Counting open negatively curved manifolds up to tangential homotopy equivalence
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python三次指数平滑法三次指数平滑法是一种时间序列预测方法,它是基于过去的三个时期的实际数据值和平滑值来预测未来的值。
这种方法比二次指数平滑法更加精确,但计算也更复杂。
下面是一个使用Python实现三次指数平滑法的简单示例。
这个例子使用了一个简化的模型,其中没有考虑季节性因素。
在实际应用中,你可能需要根据具体情况进行调整。
```pythondef triple_exponential_smoothing(data, initial_values=(alpha, beta, gamma)):# 初始平滑值actual = [data[0]]double_smooth = [data[0]]triple_smooth = [data[0]]# 第一次平滑alpha, beta, gamma = initial_valuesfor i in range(1, len(data)):actual.append((1 - alpha) * data[i] + alpha *actual[i-1])double_smooth.append((1 - beta) * actual[i] + beta *double_smooth[i-1])triple_smooth.append((1 - gamma) * double_smooth[i] + gamma * triple_smooth[i-1])# 返回预测值return triple_smooth# 示例数据data = [130, 132, 134, 136, 138, 140, 142, 144, 146, 148]initial_values = [0.5, 0.3, 0.2] # 可以根据实际情况调整这些初始值# 使用三次指数平滑法进行预测predictions = triple_exponential_smoothing(data, initial_values) print(predictions)```请注意,你需要根据实际情况调整`initial_values`参数,以及可能还需要处理季节性因素。
文章编号:1000-5404(2006)04-0311-04论著压电凝血反应频率指数衰减规律及最大曲率终点法的研究赵猛,府伟灵,陈鸣,夏涵,罗阳,赵渝徽,王丰(第三军医大学西南医院检验科,重庆400038)提要:目的建立压电凝血反应曲线拟合模型及最大曲率终点法判断反应终点。
方法用压电石英传感器检测血浆凝血酶原时间,比较指数衰减、幂函数及多项式3种拟合模型的6种曲线的拟合效果,并采用最大曲率终点法判断反应终点。
结果6条拟合曲线的非线性回归效果均有显著意义(P<0.05);同类模型内的两条曲线残差平方和(SSE)比较没有显著性差别(P>0.05),而非同类模型之间则差异显著(P<0.01),且多项式SSE最小,指数衰减模型次之,但多项式模型因其曲率存在多个峰值而不能用最大曲率法判断终点;最大曲率终点法比传统的终点频率峰法重复性更好(P<0.05)。
结论指数衰减模型y=A+B×exp(-x/C)能很好地拟合压电凝血反应曲线;最大曲率终点法能显著提高检测重复性。
关键词:生物传感器;终点判断;曲线拟合;曲率;压电凝血反应中图法分类号:R311;R319;R446.11文献标识码:AExponential decay regularity of frequency and endpoint determination by maximal-curvature in piezoelectric coagulationZHAO Meng,FU Wei-ling,CHEN Ming,XIA Han,LUO Yang,ZHAO Yu-hui,WANG Feng(Department of Clinical La-boratory Diagnosis,Southwest Hospital,Third Military Medical University,Chongqing400038,China)Abstract:Objective To establish a curve fitting model and an endpoint determination by maximal-curva-ture for the reaction curve of piezoelectric coagulation analysis.Methods A prothrombin time test in plasma was detected by piezoelectric crystal biosensor,fitting effects of six kinds of curves were compared respectively according to three fitting models including exponential decay,power and polynomial category,and the endpoint of the reaction was determined by the maximal curvature.ResuIts Non-linear regression effects of all six curves were perfect(P<0.05).The error sum of squares(SSE)among different categories had great deviation significance(P<0.01)while they showed no difference between two curves in the same category(P>0.10).Polynomial model was the best fitting curve for the piezoelectric coagulation reaction because of its minimal SSE,following by exponential decay.However,the maximal-curvature of polynomial model could not be calcu-lated for its multi-peak values.The precision of maximal-curvature judgment was better than that of the tradi-tional final frequency-peak method(P<0.05).ConcIusion The exponential decay model[y=A+B×exp (-x/C)]can fit the piezoelectric coagulation analysis process perfectly,and it is accessible to calculate the maximal-curvature for endpoint determination to improve the precision of the detection significantly.Key words:biosensor;endpoint determination;curve fitting;curvature;piezoelectric coagulation基金项目:国家“863”计划重大专项(2002AARZ2023),国家自然科学基金资助面上项目(30400107,30270388)Supported by the National“863”Key Research Project of China(2002AARZ2023)and the General Program of the National NaturalScience Foundation of China(30400107,30270388)作者简介:赵猛(1978-),男,四川省阆中市人,硕士研究生,检验师,主要从事临床免疫学及病原微生物诊断方面的研究。
慢特征分析名词解释
慢特征分析(ManifoldFeatureAnalysis,简称MFA)是一种数据挖掘算法,可以从大量包含复杂关系的高维数据中,寻找出低维潜在特征。
该算法专注于对潜在结构进行建模,从而有助于数据分析人员更有效地发现他们感兴趣的关系。
MFA的模型主要包括数据映射,维度压缩和结构学习三个主模型。
首先,MFA从输入数据中提取局部特征,将高维数据映射到一个较低维的特征空间中。
这时,数据的潜在特征更容易显现出来。
根据这些局部特征,MFA将输入特征压缩到较小的空间中,从而节省数据分析时间。
最后,MFA采用结构学习方法,将局部特征组合成潜在的全局表达形式,从而发现数据中的潜在结构。
MFA可以用于多种应用,例如可视化复杂关系,聚类,特征选择,机器学习,推荐系统,网络分析等。
它也可以被用于多种类型的数据,包括文本,图像,音频,视频等。
MFA的优点主要有三:首先,它可以将非线性关系投射到较低维空间中,从而有助于数据分析;其次,MFA涉及到数学计算比较快,大大缩短了模型构建时间;最后,它有助于发现数据中的隐藏相关性,这有助于改善机器学习的效果。
总而言之,MFA是一种有效的数据挖掘算法,可以有效地发现数据中的潜在特征,帮助数据分析人员进行更有效的数据分析。
它的主要模型包括数据映射,维度压缩和结构学习,可以用于多种应用,有助于改善机器学习的效果。
2024届山东中学联盟高三下学期5月预测热身卷英语试题+详细解析注意事项:1. 答卷前,考生务必将自己的姓名、考生号等填写在答题卡和试卷指定位置。
2. 选择题的作答:选出每小题答案后,用2B铅笔把答题卡上对应题目的答案标号涂黑。
如需改动,用橡皮擦干净后,再选涂其他答案标号。
回答非选择题时,将答案写在答题卡上。
3. 考试结束后,将本试卷和答题卡一并交回。
第一部分:阅读理解(共两节,满分50分)第一节(共15小题;每小题2.5分,满分37.5分)阅读下列短文,从每题所给的四个选项(A、B、C和D)中,选出最佳选项。
ASmall Ways You Can Donate Money To CharityThere are plenty of innovative ways that you can help people in need, even when money is tight. Here are just a few unique ways to give.Food Angel, Hong KongFood insecurity has become a global problem for families. In Hong Kong, the people behind the Food Angel program collect 45 tonnes of edible surplus food each week that grocery stores, restaurants and individuals would otherwise dispose of. That includes fresh fruits and vegetables and other perishables (易腐烂的食物) that aren’t normally accepted in food-donation boxes.The impact is significant: Volunteers make and serve around 20,000 meals and distribute more than 11,000 other meals and food packs every day.Frigos Solidaires, FranceImagine if those in need could help themselves to food with anonymity (匿名) and dignity. Frigos Solidaires, or Solidarity Fridges, was started with that aim by Dounia Mebtoul, a young restaurateur in Paris. Now, 130 fridges installed in front of places such as shops and schools offer free food to the hungry across France.Stuff A Bus, CanadaIn Edmonton, the transit service parks vehicles in front of supermarkets for its annual “Stuff a Bus” campaign each November. Volunteers collect food and cash donations from shoppers to fill buses bound for food banks. Since its start in 1995, the campaign has collected 553,000 kilograms of food and roughly half a million dollars.Rice Bucket Challenge, IndiaHeard of the Ice Bucket Challenge? You take a video of yourself dumping a bucket of ice water over your head, then nominate (指定) three more people to do the same. In some versions, the participant donates $100 if they don’t complete the challenge.“I thought it was an amazing way to raise awareness of ALS and raise funds,” recall s Manju Kalanidhi, a journalist in Hyderabad, India. But it didn’t make sense in her country, where water is too precious to waste, even for a good cause. Then in 2014, it hit her: Why not make it a Rice Bucket Challenge to fight hunger? “I gave a bucket o f rice to someone in need and clicked a photo. I shared it on Facebook and said, ‘This is a Rice Bucket Challenge.Why don’t you do it, too?’” Participants donate a bucket of rice to an individual or family —no, it’s not dumped — take a photo and post it on social media with a message encouraging others to do the same.1. Which one can help people in need get food without hurting their pride?A. Food Angel, Hong KongB. Frigos Solidaires, FranceC. Stuff A Bus, CanadaD. Rice Bucket Challenge, India2. What do you know about Rice Bucket Challenge in India?A. It is an amazing way to raise awareness of ALS.B. It was inspired by the Ice Bucket Challenge.C. A bucket of rice is given and dumped.D. A bucket of water is donated for a good cause.3. What’s the p urpose of the text?A. To explain how important to help people in need.B. To inspire readers to start a non-profit organization.C. To introduce some creative ways to give away.D. To appeal to readers to donate money to charity.【答案】1. B 2. B 3. C【解析】【导语】本文是一篇说明文。
二维ising模型蒙特卡洛算法
以下是二维 Ising 模型的蒙特卡洛算法的详细步骤:
1.初始化:生成一个二维自旋阵列,可以随机初始化每个自
旋的取值为+1或-1。
2.定义参数:设置模拟步数(或称为Monte Carlo 步数,MC
steps)、温度(T)、外部磁场(H)和相互作用强度(J)。
3.进行蒙特卡洛模拟循环:
o对于每个 MC 步:
▪对每个自旋位置(i,j)进行以下操作:
▪随机选择一个自旋(i,j)和其相邻的自
旋。
▪计算自旋翻转后的能量差ΔE。
▪如果ΔE 小于等于0,接受翻转,将自旋
翻转。
▪如果ΔE 大于0,根据Metropolis 准则以
概率 exp(-ΔE / T) 决定是否接受翻转。
o每个 MC 步结束后,记录自旋阵列的属性(例如平均磁化、能量等)。
o可以选择在一些 MC 步之后检查系统是否达到平衡状态。
如果需要,可以进行更多的 MC 步。
4.分析结果:使用模拟的自旋阵列进行统计和计算,例如计
算平均自旋、能量、磁化、磁化率、热容等。
这是基本的二维Ising 模型的蒙特卡洛算法步骤。
在实施算法时,还可以根据需要考虑边界条件(如周期性边界条件)、优化算法以提高效率等其他因素。
J. Chem. Chem. Eng. 5 (2011) 1-6.Development and Validation of a LiquidChromatography–Tandem Mass Spectrometry Method for Determination of Artemisinin in Rat PlasmaElhassan Gamal1,2, Yuen Kah1, Wong Jiawoei1, Chitneni Mallikarjun1,3, Al-Dahli Samer1, Khan Jiyauddin1 and Javed Qureshi31. School of Pharmaceutical Sciences, Universiti Sains Malaysia, Minden 11800, Penang, Malaysia2. Local Pharmaceutical Manufacturing Department, General Pharmacy Directorate, MOH, 11111, Khartoum-Sudan3. School of Pharmacy and Health Sciences, International Medical University, 5700, Kula Lumpur, MalaysiaReceived: September 03, 2010 / Accepted: October 11, 2010 / Published: January 10, 2011.Abstract: Artemisinin is a potent anti-malarial drug isolated from traditional Chinese medicinal herb, Artemisia annua. The objective of this study was to develop and validate a sensitive and specific LC-MS/MS method for the determination of artemisinin in rat plasma using amlodipine as Internal Standard. The method consist of a simple liquid-liquid extraction with methyl tertiary butyl ether (MTBE) with subsequent evaporation of the supernatant to dryness followed by the analysis of the reconstituted sample by LC-MS/MS with a Z-spray atmospheric pressure ionization (API) interface in the positive ion-multiple reaction monitoring mode to monitor precursor→product ions of m/z 282.70→m/z 209.0 for artemisinin and m/z 408.9→m/z 237.0 for amlodipine respectively. The method was linear (0.999) over the concentration range of 7.8–2000 ng/mL in rat plasma. The intra and inter-day accuracy were measured to be within 94-104.2% and precision (CV) were all less than 5%. The extraction recovery means for internal standard and all the artemisinin concentrations used were between 82-85%.Key words: Artemisinin, LC-MS/MS, amlodipine, plasma, accuracy and precision.1. IntroductionArtemsinin is the name given to the active principle of qinghaosu, an extract of the Chinese medicinal plant qinghaosu or green Artemisia (Artemisinin annua L.) which has been used for many years centuries in Chinese traditional medicine for treatment of fever and malaria [1]. In 1972, Chinese researchers isolated artemisinin from Artemisia annua L. sweet wormwood) and its structure was elucidate in 1979 as show in Fig. 1.The determination of artemisinin and its derivatives in biological matrices have previously been characterized using several analytical techniques suchCorresponding author: Gamal Osman Elhassan Ph.D., research field: pharmaceutical technology. E-mail: ******************.as LC, HPLC, GC-MS etc [3-8]. However, some of these methods suffer from few drawbacks. In particulars, interference with endogenous constituents in the plasma at the absorption wave length of the derivatized compounds may render these techniques unsatisfactory and few of them lacked the required sensitivity to be used for measurement of drugFig. 1 The chemical structure of artemisinin [2].ll Rights Reserved.Development and Validation of a Liquid Chromatography–Tandem Mass Spectrometry Method forDetermination of Artemisinin in Rat Plasma2concentration in blood sample obtained from clinical investigation [9].To increase the specificity and sensitivity of HPLC-UV method, some workers combined it with a mass spectrometry (MS) and the total system is described as LC-MS technique [10, 11]. The development of LC-tandem mass spectrometry (LC-MS/MS) has made a more specific and sensitive analysis of artemisinin and its derivatives possible [12, 13]. The objective of this study was to develop a sensitive and specific LC-MS/MS method for the determination of artemisinin in rat plasma by simple liquid-liquid extraction procedure.2. Materials and Methods2.1 MaterialsArtemisinin was purchased from Kunming Pharmaceutical Corporation (Kunming, China). Amlodipine was obtained from Sigma Chemical (Louis, USA). Acetonitrile (ACN), formic acid and methyl tertiary butyl ether (MTBE) were purchased from J.T Baker (USA).3. Methods3.1 Instrumentation and ConditionsThe instrumentation comprised of Quattro-micro tandem mass spectrometer with Z-spray atomospheric pressure ionization (API) source (Micromass, Manchester, UK) using electrospray ionization (ESI) operated at positive mode. Chromatography was performed on an Alliance 2,695 separation module (Waters, M.A, USA). The delivery system consisted of an autosampler and a column heater. The chromatographic separation was obtained using an X Terra MS C8 encapped (5 μm) (150 × 2.1 mm) analytical column (Water, USA).3.2 Sample PreparationA 250 μL aliquot of plasma was pipetted into a screw-capped culture tube, followed by 100 μL of internal standard solution (50 ng/mL). To each tube, 5 mL (MTBE) extraction solvent was then added and the mixture was vortexed for 2.5 minutes followed by centrifuging for 15 minutes at 3,500 rpm. The upper layer was transferred to a reactive vial and dried under nitrogen flow at 40 °C. The residue was then reconstituted with 250 μL of mobile phase and 20 μL was injected into the LC-MS/MS system.3.3 Assay ValidationCalibration curve at a concentration range of 7.8–2,000 ng/mL were constructed by spiking blank human plasma with a known amount of artemisinin. Plasma sample spiked with artemisinin at these concentrations 7.8, 62.5, 250, 2,000 ng/mL were used to determine the within and between-day accuracy and precision. For within-day accuracy and precision, replicates analysis (n = 6) for each concentration were performed in a single day. For between-day evaluation, analysis was carried out with a single sample of each concentration daily over 6 days, with calibration curve constructed on each day of analysis. The extraction recovery of artemisinin was estimated by comparing the peak height obtained after extraction of the samples from plasma with that of aqueous artemisinin solution of the corresponding concentration.4. Results and DiscussionBoth electrospray (TIS) and atmospheric pressure chemical ionisation (APCI) methods have been reported previously for the quantification of artemisinin derivatives in biological fluids [11, 12, 14-16]. According to the previously reported methods TIS was found to be superior to APCI for the quantification of artesunate and dihydroartemisinin (DHA) mainly because of improved linearity [16]. Therefore in this method electrospray ionization was used. When artemisinin and amlodipine were injected directly into the mass spectrometer along with mobile phase in the positive mode, the protonated molecules of artemisinin and amlodipine were set as precursorll Rights Reserved.Development and Validation of a Liquid Chromatography–Tandem Mass Spectrometry Method forDetermination of Artemisinin in Rat Plasma3(a)(b)Fig. 2 (a) Positive-ionization electrospray mass spectra of precursor ion for artemisinin; (b) Positive-ionization electrospray mass spectra of product ion for artemisinin.ions with m/z of 282.7 and 408.7, respectively. The product ion that gave the highest intensity was m/z of 209.0 for artemisinin and 237.7 for amlodipine. Fig 2(a) shows the spectra precursor ion, 2(b) production for artemisinin.Artemisinin and amlodipine have retention time of approximately 6.9 and 1.65 minutes, respectively (Fig.3). The peak was well resolved and free from interference from endogenous compounds in rat plasma (Fig. 4).ll Rights Reserved.Development and Validation of a Liquid Chromatography–Tandem Mass Spectrometry Method forDetermination of Artemisinin in Rat Plasma4Fig. 3 Plasma spiked with 500 ng/ml artemisinin and amlodipine 50 ng/mL.Fig. 4 Chromatograms for analysis of artemisinin in plasma (Rat blank plasma).Calibration curve was linear over the entire range of calibration curves with a mean correlation coefficient greater than 0.9995 (Fig. 5).The limit of quantification (LOQ) of the assay method was 7.8 ng/mL being the lowest concentration used to construct the calibration curve whereas the limit of detection (LOD) was 3.9 ng/mL at a signal to noise ratio of 3. The validation data demonstrated a good precision, accuracy and recovery. The extraction recovery means for internal standard and all artemisinin concentrations used were 75-85% (Table 1). The within-day and between-day accuracy and precision values are given in Table 2.Neither artemisinin nor the internal standard producedll Rights Reserved.Development and Validation of a Liquid Chromatography–Tandem Mass Spectrometry Method forDetermination of Artemisinin in Rat Plasma5Fig. 5 Mean calibration curve of artemisinin (ng/mL).Table 1 Extraction recovery.Concentration (ng/mL) Mean recovery (%) CV (%)7.81 75.081.5062.50 82.161.94250.00 82.03 2.072000.00 85.23 1.48Table 2 Within-day and between-day precision andaccuracy.Added (ng/mL)Within-day Between-day Accuracy (%) C.V (%) Accuracy (%) C.V (%)7.81 96.00 4.60 104.11 2.30 62.50 98.10 1.60 94.10 2.20 250.00 98.10 1.50 98.10 1.60 2000.00 96.10 2.50 97.10 1.80any detectable carry-over after three injections of upper limit of quantification. Blank rat plasma showed no interference with artemisinin. Interfering signals from blank plasma contributed less than 20% of the artemisinin signal at LOQ. There was no interference of artemisinin on the internal standard or vice versa. A small enhancement for artemisinin and the internal standard could be detected when references in neat injection solvent were compared with references in extracted blank biological matrix. The normalized matrix effects (artemisinin/internal standard) were close to 1 with a low variation in accordance with international guidelines. Post-column infusion experiments confirmed the absence of regions with severe matrix effects (i.e., no sharp drops or increases in the response) for blank human plasma extracted with the developed method.Xing et al. used artmether as an internal standard for the analysis of artemisinin [17]while for the analysis of artemisinin derivatives; artemisinin was used as internal standard [14]. In the present study amlodipine was found to be suitable because it could be separated chromatographically, ionized and fragmented under the conditions that optimized the intensity of artemisinin peak (Fig. 3).The analysis of artemisinin and its derivatives with mass spectrometry are most often performed with a different mode of ionization. Xing et al. used ESI inletin the positive ion-multiple reaction monitoring mode which relatively producing a higher sensitivity than in the SIM mode. Therefore, the mass spectrometry was operated at positive ion-MRM mode.4. ConclusionThe LC-MS/MS method described in this work is suitable for the determination of artemisinin in plasma. The assay procedure is simple with a relatively shortll Rights Reserved.Development and Validation of a Liquid Chromatography–Tandem Mass Spectrometry Method forDetermination of Artemisinin in Rat Plasma6retention time allowing sufficient sample to beprocessed to be applied to pharmacokinetic and bioavailability studies of artemisinin. The accuracy and precision of the assay method, as well as the recovery of extraction procedure were found to be satisfactory.References[1] D.L. Klayman, Qinghasou (Artemisinin): An antimalaria drug from China, Science 228 (1985) 1049-1055.[2] X.D. Luo, C.C. Shen, The chemistry, pharmacology andclinical applications of Qinghaosu (artemisinin) and it’sderivatives, Med. Res. Rev. 7 (1987) 29-52.[3] K.T. Batty, M. Ashton, K.F. Llett, G . Edwards, T.M. Davis,Selective high-performance liquid chromatography ofartesunate and α-and β-dihydroartemisinin in patients withfalciparum malaria, J. Chromatog. B 677 (2-3) (1996)345-350.[4] J. Karbwang, K. Na-Bangchang, P. Molunto, V . Banmairuroi, Determination of artemisinin and its majormetabolite, dihydroartemisinin, in plasma usinghigh-performance liquid chromatography withelectrochemical detector, J. Chromatog. B 7 (1-2) (1997)259-265.[5] K.L. Chan, K.H. Yuen, H. Takayanki, S. Jinandasa, K.K. Peh, Polymorphism of artemisinin from Artemisia annua,Phytochemistry 46 (7) (1997) 1209-1214.[6] G .Q. Li, T.O. Peggins, L.L. Fleckenstein, K. Masonic,M.H. Heiffles, T.G . Brewer, The pharmacokinetics andbiovailability of dihydroartemisinin, arteether, artemether,artesunic acid and artelinic acid in rats, J. Pharm.Pharmacol 5 (1998) 173-182.[7] B.A. Avery, K.K. Venkatesh, M.A. Avery, Rapid determination of artemisinin and related analogues usinghigh-perfomance liquid chromatography and anevaporative light scattering detector, J. Chromat. B 730 (1)(1999) 71-80.[8] S.S. Mohamed, S.A. Khalid, S.A. Ward, T.S.M. Wan,H.P.O. Tang, M. Zheng, R.K. Haynes, G . Edwards,Simultaneous determination of artemether and its majormetabolite dihydroartemisinin in plasma by gaschromatography-mass spectrometry-selected ionmonitoring, J. Chromat. B 731(1999) 251-260.[9] K.T. Batty, M. Ashton, K.F. Llett, G . Edward, T.M. Davis,The pharmacokinetics of artemisinin (ART) and artesunate (ARTS) in healthy volunteers, Am J. Trop Med. Hyg. 58(2) (1998) 125-126.[10] C. Souppart, N. Gouducheau, N. Sandenan, F. Richard,Development and validation of a high-performance liquid chromatography-mass spectrometry assay for the determination of artemisinin and its metabolite dihydraartemisinin in human plasma, J. Chromat. B 774(2002) 195-203.[11] H. Naik, D.J. Murry, L.E. Kirsch, L. Fleckenstein,Development and validation of high-performance liquid chromatography-mass spectroscopy assay for determination of artesunate and dihydrroartemisinin in human plasma, J. Chromat. B 816 (1-2) (2005) 233-242. [12] J. Xing, H. Yan, S. Zhang, G . Ren, Y . Gao, A high-performance liquid chromatography/tandem mass spectrometry method for the determination of artemisinin in rat plasma, Rapid Commun in Mass Spectro. 20 (9) (2006) 1463-1468. [13] J. Xing, H.X. Yan, R.L. Wang, L.F. Zhang, S.Q. Zhang,Liquid chromatography-tandem mass spectrometry assay for the quantitation of β-dihydroartemisinin in rat plasma, J. Chromat. B 852 (1-2) (2007) 202-207. [14] M. Rajanikanth, K.P. Madhusudanan, R.C. Gupta, An HPLC-MS method for simultaneous estimation of alpha, beta-arteether and its metabolite dihydroartemisinin, in rat plasma for application to pharmacokinetic study, J Biomed. Chromat. 17 (7) (2003) 440-446. [15] Y . Gu, Q. Li, M.V . Elendez, P. Weina, Comparison of HPLC with electrochemical detection and LC–MS/for the separation and validation of artesunate and dihydroartemisinin in animal and human plasma, J. Chromatogr B 867 (2008) 213-218. [16] W. Hanpithakpong, B. Kamanikom, A.M. Dondorp, P.Singhasivanon, N.J. White, N.P. Day, N. Lindegardh, A liquid chromatographic-tandem mass spectrometric method for determination of artesunate and its metabolite dihydroartemisinin in human plasma, J. Chromatogr. B 876 (2008) 61-68. [17] Y . Xing, H. Yan, S. Zhang, G . Ren, Y . Gao, A high-performance liquid chromatography/tandem mass spectrometry method for the determination of artemisinin rat plasma, Rapid Communication in Mass Spectrometry 20 (9) (2006) 1463-1468.ll Rights Reserved.。
PDE for Finance,Spring2003–Homework4Distributed3/10/03,due3/31/03.Problems1–4concern deterministic optimal control(Section4material);problems5–7 concern stochastic control(Section5material).Warning:this problem set is longer than usual(mainly because Problems1–4,though not especially difficult,are fairly laborious.)1)Consider thefinite-horizon utility maximization problem with discount rateρ.The dynamical law is thusdy/ds=f(y(s),α(s)),y(t)=x,and the optimal utility discounted to time0isu(x,t)=maxα∈ATte−ρs h(y(s),α(s))ds+e−ρT g(y(T)).It is often more convenient to consider,instead of u,the optimal utility discounted to time t;this isv(x,t)=eρt u(x,t)=maxα∈ATte−ρ(s−t)h(y(s),α(s))ds+e−ρ(T−t)g(y(T)).(a)Show(by a heuristic argument similar to those in the Section4notes)that v satisfiesv t−ρv+H(x,∇v)=0with HamiltonianH(x,p)=maxa∈A{f(x,a)·p+h(x,a)}andfinal-time datav(x,T)=g(x).(Notice that the PDE for v is autonomous,i.e.there is no explicit dependence on time.)(b)Now consider the analogous infinite-horizon problem,with the same equation of state,and value function¯v(x,t)=maxα∈A ∞te−ρ(s−t)h(y(s),α(s))ds.Show(by an elementary comparison argument)that¯v is independent of t,i.e.¯v=¯v(x) is a function of x alone.Conclude using part(a)that if¯v isfinite,it solves the stationary PDE−ρ¯v+H(x,∇¯v)=0.12)Recall Example1of the Section4notes:the state equation is dy/ds=ry−αwith y(t)=x,and the value function isu(x,t)=maxα≥0 τte−ρs h(α(s))dswith h(a)=aγfor some0<γ<1,andτ=first time when y=0if this occurs before time T T otherwise.(a)We obtained a formula for u(x,t)in the Section4notes,however our formula doesn’tmake sense whenρ−rγ=0.Find the correct formula in that case.(b)Let’s examine the infinite-horizon-limit T→∞.Following the lead of Problem1letus concentrate on v(x,t)=eρt u(x,t)=optimal utility discounted to time t.Show that¯v(x)=limT→∞v(x,t)=G∞xγifρ−rγ>0∞ifρ−rγ≤0with G∞=[(1−γ)/(ρ−rγ)]1−γ.(c)Use the stationary PDE of Problem1(b)(specialized to this example)to obtain thesame result.(d)What is the optimal consumption strategy,for the infinite-horizon version of thisproblem?3)Consider the analogue of Example1with the power-law utility replaced by the logarithm: h(a)=ln a.To avoid confusion let us write uγfor the value function obtained in the notes using h(a)=aγ,and u log for the value function obtained using h(a)=ln a.Recall that uγ(x,t)=gγ(t)xγwithgγ(t)=e−ρt1−γ1−γ1−γ.(a)Show,by a direct comparison argument,thatu log(λx,t)=u log(x,t)+1ρe−ρt(1−e−ρ(T−t))and g1is an as-yet unspecified function of t alone.2(b)Pursue the following scheme forfinding g1:Consider the utility h=1dγ(t)γ=0.(This leads to an explicit formula for g1but it’s messy;I’m not asking you to write it down.)(c)Indicate how g0and g1could alternatively have been found by solving appropriatePDE’s.(Hint:find the HJB equation associated with h(a)=ln a,and show that the ansatz u log=g0(t)ln x+g1(t)leads to differential equations that determine g0and g1.)4)Our Example1considers an investor who receives interest(at constant rate r)but no wages.Let’s consider what happens if the investor also receives wages at constant rate w. The equation of state becomesdy/ds=ry+w−αwith y(t)=x,and the value function isu(x,t)=maxα≥0 Tte−ρs h(α(s))dswith h(a)=aγfor some0<γ<1.Since the investor earns wages,we now permit y(s)<0, however we insist that thefinal-time wealth be nonnegative(y(T)≥0).(a)Which pairs(x,t)are acceptable?The strategy that maximizes y(T)is clearly toconsume nothing(α(s)=0for all t<s<T).Show this results in y(T)≥0exactly ifx+φ(t)w≥0whereφ(t)=1(d)In view of(a),a more careful definition of the value function for this control problemisu(x,t)=maxα≥0 τte−ρs h(α(s))dswhereτ=first time when y(s)+φ(s)w=0if this occurs before time T T otherwise.Use a verification argument to prove that the function v obtained in(c)is indeed the value function u defined this way.5)Our geometric Example2gave|∇u|=1in D(with u=0at∂D)as the HJB equation associated with starting at a point x in some domain D,traveling with speed at most1, and arriving at∂D as quickly as possible.Let’s consider what becomes of this problem when we introduce a little noise.The state equation becomesdy=α(s)ds+ dw,y(0)=x,whereα(s)is a(non-anticipating)control satisfying|α(s)|≤1,y takes values in R n,and each component of w is an independent Brownian motion.Letτx,αdenote the arrival time:τx,α=time when y(s)first hits∂D,which is of course random.The goal is now to minimize the expected arrival time at∂D,so the value function isu(x)=min|α(s)|≤1E y(0)=x{τx,α}.(a)Show,using an argument similar to that in the Section5notes,that u solves the PDE1−|∇u|+122u xx=0for−1<x<1u=0at x=±1.(a)Assuming that the solution u is unique,show it satisfies u(x)=u(−x).Conclude thatu x=0and u xx<0at x=0.Thus u has a maximum at x=0.4(b)Notice that v=u x solves1−|v|+δv x=0withδ=1。
a rX iv:mat h /98914v1[mat h.DG ]3Sep1998COUNTING OPEN NEGATIVELY CUR VED MANIFOLDS UP TO TANGENTIAL HOMOTOPY EQUIV ALENCE Igor Belegradek May 14,1998Abstract.Under mild assumptions on a group π,we prove that the class of com-plete Riemannian n –manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to πbreaks into finitely many tan-gential homotopy types.It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.§1.Introduction This paper is an attempt to understand the topology of complete infinite vol-ume Riemannian manifolds of negative sectional curvature.Every smooth open manifold admits a (possibly incomplete)Riemannian metric of negative sectional curvature [23].However,the universal cover of a complete negatively curved mani-fold is diffeomorphic to the Euclidean space,hence the homotopy type of a complete negatively curved manifold is determined by its fundamental group.For closed (or,more generally,finite volume complete)negatively curved mani-folds,the fundamental group seems to encode most of the topological information.By contrast,complete negatively curved manifolds of infinite volume with isomor-phic fundamental groups may be very different topologically.For example,the total space of any vector bundle over a closed negatively curved manifold admits a complete Riemannian metric of sectional curvature pinched between two negative constants [1].Given a group π,a positive integer n ,and real numbers a ≤b <0,consider the class M a,b,π,n of n –manifolds with fundamental groups isomorphic to πthat can be given complete Riemannian metrics of sectional curvatures within [a,b ].In this paper we discuss under what assumptions on πthe class M a,b,π,n breaksinto finitely many tangential homotopy types.Recall that a homotopy equivalence of smooth manifolds of the same dimension f :N →L is called tangential if the vector bundles f ∗T L and T N are stably isomorphic.For example,any map that is homotopic to a diffeomorphism is a tangential homotopy equivalence.Note that,unless N is a closed manifold,the bundles f ∗T L and T N are isomorphic iffthey are stably isomorphic.2IGOR BELEGRADEKIf the cohomological dimension ofπis equal to n,every manifold from the class M a,b,π,n is closed.In that case,in fact,M a,b,π,n falls intofinitely many diffeomor-phism classes[5].Here is a way to produce infinitely many homotopy equivalent negatively curved manifolds of the same dimension such that no two of them are tangentially homo-topy equivalent.Let M be a closed negatively curved manifold with H4m(M,Q)=0 for some m>0.(Such examples abound.For instance,any closed complex hy-perbolic or quaternion hyperbolic manifold is such.Most arithmetic closed real hyperbolic orientable manifolds have nonzero Betti numbers in all dimensions[30].) Then,by an elementary K-theoretic argument,for any k≥dim(M)there exists a infinite sequence of rank k vector bundles over M such that no two of them have tangentially homotopy equivalent total spaces.Yet,thanks to a theorem of An-derson[1],the total space of any vector bundle over M can be given a negatively curved metric.The following is a simplified form of our main result.Theorem 1.1.Letπbe the fundamental group of afinite aspherical complex. Suppose thatπis not virtually nilpotent and thatπdoes not split as a nontrivial amalgamated product or an HNN-extension over a virtually nilpotent group.Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks intofinitely many tangential homotopy types.Furthermore,by an easy argument we can generalize the theorem1.1to certain amalgamated products and HNN-extensions.For example,suppose thatπis the fundamental group of afinite graph of groups such that the edge groups are virtually nilpotent groups of cohomological dimension≤2.(Note that if M a,b,π,n=∅,then any nontrivial virtually nilpotent subgroup ofπof cohomological dimension≤2is isomorphic to Z,Z×Z,or the fundamental group of the Klein bottle.)Fix n anda≤b<0and suppose that,for each vertex groupπv,the class M a,b,πv,n breaksintofinitely many tangential homotopy types.Then so does the class M a,b,π,n.Applying a powerful accessibility result of Delzant and Potyagailo[16],we deduce the following.Corollary1.2.Letπbe the fundamental group of afinite aspherical complex. Assume that any nilpotent subgroup ofπhas cohomological dimension≤2.Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks intofinitely many tangential homotopy types.Any torsion free word-hyperbolic group is the fundamental group of afinite aspherical cell complex[15,5.24].Moreover,any virtually nilpotent subgroup ofπis either trivial or infinite cyclic.Corollary1.3.Letπbe a word-hyperbolic group.Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks intofinitely many tangential homotopy types.It is worth mentioning that in the locally symmetric case a different(and ele-mentary)argument yields the following.Theorem1.4.Letπbe afinitely presented torsion–free group and let X be a nonpositively curved symmetric space.COUNTING NEGATIVELY CURVED MANIFOLDS3 Then the class of manifolds of the form X/ρ(π),whereρ∈Hom(π,Isom(X)) is a faithful discrete representation,falls intofinitely many tangential homotopy types.Gromov posed a question(see[1])whether an(open)thickening of afinite as-pherical cell complex with word-hyperbolic fundamental group admits a complete negatively curved metric.The following corollary shows that most thickenings do not carry complete negatively curved metric with prescribed curvature bounds. Corollary1.5.Let K be afinite,connected,aspherical cell complex.Let a≤b<0and n>max{4,2dim(K)}.Suppose that the class M a,b,π1(K),n breaks intofinitelymany tangential homotopy types.Then the set of diffeomorphism classes of open thickenings of K that belong to the class M a,b,π1(K),nisfinite.The corollary follows from the fact that any two tangentially homotopy equiv-alent thickenings of sufficiently high dimension are diffeomorphic[29,pp.226–228]. In general,the set of diffeomorphism classes of open n-dimensional thickenings of K is infinite provided n>max{4,2dim(K)}and⊕k H4k(K,Q)=0.Thus,for such n and K,most open n-dimensional thickenings of K do not belong to the class M a,b,π1(K),n.Combining1.3and1.5,we deduce the following.Corollary1.6.Let M be a smooth aspherical manifold such thatπ1(M)is word-hyperbolic.Let a≤b<0and n>2dim(M).Then the set of isomorphism classes of vector bundles over M whose total spaces belong to M a,b,π1(K),nisfinite.Actually,we show in[6]that,in case dim(M)≥3,the assumption n>2dim(M) of the corollary1.6is redundant.Synopsis of the paper.In the section2we prove main lemmas from the global Riemannian geometry.The3rd section provides some K-theoretic background.In the4th section we define some invariant of actions.The5th section is devoted to our main results including the theorem1.1.In the section6we employ some group theory to deduce the corollary1.2.The section7deals with applications to thickenings.Applications to convex-cocompact groups are deduced in the section 8.The theorem1.4is proved in the9th section.Acknowledgments.I am grateful to Werner Ballmann,Mladen Bestvina,Thomas Delzant,Martin J.Dunwoody,Lowell E.Jones,Misha Kapovich,Vitali Kapovitch, Bernhard Leeb,John lson,Igor Mineyev,James A.Schafer,Jonathan M.Rosen-berg and Shmuel Weinberger for helpful discussions and communications.Special thanks are due to my advisor Bill Goldman for his constant interest and support.§2.Two types of convergence.By an action of an abstract groupπon a space X we mean a group homomor-phismρ:π→Homeo(X).An actionρis free ifρ(γ)(x)=x for all x∈X and all γ∈π\{id}.In particular,ifρis a free action,thenρis injective.2.1.Equivariant pointed Lipschitz topology.LetΓk be a discrete subgroup of the isometry group of a complete Riemannian manifold X k and p k be a point of X k.4IGOR BELEGRADEKThe class of all such triples{(X k,p k,Γk)}can be given the so-called equivariant pointed Lipschitz topology[20];whenΓk is trivial this reduces to the usual pointed Lipschitz topology.For convenience of the reader we give here some definitions borrowed from[20].For a groupΓacting on a pointed metric space(X,p,d)the set{γ∈Γ: d(p,γ(p))<r}is denoted byΓ(r).An open ball in X of radius r with center at p is denoted by B r(p,X).For i=1,2,let(X i,p i)be a pointed complete metric space with the distance function d i and letΓi be a discrete group of isometries of X i.In addition,assume that X i is a C∞–manifold.Take anyǫ>0.Then a quadruple(f1,f2,φ1,φ2)of maps f i:B1/ǫ(p i,X i)→B1/ǫ(p3−i,X3−i) andφi:Γi(1/3ǫ)→Γ3−i is called anǫ–Lipschitz approximation between the triples (X1,p1,Γ1)and(X2,p2,Γ2)if the following seven condition hold:•f i is a diffeomorphism onto its image;•for each x i∈B1/3ǫ(p i,X i)and everyγi∈Γi(1/3ǫ),f i(γi(x i))=φi(γi)(f i(x i));•for every x i,x′i∈B1/ǫ(p i,X i),e−ǫ<d3−i(f i(x i),f i(x′i))/d i(x i,x′i)<eǫ;•f i(B1/ǫ(p i,X i))⊃B(1/ǫ)−ǫ(p3−i,X3−i)andφi(Γi(1/3ǫ))⊃Γ3−i(1/3ǫ−ǫ);•f i(B(1/ǫ)−ǫ(p i,X i))⊃B1/ǫ(p3−i,X3−i)andφi(Γi(1/3ǫ−ǫ))⊃Γ3−i(1/3ǫ);•f3−i◦f i|B(1/ǫ)−ǫ(p i,X i)=id andφ3−i◦φi|Γi(1/3ǫ−ǫ)=id;•d3−i(f i(p i),p3−i)<ǫ.We say a sequence of triples(X k,p k,Γk)converges to(X,p,Γ)in the equivariant pointed Lipschitz topology if for anyǫ>0there is k(ǫ)such that for all k>k(ǫ), there exists anǫ–Lipschitz approximation between(X k,p k,Γk)and(X,p,Γ).Notice that if all the groupsΓk are trivial,thenΓis trivial;in this case we say that that(X k,p k)converges to(X,p)in the pointed Lipschitz topology.Note that if X k is a complete Riemannian manifold for all k,then the space X is necessarily a C∞–manifold with a complete C1,α–Riemannian metric[22].Remark2.2.For those with the Kleinian groups background we note that the equivariant pointed Lipschitz topology is closely related to the so-called Chabauty topology[7][14].Since the Kleinian group theory is an important source of exam-ples,we describe the precise relation.Let X be a complete Riemannian manifold (e.g.a hyperbolic space)with the isometry group G.The set of discrete subgroups of G has the Chabauty topology induced by a natural topology on the set of closed subsets of G,namely a sequence of closed subsets S i converges to S if for any compact K⊂G,the sets S i∩K converge to S∩K in the Hausdorfftopology on K.Consider the product topology on the set of pairs(Γ,p)whereΓis a dis-crete subgroup of G and p∈X.This product topology is in fact equivalent to the equivariant pointed Lipschitz topology where(Γ,p)is thought of as a triple (X,Γ,p).2.3.Pointwise convergence topology.Suppose that,for some p k∈X k,the sequence(X k,p k)converges to(X,p)in the pointed Lipschitz topology i.e.,for anyǫ>0there is k(ǫ)such that for all k>k(ǫ)there exists anǫ–Lipschitz approximation(f k,g k)between(X k,p k)and(X,p).(Note that if each X k is a Hadamard manifold,then for any p k∈X k,the sequence(X k,p k)is precompact in the pointed Lipschitz topology because the injectivity radius of X k at p k is uniformly bounded away from zero[20].)We say that a sequence x k∈X k converges to x∈X if for someǫd(f k(x k),x)→0as k→∞COUNTING NEGATIVELY CURVED MANIFOLDS5 where d(·,·)is the distance function on X and f k comes from theǫ–Lipschitz ap-proximation(f k,g k)between(X k,p k)and(X,p).Trivial examples:if(X k,p k) converges to(X,p)in the pointed Lipschitz topology,then p k converges to p;fur-thermore,if x∈X,the sequence g k(x)converges to x.Given a sequence of isometriesγk∈Isom(X k)we say thatγk converges,if for any sequence x k∈X k that converges to x∈X,γk(x k)converges.The limiting transformationγthat takes x to the limit ofγk(x k)of X is necessarily an isometry. Furthermore,ifγk andγ′k converge toγandγ′respectively,thenγk·γ′k convergestoγ·γ′.In particular,γ−1k converges toγ−1since the identity maps id k:X k→X kconverge to id:X→X.Letρk:π→Isom(X k)be a sequence of isometric actions of a groupπon X k.We say that a sequence of actions(X k,p k,ρk)converges in the pointwise convergence topology to an action(X,p,ρ)ifρk(γ)converges toρ(γ)for everyγ∈π.The limiting mapρ:Γ→Isom(X)that takesγto the limit ofρk(γ)is necessarily a homomorphism.Ifπis generated by afinite set S,then in order to prove that ρk converges in the pointwise convergence topology it suffices to check thatρk(γ) converges,for everyγ∈S.A sequence of actions(X k,p k,ρk)is called precompact in the pointwise conver-gence topology if every subsequence of(X k,p k,ρk)has a subsequence that converges in the pointwise convergence topology.Repeating the proof in[28,4.7],it is easy to check that a sequence of isometries γk∈Isom(X k)has a converging subsequence if,for some converging sequence x k∈X k,the sequence d k(x k,γk(x k))is bounded(where d k(·,·)is the distance function on X k).Suppose thatπis a countable group and assume that for eachγ∈πthe sequence d k(p k,ρk(γ)(p k))is bounded.Then(X k,p k,ρk)is precompact in the pointwise convergence topology.(Indeed,letγ1...γn...be the list of all elements ofπ. Take any subsequenceρk,0ofρk.Pass to subsequenceρk,1ofρk,0so thatρk,1(γ1) converges.Then pass to subsequenceρk,2ofρk,1such thatρk,2(γ2)converges,etc. Thenρk,k(γn)converges for every n.)Note that ifπis generated by afinite set S,then to prove thatρk is precompact it suffices to check that d k(p k,ρk(γ)(p k))is bounded,for allγ∈S because it implies that d k(p k,ρk(γ)(p k))is bounded,for eachγ∈π.2.4.Motivating example.Let X be a complete Riemannian manifold.Consider the isometry group Isom(X)of X and letπbe a group.The space Hom(π,Isom(X)) has a natural topology(which is usually called“algebraic topology”or“pointwise convergence topology”),namelyρk is said to converge toρif,for eachγ∈π,ρk(γ) converges toρ(γ)in the Lie group Isom(X).Note that ifπisfinitely generated, this topology on Hom(π,Isom(X))coincide with the compact-open topology.Cer-tainly,for any p∈X,the constant sequence(X,p)converges to itself in pointed Lipschitz topology.Then,obviously,the sequence(X,p,ρk)converges in the point-wise convergence topology(as defined in2.3)if and only ifρk∈Hom(π,Isom(X)) converges in the algebraic topology.Lemma2.5.Letρk:π→Isom(X k)be a sequence of isometric actions of a dis-crete groupπon complete Riemannian n-manifolds X k such thatρk(π)acts freely. If the sequence(X k,p k,ρk(π))converges in the equivariant pointed Lipschitz topol-ogy to(X,Γ,p)and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology,then6IGOR BELEGRADEK(1)Γacts freely,and(2)ρ(π)⊂Γ,and(3)ker(ρ)⊂ker(ρk),for all large k.Proof.(1)Assumeγ∈Γandγ(x)=x.Chooseǫ∈(0,1/10)so that there is anǫ–approximation(f k,g k,φk,τk)of(X k,p k,ρk)and(X,p,Γ)and x∈B(p,ǫ/10). Then g k(x)=g k(γ(x))=τk(γ)(g k(x)).Sinceρk(π)acts freely,τk(γ)=id.By the same argumentτk(id)=id.Hence id=φk(τk(id))=φk(id)=φk(τk(γ))=γas desired.(2)We need to show thatρ(γ)∈Γ,for anyγ∈π.We can assumeρ(γ)=id. Chooseǫ∈(0,1/10)so that the ball B(p,1/11ǫ)containsρ(γ)(p)and consider an ǫ–approximation(f k,g k,φk,τk)of(X k,p k,ρk)and(X,p,Γ).Then for all large enough k,ρk(γ)∈B(p k,1/10ǫ).Look atτk(ρk(γ))∈Γ(1/9ǫ). Since the setΓ(1/9ǫ)isfinite,we can pass to subsequence to assume thatτk(ρk(γ)) is equal toγǫ∈Γ(1/9ǫ);thusρk(γ)=φk(γǫ).Take an arbitrary x∈B(p,1/9ǫ).Then g k(x)→x and,hence,ρk(γ)(g k(x)) converges toρ(γ)(x).Notice thatρk(γ)(g k(x))=φk(γǫ)(g k(x))→γǫ(x).So ρ(γ)(x)=γǫ(x)for any x∈B(p,1/9ǫ).Thus,for any small enoughǫ,we have foundγǫ∈Γthat is equal toρ(γ)on the ball B(p,1/9ǫ).SinceΓacts freely,γǫ=γǫ′for allǫ′≤ǫ,that is the elementγǫ∈Γis independent ofǫ.Thusρ(γ)=γǫeverywhere and henceρ(γ)∈Γ.(3)Assumeρ(γ)=id.Fix anyǫ∈(0,1/10).Take x∈B(p,ǫ/10)and consider anǫ–approximation(f k,g k,φk,τk)of(X k,p k,ρk)and(X,p,Γ).We have g k(x)→x andρk(γ)(g k(x))→ρ(γ)(x)=x.Note that d(x,φk(ρk(γ))(x))is equal tod(f k(g k(x)),f k(ρk(γ)(g k(x))))<eǫd k(g k(x),ρk(γ)(g k(x)))−−−→0.k→∞Therefore,for all large k,φk(ρk(γ))=id,becauseΓis a discrete subgroup that acts freely.Henceρk(γ)=τk(φk(ρk(γ)))=τk(id)=id as claimed.Lemma2.6.Letρk:π→Isom(X k)be a sequence of isometric actions of a dis-crete groupπon complete Riemannian n-manifolds X k such thatρk(π)acts freely. Suppose that the sequence(X k,p k,ρk(π))converges in the equivariant pointed Lip-schitz topology to(X,Γ,p)and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.Then,for anyǫ>0and for anyfinite subset S⊂π,there is k(ǫ,S)with the property that for each k>k(ǫ,S)there exists anǫ–Lipschitz approximation (f k,g k,φk,τk)between(X k,p k,ρk(π))and(X,p,Γ)such thatφk(ρk(γ))=ρ(γ) andρk(γ)=τk(ρ(γ))for everyγ∈S.Proof.Pickδ<ǫso large that the setρ(S)(p)is contained in the ball B(1/10δ,p). Choose aδ-Lipschitz approximation(f k,g k,φk,τk)between(X k,p k,ρk(π))and (X,p,Γ).Note that for everyγ∈S,the sequenceρk(γ)(g k(p))converges toρ(γ)(p).Hence the sequence f k(ρk(γ)(g k(p)))=φk(ρk(γ))(f k(g k(p)))=φk(ρk(γ))(p)converges to ρ(γ)(p)in X.SinceΓis discrete and acts freely,φk(ρk(γ))=ρ(γ)for large enough k.Moreover, this is true for anyγ∈S,because S isfinite.Alsoρk(γ)=(τk◦φk)(ρk(γ))=τk(ρ(γ)).Thus,thisδ-Lipschitz approximation has all desired properties.COUNTING NEGATIVELY CURVED MANIFOLDS7 Proposition2.7.Let X k be a sequence of Hadamard manifolds with sectional curvatures in[a,b]for a≤b<0and letπbe afinitely generated group that is not virtually nilpotent.Letρk:π→Isom(X k)be an arbitrary sequence of free and isometric actions such that(X k,p k,ρk)converges in the pointwise convergence topology.Then the sequence(X k,p k,ρk)is precompact in the equivariant pointed Lipschitz topology.Proof.Let(X,p,ρ)be the limit of(X k,p k,ρk)in the pointwise convergence topol-ogy.Choose r so that the open ball B(p,r)⊂X contains{ρ(γ1)(p),...ρ(γm)(p)} where{γ1,...γm}generateπ.Passing to subsequence,we assume that B(p k,r) contains{ρk(γ1)(p),...ρk(γm)(p)}.Show that,for every k,there exists q k∈B(p k,r)such that for anyγ∈π\{id}, we haveρk(γ)(q k)/∈B(q k,µn/2)whereµn is the Margulis constant.Suppose not. Then for some k,the whole ball B(p k,r)projects into the thin part{InjRad<µn/2}under the projectionπk:X k→X k/ρk(π).Thus the ball B(p k,r)lies in a connected component W of theπk–preimage of the thin part of X k/ρk(π).Accord-ing to[2,p111]the stabilizer of W inρk(π)is virtually nilpotent and,moreover, the stabilizer contains every elementγ∈ρk(π)withγ(W)∩W=∅.Therefore,the whole groupρk(π)stabilizes W.Henceρk(π)must be virtually nilpotent.Asρk is injective,πis virtually nilpotent.A contradiction.Thus,(X k,q k,ρk(π))is Lipschitz precompact[20]and,hence passing to subse-quence,one can assume that(X k,q k,ρk(π))converges to some(X,q,Γ).It is a general fact that follows easily from definitions that whenever(X k,q k,Γk) converges to(X,q,Γ)in the equivariant pointed Lipschitz topology and a sequence of points p k∈X k converges to p∈X,then(X k,p k,Γk)converges to(X,p,Γ)in the equivariant pointed Lipschitz topology.Corollary2.8.Let X k be a sequence of Hadamard manifolds with sectional cur-vatures in[a,b]for a≤b<0and letπbe afinitely generated group that is not virtually nilpotent.Letρk:π→Isom(X k)be an arbitrary sequence of free and isometric such that(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.Thenρis a free action,in particularρis injective.Proof.Pass to a subsequence so that(X k,p k,ρk)converges to(X,p,Γ)in the equi-variant pointed Lipschitz topology.By2.5(3),ρis injective.Furthermore,ρ(π) acts freely because it is a subgroup ofΓ.Corollary2.9.Let X be a complete Riemannian manifold with sectional curva-tures in[a,b]where a≤b<0.Assumeπis a torsion free,finitely generated group that is not virtually nilpotent.Consider a subset of faithful discrete representations S⊂Hom(π,Isom(X))that is precompact in the pointwise convergence topology. Then,(1)for any p∈X,the set{(X,p,ρ(π)):ρ∈S}is precompact in the equivariant pointed Lipschitz topology.(2)the closure of S in Hom(π,Isom(X))consists of faithful discrete representa-tions.Proof.Proposition2.7implies(1)and Corollary2.8implies(2).8IGOR BELEGRADEKProposition2.10.Assume thatπis afinitely presented discrete group,that is not virtually nilpotent and does not have a nontrivial decomposition into an amal-gamated product or an HNN-extension over a virtually nilpotent group.Letρk:π→Isom(X k)be an arbitrary sequence of free and isometric actions of πon Hadamard n-manifolds X k.Assume that the sectional curvatures of X k lie in [a,b]for a≤b<0.Then,for some p k∈X k,the sequence(X k,p k,ρk)is precompact in the pointwise convergence topology.Proof.Let S⊂πbe afinite subset that generatesπand contains{id}.For x∈X k,we denote D k(x)the diameter of the setρk(S)(x).Set D k=inf x∈Xk D k(x).Suppose D k is unbounded.Then it follows from a work of Bestvina and Paulin [8],[32],[33](cf.[27]),that there exists an action ofπon a real tree with no proper in-variant subtree and virtually nilpotent arc stabilizers.For completeness we briefly review this construction.The rescaled pointed Hadamard manifold1D k ·X k,p k,ρk).Repeat-ing an argument of Paulin[33,§4],we can pass to subsequence that converges to a triple(X∞,p∞,ρ∞).(For the definition of the convergence see[32],[33].Paulin calls it“convergence in the Gromov topology”.)The limit space X∞is a length space of curvature−∞,that is a real tree. Because of the way we rescaled,the limit space has a natural isometric actionρ∞ofπwith no globalfixed point[32],[33].Then it is a standard fact that there exists a uniqueπ–invariant subtree T of X∞that has no properπ–invariant subtree.In fact T is the union of all the axes of all hyperbolic elements inπ.Since the sectional curvatures are uniformly bounded away from zero and−∞,the Margulis lemma implies that the stabilizer of any non-degenerate segment is virtually nilpotent (cf.[32]).Note that any increasing sequence of virtually nilpotent subgroups ofπis sta-tionary.Indeed,since a virtually nilpotent group is amenable,the union U of an increasing sequence U1⊂U2⊂U3⊂...of virtually nilpotent subgroups is also an amenable group.If the fundamental group of a complete manifold of pinched neg-ative curvature is amenable,it must befinitely generated[13],[10].In particular, U isfinitely generated,hence U n=U for some n.Thus,theπ–action on the tree T is stable[9,Proposition3.2(2)].We summarize that theπ–action on T is stable,has virtually nilpotent arc stabi-lizers and no properπ–invariant subtree.Therefore,the Rips machine[9,Theorem 9.5]produces a splitting ofπover a virtually solvable group.Any amenable sub-group ofπis virtually nilpotent[13],[10],henceπsplits over a virtually nilpotent group.This is a contradiction with the assumption that D k is unbounded.Thus the sequence D k(p k)is bounded,therefore as we observed in2.3,the se-quence(X k,p k,ρk)is precompact in the pointwise convergence topology.§3.Vector bundles andKO-theory.This section provides someKO-theoretic background.All of the facts below are well-known to experts;however it is usually not easy to locate a reference.In this paper we deal with rank n real vector bundles over open smooth n–manifolds.It is well-known that any open smooth manifold N is homotopy equiv-alent to CW–complex of dimension<dim(N),hence two rank n vector bundlesCOUNTING NEGATIVELY CURVED MANIFOLDS9 over N are isomorphic iffthey are stably isomorphic[26,8.1.5].The set of stable isomorphism classes of real vector bundles over a space X forms an abelian groupKO(X).The correspondence X→KO(X)is clearly a functor.This functor is isomorphic to the functor[−,BO]on the category of connected CW-complexes of uniformly bounded dimension[26,8.4.2].We now recall the definition of theKO∗–theory.By the Bott periodicity,Ω8(BO×Z)is weakly homotopy equivalent to BO×Z[37,11.60].We can use theKO∗on spectrumΩn(BO×Z)to define a generalized(reduced)cohomology theorythe category of all pointed connectedfinite–dimensional CW-complexes and point-KO n(X,x)=[X,x;Ωn(BO×Z),∗](cf.[37, preserving maps.In other words,we set8.42]).Let F be a forgetful map from the category of pointed connected CW–complexes to the category of connected CW–complexes.We now show that the functorsKO0(−)andKO(F(−))are isomorphic on the category of pointed connected CW-complexes of uniformly bounded dimensions.Indeed,since we deal with connectedKO0(−)=[−;BO×Z),∗]is isomorphic to the func-CW-complexes,the functorKO(−)∼=[−,BO].Therefore,it remains tor[−;BO,∗].As we mentioned aboveto show that the transformation of functors[−;BO,∗]→[F(−);BO]is an iso-morphism.In other words,it suffices to check that the map[X,x;BO,∗]→[F(X,x);BO]=[X,BO]of based homotopy classes into free homotopy classes is bijective.Indeed,any map X→BO is homotopic to a basepoint preserving map[36,7.3.Lemma2]which is unique up to based homotopy because BO is aKO(F(−)).Most path-connected H-space.[36,7.3.Theorem5].Thus,KO0(−)∼=KO(−)KO0(−)andof the time we suppress the base points and treat the functorsas isomorphic.KO i(X)can be easily computed in terms of cohomology Rationally the groupof X as explained in the lemma below.Recall that the Bott periodicity impliesKO–groups of the0–sphere are given by that,after tensoring with rationals,theKO i(S0)⊗Q=0otherwise[26,15.12.3]. KO i(S0)⊗Q∼=Q if i≡0(mod4)andLemma3.1.On the category of pointed connected CW-complexes of uniformlyKO i(−)⊗Q and⊕n>0H n(−)⊗bounded dimension the contravariant functorsKO i−n(S0)⊗Q are isomorphic.KO∗(S0)⊗Q where Proof.Consider the generalized cohomology theory H∗(−)⊗H∗(−)is the ordinary reduced cohomology.Its coefficients are isomorphic to theKO∗(−)⊗Q.Since the coefficients are rational vector coefficients of the theoryspaces,[24,3.22(ii)]implies that the theories are naturally equivalent on the cate-gory of pointedfinite-dimensional CW–complexes.For any connectedfinite-dimensional complex X, H n(X)=0,for all n≤0. Clearly for n>0,the functors H n(−)and H n(−)are isomorphic.Hence we have an isomorphism of functorsKO i(−)⊗Q∼=⊕n≥0 H n(−)⊗KO i−n(S0)⊗Q.KO i−n(S0)⊗Q∼=⊕n>0H n(−)⊗10IGOR BELEGRADEKRemark3.2.In particular,we have isomorphisms of functorsKO−1(−)⊗Q∼=⊕n>0H4n−1(−,Q).KO0(−)⊗Q∼=⊕n>0H4n(−,Q)andCorollary3.3.Let N be an open smooth manifold such that the torsion subgroup KO(N)isfinite and let n≥dim(N).ofThen the set of isomorphism classes of rank n vector bundles over N is infinite if and only if H4k(N,Q)=0for some k>0.Proof.Any open manifold N is homotopy equivalent to a CW-complex of dimension <dim(N),therefore,two rank n vector bundles over N are isomorphic iffthey are stably isomorphic.The result now follows from the above remark.Lemma3.4.Let Y be a connectedfinite-dimensional CW-complex such thatπ1(Y) isfinitely generated and dim Q H4k(Y,Q)<∞for all k>0.Then there exists a finite connected subcomplex X⊂Y such that the inclusion i:X→Y induces aKO(Y)⊗Q→KO(X)⊗Q surjection on the fundamental groups and an injectionProof.It is a standard fact[19,p117]that for any homology classα∈H i(Y,Q) there exists afinite(not necessarily connected)CW-complex Xαand a continuous map fα:Xα→Y such thatα∈fα∗(H i(Xα,Q)).For every k>0,choose afinite basis in the Q–vector space H4k(Y,Q)and construct such afinite CW–complex Xαfor every basis elementα.Let X0be the the disjoint union of all these complexes over all k and all the basis elements.Since Y isfinite-dimensional withfinite4k-th Betti numbers,the CW–complex X0isfinite.Note that X0comes with a continuous map f:X0→Y that induces a surjection on rational4k-th homology,and therefore,an injection on rational4k-cohomology for all k>0.Finally,set X to be an arbitrary connectedfinite subcomplex of Y such that f(X0)⊂ing thatπ1(Y)isfinitely generated,we add to Xfinitely many1-cells of Y to make sure the inclusion i:X֒→Y induces a surjection of fundamental groups.Moreover,by construction i induces an injection on rational4k-cohomologyKO(X)⊗Q is injective.KO(Y)⊗Q→for all k>0.By the remark3.2,§4.An invariant of actions.Let K be afinite-dimensional connected CW-complex with a reference point q. Denote˜K→K the universal cover of K;choose˜q∈˜K that is mapped to q by the covering projection.Consider a pointed contractible manifold X and an arbitrary actionρ:π1(K,q)→Diffeo(X).Since X is contractible,the X-bundle˜K×ρX over K has a section.Any two sections are homotopic through sections.We now define a certain invariant of actions ofπ1(K,q)on X.Given such an actionρ,consider the vertical vector bundle˜K×ρT X over˜K×ρX and setτ(ρ) to be the pullback of the vertical bundle via an arbitrary section s:K→˜K×ρX Clearly,two actions that are conjugate in Diffeo(X)have same invariants.Any section can be lifted to aρ-equivariant continuous map˜K→˜K×X→X.Any twoρ-equivariant continuous maps˜g,˜f:˜K→X,areρ-equivariantly homotopic.Indeed,˜f and˜g descend to sections K→˜K×ρX that must be homotopic.This homotopy lifts to aρ–equivariant homotopy of˜f and˜g.。