Let and denote respectively the functionswhere λ≥1, The author discusses the similarity transformation of the regularizing functionals of these functions and the similar property of their Fourier transformation.
Based on data obtained during the Hualhe River Basin Experiment （HUBEX） in 1999, this study intends to detect the quantitative discrepancies in the momentum （τ0）, sensible heat （H0） and latent heat （E0） fluxe...Based on data obtained during the Hualhe River Basin Experiment （HUBEX） in 1999, this study intends to detect the quantitative discrepancies in the momentum （τ0）, sensible heat （H0） and latent heat （E0） fluxes among six sets of similarity functions with the aerodynamic method. It also aims to clarify the applicability of the functions under stable conditions. The relative discrepancy was studied with the normalized transfer coefficients for τ0, H0 and E0, namely CD, CH and CQ, respectively. Except for one set of functions that adopted a rather small von Kármán＇s constant （0.365）, the relative discrepancy in τ0 among the other functions was less than 10%, while that in H0（E0） sometimes reached 25% when the bulk Richardson number （R/B） was less than 0.07. The absolute discrepancy in the fluxes was studied with statistical computations. Among the six sets of functions, the discrepancy in τ0, H0 and E0 sometimes reached 0.03 kg m^-1 s^-2, 4 W m^-2 and 10 W m^-2, respectively, and the discrepancy in the energy balance ratio sometimes exceeded 0.1. Furthermore, when RiB exceeded the critical value （Ric） for a specific set of functions, no fluxes could be derived with the functions. It is therefore suggested that RiB be compared with Ric before computing the fluxes if RiB is less than Ric. Finally, two sets of nonlinear similarity functions are recommended, due to their unlimited applicability in terms of RiB.展开更多
Modified refractivity (M) profile is an important parameter to describe the atmospheric refraction environment,as well as a key factor in uniquely evaluating electromagnetic propagation effects.In order to improve the...Modified refractivity (M) profile is an important parameter to describe the atmospheric refraction environment,as well as a key factor in uniquely evaluating electromagnetic propagation effects.In order to improve the model-derived M profile in stable (especially very stable) conditions,three nonlinear similarity functions,namely BH91,CB05,SHEBA07,are introduced in this paper to improve the original Babin_V25 model,and the performances of these modified models are verified based on the hydrometeorological observations from tower platforms,which are finally compared with the original Babin_V25 model and Local_HYQ92 model.Results show that introducing nonlinear similarity functions can significantly improve the model-derived M profile;especially,the newly developed SHEBA07 functions manage to reduce the predicted root mean square (rms) differences of M and M slope (for both 0-5m and 5-40m) by 64.5%,16.6%,and 60.4%,respectively in stable conditions.Unfortunately,this improved method reacts little on the evaporation duct height;in contrast,Local_HYQ92 model is capable of reducing the predicted rms differences of M,M slope (for both 0-5m and 5-40m),and evaporation duct height by 76.7%,40.2%,83.7%,and 58.0% respectively.Finally,a new recommendation is made to apply Local_HYQ92 and Babin_SHEBA07 in very stable conditions considering that M slope is more important than evaporation duct height and absolute M value in uniquely determining electromagnetic propagation effects.展开更多
Fifty years ago, Hans A. Panofsky published a paper entitled Determination of stress from wind and temperature measurements. In his famous paper, he presented a new profile function for the mean horizontal wind speed ...Fifty years ago, Hans A. Panofsky published a paper entitled Determination of stress from wind and temperature measurements. In his famous paper, he presented a new profile function for the mean horizontal wind speed under the condition of diabatic stratification that includes his integral similarity function. With his integral similarity function, he opened the door for Monin-Obukhov scaling in a wide range of micrometeorological and microclimatological applications. In a historic survey ranging from the sixties of the past century down to the present days, we present integral similarity functions for momentum, sensible heat, and water vapor for both unstable and stable stratification, where on the one hand free convection condition and on the other hand strongly stable stratification are addressed.展开更多
Identifying associations between microRNAs(miRNAs)and diseases is very important to understand the occurrence and development of human diseases.However,these existing methods suffer from the following limitation:first...Identifying associations between microRNAs(miRNAs)and diseases is very important to understand the occurrence and development of human diseases.However,these existing methods suffer from the following limitation:first,some disease-related miRNAs are obtained from the miRNA functional similarity networks consisting of heterogeneous data sources,i.e.,disease similarity,protein interaction network,gene expression.Second,little approaches infer disease-related miRNAs depending on the network topological features without the functional similarity of miRNAs.In this paper,we develop a novel model of Integrating Network Topology Similarity and MicroRNA Function Similarity(INTS-MFS).The integrated miRNA similarities are calculated based on miRNA functional similarity and network topological characteristics.INTS-MFS obtained AUC of 0.872 based on five-fold cross-validation and was applied to three common human diseases in case studies.As a results,30 out of top 30 predicted Prostatic Neoplasm-related miRNAs were included in the two databases of dbDEMC and PhenomiR2.0.29 out of top 30 predicted Lung Neoplasm-related miRNAs and Breast Neoplasm-related miRNAs were included in dbDEMC,PhenomiR2.0 and experimental reports.Moreover,INTS-MFS found unknown association with hsa-mir-371a in breast cancer and lung cancer,which have not been reported.It provides biologists new clues for diagnosing breast and lung cancer.展开更多
Let X= (Ω, ■, ■_t, X_t,, θ_t, p~x) be a self-similar Markov process on (0,∞) with non-decreasing path. The exact Hausdorff and Packing measure functions of the image X([0,t] ) are obtained.
In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same p...In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some weU-known examples, such as the Sierpinski gasket, the unit interval, the level 3 Sierpinski gasket, the hexagasket, the 3-dimensional Sierpinski gasket, and the Vicsek set are also considered.展开更多
In this paper we establish the oscillation inequality of harmonic functions and HOlder estimate of the functions in the domain of the Laplacian on connected post critically finite （p.c.f.） self-similar sets.
We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph model...We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.展开更多
Functional brain network (FBN) measures based on functional magnetic resonance imaging (fMRI) data, has become important biomarkers for early diagnosis and prediction of clinical outcomes in neurological diseases, suc...Functional brain network (FBN) measures based on functional magnetic resonance imaging (fMRI) data, has become important biomarkers for early diagnosis and prediction of clinical outcomes in neurological diseases, such as Alzheimer’s diseases (AD) and its prodromal state (<em>i</em>.<em>e</em>., Mild cognitive impairment, MCI). In the past decades, researchers have developed numbers of approaches for FBN estimation, including Pearson’s correction (PC), sparse representation (SR), and so on. Despite their popularity and wide applications in current studies, most of the approaches for FBN estimation only consider the dependency between the measured blood oxygen level dependent (BOLD) time series, but ignore the spatial relationships between pairs of brain regions. In practice, the strength of functional connection between brain regions will decrease as their distance increases. Inspired by this, we proposed a new approach for FBN estimation based on the assumption that the closer brain regions tend to share stronger relationships or similarities. To verify the effectiveness of the proposed method, we conduct experiments on a public dataset to identify the patients with MCIs from health controls (HCs) using the estimated FBNs. Experimental results demonstrate that the proposed approach yields statistically significant improvement in seven performance metrics over using the baseline methods.展开更多
We show that the processes described by Avrami functions are self-similar. A comparative function characterizes a self-similar process by a certain Avrami exponent. We define the self-similar categories of some well-k...We show that the processes described by Avrami functions are self-similar. A comparative function characterizes a self-similar process by a certain Avrami exponent. We define the self-similar categories of some well-known biological processes. The method to determine the Avrami exponent by choosing the comparative function is demonstrated on the diffusion model of the growth of nuclei. We generalize the results.展开更多
Fuzzy similarity measures, which are used to judge the closeness of two fuzzy sets, are presented to evaluate the water quality of the Haihe River. Based on the membership functions and coefficient of variation as the...Fuzzy similarity measures, which are used to judge the closeness of two fuzzy sets, are presented to evaluate the water quality of the Haihe River. Based on the membership functions and coefficient of variation as the weights, four fuzzy similarity measures （including Lattice similarity measure, Hamming similarity measure, Euclidean similarity measure and the max-min similarity measure） are used to classify the 299 samples into the proper water quality standard ranks. The results are compared with the traditional distance discriminant methods. The calculation of two traditional distance discriminant methods （both Euclidean distance and absolute value distance） is also based on the use of coefficients of variation as the weights. Without the Lattice similarity measure, for this method loses some information, the correct assignment of samples classified into the same water quality ranks is 75.92% with the other three similarity measures and two distance discriminant methods. This result shows the reliability of the five methods. Only considering the three similarity measures, there were only 1.01% of the samples that did not classify to the same ranks, while the corresponding ratio of the two distance discriminant methods was 5.69%. The results of leave-one-out cross validation show that more than 88% of the samples are classified to the proper ranks, which demonstrates that the similarity measures are suitable to evaluate the water quality of the Haihe River.展开更多
Animal models are increasingly gaining values by cross-comparisons of response or resistance to clinical agents used for patients.However,many disease mechanisms and drug effects generated from animal models are not t...Animal models are increasingly gaining values by cross-comparisons of response or resistance to clinical agents used for patients.However,many disease mechanisms and drug effects generated from animal models are not transferable to human.To address these issues,we developed SysFinder（http：//lifecenter.sgst.cn/SysFinder）,a platform for scientists to find appropriate animal models for translational research.SysFinder offers a ＂topic-centered＂ approach for systematic comparisons of human genes,whose functions are involved in a specific scientific topic,to the corresponding homologous genes of animal models.Scientific topic can be a certain disease,drug,gene function or biological pathway.SysFinder calculates multi-level similarity indexes to evaluate the similarities between human and animal models in specified scientific topics.Meanwhile,SysFinder offers species-specific information to investigate the differences in molecular mechanisms between humans and animal models.Furthermore,SysFinder provides a userfriendly platform for determination of short guide RNAs（sgRNAs） and homology arms to design a new animal model.Case studies illustrate the ability of SysFinder in helping experimental scientists.SysFinder is a useful platform for experimental scientists to carry out their research in the human molecular mechanisms.展开更多
A model of random particles constructed by the operation of self-similarity in fractalgeometry is presented.The correlation function of the number density has been obtained andcan be used conveniently in theoretical s...A model of random particles constructed by the operation of self-similarity in fractalgeometry is presented.The correlation function of the number density has been obtained andcan be used conveniently in theoretical study and computer simulation for wave interaction withrandom media.As an example,this model has been applied to calculate analytically the rangedependence of volume scattering in radar echoes.The result agrees with that of Rastogi andScheucher’s simulation.展开更多
Self-similar behavior for the multicomponent coagulation system is investigated analytically in this paper. Asymptotic self-similar solutions for the constant kernel, sum kernel, and product kernel are achieved by int...Self-similar behavior for the multicomponent coagulation system is investigated analytically in this paper. Asymptotic self-similar solutions for the constant kernel, sum kernel, and product kernel are achieved by introduction of different generating functions. In these solutions, two size-scale variables are introduced to characterize the asymptotic distribution of total mass and individual masses. The result of product kernel （gelling kernel） is consistent with the Vigli-Ziff conjecture to some extent. Furthermore, the steady-state solution with injection for the constant kernel is obtained, which is again the product of a normal distribution and the scaling solution for the single variable coagulation.展开更多
In this study we refer to a non-steady state, one-dimensional (on the x-axis), unconfined and saturated flow in an aquifer, described by the Boussinesq equation, combined with accretion. In accordance with the above, ...In this study we refer to a non-steady state, one-dimensional (on the x-axis), unconfined and saturated flow in an aquifer, described by the Boussinesq equation, combined with accretion. In accordance with the above, the moving boundary of the saturated area (toward x → +∝) serves as a horizontal water flux source to the unsaturated area. As time advances, the horizontally saturated zone, lying on the x-axis, becomes wider. A self-similar solution is derived that, after some mathematical manipulation, it is described in terms of Hypergeometric functions. The long-time behaviors of the solution describe the situation at which the water flux, that penetrates horizontally to the non-saturated zone, is equal to the water flux entering into the saturated zone.展开更多
文摘Let and denote respectively the functionswhere λ≥1, The author discusses the similarity transformation of the regularizing functionals of these functions and the similar property of their Fourier transformation.
基金the National Natural Science Foundation of China under Grant Nos.40175005,40275006,and 40233030 the Natural Science Foundation of Beijing under Grant No.8052012.
文摘Based on data obtained during the Hualhe River Basin Experiment （HUBEX） in 1999, this study intends to detect the quantitative discrepancies in the momentum （τ0）, sensible heat （H0） and latent heat （E0） fluxes among six sets of similarity functions with the aerodynamic method. It also aims to clarify the applicability of the functions under stable conditions. The relative discrepancy was studied with the normalized transfer coefficients for τ0, H0 and E0, namely CD, CH and CQ, respectively. Except for one set of functions that adopted a rather small von Kármán＇s constant （0.365）, the relative discrepancy in τ0 among the other functions was less than 10%, while that in H0（E0） sometimes reached 25% when the bulk Richardson number （R/B） was less than 0.07. The absolute discrepancy in the fluxes was studied with statistical computations. Among the six sets of functions, the discrepancy in τ0, H0 and E0 sometimes reached 0.03 kg m^-1 s^-2, 4 W m^-2 and 10 W m^-2, respectively, and the discrepancy in the energy balance ratio sometimes exceeded 0.1. Furthermore, when RiB exceeded the critical value （Ric） for a specific set of functions, no fluxes could be derived with the functions. It is therefore suggested that RiB be compared with Ric before computing the fluxes if RiB is less than Ric. Finally, two sets of nonlinear similarity functions are recommended, due to their unlimited applicability in terms of RiB.
基金Key project of the National Natural Science Foundation of China(4083095841005029)the "973" National Basis Research and Development Program of China (2009CB421502)
文摘Modified refractivity (M) profile is an important parameter to describe the atmospheric refraction environment,as well as a key factor in uniquely evaluating electromagnetic propagation effects.In order to improve the model-derived M profile in stable (especially very stable) conditions,three nonlinear similarity functions,namely BH91,CB05,SHEBA07,are introduced in this paper to improve the original Babin_V25 model,and the performances of these modified models are verified based on the hydrometeorological observations from tower platforms,which are finally compared with the original Babin_V25 model and Local_HYQ92 model.Results show that introducing nonlinear similarity functions can significantly improve the model-derived M profile;especially,the newly developed SHEBA07 functions manage to reduce the predicted root mean square (rms) differences of M and M slope (for both 0-5m and 5-40m) by 64.5%,16.6%,and 60.4%,respectively in stable conditions.Unfortunately,this improved method reacts little on the evaporation duct height;in contrast,Local_HYQ92 model is capable of reducing the predicted rms differences of M,M slope (for both 0-5m and 5-40m),and evaporation duct height by 76.7%,40.2%,83.7%,and 58.0% respectively.Finally,a new recommendation is made to apply Local_HYQ92 and Babin_SHEBA07 in very stable conditions considering that M slope is more important than evaporation duct height and absolute M value in uniquely determining electromagnetic propagation effects.
基金the National Science Foundation for funding Dillon Amaya’s project work in summer 2012 through the Research Experience for Undergraduates(REU)Program,grant AGS1005265
文摘Fifty years ago, Hans A. Panofsky published a paper entitled Determination of stress from wind and temperature measurements. In his famous paper, he presented a new profile function for the mean horizontal wind speed under the condition of diabatic stratification that includes his integral similarity function. With his integral similarity function, he opened the door for Monin-Obukhov scaling in a wide range of micrometeorological and microclimatological applications. In a historic survey ranging from the sixties of the past century down to the present days, we present integral similarity functions for momentum, sensible heat, and water vapor for both unstable and stable stratification, where on the one hand free convection condition and on the other hand strongly stable stratification are addressed.
基金This work was supported in part by the National Natural Science Foundation of China under Grants 61873089,62032007the Key Project of the Education Department of Hunan Province under Grant 20A087the Innovation Platform Open Fund Project of Hunan Provincial Education Department under Grant 20K025.
文摘Identifying associations between microRNAs(miRNAs)and diseases is very important to understand the occurrence and development of human diseases.However,these existing methods suffer from the following limitation:first,some disease-related miRNAs are obtained from the miRNA functional similarity networks consisting of heterogeneous data sources,i.e.,disease similarity,protein interaction network,gene expression.Second,little approaches infer disease-related miRNAs depending on the network topological features without the functional similarity of miRNAs.In this paper,we develop a novel model of Integrating Network Topology Similarity and MicroRNA Function Similarity(INTS-MFS).The integrated miRNA similarities are calculated based on miRNA functional similarity and network topological characteristics.INTS-MFS obtained AUC of 0.872 based on five-fold cross-validation and was applied to three common human diseases in case studies.As a results,30 out of top 30 predicted Prostatic Neoplasm-related miRNAs were included in the two databases of dbDEMC and PhenomiR2.0.29 out of top 30 predicted Lung Neoplasm-related miRNAs and Breast Neoplasm-related miRNAs were included in dbDEMC,PhenomiR2.0 and experimental reports.Moreover,INTS-MFS found unknown association with hsa-mir-371a in breast cancer and lung cancer,which have not been reported.It provides biologists new clues for diagnosing breast and lung cancer.
基金the National Natural Science Foundation of China
文摘Let X= (Ω, ■, ■_t, X_t,, θ_t, p~x) be a self-similar Markov process on (0,∞) with non-decreasing path. The exact Hausdorff and Packing measure functions of the image X([0,t] ) are obtained.
基金supported by the grants NSFC11201232, 12KJB110008Qing Lan Project, 13KJB110015, 12YJAZH096the Project-sponsored by SRF for ROCS, SEM
文摘In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some weU-known examples, such as the Sierpinski gasket, the unit interval, the level 3 Sierpinski gasket, the hexagasket, the 3-dimensional Sierpinski gasket, and the Vicsek set are also considered.
基金supported by the National Natural Science Foundation of China(No.11201232)Qing Lan Project of Jiangsu Province
文摘In this paper we establish the oscillation inequality of harmonic functions and HOlder estimate of the functions in the domain of the Laplacian on connected post critically finite （p.c.f.） self-similar sets.
文摘We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.
文摘Functional brain network (FBN) measures based on functional magnetic resonance imaging (fMRI) data, has become important biomarkers for early diagnosis and prediction of clinical outcomes in neurological diseases, such as Alzheimer’s diseases (AD) and its prodromal state (<em>i</em>.<em>e</em>., Mild cognitive impairment, MCI). In the past decades, researchers have developed numbers of approaches for FBN estimation, including Pearson’s correction (PC), sparse representation (SR), and so on. Despite their popularity and wide applications in current studies, most of the approaches for FBN estimation only consider the dependency between the measured blood oxygen level dependent (BOLD) time series, but ignore the spatial relationships between pairs of brain regions. In practice, the strength of functional connection between brain regions will decrease as their distance increases. Inspired by this, we proposed a new approach for FBN estimation based on the assumption that the closer brain regions tend to share stronger relationships or similarities. To verify the effectiveness of the proposed method, we conduct experiments on a public dataset to identify the patients with MCIs from health controls (HCs) using the estimated FBNs. Experimental results demonstrate that the proposed approach yields statistically significant improvement in seven performance metrics over using the baseline methods.
文摘We show that the processes described by Avrami functions are self-similar. A comparative function characterizes a self-similar process by a certain Avrami exponent. We define the self-similar categories of some well-known biological processes. The method to determine the Avrami exponent by choosing the comparative function is demonstrated on the diffusion model of the growth of nuclei. We generalize the results.
基金supported by the National Natural Science Foundation of China (No.51178018,71031001)
文摘Fuzzy similarity measures, which are used to judge the closeness of two fuzzy sets, are presented to evaluate the water quality of the Haihe River. Based on the membership functions and coefficient of variation as the weights, four fuzzy similarity measures （including Lattice similarity measure, Hamming similarity measure, Euclidean similarity measure and the max-min similarity measure） are used to classify the 299 samples into the proper water quality standard ranks. The results are compared with the traditional distance discriminant methods. The calculation of two traditional distance discriminant methods （both Euclidean distance and absolute value distance） is also based on the use of coefficients of variation as the weights. Without the Lattice similarity measure, for this method loses some information, the correct assignment of samples classified into the same water quality ranks is 75.92% with the other three similarity measures and two distance discriminant methods. This result shows the reliability of the five methods. Only considering the three similarity measures, there were only 1.01% of the samples that did not classify to the same ranks, while the corresponding ratio of the two distance discriminant methods was 5.69%. The results of leave-one-out cross validation show that more than 88% of the samples are classified to the proper ranks, which demonstrates that the similarity measures are suitable to evaluate the water quality of the Haihe River.
基金supported by the National High Technology Research and Development Program of China(No.2015AA020104)the National Key Research and Development Program on Precision Medicine(No.2016YFC0901700)+6 种基金the National Basic Research Program of China(Nos.2011CB910204,2011CB510102,and 2010CB529200)the National Key Technology Support Program (No.2013BA1101B09)the National Key Scientific Instrument and Equipment Development Project(No.2012YQ03026108)the National Grand Program on Key Infectious Diseases(No. 2015ZX10004801)the Medical-Engineering Cross Project of Shanghai Jiao Tong University(No.YG2016MS33)the Youth Innovation Promotion Association CASthe National Institutes of Health grants(Nos.R01HL117491 and R01HL129778 to Y.E.C)
文摘Animal models are increasingly gaining values by cross-comparisons of response or resistance to clinical agents used for patients.However,many disease mechanisms and drug effects generated from animal models are not transferable to human.To address these issues,we developed SysFinder（http：//lifecenter.sgst.cn/SysFinder）,a platform for scientists to find appropriate animal models for translational research.SysFinder offers a ＂topic-centered＂ approach for systematic comparisons of human genes,whose functions are involved in a specific scientific topic,to the corresponding homologous genes of animal models.Scientific topic can be a certain disease,drug,gene function or biological pathway.SysFinder calculates multi-level similarity indexes to evaluate the similarities between human and animal models in specified scientific topics.Meanwhile,SysFinder offers species-specific information to investigate the differences in molecular mechanisms between humans and animal models.Furthermore,SysFinder provides a userfriendly platform for determination of short guide RNAs（sgRNAs） and homology arms to design a new animal model.Case studies illustrate the ability of SysFinder in helping experimental scientists.SysFinder is a useful platform for experimental scientists to carry out their research in the human molecular mechanisms.
基金Supported by National Natural Science Foundation of China(68971020)
文摘A model of random particles constructed by the operation of self-similarity in fractalgeometry is presented.The correlation function of the number density has been obtained andcan be used conveniently in theoretical study and computer simulation for wave interaction withrandom media.As an example,this model has been applied to calculate analytically the rangedependence of volume scattering in radar echoes.The result agrees with that of Rastogi andScheucher’s simulation.
基金Project supported by the National Natural Science Foundation of China(Nos.11272196 and11222222)the Zhejiang Association of Science and Technology of Soft Science Research Project(No.ZJKX14C-34)
文摘Self-similar behavior for the multicomponent coagulation system is investigated analytically in this paper. Asymptotic self-similar solutions for the constant kernel, sum kernel, and product kernel are achieved by introduction of different generating functions. In these solutions, two size-scale variables are introduced to characterize the asymptotic distribution of total mass and individual masses. The result of product kernel （gelling kernel） is consistent with the Vigli-Ziff conjecture to some extent. Furthermore, the steady-state solution with injection for the constant kernel is obtained, which is again the product of a normal distribution and the scaling solution for the single variable coagulation.
文摘In this study we refer to a non-steady state, one-dimensional (on the x-axis), unconfined and saturated flow in an aquifer, described by the Boussinesq equation, combined with accretion. In accordance with the above, the moving boundary of the saturated area (toward x → +∝) serves as a horizontal water flux source to the unsaturated area. As time advances, the horizontally saturated zone, lying on the x-axis, becomes wider. A self-similar solution is derived that, after some mathematical manipulation, it is described in terms of Hypergeometric functions. The long-time behaviors of the solution describe the situation at which the water flux, that penetrates horizontally to the non-saturated zone, is equal to the water flux entering into the saturated zone.
