EFFICIENCY OF WATER REMOVAL FROM WATER/ETHANOL MIXTURES USING SUPERCRITICAL CARBON DIOXIDE

Size: px
Start display at page:

Download "EFFICIENCY OF WATER REMOVAL FROM WATER/ETHANOL MIXTURES USING SUPERCRITICAL CARBON DIOXIDE"

Transcription

1 Brazilia Joural of Chemical Egieerig ISSN Prited i Brazil Vol. 23, No. 02, pp , April - Jue, 2006 EFFICIENCY OF WATER REMOVAL FROM WATER/ETHANOL MIXTURES USING SUPERCRITICAL CARBON DIOXIDE M. A. Rodrigues 1, Ju Li 2, A. J. Almeida 3, H. A. Matos 1, ad E. Gomes de Azevedo 1* 1 Departmet of Chemical Egieerig, Istituto Superior Técico, Av. Rovisco Pais, Lisbo, Portugal. [email protected] 2 Departmet of Chemical Egieerig, Xiame Uiversity, Xiame , Chia. 3 Uidade de Ciêcias e Tecologia Farmacêuticas, School of Pharmacy, Uiversity of Lisbo, Av. Prof. Gama Pito, , Lisbo, Portugal (Received: October 20, 2004 ; Accepted: Jauary 28, 2006) Abstract - Techiques ivolvig supercritical carbo dioxide have bee successfully used for the formatio of drug particles with cotrolled size distributios. However, these processes show some limitatios, particularly i processig aqueous solutios. A diagram walkig algorithm based o available experimetal data was developed to evaluate the effect of ethaol o the efficiecy of water removal processes uder differet process coditios. Ethaol feedig was the key parameter resultig i a tefold icrease i the efficiecy of water extractio. Keywords: Water/ethaol mixtures; Supercritical carbo dioxide. INTRODUCTION New processes of particle formatio are becomig very attractive to the pharmaceutical idustry, where ihalable drug formulatios are gaiig importace. With the goal o delivery of isuli via the lugs almost accomplished by some of the major pharmaceutical compaies, it is expected that other formulatios will follow isuli i the emergig ew market of ihalable therapeutics. To eter the blood circulatio istataeously, replacig classic itraveous admiistratio, formulatios of ihalable drugs must reach deep ito the lugs, the site where these drugs act. However, lug brochioles are difficult targets because the trough ad its braches are physical barriers that ca retai most of the dose of the formulatio (Edwards, 2002). Very small particles with arrow size distributios are hard to achieve with the covetioal techologies, especially i the case of proteis due to their sesitivity to high temperatures, high shear stresses, ad orgaic solvets. Processes of particle formatio by supercritical fluids are cosistetly providig a solutio ad a competitive alterative to some of the established processes. The reasos for the relative success of supercritical fluid-based techiques are their ability to produce submicro particles with efficiet cotrol over size distributio i a sigle-step process, their exteded cotrol over the crystallie form of the products, ad their smaller evirometal impact sice this techology is potetially cleaer (Rodrigues et al., 2004). Carbo dioxide is the supercritical fluid that has bee most widely used due to its relatively low *To whom correspodece should be addressed

2 206 M. A. Rodrigues, Ju Li, A. J. Almeida, H. A. Matos, ad E. Gomes de Azevedo critical poit (7.38 MPa ad 31.1ºC), low cost, ad evirometal acceptability. However, because most of the therapeutic compouds, especially macromolecules such as proteis or ucleic acids, are sparigly soluble i supercritical (SC) CO 2, atisolvet techiques such as GAS, SAS, ad SEDS have the greatest potetial i the pharmaceutical idustry (York, 1999). Supercritical atisolvet procedures ivolve drug dissolutio i a liquid solvet that is further expaded by the dissolutio of the supercritical phase, forcig the drug to precipitate. I most cases the choice of solvet is the critical stage i processig biological macromolecules for therapeutic purposes. Ideally this solvet should dissolve large quatities of the drug without deactivatig it ad should also be highly mixable with the supercritical fluid. Such solvets have yet to be foud, but orgaic solvets such as DMSO or methylee chloride have bee reported to be able to precipitate several proteis. Orgaic solvets, however, are potetial deaturats for most therapeutic macromolecules, ad as a cosequece, the biological activity of the products ca be severely affected. Also, with this approach a dry protei reaget is required, which meas that the raw protei broth first has to be processed by other techiques, such as spray-dryig or lyophillizatio, which cosequetly raises questios o the purpose ad advatages of the supercritical fluid techology. Successful formulatios are therefore difficult to obtai without usig water i the productio process, which represets oe of the major difficulties i the processig of protei molecules with supercriticalfluid techologies. The reduced solubility of water i supercritical CO 2 makes atisolvet processes very iefficiet. The itermediate solutio of mixig water with ethaol i order to improve the miscibility with the SC-CO 2 has bee show to be somewhat reliable for may formulatios. May biological macromolecules are tolerat to moderate cocetratios of ethaol, ad small amouts of ethaol ca icrease the miscibility with the SC phase by several orders of magitude, thus icreasig the efficiecy of the process. Ethaol also shows some advatages over other solvets: low toxicity, FDA approval, ad cost. The use of water/ethaol mixtures has bee reported for GAS, SAS, ad SEDS. Briefly, the GAS process cosists i the ijectio of a drug/ethaol/water solutio ito a precipitator vessel, ad its pressurizatio with CO 2 util drug precipitatio occurs. The ethaol/water mixture is the removed by circulatig fresh SC-CO 2 util a dry powder is obtaied. I the SAS process, a aqueous solutio is sprayed cotiuously ito a flowig SC-CO 2 phase eriched with ethaol ad the particles precipitate readily as the water is dissolved i the SC phase. I SEDS the aqueous phase, the CO 2 phase, ad a ethaol phase are sprayed together through a multi-chael ozzle i such proportios that all compoets are miscible i the SC phase except the drug that precipitates. Despite the itese research i this area, supercritical-fluid techology has ot yet received widespread idustry acceptace for processes of particle formatio. To achieve a idustrial scale, the effects of SCF processig of biological macromolecules o aqueous-based therapeutic species have yet to be show ad specially most of the operatioal parameters that determie the efficiecy of the process have yet to be optimized. The aim of this work was to evaluate the effect of the water/sc-co 2 /ethaol system o the efficiecy of processes of protei microizatio with supercritical CO 2 from aqueous solutios. The low productivity associated with SC fluid atisolvet processes is due to the complete removal of water from the iitial aqueous solutio. This difficulty costrais the (aqueous solutio)/(sc phase) feed ratio to very small values, makig the process very time-cosumig. However this depeds o the process selected, GAS, SAS, or SEDS; o the ethaol feedig; ad o temperature ad pressure, amog other variables. A algorithm, referred to as diagram walkig, based o data preseted by Duarte et al. (2000) was developed to geerate the water removal profile for the GAS process depedig o the process parameters selected. Source of the Data MODEL DESCRIPTION Duarte et al. (2000) compiled available VLE data for the terary water/ethaol/co 2 system coverig a temperature rage of 35ºC to 70ºC ad a pressure rage of 10 MPa to 18.5 MPa. Uder these coditios the terary phase diagram is of type I ad, for a costat temperature ad pressure, the VLE curve is represeted by a cotiuous lie ad therefore ca be described by a polyomial fuctio with rectagular coordiates, 2 ν = au + bu + c (1) where Brazilia Joural of Chemical Egieerig

3 Efficiecy of Water Removal from Water/Ethaol Mixtures 207 v = z si( π /3); u = (2z + z )cos( π /3); C2H5OH CO2 C2H5OH ad a, b, ad c are costats obtaied by fittig the experimetal data at differet pressures ad temperatures. Duarte et al. (2000) also foud a liear relatioship betwee the tie lie-slopes (at costat temperature ad pressure) ad the tie-lie iterceptios with the VLE curve o the liquid side (u L ). A liear equatio is therefore suited to the tielie slope calculatio: m= b0 + b1ul + c (2) where m is the slope, u L is the abscissa for the liquid iterceptio betwee the tie lie ad the VLE curve, ad b 0 ad b 1 were obtaied by Duarte et al. by data fittig. Diagram Walkig Algorithm Assumig that the system is always i equilibrium (steady state), the algorithm proceeds i three steps. This three-step routie provides a path for water extractio that depeds o the coditios selected. Each path ca be observed if plotted i the terary diagram, as schematically show i Fig. 1. Figure 1: Represetatio of the calculatios with the diagram walkig algorithm for a geeric H 2 O/C 2 H 5 OH/CO 2 vapor-liquid-equilibrium curve at fixed P ad T. Step 1: Determiatio of the Overall Mole Fractio (z) of Each Compoet The routie starts with the calculatio of the overall compositio of each compoet (zco2, z H2O, zc2h5oh) ad of (u k,v k ) i rectagular coordiates. This requires a estimatio of the amout of CO 2 (loop i i Fig. 2). Because the volume of the liquid is always very small compared to the total volume, we estimate the CO 2 mole umber by V V = + = + (3) k,i k,i k,i T L k,i CO2 V,CO2 L,CO2 L,CO V 2 m,co2 I Eq. (3), CO 2, L,CO2, ad V,CO2 are respectively the total umber of moles of CO 2 ad the umber of moles of CO 2 i the liquid ad i the vapor; V T is the total volume; V L is the total liquid (water ad ethaol) volume; ad V m,co 2 is the molar volume of CO 2 at the selected pressure ad temperature. Superscript k deotes the iteratio umber whe equilibrium has bee attaied. I the first iteratio L,CO 2 is ukow. Therefore the iteratio for obtaiig the CO 2 mole fractio starts from a iitial guess for L,CO 2 = 0, as described i the block diagram show i Fig. 2. Brazilia Joural of Chemical Egieerig Vol. 23, No. 02, pp , April - Jue, 2006

4 208 M. A. Rodrigues, Ju Li, A. J. Almeida, H. A. Matos, ad E. Gomes de Azevedo Read costats: PTV,,, y, f, V, abc,,, b b T i,c2h5oh V m,co2 0, 1, ρ, H2O, C2H5OH Set: i = 0, k = 0 Set: ki, ki, ki, = 0; = ; = L,CO2 H2O H2O C2H5OH C2H5OH k k, i k, i = 0; = 0; from Eq.(3) out,co2 i,co2 CO2 Step 1 ki, z j j = CO 2,H2O,C2H5OH Step 2.1 Step 2.2 ik, x ik, j y j from Eqs.(1),(2),(4) from Eqs.(1),(5) k k, i = CO2 CO2 ki, ki, ; ; CHOH 2 5 HO 2 ki, CO2 from Eqs.(9)-(11),(3) ki, ki, ; C2H5OH L,CO2 from Eqs.(6),(7) ki, CO2 from Eq.(3) yes Is δ L,CO 2 >ξ? o k out, j from Eq.(8) Step 3 yes ki, ki, > y HO 2 HO 2 Is z? o Figure 2: Schematic represetatio of the diagram walkig algorithm. Superscript k is the iteratio umber whe equilibrium has bee attaied. Step 2: Calculatio of the Tie Lie, VLE Curve Iterceptios, ad Mole Fractios For clarificatio we separate the calculatio of the tie lie ad of the VLE curve iterceptios ito two further steps. Step 2.1 Because (u k,v k ) are already kow from step 1, the liquid-side iterceptio (u L,v L ) is determied by solvig Eqs. (1), (2), ad (4) for u L, v L ad for the tie-lie slope m: νl ν m = u u L k k The smallest solutio is chose as u L. Step 2.2 (4) With (u L,v L ) obtaied from step 2.1, the tie lie ad its vapor-phase iterceptio with the VLE curve, (u V,v V ) are determied from Eqs. (1) ad (5) with d = v L mu L : Brazilia Joural of Chemical Egieerig

5 Efficiecy of Water Removal from Water/Ethaol Mixtures 209 ν= mu + d (5) Recalculatio of the Ethaol Added Because the ethaol added ( i,c2h5oh ) depeds o the CO 2 itake, it must be recalculated for the CO obtaied i each iteratio i by usig 2 Eqs. (6) ad (7): k,i k,i k 1 k i,co2 CO2 CO2 out,co2 = + (6) k,i i,c2h5oh = k,i 1 i,co2 1 y y i,c2h5oh i,c2h5oh (7) where i,co ad 2 out,co are respectively the 2 mole umbers of CO 2 that etered ad left the system. We assume that the ethaol vapor compositio ( y i,c H OH ) i the vapor feed is costat. 2 5 Step 3: Removal of a Small Part of the Vapor A small fractio (f V ) of the vapor is removed ad replaced by fresh CO 2 ad ethaol. For each compoet the moles extracted are calculated from k k 1,i i 1 out,j V V j = f y (8) where subscript j refers to each compoet (j = H 2 O or C 2 H 5 OH or CO 2 ). The ext operatig lie is determied from Eqs. (9) ad (11); however this requires estimatio of the ethaol added to the system with Eq. (10): k,i k 1,i k H2O H2O out,h 2 O = (9) k,i k 1,i i,c2h5oh V = f y (10) V i,c2h5oh k,i k 1,i k k,i CHOH 2 5 CHOH 2 5 out,c2h5oh i,choh 2 5 = + (11) The correct amout of ethaol will be determied oly after the secod step of the ext iteratio because it depeds o the total amout of CO 2. EXPERIMENTAL To evaluate the experimetal reproducibility ad usefuless of the diagram walkig algorithm predictios, a GAS apparatus was used to obtai experimetal data. Each experimetal ru starts by ijectig a ethaol/water solutio of kow compositio ito a 460 cm 3 high-pressure visual cell, placed iside a temperature-cotrolled air box, as schematically show i Fig. 3. The cell is the pressurized with CO 2 (99.998% pure, supplied by Ar Líquido, Portugal) usig a motor-drive gas compressor (Newport Scietific, model ). The gas circulates first through a temperature-cotrolled cylider for temperature equilibratio. The workig pressure is cotrolled by a back-pressure regulator (Tescom, model ). Whe the workig pressure is reached, the backpressure regulator opes ad the liquid is extracted by the circulatig CO 2. The total amout of CO 2 cosumed is calculated based o flow meter readigs (Omega, model FLR 1000). Extractio is stopped whe oly oe phase is observed iside the cell. All the extractio rus were carried out at 10 MPa ad 40ºC with CO 2 flow rates from 3 to 7 g mi -1. Figure 3: GAS apparatus used i the determiatio of the efficiecy of water extractio by supercritical CO 2. Brazilia Joural of Chemical Egieerig Vol. 23, No. 02, pp , April - Jue, 2006

6 210 M. A. Rodrigues, Ju Li, A. J. Almeida, H. A. Matos, ad E. Gomes de Azevedo RESULTS AND DISCUSSION The Effect of Ethaol Fractio o Extractio Efficiecy Several extractio pathways were obtaied with the diagram walkig algorithm for differet iitial mass fractios of ethaol ad differet ethaol feeds. I Table 1 the variables that were kept costat are show. I Fig. 4 the higher selectivity of the CO 2 for ethaol i the terary system ca be see. It is evidet that without ethaol feed the icrease i extractio produced by the ethaol iitial fractio eds rapidly ad the system teds towards a biary CO 2 /water behavior. It ca also be observed that the efficiecy of complete water removal is little affected by iitial ethaol cotet. Efficiecy is defied here as the moles of water removed per mole of CO 2 cosumed. As Table 2 shows, a 35% icrease i efficiecy is achieved oly for ethaol mass fractios greater tha 0.7; however, these high-ethaol cotets are frequetly usuited for the processig of biological molecules ad i most cases the advatages of addig ethaol to the liquid feed do ot compesate the advatages of usig ethaol-free solutios. Table 2 shows that 5% (mass) ethaol i the CO 2 feed icreases the extractio efficiecy by over 500%, revealig that this is the key parameter for successful water removal. A importat drawback may however result from the ethaol feed; the ethaol cotet i the liquid fractio icreases durig extractio ad may reach values usuitable for successful formulatios. Stability of the active molecules i the terary liquid mixture is therefore the limitig parameter for ethaol feed ad cosequetly for extractio efficiecy. Table 1: Fixed variables. Pressure Temperature Vessel volume CO 2 desity Liquid desity Water cotet 10 MPa 40ºC 200 cm g/cm g/cm3 0.5 mol C 2 H 5 OH C 2 H 5 OH H 2 O CO H 2 O CO 2 Figure 4: Extractio pathways geerated by the diagram walkig algorithm for solutios with differet iitial ethaol mass fractios (0.7:, 0.5: ; 0.1: - - -) without ethaol feed (left) ad with ethaol feed (right). The straight lies are the iitial tie lies geerated for each mixture. Table 2: Compariso of efficiecies of water extractio from solutios with differet iitial ethaol mass fractios usig a CO 2 feed with or without ethaol at 10 MPa ad 40ºC. Ethaol mass fractio Without ethaol i CO 2 feed Efficiecy 10 3 Icrease i efficiecy (%) With 5% ethaol i CO 2 feed Efficiecy 10 3 Overall icrease i efficiecy (%) Brazilia Joural of Chemical Egieerig

7 Efficiecy of Water Removal from Water/Ethaol Mixtures 211 The Effect of P ad T o the Extractio of Water with Ethaol-Saturated CO 2 As described i Table 3, pressure ad temperature have a sigificat effect o process efficiecy. Depedig o the iitial ethaol cotet, a icrease of 4 MPa (or of 20ºC) results i efficiecies that are 20-30% higher or eve much higher with a simultaeous icrease i both pressure ad temperature. Experimetal Reproducibility of the Algorithm Table 4 compares the efficiecies of water extractio obtaied experimetally (at 10 MPa ad 40ºC) with those predicted by the diagram walkig algorithm. There is good agreemet betwee the experimetal ad predicted values, with a deviatio ot larger tha 10%. Moreover, we foud that this agreemet was slightly sesitive to chages i the solutio mass or i the iitial mass fractio of ethaol. Table 3: Compariso of efficiecies of water extractio from solutios with differet iitial ethaol mass fractios at differet temperatures ad pressures usig a CO 2 feed with 5% ethaol. 10 MPa, 40ºC 14 MPa, 40ºC 14 MPa, 60ºC Ethaol mass fractio Efficiecy 103 Efficiecy 103 Icrease i efficiecy at 40ºC (%) Efficiecy 103 Icrease i efficiecy at 14 MPa (%) Table 4: Compariso of efficiecies of water extractio obtaied experimetally (at 10 MPa, 40ºC) with those predicted with the diagram walkig algorithm. Mass of hydroalcoholic solutio (g) Mass fractio of ethaol Experimetal CO 2 Cosumed (mol) Predicted CONCLUSIONS A diagram walkig algorithm based o available experimetal data was developed to evaluate the effect of ethaol o the efficiecy of water extractio uder differet process coditios i supercritical fluid-based techiques. Ethaol feed was the key parameter for icreasig water extractio. The combiatio of the use of ethaol feed with a icrease of 4 MPa ad 20ºC icreased extractio efficiecy tefold, revealig that water extractio limitatios i supercritical atisolvet processes ca be overcome with the use of a third compoet such as ethaol. Compariso with experimetal results showed that the diagram walkig algorithm preseted here is a reliable tool for evaluatio of the effect of ethaol o the efficiecy of water removal with supercritical CO 2. Further work is beig carried out for a complete itegratio of the phase equilibria for the terary water/ethaol/co 2 system with the algorithm preseted here. Together they will allow a more geeral approach because this techique ca the be applied at ay pressure ad temperature. ACKNOWLEDGMENTS Fiacial support from FCT, Lisbo (Projects POCTI//EQU/34961/1999, POCI/EQU/55911/2004, ad postdoctoral grat BPD/5526/2001) ad the Europea Uio Programme FEDER is gratefully ackowledged. Brazilia Joural of Chemical Egieerig Vol. 23, No. 02, pp , April - Jue, 2006

8 212 M. A. Rodrigues, Ju Li, A. J. Almeida, H. A. Matos, ad E. Gomes de Azevedo REFERENCES Duarte, C., Aguiar-Ricardo, A., Ribeiro, N., Casimiro, T., ad Nues da Pote, M. N., Correlatio of Vapor-Liquid Equilibrium for Carbo Dioxide+Ethaol+Water at Temperatures from 35 to 70ºC. Separatio Sciece ad Techology, 35, 2187 (2000). Edwards, D. A., Delivery of Biological Agets by Aerosols. AIChE Joural, 48, 2 (2002). Rodrigues, M. A., Peiriço, N. M., Matos, H. A., Gomes de Azevedo, E., Lobato, M. R., ad Almeida, A., Microcomposites Theophyllie/Hydrogeated Palm Oil from a PGSS Process for Cotrolled Drug Delivery Systems. J. Supercritical Fluids, 29, 181 (2004). York, P., Strategies for Particle Desig Usig Supercritical Fluids. Pharmaceutical Sciece & Techology Today, 2, 430 (1999). Brazilia Joural of Chemical Egieerig

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

CHAPTER 3 The Simple Surface Area Measurement Module

CHAPTER 3 The Simple Surface Area Measurement Module CHAPTER 3 The Simple Surface Area Measuremet Module I chapter 2, the quality of charcoal i each batch might chage due to traditioal operatio. The quality test shall be performed before usig it as a adsorbet.

More information

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 [email protected] Abstract:

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Design Project: Indoor Location of Wireless Devices Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: [email protected] Supervised

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC

ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC 8 th Iteratioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS S u c e a v a, R o m a i a, M a y 25 27, 2 6 ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC Vadim MUKHIN 1, Elea PAVLENKO 2 Natioal Techical

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

DAME - Microsoft Excel add-in for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2

DAME - Microsoft Excel add-in for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2 Itroductio DAME - Microsoft Excel add-i for solvig multicriteria decisio problems with scearios Radomir Perzia, Jaroslav Ramik 2 Abstract. The mai goal of every ecoomic aget is to make a good decisio,

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

Convention Paper 6764

Convention Paper 6764 Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or

More information

3 Energy. 3.3. Non-Flow Energy Equation (NFEE) Internal Energy. MECH 225 Engineering Science 2

3 Energy. 3.3. Non-Flow Energy Equation (NFEE) Internal Energy. MECH 225 Engineering Science 2 MECH 5 Egieerig Sciece 3 Eergy 3.3. No-Flow Eergy Equatio (NFEE) You may have oticed that the term system kees croig u. It is ecessary, therefore, that before we start ay aalysis we defie the system that

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff, NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Data Analysis and Statistical Behaviors of Stock Market Fluctuations

Data Analysis and Statistical Behaviors of Stock Market Fluctuations 44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:

More information

Quadrat Sampling in Population Ecology

Quadrat Sampling in Population Ecology Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

P 1 2 V V V T V V. AP Chemistry A. Allan Chapter 5 - Gases

P 1 2 V V V T V V. AP Chemistry A. Allan Chapter 5 - Gases A Chemistry A. Alla Chapter 5 - Gases 5. ressure A. roperties of gases. Gases uiformly fill ay cotaier. Gases are easily compressed 3. Gases mix completely with ay other gas 4. Gases exert pressure o their

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Swaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps

Swaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while

More information

Using Four Types Of Notches For Comparison Between Chezy s Constant(C) And Manning s Constant (N)

Using Four Types Of Notches For Comparison Between Chezy s Constant(C) And Manning s Constant (N) INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH OLUME, ISSUE, OCTOBER ISSN - Usig Four Types Of Notches For Compariso Betwee Chezy s Costat(C) Ad Maig s Costat (N) Joyce Edwi Bategeleza, Deepak

More information

Electrostatic solutions for better efficiency

Electrostatic solutions for better efficiency Electrostatic solutios for better efficiecy idustry for egieers, professioals ad techicias i developmet, productio ad istallatio. www.kerste.de/e Electrostatic solutios kerste has bee the leadig supplier

More information

How to read A Mutual Fund shareholder report

How to read A Mutual Fund shareholder report Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

HCL Dynamic Spiking Protocol

HCL Dynamic Spiking Protocol ELI LILLY AND COMPANY TIPPECANOE LABORATORIES LAFAYETTE, IN Revisio 2.0 TABLE OF CONTENTS REVISION HISTORY... 2. REVISION.0... 2.2 REVISION 2.0... 2 2 OVERVIEW... 3 3 DEFINITIONS... 5 4 EQUIPMENT... 7

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Cantilever Beam Experiment

Cantilever Beam Experiment Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Catilever Beam Experimet Backgroud A disk drive maufacturer is redesigig several disk drive armature mechaisms. This is the result of evaluatio

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

How To Improve Software Reliability

How To Improve Software Reliability 2 Iteratioal Joural of Computer Applicatios (975 8887) A Software Reliability Growth Model for Three-Tier Cliet Server System Pradeep Kumar Iformatio Techology Departmet ABES Egieerig College, Ghaziabad

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

More information

Problem Solving with Mathematical Software Packages 1

Problem Solving with Mathematical Software Packages 1 C H A P T E R 1 Problem Solvig with Mathematical Software Packages 1 1.1 EFFICIENT PROBLEM SOLVING THE OBJECTIVE OF THIS BOOK As a egieerig studet or professioal, you are almost always ivolved i umerical

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand [email protected]

Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTI-TRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS

Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS 1 Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS 2 Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS 1.0 MAINTENANCE Maiteace is a routie ad recurrig activity of keepig a particular

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

Overview on S-Box Design Principles

Overview on S-Box Design Principles Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea

More information

Patentability of Computer Software and Business Methods

Patentability of Computer Software and Business Methods WIPO-MOST Itermediate Traiig Course o Practical Itellectual Property Issues i Busiess November 10 to 14, 2003 Patetability of Computer Software ad Busiess Methods Tomoko Miyamoto Patet Law Sectio Patet

More information

AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 2006 Scoring Guidelines Form B AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

More information

Partial Di erential Equations

Partial Di erential Equations Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Queuing Systems: Lecture 1. Amedeo R. Odoni October 10, 2001

Queuing Systems: Lecture 1. Amedeo R. Odoni October 10, 2001 Queuig Systems: Lecture Amedeo R. Odoi October, 2 Topics i Queuig Theory 9. Itroductio to Queues; Little s Law; M/M/. Markovia Birth-ad-Death Queues. The M/G/ Queue ad Extesios 2. riority Queues; State

More information

MARKET STRUCTURE AND EFFICIENCY OF INSURANCE COMPANIES IN AZERBAIJAN REPUBLIC

MARKET STRUCTURE AND EFFICIENCY OF INSURANCE COMPANIES IN AZERBAIJAN REPUBLIC Idia J.Sci.Res. 7 (1): 114-120, 2014 ISSN: 0976-2876 (Prit) ISSN: 2250-0138(Olie) MARKET STRUCTURE AND EFFICIENCY OF INSURANCE COMPANIES IN AZERBAIJAN REPUBLIC 1 HANIFEHZADEH LATIF PhD, Ardabil Brach,

More information

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

More information

Chair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics

Chair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

RECIPROCATING COMPRESSORS

RECIPROCATING COMPRESSORS RECIPROCATING COMPRESSORS There are various compressor desigs: Rotary vae; Cetrifugal & Axial flow (typically used o gas turbies); Lobe (Roots blowers), ad Reciprocatig. The mai advatages of the reciprocatig

More information

(VCP-310) 1-800-418-6789

(VCP-310) 1-800-418-6789 Maual VMware Lesso 1: Uderstadig the VMware Product Lie I this lesso, you will first lear what virtualizatio is. Next, you ll explore the products offered by VMware that provide virtualizatio services.

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Evaluating Model for B2C E- commerce Enterprise Development Based on DEA

Evaluating Model for B2C E- commerce Enterprise Development Based on DEA , pp.180-184 http://dx.doi.org/10.14257/astl.2014.53.39 Evaluatig Model for B2C E- commerce Eterprise Developmet Based o DEA Weli Geg, Jig Ta Computer ad iformatio egieerig Istitute, Harbi Uiversity of

More information

A Fuzzy Model of Software Project Effort Estimation

A Fuzzy Model of Software Project Effort Estimation TJFS: Turkish Joural of Fuzzy Systems (eissn: 309 90) A Official Joural of Turkish Fuzzy Systems Associatio Vol.4, No.2, pp. 68-76, 203 A Fuzzy Model of Software Project Effort Estimatio Oumout Chouseioglou

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

Engineering Data Management

Engineering Data Management BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

Log-Logistic Software Reliability Growth Model

Log-Logistic Software Reliability Growth Model Log-Logistic Software Reliability Growth Model Swapa S. Gokhale ad Kishor S. Trivedi 2y Bours College of Egg. CACC, Dept. of ECE Uiversity of Califoria Duke Uiversity Riverside, CA 9252 Durham, NC 2778-29

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece

More information

France caters to innovative companies and offers the best research tax credit in Europe

France caters to innovative companies and offers the best research tax credit in Europe 1/5 The Frech Govermet has three objectives : > improve Frace s fiscal competitiveess > cosolidate R&D activities > make Frace a attractive coutry for iovatio Tax icetives have become a key elemet of public

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College

More information

Domain 1: Designing a SQL Server Instance and a Database Solution

Domain 1: Designing a SQL Server Instance and a Database Solution Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information