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Random fields: Applications in cell biology

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Stochastic Spatial Processes

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References

  • ADLER, R.(1981) The Geometry of Random Fields. New York:Wiley

    MATH  Google Scholar 

  • ADLER, R.,FEIGIN, P.D.(1984) On the cadlaguity of random measures. Ann.Probab.12,615–630

    Article  MathSciNet  MATH  Google Scholar 

  • ADLER, R.,MONRAD, D.,SCISSORS, R.H.,WILSON, R.J.(1983) Representations,decompositions,and sample function continuity of random fields with independent increments. Stoch.Proc.Appl.15,3–30

    Article  MathSciNet  MATH  Google Scholar 

  • AIZENMAN, M.,GOLDSTEIN, S.,GRUBERG, C.,LEBOWITZ, J.L.,MARTIN, P.(1977) On the equivalence between KMS-states and equilibrium states for classical systems. Commun.Math.Phys.,53,209–220

    Article  MathSciNet  Google Scholar 

  • ALBEVERIO, S.,HØEGH-KROHN, R.(1984) Local and global Markoff fields. Rep.Math.Phys.,19,225–248

    Article  MathSciNet  MATH  Google Scholar 

  • ALBEVERIO, S.,HØEGH-KROHN, R.,OLSEN, G. (1981) The global Markov property J.Multivar.Anal.,11,599–607

    Article  MathSciNet  MATH  Google Scholar 

  • ANDERSON, R.M.(1982) Star-finite representations of measure spaces. Trans.Amer.Math.Soc.,271,667–687

    Article  MathSciNet  MATH  Google Scholar 

  • BASS, R.F.,PYKE, R.(1984a) The existence of set-indexed Lévy processes. Z.Wahrscheinlichkeitstheorie verw.Geb.,66,157–172; (1984b) Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets. Ann.Probab.,12,13–34

    Article  MathSciNet  MATH  Google Scholar 

  • BASS, R.F.,PYKE, R.(1985) The space D(A) and weak convergence for set-indexed processes. Ann.Probab.,13,860–884

    Article  MathSciNet  MATH  Google Scholar 

  • BICKEL, P.J.,WICHURA, M.J.(1971) Convergence criteria for multiparameter stochastic processes and some applications. Ann.Math.Statist.,42,1656–1670

    Article  MathSciNet  MATH  Google Scholar 

  • BLACKWELL, D.,RYLL-NARDZEWSKI, C.(1963) Non-existence of everywhere proper conditional distributions. Ann.Math.Statist.,34,223–225

    Article  MathSciNet  MATH  Google Scholar 

  • BLACKWELL, D.,DUBINS, L.E.(1975) On existence and non-existence proper regular conditional distributions. Ann.Probab.,3,741–752

    Article  MathSciNet  MATH  Google Scholar 

  • BOLTHAUSEN, E.(1982) On the central limit theorem for stationary mixing random fields. Ann.Probab.,10,1047–1050

    Article  MathSciNet  MATH  Google Scholar 

  • BRADLEY, R.C.(1984) Some remarks on strong mixing conditions. Proc.7th Conf.Probability Theory,Brasov 1982,pp.65–72. Bucuresti:Ed.Academiei

    Google Scholar 

  • CAHN, J.W.(1972) The generation and characterization of shape. Adv.Appl.Probab.,4 (Suppl.),221–242

    Article  Google Scholar 

  • CASSANDRO, M.,OLIVIERI, E.,PELLEGRINOTTI, A.,PRESUTTI, E.(1978) Existence and uniqueness of DLR measures for unbounded spin systems. Z.Wahrscheinlichkeitstheorie verw.Geb.,41,313–334

    Article  MathSciNet  MATH  Google Scholar 

  • CHERNAVSKII, D.S.,EIDUS, V.L.,POLEZHAEV, A.A. (1981) On kinetics of phase transitions in cell membranes. BioSystems 13,171–179

    Article  Google Scholar 

  • CHOQUET, G.(1969) Lecture on Analysis.Vol.II.Representation Theory. Reading(Mass.):W.A.Benjamin

    Google Scholar 

  • CHOW, Y.S.,TEICHER, H.(1978) Probability Theory.Independence,Interchange-ability,Martingales. New York-Heidelberg-Berlin:Springer

    Google Scholar 

  • COX, T.(1977) An example of phase transition in countable one-dimensional Markov random fields. J.Appl.Probab.,14,205–211

    Article  MathSciNet  MATH  Google Scholar 

  • COX, T.(1979) An alternate proof of a theorem of Kesten concerning Markov random fields. Ann.Probab.,7,377–378

    Article  MathSciNet  MATH  Google Scholar 

  • CURTIS, A.S.G.(1978) Cell-cell recognition:Positioning and patterning systems. Symp.Soc.Exper.Biol.,32,51–82. Cambridge:Cambridge Univ. Press

    Google Scholar 

  • DALETSKII, Y.L.,SMOLYANOV, O.G.(1984) On the weak sequential completeness of the spaces of Radon measures. Theor.Probab.Appl.,29,142–147

    Article  MathSciNet  Google Scholar 

  • DANG-NGOC, N.,YOR, M.(1978) Champs markoviens et mesures de Gibbs sur R. Ann.Sci.Ecole Norm.Sup.,11,29–69

    MathSciNet  MATH  Google Scholar 

  • DARST, R.B.(1971) On universal measurability and perfect probability. Ann.Math.Statist.,42,352–354

    Article  MATH  Google Scholar 

  • DeHOFF, R.T.(1972) The evolution of particulate structures. Adv.Appl.Probab.,4 (Suppl.),188–198

    Article  Google Scholar 

  • DIEUDONNE, J.(1970) Treatise on Analysis.Vol.I. New York:Academic Press

    Google Scholar 

  • DOBRUSHIN, R.L.(1965) Existence of a phase transition in two-dimensional and three-dimensional Ising models. Theor.Probab.Appl.,10,193–213

    Article  MathSciNet  MATH  Google Scholar 

  • DOBRUSHIN, R.L.(1967) Existence of phase transitions in models of a lattice gas. Proc.5th Berkeley Symp.Math.Statist.Probability,Vol.III, pp.73–87. Berkeley:Univ.California Press

    Google Scholar 

  • DOBRUSHIN, R.L.(1968a) The description of a random field by means of conditional probabilities and conditions of its regularity. Theor.Probab.Appl.,13,197–224; (1984b) Gibbsian random fields for lattice systems with pairwise interactions. Funct.Anal.Appl.,2,292–301; (1968c) The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct.Anal.Appl.,2, 302–312

    Article  MathSciNet  MATH  Google Scholar 

  • DOBRUSHIN, R.L.,MAJOR, P.(1981) On the asymptotic behavior of some self-similar random fields. Sel.Math.Sov.,1,265–291

    MathSciNet  MATH  Google Scholar 

  • DOBRUSHIN, R.L.,PECHERSKI, E.A.(1983) A criterion of the uniqueness of Gibbsian fields in the non-compact case. Proc.4th USSR-Japan Symp. (Lecture Notes in Math.,Vol.1021),pp.97–110. Berlin-Heidelberg-New York:Springer

    Google Scholar 

  • DOBRUSHIN, R.L.,SINAI, Y.G.(1980) Mathematical problems in statistical mechanics. Math.Physical Rev.,1,55–106.

    MathSciNet  MATH  Google Scholar 

  • DÖHLER, R.(1980) On the conditional independence of random events. Theor.Probab.Appl.,25,628–634

    Article  MATH  Google Scholar 

  • DOOB, J.L.(1953) Stochastic Processes. New York:Wiley

    MATH  Google Scholar 

  • DUDLEY, R.M.(1973) Sample functions of the Gaussian process. Ann.Probab.1,66–103

    Article  MathSciNet  MATH  Google Scholar 

  • DYNKIN, E.B.(1978) Sufficient statistics and extreme points. Ann.Probab.6,705–730

    Article  MathSciNet  MATH  Google Scholar 

  • DYNKIN, E.B.(1980) Markov processes and random fields. Bull.Amer.Math.Soc.,3,975–999

    Article  MathSciNet  MATH  Google Scholar 

  • FISHER, M.E.,RUELLE, D.(1966) The stability of many-particle systems. J.Math.Phys.,7,260–270

    Article  MathSciNet  Google Scholar 

  • FLOOD, R.G.,SULLIVAN, W.G.(1980) Consistency of random field specifications Z.Wahrscheinlichkeitstheorie verw.Geb.,53,147–156

    Article  MathSciNet  MATH  Google Scholar 

  • FÖLLMER, H.(1975) Phase transition and Markov boundary. Lecture Notes in Math.,Vol.465(Séminaire de Probabilités,IX),pp.305–317. Berlin-Heidelberg-New York:Springer

    MATH  Google Scholar 

  • FÖLLMER, H.(1979) Macroscopic convergence of Markov chains on infinite product spaces. In:Random Fields.Rigorous Results in Statistical Mechanics and Quantum Field Theory (J. Fritz,J.L. Lebowitz,D. Szász eds.),pp.363–371. Amsterdam:North-Holland

    Google Scholar 

  • FÖLLMER, H.(1980) On the global Markov property. In:Quantum Fields,Algebras,Proccesses (L. Streit ed.),pp.293–302. Wien-New York:Springer

    Chapter  Google Scholar 

  • FÖLLMER, H.(1982) A covariance estimate for Gibbs measures. J.Funct.Anal.46,387–395

    Article  MathSciNet  MATH  Google Scholar 

  • FÖLLMER, H.(1984) Almost sure convergence of multiparameter martingales for Markov random fields. Ann.Probab.,12,133–140

    Article  MathSciNet  MATH  Google Scholar 

  • GARSTENS, M.A.(1970) Remarks on statistical mechanics and theoretical biology. In:Towards a Theoretical Biology (C.H. Waddington ed.), Vol.3,pp.167–173. Edinburgh:Edinburgh Univ.Press

    Google Scholar 

  • GAUNT, S.J.,SUBAK-SHARPE, J.H.(1979) Selectivity in metabolic cooperation between cultured mammalian cells. Exper.Cell Res.,120,307–320

    Article  Google Scholar 

  • GLÖTZL, E.(1978) Gibbsian description of point processes. In:Point Processes and Queuing Problems (P. Bártfai,J. Tomkó eds.),pp.69–84. Amsterdam:North-Holland

    Google Scholar 

  • GOODWIN, B.C.(1971) A model of early amphibian development. Symp.Soc.Exp.Biol.,25,417–428. Cambridge:Cambridge Univ.Press

    Google Scholar 

  • GROBSTEIN, C.(1956) Inductive tissue interaction in development.Adv.Cancer Res.,4,187–236

    Article  Google Scholar 

  • GUT, A.(1978) Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann.Probab.,6,469–482

    Article  MathSciNet  MATH  Google Scholar 

  • GUT, A.(1979) Moments of the maximum of normed partial sums of random variables with multidimensional indices. Z.Wahrscheinlichkeitstheorie verw.Geb.,46,205–220

    Article  MathSciNet  MATH  Google Scholar 

  • HAMMERSLEY, J.M.(1972) Stochastic models for the distribution of particles in space. Adv.Appl.Probab.,4,(Suppl.),47–68

    Article  MATH  Google Scholar 

  • HAMMERSLEY, J.M.,MAZZARINO, G.(1983) Markov fields,correlated percolation, and the Ising model. In:The Mathematics and Physics of Disordered Media (B.D. Hughes,B.W. Ninham eds.),pp.201–245. Berlin-Heidelberg-New York-Tokyo:Springer

    Google Scholar 

  • HARRIS, A.K.,STOPAK, D.,WARNER, P.(1984) Generation of spatially periodic patterns by a mechanical instability:A mechanical alternative to the Turing model. J.Embryol.Exp.Morphol.,80,1–20

    Google Scholar 

  • HEGERFELDT, G.C.,NAPPI, C.R.(1977) Mixing properties in lattice systems. Commun.Math.Phys.,53,1–7

    Article  MathSciNet  MATH  Google Scholar 

  • HELMS, L.L.(1983) Hyperfinite spin models. In:Nonstandard Analysis-Recent Developments (A.E. Hurd ed.),pp.15–26. Berlin-Heidelberg-New York-Tokyo:Springer

    Chapter  Google Scholar 

  • HELMS, L.L.,LOEB, P.A.(1979) Applications of nonstandard analysis to spin models. J.Math.Anal.Appl.,69,341–352

    Article  MathSciNet  MATH  Google Scholar 

  • HIGUCHI, Y.(1977) Remarks on the limiting Gibbs states on a (d+1)-tree. Publ.RIMS Kyoto Univ.,13,335–348

    Article  MathSciNet  MATH  Google Scholar 

  • HURD, A.E.(1981) Nonstandard analysis and lattice statistical mechanics: A variational principle. Trans.Amer.Math.Soc.,263,89–110

    Article  MathSciNet  MATH  Google Scholar 

  • IBRAGIMOV, I.A.,ROZANOV, Y.A.(1978) Gaussian Random Processes. New York-Heidelberg-Berlin:Springer

    Book  MATH  Google Scholar 

  • IVANOFF, G.(1980) The branching random field. Adv.Appl.Probab.,12,825–847

    Article  MathSciNet  MATH  Google Scholar 

  • KALLIANPUR, G.,MANDREKAR, V.(1974) The Markov property for generalized Gaussian random fields. Ann.Inst.Fourier 24,143–167

    Article  MathSciNet  MATH  Google Scholar 

  • KARR, A.F.(1978) Lévy random measures. Ann.Probab.,6,57–71 (Correction: 1979,7,1098)

    Article  MathSciNet  MATH  Google Scholar 

  • KARR, A.F.(1979) Classical limit theorems for measure-valued Markov processes. J.Multivar.Anal.,9,234–247

    Article  MathSciNet  MATH  Google Scholar 

  • KESTEN, H.(1976) Existence and uniqueness of countable one-dimensional Markov random fields. Ann.Probab.,4,557–569

    Article  MathSciNet  MATH  Google Scholar 

  • KNIGHT, F.B.(1970) A remark on Markovian germ fields. Z.Wahrscheinlichkeitstheorie verw.Geb.,15,291–296

    Article  MathSciNet  MATH  Google Scholar 

  • KNIGHT, F.B.(1979) Prediction processes and an autonomous germ-Markov property. Ann.Probab.,7,385–405

    Article  MathSciNet  MATH  Google Scholar 

  • KOREZLIOGLU, H.,MAZZIOTTO, G.,SZPIRGLAS, J. (eds.):Processus Aléatoires à Deux Indices (Lect.Notes in Math.,Vol.863), Berlin-Heidelberg-New York:Springer

    Google Scholar 

  • KOTANI, S.(1973) On a Markov property for stationary Gaussian processes with a multidimensional parameter. Proc.2nd Japan-USSR Symp.on Probab.Theory (Lect.Notes in Math.,Vol.330),pp.239–250. Berlin-Heidelberg-New York:Springer

    MATH  Google Scholar 

  • KRENGEL, U.(1985) Ergodic Theorems. Berlin-New York:de Gruyter

    Book  MATH  Google Scholar 

  • KUZNETSOV, S.E.(1984) Specifications and a stopping theorem for random fields. Theor.Probab.Appl.,29,66–78

    Article  MathSciNet  MATH  Google Scholar 

  • LANFORD III, O.E.(1973) Entropy and equilibrium states in classical statistical mechanics. Lect.Notes in Physics,Vol.20,pp.1–113. Berlin-Heidelberg-New York:Springer

    Google Scholar 

  • LANFORD III, O.E.,RUELLE, D.(1967) Integral representations of invariant states on B* algebras. J.Math.Phys.,8,1460–1463

    Article  MathSciNet  MATH  Google Scholar 

  • LEVENSON, R.,HOUSMAN, D.(1981) Commitment:How do cells make the decision to differentiate? Cell 25,5–6

    Article  Google Scholar 

  • LEVINS, R.(1970) Complex systems. In:Towards a Theoretical Biology (C.H. Waddington ed.),Vol.3,pp.73–88. Edinburgh:Edinburg Univ.Press

    Google Scholar 

  • LLOYD, S.P.(1962) On a measure of stochastic dependence. Theor.Probab.Appl.,7,301–312

    Article  MathSciNet  MATH  Google Scholar 

  • LO, C.W.,GILULA, N.B.(1980) PCC4azal teratocarcinoma stem cell differentiation in culture.III.Cell-to-cell communication properties. Devel.Biol.,75,112–120

    Article  Google Scholar 

  • LOEVE, M.(1973) Paul Lévy,1886–1971. Ann.Probab.,1,1–8

    Article  MathSciNet  MATH  Google Scholar 

  • MANDREKAR, V.(1976) Germ-field Markov property for multiparameter processes. Lect.Notes in Math.,Vol.511(Sémin.de Probabilités X), pp. 78–85. Berlin-Heidelberg-New York:Springer

    MATH  Google Scholar 

  • MANDREKAR, V.(1983) Markov properties for random fields. In:Probabilistic Analysis and Related Topics (A.T. Bharucha-Reid ed.),Vol.3, pp.161–193. New York:Academic Press

    Google Scholar 

  • MARUYAMA, G.(1970) Infinitely divisible processes. Theor.Probab.Appl., 15,1–22

    Article  MathSciNet  MATH  Google Scholar 

  • McKEAN(1963) Brownian motion with a several-dimensional time. Theor.Probab.Appl.,8,335–354

    Article  MathSciNet  MATH  Google Scholar 

  • MEYER, P.A.(1966) Probability and Potentials. Waltham(Mass.):Blaisdell

    MATH  Google Scholar 

  • MINLOS, R.A.(1967a) Limiting Gibbs distributions. Funct.Anal.Appl.,1,140–150; (1967b) Regularity of the Gibbs limit distributions. Funct.Anal.Appl.,1,206–217

    Article  MATH  Google Scholar 

  • MITTENTHAL, J.E.(1981) The rule of normal neighbors:A hypothesis for morphogenetic pattern regulation. Devel.Biol.,88,15–26

    Article  Google Scholar 

  • MIYAMOTO, M.(1982) Spitzr's Markov chains with measurable potentials. J.Math.Kyoto Univ.,22,41–69

    MathSciNet  MATH  Google Scholar 

  • MOULIN OLLAGNIER, J.M.(1985) Ergodic Theory and Statistical Mechanics. (Lect.Notes in Math.,Vol.1115) Berlin-Heidelberg-New York-Tokyo:Springer

    MATH  Google Scholar 

  • MOULIN OLLAGNIER, J.,PINCHON, D.(1981) Mesures de Gibbs invariantes et mesures d'equilibre. Z.Wahrscheinlichkeitstheorie verw.Geb.,55,11–23

    Article  MathSciNet  MATH  Google Scholar 

  • MOUSSOURIS, J.(1974) Gibbs and Markov random systems with constraints. J.Statist.Phys.,10,11–33

    Article  MathSciNet  Google Scholar 

  • NAORA, H.,DEACON, N.J.(1982) Implication of the effect of extragenic territorial DNA sequences on a mechanism involving switch-onn/off of neighbouring genes by transposable elements in eukaryotes. J.Theor.Biol.,95,601–606

    Article  Google Scholar 

  • NEADERHOUSER, C.C.(1980) Convergence of block spins defined by a random field. J.Statist.Phys.,22,673–684

    Article  MathSciNet  Google Scholar 

  • NGUYEN, X.X.,ZESSIN, H.(1979) Integral and differential characterizations of the Gibbs process. Math.Nachr.,88,105–115

    Article  MathSciNet  MATH  Google Scholar 

  • OGATA, Y.,TANEMURA, M.(1981) Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann.Inst.Statist.Math.,33,315–338; Likelihood analysis of spatial point patterns. J.Roy.Statist.Soc.Ser.B,46,496–518

    Article  MATH  Google Scholar 

  • ODELL, G.M.(1984) A mathematically modelled cytogel cortex exhibits periodic Ca++-modulated contraction cycles seen in Physarum shuttle streaming. J.Embryol.Exp.Morphol.,83(Suppl.),261–287

    Google Scholar 

  • ODELL, G.M.,OSTER, G.,ALBERCH, P.,BURNSIDE, B. (1981) The mechanical basis of morphogenesis.I.Epithelial folding and invagination. Devel.Biol. 85,446–462

    Article  Google Scholar 

  • OKABE, Y.(1973) On a Markovian property of Gaussian processes. Proc.2nd Japan-USSR Symp.Probab.Theory (Lect.Notes in Math.,Vol.330),pp. 340–354. Berlin-Heidelberg-New York:Springer

    MATH  Google Scholar 

  • OKADA, T.S.(1980) Cellular metaplasia or transdifferentiation as a model for retinal cell differentiation. Curr.Topics in Devel.Biol.,16,349–380

    Article  Google Scholar 

  • PAPANGELOU, F.(1983) Stationary one-dimensional Markov random fields with a continuous state space. In:Probability,Statistics and Analysis (J.F.C. Kingman,G.E.H. Reuter eds.),pp.199–218. Cambridge:Cambridge Univ.Press

    Chapter  Google Scholar 

  • PARTHASARATHY, K.R.(1967) Probability Measures on Metric Spaces. New York:Academic Press

    Book  MATH  Google Scholar 

  • PELIGRAD, M.(1981) An invariance principle for dependent random variables. Z.Wahrscheinlichkeitstheorie verw.Geb.,57,495–507; (1985) An invariance principle for ø-mixing sequences. Ann.Probab.,13,1304–1313

    Article  MathSciNet  MATH  Google Scholar 

  • PINK, D.A.(1984) Theoretical models for monolayers,bilayers,and biological membranes. In:Biomembrane Structure and Function (D. Chapman ed.),pp.319–354. Weinheim:Verlag Chemie

    Google Scholar 

  • PINK, D.A.,CHAPMAN, D.(1979) Protein-lipid interactions in bilayer membranes:A lattice model. Proc.Natl.Acad.Sci.USA,76,1542–1546

    Article  Google Scholar 

  • PITT, L.D.(1971) A Markov property for Gaussian processes with a multi-dimensional parameter. Arch.Rational Mech.Anal.,43,367–391

    Article  MathSciNet  MATH  Google Scholar 

  • PITTS, J.D.(1972) Direct interaction between animal cells. In:Cell Interactions (L.G. Silvestri ed.),pp.277–285. Amsterdam:North-Holland

    Google Scholar 

  • PRESTON, C.J.(1974) Gibbs States on Countable Sets. London:Cambridge Univ.Press

    Book  MATH  Google Scholar 

  • PRESTON, C.J.(1975) Spatial birth-and-death processes. Bull.Intern.Stat.Inst.,46(book 2),371–391

    MathSciNet  MATH  Google Scholar 

  • PRESTON, C.J.(1976) Random Fields. (Lect.Notes in Math.,Vol.534) Berlin-Heidelberg-New York:Springer

    MATH  Google Scholar 

  • PRESTON, C.J.(1980) Construction of specifications. In:Quantum Fields, Algebras,Processes (L. Streit ed.),pp.268–292. Wien-New York:Springer

    Google Scholar 

  • PRIGOGINE, I.(1980) From Being to Becoming. San Francisco:W.H.Freeman

    Google Scholar 

  • PYKE, R.(1973) Partial sums of matrix arrays and Brownian shhets. In: Stochastic Analysis (D.G. Kendall,E.F. Harding eds.),pp.331–348. London:Wiley

    Google Scholar 

  • PYKE, R.(1983) A uniform central limit theorem for partial-sum processes indexed by sets. In:Probability,Statistics and Analysis (J.F.C. Kingman,G.E.H. Reuter eds.),pp.219–240. Cambridge:Cambridge Univ.Press

    Chapter  Google Scholar 

  • RAMACHANDRAN, D.(1981) A note on regular conditional probabilities in Doob's sense. Ann.Probab.,9,907–908

    Article  MathSciNet  MATH  Google Scholar 

  • READY, D.F.,HANSON, T.E.,BENZER, S.(1976) Development of the Drosophila retina,a neurocrystalline lattice. Devel.Biol.,53,217–240

    Article  Google Scholar 

  • ROBERTSON, A.,COHEN, M.H.(1972) Control of developing fields. Ann.Rev. Biophys.Bioeng.,1,409–464

    Article  Google Scholar 

  • RÖCKNER, M.(1983) Markov property of generalized fields and axiomatic potential theory. Math.Ann.,264,153–177; (1985) Generalized Markov fields and Dirichlet forms. Acta Applicandae Mathematicae 3,285–311

    Article  MathSciNet  Google Scholar 

  • ROSE, M.R., DOOLITTLE, W.F.(1983) Molecular biological mechanisms of speciation. Science 220,157–162

    Article  Google Scholar 

  • ROSENBLATT, M.(1979) Some remarks on a mixing condition. Ann.Probab.,7,170–172

    Article  MathSciNet  MATH  Google Scholar 

  • ROSENSTRAUS, M.J., SPADORO, J.P., NILSSON, J.(1983) Cell position regulates endodermai differentiation in embryonal carcinoma cell aggregates. Devel.Biol.,98,110–116

    Article  Google Scholar 

  • ROZANOV, Y.A.(1982) Markov Random Fields. Berlin-Heidelberg-New York: Springer

    Book  MATH  Google Scholar 

  • RUELLE, D.(1969) Statistical Mechanics.Rigorous Results. Reading(Mass.): W.A.Benjamin

    Google Scholar 

  • RUELLE, D.(1978) Themodynamic Formalism.The Mathematical Structures of Classical Equilibrium Statistical Mechanics. Reading(Mass.): Addison-Wesley

    Google Scholar 

  • RUELLE, D.(1981) A mechanism for speciation based on the theory of phase transitions. Math.Biosci.,56,71–75

    Article  MathSciNet  MATH  Google Scholar 

  • SAZONOV, V.(1962) On perfect measures. Amer.Math.Soc.Transl.,248,229–254

    MATH  Google Scholar 

  • SCHWARTZ, L.(1973) Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. London:Oxford Univ.Press

    MATH  Google Scholar 

  • SCOTT, H.L.(1974) A model of phase transitions in lipid bilayers and biologcal membranes. J.Theor.Biol.,46,241–253

    Article  Google Scholar 

  • SINAI, Y.G.(1982) Theory of Phase Transitions:Rigorous Results. Oxford: Pergamon Press

    Google Scholar 

  • SMYTHE, R.T.(1976) Multiparameter subadditive processes. Ann.Probab.,4,772–782

    Article  MathSciNet  MATH  Google Scholar 

  • SOKAL, A.D.(1981) Existence and compatible families of proper regular conditional probabilities. Z.Wahrscheinlichkeitstheorie verw. Geb.,56,537–548

    Article  MathSciNet  MATH  Google Scholar 

  • SPITZER,F.(1971) Random Fields and Interacting Particle Systems. Mathematical Assoc.America

    Google Scholar 

  • SPITZER,(1975a) Markov random fields on an infinite tree. Ann.Probab., 3,387–398; (1975b) Phase transition in one dimensional nearest neighbor system. J.Funct.Anal.,20,240–254

    Article  MathSciNet  MATH  Google Scholar 

  • STATULYAVICHUS, V.A.(1983) On a condition of almost Markov regularity. Theor.Probab.Appl.,28,379–383

    Article  MathSciNet  MATH  Google Scholar 

  • STOLL, A.(1986) A nonstandard construction of Lévy Brownian motion. Probab.Theor.Rel.Fields,71,321–334

    Article  MathSciNet  MATH  Google Scholar 

  • SURGAILIS, D.(1981) On infinitely divisible self-similar random fields. Z.Wahrscheinlichkeitstheorie verw.Geb.,58,453–477

    Article  MathSciNet  MATH  Google Scholar 

  • TJUR, T.(1980) Probability Based on Radon Measures. Chichester:Wiley

    MATH  Google Scholar 

  • URBANIK, K.(1975) Extreme point method in probability theory. Lect.Notes in Math.,Vol.472,pp.169–194. Berlin-Heidelberg-New York:Springer

    MATH  Google Scholar 

  • van den HOEVEN, P.C.T.(1983) On Point Processes (Mathematical Centre Tracts,Nr.165) Amsterdam:Mathematisch Centrum

    Google Scholar 

  • VANMARCKE, E.(1983) Random Fields:Analysis and Synthesis. Cambridge (Mass.) MIT Press

    MATH  Google Scholar 

  • van PUTTEN, C., van SCHUPPEN, J.H.(1985) Invariance properties of the conditional independence relation. Ann.Probab.,13,934–945

    Article  MathSciNet  MATH  Google Scholar 

  • WADDINGTON, C.H.(1972) Form and information. In:Towards a Theoretical Biology (C.H. Waddington ed.),Vol.4,pp.109–145. Edinburgh:Edinburgh Univ.Press

    Google Scholar 

  • WADDINGTON, C.H.(1973) The morphogenesis of patterns in Drosophila. In: Developmental Systems:Insects (S.J. Counce,C.H. Waddington eds.), Vol.2,pp.499–535. London:Academic Press

    Google Scholar 

  • WALSH, J.B.(1979) Convergence and regularity of multiparameter strong martingales. Z.Wahrscheinlichkeitstheorie verw.Geb.,46,177–192

    Article  MathSciNet  MATH  Google Scholar 

  • WINKLER, G.(1981) The number of phases of inhomogeneous Markov fields with finite state spaces on N and Z and their behaviour at infinity Math.Nachr.,104,101–117

    Article  MathSciNet  MATH  Google Scholar 

  • WSCHEBOR, M.(1985) Surfaces Aléatoires.Mesure géométrique des Ensembles de niveau (Lect.Notes in Math.,Vol.1147).Berlin-Heidelberg-New York-Tokyo:Springer

    MATH  Google Scholar 

  • YANG, J.,RICHARDS, J.,BOWMAN, P.,GUZMAN, R.,ENAMI, J.,McCORMICK, K.,HAMAMOTO, S.,PITELKA, D.,NANDI, S.(1979) Sustained growth and three-dimensional organization of primary mammary tumor epithelial cells embedded in collagen gels. Proc.Natl.Acad.Sci.USA,76,3401–3405

    Article  Google Scholar 

  • ZACHARY, S.(1983) Countable state space Markov random fields and Markov chains on trees. Ann.Probab.,11,894–903

    Article  MathSciNet  MATH  Google Scholar 

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  1. German Cancer Research Center, Heidelberg

    Petre Tautu

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  1. Petre Tautu
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© 1986 Springer-Verlag

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Tautu, P. (1986). Random fields: Applications in cell biology. In: Tautu, P. (eds) Stochastic Spatial Processes. Lecture Notes in Mathematics, vol 1212. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076254

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  • DOI: https://doi.org/10.1007/BFb0076254

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  • Print ISBN: 978-3-540-16803-4

  • Online ISBN: 978-3-540-47053-3

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