Hecke C-algebras and semidirect products

《高级宏观经济学》教学大纲（硕士研究生） - RonaldoCarpio《高级微观经济分析》教学大纲(博士研究生)课程代码:(按本专业或方向培养方案填写)课程名称:(按本专业或方向培养方案填写)英文名称:Advanced Microeconomic Analysis课程性质:(按本专业或方向培养方案填写)学分学时:3学分，48学时授课对象:金融学院一年级博士研究生课程简介:Based on Microeconomics I (for master students), the course will discuss thecontemporary development in microeconomics. This course is also designed to develop andextend the students’ analytical and reading skills in modern microeconomics. A student who haspassed the course should be able to read typical articles in the mainline journals, understand theanalytical derivations and arguments commonly used in the literature, and know how to solve themore widely used models.先修课程:Microeconomics for master students选用教材:1、 Mas-Colell, A., M. D. Whinston, and J. Green, Microeconomic Theory. (MWG)2、 Jehle, Geoffrey A. and Philip J. Reny, Advanced Microeconomic Theory. (JR)考核方式与成绩评定:Final Exam %; Midterm Exam %; Class Participation % 主讲教师:Carpio Ronaldo、颜建晔所属院系:金融学院联系方式:******************、*******************答疑时间及地点:求索楼123,Wednesday 13:30-14:30 (Carpio)，Tuesday 15:00-17:00(颜)第一章:Consumer Theory教学目标和要求:Understand the consumer’s problem and consumer demand.教学时数:6学时教学方式:讲授准备知识:calculus教学内容:Preferences, Utility, and Consumer’s Problem第一节:Consumer’s Problem第二节:Indirect Utility, Demand作业与思考题:JR Ch 1.6参考资料:JR Ch 1, Appendix A1, A21第二章: Topics in Consumer Theory教学目标和要求:Understand duality, integrability, and uncertainty.教学时数:6 学时教学方式:讲授准备知识:statistics教学内容:Duality, Integrability, and Uncertainty 第一节:Duality of Consumer’s Problem第二节:Revealed Preferences & Uncertainty 作业与思考题:JR Ch 2.5 参考资料:JR Ch 2第三章: Theory of the Firm教学目标和要求:Understand the firm’s profit maximization problem.教学时数:6 学时教学方式:讲授准备知识:Chapter 1,2教学内容:Production, Cost, Profit Maximization 第一节:Production Functions & Cost第二节:Duality in Production, Competitive Firms 作业与思考题:JR Ch 3.6参考资料:JR Ch 3第四章: Partial Equilibrium教学目标和要求:Understand partial equilibrium markets. 教学时数:3学时教学方式:讲授准备知识:Chapter 3教学内容:Perfect & Imperfect Competition, Welfare 第一节:Competition 第二节:Equilibrium & Welfare作业与思考题:JR Ch 4.4参考资料:JR Ch 4第五章: Walras’/competitive equilibrium2教学目标和要求:competitive market economies from a Walrasian (general) equilibrium perspective.Let students understand “why the competitive market/equilibrium may work or fail?”教学时数:6学时方式:讲授教学准备知识:consumer theory, production theory教学内容:第一节:Walrasian economy and mathematical language of microeconomics 第二节:competitive equilibria of pure exchange and with production 作业与思考题:JR5.5, exercises of MWG Ch15, 18, 教师自编习题集参考资料:MWG Mathematical Appendix, Ch15, 18; JR5.4第六章: Social choice function/theory and social welfare: normative aspect of microeconomics教学目标和要求:When we judge some situation, such as a market equilibrium, as “good”or “bad”, or “better” or “worse” than another, we necessarily make at least implicit appeal to some underlying ethical standard. Welfare economics helps to inform the debate on social issues by forcingus to confront the ethical premises underlying our arguments as well as helping us to seetheir logical implications.Let students have a systematic framework for thinking about normative and social welfare topics.教学时数:3学时教学方式:讲授准备知识:Walrasian equilibrium教学内容:第一节:social choice, comparability, and some possibilities第二节:Rawlsian, Utiliterian, and flexible forms作业与思考题:JR6.5, exercises of MWG Ch21, 22, 教师自编习题集参考资料:MWG Ch21.A, Ch21.E, Ch22.C; JR Ch6第七章: Strategic Behavior and Asymmetric Information教学目标和要求:A central feature of contemporary microeconomicsafter Walrasian economy is the multi-agent interaction which represents the potential for the presence of strategicinterdependence. Let students grasp classic models of imperfect competition under symmetric and asymmetric information.3教学时数:3学时教学方式:讲授准备知识:perfect competition教学内容:第一节:monopoly and oligopoly under symmetric information第二节:oligopoly under asymmetric information作业与思考题:教师自编习题集参考资料:MWG Ch12; JR Ch4第八章: Theory of Incentives教学目标和要求:The strategic opportunities that arise in the presence of asymmetricinformation typically lead to inefficient market outcomes, a form of market failure. Underasymmetric information, the first welfare theorem no longer holds generally. Thus, the main themeto be explored is to stimulate different agents’ optimal/efficient behaviors in differentinformational settings to achieve the “second-best” market outcomes.教学时数:9学时教学方式:讲授准备知识:Strategic Behavior and Asymmetric Information教学内容:第一节:Adverse selection第二节:Moral hazard*第三节:Task separation/integration,第三节:Career concern作业与思考题:exercises of MWG Ch13, 14, 教师自编习题集参考资料:JR Ch8; MWG Ch13, 14第九章前沿研究讲座:待定邀请校外老师(待定)给学生们讲演最新研究，引导学生讨论;在学生掌握现代微观经济学基本模型之后能够接触到前沿研究。

Compare drug release profiles of water poor soluble drugs from a novel chitosan and polycarbophil interpolyelectrolytes complexation (PCC) and hydroxylpropyl - methylcellulose (HPMC) based matrix tabletsZhilei Lu*, Weiyang Chen, Eugene Olivier, Josias H., HammanDepartment of Pharmaceutical Sciences, Tshwane University of Technology, Private Bag X680, P retoria, 0001, South Africa资料个人收集整理，勿做商业用途*Corresponding author:Zhilei Lu (Dr.)Department of Pharmaceutical Sciences,Tshwane University of Technology,Private Bag X680,Pretoria, 0001,South Africa (e-mail: luzj@tut.ac.za)AbstractThe aim of this study was to compare the drug release behaviours of water poor soluble drugs from an interpolyelectrolyte complex (IPEC) of chitosan with polycarbophil (PCC) and hydroxylpropylmethylcellulose (HPMC) based matrix tablets. A novel interpoly - electrolyte complex (IPEC) of chitosan with polycarbophil (PCC) was synthesized and characterized. Water poor soluble drugsHydrochlorothiazide and Ketoprofen were used in this study as model drugs.Polymers (including PCC, HPMC K100M and HPMC K100LV) based matrix tablets drug controlled release system were prepared using direct compression method.The results illustrate PCC based-matrix tablets offer a swelling controlled release system for water poor soluble drug and drug release mechanism from this matrix drug delivery system can be improved by addition of microcrystalline cellulose (Avicel).Analysis of the in vitro release kinetic parametersof the matrix tablets, PCC based matrix tablets exhibited similar or higher drug release exponent (n) and mean dissolution time (MDT) values compared to the HPMC based matrix tablets. It demonstrated that PCC polymer can be successfully used as a matrixcontrolled release system for the water poor soluble model drugs such as hydroxylpropylmethylcellulose (HPMC). 资料个人收集整理，勿做商业用途1 IntroductionOver the last three decades years, as the expense and complications involved in marketing new drug entities have increased, with concomitant recognition of the therapeutic advantages of controlled drug delivery, greater attention has been focused on the development of novel and controlled release drug delivery systemsto provide a long-term therapeutic of drugs at the site of action following a single dose (Mandal, 2000; Jantzen and Robinson, 2002). Many formulation techniques have been used tobuild t”he barrier into the peroral dosage form to provide slow release of the maintenance dose. These techniques include the use of coatings, embedding of the drug in a wax, polymeric or plastic matrix, microencapsulation, chemical bindingto ion-exchange resins and incorporation into an osmotic pump (Collett and Moreton, 2002:293). Among different technologies used in controlled drug delivery, polymeric matrix systems are the most majority because of the simplicity of formulation, ease of manufacturing, low cost and applicability to drugs with wide range of solubility (Colombo, et al., 2000; Jamzad and Fassihi, 2006). 资料个人收集整理，勿做商业用途Drugs release profiles from polymeric matrix system can influence by different factors, but the type, amount, and physicochemical properties of the polymers used play a primary role (Jamzad and Fassihi, 2006). Hydroxylpropyl-methylcellulose (HPMC) is the most important hydrophilic carrier material used for oral drug sustained delivery systems (Colombo, 1993; Siepmann and Peppas, 2001). BecauseHPMC is water soluble polymer, it is generally recognized that drug release fromHPMC matrices follows two mechanisms, drug diffusion through the swelling gellayer and release by matrix erosion of the swollen layer (Ford et al., 1987; Raoet al., 1990; Colombo, 1993; Tahara et al., 1995; Reynolds, Gehrke et al., 1998; Siepmann et al., 1999; Siepmann and Peppas, 2001). However, diffusion, swelling and erosion are most important rate-controlling mechanisms of commercial available controlled release products (Langer and Peppas, 1983), the major advantages of swelling/erosionHPMC based matrix drug delivery system are: (i) minimum the drug burst release; (ii) the different physicochemical drugs release rate approach a constant; (iii) the possibility to predict the effect of the device design parameters (e.g. shape, size and composition of HPMC-based matrix tablets) on the resulting drug release rate, thus facilitating the development of new pharmaceutical products (Colombo, 1993;Siepmann and Peppas, 2001资).料个人收集整理，勿做商业用途Interpolyelectrolyte complexes (IPEC) are formed as precipitates by two oppositely charged polyelectrolytes in an aqueous solution, have been reported as a new class of polymer carriers, which play an important role in creating new oral drug delivery systems (Peppas and Khare, 1993; Berger et al., 2004). A variety chemical structure and stoichiometry of both components in interpolyelectrolyte complexes depends onthe pH values of the media, ionic strength, concentration, mixing ratio, and temperature (Peppas, 1986; Dumitriu and Chornet, 1998; Berger et al., 2004;Moustafine et al., 2005a). Chitosan is a positively charged (amino groups) deacetylated derivative of the natural polysaccharide, chitin (Paul and Sharma, 2000).Chitosan has already been successfully used to form complexes with natural anionic polymers such as carboxymethylcellulose, alginic acid, dextran sulfate,carboxymethyl dextran, heparin, carrageenan,pectin methacrylic acid copolymers ? (Eudragit polymers) and xanthan (Dumitriu and Chornet, 1998, Berger et al., 2004,Sankalia et al., 2007, Margulis and Moustafine, 2006)资. 料个人收集整理，勿做商业用途In this study, a novel polymer - IPEC between chitosan and polycarbophil (PCC) was synthesized, characterized and used as direct compressedexcipients in the matrix tablet. Although it have been well known that various IPEC have been used as a polymer carriers in drug controlled release system (Peppas and Khare, 1993, Garcia and Ghaly, 1996, Lorenzo-Lamoza et al., 1998, Soppirnath and Aminabhavi, 2002,Chen et al., 2004, Nam et al., 2004, Moustafine et al., 2005b), IPEC chitosan and polycarbophil was used as a polymer carriers have been investigated by Lu et al., (2006, 2007a, 2007 b, 2008a; 2008b资料个人收集整理，勿做商业用途The aim of this study was to comparein vitro water poor soluble drugs release profile of HPMC based matrices system to PCC based matrices system at same formulation.Water poor soluble model drugs Hydrochlorothiazide and ketoprofen were used in thisstudy. Two types HPMC (K100M and K100LV) and PCC polymers were used indirect compressedpolymers based matrix drug release system. The results of the hydration and erosion studies showed PCC based matrix systems have superior swelling properties. Drug release exponent (n) of each formulation PCC based matrices tablets are higher than HPMC based matrices tablets at pH 7.4 buffer solutions. It demonstrated that PCC has high potential to use in polymer based matrix drug con trolled released delivery for water poor soluble drugs资料个人收集整理，勿做商业用途2. Materials and methods2.1 MaterialsChitosan (Warren Chem Specialities, South Africa, Deacetylation Degree =91.25%),Polycarbophil (Noveon, Cleveland, USA), Hydroxylpropylmethylcellulose (MethocelK100M, K100LV Premium, Colorcon Limited, Kent, England), Ketoprofen (Changzhou Siyao Pharma. China), Hydrochlorothiazide (Huzhou Konch Pharmaceutical Co., Ltd. China), Microcrystalline cellulose (Avicel, pH101, FMC corporation NV, Brussels, Belgium), Sodium carboxymethyl starch (Explotab, Edward Mendell Co., Inc New York, USA). All other chemicals were of analytical grade and used as receive资料个人收集整理，勿做商业用途2.2 Preparation of interpolyelectrolytes complexation between chitosan and P olycarb op hil (PCC)资料个人收集整理，勿做商业用途Chitosan (30 g) was dissolved in 1000 ml of a 2% v/v acetic acid solution andpolycarbophil (30 g) was dissolved in 1000 ml of a 2% v/v acetic acid solution. Thechitosan solution was slowly added to the polycarbophil solution underhomogenisation (5200 rpm, ZKA , Germany) over a period of 20 minutes. Themixture was then mechanically stirred for a period of 1 hour at a speed of 1200 rpm(Heidolph RZR2021, Germany). The gel formed was separated by centrifuging for 5 min at 3000 rpm and then washed several times with a 2% v/v acetic acidsolution toremove any unreacted polymeric material. The gel was freeze dried for a period of 48 hours (Jouan LP3, France) and the lyophilised powder was screened through a 300prn sieve资料个人收集整理，勿做商业用途2.3Differential scanning calorimetry (DSC)DSC thernograns of the PCC were recorded with a Shinadzu DSC50 (Kyoto, Japan) instrument. The thermal behaviour was studied by sealing 2 mg samples of the material in aluminium crimp cells and heating it at a heating rate of 10o C per min under the flow of nitrogen at a flow rate of 20 ml/min. The calorimeter was calibrated with 2 mg of indium (Kyoto, Japan, melting point 156.4o C) at a heating rate of 10o C per min.资料个人收集整理，勿做商业用途2.4Fourier transforn infrared (FT-IR)Fourier transforn infrared (FT-IR) spectral data of the PCC polyner was obtained on a FTS-175C spectrophotoneter (BIO-RAD, USA) using the KBr disk nethod. 资料个人收集整理，勿做商业用途2.5Preparation of the natrix tabletsIn order to conpare the release profiles of water poor soluble drugs fron polyner based natrix tablets, nonolithic natrix type tablets containing hydrochlorothiazide or ketoprofen were prepared by conpressing a nixture of the ingredients with varying concentrations of the PCC, HPMC K100M and HPMC K100LV as indicated in Table 1. The ingredients of the different fornulationswere nanually pre-nixed by stirring in a 1000 nl glass beaker for 30 ninutes with a spatula. After the addition of 0.05 g of nagnesiun stearate (0.5% w/w), the powder nass was nixed for 10 nin. The powdernixture was conpressed using a rotating tablet press (Cadnach, India) fitted with round, shallow pun ches to p roduce matrix type tablets with a 6 mm diameter资料个人收集整理，勿做商业用途26 Weight, hard ness, thick ness and friability of tablets资料个人收集整理，勿做商业用途Weight variation was tested by weighing 20 randomly selected tablets individually, the n calculati ng the average weight and comparing the in dividual tablet weights to the average. The specification for weight variation is 10% from the average weight if the average weight < 0.08 g (USP 2006资料个人收集整理，勿做商业用途The hardnessof ten randomly selected matrix type tablets of each formulation was determined using a hardness tester (TBH 220 ERWE K A, Germany). The force (N) n eeded to break each tablet was recorded料个人收集整理，勿做商业用途The thick ness of each of 10 ran domly selected matrix type tablets were measured witha vernier calliper (accuracy = 0.02 mm). The thickness of the tablet should be within 5% variation of the average value资料个人收集整理，勿做商业用途A friability test was con ducted on the tablets using an Erweka Friabilator (TA3R,Germany). Twenty matrices were randomly selected from each formulation and any loose dust was removed with the aid of a soft brush. The selected tablets were weighed accurately and p laced in the drum of the friabilator. The drum was rotated at 25 rpm for 4 minu tes after which the matrices were removed. Any loose dust was removed from the matrices before they were weighed again. The friability maximal limit is 1% (USP 2006) was calculated using the following equation资料个人收集整理，勿做商业用途F (%) = W before (g)「W曲(g)X 100%(1)W after (g)Where F is the friability, W before is the initial weight of the matrices and W after is the weight of the matrices after test ing资料个人收集整理，勿做商业用途2.7 Swelli ng and erosi on studiesSwelling and erosion studies were carried out for all formulations matrix tablets. The matrices were weighed in dividually before they were pl aced in 900 ml p hos phate buffer (pH 7.4) at 37.0 0.寸C.± The medium was stirred with a paddle at a rotation speed of 50 rpm in a USP dissolution flask. At each time point, three tablets of each formulatio n were removed from the dissoluti on flask and gen tly wiped with a tissue toremove surface water, weighed and then placed into a plastic bowel. The matrix tablets were dried at 60°C until constant weight was achieved. The mean weights were determ ined for the three tablets at each time in terval. The data obta ined from this exp erime nt was used to calculate the swelli ng in dex and p erce ntage mass loss.料个人收集整理，勿做商业用途2.7.1Swelli ng indexThe swelli ng in dex (or degree of swelli ng) was calculated accord ing to the followi ngequati on资料个人收集整理，勿做商业用途s,=WJ—00W dWhere SI is the swelling index, W s and W d are the swollen and dry matrix weights, resp ectively, at immersio n timet in the buffer soluti on.资料个人收集整理，勿做商业用途2.7.2P erce ntage of matrix erosi onThe p erce ntage of matrix erosi on is calculated in relatio n to the in itial dry weight of the matrices, accord ing to the followi ng equation 资料个人收集整理，勿做商业用途Erow 册件"00%Where: dry weight (t) is the weight at time t.28 Assay of hydrochlorothiazide and ket oprofen in matrix tablets.料个人收集整理，勿做商业用途The drug content of the matrix type tablets was determ ined by crush ing 10 ran domly selected tablets from each formulatio n in a mortar and p estle. App roximately 80 mg po wder from the hydrochlorothiazide or ket oprofen containing matrices were weighed accurately and individually transferred into a 200 ml volumetric flasks, which were then made up to volume with p hos phate buffer soluti on (pH 7.4). This mixture was stirred for 30 minutes to allow compiete release of the drug. After filtration through a 0.45 阿filter membrane, the solution was assayed using ultraviolet (UV) spectrophotometry (Helios a Thermo , England) at a wavelength of 271 nm for hydrochlorothiazide and 261 nm for ketoprofen. The assay for drug content wasperformed in triplicate for each formulation. The percentage drug content of the tablets was calculated by mea ns of the followi ng equation:料个人收集整理，勿做商业用途DC (% w/w^W dru^x100%WmtWhere DC is the drug content, W drug is the weight of the drug and W mt is the weight of the matrix tablet .资料个人收集整理，勿做商业用途2.9 Release an alysisThe USP (2006) dissoluti on app aratus 2 (i.e. p addle) was used to determ ine the in vitro drug release from the different polymers based matrix tablets. The dissolution medium (900 ml) consisted of phosphate buffer solution (pH 7.4) at 37 0.5 o C and a ± rotation speed of 50 rpm was used. Three hydrochlorothiazide or ketoprofen matrix tablets of each formulatio n were in troduced into each of three dissoluti on vessels (i.e.?in triplicate) in a six station dissolution apparatus (TDT-08L, Electrolab , India).Samp les (5 ml) were withdraw n at sp ecially in tervals, and 5 ml of p reheated dissolution medium was replaced immediately. Sink conditions were maintained throughout the study. The samp les were filtered through a 0.45 阿membra ne, hydrochlorothiazide or ketoprofen content in the solution was determined using ultraviolet (UV) spectrop hotometry at a wavele ngth of 271 or 261 nm, res pectively.An alyses were p erformed in tripli cate资料个人收集整理，勿做商业用途2.9.1 Kin eticCon trolled release drug delivery systems may be classified accord ing to their mecha ni sms of drug release, which in cludes diffusi on-con trolled, dissoluti on con trolled, swelli ng con trolled and chemically con trolled systems (La nger et al., 1983). Drug release from sim pie swellable and erosi on systems may be described by the well-known power law expression and is defined by the following equation(Ritger and Pepp as, 1987; P illay and Fassihi, 1999资料个人收集整理，勿做商业用途Where M t is the amount of drug released at time t, M is the overall amount of drug released, K is the release con sta nt; n is the release or diffusi onal exponent; M/M is the cumulative drug concen trati on released at time t (or fractio nal drug release)料个人收集整理，勿做商业用途The release exponent (n) is used for the in terpretati on of the release mecha nism from poly meric matrix con trolled drug release systems (Peppas 1985). For the case of < 0.45 corrosFickdantdiffusi on release (Case I<an89homalous (non-Fickia n) transport, n = 0.89 toa zero-order (Case II) release kin etics, and n > 0.89to a super Case II transport (Ritger and Pepp as, 1987资料个人收集整理，勿做商业用途 The dissoluti on data were modelled by using the Po wer law equati on (Eq 7) withgraphs analysis software (Origin Scientific Graphing and Analysis software, Version 7, Origi nLab Corpo rati on, USA) using the Gaussia n-Newt on (Leve nberg-Hartely) app roach 资料个人收集整理，勿做商业用途2.9.2 Mea n dissolutio n time (MDT)MDT is a statistical moment that describes the cumulative dissolution process and provides a quantitative estimation of the drug release rate. It is defined by the following equation (Reppas and Nicolaides, 2000; Sousa^t al., 2002):资料个人收集整理，勿 做商业用途nMDTt i M t/M^ i =±Where MDT is the mean dissolution time, M t is the amount of the drug released at time t; t i is the time (min) at the midpoint between i and i-1 and M 乂 is the overall amount of the drug released.料个人收集整理，勿做商业用途cyli ndrical, i n sp ecially, n diffusi on al), 0.45 < n2.9.3 Differe nt factor f i and Similarityfactor f 2 The different factor f i is a measure of the relative error between two dissoluti on curves and the similarity factor f 2 is a measure of similarity in the p erce ntage dissoluti on betwee n two dissoluti on curves (Moore and Fla nn er, 1996). Assu ming that the p erce ntage dissolved values for two p rofiles cannot be higher tha n 100, the differe nt factor f 1 can have values from 0 (whe n no differe nee the two curves exists) to 100 (when maximum differenee exists). With the same assumption holding, the similarity factor f 2 can have values from 100 (when no differenee between the two curves exists) to 0 (when maximum differenee exists) (Pillay and Fassihi, 1999;Moore and Fla nn er, 1996; Re ppas and Nicolaides, 2000). In this study, these factors were used to confirm the relative of release p rofiles of water poor soluble model drugs from poly mers based matrix tablets of the same formulati ons. They are defi ned bythe followi ng equati ons:资料个人收集整理，勿做商业用途 f^100xn Z |Rt —Tt t 吕 n z R f2"0^「100hXG (Rt -T , I V ny 丿］Where n is the number of sample withdrawal points, R t is the percentage of the refere nee dissolved at time t, T is the p erce ntage of the test dissolved at time 资料个人 收集整理，勿做商业用途 3 Results and discussion 3.1 Prep arati on and characterisati on ofPCC The ion ic bond of the interpo "electrolyte comp lex (IP EC) betwee n chitosa n andpo lycarb op hil was con firmed by means p reviously p ublished differe ntial sea nning calorimetry (DSC) (Lu et al., 2007b) and Fourier tran sform in frared (FT-IR). Fig.1shows the FT-IR sp ectra of chitosa n, po lycarb op hil and the PCC poly mer.资料个人收集整理，勿做商业用途-1A peak that appears at 1561 cm on the IR spectrum of the PCC, which might be assigned to the carboxylate groups that formed ionic bonds with the protonated amino groups of chitosan as previously illustrated for the interaction between Eudragit E andEudragit L (Moustafine at al., 2005). This ionic bond seems to be the primary bin di ng force for the formatio n of a comp lex betwee n chitosa n and po lycarb op hil.资料个人收集整理，勿做商业用途Chitosan is a cationic polymer of natural origin with excellent gel and film forming properties. Polycarbophil can also be considered as polyanions with negatively charged carboxylate groups. Mixing chitosan and polycarbophil in acidic solution (2% acetic acid solution was used in the study), ionic bonds should form between the protonated free amino groups of chitosan and carboxylate groups of polycarbophil.According to the results obtained from DSC and FT-IR, the possible process of formatio n of interpo "electrolyte comp lexes may be described as illustrated in Fig.2资料个人收集整理，勿做商业用途3.2Physical characteristics and drug content of poly mers based matrix tablets^ 料个人收集整理，勿做商业用途As summarised in Table 2, the physical characteristics of matrix tablets showed the good thickness uniformity, as ranged from 3.40 0.04 to 4.12± 0.0±4mm, a variationof matrix tablets weight from 73.3 2.4 mg to±87.9 4.0±mg, furthermore the weight variation of all formulation tablets is very low (< 10% from the average weight) (USP 2006). Hardness of the matrix tablets shows a range from 68 ±14 to 94 ±12N.The tablets also pasted the friability test (<1%), confirm that all formulations tablets are within USP (2006) limits. Drugs content of all formulations ranged from 4.60 0.65 to 5.01 0.1±1%.资料个人收集整理，勿做商业用途3.3Swelli ng and erosi on prop erties of the poly mers based matrices tablets资料个人收集整理，勿做商业用途Investigation of matrix hydration and erosion by gravimetrical analysis is a valuable exercise to better understand the mechanism of release and the relative importance of participating parameters (Jamzad and Fassihi, 2006). Fig.3 and Fig.4 illustrate the water uptake profiles and Fig.5， Fig.6 illustrate percentage of matrix erosion of all formulation tablets, respectively. Swelling properties of the all formulation matrix tablets based on the content of PCC, HPMC K100M and HPMC K100LV in the matrices tablets. Water uptake and percentage of matrix erosion values of these matrix tablets show superior swelling characteristics either HPMC K100LV based matrix tablets, or PCC based matrix tablets. 资料个人收集整理，勿做商业用途IPEC betwee n chitosa n and po lycarb op hil is a three -dime nsional n etworks water insoluble poly (acrylic acid) polymer with free hydroxy groups. Hydroxy groups ofPCC contribute hydrophilic capacity significantly and polymer erosion characteristics depend on the reaction ratio of chitosan and polycarbophil while polymer synthesis.While the PCC based matrixes were put into the buffer solution, the electrostatic repulsion between fixed charges (hydroxy groups) uncoiled the polymers chains.The counterion diffusion inside the PCC gel creates an additional osmotic pressure difference across the gel, consequently lead to higher water uptake (Peppas and Khare, 1993; Lu, et al., 2007b). During the matrix erosion, the ionic bonds between chitosan and polycarbophil were not broken by the matrix swelling. PCC based matrix tablets (F1 and F7 formulation) have superior swelling behaviors compare to the HPMC based matrices. Swelling index values of F1 and F7 formulation matrix tablets are 1599.62±216.68 % and 1579.82 ±118.05 % at 12 hours, respectively.Furthermore, addition of microcrystalline cellulose (Avicel) can increase matrices erosion significantly. Compare the erosion behaviors of F1, F7 and F2, F8formulation (containing 20% Avicel), F1 and F7 matrix tablets erode 5.74 1.62 % and 6.59 1±.18 % on 12 hours only, cont rary F2 and F8 matrix tablets erode 55.59 1.43 and 100 % respectively. Microcrystalline cellulose (Avicel) is widely used in pharmaceutical, primarily as a binder/diluent, also has some disintegrant properties on oral tablet and capsule formulations where it is used in both wet granulation and direct-compression process (Wheatley, 2000). In this study, matrix erosion behaviours were act by microcrystalline cellulose facilitating the transport of liquid into the pore of matrix tablets. It demonstrates that PCC polymer have capacity to form swelli ng only or swelli ng-erosi on matrix drug delivery system.资料个人收集整理，勿做商业用途It also was confirmed that PCC based matrix tablets have much better swelling behaviors than HPMC based matrix tablets by comparing swelling curves in Fig.3 andFig.4. Swelling index values of F1 and F7 formulation matrix tablets are 1599.62216.68 % and 1579.82 11±8.05 % at 12 hours, contrary F3 and F9 matrix tablets are545.96 ±4.32% and 547.72 2±6.27%. HPMC K100LV based matrix tablets have excellence erosion curves in this study, F5, F6, F11 and F12 formulation matrixtablets eroded 100% on 12 hours, but F2 and F8 (PCC based tablets) formulation matrix tablets can eroded 55.59 ±1.43 and 100 % with microcrystalline cellulosefacilitating. 资料个人收集整理，勿做商业用途3.4Drug releaseIn vitro drug release was performed in pH 7.4 phosphate buffer solution for 12 hours.Results of percentage drug release versus time for hydrochlorothiazide and ketoprofen in different formulations matrices tablets are presented in Fig.7 and Fig. 8, while theMDT and drug release kinetics values were present in Table 3资.料个人收集整理，勿做商业用In this study, water poor soluble model drugs hydrochlorothiazide and ketoprofen release from polymers based matrix tablets was controlled by the polymer matrices swelling or swelling combination with erosion. Percentage of drug release, matrix swelling and erosion of F7 were summarised in Fig 9. The percentageketoprofen release curve follows the percentagematrix tablets swelling curve, it demonstrates that PCC based matrix drug delivery system is the swelling dependent drug release system for water poor soluble model drugs. Same as F7 matrix tablets, F1 matrix tablets is also a swelling only drug delivery system, in these matrix systems drugs release behaviour primarily depend on the matrix swelling characteristics. Because as the superior swelling capacity of PCC based matrix tablets, liquid environments inside of the matrix provide that the model drugs release are zero order drug release.As described in Table 3, release exponentsn)( of F1 and F7 are 0.83 0.03 a±nd 0.99 ± 0.02 during the experimental time, respectively. 资料个人收集整理，勿做商业用途Addition of microcrystalline cellulose (Avicel) influence the model drugs release profiles from PCC based matrix tablets significantly. Cumulative drug release of F2 and F8 formulation tablets is 93.7 4.13 % a±nd 99.6 4.2±5% at 12 hours, relativelyF1 and F7 formulation tablets is 73.8 1.13 % an±d 47.2 4.5±3 % only. This can be explained by drugs release mechanism were swelling and erosion instead of swelling only, consequently accelerate the drugs release. The adjustable capacity of PCC based matrix drug delivery system by addition of microcrystalline cellulose (Avicel) dem on strates the poten tial useful of PCC poly mer in drug con trolled release field 资料个人收集整理，勿做商业用途Compare to the PCC based matrix tablets, the drugs release profiles of HPMC based matrix tablets were adjusted difficultly. The relatives f1 and f2 values of difference polymers including PCC, HPMC 100M, HPMC 100LV based matrix tablets containing hydrochlorothiazide under same formulation were show in Table 4. As describedf1 and f2 values in Table 4, F3 and F4, F5 and F6 formulation tablets have similar drug release behaviours, but F1 and F2 formulation tablets illustrate different drug release behaviours. This phenomenacan be explained by the superior water uptake capacity of PCC polymer, more water containing can easier broken the physical tensility between the polymer particles. 资料个人收集整理，勿做商业用途However, HPMC 100LV polymer has excellence erosion characteristics, in this study model drugs release from HPMC 100LV based matrix tablets illustrate matrix erosion dependent properties. In generally, drug release from swelling and erosion matrix system shows zero order release pare the drug release exponentsn(), release constant (k1), and mean dissolution time (MDT) of F2 to F5, F6, they have not significantly different as described in Tablet 3, furthermore the relatives f1 and f2 values between F2 and F5, F6 in Table 4 show they are similar release profiles. It imply PCC based matrix tablets can become a swelling and erosion drug delivery system by the addition of microcrystalline cellulose (Avicel), this drug delivery system illustrate similar drug release p rofiles as HPMC 100LV based matrix tablets资料个人收集整理，勿做商业用途Although it is very complex process that the model drugs release from swelling and。

sigmaaldrich引用英文全名Sigma-Aldrich is a leading supplier of chemicals, biochemicals, and other research tools for the scientific community. The company, which is now part of Merck KGaA, offers a wide range of products to support research in fields such as life sciences, material sciences, and analytical chemistry.Sigma-Aldrich is well-known for its high-quality products, which are used by researchers in academia, government laboratories, and industry around the world. The company's catalog includes over 300,000 products, including organic and inorganic chemicals, solvents, biochemicals, and research kits.One of the key advantages of Sigma-Aldrich is its commitment to maintaining the highest standards of quality and purity in its products. The company's products are tested rigorously to ensure that they meet the specifications and performance criteria demanded by researchers. Sigma-Aldrich is ISO 9001 certified, and many of its products are manufactured in facilities that comply with Good Manufacturing Practice (GMP) standards.In addition to its extensive product catalog, Sigma-Aldrich also provides a range of services to support researchers in theirwork. The company's website offers a wealth of information, including technical data, safety data sheets, and application notes for many of its products. Customers can also access training resources, webinars, and other educational materials to help them get the most out of Sigma-Aldrich's products.Sigma-Aldrich is also committed to sustainability and social responsibility. The company works to minimize its environmental impact by reducing waste, conserving resources, and promoting the use of renewable energy. Sigma-Aldrich also supports charitable organizations and community initiatives to give back to the communities in which it operates.In conclusion, Sigma-Aldrich is a trusted supplier ofhigh-quality chemicals and research tools for the scientific community. With its extensive product catalog, commitment to quality, and focus on sustainability, Sigma-Aldrich is a valuable partner for researchers in a wide range of fields.。

《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。

基础数学专业硕士研究生培养方案Pure Mathematics一、培养目标和要求（一）努力学习马列主义、毛泽东思想和邓小平理论，坚持党的基本路线，热爱祖国，遵纪守法，品德良好，学风严谨，具有较强的事业心和献身精神，积极为社会主义现代化建设服务。

（二）掌握坚实宽广的理论基础和系统深入的专门知识，具有独立从事科学研究工作的能力和社会管理方面的适应性，在科学和管理上能做出创造性的研究成果。

（三）积极参加体育锻炼，身心健康。

（四）硕士应达到的要求：①掌握本学科的基础理论和相关学科的基础知识，有较强的自学能力，及时跟踪学科发展动态。

②具有项目组织综合能力和团队工作精神，具有和谐的人际关系。

③具有强烈的责任心和敬业精神。

④广泛获取各类相关知识，对科技发展具有敏感性。

⑤有扎实的英语基础知识，能流利阅读专业文献，有较好的听说写译综合技能。

（五）本专业主要学习分析学（实分析、泛函分析、C*-代数、算子代数、调和分析、函数逼近论等），代数学（代数学基础、代数学、Lie代数与代数群、环与代数，交换代数，半群理论等），微分方程（（线性）偏微分方程、非线性偏微分方程，Euler方程组，Navier-Stokes方程组等），组合学(组合论、图论)和几何学(拓扑学，微分几何，代数几何)等方面的数学基础知识。

本专业毕业生要具有扎实宽广的数学基础，毕业后主要从事与数学相关的科研、教学工作，或在工程技术、经济、金融等部门中利用数学和计算机解决实际问题的工作，为高等院校、中学及相关领域培养合格的专门人才。

二、学习年限学制3年，学习年限最长不超过5年。

三、研究方向本学科专业主要研究方向有泛函分析、调和分析与函数逼近、交换代数与代数几何、Lie代数与线性群、一般代数学、组合数学、偏微分方程等。

主要导师有王军、许庆祥、周才军、李中凯、王宇、张建刚、裴玉峰、王丽、徐本龙、戴文荣等教授和副教授。

每年招生导师和研究方向，详见招生简章。

（一）泛函分析本方向研究离散群上的Toeplitz算子、算子广义逆的理论及其应用等。

rocheRoche: A Global Leader in Biotechnology and Pharmaceutical ResearchIntroductionRoche is a multinational pharmaceutical and biotechnology company headquartered in Basel, Switzerland. With a history spanning over 120 years, Roche has emerged as a global leader in pharmaceutical research and development, focusing on innovative treatments for various diseases and conditions. This document explores the history, core values, research areas, and contributions of Roche to the field of biotechnology and pharmaceuticals.1. HistoryRoche was founded in 1896 by Fritz Hoffmann-La Roche with the aim of developing new treatments for various ailments. The company started as a small vitamin-producing laboratory, but soon expanded its scope to include a wide range of pharmaceutical products. Over the years, Roche has made significant strides in medical research and innovation, leadingto breakthrough discoveries in the treatment of diseases like cancer, viral infections, and rare genetic disorders.2. Core ValuesRoche operates based on a set of core values that guide its actions and decisions. These values include integrity, courage, and passion for science. By maintaining the highest ethical standards, Roche strives to build trust with its stakeholders and ensure the safety and efficacy of its products. The company's commitment to scientific excellence and curiosity has driven its success in developing cutting-edge therapies.3. Research AreasRoche has established itself as a leader in various research areas, including oncology, neuroscience, immunology, and infectious diseases. The company invests heavily in research and development, continuously striving to improve existing treatments and discover novel therapeutic approaches. For instance, Roche has pioneered the development of personalized medicine, tailoring treatments to individual patients based on their unique genetic profiles.In the field of oncology, Roche has made groundbreaking advancements with targeted therapies and immunotherapies. Drugs like Herceptin and Avastin have revolutionized the treatment of breast and colorectal cancer, respectively. Roche's commitment to oncology research has also led to the development of breakthrough therapies for lung cancer, melanoma, and lymphoma.In neuroscience, Roche focuses on developing treatments for neurological disorders such as Alzheimer's disease, Parkinson's disease, and multiple sclerosis. The company's innovative approach involves identifying and targeting specific molecular pathways implicated in these diseases, aiming to slow down or halt their progression.Roche's research efforts in immunology have resulted in the development of drugs like Actemra, a treatment for rheumatoid arthritis, and Ocrevus, a therapy for multiple sclerosis. These medications have improved the quality of life for patients and addressed unmet medical needs.4. Contributions to Public HealthRoche's contributions to public health extend beyond the development of life-saving medications. The companyactively engages in healthcare initiatives aimed at improving patient outcomes and access to healthcare globally. Through partnerships with governments, non-profit organizations, and healthcare professionals, Roche works to strengthen healthcare systems, increase disease awareness, and improve patient education.Roche also fosters a culture of sustainability and environmental responsibility. The company has implemented various measures to minimize its environmental footprint, reduce waste generation, and promote renewable energy sources. Roche recognizes the importance of protecting the planet for future generations and aims to integrate sustainability into all aspects of its operations.5. ConclusionRoche's commitment to scientific excellence, innovation, and patient well-being has made it a global leader in biotechnology and pharmaceutical research. With a diverse portfolio of life-saving medications and a strong focus on research and development, Roche continues to make significant contributions to the field of healthcare. By addressing unmet medical needs and pioneering new treatment approaches, Roche positions itself at the forefront of the fight against diseases around the world.。

Kronecker productFrom Wikipedia, the free encyclopediaIn mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.The Kronecker product is named after Leopold Kronecker, even though there is little evidence that he was the first to define and use it. Indeed, in the past the Kronecker product was sometimes called the Zehfuss matrix, after Johann Georg Zehfuss.Contents1 Definition1.1 Examples2 Properties2.1 Relations to other matrix operations2.2 Abstract properties3 Matrix equations4 Related matrix operations4.1 Tracy-Singh product4.2 Khatri-Rao product5 See also6 Notes7 References8 External linksDefinitionIf A is an m × n matrix and B is a p × q matrix, then the Kronecker product A⊗B is the mp × nq block matrix:more explicitly:If A and B represent linear transformations V1 → W1 and V2 → W2, respectively, then A⊗B represents the tensor product of the two maps, V1⊗V2 → W1⊗W2.ExamplesPropertiesRelations to other matrix operations1. Bilinearity and associativity: The Kronecker product is a special case of the tensorproduct, so it is bilinear and associative:where A, B and C are matrices and k is a scalar.2. Non-commutative: In general A⊗B and B⊗A are different matrices. However, A⊗Band B⊗A are permutation equivalent, meaning that there exist permutation matrices P and Q such thatIf A and B are square matrices, then A⊗B and B⊗A are even permutation similar,meaning that we can take P = Q T.3. The mixed-product property and the inverse of a Kronecker product: If A, B, C andD are matrices of such size that one can form the matrix products AC and BD, thenThis is called the mixed-product property, because it mixes the ordinary matrix product and the Kronecker product. It follows that A⊗B is invertible if and only if A and B are invertible, in which case the inverse is given by4. Transpose: The operation of transposition is distributive over the Kronecker product:5. Determinant: Let A be an n × n matrix and let B be a p × p matrix. ThenThe exponent in |A| is the order of B and the exponent in |B| is the order of A.6. Kronecker sum and exponentiation If A is n × n, B is m × m and I k denotes the k ×k identity matrix then we can define what is sometimes called the Kronecker sum, ⊕, byNote that this is different from the direct sum of two matrices. This operation isrelated to the tensor product on Lie algebras. We have the following formula for thematrix exponential which is useful in the numerical evaluation of certain continuous-time Markov processes[citation needed],Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let H i be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the ensemble is.Abstract properties1. Spectrum: Suppose that A and B are square matrices of size n and m respectively. Letλ1, ..., λn be the eigenvalues of A and μ1, ..., μm be those of B (listed according to multiplicity). Then the eigenvalues of A⊗B areIt follows that the trace and determinant of a Kronecker product are given by2. Singular values: If A and B are rectangular matrices, then one can consider theirsingular values. Suppose that A has r A nonzero singular values, namelySimilarly, denote the nonzero singular values of B byThen the Kronecker product A⊗B has r A r B nonzero singular values, namelySince the rank of a matrix equals the number of nonzero singular values, we find that3. Relation to the abstract tensor product: The Kronecker product of matricescorresponds to the abstract tensor product of linear maps. Specifically, if the vector spaces V, W, X, and Y have bases {v1, ..., v m}, {w1, ..., w n}, {x1, ..., x d}, and {y1,..., y e}, respectively, and if the matrices A and B represent the linear transformations S : V → X and T : W → Y, respectively in the appropriate bases, then the matrix A⊗B represents the tensor product of the two maps, S⊗T : V⊗W → X⊗Y with respect to the basis {v1⊗ w1, v1⊗ w2, ..., v2⊗ w1, ..., v m⊗ w n} of V⊗W and the similarlydefined basis of X⊗Y with the property that A⊗B(v i⊗ w j) = (A v i)⊗(B w j), where i and j are integers in the proper range.[1] When V and W are Lie algebras, and S : V → V and T : W → W are Lie algebra homomorphisms, the Kronecker sum of A and B represents theinduced Lie algebra homomorphisms V⊗W → V⊗W.4. Relation to products of graphs: The Kronecker product of the adjacency matrices oftwo graphs is the adjacency matrix of the tensor product graph. The Kronecker sum of the adjacency matrices of two graphs is the adjacency matrix of the Cartesian product graph.See,[2] answer to Exercise 96.Matrix equationsThe Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation AXB = C, where A, B and C are given matrices and the matrix X is the unknown. We can rewrite this equation asHere, vec(X) denotes the vectorization of the matrix X formed by stacking the columns of Xinto a single column vector. It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution if and only if A and B are nonsingular (Horn & Johnson 1991, Lemma 4.3.1).If X is row-ordered into the column vector x then AXB can be also be written as (Jain 1989, 2.8 Block Matrices and Kronecker Products) (A⊗B T)x.Related matrix operationsTwo related matrix operations are the Tracy-Singh and Khatri-Rao products which operate on partitioned matrices. Let the m × n matrix A be partitioned into the m i × n j blocks A ij and p × q matrix B into the p k × qℓ blocks B kl with of course Σi m i = m, Σj n j = n, Σk p k = p and Σℓ qℓ = q.Tracy-Singh productThe Tracy-Singh product[3][4] is defined aswhich means that the (ij)-th subblock of the mp × nq product A ○ B is the m i p × n j q matrix A ij ○ B, of which the (kℓ)-th subblock equals the m i p k × n j qℓ matrix A ij⊗B kℓ. Essentially the Tracy-Singh product is the pairwise Kronecker product for each pair of partitions in the two matrices.For example, if A and B both are 2 × 2 partitioned matrices e.g.:we get:Khatri-Rao productThe Khatri-Rao product[5][6] is defined asin which the ij-th block is the m i p i × n j q j sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σi m i p i) × (Σj n j q j). Proceeding with the same matrices as the previous example we obtain:This is a submatrix of the Tracy-Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy-Singh product).A column-wise Kronecker product of two matrices may also be called the Khatri-Rao product.This product assumes the partitions of the matrices are their columns. In this case m1 = m, p1 = p, n = q and for each j: n j = p j = 1. The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:so that:See alsoGeneralized linear array modelMatrix productNotes1. ^ Pages 401–402 of Dummit, David S.; Foote, Richard M. (1999), Abstract Algebra (2 ed.), New York:John Wiley and Sons, Inc., ISBN 0-471-36857-12. ^ D. E. Knuth: "Pre-Fascicle 0a: Introduction to Combinatorial Algorithms" (http://www-cs-/~knuth/fasc0a.ps.gz), zeroth printing (revision 2), to appear as part of D.E.Knuth: The Art of Computer Programming Vol. 4A3. ^ Tracy, DS, Singh RP. 1972. A new matrix product and its applications in matrix differentiation.Statistica Neerlandica 26: 143–157.4. ^ Liu S. 1999. Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra and itsApplications 289: 267–277. (pdf (/science?_ob=MImg&_imagekey=B6V0R-3YVMNR9-R-1&_cdi=5653&_user=877992&_orig=na&_coverDate=03%2F01%2F1999&_sk=997109998&view=c&wchp=dGLbVlb-zSkWb&md5=21c8c66f17da8d1bab45304a29cc96ac&ie=/sdarticle.pdf))5. ^ Khatri C. G., C. R. Rao (1968), "Solutions to some functional equations and their applications tocharacterization of probability distributions" (http://sankhya.isical.ac.in/search/30a2/30a2019.html), Sankhya30: 167–180.6. ^ Zhang X, Yang Z, Cao C. (2002), "Inequalities involving Khatri-Rao products of positive semi-definite matrices", Applied Mathematics E-notes2: 117–124.ReferencesHorn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, CambridgeUniversity Press, ISBN 0-521-46713-6.Jain, Anil K. (1989), Fundamentals of Digital Image Processing, Prentice Hall, ISBN 0-13-336165-9.Steeb, Willi-Hans (1997), Matrix Calculus and Kronecker Product with Applications and C++ Programs, World Scientific Publishing, ISBN 981-02-3241-1Steeb, Willi-Hans (2006), Problems and Solutions in Introductory and Advanced MatrixCalculus, World Scientific Publishing, ISBN 981-256-916-2External linksHazewinkel, Michiel, ed. (2001), "Tensor product"(/index.php?title=p/t092410), Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4Kronecker product (/?op=getobj&from=objects&id=4163),.MathWorld Kronecker Product (/KroneckerProduct.html)New Kronecker product problems (http://issc.uj.ac.za/downloads/problems/newkronecker.pdf) Earliest Uses: The entry on The Kronecker, Zehfuss or Direct Product of matrices hashistorical information. (/k.html)Generic C++ and Fortran 90 codes for calculating Kronecker products of two matrices.(https:///projects/kronecker/)Retrieved from "/w/index.php?title=Kronecker_product&oldid=556239113" Categories: Matrix theoryThis page was last modified on 22 May 2013 at 09:24.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.。