This paper first presents a new method to improve the pseudorandomness of pseudorandom sequences in the meaning defined by Golomb's assumptions. Next, using 4 chaotic maps constructs a new 8-dimensional generalized synchronization system (8DCGSS). Third using the 8DCGSS introduces a chaotic pseudorandom number generator (chaotic PRNG) with keyspace larger than 2(1117). Then, using FIPS 140-2 test criteria and generalized FIPS 140-2 test criteria tests the pseudorandomness of the keystreams with length 20 000 bits, 100 000 bits and 1 000 000 bits generated by the chaotic PRNG, an Matlab PRNG, an RC4 PRNG, and an m-sequence PRNG with period 2(20) - 1. The corresponding improved keystreams by our method are also tested. The results show that the presented method significantly increases the pseudorandomness of keystreams generated by the 4 PRNGs. However, the keystreams generated by the m-sequence PRNG do not exhibit strong pseudorandomness when their lengths are less than 100 000.

Background In December 2019, a novel coronavirus-induced pneumonia (COVID-19) broke out in Wuhan, China. On 19 January and on 8 June 2020, there were two wave COVID-19 epidemics happened in Beijing. Modelling, simulations and analysis for the two wave epidemics are important issues. Methods This study introduces a symptomatic-asymptomatic-recoverer-death differential equation model (SARDDE). It presents the conditions of the asymptotical stability on the disease-free equilibrium of the SARDDE. It proposes the necessary conditions of disease spreading for the SARDDE. Based on the reported data of the first and the second COVID-19 epidemics in Beijing and numerical simulations, it determines the parameters of the SARDDE, respectively. Results Numerical simulations of the SARDDE describe well the outcomes of the current symptomatic and asymptomatic individuals, the recovered symptomatic and asymptomatic individuals, and the died infected individuals, respectively. The numerical simulations obtain the following results. Conclusions Virtual simulations suggest that the strict prevention and control strategies implemented by Beijing government are effective and necessary; using the data from the beginning to the days after about two weeks after the turning points can estimate well and approximately the following outcomes of the two COVID-19 epidemics, respectively. To avoid multiple epidemic outbreaks, a recommendation is that the authorities need to have maintained the prevention and control measures implemented, at least, 7 days after reaching the turning point until new current infection cases disappear. It is expected that the research can provide better understanding, explaining, and dominating for epidemic spreads, prevention and control measures. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement No funding supported in the work. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: No IRB/oversight body that provided approval or exemption for the research. I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable. Yes All data cite in this article selected from the web ref. \[18\] (

Background: After successfully preventing the spread of five wave COVID-19 epidemics in Shanghai, Omicron and Delta variants have been causing a surge COVID-19 infection in this city recently. Summaries, analysis and simulations for this wave epidemic are important issues. Methods: Using differential equations and real word data, this study modelings and simulates the recent COVID-19 epidemic in Shanghai, estimates transmission rates, recovery rates, and blocking rates to symptomatic and asymptomatic infections, and symptomatic (infected) individuals' death rates. Visual simulations predict the outcomes of this wave Shanghai epidemic. It compares parallely with the recent mainland China COVID-19 epidemics (RMCE). Results: The simulation results were in good agreement with the real word data at the end points of 11 investigated time-intervals. Visual simulation results showed that on the day 90, the number of the current symptomatic (infected) individuals may be between 852 and 7314, the number of the current asymptomatic (infected) individuals charged in the observations may be between 10066 and 50292, the number of the current cumulative recovered symptomatic infected individuals may be between 52070 and 74687, the number of the current cumulative asymptomatic individuals discharged from the medical observations may be between 63509 and 5164535. The number of the died symptomatic individuals may be between 801 and 1226. (a) The transmission rate of the symptomatic infections caused by the symptomatic individuals was much lower than the corresponding average transmission rate of the RMCE. (b) The transmission rate of the asymptomatic infections caused by the symptomatic individuals was much higher than the first 90 day's average transmission rate of RMCE. (c) The transmission rate of the symptomatic infections caused by the asymptomatic individuals was much lower than the first 60 day's average transmission rate of RMCE, and was much higher than the last 60 day's average transmission rate of RMCE. (d) The transmission rate to the asymptomatic infections caused by the asymptomatic individuals was much higher than the corresponding average transmission rate of RMCE. (e) The last30 days' average blocking rate to the symptomatic infections were lower than the last 30 days' average blocking rates of RMCE (f) The last 30 days' average blocking rate to the asymptomatic infections were much higher than the last 30 days' average blocking rate of RMCE. However the first 30 days' average blocking rate to the asymptomatic infections were much lower than the first 30 days' average blocking rate of RMCE. (g) The first 37 days' recovery rates of the symptomatic individuals were much lower than the corresponding first 70 days' recovery rates of the symptomatic individuals of RMCE. The recovery rates between 38- and 52-days of the symptomatic individuals were much lower than the corresponding the recovery rates between 91-and 115-days of the symptomatic individuals of RMCE. The last week's recovery rate was similar to the last week's recovery rate of RMCE. (h) The first 30 days' average recovery rate recovery rate to the symptomatic individuals were much lower than the first 30 days' average recovery rate recovery rate of RMCE . The last 30 days' average recovery rate recovery rate of the symptomatic individuals were still much lower than the last 30 days' average recovery rate of RMCE. Conclusions: The last 30 days' low blocking rates to the symptomatic infections, the first 30 day's low blocking rates to the symptomatic infections to asymptomatic infections, the low recovery rates of the symptomatic and asymptomatic individuals, and the high transmission rate of the asymptomatic infections may be the reasons to cause the rapid spread of the recent Shanghai epidemic. It needs to implement more strict prevention and control strategies, rise the recovery rates of symptomatic and asymptomatic infections, and reduce the death rates for preventing the spread of this wave COVID-19 epidemic in Shanghai.

Abstract： Pseudorandom number generators (PRNGs) are widely used in many fields, especially in cryptographic applications. Pseudorandom number sequences generated via a poor PRNG will lead to weak or guessable its keys, and leak the information which is prevented. Based on the Golomb's assumptions on idea pseudorandom sequences and FIPS 140-2 randomness test criteria, this paper first presents a new approach for improving the randomness of pseudorandom sequences. Second, using a generalized synchronization theorem, the Henon map, the logistic map and a tube map constructs a new 8-dimensional chaotic generalized synchronization system (8DCGSS). Then using the 8DCGSS designs a chaotic PRNG (CPRN). The keyspace of the CPRNG is larger than 21117. Finally, using the FIPS 140-2 randomness test and a generalized FIPS 140-2 randomness test measures the randomness of the keystreams generated via the CPRNG, the Matlab PRNG, the RC4 algorithm, and the m-sequence, and the improved keystreams in term of our approach. The results show that our approach are able to increase significantly the randomness of the keystreams generated by the four PRNGs.

Background The recent COVID-19 epidemic in mainland China is an important issue for studying the prevention and disease control measures and the spread of the COVID-19 epidemic. Following our previous study for the mainland China epidemic during December 31 2021 to 30 April 2022, this paper studies and compares the the mainland China epidemic during December 31 2021 to December 6, 2022. Methods Using differential equations and real word data (both domestic and foreign input infected individuals) modelings and simulates COVID-19 epidemic in mainland China during May 1 2022 to December 6 2022, estimates the transmission rates, the recovery rates, the blocking rates to the symptomatic and the asymptomatic infections, and the died rate of the symptomatic infected individuals. The transmission rates and the recovery rates of the foreign input COVID-19 infected individuals in mainland China have also been studied. Using virtual simulations predict the outcomes of the epidemics. Results The simulation results were in good agreement with the real word data. For the mainland epidemic simulations: For the foreign input epidemic simulations: Conclusions ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement This study did not receive any funding ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: . I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes .

BackgroundGlobally COVID-19 epidemics have caused tremendous disasters. China prevented effectively the spread of COVID-19 epidemics before 2022. Recently Omicron and Delta variants cause a surge in reported COVID-19 infections. MethodsUsing differential equations and real word data, this study modelings and simulates COVID-19 epidemic in mainland China, estimates transmission rates, recovery rates, and blocking rates to symptomatic and asymptomatic infections. The transmission rates and recovery rates of the foreign input COVID-19 infected individuals in mainland China have also been studied. ResultsThe simulation results were in good agreement with the real word data. The recovery rates of the foreign input symptomatic and asymptomatic infected individuals are higher than those of the mainland COVID-19 infected individuals. The blocking rates to symptomatic and asymptomatic mainland infections are lower than those of previous epidemics. The blocking rate implemented between March 24-31, 2022 may not prevent the rapid spreads of COVID-19 epidemics in mainland China. For the foreign input COVID-19 epidemics, the numbers of the current symptomatic individuals and the asymptomatic individuals charged in medical observations have decreased significantly after March 17 2022. ConclusionsNeed to implement more strict prevention and control strategies to prevent the spread of COVID-19 mainland infected individuals. Need to keep the present prevention and therapy measures to foreign input COVID-19 infected infections until infected individuals to be cleared.

Shanghai is the best city to prevent the spread of COVID-19 infection in China. Since February 20, 2020, Shanghai has experienced five waves of COVID-19. Out of a total of 388 patients with COVID-19 symptoms, 381 were cured and seven died. Medical staff achieved zero infection. This paper summarizes, analyzes and simulates COVID-19 epidemics in Shanghai. The simulation results show that for five waves of epidemics, after reaching the infection turning point, the blocking rate of symptomatic infection is over 99%. The administration needs to maintain the prevention and control implemented 7 days after reaching the infection turning point until the new infection goes away.

To date, over 178 million people on infected with COVID-19. It causes more 3.8 millions deaths. Based on a previous symptomatic-asymptomatic-recoverer-dead differential equation model (SARDDE) and the clinic data of the first COVID-19 epidemic in Shanghai, this paper determines the parameters of SARDDE. Numerical simulations of SARDDE describe well the outcomes of current symptomatic individuals, recovered symptomatic individuals, and died individuals, respectively. The numerical simulations suggest that both symptomatic and asymptomatic individuals cause lesser asymptomatic spread than symptomatic spread; blocking rate of about 95.5% cannot prevent the spread of the COVID19 epidemic in Shanghai. The strict prevention and control strategies implemented by Shanghai government is not only very effective but also completely necessary. The numerical simulations suggest also that using the data from the beginning to the day after about 19 days at the turning point can estimate well the following outcomes of the COVID-19 academic. It is expected that the research can provide better understanding, explaining, and dominating for epidemic spreads, prevention and control measures.

Based on the robust chaos theorem of S-unimodal maps, this paper studies a kind of cubic polynomial discrete maps (CPDMs) and sets up a novel theorem. This theorem gives general conditions for the occurrence of robust chaos in the CPDMs. By using the theorem, we construct a CPDM. The parameter regions of chaotic robustness of the CPDM are larger than these of Logistic map. By using a fixed point arithmetic, we investigate the cycle lengths of the CPDM and a Logistic map. The results show that the maximum cycle lengths of 1000 chaotic sequences with length 3×107 generated by different initial value conditions exponentially increase with the resolutions. When the resolutions reach 10-7~10-13, the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches 10-14, there is the situation without cycle for 1000 cubic polynomial chaotic sequences with length 3×107. By using the CPDM and Logistic map, we design four chaos-based pseudorandom number generators (CPRNGs): CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV. The randomness of two 1000 key streams consisting of 20000 bits is tested, respectively, generated by the four CPRNGs. The result suggests that CPRNGIII based on the cubic polynomial chaotic generalized synchronic system has better performance.

Recently chaos-based encryption has been obtained more and more attention. Chaotic systems without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs) because there does not exist corresponding chaos criterion theorem on such systems. This paper proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions introduces a chaotic system without equilibria. Using this system and the generalized chaos synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization (8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results show that there are no significant correlations between the key stream and the perturbed key streams generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image. The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar to those generated via ideal chaotic PRNGs.

Liver damage caused by hepatitis B virus (HBV) infections is diffuse. Live medical images and FibroScan images show that it is inhomogeneous. Modeling and interpreting such phenomena are both important for theoretical research and practical application. The cellular neural networks (CNNs) introduced by Chua and Yang can model widely observed patterns in both nonbiological and biological media. The sawtooth and rectangle (SR) CNN introduced by Chua and colleagues is able to generate SR-shaped patterns for any random-input grayscale patterns, which are also similar to some chronic HBV-infected patients' liver B-scan images. This paper will mathematically study the dynamic behaviors of the SR CNN, numerically simulate the output images of the SR CNN, and biologically interpret the relationships between the output images of the SR CNN and the patterns in HBV-infected patients' liver B-scan images. Our research results show that the SR CNN may be used as a candidate for modeling liver infections.

This paper introduces nine four-dimensional discrete chaotic systems with one-line equilibria (DCSLE), consisting of some simple sine functions. Based on the generalized chaos synchronization (GCS) theorem, a DCSLE is used to construct an eight-dimensional DCSLE GCS system. The new DCSLE GCS system is verified by numerical simulation and then used to design a chaotic pseudorandom number generator (CPRNG). The randomness of ten 100-key streams generated by the CPRNG, two GCS-based CPRNGs, the RC4 PRNG and the ZUC PRNG are tested by the SP800-22/FIPS 140-2 tests. The test results confirm that the randomness performances of the three CPRNGs are promising, for there are no significant correlations between a keystream and any perturbed keystream generated by such CPRNG. Also, the key space of the CPRNG is larger than 2(1170). Finally, the CPRNG is used with an avalanche-effect encryption scheme to encrypt an RGB image, demonstrating that the CPRNG is able to generate the avalanche effects which are similar to those generated by ideal CPRNGs.

Hepatitis B is one of serious infectious diseases in the wold. Most chronical Hepatitis B patients need lifelong time's therapy. Evaluating the efficacy of anti-hepatitis virus infection treatment is very important. This paper uses the multiple linear regression with normalization approach to predict the efficacy of the traditional Chinese medicine and Adefovior anti-hepatitis B virus infection therapy for poor response and complete response patients. Results showed that based on the patients’ clinic HBV DNA, ALT and HBeAg data collected at the week 24, the multiple linear regression models with normalization are able to predict the poor response and the complete response patients’ outcome half years earlier in more than 90% accuracy

Epidemic Spreading is a major global health problem. Modeling epidemic spreading dynamics is important for understanding and controlling epidemic spreading, providing prevention strategies. This paper points out some flaws existing in the susceptible-infected - susceptible (SIS) model proposed by Safan and Rihan, and proposes a modified susceptible-infected-recovered (SIR) model on homogenous networks. It is proved that if the basic reproduction number Rv of the model is less than one, then the infection-free equilibrium of the model is globally asymptotically stable. On the other hand, if Rv of the model is more than one, the endemic equilibrium of the model is globally asymptotically stable. This paper also numerically predicts the effect of vaccination ratio on the size of HBV infected mainland Chinese population.

泛函分析是在20世纪初期从数学分析中独立并发展起来的新兴学科,是整个分析数学中最年轻的学科之一 [1-4].它是研究无限维线性空间上的泛函和算子理论的一门数学,被誉为20世纪的微积分.但因其高度的抽象性,使得在教学过程中常常出现"学生难学、教师难教"的情形,再加上全英文教学使得该课程的讲授成为一项具有挑战性的工作.经过近年教学实践,对教学理念的更新做出如下几个方面的探讨.

OBJECTIVE:To observe the clinical efficacy and safety of Shuanghu Qinggan Granule ( , SQG) plus Yigan Yiqi Jieyu Granule (, YYJG) combined with lamivudine (LAM) on chronic hepatitis B (CHB) patients.METHODS:The study was a multicenter, randomized, double-blinded and parallel controlled trial. A total of 320 patients were randomly allocated into 2 groups equally: 160 patients (treatment group) were given SQG and YYJG combined with LAM; and 160 patients (control group) were given LAM plus Chinese herb placebo, respectively. Liver functions, hepatitis B envelop antigen (HBeAg) titer levels, and hepatitis B virus DNA (HBV-DNA) load were monitored.RESULTS:(1) In the 48th week, the treatment group showed superior HBeAg seroconversion rate than that in the control group (38.0% vs. 24.0%, P<0.05). (2) In the 48th week, the treatment group demonstrated lower HBeAg titer than that in the control group (P<0.05). (3) In the 12th, 24th, 48th week, there was no statistical significance in HBV-DNA response rate between the two groups. (4) In the 12th week, the level of glutamyl transpeptidase (GGT) was significantly decreased in the treatment group compared with the control group (P<0.05); in the 36th week, the levels of alanine aminotransferase and aspartate transaminase were significantly lower in the treatment group than those in the control group (P<0.05).CONCLUSION:The protocol of SQG and YYJG combined with LAM to treat CHB showed superior efficacy than LAM monotherapy.