Computational modeling techniques
|
|
- Andra Bell
- 5 years ago
- Views:
Transcription
1 Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi
2 Cntent Stichimetric matrix Calculating the mass cnservatin relatins Calculating the steady state Elementary fluxes Sensitivity cefficients 5..4 Advanced cmputatinal mdeling
3 Define all these cncepts in a general framewr, n the level f reactin-based mdels The discussin abut the mathematical details f these cncepts is fr ODE-based mdels Similar definitins can als be given fr ther math framewrs Recall T each reactin we assciate a reactin rate it tells hw fast reactants are cnsumed, prducts are prduced The frm f the reactin rate depends n the inetic law chsen fr the reactin Mass-actin Michaelis-Menten Inhibitin f varius types Hill 5..4 Advanced cmputatinal mdeling 3
4 Stichimetry The stichimetric cefficients dente the quantitative prprtin in which substrate and prduct mlecules are invlved in a reactin. Examples: Fr an irreversible reactin S +S P, the stichimetric cefficients f S, S, and P are -, -, and, respectively. In general, the stichimetric cefficients are psitive fr prducts and negative fr reactants. Fr a reversible reactin S +S P, the stichimetric cefficients f S, S, and P are -, -, and, respectively. Fr a reactin S +S P+S the stichimetric cefficients are -,, and, respectively Advanced cmputatinal mdeling 4
5 Stichimetric matrix The stichimetric cefficients f a reactin netwr cnsisting f s species and r reactins are rganized in a s-called stichimetric matrix, dented N=(n ij ) sxr, where n ij dentes the stichimetric cefficient f species S i in reactin R j. Example: Reactin netwr Stichimetric matrix r : A B r : A+C D r 3 : D B+E r 4 : A+B D+B A B C D E 5..4 Advanced cmputatinal mdeling 6
6 Stichimetric matrix Fr a reactin netwr cnsisting f s substances and r reactins the dynamics, in particular the change f cncentratins in time, is described by the fllwing system f equatins: S dt i = n ii v j fr all i s, where v j is the rate f reactin j d r j=, This can be rewritten in the matrix ntatin: ds dt = NN, where S is a vectr f the cncentratins f all the substances in the reactin netwr, i.e. S=([S ],,[S s ]) T, and v= (v,,v r ) T is the vectr f the reactin rates Advanced cmputatinal mdeling 7
7 Stichimetric matrix Example A B ( ) A+B C ( +, - ) Stichimetric matrix: Reactin rates (mass actin inetics): 8 = N [C] [A][B], [A] + = = ν ν = + [C] [A][B] [A] d d[c] d d[b] d d[a] t t t 5..4 Advanced cmputatinal mdeling
8 Stichimetric matrix The stichimetric matrix cntains valuable infrmatin abut the structure f the netwr Mass cnservatin relatins Steady states Elementary fluxes Sensitivity cefficients Discuss each f them in the rest f this lecture 5..4 Advanced cmputatinal mdeling 9
9 Mass-cnservatin relatins 5..4 Advanced cmputatinal mdeling
10 Mass cnservatin relatins Frequently, the cncentratins f several species invlved in bichemical reactin netwrs are included in s-called cnservatin sums. A characteristic feature f such species is that their gain and lss rates are equal; they can frm cmplexes with ther species r be part f ther species. A mass cnservatin relatin is a cnstant linear cmbinatin f sme f the species f the mdel 5..4 Advanced cmputatinal mdeling
11 Mass cnservatin relatins Example reactins: A A A + B A :B A :B C + A :B C N = species: A, A, B, A :B, C - The ttal amunts f A and B are cnserved in time. Neither f them is prduced nr degraded. #A + #A + #A :B = cnst. #B + #A :B = cnst Advanced cmputatinal mdeling
12 Mass cnservatin relatins T identify the cnservatin relatins we slve the fllwing equatin in matrix G: where N is the stichimetric matrix Indeed, fr such G: Example (cntinued): 3 GN = =. = GNv dt ds G GN G N S = = = = C B : A B A A 5..4 Advanced cmputatinal mdeling
13 Recall frm linear algebra The number f independent rws f G, i.e. the number f cnservatin relatins, is equal t s-ran(n). In the example s=5 and Ran(N)=3. It fllws that G cntains independent rws, i.e., there are tw mass cnservatin relatins. Observatin: if the stichimetric matrix is full ran, it fllws that the system has n cnservatin relatins Advanced cmputatinal mdeling 4
14 Mass cnservatin relatins Cnservatin relatins can be used t reduce the system f differential equatins ds/dt=nv describing the dynamics f a reactin netwr. Each cnservatin relatin leads t ne mre dependent variable, that can be expressed in terms f the independent variables and eliminated frm the system f ODEs 5..4 Advanced cmputatinal mdeling 5
15 Mass cnservatin relatins Example reactins: A A A + B A :B A :B C + A :B C The mass cnservatin relatins (based n G): [A]+[A ]+[A :B]=K [B]+[A :B]=K In ther wrds: [A]=K-[A ]-[A :B] [B]=K -[A :B] The initial system f 5 ODEs in [A], [A ], [B], [A :B], [C] is reduced t a system f 3 independent ODEs in [A ], [A :B], [C] GN G N S = = = = C B : A B A A Advanced cmputatinal mdeling
16 Steady states 5..4 Advanced cmputatinal mdeling 7
17 Steady state Steady state ne f the basic cncepts f dynamical systems thery, extensively utilized in mdelling. Steady states (statinary states, fixed pints, equilibrium pints) are determined by the fact that the values f all state variables remain cnstant in time: S(t)=S If S()=S, then ds dt = NN = Slve the equatin in the unnwn S (a vectr with s cmpnents, ne fr each variable) N is the stichimetric matrix v is a functin f the cmpnents f S An algebraic (system f) equatin(s) fr the typical inetics, e.g. mass-actin r MM 5..4 Advanced cmputatinal mdeling 8
18 Steady state Example (mass actin inetics) A B ( ) A+B C ( +, - ) Steady state algebraic equatins ([A], [B], and [C] are unnwns) r 9 = + ] [ ] [ ] [ ] [ C B A A = + = + = [C] [B] [A] [C] [B] [A] [A] [C] [B] [A] [A] N ν 5..4 Advanced cmputatinal mdeling
19 Elementary fluxes 5..4 Advanced cmputatinal mdeling
20 Elementary flux mdes Cncept f elementary flux mde a minimal set f enzymes (r, in ther wrds, reactins) that can perate at steady state the smallest sub-netwrs that allw a binetwr t functin at steady state a minimal cmbinatin f reactins whse cmbined effect maintains the netwr in steady state any subset f it des nt maintain the steady state they ffer a ey insight int the bjectives f the netwr each elementary flux mde shuld have a clear bilgical interpretatin in terms f the bjectives f the netwr determines whether a given set f enzymes/reactins are feasible at steady states Larger flux mdes can be btained by cmpsing several flux mdes: steady-state flux distributins 5..4 Advanced cmputatinal mdeling
21 Calculating the elementary flux mdes We are interested in linear cmbinatins f reactins whse cmbined effect is t preserve the steady state dente w i the weight f reactin i in the flux mde Recall: ds dt f fluxes = NN, where N is the stichimetric matrix and v is the vectr We are interested in cmbinatins f fluxes (w,,w r ) that ensure dd dd = In ther wrds, slve the equatin Nw= in the unnwn w Recall frm linear algebra: the slutin is called the ernel (r the null space) f matrix N In general, the slutin is a vectrial space we are interested in its base; all ther slutins are linear cmbinatins f the elements in the base 5..4 Advanced cmputatinal mdeling
22 Example v v v 3 S S v 4 S 3 Stichimetric matrix: NK= yields slutin K = v 4 = means that in any steady state, the rates f prductin and degradatin f S 3 must be equal 5..4 Advanced cmputatinal mdeling 3
23 Sensitivity cefficients 5..4 Advanced cmputatinal mdeling 6
24 Lcal sensitivity analysis Lcal sensitivity analysis is a methd t estimate the changes brught int the system thrugh small perturbatins in the parameters f the mdel. It prvides means t: estimate the rbustness f the mdel against small changes in the mdel, identify pssibilities fr bringing a certain desired change in the system. We write the system f ODEs describing the dynamics f a reactin netwr in the fllwing frm: where κ=(,, M ) T is the parameter vectr Advanced cmputatinal mdeling 7
25 Lcal sensitivity analysis First rder lcal cncentratin sensitivity cefficients: hw the slutin depends n small variatins in the parameter values We dente S(t,κ)=([S ](t,κ),[s ](t,κ),,[s N ](t,κ)) T the slutin f the system with respect t the parameter vectr κ. The cncentratin sensitivity cefficients are the time functins fr all i N, j M Advanced cmputatinal mdeling 8
26 Lcal sensitivity analysis Very ften hwever, the fcus is n sensitivity analysis arund steady states. In case f asympttically stable steady states, cnsider lim t ( S/ j )(t) statinary sensitivity cefficients. reflects the dependency f the steady state n the parameters f the mdel Advanced cmputatinal mdeling 3
27 Scaled sensitivity cefficients When used fr cmparing the relative effect f a parameter change in tw r mre variables, the sensitivity cefficients must have the same physical dimensin r be dimensinless. One simply cnsiders the matrix C f (dimensinless) nrmalized (als called scaled) sensitivity cefficients: Numerical estimatins f the nrmalized sensitivity cefficients fr a steady state may be cmputed in sftware applicatins such as COPASI ( r SBML-SAT ( a tl fr MATLAB ( Advanced cmputatinal mdeling 33
28 Nte: a similar analysis can als be dne fr the dependency with respect t the initial cnditins Sip it here 5..4 Advanced cmputatinal mdeling 34
29 Lcal sensitivity analysis example Mdel: A -> B, B -> A ODEs: d[a]/dt = - [A] + [B] d[b]/dt = [A] - [B] Numerical setup: [A]() = mml/ml, [B]() = mml/ml, =.5 s -, =.3 s - Steady state cncentratins: [A] ss = 7.5 mml/ml, [B] ss =.5 mml/ml 5..4 Advanced cmputatinal mdeling 36
30 Lcal sensitivity analysis Scaled statinary sensitivity cefficients: C ij [A] [B] Interpretatin: increasing frm.5 by % yields a decrease in the level f [A] at the steady state by ~.6%. Nte: these results are accurate nly fr infinitesimally small changes! 5..4 Advanced cmputatinal mdeling 37
31 Lcal sensitivity analysis Verificatin (% change): set t.55 (increase by % with respect t the riginal value) new steady state: [A] new_ss = mml/ml, [B] new_ss =.5466 mml/ml ([A] new_ss -[A] ss )/[A] ss =7.4534/7.5-» » -.6 = -.6% ([B] new_ss -[B] ss )/[B] ss =.5466/.5-=,378-».37% T be cmpared with C = -.646(%) and C = (%) Verificatin (% change): set t.33 (increase by % with respect t the riginal value) new steady state: [A] new_ss =7.958 mml/ml, [B] new_ss =.48 mml/ml ([A] new_ss -[A] ss )/[A] ss =7.958/7.5-»,64-=6,4% ([B] new_ss -[B] ss )/[B] ss =.48/.5-=, =-3,6% T be cmpared with C = (%) and C = (%) 5..4 Advanced cmputatinal mdeling 38
Computational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More informationChE 471: LECTURE 4 Fall 2003
ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.
More informationThermodynamics Partial Outline of Topics
Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)
More informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
More informationPattern Recognition 2014 Support Vector Machines
Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft
More informationBootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More information1996 Engineering Systems Design and Analysis Conference, Montpellier, France, July 1-4, 1996, Vol. 7, pp
THE POWER AND LIMIT OF NEURAL NETWORKS T. Y. Lin Department f Mathematics and Cmputer Science San Jse State University San Jse, Califrnia 959-003 tylin@cs.ssu.edu and Bereley Initiative in Sft Cmputing*
More informationALE 21. Gibbs Free Energy. At what temperature does the spontaneity of a reaction change?
Name Chem 163 Sectin: Team Number: ALE 21. Gibbs Free Energy (Reference: 20.3 Silberberg 5 th editin) At what temperature des the spntaneity f a reactin change? The Mdel: The Definitin f Free Energy S
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationMath 0310 Final Exam Review Problems
Math 0310 Final Exam Review Prblems Slve the fllwing equatins. 1. 4dd + 2 = 6 2. 2 3 h 5 = 7 3. 2 + (18 xx) + 2(xx 1) = 4(xx + 2) 8 4. 1 4 yy 3 4 = 1 2 yy + 1 5. 5.74aa + 9.28 = 2.24aa 5.42 Slve the fllwing
More informationLecture 13: Electrochemical Equilibria
3.012 Fundamentals f Materials Science Fall 2005 Lecture 13: 10.21.05 Electrchemical Equilibria Tday: LAST TIME...2 An example calculatin...3 THE ELECTROCHEMICAL POTENTIAL...4 Electrstatic energy cntributins
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationT Algorithmic methods for data mining. Slide set 6: dimensionality reduction
T-61.5060 Algrithmic methds fr data mining Slide set 6: dimensinality reductin reading assignment LRU bk: 11.1 11.3 PCA tutrial in mycurses (ptinal) ptinal: An Elementary Prf f a Therem f Jhnsn and Lindenstrauss,
More informationThermodynamics and Equilibrium
Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,
More informationCompressibility Effects
Definitin f Cmpressibility All real substances are cmpressible t sme greater r lesser extent; that is, when yu squeeze r press n them, their density will change The amunt by which a substance can be cmpressed
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 11: Mdeling with systems f ODEs In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ Mdeling with differential equatins Mdeling strategy Fcus
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationAP CHEMISTRY CHAPTER 6 NOTES THERMOCHEMISTRY
AP CHEMISTRY CHAPTER 6 NOTES THERMOCHEMISTRY Energy- the capacity t d wrk r t prduce heat 1 st Law f Thermdynamics: Law f Cnservatin f Energy- energy can be cnverted frm ne frm t anther but it can be neither
More informationECEN 4872/5827 Lecture Notes
ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationPart 3 Introduction to statistical classification techniques
Part 3 Intrductin t statistical classificatin techniques Machine Learning, Part 3, March 07 Fabi Rli Preamble ØIn Part we have seen that if we knw: Psterir prbabilities P(ω i / ) Or the equivalent terms
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More informationThe blessing of dimensionality for kernel methods
fr kernel methds Building classifiers in high dimensinal space Pierre Dupnt Pierre.Dupnt@ucluvain.be Classifiers define decisin surfaces in sme feature space where the data is either initially represented
More informationRigid Body Dynamics (continued)
Last time: Rigid dy Dynamics (cntinued) Discussin f pint mass, rigid bdy as useful abstractins f reality Many-particle apprach t rigid bdy mdeling: Newtn s Secnd Law, Euler s Law Cntinuus bdy apprach t
More information4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression
4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw
More informationA Few Basic Facts About Isothermal Mass Transfer in a Binary Mixture
Few asic Facts but Isthermal Mass Transfer in a inary Miture David Keffer Department f Chemical Engineering University f Tennessee first begun: pril 22, 2004 last updated: January 13, 2006 dkeffer@utk.edu
More informationChapter 17 Free Energy and Thermodynamics
Chemistry: A Mlecular Apprach, 1 st Ed. Nivald Tr Chapter 17 Free Energy and Thermdynamics Ry Kennedy Massachusetts Bay Cmmunity Cllege Wellesley Hills, MA 2008, Prentice Hall First Law f Thermdynamics
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More information[ ] [ ] [ ] [ ] [ ] [ J] dt x x hard to solve in general solve it numerically. If there is no convection. is in the absence of reaction n
.3 The material balance equatin Net change f [J] due t diffusin, cnvectin, and reactin [ ] [ ] [ ] d J J J n = D v k [ J ] fr n - th reactin dt x x hard t slve in general slve it numerically If there is
More informationSurface and Contact Stress
Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 3: Mdeling change (2) Mdeling using prprtinality Mdeling using gemetric similarity In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ http://users.ab.fi/ipetre/cmpmd/
More informationENGI 4430 Parametric Vector Functions Page 2-01
ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
More informationLecture 12: Chemical reaction equilibria
3.012 Fundamentals f Materials Science Fall 2005 Lecture 12: 10.19.05 Chemical reactin equilibria Tday: LAST TIME...2 EQUATING CHEMICAL POTENTIALS DURING REACTIONS...3 The extent f reactin...3 The simplest
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationComplex Reactions and Mechanisms (continued)
5.60 Spring 2005 Lecture #29 page 1 Cmplex Reactins and Mechanisms (cntinued) Sme cmments abut analyzing kinetic data A) Reactins with ne reactant: A prducts a) Plt r analyze [A vs. t ln[a vs. t 1/[A vs.
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationChapters 29 and 35 Thermochemistry and Chemical Thermodynamics
Chapters 9 and 35 Thermchemistry and Chemical Thermdynamics 1 Cpyright (c) 011 by Michael A. Janusa, PhD. All rights reserved. Thermchemistry Thermchemistry is the study f the energy effects that accmpany
More informationChem 115 POGIL Worksheet - Week 8 Thermochemistry (Continued), Electromagnetic Radiation, and Line Spectra
Chem 115 POGIL Wrksheet - Week 8 Thermchemistry (Cntinued), Electrmagnetic Radiatin, and Line Spectra Why? As we saw last week, enthalpy and internal energy are state functins, which means that the sum
More informationA Matrix Representation of Panel Data
web Extensin 6 Appendix 6.A A Matrix Representatin f Panel Data Panel data mdels cme in tw brad varieties, distinct intercept DGPs and errr cmpnent DGPs. his appendix presents matrix algebra representatins
More informationModeling of ship structural systems by events
UNIVERISTY OF SPLIT FACULTY OF ELECTRICAL ENGINEERING, MECHANICAL ENGINEERING AND NAVAL ARCHITECTURE Mdeling ship structural systems by events Brank Blagjević Mtivatin Inclusin the cncept entrpy rm inrmatin
More informationTypes of Energy COMMON MISCONCEPTIONS CHEMICAL REACTIONS INVOLVE ENERGY
CHEMICAL REACTIONS INVOLVE ENERGY The study energy and its transrmatins is knwn as thermdynamics. The discussin thermdynamics invlve the cncepts energy, wrk, and heat. Types Energy Ptential energy is stred
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More informationChem 75 February 16, 2017 Exam 2 Solutions
1. (6 + 6 pints) Tw quick questins: (a) The Handbk f Chemistry and Physics tells us, crrectly, that CCl 4 bils nrmally at 76.7 C, but its mlar enthalpy f vaprizatin is listed in ne place as 34.6 kj ml
More information5.60 Thermodynamics & Kinetics Spring 2008
MIT OpenCurseWare http://cw.mit.edu 5.60 Thermdynamics & Kinetics Spring 2008 Fr infrmatin abut citing these materials r ur Terms f Use, visit: http://cw.mit.edu/terms. 5.60 Spring 2008 Lecture #17 page
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationWhen a substance heats up (absorbs heat) it is an endothermic reaction with a (+)q
Chemistry Ntes Lecture 15 [st] 3/6/09 IMPORTANT NOTES: -( We finished using the lecture slides frm lecture 14) -In class the challenge prblem was passed ut, it is due Tuesday at :00 P.M. SHARP, :01 is
More informationBiocomputers. [edit]scientific Background
Bicmputers Frm Wikipedia, the free encyclpedia Bicmputers use systems f bilgically derived mlecules, such as DNA and prteins, t perfrm cmputatinal calculatins invlving string, retrieving, and prcessing
More information7 TH GRADE MATH STANDARDS
ALGEBRA STANDARDS Gal 1: Students will use the language f algebra t explre, describe, represent, and analyze number expressins and relatins 7 TH GRADE MATH STANDARDS 7.M.1.1: (Cmprehensin) Select, use,
More informationLeast Squares Optimal Filtering with Multirate Observations
Prc. 36th Asilmar Cnf. n Signals, Systems, and Cmputers, Pacific Grve, CA, Nvember 2002 Least Squares Optimal Filtering with Multirate Observatins Charles W. herrien and Anthny H. Hawes Department f Electrical
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationLim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)?
THESE ARE SAMPLE QUESTIONS FOR EACH OF THE STUDENT LEARNING OUTCOMES (SLO) SET FOR THIS COURSE. SLO 1: Understand and use the cncept f the limit f a functin i. Use prperties f limits and ther techniques,
More informationDrought damaged area
ESTIMATE OF THE AMOUNT OF GRAVEL CO~TENT IN THE SOIL BY A I R B O'RN EMS S D A T A Y. GOMI, H. YAMAMOTO, AND S. SATO ASIA AIR SURVEY CO., l d. KANAGAWA,JAPAN S.ISHIGURO HOKKAIDO TOKACHI UBPREFECTRAl OffICE
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More informationUniversity Chemistry Quiz /04/21 1. (10%) Consider the oxidation of ammonia:
University Chemistry Quiz 3 2015/04/21 1. (10%) Cnsider the xidatin f ammnia: 4NH 3 (g) + 3O 2 (g) 2N 2 (g) + 6H 2 O(l) (a) Calculate the ΔG fr the reactin. (b) If this reactin were used in a fuel cell,
More informationOF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
U. S. FOREST SERVICE RESEARCH PAPER FPL 50 DECEMBER U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More information8 th Grade Math: Pre-Algebra
Hardin Cunty Middle Schl (2013-2014) 1 8 th Grade Math: Pre-Algebra Curse Descriptin The purpse f this curse is t enhance student understanding, participatin, and real-life applicatin f middle-schl mathematics
More informationDetermining the Accuracy of Modal Parameter Estimation Methods
Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system
More information5 th grade Common Core Standards
5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin
More informationPipetting 101 Developed by BSU CityLab
Discver the Micrbes Within: The Wlbachia Prject Pipetting 101 Develped by BSU CityLab Clr Cmparisns Pipetting Exercise #1 STUDENT OBJECTIVES Students will be able t: Chse the crrect size micrpipette fr
More informationSemester 2 AP Chemistry Unit 12
Cmmn In Effect and Buffers PwerPint The cmmn in effect The shift in equilibrium caused by the additin f a cmpund having an in in cmmn with the disslved substance The presence f the excess ins frm the disslved
More informationChapter 3: Cluster Analysis
Chapter 3: Cluster Analysis } 3.1 Basic Cncepts f Clustering 3.1.1 Cluster Analysis 3.1. Clustering Categries } 3. Partitining Methds 3..1 The principle 3.. K-Means Methd 3..3 K-Medids Methd 3..4 CLARA
More informationMath 105: Review for Exam I - Solutions
1. Let f(x) = 3 + x + 5. Math 105: Review fr Exam I - Slutins (a) What is the natural dmain f f? [ 5, ), which means all reals greater than r equal t 5 (b) What is the range f f? [3, ), which means all
More informationSection 6-2: Simplex Method: Maximization with Problem Constraints of the Form ~
Sectin 6-2: Simplex Methd: Maximizatin with Prblem Cnstraints f the Frm ~ Nte: This methd was develped by Gerge B. Dantzig in 1947 while n assignment t the U.S. Department f the Air Frce. Definitin: Standard
More informationLecture 10, Principal Component Analysis
Principal Cmpnent Analysis Lecture 10, Principal Cmpnent Analysis Ha Helen Zhang Fall 2017 Ha Helen Zhang Lecture 10, Principal Cmpnent Analysis 1 / 16 Principal Cmpnent Analysis Lecture 10, Principal
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More information(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f
1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationChapter 4. Unsteady State Conduction
Chapter 4 Unsteady State Cnductin Chapter 5 Steady State Cnductin Chee 318 1 4-1 Intrductin ransient Cnductin Many heat transfer prblems are time dependent Changes in perating cnditins in a system cause
More informationNGSS High School Physics Domain Model
NGSS High Schl Physics Dmain Mdel Mtin and Stability: Frces and Interactins HS-PS2-1: Students will be able t analyze data t supprt the claim that Newtn s secnd law f mtin describes the mathematical relatinship
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationMaterials Engineering 272-C Fall 2001, Lecture 7 & 8 Fundamentals of Diffusion
Materials Engineering 272-C Fall 2001, Lecture 7 & 8 Fundamentals f Diffusin Diffusin: Transprt in a slid, liquid, r gas driven by a cncentratin gradient (r, in the case f mass transprt, a chemical ptential
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More informationSUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical model for microarray data analysis
SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical mdel fr micrarray data analysis David Rssell Department f Bistatistics M.D. Andersn Cancer Center, Hustn, TX 77030, USA rsselldavid@gmail.cm
More informationPhysics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1
Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationMATCHING TECHNIQUES Technical Track Session VI Céline Ferré The World Bank
MATCHING TECHNIQUES Technical Track Sessin VI Céline Ferré The Wrld Bank When can we use matching? What if the assignment t the treatment is nt dne randmly r based n an eligibility index, but n the basis
More informationMATCHING TECHNIQUES. Technical Track Session VI. Emanuela Galasso. The World Bank
MATCHING TECHNIQUES Technical Track Sessin VI Emanuela Galass The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Emanuela Galass fr the purpse f this wrkshp When can we use
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationMathematics and Computer Sciences Department. o Work Experience, General. o Open Entry/Exit. Distance (Hybrid Online) for online supported courses
SECTION A - Curse Infrmatin 1. Curse ID: 2. Curse Title: 3. Divisin: 4. Department: 5. Subject: 6. Shrt Curse Title: 7. Effective Term:: MATH 70S Integrated Intermediate Algebra Natural Sciences Divisin
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid
More informationHeat Effects of Chemical Reactions
eat Effects f hemical Reactins Enthalpy change fr reactins invlving cmpunds Enthalpy f frmatin f a cmpund at standard cnditins is btained frm the literature as standard enthalpy f frmatin Δ (O (g = -9690
More informationAppendix I: Derivation of the Toy Model
SPEA ET AL.: DYNAMICS AND THEMODYNAMICS OF MAGMA HYBIDIZATION Thermdynamic Parameters Appendix I: Derivatin f the Ty Mdel The ty mdel is based upn the thermdynamics f an isbaric twcmpnent (A and B) phase
More informationComparing Several Means: ANOVA. Group Means and Grand Mean
STAT 511 ANOVA and Regressin 1 Cmparing Several Means: ANOVA Slide 1 Blue Lake snap beans were grwn in 12 pen-tp chambers which are subject t 4 treatments 3 each with O 3 and SO 2 present/absent. The ttal
More informationCHAPTER Read Chapter 17, sections 1,2,3. End of Chapter problems: 25
CHAPTER 17 1. Read Chapter 17, sectins 1,2,3. End f Chapter prblems: 25 2. Suppse yu are playing a game that uses tw dice. If yu cunt the dts n the dice, yu culd have anywhere frm 2 t 12. The ways f prducing
More informationIntroduction to Smith Charts
Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary
More informationMore Tutorial at
Answer each questin in the space prvided; use back f page if extra space is needed. Answer questins s the grader can READILY understand yur wrk; nly wrk n the exam sheet will be cnsidered. Write answers,
More informationMatter Content from State Frameworks and Other State Documents
Atms and Mlecules Mlecules are made f smaller entities (atms) which are bnded tgether. Therefre mlecules are divisible. Miscnceptin: Element and atm are synnyms. Prper cnceptin: Elements are atms with
More informationFIELD QUALITY IN ACCELERATOR MAGNETS
FIELD QUALITY IN ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract The field quality in the supercnducting magnets is expressed in terms f the cefficients f the Furier series
More informationPart One: Heat Changes and Thermochemistry. This aspect of Thermodynamics was dealt with in Chapter 6. (Review)
CHAPTER 18: THERMODYNAMICS AND EQUILIBRIUM Part One: Heat Changes and Thermchemistry This aspect f Thermdynamics was dealt with in Chapter 6. (Review) A. Statement f First Law. (Sectin 18.1) 1. U ttal
More informationNUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED CONCRETE STRUCTURES WITH CRACKS
VIII Internatinal Cnference n Fracture Mechanics f Cnete and Cnete Structures FraMCS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds) NUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED
More informationDepartment of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets
Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0
More informationSubject description processes
Subject representatin 6.1.2. Subject descriptin prcesses Overview Fur majr prcesses r areas f practice fr representing subjects are classificatin, subject catalging, indexing, and abstracting. The prcesses
More informationA Chemical Reaction occurs when the of a substance changes.
Perid: Unit 8 Chemical Reactin- Guided Ntes Chemical Reactins A Chemical Reactin ccurs when the f a substance changes. Chemical Reactin: ne r mre substances are changed int ne r mre new substances by the
More informationCHEM Thermodynamics. Change in Gibbs Free Energy, G. Review. Gibbs Free Energy, G. Review
Review Accrding t the nd law f Thermdynamics, a prcess is spntaneus if S universe = S system + S surrundings > 0 Even thugh S system
More informations of the two electrons are strongly coupled together to give a l couple together to give a resultant APPENDIX I
APPENDIX I Cupling Schemes and Ntatin An extensive treatment f cupling schemes and ntatin is given by White r Kuhn. A brief review is given here t allw ne t read this manual with sme insight. The mtins
More information**DO NOT ONLY RELY ON THIS STUDY GUIDE!!!**
Tpics lists: UV-Vis Absrbance Spectrscpy Lab & ChemActivity 3-6 (nly thrugh 4) I. UV-Vis Absrbance Spectrscpy Lab Beer s law Relates cncentratin f a chemical species in a slutin and the absrbance f that
More information