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Lecture 4

Lecture 4. Solving simple stoichiometric equations. A linear system of equations. The Gauß scheme. Multiplicative elements . A non-linear system. Matrix algebra deals essentially with linear linear systems. Solving a linear system.

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Lecture 4

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  1. Lecture 4 Solving simple stoichiometric equations A linear system of equations TheGaußscheme Multiplicativeelements. A non-linear system Matrix algebra dealsessentiallywithlinearlinear systems.

  2. Solving a linear system Thedivisionthrough a vectoror a matrixis not defined! 2 equations and fourunknowns

  3. For a non-singularsquarematrixtheinverseisdefined as Singularmatricesarethosewheresomerowsorcolumnscan be expressed by a linearcombination of others. Suchcolumnsorrows do not containadditionalinformation. Theyareredundant. A matrixissingularifit’s determinant is zero. r2=2r1 r3=2r1+r2 A linearcombination of vectors Det A: determinant of A A matrixissingularifatleast one of theparameters k is not zero.

  4. Theinverse of a 2x2 matrix Theinverse of a diagonal matrix Determinant Theinverse of a squarematrixonlyexistsifits determinant differsfrom zero. Singularmatrices do not have an inverse (A•B)-1 = B-1 •A-1 ≠ A-1 •B-1 Theinversecan be unequivocallycalculated by the Gauss-Jordan algorithm

  5. Solving a simplelinear system

  6. The general solution of a linear system Identitymatrix OnlypossibleifAis not singular. IfAissingularthe system has no solution. Systems with a uniquesolution Thenumber of independent equationsequalsthenumber of unknowns. X: Not singular TheaugmentedmatrixXaugis not singular and hasthe same rank as X. Therank of a matrixis minimum number of rows/columns of thelargestnon-singularsubmatrix

  7. Consistent Rank(A) = rank(A:B) = n Infinitenumber of solutions Consistent Rank(A) = rank(A:B) < n No solution Inconsistent Rank(A) < rank(A:B) Infinitenumber of solutions Consistent Rank(A) = rank(A:B) < n Inconsistent Rank(A) < rank(A:B) No solution Consistent Rank(A) = rank(A:B) = n Infinitenumber of solutions

  8. We haveonlyfourequations but five unknowns. The system isunderdetermined. Themissingvalueisfound by dividingthevectorthroughitssmallestvalues to findthesmallestsolution for natural numbers.

  9. Includinginformation on thevalences of elements Equality of atomsinvolved We have 16 unknows but withoutexperminetnalinformationonly 11 equations. Such a system isunderdefined. A system with n unknownsneedsatleast n independent and non-contradictoryequationsfor a uniquesolution. If ni and aiareunknowns we have a non-linearsituation. We eitherdetermine ni oraiormixedvariablessuchthat no multiplicationsoccur.

  10. Thematrixissingularbecause a1, a7, and a10 do not containnewinformation Matrix algebra helps to determinewhatinformationisneeded for an unequivocalinformation. Fromtheknowledge of thesalts we get n1 to n5

  11. We havesixvariables and sixequationsthatare not contradictory and containdifferentinformation. Thematrixistherefore not singular.

  12. Linearmodelsinbiology The logistic model of population growth K denotesthemaximumpossibledensity under resourcelimitation, thecarryingcapacity. rdenotestheintrinsicpopulation growth rate. Ifr > 1 thepopulationgrowths, atr < 1 thepopulationshrinks. t N 1 1 2 5 3 15 4 45 We needfourmeasurements

  13. Population growth We have an overshot. In thenext time step thepopulationshoulddecreasebelowthecarryingcapacity. Overshot K N K/2 t Fastestpopulation growth

  14. Thetransitionmatrix Assume a genewithfourdifferentalleles. Each allele canmutateintoanther allele. Themutationprobabilitiescan be measured. Initial allele frequencies A→A B→A C→A D→A A→A Whatarethefrequenciesinthenextgeneration? A→B A→C A→D Transitionmatrix Probabilitymatrix Sum 1 1 1 1 Σ = 1 Thefrequenciesat time t+1 do onlydepent on thefrequenciesat time t but not on earlierones. Markovprocess

  15. Doesthemutationprocessresultinstable allele frequencies? Stable state vector Eigenvector of A Eigenvalue Unit matrix Eigenvector Everyprobabilitymatrixhasatleast one eigenvalue = 1. Thelargesteigenvaluedefinesthestable state vector

  16. The insulin – glycogen system At high bloodglucoselevels insulin stimulatesglycogensynthesis and inhibitsglycogenbreakdown. ThechangeinglycogenconcentrationDN can be modelled by the sum of constantproductiong and concentration dependent breakdownfN. Atequilibrium we have Thesymmetric and squarematrixDthatcontainssquaredvaluesiscalledthedispersionmatrix Thevector {-f,g} isthestationary state vector (thelargesteigenvector) of thedispersionmatrix and givestheequilibriumconditions (stationary point). Theglycogenconcentrationatequilibrium: Theequilbriumconcentrationdoes not depend on theinitialconcentrations Thevalue -1 istheeigenvalue of this system.

  17. A matrixwithncolumnshasneigenvalues and neigenvectors.

  18. Someproperties of eigenvectors IfL isthe diagonal matrix of eigenvalues: Theeigenvectors of symmetricmatricesareorthogonal Eigenvectors do not changeafter a matrixismultiplied by a scalar k. Eigenvaluesarealsomultiplied by k. Theproduct of alleigenvaluesequalsthe determinant of a matrix. The determinant is zero ifatleast one of theeigenvaluesis zero. In thiscasethematrixissingular. If A istrianagularor diagonal theeigenvalues of A arethe diagonal entries of A.

  19. Page Rank Google sortsinternetpagesaccording to a ranking of websitesbased on theprobablitites to be directled to thispage. Assume a surferclickswithprobability d to a certainwebsite A. Having N sitesintheworld (30 to 50 bilion) theprobability to reach A is d/N. Assumefurther we havefoursite A, B, C, D, withlinks to A. AssumefurtherthefoursiteshavecA, cB, cC, and cDlinks and kA, kB, kC, and kDlinks to A. Iftheprobability to be on one of thesesitesispA, pB, pC, and pD, theprobability to reach A fromany of thesitesistherefore

  20. Thetotalprobability to reach A is Google uses a fixedvalue of d=0.15. Neededisthenumber of links per website. In reality we have a linear system of 30-50 bilion equations!!! ProbabilitymatrixP Rankvectoru Internet pagesarerankedaccording to probability to be reached

  21. A B D C Larry Page (1973- SergejBrin (1973-

  22. Page Rank as an eigenvector problem In reality theconstantisverysmall Thefinalpagerankisgiven by thestationary state vector(thevector of thelargesteigenvalue).

  23. Home work and literature • Refresh: • Linearequations • Inverse • Stochiometricequations • Prepare to thenextlecture: • Arithmetic, geometricseries • Limits of functions • Sums of series • Asymptotes Literature: Mathe-online Asymptotes: www.nvcc.edu/home/.../MTH%20163%20Asymptotes%20Tutorial.pp http://www.freemathhelp.com/asymptotes.html Limits: Pauls’sonlinemath http://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspx Sums of series: http://en.wikipedia.org/wiki/List_of_mathematical_series http://en.wikipedia.org/wiki/Series_(mathematics)

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