Reachable set estimation for discrete-time Markovian jump neural networks with unified uncertain transition probability

This paper focuses on the reachable set estimation for Markovian jump neural networks with time delay.By allowing uncertainty in the transition probabilities,a framework unifies and enhances the generality and realism of these systems.To fully exploit the unified uncertain transition probabilities,an equivalent transformation technique is introduced as an alternative to traditional estimation methods,effectively utilizing the information of transition probabilities.Furthermore,a vector Wirtinger-based summation inequality is proposed,which captures more system information compared to existing ones.Building upon these components,a novel condition that guarantees a reachable set estimation is presented for Markovian jump neural networks with unified uncertain transition probabilities.A numerical example is illustrated to demonstrate the superiority of the approaches.

Some New Results on Transition Probability

In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.

Development of diabetic complications and influencing factors among 32653 type 2 diabetes patients:retrospective cohort study using a multi-state Markov

Background:Patients with type 2 diabetes are at high risk for developing multiple chronic complications.However,there is a lack of studies of the cumulative number of diabetic complications in China.Methods:A retrospective cohort study was performed from 2009 to 2021.Type 2 diabetes patients who were first diagnosed after the age of 35 years between January 1,2009,and December 31,2017,were included.Five states were defined according to the number of chronic complications:no(S0),one(S1),two(S2),three(S3),and four or more complications(S4).A multi-state Markov model was constructed to estimate transition probability,transition intensity,mean sojourn time,and the possible factors for each state.Results:The study included 32653 type 2 diabetes patients(mean age,59.59 years;15929(48.8%)male),and mean follow-up time of 7.75 years.In all,4375 transitions were observed.The 12-year transition probability of from state S0 to S1 was the lowest at 16.4%,while that from S2 to S3 was the highest,at 45.6%.Higher fasting blood glucose,lower high-density lipoprotein cholesterol,higher total cholesterol,and an unhealthy diet were associated with higher risk of progression from S0 to S1.Being female,less than 60 years old,weekly physical activity,and vegetarian diet decreased this risk.Being female and less than 60 years old reduced the likelihood of transition from S1 to S2,whereas lower high-density lipoprotein cholesterol increased this likelihood.Conclusions:Following the occurrence of two complications in type 2 diabetes patients,the risk for accumulating a third complication within a short time is significantly increased.It is important to take advantage of the stable window period when patients have fewer than two complications,strengthen the monitoring of blood glucose and blood lipids,and encourage patients to maintain good living habits to prevent further deterioration.

Transition probabilities for NII 2p4f-2p3d and 2s2p^23d-2s2p^23p obtained by a semiclassical method

Based on NII spectra, some transition probabilities for 2p4f-2p3d and 2s2p23d-2s2p23p are obtained by a semi- classical method. The results are in good agreement with other measurements and the data reported by the National Institute of Standards and Technology. The transition probability for a line of 424.18nm is reported for the first time. Meanwhile, a feasible method of calculating transition parameters related to special excited configurations or highly excited states is provided.

Electric-dipole allowed(E1) and forbidden(E2,M1 and M2) transition probabilities of 4f for N^+

By applying systematically enlarged multi-configuration Dirac-Fock wavefunction, the transitions for electricdipole allowed （E1） and forbidden （E2, M1 and M2） lines are studied among 4f pair coupling and low-lying configurations for singly ionized nitrogen. Most important effects of relativity, electron correlation, the rearrangement of electron density, Breit interaction, and quantum electrodynamic effects are included in the computation. Then, allowed （E1） and forbidden （E2, M1 and M2） transition probabilities of 4f for N＋ are obtained and compared with experimental results. Good agreement with available experimental results is found and most of the data of 4f are presented for the first time.

Extreme ultraviolet and x-ray transition wavelengths in Rb XXIV

Energy levels, radiative rates, oscillator strengths and line strengths are reported for transitions among the lowest 97 levels of the（1s22s22p6） 3s23p2, 3s23p3 d, 3s3p3, 3p4, 3s3p23 d, and 3s23d2 configurations of Rb XXIV. A multiconfiguration Dirac-Fock（MCDF） method is adopted for the calculations. Radiative rates, oscillator strengths, and line strengths are provided for all electric dipole（E1）, magnetic dipole（M1）, electric quadrupole（E2）, and magnetic quadrupole（M2）transitions from the ground level to all 97 levels, although calculations are performed for a much larger number of levels.To achieve the accuracy of the data, comparisons are provided with similar data obtained from the Flexible Atomic Code（FAC） and also with the available theoretical and experimental results. Our energy levels are found to be accurate to better than 1.2%. Wavelengths calculated are found to lie in EUV（extreme ultraviolet） and x-ray regions. Additionally, lifetimes for all 97 levels are obtained for the first time.

Transition probabilities and lifetimes of the low-lying levels of Fe XIV

The multi-configuration Dirac-Fock method is employed to calculate the transition energies, probabilities, and oscillator strengths for electric dipole allowed （El） and forbidden （M1, E2, M2） lines for the 3s^23p, 3s3p^2, 3s^23d, 3p^3, and 3s3p3d configurations of Fe XIV. The lifetimes of all 40 levels of these low-lying configurations are also derived. The valence valence and core-valence correlation effects are accounted for in a systematic way. Breit interactions and quantum electrodynamics （QED） effects are estimated in subsequent relativistic configuration interaction （CI） calculations. The present results are in good agreement with other available theoretical and experimental values, and therefore can be used for the further astrophysical investigations.

Upward Transition Probabilities B(E2)↑ Properties Study of Even-Even ^104-114Ru Nuclei

In this research work, the upward transition probabilities for the transition levels, 0+ → 2+, 2+ → 4+, 4+ → 6+ and 6+ → 8+ levels of even-even neutron rich 104-114Ru isotopes have been calculated by using the Global Best Fit (GBF) method. In addition, the associated parameters such as, Quadrupole moment and Deformation parameter of even-even 104-114Ru have been calculated. The dependency of these nuclear parameters shows the nuclear magic number tendency.

Robust analysis of discounted Markov decision processes with uncertain transition probabilities

Optimal policies in Markov decision problems may be quite sensitive with regard to transition probabilities.In practice,some transition probabilities may be uncertain.The goals of the present study are to find the robust range for a certain optimal policy and to obtain value intervals of exact transition probabilities.Our research yields powerful contributions for Markov decision processes(MDPs)with uncertain transition probabilities.We first propose a method for estimating unknown transition probabilities based on maximum likelihood.Since the estimation may be far from accurate,and the highest expected total reward of the MDP may be sensitive to these transition probabilities,we analyze the robustness of an optimal policy and propose an approach for robust analysis.After giving the definition of a robust optimal policy with uncertain transition probabilities represented as sets of numbers,we formulate a model to obtain the optimal policy.Finally,we define the value intervals of the exact transition probabilities and construct models to determine the lower and upper bounds.Numerical examples are given to show the practicability of our methods.

Emission Intensity in the Hydrogen Atom Calculated from a Non-Probabilistic Approach to the Electron Transitions

Quantum aspects of the Joule-Lenz law for the transmission of energy allowed us to calculate the time rate of energy transitions between the quantum states of the hydrogen atom in a fully non-probabilistic way. The calculation has been extended to all transitions between p and s states having main quantum numbers not exceeding 6. An evident similarity between the intensity pattern obtained from the Joule-Lenz law and the corresponding quantum-mechanical transition pro-babilities has been shown.

Forbidden transition properties of fine-structure 2p^3 ~4S_(3/2)–2p^3 ~2D_(3/2,5/2) for nitrogen-like ions

Based on relativistic wave functions from multiconfiguration Dirac-Hartree-Fock and configuration interaction cal- culations, E2 and M1 transition probabilities of 2p3 4S3/2-2p3 2D3/2,5/2 are investigated in the nitrogen-like sequence with 7 〈 Z 〈 16. The contributions of the electron correlations, Breit interaction, and the quantum electrodynamic （QED） effects on the transition properties are analyzed. The present results can be used for diagnosing plasma. In addition, several N-like ions can also be recommended as a promising candidate for a highly charged ion （HCI） clock with a quality factor （Q） of trnnsition as hitch as 1020.

Transition parameters of Li-like ions(Z=7-11)in dense plasmas

The energy levels,transition energies,transition probabilities,weighted oscillator strengths,and line strengths of Li-like ions(Z=7-11)in dense plasmas are investigated in this work.The relativistic effects and electron correlation effects are described by the MCDHF method.The ion sphere model is applied to include the dense plasma screening effect.The ground configuration 1s^(2)2s and the excited 1s^(2)2p,1s^(2)3l(l=0-2)are considered.The configuration sets are enlarged until n=7 where the calculated energy levels have converged.The critical free electron densities of 1s^(2)3d states are estimated.Except for 1s^(2)3s-1s^(2)3p transitions,the transition energies for△n=0 increase,and for△n≠0 decrease with increasing free electron densities.For 1s^(2)3s-1s^(2)3p transitions,the spectra show blue-shift at lower free electron densities and red-shift at higher free electron densities,and the energy level crossing phenomens are observed at higher free electron densities.

Theoretical investigation on forbidden transition properties of fine-structure splitting in ~2D state for K-like ions with 26≤Z≤36

Excitation energies, magnetic dipole, and electric quadrupole transition probabilities of the 3d ^2D3/2-3d ^2D5/2 transition in the potassium-like （K-like） sequence with 26 ≤ Z ≤ 36 are investigated by using the multi-configuration Dirac-Hartree-Fock （MCDHF） method. The contributions of the electron correlations, Breit interaction, and the leading-order quantum electrodynamic （QED） effects on the transition properties are analyzed. The present results are interested in the laboratory tokamak and the astronomical observations. Furthermore, the feasibility of these ions for the highly charged ion （HCI） clocks is discussed. Considering the wavelength of lasers and manipulation process of the atomic clocks, Cu^10＋ and Zn^11＋ are recommended as promising candidates with achievable quality factors at the 10^15 level.

Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force

The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored.Unlike the existing work in detecting early warning signals,this paper presents the most probable transition paths to characterize the regime shifts.The most probable transition paths are obtained by minimizing the Freidlin-Wentzell(FW)action functional and Onsager-Machlup(OM)action functional,respectively.The most probable path shows the movement trend of the lake eutrophication system under noise excitation,and describes the global transition behavior of the system.Under the excitation of Gaussian noise,the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and the most probable transition paths.Under the excitation of Gaussian white noise and periodic force,we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.

A new formula of recovery factor for non-equilibrium transport of graded suspended sediment in the Middle Yangtze River

Suspended sediment concentrations in the Middle Yangtze River(MYR)reduced greatly after the Three Gorges Project operation,causing the composition of bed material to coarsen continuously.However,little is known about the non-equilibrium transport of graded suspended sediment owing to different bed material compositions(BMCs)along the MYR,and it is necessary to determine the magnitude of recovery factor.Using the Markov stochastic process in conjunction with the hiding-exposure effect of non-uniform bed-material,a new formula is proposed for calculating the recovery factor including the effect of different BMCs,and it is incorporated into the non-equilibrium transport equation to simulate the recovery processes of suspended load in both sand-gravel bed and sand bed reaches of the MYR.The results show that:(i)the recovery rate of graded sediment concentrations at Zhicheng was slower than that at Shashi during the period 2003-2007;(ii)the mean recovery factors of the coarse,medium,and fine sediment fractions in the ZhichengShashi reach were 0.152,0.0012,and 0.0005,respectively,and the coarse sediment recovered up to the maximum sediment concentration of 0.138 kg/m3over a distance of 15 km;and(iii)the results of the new formula that can consider the effect of bed material composition are in general agreement with the field observations,and the spatial and temporal delay effects are inversely related to particle size and BMC.Consequently,the BMC effect on the nonequilibrium sediment transport in different reaches of the MYR needs to be considered for higher simulation accuracy.

Passengers gathering-scattering analysis at the corners of subway stations

During both daily operation and emergency evacuation,the corners of walking facilities in subway stations play an important role in efficient circulation.However,the effectiveness of the corner is difficult to assess.In this paper,a method of passenger gathering and scattering analysis based on queueing models was proposed to investigate the corner performance in subway stations.Firstly,we constructed a set of state spaces of passenger flow according to passenger density and proposed the state transition model of passenger flow.Moreover,the model of passenger flow blocking and unblocking probability were also presented.Then,to illustrate the validity of the method and model,several passenger gathering-scattering scenarios and were simulated to verify the influence of passenger distribution and facility width on passenger walking,and the blocking probability,throughput,and expected time were also analyzed under various widths of the target corridor and arrival rates.Results showed that the proposed model can reproduce the trend of walking parameters changing and the self-organizing phenomenon of'faster is lower'.With the increase of arrival rates of passengers,walking speeds of passengers decrease and the expected walking time is prolonged,and the blocking probability sharply increased when the arrival rate exceeded 7 peds/s.In addition,with change of width of the target facility,efficiency of capacity of walking circulation facility fluctuated.With the width of the target corridor enlarged by 10%,the steady state of passenger flow was less crowded.Therefore,corridor width is critical to the circulation efficiency of passengers in subway stations.The conclusions will help to develop reasonable passenger flow control plans to ease the jam and keep passengers walking safely.

State and Mode Feedback Control for Discrete-time Markovian Jump Linear Systems With Controllable MTPM

In this note, the state and mode feedback control problems for a class of discrete-time Markovian jump linear systems(MJLSs) with controllable mode transition probability matrix(MTPM) are investigated. In most achievements, controller design of MJLSs pays more attention to state/output feedback control for stability, while the system cost in practice is out of consideration. In this paper, we propose a control mechanism consisting of two parts: finite-path-dependent state feedback controller design with which uniform stability of MJLSs can be ensured, and mode feedback control which aims to decrease system cost. Differing from the traditional state/output feedback controller design, the main novelty is that the proposed control mechanism not only guarantees system stability, but also decreases system cost effectively by adjusting the occurrence probability of system modes. The effectiveness of the proposed mechanism is illustrated via numerical examples.

SOME PROPERTIES OF GALTON-WATSON BRANCHING PROCESSES IN VARYING ENVIRONMENTS

This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.

Analysis on Stochastic Passage Characteristics for Gun Barrel in Shooting Domain

Aiming at some weapon systems with shooting domain,the stochastic passage characteristics of the barrel were studied.On the basis of the exact definition of the stochastic passage characteristics,its opportunity-awaiting time,residence time and stochastic passage period were given by using the transition probability matrix,and they all obeyed the geometry distributions.Their means and variances were also derived,and the relations between the time indexes and the structure and parameters of weapon control system were established.Finally,the creditability of the conclusions was verified by the test data of weapon system in proving ground.

Stark spectra of Rydberg states in atomic cesium in the vicinity of n=18

The Stark structures in a cesium atom around n=18 are numerically calculated. The results show that the components of 20D states with a small azimuthal quantum number ｜m｜ shift upward a lot, and those with a large ｜m｜ shift downward a little within 1100 V/cm. All components of P states shift downward. Experimental work has been performed in ultracold atomic cesium. Atoms initially in 6P3/2 state are excited to high-n Rydberg states by a polarization light perpendicular to the field, and Stark spectra with ｜m｜=1/2,3/2,5/2 are simultaneously observed with a large linewidth for the first time. The observed spectra are analyzed in detail. The relative transition probability is calculated. The experimental results are in good agreement with our numerical computation.