Exact Renormalization Group for O(4) Gauged Supergravity

药物与临床China &Foreign Medical Treatment 中外医疗米非司酮周期疗法治疗对围绝经期异常子宫出血患者的影响分析梅娟，戴淑婷，张雪芳漳州招商局经济技术开发区第一医院妇产科，福建漳州 353100［摘要］ 目的 研讨米非司酮周期疗法治疗对围绝经期异常子宫出血（Abnormal Uterine Bleeding, AUB ）患者的影响。

方法 随机选取2021年5月—2023年5月漳州招商局经济技术开发区第一医院确诊为AUB 的100例患者为研究对象，依据随机数表法分组，对照组（n =50）接受米非司酮持续口服，观察组（n =50）接受米非司酮周期疗法治疗，比较两组方案疗效、性激素水平、临床相关指标以及用药安全性。

结果 观察组经3个周期用药后的疗效（96.00%）高于对照组（80.00%），差异有统计学意义（χ2=6.060，P <0.05）。

用药后，观察组各项性激素指标水平均低于对照组，差异有统计学意义（P 均<0.05）。

观察组用药后血红蛋白值高于对照组，子宫内膜厚度小于对照组，差异有统计学意义（P 均<0.05）。

两组间发生药物不良反应基本相当，差异无统计学意义（P >0.05）。

结论 对围绝经期AUB 患者实施米非司酮周期疗法用药，能够明显提升用药效果，调节患者机体性激素水平，提高血红蛋白值，缩小子宫内膜厚度，同时保障用药安全。

［关键词］ 围绝经期异常子宫出血；米非司酮；周期疗法；性激素；子宫内膜厚度［中图分类号］ R5 ［文献标识码］ A ［文章编号］ 1674-0742（2024）02(a)-0103-04Analysis of the Effect of Mifepristone Cycle Therapy on Perimenopausal Abnormal Uterine Bleeding PatientsMEI Juan, DAI Shuting, ZHANG XuefangDepartment of Obstetrics and Gynecology, Zhangzhou China Merchants Economic and Technological Development Zone First Hospital, Zhangzhou, Fujian Province, 353100 China[Abstract] Objective To study the effect of mifepristone cycle therapy on patients with perimenopausal abnormal uter⁃ine bleeding (AUB). Methods 100 patients diagnosed as perimenopausal AUB by Zhangzhou China Merchants Eco⁃nomic and Technological Development Zone First Hospital from May 2021 to May 2023 were randomly selected as the study objects and the group design was completed based on the random number table method. 50 cases in the control group received continuous oral mifepristone, and 50 cases in the observation group were treated with mifepristonecycle therapy. The efficacy, sex hormone levels, clinical indexes and drug safety of the two groups were compared. Results After 3 cycles of treatment, the therapeutic effect of the regimen in the observation group (96.00%) was higherthan that in the control group (80.00%), and the difference was statistically significant (χ2=6.060, P <0.05). After treat⁃ment, the levels of various sex hormone indexes in the observation group were lower than those in the control group, and the differences were statistically significant (all P <0.05). The hemoglobin value of the observation group was higher than that of the control group, and the endometrial thickness was lower than that of the control group, the differ⁃ences were statistically significant (both P <0.05). The incidence of adverse drug reactions between the two groups wassimilar, and there was no statistical significance difference (P >0.05). Conclusion The implementation of mifepristonecycle therapy for perimenopausal AUB patients can significantly enhance the effect of medication, regulate the level ofsex hormones in the patient's body, increase hemoglobin value, and reduce the thickness of the endometrium, as well DOI ：10.16662/ki.1674-0742.2024.04.103[作者简介] 梅娟（1983-），女，本科，主治医师，研究方向为妇产科临床。

Inventory Modeling in Supply Chain Management: AReviewCheng Tiexin Yue Jingbo Guo TaoCollege of Management, Tianjin Polytechnic University, Tianjin, China 300384tiexincheng@Abstract—In supply chain management, the inventory management of materials, semi-manufactured goods and products is often concerned and attracts a lot of scholars’ attentions. With the economy globalization, three new trends appeared in the supply chain management: materials procurement globalization, manufacture globalization and products distribution globalization. Consequently, three new areas in inventory modeling were paid more attentions to: 1. Multiple supplier and multi-product inventory models from the point of the upstream of the supply chain; 2. Multi-echelon inventory models including manufacturers, dealers and retailers from the point of the interior structure of the supply chain; 3. Stochastic multi-product demand inventory models from the point of the downstream of the supply chain. In this paper, the three areas mentioned above were discussed in detail and some new inventory models and researches were reviewed, and at the end of the paper, the research directions of the inventory management in supply chain management were given.Keywords-Supply chain, Inventory management, Multi-echelon inventory, forecastingI.I NTRODUCTIONIn supply chain management, inventory management about materials, semi-manufactured goods and products was widely focused on, and specialists and scholars all over the world have made a lot of researches on this area, especially for establishment of the inventory model. In 1915, when the first constant inventory model for the single product was set up, Ford. W. Harris established the model of EOQ (Economic Order Quantity), subsequently the researches on this area proceeded rapidly. With the economy globalization, three new trends appeared in supply chain management: materials procurement globalization, manufacture globalization, and products distribution globalization. Consequently, three new areas in inventory modeling were paid more attentions to: 1. Multiple supplier and multi-product inventory models from the point of the upstream of the supply chain; 2. Multi-echelon inventory models including manufacturers, dealers and retailers from the point of the interior structure of the supply chain; 3. Stochastic multi-product demand inventory models from the point of the downstream of the supply chain. In this paper, the new status and results of the research in this area will be reviewed in detail from the mentioned three trends.II.M ULTI-SUPPLIER AND MULTI-PRODUCT INVENTORYMODELSAbout the research on multi-supplier, Sculli & Wu[1] set up one model, in which two suppliers were introduced, and assumed that the lead time of the demand of products was Normal Distribution. According to this model, it was argued that the two suppliers had the same the replenishment quantity. Moinzadeh & Nahmias[2] established the inventory model with the continuing lead time for two suppliers, in which it was assumed (1) the two suppliers had the samecontinuous lead time (120ll<<) and different prices(21pp>); (2) the order costs: 21,CC; (3) shortages were allowed but the loss were aroused. The goal of the model was minimize the average inventory cost, which was determined by the order cost, storage costs and shortage loss, for long term, and according to that model the optimal replenishment policy ),,,(2121QQss was obtained. Based on the storage quantity t x at time t, the strategy is to order the quantity of goods (Q1) to supplier 1 when the t x was equal to the trigger level (s1) at time t, and to order the quantity of goods (Q2) to supplier 2 when the t x was equal to the emergency trigger level (s2) within the time l1, which was the lead time of Q1, before the Q1 occurred.Moinzadeh & Schmidit[3] studied the model set up by Moinzadeh & Nahmias[2], and modified it. They divided the optimal replenishment policy into two ways: the regular order (Q1) and the emergency order (Q2). In their revised model, when the demand occurs, (1) if 1+≥etSx (where: t x denotes the storage quantity of goods at time t;e S denotes the emergency trigger level), then the regular order (Q1) isapplied, (2) if etSx≤and the replenishment time of the regular order is less than the lead time of supplier 2, then theregular order (Q1) is applied, too, and (3) if etSx≤ and the replenishment time of the regular order is less than the lead time of supplier 2, then the emergency order is applied, that is to order the quantity of goods (Q2) to supplier 2.Chiang & Gutierrez[4] set up one model of two supply modes with periodic check for supplier, in their model, the cost of the emergence order was divided into two situations: C=0 and C>0, they applied dynamic programming to optimize the model, and the optimized policy was: when the inventorySponsored by Tianjin Municipal Science and Technology Commission, Project ID: 08JCZDJC24200.Cheng Tiexin: Ph.D of Management Science and Engineering, Associate Professor, Research areas: Project Management, Knowledge Management, Supply Chain Management Tel: 0086-22-83956951.was checked, if e t S x ≥, the regular order was applied; orelse if e t S x <, the emergency order was applied. In 1998, Chiang & Gutierrez extended the model above. In the new model, it assumed that the check cycle was continuous other than periodic and the cost of order was alterable not fixed. The models mentioned above belong to the models of two suppliers or two supply modes, however, Dayani Sedarage, Okitsugu Fujiwara & Huynh Trung Luong[5], and Ram Ganeshan[6] introduced the multi-supplier (N suppliers) in their inventory models, discussed the models of the multi-supplier (N suppliers) in detail and gave the optimized policy for inventory control.Considering that the uncertainties, such as the failure of the equipments, the strike of workers, the adverse climatic conditions and so on, would affect the suppliers to supply the goods on time, in 1996, Parlar & Perry[7] set up the model, in which they assumed that the goods were exchangeable and put one status variable, the value of which was ON or OFF (ON means to supply and OFF means not to supply), to every supplier. If the number of the suppliers was n , then there would be 2n combinations of suppliers, then they optimized the model to get one policy for ordering: (s i ,Q i ), where the reorder point s i was related with the order quantity Q i and the status variable of the suppliers.III.M ULTI -ECHELON INVENTORY MODELSThe economy globalization resulted in the globalization of manufacture and sale, therefore the multi-echelon inventory management was paid more and more attentions. According to the features of the multi-echelon inventory, we divided it into 3 types: (1) Serial inventory systems, (2) Assembly inventory systems and (3) Distribution inventory systems. About the multi-echelon inventory, Clark and Scarf[8] introduced the concept of echelon stock as opposed to installation stock. In echelon stock policies, ordering decisions at a given stage are based on the echelon inventory position, which is the sum of the inventory position at the considered stage and at all the downstream stages. They proved that there existed the optimal base stock ordering policy in the pure serial inventory systems, and developed one effective decomposing method to compute the optimal base stock policy. In addition, they also discussed the distribution inventory system and gave an approximate method for it. Federgruen & Zipkin[9] extended the model established by Clark & Scarf from the finite horizon to the infinite horizon with stationary parameters and developed an efficient computational method. Hochstaedter extended the model established by Clark & Scarf from the pure serial inventory system to the distribution inventory system, and Rosling[10] extended the model established by Clark & Scarf to the assembly inventory system and gave the method to get the optimal base stock policy of it.Generally, there are different ways to manage multi-echelon inventory systems. When the strategy of “one for one” is taken, installation stock policies can be proper, which means that the multi-echelon inventory control policy is the same as the installation stock policy. According to the current storage quantity of goods, the order policy of the multi-echeloninventory system can be obtained through calculating the total order quantities of the suppliers. However, if the times and quantities of the orders are very large, the strategy of “one for one” usually can not ensure the optimization for the inventory management. Axsäter & Rosling[11] proved that in serial and assembly inventory systems echelon stock policies achieved better performance than installation ones do, but in the distribution inventory system, these two policies had different advantages respectively. Axsäter & Rosling considered a two-echelon distribution inventory system with stochastic demand, proved that optimization of continuous review (R ,Q )-policies were usually very efficient in case of relatively low demand, and gave a method by which a high-demand system was approximated by a low-demand system. Tetsuo Iida[12] studied a dynamic multi-echelon inventory problem in the finite horizon, and gave the near-myopic policies which were sufficiently close to the optimal one and also could be applied to the distribution system.IV.T HE INVENTORY MODELS FOR STOCHASTIC MULTI -PRODUCT DEMANDThe classical EOQ model is based on the constant demand, and it is assumed that demand is continuous and even. If R stands for the rate of demand (demand quantity per time), which is constant, then the demand quantity in the time of t is Rt . But, in the real market, the product demand is dynamic and stochastic with the respect of the change of the price and time etc. At present, there are several kinds of the methods to forecast the products demand mainly as follow:A. The model of EconometricsThe demand is the function of the time (t ) in common econometrical forecasting models. In the classical EOQ model as mentioned above, the assumption is that demand function (Q =Rt ) is the linear function of time variable. Silver and Meal[13] studied the inventory model of the demand function of time (t ), and they proposed a heuristic algorithm which can be applied in most EOQ models. Donaldson[14] discussed the conditions in detail that the inventory horizon is finite and demand function is the linear function of time variable, and proposed the optimal reorder point. In addition, other scholars, such as Ritchie[15], Buchanan[16], Mitra et al.[17], Goyal[18], studied this kind of inventory models. The inventory model discussed above possessed the linear demand function which can change along with the time variable, however in the real market, the assumption of linear demand is so simple that it is far from the actual situation, hence some scholars turned to the non-linear demand function. Hariga and Benkherouf[19] established the inventory model in which demand function is the exponential function of time variable, considered the loss of shortages, and proposed the optimal policy of inventory replenishment on the condition that shortages were not allowed. Wee[20] also established the inventory model with the demand of the exponential function of time, it’s different from Hariga and Benkherouf, he proposed the optimal policy of inventory replenishment on condition that shortages were allowed. Considering that some demands of products (like computer chips and the aircraft components) grow rapidly for the new products and drop rapidly for the outdated products,S. Khanra and K.S. Chaudhuri[21] took the quadratic function of the time variable as the demand function, and established the corresponding inventory model.B.The model of Time SeriesThe model of time series is one common model, which can be applied to forecast most of products’ demand. The classical Gaussian Automatic Regressive Model (G.E.P. Box, G.M. Jenkins[22]), which is usually called the AR model, is applied widely in the commercial forecasting. Holt-Winters (HW) model (Holt, 1957; Winters, 1960) introduced exponentially weighted Moving Average models, which is usually called the MA model, to forecast the inventory demand, whereas Don M. Miller and Dan Williams[23] introduced the method of Ratio-to-Moving-Average Decomposition to do it, which can eliminate the seasonal influence to the demand. Lisa Bianchi, Jeffrey Jarrett and R. Choudary Hanumara[24] forecasted demand of the telecommunication market with Automatic Regressive Integrated Moving Average models (ARIMA), and contrasted the results with the Holt-Winters(HW) model. Moreover, S. L. Ho and M. Xie[25] analyzed the reliability of ARIMA model, Xiaolong Zhang[26] discussed how to eliminate the bullwhip effect in supply chain with different forecasting methods of time series, and proposed a simple rule to select different forecasting model.A combined forecast might improve upon the better of the two individual forecasts. Alternatively, combinations with other statistical forecasting methods might be advantageous. The concept of combining forecasts started with the seminal work of Bates and Granger[27]. Given two individual forecasts of a time series, they demonstrated that a suitable linear combination of the two forecasts may result in a better forecast than the two original ones, in the sense of a smaller error variance. Newbold and Granger[28], Makridakis et al.[29], have reported empirical results that showed that combinations of forecasts outperformed individual methods. Throughout the years, applications of combined forecasts have been found in many fields such as meteorology, economics, insurance and forecasting sales and price, see Clemen[30]. Chi Kin Chan, Brian G. Kingsman and H. Wong[31] described a case study of demand combining forecasting for inventory management, besides comparing performances between combination forecasts and individual forecasts. They also investigated the differences between regular changing weights and constant weights for a certain forecast horizon, finally gave the optimal stock policy.C.The Stochastic demand modelsThe classical newspaper boy model belongs to the inventory models of the stochastic demand. It is often assumed that the product demand is one kind of probability distributions, for example, the demand of discrete products is often supposed to obey the Possion distribution, and the demand of continuous products obeys the Normal distribution (Chiang and Benton[4]) etc. Ignall and Veinot proposed the inventory problem of stochastic multi-products during 1960's; subsequently, Goyal[18], Rosenblatt and Rothblum[32], Anily[33] did further researches for the inventory problem of stochastic multi-product demands and established some mathematical models, most of their researches are based on the classical EOQ model. Canadian scholar Dirk Beyer, Suresh P. Sethi and R. Sridhar[34] proposed the stochastic multi-products inventory model, which was set limit to the capacity of the inventory on the foundation of aforementioned researches, this model was the improvement of hereinbefore models.In addition, some scholars applied the Bayes method to revise the forecasting outcome of the stochastic product demand, e.g. K. Surekha and Moheb Ghali[35], K. Rajashree Kamath and T. P. M. Pakkala[36] analyzed the problem of inventory demand of Stationary and Non-Stationary, and obtained more reasonable optimized inventory policy with Bayes forecasting method.V.T HE NEW RESEARCH DIRECTIONS FOR INVENTORYMODELINGA.Integrated inventory modelingAt present, supply chain management has the trend to integration; more and more manufacturers in the supply chain form the strategic alliance. Inventory management is also in the direction of integration. Consequently, the single manufacturer or supplier has to establish integrated inventory model in view of the supply chain from upstream to downstream when they make decision of the inventory management, and multi-echelon inventory modeling needs to be applied.B.Internet and E-business based virtual inventory modelingWith the development and application of the Internet and E-business in supply chain, the purchase and order costs between buyers and sellers are decreasing, and the risks of suppliers are being reduced. This results in that multiple suppliers’ pattern is superior to single supplier pattern. The development of IT causes information-sharing between the buyers and the sellers. Buyer’s demand can be forecasted based on the information of the venders, however, suppliers should deal with a great deal of data. Therefore, Data Mining and Knowledge Discovery in Database have the wide application prospects in inventory management; in the meanwhile, more attentions will be paid to virtual inventory modeling.C.Inventory modeling under asymmetric informationIn the real supply chain, every partner (manufacturer or supplier) makes his decision independently, hence there exits asymmetric information. Even if the coordination has been set up in supply chain, the partner usually keeps his commercial information, such as costs, profits and so on, in secret, which leads to that it is difficult to get this information for other partners. Therefore, inventory modeling under asymmetric information is more valuable and practicable. There are some researches on this field, in which the game theory was often applied to decision-making under asymmetric information; however it needs to be studied more intensively and extensively.VI.C ONCLUSIONIn this paper, three types of inventory models were discussed from the aspects of supply chain management:(1)Multi-supplier and multi-product inventory model;(2)Multi-echelon inventory model; (3)Stochastic multi-product demand inventory model. The research history and development of the inventory models were reviewed and the latest research results were discussed. Finally, the future research directions of the inventory management in supply chain management were given. The inventory management was under way of integration, the IT and Internet will be considered and paid more and more attentions to inventory modeling, and inventory modeling under asymmetric information will become more valuable and practicable.R EFERENCES[1]Sculli, D., Wu, S.Y.,. Stock control with two suppliers and normal leadtimes. Journal of the Operational Research Society, 32(11), 1981, 1003-1009.[2]Moinzadeh, K., Nahmias, S., A continuous review model for aninventory system with two supply modes. Management Science,1988(34): 761–773.[3]Moinzadeh, K., Schmidt, C.P., An (S-1, S) inventory system withemergency orders. Operations Research, 1991(39): 308-321.[4]Chiang, C., Beton, W.C., Sole souring versus dual souring understochastic demands and lead times, Naval Research Logistics 41,1994,609-624.[5]Dayani Sedarage, Okitsugu Fujiwara, Huynh Trung Luong,Determining optimal order splitting and reorder level for N-supplierinventory systems, European Journal of Operational Research 116(1999) 389-404.[6]Ram Ganeshan, Managing supply chain inventories:A multiple retailer,one warehouse, multiple supplier model, Int. J. Production Economics,59 (1999) 341-354.[7]M. Parlar and D. Perry, Inventory models of future supply uncertaintywith single and multiple sources. Naval Research Logistics,1996(43):191-210.[8] A.J. Clark, H.E. Scarf, Optimal policies for a multi-echelon inventoryproblem, Management Science, 6(1960) 475-490.[9]Federgruen, P. Zipkin, Computional issues in an infinite-horizon multi-echelon inventory model, Operations Research, 32 (1984)818-836. [10]K. Rosling, Optimal inventory policies for assembly systems underrandom demands, Operations Research, 37(1989)565-579.[11]Axsäter and Rosling, Installation vs. echelon stock policies for multi-level inventory control, Management Science, 39 (1993) 1274-1280. [12]Tetsuo Iida, The infinite horizon non-stationary stochastic multi-echelon inventory problem and near-myopic polices, European Journalof Operational Research, 134(2001)525-539.[13]Silver EA, Meal HC. A simple modification of the EOQ for the case ofa varying demand rate. Production and Inventory Management,1969;10(4):52-65.[14]Donaldson WA. Inventory replenishment policy for a linear trend indemand—an analytical solution. Operational Research Quarterly, 1977;28:663-70.[15]Ritchie E. Practical inventory replenishment policies for a linear trendin demand followed by a period of steady demand. Journal ofOperational Research Society, 1980;31:605-13.[16]Buchanan JT. Alternative solution methods for the inventoryreplenishment problem under increasing demand. Journal ofOperational Research Society, 1980; 31:615-20.[17]Mitra A, Fox JF, Jessejr RR. A note on determining order quantitieswith a linear trend in demand. Journal of Operational Research Society,1984;35:141-4. [18]Goyal SK. On improving replenishment policies for linear trend indemand. Engineering Costs and Production Economics, 1986; 10:73-6.[19]Hariga MA, Benkherouf L. Optimal and heuristic inventoryreplenishment models for deteriorating items with exponential time-varying demand. European Journal of Operational Research,1994;79:123-37.[20]Wee HM. A deterministic lot-size inventory model for deterioratingitems with shortages and a declining puter and Operations Research, 1995; 22(3):345-56.[21]S. Khanra, K.S. Chaudhuri, A note on an order-level inventory modelfor a deteriorating item with time-dependent quadratic demand,Computers & Operations Research, 30 (2003) 1901-1916.[22]G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting andControl, Seconded Edition., Holden-Day, San Francisco, 1976. [23]Don M. Miller , Dan Williams, Shrinkage estimators of time seriesseasonal factors and their effect on forecasting accuracy, International Journal of Forecasting, 2002.[24]Lisa Bianchi, Jeffrey Jarrett, R. Choudary Hanumara, Improvingforecasting for telemarketing centers by ARIMA modeling withintervention, International Journal of Forecasting, 14 (1998) 497-504.[25]S.L. Ho and M. Xie, The use of ARIMA models for relliablityforcasting and analysis, Computers and Electrical Engineering, 1998,Vol. 35, 213-216.[26]Xiaolong Zhang, The impact of forecasting methods on the bullwhipeffect, Int. J. Production Economics, 2004(88):15-27.[27]Bates, J.M., Granger, C.W.J., The combination of forecasts.Operational Research Quarterly, 20(1969.) 451-468.[28]Newbold, P., Granger, C.W.J., Experience with forecasting univariatetime series and the combination of forecasts (with discussion). Journal of the Royal Statistical Society Series A, 137(1974), 131-149. [29]Makridakis, S., Winkler, R.L., Averages of forecasts: Some empiricalresults, Management Science, 29(1983), 987-996.[30]Clemen, R.T., Combining forecasts: A review and annotatedbibliography. International Journal of Forecasting, 5(1989), 559±583.[31]Chi Kin Chan, Brian G. Kingsman, H. Wong, The value of combiningforecasts in inventory management-a case study in banking, European Journal of Operational Research, 117 (1999) 199-210.[32]Rosenblatt, M. J. and Uriel G. Rothblum, The Single Resource Multi-item Inventory Systems, Operational Research, 1990, 38, 686-693. [33]Anily S, Multi-Item Replenishment and Storage Problems(MIRSP):Heuristics and Bounds, Operational Research, 1991, 39, 233-239. [34]Dirk Beyer, Suresh P. Sethi, R. Sridhar, Stochastic Multi-ProductInventory Models with Limited Storage, work paper, University ofToronto, Ontario, Canada,1997.[35]K. Surekha, Moheb Ghali, The speed of adjustment and productionsmoothing: Bayes estimation, Int. J. Production Economics, 71(2001)55-65.[36]K. Rajashree Kamath,T. P. M. Pakkala, A Bayesian approach todynamic inventory model under an unknown demand distribution,Computers & Operations Research, 29(2002): 403-422.Inventory Modeling in Supply Chain Management: A Review作者：Cheng Tiexin， Yue Jingbo， Guo Tao作者单位：College of Management, Tianjin Polytechnic University, Tianjin, China 300384本文链接：/Conference_WFHYXW331993.aspx。

密度矩阵重整化群探究作者：郭丰齐来源：《中国新通信》 2018年第23期一、引言在人们最早研究很多物理问题中的时候，大家发现了在低温下的稀磁合金中电子表现出了一些反常现象，这就是kondo 效应[1]。

为了解决 kondo 效应。

研究者首先用最常规的蒙特卡洛的方法去数值计算 kondo 模型，但是误差很大，并不能很好的刻画 kondo 模型，而 kondo 模型后来被 Wilson利用 NRG 方法成功的刻画。

但是当把 NRG 模型移到其它模型中的时候，如 1DHeisenberg ，Hubbard 模型中，却发现误差特别大，所以1992 年的 DMRG 方法提出，对这个问题的研究起到了一个很大的突破。

二、密度矩阵重整化群算法在强关联系统模型的研究学习中我们发现，解析解能用于极个别模型中的一维情况[3]。

目前不能严格求二维及更高维度的强关联系统的解析解。

因此大部分系统只能用数值计算进行近似求解[2]，所以只能靠数值计算的方法来研究大部分系统的关联函数、基态能量和低能激发态等这些物理性质。

主要有 NRG （Numerical renormalization group, 数值重整化群）、DMRG （Density matrix renormalization group, 密度矩阵重整化群）、ED（exact diagonalization，严格对角化）及 QMC（Quantum Monte Carlo method，量子蒙特卡洛）等这些方法来研究大部分系统的低能级激发态和关联函数，基态能量这些物理性质。

其中，较大的格点系统通常用 QMC 和DMRG 来进行数值化计算。

2.1 历史背景为得到一个量子多体系统的所有能量本征值和与本征值对应的本征波函数。

解析计算很复杂，随着数值计算的应用，使其过程相对简化，但要想求这个系统的数值解，我们通常需要对这个系统的 H（Hamiltonian，哈密顿量）进行对角化，对于数值计算的方法来说，主要困难在于随着系统的尺寸增大，系统的 Hilbert 空间维度是呈指数增长的。

八卦一下量子机器学习的历史人工智能和量子信息在讲量子机器学习之前我们先来八卦一下人工智能和量子信息。

1956，达特茅斯，十位大牛聚集于此，麦卡锡（John McCarthy）给这个活动起了个别出心裁的名字：“人工智能夏季研讨会”（Summer Research Project on Artificial Intelligence），现在被普遍认为是人工智能的起点。

AI的历史是非常曲折的，从符号派到联结派，从逻辑推理到统计学习，从经历70年代和80年代两次大规模的政府经费削减，到90年代开始提出神经网络，默默无闻直到2006年Hinton提出深层神经网络的层级预训练方法，从专注于算法到李飞飞引入ImageNet，大家开始注意到数据的重要性，大数据的土壤加上计算力的摩尔定律迎来了现在深度学习的火热。

量子信息的历史则更为悠久和艰难。

这一切都可以归结到1935年，爱因斯坦，波多尔斯基和罗森在“Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”一文中提出了EPR悖论，从而引出了量子纠缠这个概念。

回溯到更早一点，1927年第五次索尔维会议，世界上最主要的物理学家聚在一起讨论新近表述的量子理论。

会议上爱因斯坦和波尔起了争执，爱因斯坦用“上帝不会掷骰子”的观点来反对海森堡的不确定性原理，而玻尔反驳道，“爱因斯坦，不要告诉上帝怎么做”。

这一论战持续了很多年，伴随着量子力学的发展，直到爱因斯坦在1955年去世。

爱因斯坦直到去世也还一直坚持这个世界没有随机性这种东西，所有的物理规律都是确定性的，给定初态和演化规律，物理学家就能推算出任意时刻系统的状态。

而量子力学生来就伴随了不确定性，一只猫在没测量前可以同时“生”和'死'，不具备一个确定的状态，只有测量后这只猫才具备“生”和'死'其中的一种状态，至于具体是哪一种状态量子力学只能告诉我们每一种态的概率，给不出一个确定的结果。

角转移矩阵重整化群方法及其应用何春山【摘要】Renormalization group ( RG) theory is a very important theory to research phase transition and critical phenomenon. With the development of the computing technology, numerical simulation methods based on the RG are used to compute the physical parameters. The corner transfer matrix renormalization group (CTMRG) method can get high precision results even if the physical system is in the critical status. CTMRG method is used to find the critical point of the two-dimensional Ising model. The numerical critical coupling constant is consistent with the exact result with good precision ( 10-5 ).%相变和临界现象在自然界普遍存在,研究的主要手段是重整化群理论.随着计算机技术的发展,基于重整化群思想的数值模拟也得到了广泛的应用,它能够精确地计算系统处于临界状态时的物理参数.该文采用角转移矩阵重化群方法计算了无外场二维伊辛模型的临界耦合常数,得到了准确度为10-5的数值计算结果.【期刊名称】《中山大学学报（自然科学版）》【年(卷),期】2011(050)006【总页数】5页(P30-34)【关键词】角转移矩阵重整化群;二维伊辛模型;临界点【作者】何春山【作者单位】中山大学物理科学与工程技术学院,广东广州 510275【正文语种】中文【中图分类】O414.21重整化群理论的出现，翻开了现代临界现象研究新的一页。

a r X i v :c o n d -m a t /0509160v 2 [c o n d -m a t .s t a t -m e c h ] 17 J a n 2006Renormalization group approach to satisfiabilityS.N.CoppersmithDepartment of Physics,University of Wisconsin,1150University Avenue,Madison,WI 53706USA(Dated:February 2,2008)Satisfiability is a classic problem in computational complexity theory,in which one wishes to determine whether an assignment of values to a collection of Boolean variables exists in which all of a collection of clauses composed of logical OR’s of these variables is true.Here,a renormalization group transformation is constructed and used to relate the properties of satisfiability problems with different numbers of variables in each clause.The transformation yields new insight into phase transitions delineating “hard”and “easy”satisfiability problems.PACS numbers:,89.20Ff,75.10.NrComputational complexity theory addresses the question of how fast the resources required to solve a given problem grow with the size of the input needed to specify the problem.[1]P is the class of problems that can be solved in polynomial time,which means a time that grows as a polynomial of the size of the prob-lem specification,while NP is the class of problems for which a solution can be verified in polynomial time.Whether or not P is distinct from NP has been a cen-tral unanswered question in computational complexity theory for decades.[2]Satisfiability (SAT)is a classic problem in compu-tational complexity.An often-studied type of SAT is K -SAT,in which one attempts to find assignment of N variables such that the conjunction (AND)of M constraints,or clauses,each of which is the disjunction (OR)of K literals,each literal being either a negated or un-negated variable,is true.(This way of writing the problem,as a conjunction of clauses that are disjunc-tions,is called conjunctive normal form.)For example,the 3-SAT instance with the 4variables x 1,x 2,x 3,and x 4and the four clauses(x 1=1OR x 2=0OR x 4=1)AND (x 1=0OR x 3=1OR x 4=0)AND (x 2=1OR x 3=1OR x 4=1)AND(x 1=0OR x 3=0OR x 4=1),(1)is satisfiable because it it is true for the assignments x 1=1,x 2=1,x 3=1,x 4=1.Below,we will write satisfiability problems in conjunctive normal form us-ing the notation of [3],where the ANDs and ORs are implied and the literals have positive or negative signs depending on whether or not they are negated.For ex-ample,the expression of Eq.(1)is written(1-24),(-13-4),(234),(-1-34).2-SAT can be solved in polynomial time [1],while K -SAT with K ≥3is known to be NP-complete [4]:if a polynomial algorithm for 3-SAT exists,then P is equal to NP.The complexity of SAT is intimately re-lated to the presence of phase transitions [5,6,7,8,9,10,11].For random problems with N variables,M clauses,and K literals per clause,as M is increased there is a phase transition from a satisfiable phase,in which almost all random instances are satisfiable,to an unsatisfiable phase,in which almost all random in-stances are unsatisfiable.The most difficult instances are near this SAT-unSAT transition.It has also been shown that there is a transition as the parameter K is changed between 2and 3,at K c ∼2.4,at which the nature of the SAT-unSAT transition changes [5].Here,we investigate the relationship between satis-fiability problems with different values of K by con-structing a renormalization group transformation,sim-ilar to those used for phase transition problems [12,13,14,15],that reduces the number of degrees of freedom,while possibly increasing the number and range of in-teractions [16](which in this context is the number of literals per clause).To do this,we note that the expres-sion((A 1x ),(A 2x ),...(A P x ),(B 1−x ),(B 2−x ))...,(B Q −x ))is satisfiable if and only if((A 1B 1),(A 1B 2),...(A 1B Q ),(A 2B 1),(A 2B 2),...(A 2B Q ),...,(A P B 1),(A P B 2),...,(A P B Q ))2is satisfiable.Here,the A i’s and B i’s are arbitrary clauses and x is a variable.(The easiest way to see the equivalence is to note that both expressions are satisfi-able if and only(A1AND A2AND...A P)OR(B1 AND B2AND...B Q)is.)Thefirst step of the renor-malization procedure is to use this identity to elimi-nate a given variable.In this step,P clauses in which a given variable comes in un-negated and Q clauses in which the same variable comes in negated are elimi-nated and replaced with PQ“resolution”[17]clauses. Thus,eliminating a“frustrated”[18]variable(one that enters into different clauses negated and un-negated) increases the number of clauses if PQ-(P+Q)>0.The resolution of two clauses of length K i and K j has length K i+K j−2.Note that resolving two2-clauses yields a2-clause,resolving a2-clause with a clause of length K≥3yields a clause length K,and resolving two clauses of with lengths K1≥3and K2≥3yields a clause with length greater than both K1and K2.One then simplifies the resulting satisfiability ex-pression by noting that1.Duplicate clauses are redundant,2.Duplicate literals in a given clause are redundant,3.If a variable enters into one clause both negatedand un-negated,then the clause must be true andcan be removed,4.If a clause has one literal,then the value of thecorresponding variable is determined,and5.If a subset of the literals in a clause comprise adifferent clause,then the clause with more liter-als is redundant.This last point means,for example,that if an expres-sion contains both(13-45)and(13),then(13-45) can be removed,because it is satisfied automatically if (13)is satisfied.This procedure is known in computer science as“the Davis-Putnam procedure of1960[19]with subsump-tion[20],”and was originally proposed as a method for solving satisfiability instances.It does not perform well in practice[21],and has been proven to require expo-nential time on some instances[22,23].However,here the aim is not to solve a given instance,but rather to investigate the“flow”of the problem itself as variables are eliminated[12,24].In particular,this renormal-ization group(RG)transformation provides a natural framework for understanding a phase transitions be-tween“easy”and“hard”satisfiability problems iden-tified in[5].We present evidence that the change in the nature of the SAT-unSAT phase transition critical value K c∼2.4[5]is intimately related to whether or not the num-ber of clauses proliferates exponentially upon repeated application of the renormalization group(RG)trans-formation.Note that when K=2the clause length decreases upon renormalization,since the resolution of two2-clauses is a2-clause,so no clause gets longer, and some of the resulting clauses have a duplicate lit-eral and so get shorter.Having a large number of2-clauses limits the growth in the number of long clauses because of subsumption,so there is a qualitative dif-ference in the behavior depending on whether the ratio of the number of2-clauses to the number of variables grows or shrinks upon renormalization.We show numerical data for an RG implementa-tion in which successive variables are chosen randomly and eliminated if they occur in a clause of minimum length.This procedure is used because it focuses on short clauses,which are much more restrictive than long clauses.Figure1showsαK,the ratio of M K, the number of clauses of length K to N,the number of variables remaining in the problem,as a function of K,as the RG proceeds.The average and standard deviation of numerical data from5realizations at the SAT-unSAT transition with p=0.2and p=0.6are shown(using parameter values for the transition loca-tions from[5]).Large numbers of long clauses are gen-erated when p=0.6>p c and not when p=0.2<p c.It has been proven that the SAT-unSAT transition for 2-SAT occurs atα=1[25],and for2≤K<2.4,the SAT-unSAT transition is believed to occur whenα2= 1[5].Figure2(left)shows that when K=2.2,α2 increases upon renormalization whenα2>1and de-creases upon renormalization whenα2<1.Because adding additional three-clauses does not affect the be-havior of the two-clauses,and because two-clauses are much more restrictive than longer clauses,the two-clauses dominate the problem wheneverα2>1.One would expect that3-clauses by themselves would pro-liferate when N initial,the initial number of variables, was equal to3M3/2.(This estimate,the analog of the result for two-clauses,follows from setting the number of literals in all3-clauses to twice the number of vari-ables,which means that on average each variable enters3FIG.1:Plot ofαK,the ratio of M K,the number of clauses of length K,to N,the number of variables,at different stages of the renormalization process.The points plotted are the mean and standard deviation of the results fromfive indepen-dent system realizations.The parameters are chosen to be at the SAT-unSAT transition with p=0.2<p c(left panel) and p=0.6>p c(right panel).When p>p c the clause length increases markedly and the number of clauses grows enormously.into one3-clause negated and one3-clause un-negated. Moreover,eliminating one variable on average yields two less literals and one less variable,so that,ignoring fluctuations,the relationship remains true.)However, because the2-clauses prevent the3-clauses from prolif-erating,adding3-clauses to the2-clauses changes the nature of the SAT-unSAT transition only when enough 3-clauses have been added so that the SAT-unSAT tran-sition occurs withα2<1.In this regime,under renor-malization the2-clauses disappear and so the clauses all become longer.Since very long clauses are ORs of many literals and hence easy to satisfy,the number of clauses must go up sufficiently fast for the problem to be difficult to solve—at large K,the SAT-unSAT tran-sition occurs when the ratio of the number of clauses to the number of variables is∝2K[11].When K>K c, near the SAT-unSAT transition we expect the number of clauses to grow geometrically with iteration number, and the numerical data are consistent with the maxi-mum number of clauses obtained during the renormal-ization process increasing exponentially with the initial number of variables,with an exponent that increases monotonically with(M/N)initial,the initial ratio of the number of clauses to the number of variables.Fig-ure2(right)shows a phase boundary lines for the SAT-FIG.2:Left:plot ofα2,the ratio of the number of2-clauses to the number of variables,versus N,the number of undeci-mated variables,for instances with p=0.2and different ini-tial values ofα.The numerical data are averages and standard deviations offive realizations of systems of size500when the initialα=1,size400when the initialα=αc=1.2, and size300when the initialα=1.67.The numerical data are consistent with the hypothesis that when K<K c the SAT-unSAT transition occurs whenα2neither decreases nor increases upon renormalization.Right:Schematic phase diagram showing the SAT-unSAT transition(using data of Ref.[5]),the region in which the number of2-clauses in-creases upon renormalization(the red hatched region in the left of thefigure)and an estimate of the region in which the number of clauses with K≥3increases upon renormaliza-tion(the blue hatched region in the right of thefigure).The SAT-unSAT transition line crosses into the region in which long clauses proliferate exponentially at the intersection of the three lines.The estimate for3-clause proliferation given in the text yields an intersection at(K=2.4,α=1.25). unSAT transition,for the onset of increase in the num-ber of2-clauses,and our estimate for the onset of pro-liferation of clauses of length greater than or equal to three.When K>K c,the SAT-unSAT transition[5,6,7, 8,9,10,26]occurs in a regime in which the renormal-ization transformation causes both the typical clause length and the total number of clauses to grow.We conjecture that the SAT-UNSAT transition occurs at the value of M/N at which the rate of exponential growth is a critical value.A“replica-symmetry break-ing”transition[7,9,10,26]at a somewhat smaller value ofαcan be interpreted in terms of propagation of constraints on eliminated literals,as will be discussed elsewhere.[27]Because of the exponential clause pro-4liferation,numerical investigation of these transitions using this renormalization group is limited to small sizes.However,the renormalization group may still be useful for investigating these transitions using an-alytic techniques appropriate for large K[10,11]for any K>K c,though it will be necessary to understand how to account for possible RG-induced correlations between clauses.When K>K c,the number of clauses continues to increase under renormalization until it is no longer un-likely that a given compound clause contains a repeated variable(in addition to the decimated one),which we expect to occur when the renormalized clause length is of order√N,where the problem specifica-tion itself is exponentially large in N,it appears that the renormalization group procedure can transform the problem out of NP and even PSPACE altogether.This property may indicate that whether or not a given com-putational problem has a solution that can be verified using polynomially-bounded resources has no funda-mental effect on the difficulty of solving the problem. 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