We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time-homogeneity of the problem considered. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J.2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from an arbitrary smooth-enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies—payout on the maximal rate, and constant barrier strategies—by measuring the distance from their performance functions to the value function.
{"title":"Measuring the suboptimality of dividend controls in a Brownian risk model","authors":"J. Eisenberg, Paul Krühner","doi":"10.1017/apr.2023.6","DOIUrl":"https://doi.org/10.1017/apr.2023.6","url":null,"abstract":"\u0000 We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time-homogeneity of the problem considered. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J.2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from an arbitrary smooth-enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies—payout on the maximal rate, and constant barrier strategies—by measuring the distance from their performance functions to the value function.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45027262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored $delta$ -shock model, $delta ge 1$ , for which the system fails whenever no shock occurs within a $delta$ -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob.32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order $delta$ (a geometric distribution of order $delta$ under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored $delta$ -shock model, for which the system fails when no shock occurs within a $delta$ -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold $gamma >0$ . Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.
假设一个系统受到一系列随机冲击的影响，这些冲击在一定时期内发生。本文研究了离散截尾$delta$ -冲击模型$delta ge 1$，该模型假定连续冲击之间的到达时间由一阶马尔可夫链描述(以及在二项冲击过程下，即连续冲击之间的到达时间具有几何分布)，当从最后一次冲击到$delta$ -长度的时间段内没有发生冲击时，系统失效。利用Chadjiconstantinidis等人引入的马尔可夫链嵌入技术。prop .32, 2000)，我们研究了系统寿命、冲击次数和无冲击发生的周期数的联合分布和边际分布，直至系统失效。得到了这些随机变量的联合概率和边际概率生成函数，并给出了它们的概率质量函数和矩的几个递推式和精确公式。证明了系统的寿命服从阶为$delta$的马尔可夫几何分布(二项设置下阶为$delta$的几何分布)，并服从矩阵几何分布。通过证明系统寿命随机变量的位移服从复合几何分布，给出了系统在二项冲击过程下的一些可靠性特性。最后，我们引入了一种新的混合离散截尾$delta$ -冲击模型，当最后一次冲击在$delta$ -长度的时间内没有发生冲击，或者冲击的幅度大于给定的临界阈值$gamma >0$时，系统就会失效。同样地，对于该混合模型，我们研究了在二项冲击过程下，系统的寿命、冲击次数和不发生冲击的周期数的联合分布和边际分布，直至系统失效。
{"title":"Distributions of random variables involved in discrete censored δ-shock models","authors":"S. Chadjiconstantinidis, S. Eryilmaz","doi":"10.1017/apr.2022.72","DOIUrl":"https://doi.org/10.1017/apr.2022.72","url":null,"abstract":"\u0000 Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -shock model, \u0000 \u0000 \u0000 \u0000$delta ge 1$\u0000\u0000 \u0000 , for which the system fails whenever no shock occurs within a \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob.32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 (a geometric distribution of order \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -shock model, for which the system fails when no shock occurs within a \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold \u0000 \u0000 \u0000 \u0000$gamma >0$\u0000\u0000 \u0000 . Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46101365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.
{"title":"Asymptotic results on tail moment and tail central moment for dependent risks","authors":"Jinzhu Li","doi":"10.1017/apr.2022.74","DOIUrl":"https://doi.org/10.1017/apr.2022.74","url":null,"abstract":"\u0000 In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44091059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.
{"title":"Stochastic comparison on active redundancy allocation to k-out-of-n systems with statistically dependent component and redundancy lifetimes","authors":"Yinping You, Xiaohu Li, Xiaoqin Li","doi":"10.1017/apr.2022.70","DOIUrl":"https://doi.org/10.1017/apr.2022.70","url":null,"abstract":"\u0000 In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43527329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"APR volume 55 issue 2 Cover and Back matter","authors":"","doi":"10.1017/apr.2023.12","DOIUrl":"https://doi.org/10.1017/apr.2023.12","url":null,"abstract":"","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43814903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"APR volume 55 issue 2 Cover and Front matter","authors":"","doi":"10.1017/apr.2023.11","DOIUrl":"https://doi.org/10.1017/apr.2023.11","url":null,"abstract":"","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49148117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study (asymmetric) $U$ -statistics based on a stationary sequence of $m$ -dependent variables; moreover, we consider constrained $U$ -statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem, together with results on rate of convergence, moment convergence, functional convergence, and a renewal theory version. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and permutations. We obtain both new results and new proofs of old results.
{"title":"Asymptotic normality for \u0000$boldsymbol{m}$\u0000 -dependent and constrained \u0000$boldsymbol{U}$\u0000 -statistics, with applications to pattern matching in random strings and permutations","authors":"S. Janson","doi":"10.1017/apr.2022.51","DOIUrl":"https://doi.org/10.1017/apr.2022.51","url":null,"abstract":"Abstract We study (asymmetric) \u0000$U$\u0000 -statistics based on a stationary sequence of \u0000$m$\u0000 -dependent variables; moreover, we consider constrained \u0000$U$\u0000 -statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem, together with results on rate of convergence, moment convergence, functional convergence, and a renewal theory version. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and permutations. We obtain both new results and new proofs of old results.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45373391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gabriel Zayas-Cabán, Jiaxin Liang, Stefanus Jasin, Guihua Wang
The above lemma is used to prove Theorems 1–2 and Propositions 1–3 in Sections 4 and 6 of [1]. It has been graciously pointed out to us that the bound in the lemma may not be correct in general. The original proof of this lemma uses a combination of linear program (LP) duality and sensitivity analysis results. The mistake is in the application of a known sensitivity analysis result under a certain assumption that happens to be not necessarily satisfied by our LP. Fortunately, it is possible to correct the bound in the above lemma. The new bound that we will prove in this correction note is as follows:
{"title":"An asymptotically optimal heuristic for general nonstationary finite-horizon restless multi-armed, multi-action bandits: Corrigendum","authors":"Gabriel Zayas-Cabán, Jiaxin Liang, Stefanus Jasin, Guihua Wang","doi":"10.1017/apr.2022.59","DOIUrl":"https://doi.org/10.1017/apr.2022.59","url":null,"abstract":"The above lemma is used to prove Theorems 1–2 and Propositions 1–3 in Sections 4 and 6 of [1]. It has been graciously pointed out to us that the bound in the lemma may not be correct in general. The original proof of this lemma uses a combination of linear program (LP) duality and sensitivity analysis results. The mistake is in the application of a known sensitivity analysis result under a certain assumption that happens to be not necessarily satisfied by our LP. Fortunately, it is possible to correct the bound in the above lemma. The new bound that we will prove in this correction note is as follows:","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46883874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t geq 0$ , given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.
{"title":"The size of a Markovian SIR epidemic given only removal data","authors":"F. Ball, P. Neal","doi":"10.1017/apr.2022.58","DOIUrl":"https://doi.org/10.1017/apr.2022.58","url":null,"abstract":"Abstract During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time \u0000$t geq 0$\u0000 , given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41387185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Consider Bernoulli bond percolation on a graph nicely embedded in hyperbolic space $mathbb{H}^d$ in such a way that it admits a transitive action by isometries of $mathbb{H}^d$ . Let $p_{text{a}}$ be the supremum of all percolation parameters such that no point at infinity of $mathbb{H}^d$ lies in the boundary of the cluster of a fixed vertex with positive probability. Then for any parameter $p < p_{text{a}}$ , almost surely every percolation cluster is thin-ended, i.e. has only one-point boundaries of ends.
{"title":"Thin-ended clusters in percolation in \u0000$mathbb{H}^d$","authors":"J. Czajkowski","doi":"10.1017/apr.2022.43","DOIUrl":"https://doi.org/10.1017/apr.2022.43","url":null,"abstract":"Abstract Consider Bernoulli bond percolation on a graph nicely embedded in hyperbolic space \u0000$mathbb{H}^d$\u0000 in such a way that it admits a transitive action by isometries of \u0000$mathbb{H}^d$\u0000 . Let \u0000$p_{text{a}}$\u0000 be the supremum of all percolation parameters such that no point at infinity of \u0000$mathbb{H}^d$\u0000 lies in the boundary of the cluster of a fixed vertex with positive probability. Then for any parameter \u0000$p < p_{text{a}}$\u0000 , almost surely every percolation cluster is thin-ended, i.e. has only one-point boundaries of ends.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49120931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：