System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den
|
|
- Maurice Glenn
- 6 years ago
- Views:
Transcription
1 University of Groningen System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1996 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Hof, J. M. V. D. (1996). System theory and system identification of compartmental systems s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 1 maximum. Download date:
2 15 Chapter 2 Compartmental Systems 2.1 Introduction Compartmental systems are mathematical systems that are frequently used in biology and mathematics. Also a subclass of the class of chemical processes can be modelled as compartmental systems. A compartmental system consists of several compartments with more or less homogeneous amounts of material. The compartments interact by processes of transportation and diusion. The dynamics of a compartmental system is derived from mass balance considerations. In this thesis linear compartmental systems consisting of inputs, states, and outputs are studied. The outputs of these systems are not the real outputs, i.e., material leaving the system, but the observations of the amount or concentration of material, for example in one or more compartments. The inputs, states, and outputs are positive, so these systems are in system theory called positive linear systems. In this chapter some properties of compartmental systems that are needed in this thesis are presented. For other properties of compartmental systems, see for example [2, 14, 55, 76, 77, 143]. For examples of positive systems that are not necessarily compartmental, see [123]. The outline of the chapter is as follows. In Section 2.2 continuous-time linear compartmental systems are considered and in Section 2.3 the discrete-time case is treated. A graphical representation of compartmental systems is presented in Section 2.4.
3 16 Chapter 2. Compartmental Systems Notation In this thesis the set R + =[;1) is called the set of positive real numbers and (; 1) the set of strictly positive real numbers. This terminology is used in [35, Section 2.2]. Let Z + = f1; 2; 3;:::g denote the set of the positive integers and NI = f; 1; 2;:::g the set of the natural numbers. Denote by R n + the set of n- tuples of the positive real numbers. Note that R n + is not a vector space over R because it does not admit an inverse with respect to addition. For n 2 Z + let Z n = f1; 2; 3;:::;ng and NI n = f; 1; 2;:::;ng. The set R km + of matrices over R + will be called the set of positive matrices of size k by m. For A 2 R km, A T denotes its transpose. For matrices A; B 2 R nm, write A B if a ij b ij for all i 2 Z n, j 2 Z m, and A>Bif A B and A 6= B. 2.2 Continuous-time compartmental systems A compartmental system is a system consisting of a nite number of subsystems, which are called compartments. Each compartment is kinetically homogeneous, i.e., any material entering the compartment is instantaneously mixed with the material of the compartment. Compartmental systems are dominated by the law of conservation of mass. They form also natural models for other areas of applications that are subject to conservation laws. Throughout this thesis it is assumed that the structure of the system is known, but that some parameters are unknown, for example the elements of F, I, and C, mentioned below. Consider an n-compartmental system. The behaviour of the ith compartment can be represented as in Figure 2.1. F ji I i q i F ij F i Figure 2.1: One compartment with possible ows. In this gure, q i denotes the amount of material considered in compartment i. The arrows represent the ows into and out of the compartment. The ow into compartment i from outside the system is denoted by I i and it is called the inow. Usually I i is a function of time or just a constant, but occasionally may be dependent onqas well. The symbols F ij and F ji represent the ow
4 2.2. Continuous-time compartmental systems 17 from compartment j into compartment i and the ow from compartment i into compartment j, respectively. Finally, F i istheoutow to the environment from compartment i. The mass balance equations for every compartment can be written as _q i = X j6=i( F ji + F ij )+I i F i : (2.2.1) We assume that the ows F ij can be written as: F ij = f ij q j ; i =;:::;n; j =1;:::;n; i6=j; in which the functions f ij are called the fractional transfer coecients, see [77, Section 2.1]. In general, f ij are functions of q and time t. If f ij is independent of q, the system is a linear system. In this thesis it is assumed that f ij is also independent of the time t, i.e., the system is a time-invariant linear system. Using this, (2.2.1) can be written as _q = Fq+I; where q = q 1 q n T 2 R n +, F = (f ij ) 2 R nn, with f ij constants for i 6= j, f ii = (f i + P j6=i f ji), and I denotes the inow from outside the system. Since I i and q i, this system is easily seen to be a positive linear system, i.e., a system with positive input, state, and output, if for the output is taken y = Cq; y 2 R k ; C 2 R kn + ; where y denotes the vector of the observations. Note that the output is not the outow of the compartmental system. The outow, which is sometimes also called excretion, represents the ow of material leaving the system. The outputs of an experiment are measurements and usually dier from the material outows. On the other hand, the terms inow and input can be used interchangeably. Another property of compartmental systems is that the total ow out of a compartmentover any time interval cannot be larger than the amount that was initially present plus the amount that owed into the compartment during that interval. Together with the constraints on positive linear systems, this comes down to 1. f ij ; for all i; j 2 Z n ; i 6= j; 2. i=1 f ij ; for all j 2 Z n : The rst condition guarantees that positive solutions stay positive. The second condition stems from the inequality f j. A matrix F satisfying Conditions 1 and2above is said to be a compartmental matrix. Condition 2 states that all column sums of the matrix F 2 R nn are less than or equal to zero. Below some properties of compartmental matrices from the literature will be discussed that are needed in this thesis. References are [46, 48, 77, 136].
5 18 Chapter 2. Compartmental Systems Denition A matrix A 2 R nn is said to be reducible if there exists a permutation matrix P 2 R nn such that PAP T U = ; Q R with U and R square matrices. The matrix A is said to be irreducible if A is not reducible. 2 Let F 2 R nn be a compartmental matrix. Then it follows from [11, Theorem 6.4.6] that (F ) f2cjre() < or=g. Here (F ) denotes the spectrum of F, and Re() means the real part of. Since a system _x = Fx is asymptotically stable if and only if (F ) f 2 C j Re() < g, a compartmental matrix is asymptotically stable if and only if =2 (F ). In the rest of this section compartmental matrices having zero as an eigenvalue are characterized. Proposition (Adapted from [136, Theorem III].) Let F 2 R nn beanirreducible compartmental matrix. Then 2 (F ) if and only if P n i=1 f ij =for all j 2 Z n An important concept for the analysis of compartmental systems is that of a trap, dened below. Denition Consider an n-compartmental system. A trap is a compartment or a set of compartments from which there are no transfers or ows to the environment nor to compartments that are not in that set. A trap is said to be simple if it does not strictly contain a trap. 2 In the physical literature traps are usually referred to as sinks. Let S be a linear compartmental system consisting of compartments C 1, C 2 ;:::;C n and let q j be the amount of material in C j. Let T S be a subsystem of S. Renumbering the compartments, assume T consists of the compartments C m ;C m+1 ;:::;C n, for m n. Let F 2 R nn be the compartmental matrix corresponding to S, consistent with this renumbering. Then T is a trap if and only if f ij =; for all (i; j) such that j = m; m +1;:::;n; i=;1;:::;m 1: The following two theorems are due to Fife [46]. (2.2.2) Theorem The linear compartmental system S contains a trap if and only if one of the following conditions holds: 1. i=1 f ij =; for all j 2 Z n ; 2. there exists a permutation matrix P 2 R nn such that PFP T U = ; Q R with U, R square matrices and the sum of every column of R is zero.
6 2.3. Discrete-time compartmental systems 19 Theorem The linear compartmental system S contains a trap if and only if 2 (F ). In response of Fife [46], Foster and Jacquez [48] derived the following result. See also Theorems 1 and 2 together with their proofs in [77]. Theorem Let S be a compartmental system with system matrix F. 1. Zero is an eigenvalue of F of multiplicity m 2 Z + if and only if S contains m simple traps. 2. Assume zero is an eigenvalue of F of multiplicity m 2 Z +. Then there exists a partition of S into a disjoint union of subsystems S = S 1 [ S 2 [[S p ; such that S i receives no input from S i+1 ;:::;S p, i = 1;:::;p 1, and S p m+1 ;:::;S p are traps. Relative to this partition the system matrix is given by PFP T = F, e with ef = F F p m;1 F p m;p m F p m+1;1 F p m+1;p m F p m+1;p m F p1 F p;p m F pp where F ii is irreducible for all i 2 Z p and zero is an eigenvalue of F ii of multiplicity 1 for i = p m +1;:::;p, and the sum of every column of F ii, i = p m +1;:::;p,iszero. An additional consequence of this theorem is that if zero is an eigenvalue of a compartmental matrix of (algebraic) multiplicity m, the geometric multiplicity is also m, so there are always m independent eigenvectors for the eigenvalue zero. 1 C A ; 2.3 Discrete-time compartmental systems In this section discrete-time compartmental systems will be considered. For that purpose it is assumed that transfer of material occurs at discrete times t 1 ;t 2 ;:::; or a continuous-time system is sampled at discrete times, in which case the state at time t k has been changed into the state at time t k+1. What happens in between will not be considered explicitly. So this can also be seen as if a transfer has occurred at time t k+1. The discrete times will be assumed to be equally spaced, to obtain a time-invariant system. Let this space be the unit time, so t k+1 = t k +1.
7 2 Chapter 2. Compartmental Systems Let q i (t) be the amount of material in the ith compartment at time t. The amount transferred from the jth to the ith compartment between time t and time t +1 is G ij (t). This transferred material will be assumed to be linear dependent onq j, i.e., G ij (t) =g ij q j (t). The state at time t + 1 will be given by q i (t +1)= X j6=i g ij q j (t)+i i (t)+g ii q i (t); where g ii q i (t) is the amount of material that was in compartment i at time t and is still (or again) in compartment i at time t +1. This amount g ii q i (t) is equal to q i (t) minus the amount that left compartment j: X n g ii q i (t) =q i (t) g oi q i (t) g ji q i (t) = 1 g oi g ji qi (t): Hence dene g ii =1 g oi n X j=1;j6=i g ji : j=1;j6=i j=1;j6=i The total outow of a compartment at time t +1 cannot be larger than the amount that was present at time t, if the inow from outside is assumed to be zero. Together with the constraints on positive linear systems (in discretetime), this comes down to 1. g ij ; for all i; j 2 Z n ; 2. i=1 g ij 1; for all j 2 Z n : A matrix G satisfying Conditions 1 and 2 above is said to be a compartmental matrix (in the discrete-time case). Condition 2 states that all column sums of G =(g ij ) 2 R nn + are less than or equal to one. Below properties of compartmental matrices in discrete-time will be discussed, analogous to the continuous-time case. In the rest of this section, G refers to a discrete-time compartmental matrix, whereas F refers to a continuous-time compartmental matrix. Let G 2 R nn + be a compartmental matrix. Then (G) f2cjjj1g, since the sum of every column of G is less than or equal to one, see [97, Section 6.2], or [11, Chapter 2]. Because a system x(t +1) = Gx(t) is asymptotically stable if and only if (G) f 2 C j jj < 1g, it follows from the Perron-Frobenius Theorem, see [97], that a compartmental system is asymptotically stable if and only if the spectral radius (G) 6= 1, which is equivalent to 1 =2(G). Analogously to the continuous-time case, compartmental matrices having spectral radius one are characterized. Proposition Let G 2 P R nn + be an irreducible compartmental matrix. Then n (G) =1if and only if i=1 g ij =1for all j 2 Z n.
8 2.3. Discrete-time compartmental systems 21 Proof. This follows from [11, Theorem ]. 2 A trap in an n-compartmental system is dened in the same way as for continuous-time systems, see Denition As in the continuous-time case, let C 1 ;C 2 ;:::;C n be the compartments of a linear compartmental system S. After renumbering, let T S consist of the compartments C m ;C m+1 ;:::;C n, for m n. Then T is a trap if and only if g ij =; for all (i; j) such that j = m; m +1;:::;n; i=;1;:::;m 1; (2.3.1) where G = (g ij ) 2 R nn + is the compartmental matrix corresponding to S. Consider F = G I 2 R nn. Since 1. f ij = g ij ; for i; j 2 Z n ; i 6= j; 2. j=1 f ji = g ii 1+ j=1;j6=i g ji = j=1 g ji 1 ; F is a continuous-time compartmental matrix. Assume F is the system matrix for a continuous-time compartmental system S F and let T F S F consist of the last n m + 1 compartments e Cm ;:::; e Cn. Proposition Consider T S and T F S F dened above. Then T is a (simple) trap if and only if T F is a (simple) trap. Proof. The subsystem T is a trap if and only if (2.3.1) holds, which is equivalent to g ij =; for all (i; j) such that j = m; m +1;:::;n; i=1;2;:::;m 1; and P g jj =1 n i=1;i6=j g ij; for all j = m; m +1;:::;n; since g j =. Because f ij = g ij for i 6= j and f jj = g jj to f ij =; for all (i; j) such that j = m; m +1;:::;n; i=1;2;:::;m 1; and P n f jj = for all j = m; m +1;:::;n; i=1;i6=j f ij; P n 9 >= >; (2.3.2) 1, (2.3.2) is equivalent which is, because f j = i=1 f ij, equivalent to (2.2.2), i.e., T F is a trap. In the same way itcan be proved that T is a simple trap if and only if T F is a simple trap. 2 Using Proposition 2.3.2, the following theorems, analogous to Theorems 2.2.4, 2.2.5, and 2.2.6, can be proved. Theorem The linear compartmental system S contains a trap if and only if one of the following conditions holds. 9 >= >;
9 22 Chapter 2. Compartmental Systems 1. i=1 g ij =1; for all j 2 Z n ; 2. there exists a permutation matrix P 2 R nn such that PGP T U1 = ; R 1 Q 1 with U 1, R 1 square matrices and the sum of every column of R 1 is one. Proof. Since i=1 f ij = n X i=1 g ij 1 and PFP T = P(G I)P T = PGP T I = U =: ; Q R U1 I Q 1 R 1 I in which the sum of every column of R = R 1 I is equal to the sum of every column of R 1 minus one, it follows that the conditions stated in the theorem are equivalent to the conditions stated in Theorem The theorem now follows using Proposition Theorem The linear compartmental system S contains a trap if and only if 1 2 (G). Proof. The following statements are equivalent: (i) 2 (F ); (ii) det(f ) = ; (iii) det(g I) = ; (iv) 1 2 (G); (v) (G) = 1. The last equivalence relation follows from the Perron-Frobenius Theorem. With Theorem and Proposition 2.3.2, the theorem is proved. 2 Theorem Let S be a compartmental system with system matrix G. 1. One is an eigenvalue of G of multiplicity m 2 Z + if and only if S contains m simple traps. 2. Assume one is an eigenvalue of G of multiplicity m 2 Z +. Then there exists a partition of S into a disjoint union of subsystems S = S 1 [ S 2 [[S p ; such that S i receives no input from S i+1 ;:::;S p, i = 1;:::;p 1, and S p m+1 ;:::;S p are traps. Relative to this partition the system matrix is
10 2.4. Graphical representations of compartmental systems 23 given by PGP T = e G, with eg = G G p m;1 G p m;p m G p m+1;1 G p m+1;p m G p m+1;p m G p1 G p;p m G pp where G ii is irreducible for all i 2 Z p and one is an eigenvalue of G ii of multiplicity 1 for i = p m +1;:::;p, and the sum of every column of G ii, i = p m +1;:::;p, is one. Proof. 1. The following statements are equivalent one is an eigenvalue of G of multiplicity m 2 Z + ; det(g I) =( 1) m p() with p(1) 6= ; det(f I) = m p 1 () with p 1 () = p(1) 6= ; zero is an eigenvalue of F of multiplicity m 2 Z +. The equivalence between the second and the third statement follows from det(f I) = det(g I I) = det(g ( +1)I)=((+1) 1) m p( +1)= m p 1 () with p 1 () =p(+1). Now statement 1 follows from Proposition and statement 1 of Theorem Consider the following statements. a. one is an eigenvalue of G of multiplicity m 2 Z + ; b. zero is an eigenvalue of F of multiplicity m 2 Z + ; c. statement 2 in Theorem 2.2.6; d. statement 2 in Theorem From 1 it follows that (a, b), and Theorem provides (b ) c). Noting that PGP T = PFP T +I, G ij = F ij for i 6= j, and G ii = F ii + I, where the sum of every column of G ii is equal to the sum of every column of F ii plus 1, the implication (c ) d) follows from the statements of the proof of part 1 for m =1. This completes the proof of part C A ; 2.4 Graphical representations of compartmental systems A compartmental system can also be represented by a directed graph, see for example [34] or [76, Chapter 3]. Every compartment is represented by a vertex and there is a directed arc from q j to q i if and only if f ij >. To incorporate outows (f j > ) and inows into this graphical representation,
11 24 Chapter 2. Compartmental Systems the following vertices and arcs can be added. The environment is treated as a single compartment q, so all outows go to this compartment. Then there is a directed arc from P q j to q if and only if f j >. Note that in the continuoustime case f j = n P i=1 f ij and in the discrete-time case f j =1 n i=1 f ij. Every inow can be treated as a separate source point, or all inows can be lumped into one single source point. The choice between these two possibilities, or even a combination of the two, depends on the nature of the inows and on the problem. In any case, there is an arc between a source point andavertex q j if I j 6=. Example Consider the compartmental system with input I 2 R 4 and system matrix F 2 R 44, F = f 21 f 12 f 21 f 12 f 32 f 23 f 32 f 23 f 43 f 34 f 43 f 34 f 4 1 C A ; I = I 1 1 C A : f21 f32 f43 I1 f4 I q1 q2 q3 q4 q f12 f23 f34 Figure 2.2: Example of catenary system. Figure 2.2 represents its graph. An n-compartmental system of such a form is called a catenary system. In a catenary system the n compartments are arranged in a linear array such that every compartment exchanges only with its immediate neighbours. This particular example, in which there is only inow into the rst compartment and only outow from the last compartment, is a special case. The system matrix for a catenary system has nonzero entries only on the main diagonal and on the rst super-diagonal and the rst sub-diagonal. 2 Example Consider the compartmental system with system matrix F = 1 f 41 f 14 f 42 f 24 C f 43 f 34 A 2 R44 ; f 41 f 42 f 43 f 44 with f 44 = f 14 f 24 f 34 f 4. The corresponding graph is Figure 2.3. An n-compartmental system of this form, with a central compartment, is called a mammillary system. Its central compartment is called the mother compartment. There exists only exchange between the mother compartment and
12 2.4. Graphical representations of compartmental systems 25 q2 f24 f42 f43 f14 q3 q1 f34 q4 f41 f4 q Figure 2.3: Example of mammillary system. another compartment, not directly between the other compartments. The excretion and the inow can in principle be in any compartment. In Chapter 8 examples of mammillary systems will be treated.
University of Groningen. Statistical Auditing and the AOQL-method Talens, Erik
University of Groningen Statistical Auditing and the AOQL-method Talens, Erik IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check
More informationCitation for published version (APA): Ruíz Duarte, E. An invitation to algebraic number theory and class field theory
University of Groningen An invitation to algebraic number theory and class field theory Ruíz Duarte, Eduardo IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationUniversity of Groningen. Statistical inference via fiducial methods Salomé, Diemer
University of Groningen Statistical inference via fiducial methods Salomé, Diemer IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationCitation for published version (APA): Shen, C. (2006). Wave Propagation through Photonic Crystal Slabs: Imaging and Localization. [S.l.]: s.n.
University of Groningen Wave Propagation through Photonic Crystal Slabs Shen, Chuanjian IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it.
More informationUniversity of Groningen. Morphological design of Discrete-Time Cellular Neural Networks Brugge, Mark Harm ter
University of Groningen Morphological design of Discrete-Time Cellular Neural Networks Brugge, Mark Harm ter IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationCitation for published version (APA): Sok, R. M. (1994). Permeation of small molecules across a polymer membrane: a computer simulation study s.n.
University of Groningen Permeation of small molecules across a polymer membrane Sok, Robert Martin IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite
More informationCitation for published version (APA): Halbersma, R. S. (2002). Geometry of strings and branes. Groningen: s.n.
University of Groningen Geometry of strings and branes Halbersma, Reinder Simon IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationUniversity of Groningen. Laser Spectroscopy of Trapped Ra+ Ion Versolato, Oscar Oreste
University of Groningen Laser Spectroscopy of Trapped Ra+ Ion Versolato, Oscar Oreste IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationCitation for published version (APA): Martinus, G. H. (1998). Proton-proton bremsstrahlung in a relativistic covariant model s.n.
University of Groningen Proton-proton bremsstrahlung in a relativistic covariant model Martinus, Gerard Henk IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationCitation for published version (APA): Fathi, K. (2004). Dynamics and morphology in the inner regions of spiral galaxies Groningen: s.n.
University of Groningen Dynamics and morphology in the inner regions of spiral galaxies Fathi, Kambiz IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to
More informationUniversity of Groningen. Extraction and transport of ion beams from an ECR ion source Saminathan, Suresh
University of Groningen Extraction and transport of ion beams from an ECR ion source Saminathan, Suresh IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationCitation for published version (APA): Shen, C. (2006). Wave Propagation through Photonic Crystal Slabs: Imaging and Localization. [S.l.]: s.n.
University of Groningen Wave Propagation through Photonic Crystal Slabs Shen, Chuanjian IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it.
More informationSUPPLEMENTARY INFORMATION
University of Groningen Direct observation of the spin-dependent Peltier effect Flipse, J.; Bakker, F. L.; Slachter, A.; Dejene, F. K.; van Wees, Bart Published in: Nature Nanotechnology DOI: 10.1038/NNANO.2012.2
More informationCitation for published version (APA): Andogah, G. (2010). Geographically constrained information retrieval Groningen: s.n.
University of Groningen Geographically constrained information retrieval Andogah, Geoffrey IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
More informationUniversity of Groningen. Bifurcations in Hamiltonian systems Lunter, Gerard Anton
University of Groningen Bifurcations in Hamiltonian systems Lunter, Gerard Anton IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationUniversity of Groningen. Enantioselective liquid-liquid extraction in microreactors Susanti, Susanti
University of Groningen Enantioselective liquid-liquid extraction in microreactors Susanti, Susanti IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite
More informationCitation for published version (APA): Kootstra, F. (2001). Time-dependent density functional theory for periodic systems s.n.
University of Groningen Time-dependent density functional theory for periodic systems Kootstra, Freddie IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationGeometric approximation of curves and singularities of secant maps Ghosh, Sunayana
University of Groningen Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationUniversity of Groningen. Hollow-atom probing of surfaces Limburg, Johannes
University of Groningen Hollow-atom probing of surfaces Limburg, Johannes IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check
More informationCitation for published version (APA): Kooistra, F. B. (2007). Fullerenes for organic electronics [Groningen]: s.n.
University of Groningen Fullerenes for organic electronics Kooistra, Floris Berend IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationDetailed Proof of The PerronFrobenius Theorem
Detailed Proof of The PerronFrobenius Theorem Arseny M Shur Ural Federal University October 30, 2016 1 Introduction This famous theorem has numerous applications, but to apply it you should understand
More informationSystem-theoretic properties of port-controlled Hamiltonian systems Maschke, B.M.; van der Schaft, Arjan
University of Groningen System-theoretic properties of port-controlled Hamiltonian systems Maschke, B.M.; van der Schaft, Arjan Published in: Proceedings of the Eleventh International Symposium on Mathematical
More informationCitation for published version (APA): Raimond, J. J. (1934). The coefficient of differential galactic absorption Groningen: s.n.
University of Groningen The coefficient of differential galactic absorption Raimond, Jean Jacques IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite
More informationCitation for published version (APA): Kooistra, F. B. (2007). Fullerenes for organic electronics [Groningen]: s.n.
University of Groningen Fullerenes for organic electronics Kooistra, Floris Berend IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationCitation for published version (APA): Hoekstra, S. (2005). Atom Trap Trace Analysis of Calcium Isotopes s.n.
University of Groningen Atom Trap Trace Analysis of Calcium Isotopes Hoekstra, Steven IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationUniversity of Groningen. Event-based simulation of quantum phenomena Zhao, Shuang
University of Groningen Event-based simulation of quantum phenomena Zhao, Shuang IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationUniversity of Groningen. Levulinic acid from lignocellulosic biomass Girisuta, Buana
University of Groningen Levulinic acid from lignocellulosic biomass Girisuta, Buana IMPRTANT NTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationMarkov Chains, Stochastic Processes, and Matrix Decompositions
Markov Chains, Stochastic Processes, and Matrix Decompositions 5 May 2014 Outline 1 Markov Chains Outline 1 Markov Chains 2 Introduction Perron-Frobenius Matrix Decompositions and Markov Chains Spectral
More informationCitation for published version (APA): Wang, Y. (2018). Disc reflection in low-mass X-ray binaries. [Groningen]: Rijksuniversiteit Groningen.
University of Groningen Disc reflection in low-mass X-ray binaries Wang, Yanan IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check
More informationTheoretical simulation of nonlinear spectroscopy in the liquid phase La Cour Jansen, Thomas
University of Groningen Theoretical simulation of nonlinear spectroscopy in the liquid phase La Cour Jansen, Thomas IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF)
More informationSpectrally arbitrary star sign patterns
Linear Algebra and its Applications 400 (2005) 99 119 wwwelseviercom/locate/laa Spectrally arbitrary star sign patterns G MacGillivray, RM Tifenbach, P van den Driessche Department of Mathematics and Statistics,
More informationThe role of camp-dependent protein kinase A in bile canalicular plasma membrane biogenesis in hepatocytes Wojtal, Kacper Andrze
University of Groningen The role of camp-dependent protein kinase A in bile canalicular plasma membrane biogenesis in hepatocytes Wojtal, Kacper Andrze IMPORTANT NOTE: You are advised to consult the publisher's
More informationCitation for published version (APA): Brienza, M. (2018). The life cycle of radio galaxies as seen by LOFAR [Groningen]: Rijksuniversiteit Groningen
University of Groningen The life cycle of radio galaxies as seen by LOFAR Brienza, Marisa IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it.
More informationCan a Hexapole magnet of an ECR Ion Source be too strong? Drentje, A. G.; Barzangy, F.; Kremers, Herman; Meyer, D.; Mulder, J.; Sijbring, J.
University of Groningen Can a Hexapole magnet of an ECR Ion Source be too strong? Drentje, A. G.; Barzangy, F.; Kremers, Herman; Meyer, D.; Mulder, J.; Sijbring, J. Published in: Default journal IMPORTANT
More informationUniversity of Groningen. Photophysics of nanomaterials for opto-electronic applications Kahmann, Simon
University of Groningen Photophysics of nanomaterials for opto-electronic applications Kahmann, Simon IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to
More informationSuperfluid helium and cryogenic noble gases as stopping media for ion catchers Purushothaman, Sivaji
University of Groningen Superfluid helium and cryogenic noble gases as stopping media for ion catchers Purushothaman, Sivaji IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationCarbon dioxide removal processes by alkanolamines in aqueous organic solvents Hamborg, Espen Steinseth
University of Groningen Carbon dioxide removal processes by alkanolamines in aqueous organic solvents Hamborg, Espen Steinseth IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's
More informationPermutation transformations of tensors with an application
DOI 10.1186/s40064-016-3720-1 RESEARCH Open Access Permutation transformations of tensors with an application Yao Tang Li *, Zheng Bo Li, Qi Long Liu and Qiong Liu *Correspondence: liyaotang@ynu.edu.cn
More informationSensitized solar cells with colloidal PbS-CdS core-shell quantum dots Lai, Lai-Hung; Protesescu, Loredana; Kovalenko, Maksym V.
University of Groningen Sensitized solar cells with colloidal PbS-CdS core-shell quantum dots Lai, Lai-Hung; Protesescu, Loredana; Kovalenko, Maksym V.; Loi, Maria Published in: Physical Chemistry Chemical
More informationSubstrate and Cation Binding Mechanism of Glutamate Transporter Homologs Jensen, Sonja
University of Groningen Substrate and Cation Binding Mechanism of Glutamate Transporter Homologs Jensen, Sonja IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationCitation for published version (APA): Nouri-Nigjeh, E. (2011). Electrochemistry in the mimicry of oxidative drug metabolism Groningen: s.n.
University of Groningen Electrochemistry in the mimicry of oxidative drug metabolism Nouri-Nigjeh, Eslam IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationUniversity of Groningen. Agreeing asynchronously Cao, Ming; Morse, A. Stephen; Anderson, Brian D. O.
University of Groningen Agreeing asynchronously Cao, Ming; Morse, A. Stephen; Anderson, Brian D. O. Published in: IEEE Transactions on Automatic Control DOI: 10.1109/TAC.2008.929387 IMPORTANT NOTE: You
More informationUniversity of Groningen
University of Groningen Identification of Early Intermediates of Caspase Activation Using Selective Inhibitors and Activity-Based Probes Berger, Alicia B.; Witte, Martin; Denault, Jean-Bernard; Sadaghiani,
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationCitation for published version (APA): Hoefman, M. (1999). A study of coherent bremsstrahlung and radiative capture s.n.
University of Groningen A study of coherent bremsstrahlung and radiative capture Hoefman, Marieke IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite
More informationUniversity of Groningen
University of Groningen Nature-inspired microfluidic propulsion using magnetic actuation Khaderi, S. N.; Baltussen, M. G. H. M.; Anderson, P. D.; Ioan, D.; den Toonder, J.M.J.; Onck, Patrick Published
More informationSpectral Properties of Matrix Polynomials in the Max Algebra
Spectral Properties of Matrix Polynomials in the Max Algebra Buket Benek Gursoy 1,1, Oliver Mason a,1, a Hamilton Institute, National University of Ireland, Maynooth Maynooth, Co Kildare, Ireland Abstract
More informationCitation for published version (APA): Boomsma, R. (2007). The disk-halo connection in NGC 6946 and NGC 253 s.n.
University of Groningen The disk-halo connection in NGC 6946 and NGC 253 Boomsma, Rense IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it.
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationCompartmental modeling
Compartmental modeling This is a very short summary of the notes from my two-hour lecture. These notes were not originally meant to be distributed, and so they are far from being complete. A comprehensive
More informationUniversity of Groningen
University of Groningen Who is protecting tourists in New Zealand from severe weather hazards?: an exploration of the role of locus of responsibility in protective behaviour decisions Jeuring, Jelmer;
More informationStructural identifiability of linear mamillary compartmental systems
entrum voor Wiskunde en Informatica REPORTRAPPORT Structural identifiability of linear mamillary compartmental systems JM van den Hof Department of Operations Reasearch, Statistics, and System Theory S-R952
More informationUniversity of Groningen
University of Groningen Photosynthetic Quantum Yield Dynamics Hogewoning, Sander W.; Wientjes, Emilie; Douwstra, Peter; Trouwborst, Govert; van Ieperen, Wim; Croce, Roberta; Harbinson, Jeremy Published
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Centre for Research on Inner City Health St Michael s Hospital Toronto On leave from Department of Mathematics University of Manitoba Julien Arino@umanitoba.ca
More informationNotes on Linear Algebra and Matrix Theory
Massimo Franceschet featuring Enrico Bozzo Scalar product The scalar product (a.k.a. dot product or inner product) of two real vectors x = (x 1,..., x n ) and y = (y 1,..., y n ) is not a vector but a
More informationCitation for published version (APA): Sarma Chandramouli, V. V. M. (2008). Renormalization and non-rigidity s.n.
University of Groningen Renormalization and non-rigidity Sarma Chandramouli, Vasu Venkata Mohana IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
More informationUniversity of Groningen. Water in protoplanetary disks Antonellini, Stefano
University of Groningen Water in protoplanetary disks Antonellini, Stefano IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check
More informationCitation for published version (APA): van der Vlerk, M. H. (1995). Stochastic programming with integer recourse [Groningen]: University of Groningen
University of Groningen Stochastic programming with integer recourse van der Vlerk, Maarten Hendrikus IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to
More informationUniversity of Groningen. Managing time in a changing world Mizumo Tomotani, Barbara
University of Groningen Managing time in a changing world Mizumo Tomotani, Barbara IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationCitation for published version (APA): Borensztajn, K. S. (2009). Action and Function of coagulation FXa on cellular signaling. s.n.
University of Groningen Action and Function of coagulation FXa on cellular signaling Borensztajn, Keren Sarah IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More information9.1 Eigenvectors and Eigenvalues of a Linear Map
Chapter 9 Eigenvectors and Eigenvalues 9.1 Eigenvectors and Eigenvalues of a Linear Map Given a finite-dimensional vector space E, letf : E! E be any linear map. If, by luck, there is a basis (e 1,...,e
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Department of Mathematics University of Manitoba Winnipeg Julien Arino@umanitoba.ca 19 May 2012 1 Introduction 2 Stochastic processes 3 The SIS model
More informationUniversity of Groningen
University of Groningen Nonlinear optical properties of one-dimensional organic molecular aggregates in nanometer films Markov, R.V.; Plekhanov, A.I.; Shelkovnikov, V.V.; Knoester, Jasper Published in:
More informationUniversity of Groningen. Taking topological insulators for a spin de Vries, Eric Kornelis
University of Groningen Taking topological insulators for a spin de Vries, Eric Kornelis IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it.
More informationZ-Pencils. November 20, Abstract
Z-Pencils J. J. McDonald D. D. Olesky H. Schneider M. J. Tsatsomeros P. van den Driessche November 20, 2006 Abstract The matrix pencil (A, B) = {tb A t C} is considered under the assumptions that A is
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationPeptide folding in non-aqueous environments investigated with molecular dynamics simulations Soto Becerra, Patricia
University of Groningen Peptide folding in non-aqueous environments investigated with molecular dynamics simulations Soto Becerra, Patricia IMPORTANT NOTE: You are advised to consult the publisher's version
More informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Simulated Annealing Barnabás Póczos & Ryan Tibshirani Andrey Markov Markov Chains 2 Markov Chains Markov chain: Homogen Markov chain: 3 Markov Chains Assume that the state
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationIn particular, if A is a square matrix and λ is one of its eigenvalues, then we can find a non-zero column vector X with
Appendix: Matrix Estimates and the Perron-Frobenius Theorem. This Appendix will first present some well known estimates. For any m n matrix A = [a ij ] over the real or complex numbers, it will be convenient
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationUniversity of Groningen
University of Groningen Agmatine deiminase pathway genes in Lactobacillus brevis are linked to the tyrosine decarboxylation operon in a putative acid resistance locus Lucas, Patrick M.; Blancato, Victor
More informationLinear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.
POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems
More information--------------------------------------------------------------------------------------------- Math 6023 Topics: Design and Graph Theory ---------------------------------------------------------------------------------------------
More informationChapter 3. Linear and Nonlinear Systems
59 An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them Werner Heisenberg (1901-1976) Chapter 3 Linear and Nonlinear Systems In this chapter
More informationOptical hole burning and -free induction decay of molecular mixed crystals Vries, Harmen de
University of Groningen Optical hole burning and -free induction decay of molecular mixed crystals Vries, Harmen de IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF)
More informationSpin caloritronics in magnetic/non-magnetic nanostructures and graphene field effect devices Dejene, Fasil
University of Groningen Spin caloritronics in magnetic/non-magnetic nanostructures and graphene field effect devices Dejene, Fasil DOI: 10.1038/nphys2743 IMPORTANT NOTE: You are advised to consult the
More informationLinear Compartmental Systems-The Basics
Linear Compartmental Systems-The Basics Hal Smith October 3, 26.1 The Model Consider material flowing among n compartments labeled 1, 2,, n. Assume to begin that no material flows into any compartment
More informationSection 1.7: Properties of the Leslie Matrix
Section 1.7: Properties of the Leslie Matrix Definition: A matrix A whose entries are nonnegative (positive) is called a nonnegative (positive) matrix, denoted as A 0 (A > 0). Definition: A square m m
More informationMath 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank
Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank David Glickenstein November 3, 4 Representing graphs as matrices It will sometimes be useful to represent graphs
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationPublished in: ELECTRONIC PROPERTIES OF NOVEL MATERIALS - PROGRESS IN MOLECULAR NANOSTRUCTURES
University of Groningen Fullerenes and nanostructured plastic solar cells Knol, Joop; Hummelen, Jan Published in: ELECTRONIC PROPERTIES OF NOVEL MATERIALS - PROGRESS IN MOLECULAR NANOSTRUCTURES IMPORTANT
More informationUniversity of Groningen
University of Groningen Role of mitochondrial inner membrane organizing system in protein biogenesis of the mitochondrial outer membrane Bohnert, Maria; Wenz, Lena-Sophie; Zerbes, Ralf M.; Horvath, Susanne
More informationTHREE GENERALIZATIONS OF WEYL'S DENOMINATOR FORMULA. Todd Simpson Tred Avon Circle, Easton, MD 21601, USA.
THREE GENERALIZATIONS OF WEYL'S DENOMINATOR FORMULA Todd Simpson 7661 Tred Avon Circle, Easton, MD 1601, USA todo@ora.nobis.com Submitted: July 8, 1995; Accepted: March 15, 1996 Abstract. We give combinatorial
More information642:550, Summer 2004, Supplement 6 The Perron-Frobenius Theorem. Summer 2004
642:550, Summer 2004, Supplement 6 The Perron-Frobenius Theorem. Summer 2004 Introduction Square matrices whose entries are all nonnegative have special properties. This was mentioned briefly in Section
More informationContributions to multivariate analysis with applications in marketing Perlo-ten Kleij, Frederieke van
University of Groningen Contributions to multivariate analysis with applications in marketing Perlo-ten Kleij, Frederieke van IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's
More informationJordan normal form notes (version date: 11/21/07)
Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let
More informationCitation for published version (APA): Kole, J. S. (2003). New methods for the numerical solution of Maxwell's equations s.n.
University of Groningen New methods for the numerical solution of Maxwell's equations Kole, Joost Sebastiaan IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More information10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )
c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side
More informationDual photo- and redox- active molecular switches for smart surfaces Ivashenko, Oleksii
University of Groningen Dual photo- and redox- active molecular switches for smart surfaces Ivashenko, Oleksii IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationDEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix to upper-triangular
form) Given: matrix C = (c i,j ) n,m i,j=1 ODE and num math: Linear algebra (N) [lectures] c phabala 2016 DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix
More informationIt is well-known (cf. [2,4,5,9]) that the generating function P w() summed over all tableaux of shape = where the parts in row i are at most a i and a
Counting tableaux with row and column bounds C. Krattenthalery S. G. Mohantyz Abstract. It is well-known that the generating function for tableaux of a given skew shape with r rows where the parts in the
More informationMath 314 Lecture Notes Section 006 Fall 2006
Math 314 Lecture Notes Section 006 Fall 2006 CHAPTER 1 Linear Systems of Equations First Day: (1) Welcome (2) Pass out information sheets (3) Take roll (4) Open up home page and have students do same
More informationUniversity of Groningen
University of Groningen "Giant Surfactants" Created by the Fast and Efficient Functionalization of a DNA Tetrahedron with a Temperature-Responsive Polymer Wilks, Thomas R.; Bath, Jonathan; de Vries, Jan
More informationMaximizing the numerical radii of matrices by permuting their entries
Maximizing the numerical radii of matrices by permuting their entries Wai-Shun Cheung and Chi-Kwong Li Dedicated to Professor Pei Yuan Wu. Abstract Let A be an n n complex matrix such that every row and
More informationKernels of Directed Graph Laplacians. J. S. Caughman and J.J.P. Veerman
Kernels of Directed Graph Laplacians J. S. Caughman and J.J.P. Veerman Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207. caughman@pdx.edu, veerman@pdx.edu
More informationPY 351 Modern Physics Short assignment 4, Nov. 9, 2018, to be returned in class on Nov. 15.
PY 351 Modern Physics Short assignment 4, Nov. 9, 2018, to be returned in class on Nov. 15. You may write your answers on this sheet or on a separate paper. Remember to write your name on top. Please note:
More information