Supplement 8: Conservative and non-conservative partitioned systems: equivalences and interconversions

Transcription

1 Research The quantitatin f buffering actin. I. A frmal and general apprach. Bernhard M. Schmitt Supplement 8: Cnservative and nn-cnservative partitined systems: equivalences and intercnversins The aims define the basic data structure in which buffering terminlgy makes sense. Apparently, the buffering cncept and the assciated terminlgy are applicable if and nly if the system under study has been structured in a particular way. Namely, the elements f a whle must be gruped int eactly tw separate grups bth f which can be cndensed int a single functin f a cmmn variable. In the aimatic definitin f the signed prbability measure t and the ther three buffering parameters b, T, B (Supplement 7), this gruping is reflected in an rdered system f a subbag and its unique cmplement. Fr instance, if F is a bag f functins f i, then chsing a particular subbag S ut f F induces a cmplement S C. Tgether, S and S C define a unique rdered tw-partitined system {S, S C }, where the tw partitins S and S C each pssess unique partitining functins τ() and β(), respectively. The rdered pair f functins can be represented as a buffered system {τ(), β()}. Analgusly, if M is a bag f triples (S,, 2 ), where S is a subbag frm F, and, 2 are tw elements f the cmmn dmain f the functins f i, then a particular subbag A ut f M induces, tgether with its cmplement A c, a buffered system {A,A c } in M. In ne etreme case, the functins τ() and β() have a single cnstant value; then, hwever, their derivatives are zer, and the buffering parameters are nt defined. In anther special case, transfer and buffering functins are linear functins f the type f i : a i (where ai is a cnstant with a i D and a i ); in that case, the buffering parameters are cnstant ver the entire dmain. This case crrespnds t the standard situatin in prbability thery (if ne chses t interpret the measures t, b, T, B in terms f prbabilities, with the independent variable representing the number f events): Every pssible utcme is assciated with a prbability that is assumed t be cnstant and independent frm the number f events. Fr instance, the prbability f the utcme head is suppsed t remain the same, hw ever ften the cin is flipped. Page

2 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 This assumptin f cnstant partitining is usually inadequate when the measures t, b, T, and B are interpreted in terms f chemical buffering, with representing the ttal quantity in questin t be buffered. Fr instance, in a slutin f a weak acid, the additin f H + ins is assciated with prgressively decreasing buffering strength. The basic dichtmus structure is bvius in classic buffering phenmena. Multi-partitined system can be transfrmed in multiple ways int a frmally crrect buffered systems. The resulting tw functins (transfer and buffering functin) psit tw cmplementary rles, namely f that which is being buffered, and f that which is buffering. Whether these rles are meaningful and useful depends n etrinsic criteria, nt n frmal nes. In H + buffering, fr instance, this basic dichtmic structure is evident as the dichtmy between bund vs. free H + ins. In physilgical slutins, several H + buffer species are usually present simultaneusly, each binding H + ins in its wn way. In principle, these binding prcesses may be described individually by respective functins, each with ttal H + ins as the independent variable. Mrever, the amunt f free H + ins (i.e., H + ins bund t water) is described by anther functin f ttal H + ins. Of the buffer species, tw r mre may happen t have identical binding characteristics and cncentratins; then, the cllectin f functins cntains duplicates and therefre cnstitutes a bag, nt a set. Alternatively, we may wish t describe the buffering prcess in terms f individual buffer mlecules, which eist in multiple cpies with identical prperties. These cpies are described by identical functins, and thus the resulting cllectin f functins is again nt a set. Bags (r multisets) ffer themselves naturally as a cnvenient data structure in rder t frmalize the physical r chemical prcesses in these ways fr the cmputatin f H + buffering strength. The functins are gruped int tw bags f functins that describe H + binding t water and t ther mlecules, and are lumped tgether int transfer functin and buffering functin, respectively. Buffering f Ca ++ r ther ins is treated analgusly. Buffering phenmena can present as nn-cnservative, multi-partitined systems. Fr H + buffering, the classic case f buffering, the dichtmy f bund vs. free prvided an intuitive criterin fr gruping the elements f the whle int tw meaningful, cmplementary grups. Fr ther buffering phenmena, such as red buffering r bld pressure buffering, analgus criteria are less bvius. Mrever, buffering phenmena d nt necessarily crrespnd t the partitining f a cnserved quantity int tw partitins, i.e., t cnservative tw-partitined systems. Rather, many scientifically relevant buffering phenmena present in sme ther, nn-cnservative frm. Fur basic categries f partitined systems can be distinguished: i) Cnservative, tw-partitined systems, dented 2 Π c. An eample are H + ins in an aqueus slutin f a single weak acid; ii) Cnservative multi- r n-partitined systems; an eample are H + ins in a slutin f several buffer species f different cncentratin and with different Kd values. Fr these systems, we use the shrthand n Π c ; iii) General tw-partitined systems, including bth cnservative and nn-cnservative nes; thse systems are dented by 2 Π ; iii) Multi-partitined systems in the general sense, including bth cnservative and nn-cnservative nes; they are dented by n Π. The terms multi-partitined system and cllectin f functins f a cmmmn independent variable are synnymus. Equivalences between cnservative and nn-cnservative, r between multi-partitined and tw-partitined systems Imprtantly, there are certain transfrmatins r intercnversins between these fur classes f partitined systems that are invariant with respect t the buffering parameters t, b, T, and B. Multipartitined systems ( n Π r n Π c ) can be transfrmed int tw-partitined systems ( 2 Π r 2 Πc) and vice versa; tw-partitined systems ( 2 Π) can be transfrmed int cnservative ( 2 Π c ) r even int standardized tw-partitined systems and vice versa (see belw). Fr any nn-cnservative tw-partitined system there eists at least ne cnservative twpartitined system system that is equivalent in terms f the fur buffering parameters. Cnservative systems, n the ne hand, are simple Page 2

3 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 and prvide a kind f standard which makes it easy t cmpare different systems. Nn-cnservative systems, n the ther hand, may ffer imprtant advantages f their wn. It is f practical interest t knw hw the systems f these varius classes are related and can be intercnverted. Frm multi-partitined t tw-partitined systems: n Π 2 Π transfrmatins Buffering and the fur buffering parameters t, b, T, and B are defined fr systems f eactly tw functins, but nt fr systems f mre functins. We can transfrm, hwever, any n-partitined system, cnservative r nt, int a tw-partitined system. Reductin f n functins t 2 functins is achieved by cncatenating ( fusing r merging ) the many partitins until there are nly tw, nt necessarily nnempty, partitins left. Their rdered cmbinatin yields a buffered system. Obviusly, cncatenatin and rdering can be dne in several, different ways, crrespnding t different views f what is cnsidered that which is buffering as ppsed that which is being buffered. In general, the maimum number f twpartitined systems that can be frmed frm ne n- partitined system is 2 n ; this number is smaller than 2 n if nt all n elements f the system are different, i.e., if the bag F is nt a set. Such n Π 2 Π transfrmatins are pssible and necessary befre buffering terminlgy can be applied in a meaningful way t a multi-partitined system. It is imprtant t keep in mind that this transfrmatin is nt unique: several equally valid tw-partitined systems can be derived frm ne n- partitined system, and these will have different buffering prperties. The aimatic definitin f the buffering parameters is cmpatible with any f these tw-partitined systems, and preference f a particular ne must therefre be justified n ther grunds. Fr instance, it is usually meaningful t lump tgether all the chemically hetergeneus buffer species in physilgical slutins int a single virtual buffer as ppsed t the slvent water in rder t epress verall H + r Ca ++ buffering strength. Buffering in nn-cnservative tw-partitined systems Buffered systems derived frm cnservative tw-partitined systems satisfied the cnservatin cnditin σ ()=; they thus crrespnded t plane space curves in R 3 that lie inside the standard plane -y-z=, r in planes that are parallel t the standard plane (Figure D). In principle, hwever, ur measures f buffering can be cmputed fr any rdered pair f differentiatable functins, cnservative r nt. Fr instance, the buffering rati B is given by rati f tw first derivatives τ ()/β (), and that rati clearly eists fr any cmbinatin f differentiable functins. Similarly, we can always cmpute the parameters t, b, and T. The eistence f these parameters suggests that we can talk abut buffering in nncnservative systems in the same meaningful and eact way as in cnservative systems. In terms f space curves, buffering parameters are defined, meaningful and measurable in systems that lie within the standard plane, in parallel planes, in nnparallel planes, r in fact even in certain nn-planar surfaces. In a particular scientific cntet, such nncnservative systems f functins can represent the partitining f a nn-cnserved quantity. This nvel and therefre smewhat unfamiliar cncept will becme very useful fr the treatment f dimensinally hetergeneus systems such as bld pressure buffering (Buffering II). Linear r nn-linear distrtin engender nncnservative systems. The simplest way t btain a space curve that falls utside the standard plane is t add a nnzer cnstant c as an ffset t the sigma functin, such that σ()=+c and σ()=c. By definitin, a system is nn-cnservative if σ (). In ur case, we find σ ()=, and therefre the system again cnservative in the strict sense. Simple nn-cnservative systems are btained by scaling up r dwn all y and z values f a standardized partitined system by a cnstant factr. We speak f a linearly distrted system if the crrespnding sigma functin has the frm σ()=d, where d is a cnstant real number different frm zer and frm. The crrespnding Page 3

4 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 Figure : Equivalences between different partitined systems. A, -equivalents (clumn) and yz-equivalents (rw), including their respective standard frms Varius tw-partitined systems f functins are represented as area plts shwing the values f tw partitining functins π and π 2 (rdinate) f a cmmn independent variable (abscissa). The partitining functins f a given system (center panel) can be transfrmed in multiple ways such that the prprtin between their individual rates f change (i.e., buffering) is preserved fr any value f (tp and bttm panels), r alternatively fr any y r z r pair (y,z) (left and right panels). One particular - equivalent, termed the cannical r standard partitined system, has a sigma functin σ()=. Analgusly, there are cannical y- and the z-equivalents fr which σ()=. Amng the systems that are equivalent in bth y and z, the cannical ne satisfies the slightly mre general cnditin σ()=+c, where c is a cnstant. C, Intercnversins and standardizatin f partitined systems Multi-partitined systems (upper rw) need t be cnverted (vertical arrws) int tw-partitined systems in rder t frm buffered systems (lwer rw) in which buffering can be quantitated. A given n- r 2- partitined system can be transfrmed int a cannical r standardized system that is equivalent with respect t the prprtin between the rates f change f the individual partitining functins at all values f r y r z r sme ther parameter h. The rder in which the tw transfrmatins are carried ut that lead frm a n- partitined system t a standardized twpartitined system des nt affect the result, i.e., these transfrmatins are cmmutative. B, Parametric frms, r h-equivalents Partitined systems may be cmpared directly at similar values, but als fr pints that crrespnd with respect t any arbitrary parameter h. In this figure, tw-partitined systems are visualized as systems f cmmunicating vessels. Ttal fluid vlume is represented by the independent variable, and individual fluid vlumes in A and B by the values f the partitining functins π and π 2. Rates f change becme equal t vlume flws. When there is a net flw int the entire system, the prprtin between flw int A vs. B at similar ttal vlumes () r similar individual vlumes (y r z) is different between the three systems. Hwever, when epressed in parametric frm as functins f fluid level h, the fur buffering parameters t(h), b(h), T(h) and B(h) are identical in the three systems fr a given value f h. D, Standard plane Within each equivalence class frmed by thse partitined systems that ehibit similar buffering behavir at same values f (r y r z r h), there is eactly ne partitined system crrespnding t a space curve in R 3 that crsses the rigin and falls int the plane described by the equatin -y-z=. This curve is called the cannical r standard representative f the entire class, and the plane is termed standard plane. Page 4

5 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 space curves lie within a plane given by the equatin [d y z] =. In mre invlved distrted systems, the factr d may nt be cnstant, but vary as a functin d() f. Then, the sigma functin becmes a nn-linear functin f, and we speak f a nn-linearly distrted system. Space curves crrespnding t nn-linearly distrted systems lie within a nn-planar surface given by the equatin [σ() y z] = ; this equatin describes a surface that is symmetrical with respect t the plane that ges thrugh the rigin f the aes and is perpendicular t the standard plane. In principle, the sigma functin may be any differentiable functin, including peridic and chatic nes. Thus, a tw-partitined system has a plane space curve as gemetrical equivalent if it is cnservative. On the ther hand, if σ() is a sine wave, fr instance, the tw-partitined system has as gemetrical equivalent a space curve within a surface that resembles a tilted tin rf. Parametrizatin can thus turn cnservative systems int cmplicated nn-cnservative systems withut altering buffering behavir. Reversly, partitined systems that present initially in a rather invlved frm can be transfrmed int the much simpler cnservative systems. Dimensinal hetergeneity engenders nn-cnservative systems. Anther factr cmes int play as ne mves frm pure numbers t real wrld phenmena where it is ften necessary t emply scientific units f a particular physical dimensin. The case is simple if, τ(), and β() are all f identical physical dimensin. Such systems are termed here dimensinally hmgeneus systems. In ther cases, hwever, the independent variable is f ne dimensin, while transfer and buffering functins are f anther. Then, we say the system is dimensinally hetergeneus (yet ther systems that are f different dimensins in transfer vs. buffering functin rarely make sense and are nt cnsidered here). In dimensinally hetergeneus systems, transfer, buffering, and sigma functins implicitly include a cnversin factr K that brings abut the difference between their physical dimensin and particular unit n the ne hand, and dimensin and unit f the independent variable n the ther. In simple cases, this factr is a cnstant K: σ() = K. Then, K can be written as the prduct f a dimensinless scaling factr d and a unit {D} f a certain physical dimensin: K=d {D}. Fr instance, epressing distance l as a functin l:t l(t) f time requires a cnversin factr K with the dimensins f a velcity={length/time}. Varius cmbinatins f scaling factr d and unit {D} may yield a given cnversin factr K; fr instance, K= meter/secnd.93 {yards/secnd} {inches/secnd}. Imprtantly, the fur buffering parameters t, b, T, and B remain dimensinless even in dimensinally hetergeneus systems. This prperty fllws frm the way in which these parameters were defined, and turns the parameters t, b, T and B int etremely versatile scientific units. Fr instance, the changes f ttal, bund and free H + ins that cnstitute classical acid-base buffering can be epressed either eclusively in terms f mlar cncentratins, r ne may cmbine mlar units (fr added H + ins) with grams, equivalent charges, r abslute numbers (fr free and bund H + ins). The fur measures will invariably assume the same values, irrespective f the particular representatin chsen. On the ther hand, the pssibility t handle dimensinally hetergeneus systems prvides the key t a quantitative treatment f several imprtant buffering phenmena, such as bld pressure buffering and systems level buffering (Buffering II). Nn-cnservative systems that are bth distrted and dimensinally hetergeneus In the simplest case f a dimensinally hetergeneus system, the cnversin factr K is equal t a physical unit {D}, with the scaling factr d being equal t. The sigma functin is then σ()=k ={D}, and its derivative σ ()=K={D}. If {D} is a genuine physical unit (and nt, trivially, a dimensinless number), then the first derivatives τ (), β (), r σ () can never equal unity. Therefre, dimensinally hetergeneus systems are always Page 5

6 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 nn-cnservative. The cnverse statement is nt true, hwever: nt all nn-cnservative systems are dimensinally hetergeneus. As with dimensinally hmgeneus sytems, we can carry the generalizatin f hetergeneus systems even further and drp the cnstraint that the scaling factr d shuld be a cnstant. Thus, such a system nly needs t satisfy the cnditin σ ()= d() {D} where d() may stand fr any dimensinless, cntinuus, differentiatable functin. Such systems may be termed dimensinally hetergeneus, nnlinearly distrted systems. Again, bld pressure buffering and systems level buffering are eamples fr such systems (Buffering II). Dimensinally hmgeneus and hetergeneus systems have similar gemetrical equivalents. Intercnversins between cnservative and nn-cnservative systems Cnservative and nn-cnservative systems may ehibit identical buffering behavir. The parameters t, b, T r B put a cnstraint nly n the prprtin between τ () and β (), i.e., the rates f change f transfer and buffering functin, but nt n the abslute rates. Thus, scaling these rates f change jintly up r dwn will change σ () prprtinally t a value different frm, but will nt change the buffering parameters. Reversly, tw functins f a nn-cnservative system (i.e., with σ () ) can always be rescaled t yield a crrespnding cnservative system with identical buffering parameters. Put gemetrically, fr any arbitrary space curve there is a crrespnding curve within the standard plane thus that the prprtin between the tw slpes dy/d and dz/d is identical at any tw crrespnding pints f the tw curves. Cnservative tw-partitined systems prvide the simplest pssible representatin f buffered systems. We can thus think f a given cnservative twpartitined system as the riginal frm f a twpartitined system with particular buffering prperties, and f the equivalent nn-cnservative partitined systems as the results f a defrmatin f that system. Amng these equivalent systems, cnservative tw-partitined systems are distinguished by the fact that the crrespnding space curves are cnfined t a plane in R 3. In fact, they are purely tw-dimensinal entities inasmuch they can be represented by a linear cmbinatin f nly tw independent unit vectrs, e.g. λ =(,,) and λ 2 =(,,). The plane defined by these tw vectrs was termed standard plane. This mapping frm nn-cnservative t cnservative system is a prjectin because it maps a three-dimensinal system int tw dimensins (i.e., a map R 3 R 2 ). With respect t the number f dimensins, the tw-dimensinal representatin prvides the mst simple way t frmulate a tw-partitined systems f certain buffering prperties. On the ther hand, there eists an infinite number f such prjectins. In rder t be useful, this cllectin f systems needs t be structured further. Partitined systems can be cmpared under different aspects. Cnsider the cncept f a vectr : Amng all pssible arrws in a particular space, all thse are said t be identical whse length and directin are identical. This definitin is an abstractin that ignres sme aspects (psitin), and cmpares different elements slely under selected aspects (length and directin). Elements that are similar with respect t these selected aspects frm an equivalence class that is induced by an equivalence relatin. Fr vectrs, this equivalence relatin is A has same length and directin as B. Nte that the equivalence classes induced by a particular equivalence relatin in a set f elements frm a partitined system. Similarly, the cmparisn f partitined systems f functins under a particular aspect is a fundamental and useful scientific peratin. Fr instance, ne usually cmpares H + buffering strength in different samples at similar values f phi r [H + ]i, crrespnding t similar values f the transfer functin τ(), as elabrated in detail in the accmpanying paper (Buffering II). In ther situatins, it is mre apprpriate t cmpare different systems at similar values f the independent variable. Fr instance, it makes sense t cmpare the buffering f rgan perfusin in the Page 6

7 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 face f variable perfusin pressure in different rgans (e.g. in kidney and brain) at similar perfusin pressures (i.e., similar values) rather than at similar bld flws (i.e., similar values f the transfer functin τ()) (Buffering II). In yet ther situatins, it may be mst natural r useful t cmpare systems under an aspect that is nt reflected by any ne f the parameters, τ() r β(), but by sme further parameter. Fr instance, we may want t cmpare the buffering behavir f different systems f cmmunicating vessels at similar fluid levels h which is related nly indirectly t. Cmparing partitined systems under a particular aspect induces equivalence classes. Obviusly, systems may be similar under ne aspect, and different under anther ne. The questin whether buffering is similar fr crrespnding pints f different systems can thus nly be answered after specifiying a criterin that specifies unambiguusly which are the crrespnding pints. The systems that ehibit similar buffering at crrespnding pints frm an equivalence class. Fr instance, the criterin same buffering behavir at same (e.g. perfusin pressure) induces equivalence classes which we term -equivalents (Figure A, panels in same clumn), the criterin same buffering at same y (e.g. ph) induces equivalence classes which we term yequivalents (Figure A, panels in same rw), and the criterin similar buffering behavir at similar values f parameter h (e.g. fluid level) induces equivalence classes which can be termed h-equivalents (Figure B). Analgusly, ther parameters used t epress a system in parameteric frm will induce ther equivalence classes. An entire equivalence class can be represented by a unique cannical element. Vectrs that are similar have similar directin and length, but may be lcated at different psitins in space. Amng all similar vectrs, the ne that starts at the rigin f the aes can be represented in a particularly clear and simple frm, bth algebraically and graphically. This cannical frm allws ne t study the relevant prperties f all members f the equivalence class in a cnvenient way. In tplgy, the reductin f an equivalence class int a single representative element is termed identificatin. Similarly, we can identify classes f equivalent partitined systems with a single cannical r standard system. We define as cannical that element Π f a class f equivalent partitined systems Π whse sigma functin has a unity slpe and whse individual partitining functins have zer value at =: Π = {Π σ ()= π i ()=} Put in gemetrical terms and fr a tw-partitined system, f all space curves in an equivalence class, the standard r cannical ne is that unique space curve that falls int the standard plane and crsses the rigin f the three aes. The secnd cnditin is nt applicable, f curse, t thse partitining functins whse abslute values are used t define the class. This is the case, fr instance, with y- and z-equivalents; the cannical system then is the ne that satisfies the weaker cnditin σ()=. Furthermre, the y-and-zequivalents f a tw-partitined system are fully cnstrained in bth the y- and z-values, such that σ() may have any value. The same cnstraints guarantee, hwever, that there is nly a single cnservative system that can be used as standard equivalent in this case. The cnversin f nn-cnservative systems int standard cnservative systems is termed a Π Π transfrmatin. Nte that standardizatin f dimensinally hetergeneus systems will turn them int dimensinally hmgeneus systems. Eample: finding the cannical -equivalent, r standardizatin in T illustrate the basic principles, cnsider a simple 2 Π 2 Π transfrmatin, namely the transfrmatin f a cnservative tw-partitined system 2 Π with ffset int a cnservative twpartitined system 2 Π withut ffset. Gemetrically, this means translating a plane space curve int the standard plane frm sme ther, parallel plane. In rder t btain an -equivalent, Page 7

8 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 this translatin furthermre must nt change the fur buffering parameters at any value f the independent variable, such that nly the dependent variables y and z may be changed. Analytically, this type f standardizatin is achieved by replacing each partitining functin π i () by a crrespnding standard partitining functin π i () that satisfies the equatin π i () = π i () - π i (). In gemetrical terms, this standardizatin simply remves an ffset by translating the entire curve parallel t the yz-plane twards the standard plane such that the curve crsses the rigin. Such a translatin is defined by a vectr v r with v r y π ( ) = = z π (. 2 ) The resulting unique tw-partitined system Π ={π (), π 2 ()} is said t be the cannical r standard -equivalent f 2 Π, f which there is eactly ne. Carrying ut this transfrmatin in the r reverse directin (i.e., by using a vectr v f similar length and ppsite directin) transfrms the standard system Π int a nn-standard - equivalent f Π. Fr each standard partitined system there is, hwever, an infinite number f 2 Π systems that are, with respect t the buffering parameters at a given, equivalent t Π as well as t each ther. Eample: finding the cannical y,z-equivalent, r standardizatin in the dependent variables Alternatively, ne can carry ut the standardizatin such that the buffering parameters are preserved fr any given pair (y,z). Analytically, this is achieved by substituting a given nnstandardized partitining functin π i () with a standard partitining functin π i () fr which π i () = π i [+π ()+ π 2 ()] = π i [σ()] (t be read as a functin π i f the argument [σ()] a nested functin, nt a prduct). Gemetrically, this crrespnds t a classical parallel prjectin f the riginal space curve nt the standard plane alng lines parallel t the ais. The resulting unique twpartitined system Π {π (), π 2 ()} is said t be y,z the standard y,z-equivalent. Again, carrying ut such transfrmatins in the reverse directin results in a family f nn-standardized 2 Π systems that are nn-standard equivalents in f the standard system as well as f each ther. Standardizatin in the dependent variables: nn-linearly distrted systems & general frm Analytically, the Π y,z transfrmatin f a system f tw arbitrary functins y=π () and z=π 2 () is achieved by substituting fr π a partitining functin π that satisfies the cnditin π () = π [π ()+ π 2 ()] = π [σ()], and fr π 2 a partitining functin π 2 fr which π2() = π 2[π()+ π2()] = π 2[σ()]. Standardizatin in the dependent variables fr general multi-partitined systems is achieved by substituting each functin π i with a standard partitining functin π i that satisfies the cnditin π () = π [π ()+ π 2 () + π n ()] = π [σ()]. Gemetrically, nly standardizatins f twpartitined systems (i.e., Π y,z transfrmatins) have equivalents in R 3. Here, this standardizatin crrespnds t a classical parallel prjectin f the space curve nt the nrmal plane parallel t the ais. In vectr ntatin, this prjectin is given as σ() π () π() π2() π2(). When n>2, the n Π n Π standardizatins in the dependent variables lack a gemetrical equivalent in R 3. Standardizatin in : Linearily distrted systems withut and with ffset Net, cnsider a tw-partitined system fr which σ()=a. Ecept fr the trivial case f a=, such a system is nn-cnservative because σ (), and can thus be classified as a linearily distrted partitined system withut ffset. Standardizatin f such systems in is achieved by replacing each Page 8

9 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 partitining functin π i by a crrespnding standardized partitining functin π i fr which πi() π i ( ) =. a Gemetrically, this standardizatin is equivalent t prjecting each pint f the riginal curve nt the standard plane alng it s shrtest cnnectin with the -ais. Fr a linearily distrted partitined systems with ffset (i.e., with σ()=a +c and c ), the standard -equivalent is btained by substituting fr any given functin π i a standard partitining functin π i fr which πi() πi( ) π i ( ) =. a Gemetrically, this crrespnds t the cmpsitin f a translatin (frcing the entire curve thrugh the rigin) and prjectin f each pint nt the nrmal plane alng it s shrtest cnnectin with the -ais. Standardizatin in : nn-linearly distrted systems & general frm Net, we need t cnsider the standardizatin in f a general, nn-cnservative tw-partitined systems with an arbitrary nnlinear sigma functin σ(); in brief, these are called 2 Π Π transfrmatins. Analytically: In general, the standardizatin in f a tw-partitined system can be achieved by transfrming π int a standardized partitining functin π such that π () = π' () d = ' () ' () π + π 2 and π 2 int a functin π 2 () such that π () = π' () 2 d = ' () ' () π + π 2 π' () d σ' () π' 2 () d σ' (). 2 Multipartitined, r n Π systems, with mre than tw functins can be standardized analgusly. Thus, n Π n Π transfrmatins in are carried ut by substituting each functin π i () with a standard partitining functin π' () i π' i () i() d d ' () ' ()... ' n () '() π = =. π + π + π 2 σ Gemetrically: Π Π transfrmatins d nt crrespnd t any f the classical prjectins. Fr tw-partitined systems represented gemetrically as space curves in R 3, they can be visualized in tw ways. Firstly, a 2 Π 2 Π transfrmatin can be achieved by dividing the nn-standardized space curve int n linearized segments f length, rescaling them individually in y and z directins by a cmmn factr such that ( y+ z)= while keeping the rati y/ z cnstant, jining the rescaled elements tip t end, and the frcing the resulting space curve thrugh the rigin by translating it alng a vectr that is perpendicular t the -ais. This prcedure is carried ut with infinitely small and infinitely many segments (n and ). Secndly, 2 2 Π Π transfrmatins may be thught f as an iterative prjectin, illustrated here fr a buffered system fr which R + : i) Divisin f the curve int n individual pints P i =P( ), P( 2 ) with i =,, 2 and i < i+ and n ; ii) Translatin f the entire space curve alng a vectr v r y π ( ) = = z π ( ; 2 ) this frces the curve thrugh the rigin, and P( ) int the standard plane; iii) Translatin f all pints {P( i ) i > } alng a line that is parallel t the yzplane and has a slpe dπ 2 ( )/dπ ( ) until the pint P( ) falls int the standard plane; iv) Prjectin f all pints {P( i ) i > } (these pints have been translated twice by nw) alng alng a line that is parallel t the yz-plane and has a slpe dπ 2 ( 2 )/dπ ( 2 ) until the P( 2 ) falls int the standard plane; v) This prcedure is cntinued t the end f the curve. The prcess thus successively unrlls and cmpresses (r stretches) the space curve nt the standard plane. Sme rules fr transfrmatins Standardizatin in always yields singlevalued partitining functins, but standardizatin in the dependent variables nt necessarily s, even Page 9

10 Theretical Bilgy and Medical Mdelling, 25 B.M. Schmitt Buffering I Supplement 8 when the individual partitining functins π i () are all single-valued. Mrever, standardizatin in y,z des nt frce the space curve thrugh the rigin. n Π 2 Π and Π Π transfrmatins f systems f functins can be carried ut sequentially, either in the rder n Π n Π 2 Π r n Π 2 Π 2 Π. In either case, the verall result is a n Π 2 Π transfrmatin. Prvided that the tw transfrmatins are either bth in, r bth in y and z, the result is independent frm the path taken. In ther wrds, these transfrmatins are cmmutative. Page