The bipartite containment control problem for heterogeneous nonlinear multi-agent systems(HNMASs)within multi-group networks under signed digraphs is investigated,where the first-order and second-order nonlinear dynam...The bipartite containment control problem for heterogeneous nonlinear multi-agent systems(HNMASs)within multi-group networks under signed digraphs is investigated,where the first-order and second-order nonlinear dynamic agents belong to distinct groups.Interactions are cooperative-antagonistic within each group and sign-in-degree balanced across the inter-groups.Firstly,a state feedback control protocol is designed to ensure that the trajectories of followers in diverse groups can converge to distinct convex hulls formed by their corresponding leaders,respectively.As an extension,the bipartite control problem with time-variant formation for the multi-agent system(MAS)is also considered,and a corresponding control protocol with formation compensation vectors is given.Finally,in view of Lyapunov stability theory and matrix inequality,the sufficient conditions for realizing the bipartite containment control are obtained,and several simulations are provided to verify the validity of the above methods.展开更多
In this article, we establish the global stability of an endemic equilibrium of multi-group SIR epidemic models, which have not only an exchange of individuals between patches through migration but also cross patch in...In this article, we establish the global stability of an endemic equilibrium of multi-group SIR epidemic models, which have not only an exchange of individuals between patches through migration but also cross patch infection between different groups. As a result, we partially generalize the recent result in the article [16].展开更多
Muon tomography is a capable imaging technique to measure the geometry of high-Z objects. However,most existed algorithms used in muon tomography have obscured the effects of angular distribution and momentum spectra ...Muon tomography is a capable imaging technique to measure the geometry of high-Z objects. However,most existed algorithms used in muon tomography have obscured the effects of angular distribution and momentum spectra of cosmic ray muons and reduced the spatial resolution. We present a modified multi-group model that takes into account these effects and calibrates the model by the material of lead. Performance tests establish that the model is capable of measuring the thickness of a Pb slab and identifying the material of an unknown slab on a reasonable exposure timescale, in both cases of complete and incomplete angular data. Results show that the modified multi-group model is helpful for improvements in image resolution in real applications.展开更多
In this paper, we study worm dynamics in computer networks composed of many autonomous systems. A novel multigroup SIQR （susceptible-infected-quarantined-removed） model is proposed for computer worms by explicitly c...In this paper, we study worm dynamics in computer networks composed of many autonomous systems. A novel multigroup SIQR （susceptible-infected-quarantined-removed） model is proposed for computer worms by explicitly considering anti-virus measures and the network infrastructure. Then, the basic reproduction number of worm R0 is derived and the global dynamics of the model are established. It is shown that if R0 is less than or equal to 1, the disease-free equilibrium is globally asymptotically stable and the worm dies out eventually, whereas, if R0 is greater than 1, one unique endemic equilibrium exists and it is globally asymptotically stable, thus the worm persists in the network. Finally, numerical simulations are given to illustrate the theoretical results.展开更多
To overcome the default of single search tendency, the ants in the colony are divided into several sub-groups. The ants in different subgroups have different trail information and expectation coefficients. The simulat...To overcome the default of single search tendency, the ants in the colony are divided into several sub-groups. The ants in different subgroups have different trail information and expectation coefficients. The simulated annealing method is introduced to the algorithm. Through setting the temperature changing with the iterations, after each turn of tours, the solution set obtained by the ants is taken as the candidate set. The update set is obtained by adding the solutions in the candidate set to the previous update set with the probability determined by the temperature. The solutions in the candidate set are used to update the trail information. In each turn of updating, the current best solution is also used to enhance the trail information on the current best route. The trail information is reset when the algorithm is in stagnation state. The computer experiments demonstrate that the proposed algorithm has higher stability and convergence speed.展开更多
The accurate modeling of depletion,intricately tied to the solution of the neutron transport equation,is crucial for the design,analysis,and licensing of nuclear reactors and their fuel cycles.This paper introduces a ...The accurate modeling of depletion,intricately tied to the solution of the neutron transport equation,is crucial for the design,analysis,and licensing of nuclear reactors and their fuel cycles.This paper introduces a novel multi-group Monte-Carlo depletion calculation approach.Multi-group cross-sections(MGXS)are derived from both 3D whole-core model and 2D fuel subassembly model using the continuous-energy Monte-Carlo method.Core calculations employ the multi-group Monte-Carlo method,accommodating both homogeneous and specific local heterogeneous geometries.The proposed method has been validated against the MET-1000 metal-fueled fast reactors,using both the OECD/NEA benchmark and a new refueling benchmark introduced in this paper.Our findings suggest that microscopic MGXS,produced via the Monte-Carlo method,are viable for fast reactor depletion analyses.Furthermore,the locally heterogeneous model with angular-dependent MGXS offers robust predictions for core reactivity,control rod value,sodium void value,Doppler constants,power distribution,and concentration levels.展开更多
In this paper,a stochastic multi-group AIDS model with saturated incidence rate is studied.We prove that the system is persistent in the mean under some parametric restrictions.We also obtain the sufficient condition ...In this paper,a stochastic multi-group AIDS model with saturated incidence rate is studied.We prove that the system is persistent in the mean under some parametric restrictions.We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function.Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system,which greatly improves upon previous results.展开更多
A multi-group cross-section library is fundamental for deterministic lattice physics calculations.Most existing multi-group cross-section libraries are customized for particular computer codes,as well as for particula...A multi-group cross-section library is fundamental for deterministic lattice physics calculations.Most existing multi-group cross-section libraries are customized for particular computer codes,as well as for particular types of nuclear reactors.This paper presents an HDF5-format multi-group cross-section library named XPZLIB.XPZLIB was produced using a selfdeveloped XPZR module integrated into the NJOY2016 code,and an in-house PyNjoy2022 system was developed for autoprocessing.XPZLIB contains detailed data content and well-organized data structures that are user-and developer-friendly.Three typical XPZLIBs with different numbers of energy groups,nuclides,and depletion reaction types were released via the Tsinghua cloud website.Furthermore,the applicability of the released XPZLIBs was investigated using HTGR and PWR lattice calculations,which can provide guidance for applying XPZLIB under different scenarios.展开更多
Computed tomography(CT) reconstruction with a well-registered priori magnetic resonance imaging(MRI) image can improve reconstruction results with low-dose CT, because well-registered CT and MRI images have similar st...Computed tomography(CT) reconstruction with a well-registered priori magnetic resonance imaging(MRI) image can improve reconstruction results with low-dose CT, because well-registered CT and MRI images have similar structures. However, in clinical settings, the CT image of patients does not always match the priori MRI image because of breathing and movement of patients during CT scanning. To improve the image quality in this case, multi-group datasets expansion is proposed in this paper. In our method, multi-group CT-MRI datasets are formed by expanding CT-MRI datasets. These expanded datasets can also be used by most existing CT-MRI algorithms and improve the reconstructed image quality when the CT image of a patient is not registered with the priori MRI image. In the experiments, we evaluate the performance of the algorithm by using multi-group CT-MRI datasets in several unregistered situations. Experiments show that when the CT and priori MRI images are not registered, the reconstruction results of using multi-group dataset expansion are better than those obtained without using the expansion.展开更多
A multi-group epidemic model with a variables separated incidence rate and delays is analyzed. For strongly and non-strongly connected networks, the basic reproductive number Ro is calculated, respectively. By applyin...A multi-group epidemic model with a variables separated incidence rate and delays is analyzed. For strongly and non-strongly connected networks, the basic reproductive number Ro is calculated, respectively. By applying the Lyapunov functionals and the LaSalle invariance principle, we prove the global asymptotic stability of infection-free equilibrium P0 when Ro 〈 1 and the endemic equilibrium P＊ when Ro 〉 1.展开更多
We establish a stochastic differential equation epidemic model of multi-group SIR type based on the deterministic multi-group SIR mode. Then, we define the basic reproduction number R0^S and show that it is a sharp th...We establish a stochastic differential equation epidemic model of multi-group SIR type based on the deterministic multi-group SIR mode. Then, we define the basic reproduction number R0^S and show that it is a sharp threshold for the dynamic of the stochastic multi-group SIR model. More specially, if R0^S 〈 1, then the disease-free equilibrium will be asymptotically stable which means the disease will die out, if R0^S 〉 1, the disease-free equilibrium will unstable, and our model will positively recurrence to a positive domain which implies the persistence of our model. Numerical simulation examples are carried out to substantiate the analytical results.展开更多
In this study, based differential equations methods are used to solve equations because these methods are dependent on boundary value data more than other mathematical equations. We have calculated neutron flux, criti...In this study, based differential equations methods are used to solve equations because these methods are dependent on boundary value data more than other mathematical equations. We have calculated neutron flux, criticality and geometrical eigenvalue by using multi-group method and solving the neutron diffusion equation for finite and infinite cylindrical and spherical reactors in this study. For the calculation of the total neutron flux cross sections, we need the neutron diffusion equation. Thus, we have established the relationship between neuron flow and cross-section of neuron depending on neutron energy. Critical calculations have been made by comparing the results with MNCP (montecarlo n-partical) simulation methods. For necessary computer calculations, the programme, Wolfram-Matematica-7 has been used.展开更多
A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-ana...A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-analytic basis functions,and a colorset based form function generating method is proposed,which can accurately model the spectral interaction occurring at assembly interface.To demonstrate its accuracy and applicability to realistic problems, the new method is tested against two benchmark problems,including a mixed-oxide fuel problem.The results show that the new method is comparable in accuracy to fine-mesh methods.展开更多
Consider the heterogeneity(e.g.,heterogeneous social behaviour,heterogeneity due to different geography,contrasting contact patterns and different numbers of sexual partners etc.)of host population,in this paper,the a...Consider the heterogeneity(e.g.,heterogeneous social behaviour,heterogeneity due to different geography,contrasting contact patterns and different numbers of sexual partners etc.)of host population,in this paper,the authors propose an infection age multigroup SEIR epidemic model.The model system also incorporates the feedback variables,where the infectivity of infected individuals may depend on the infection age.In the direction of mathematical analysis of model,the basic reproduction number R_0 has been computed.The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R_(0).More precisely,for R_(0)≤1,the disease-free equilibrium is globally asymptotically stable and for R_(0)>1,they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method.By considering a numerical example,they investigate the effects of infection age and feedback on the prevalence of the disease.Their result shows that feedback parameters have different and even opposite effects on different groups.However,by choosing an appropriate value of feedback parameters,the disease could be eradicated or maintained at endemic level.Besides,the infection age of infected individuals may also change the behaviour of the disease,global stable to damped oscillations or damped oscillations to global stable.展开更多
This paper presents a law of large numbers result,as the size of the population tends to infinity,of SIR stochastic epidemic models,for a population distributed over distinct patches(with migrations between them)and d...This paper presents a law of large numbers result,as the size of the population tends to infinity,of SIR stochastic epidemic models,for a population distributed over distinct patches(with migrations between them)and distinct groups(possibly age groups).The limit is a set of Volterra-type integral equations,and the result shows the effects of both spatial and population heterogeneity.The novelty of the model is that the infectivity of an infected individual is infection age dependent.More precisely,to each infected individual is attached a random infection-age dependent infectivity function,such that the various random functions attached to distinct individuals are i.i.d.The proof involves a novel construction of a sequence of i.i.d.processes to invoke the law of large numbers for processes in,by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation(as in the propagation of chaos theory).We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE.The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in[20],where standard tightness criteria for convergence of stochastic processes were employed.To illustrate this new approach,we first explain the new proof under the weak assumptions for the homogeneous model,and then describe the multipatch-multigroup model and prove the law of large numbers for that model.展开更多
基金National Natural Science Foundation of China(No.12071370)。
文摘The bipartite containment control problem for heterogeneous nonlinear multi-agent systems(HNMASs)within multi-group networks under signed digraphs is investigated,where the first-order and second-order nonlinear dynamic agents belong to distinct groups.Interactions are cooperative-antagonistic within each group and sign-in-degree balanced across the inter-groups.Firstly,a state feedback control protocol is designed to ensure that the trajectories of followers in diverse groups can converge to distinct convex hulls formed by their corresponding leaders,respectively.As an extension,the bipartite control problem with time-variant formation for the multi-agent system(MAS)is also considered,and a corresponding control protocol with formation compensation vectors is given.Finally,in view of Lyapunov stability theory and matrix inequality,the sufficient conditions for realizing the bipartite containment control are obtained,and several simulations are provided to verify the validity of the above methods.
基金supported by Japan Society for the Promotion of Science (Grant Scientific Research (c), No. 24540219 to the first author, JSPS Fellows, No.237213 to the second author, and No. 222176 to the third author)
文摘In this article, we establish the global stability of an endemic equilibrium of multi-group SIR epidemic models, which have not only an exchange of individuals between patches through migration but also cross patch infection between different groups. As a result, we partially generalize the recent result in the article [16].
基金supported by the Science and Technology Development Foundation of CAEP(No.2015B0103014)the National Natural Science Foundation of China(No.11605163)
文摘Muon tomography is a capable imaging technique to measure the geometry of high-Z objects. However,most existed algorithms used in muon tomography have obscured the effects of angular distribution and momentum spectra of cosmic ray muons and reduced the spatial resolution. We present a modified multi-group model that takes into account these effects and calibrates the model by the material of lead. Performance tests establish that the model is capable of measuring the thickness of a Pb slab and identifying the material of an unknown slab on a reasonable exposure timescale, in both cases of complete and incomplete angular data. Results show that the modified multi-group model is helpful for improvements in image resolution in real applications.
基金Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK2010526)the Six Projects Sponsoring Talent Summits of Jiangsu Province, China (Grant No. SJ209006)+2 种基金the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20103223110003)the Ministry of Education Research in the Humanities and Social Sciences Planning Fund of China (Grant No. 12YJAZH120)the Graduate Student Innovation Research Project of Jiangsu Province, China (Grant Nos. CXLX11 0417 and CXLX11 0404)
文摘In this paper, we study worm dynamics in computer networks composed of many autonomous systems. A novel multigroup SIQR （susceptible-infected-quarantined-removed） model is proposed for computer worms by explicitly considering anti-virus measures and the network infrastructure. Then, the basic reproduction number of worm R0 is derived and the global dynamics of the model are established. It is shown that if R0 is less than or equal to 1, the disease-free equilibrium is globally asymptotically stable and the worm dies out eventually, whereas, if R0 is greater than 1, one unique endemic equilibrium exists and it is globally asymptotically stable, thus the worm persists in the network. Finally, numerical simulations are given to illustrate the theoretical results.
基金Project supported by the National Natural Science Foundation of China (Grant No.50608069)
文摘To overcome the default of single search tendency, the ants in the colony are divided into several sub-groups. The ants in different subgroups have different trail information and expectation coefficients. The simulated annealing method is introduced to the algorithm. Through setting the temperature changing with the iterations, after each turn of tours, the solution set obtained by the ants is taken as the candidate set. The update set is obtained by adding the solutions in the candidate set to the previous update set with the probability determined by the temperature. The solutions in the candidate set are used to update the trail information. In each turn of updating, the current best solution is also used to enhance the trail information on the current best route. The trail information is reset when the algorithm is in stagnation state. The computer experiments demonstrate that the proposed algorithm has higher stability and convergence speed.
基金supported by the National Natural Science Foundation of China(Nos.12105170,12135008)Science and Technology on Reactor System Design Technology Laboratory.
文摘The accurate modeling of depletion,intricately tied to the solution of the neutron transport equation,is crucial for the design,analysis,and licensing of nuclear reactors and their fuel cycles.This paper introduces a novel multi-group Monte-Carlo depletion calculation approach.Multi-group cross-sections(MGXS)are derived from both 3D whole-core model and 2D fuel subassembly model using the continuous-energy Monte-Carlo method.Core calculations employ the multi-group Monte-Carlo method,accommodating both homogeneous and specific local heterogeneous geometries.The proposed method has been validated against the MET-1000 metal-fueled fast reactors,using both the OECD/NEA benchmark and a new refueling benchmark introduced in this paper.Our findings suggest that microscopic MGXS,produced via the Monte-Carlo method,are viable for fast reactor depletion analyses.Furthermore,the locally heterogeneous model with angular-dependent MGXS offers robust predictions for core reactivity,control rod value,sodium void value,Doppler constants,power distribution,and concentration levels.
基金The work was supported by NSF of China(11801041,11871473)Foudation of Jilin Province Science and Technology Development(20190201130JC)+1 种基金Scientific Rsearch Foundation of Jilin Provincial Education Department(JJKH20181172KJ,JJKH20190503KJ)Natural Science Foundation of Changchun Normal University.
文摘In this paper,a stochastic multi-group AIDS model with saturated incidence rate is studied.We prove that the system is persistent in the mean under some parametric restrictions.We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function.Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system,which greatly improves upon previous results.
基金supported by the National Key R&D Program of China(2020YFE0202500).
文摘A multi-group cross-section library is fundamental for deterministic lattice physics calculations.Most existing multi-group cross-section libraries are customized for particular computer codes,as well as for particular types of nuclear reactors.This paper presents an HDF5-format multi-group cross-section library named XPZLIB.XPZLIB was produced using a selfdeveloped XPZR module integrated into the NJOY2016 code,and an in-house PyNjoy2022 system was developed for autoprocessing.XPZLIB contains detailed data content and well-organized data structures that are user-and developer-friendly.Three typical XPZLIBs with different numbers of energy groups,nuclides,and depletion reaction types were released via the Tsinghua cloud website.Furthermore,the applicability of the released XPZLIBs was investigated using HTGR and PWR lattice calculations,which can provide guidance for applying XPZLIB under different scenarios.
基金the National Natural Science Foundation of China(No.813716234)the National Basic Research Program(973)of China(No.2010CB834302)Shanghai Jiao Tong University Medical Engineering Cross Research Funds(Nos.YG2013MS30 and YG2014ZD05)
文摘Computed tomography(CT) reconstruction with a well-registered priori magnetic resonance imaging(MRI) image can improve reconstruction results with low-dose CT, because well-registered CT and MRI images have similar structures. However, in clinical settings, the CT image of patients does not always match the priori MRI image because of breathing and movement of patients during CT scanning. To improve the image quality in this case, multi-group datasets expansion is proposed in this paper. In our method, multi-group CT-MRI datasets are formed by expanding CT-MRI datasets. These expanded datasets can also be used by most existing CT-MRI algorithms and improve the reconstructed image quality when the CT image of a patient is not registered with the priori MRI image. In the experiments, we evaluate the performance of the algorithm by using multi-group CT-MRI datasets in several unregistered situations. Experiments show that when the CT and priori MRI images are not registered, the reconstruction results of using multi-group dataset expansion are better than those obtained without using the expansion.
基金Supported by Weihai Science and Technology Development Plan Project(Grant No.2013DXGJ06)the Natural Science Foundation of Shandong Province(Grant No.ZR2015AM018)
文摘A multi-group epidemic model with a variables separated incidence rate and delays is analyzed. For strongly and non-strongly connected networks, the basic reproductive number Ro is calculated, respectively. By applying the Lyapunov functionals and the LaSalle invariance principle, we prove the global asymptotic stability of infection-free equilibrium P0 when Ro 〈 1 and the endemic equilibrium P＊ when Ro 〉 1.
基金Acknowledgments This work was supported by the National Natural Science Foundation of China Grant 61273126, and the Natural Science Foundation of Guangdong Province Under Grants 10251064101000008 and S201210009675, the Fundamental Research Funds for the Central Universities 2012ZM0059, and Research Fund for the Doctoral Program of Higher Education of China under grant 20130172110027.
文摘We establish a stochastic differential equation epidemic model of multi-group SIR type based on the deterministic multi-group SIR mode. Then, we define the basic reproduction number R0^S and show that it is a sharp threshold for the dynamic of the stochastic multi-group SIR model. More specially, if R0^S 〈 1, then the disease-free equilibrium will be asymptotically stable which means the disease will die out, if R0^S 〉 1, the disease-free equilibrium will unstable, and our model will positively recurrence to a positive domain which implies the persistence of our model. Numerical simulation examples are carried out to substantiate the analytical results.
文摘In this study, based differential equations methods are used to solve equations because these methods are dependent on boundary value data more than other mathematical equations. We have calculated neutron flux, criticality and geometrical eigenvalue by using multi-group method and solving the neutron diffusion equation for finite and infinite cylindrical and spherical reactors in this study. For the calculation of the total neutron flux cross sections, we need the neutron diffusion equation. Thus, we have established the relationship between neuron flow and cross-section of neuron depending on neutron energy. Critical calculations have been made by comparing the results with MNCP (montecarlo n-partical) simulation methods. For necessary computer calculations, the programme, Wolfram-Matematica-7 has been used.
基金Partially supported by the National Natural Science Foundation of China via research project 10605016
文摘A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-analytic basis functions,and a colorset based form function generating method is proposed,which can accurately model the spectral interaction occurring at assembly interface.To demonstrate its accuracy and applicability to realistic problems, the new method is tested against two benchmark problems,including a mixed-oxide fuel problem.The results show that the new method is comparable in accuracy to fine-mesh methods.
基金supported by the National Natural Science Foundation of China(No.12022113)Henry Fok Foundation for Young Teachers,China(No.171002)+2 种基金Outstanding Young Talents Support Plan of Shanxi Province,Science and Engineering Research Board(SERB for short),India(No.ECR/2017/002786)UGC-BSR Research Start-Up-Grant,India(No.F.30-356/2017(BSR))Senior Research Fellowship from the Council of Scientific and Industrial Research(CSIR for short),India(No.09/1131(0006)/2017-EMR-I)。
文摘Consider the heterogeneity(e.g.,heterogeneous social behaviour,heterogeneity due to different geography,contrasting contact patterns and different numbers of sexual partners etc.)of host population,in this paper,the authors propose an infection age multigroup SEIR epidemic model.The model system also incorporates the feedback variables,where the infectivity of infected individuals may depend on the infection age.In the direction of mathematical analysis of model,the basic reproduction number R_0 has been computed.The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R_(0).More precisely,for R_(0)≤1,the disease-free equilibrium is globally asymptotically stable and for R_(0)>1,they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method.By considering a numerical example,they investigate the effects of infection age and feedback on the prevalence of the disease.Their result shows that feedback parameters have different and even opposite effects on different groups.However,by choosing an appropriate value of feedback parameters,the disease could be eradicated or maintained at endemic level.Besides,the infection age of infected individuals may also change the behaviour of the disease,global stable to damped oscillations or damped oscillations to global stable.
文摘This paper presents a law of large numbers result,as the size of the population tends to infinity,of SIR stochastic epidemic models,for a population distributed over distinct patches(with migrations between them)and distinct groups(possibly age groups).The limit is a set of Volterra-type integral equations,and the result shows the effects of both spatial and population heterogeneity.The novelty of the model is that the infectivity of an infected individual is infection age dependent.More precisely,to each infected individual is attached a random infection-age dependent infectivity function,such that the various random functions attached to distinct individuals are i.i.d.The proof involves a novel construction of a sequence of i.i.d.processes to invoke the law of large numbers for processes in,by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation(as in the propagation of chaos theory).We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE.The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in[20],where standard tightness criteria for convergence of stochastic processes were employed.To illustrate this new approach,we first explain the new proof under the weak assumptions for the homogeneous model,and then describe the multipatch-multigroup model and prove the law of large numbers for that model.
