all assets are the same. By this way, we derive an explicit expression for the bond valuation and provide an approach to recover the benchmark interest rate from an effective bond market. Key words: Evolutionary finance, evolutionary bond market, continuous-time model, time-dependent strategies.

Continuous-Time Evolutionary Stock and Bond Markets with Time-Dependent Strategies

This paper develops a general continuous-time evolutionary finance model with time-dependent strategies. It is shown that the continuous model, which is a limit of a general discrete model, is well-defined and if there exists one completely diversified strategy in the market, then there is no sudden bankruptcy. After that a deterministic evolutionary bond market is studied in detail. It is certified that a bond market is evolutionary stable, which is equal to arbitrage-free if and only if the total returns defined in this paper across all the assets are the same, or each bond is evaluated by an improper integral in which the integrand is a discounted value of the dividend payoff with the discount rate being market consumption parameter. Last an approach to compute the benchmark interest rate is provided.