Predrag Cvitanović publications
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Periodic orbit theory publications

	Periodic orbit research overview
	Abstract of a general colloquium on the periodic orbit theory

Program whose goal is a theory of turbulent dynamics of classical, stochastic and quantum fields. Chaotic field theory is developed in two directions: 1) explorations of the feasibility of describing weak turbulence in terms of spatiotemporally recurrent patterns, and 2) new perturbation theory methods for computing corrections about nontrivial saddles of path integrals.

Read the sketch paper first, perhaps click through the trace formulas seminar next.

Predrag Cvitanović Chaotic Field Theory: a Sketch, Physica A 288 , 61 (2000) [ ps.gz , arXiv:nlin.CD/0001034 , source files , NSF critique ]

Explorations of weak turbulence described in terms of spatiotemporally recurrent patterns.

	Predrag Cvitanović, R. L. Davidchack and E. Siminos State space geometry of a spatio-temporally chaotic Kuramoto-Sivashinsky flow [ arXiv:0709.2944 ]
	Yueheng Lan and Predrag Cvitanović Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics [ arXiv:0804.2474 ]
	Predrag Cvitanović, F. Christiansen and V. Putkaradze Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns Nonlinearity 10, 50 (1997) [ ps.gz , chao-dyn/9606016 , NSF critique ]
	Predrag Cvitanović and Y. Lan Turbulence and what to do about it (Bristol, Gottingen 2004 - PDF overheads) Predrag Cvitanović Spatiotemporal chaos in terms of unstable recurrent patterns (overheads + streaming video (!), seminar abstract) Related links: Vachtang Putkaradze PhD thesis The initial attempt: Vachtang's term paper Why ``Hopf's last hope'' in the title?

State space geometry of moderate Re turbulence

Getting hang of turbulence for plumbers: pipes and planes

	Divakar Viswanath and Predrag Cvitanović Stable manifolds and the transition to turbulence in pipe flow J. Fluid Mech. (2008) , submitted [ arXiv:0801.1918 ]
	John F. Gibson, Jonathan Halcrow and Predrag Cvitanović Visualizing the geometry of state space in plane Couette flow J. Fluid Mech. (2008), to appear; [ arXiv:0705.3957 ]
	John F. Gibson Movies of plane Couette flow - recommended

Getting hang of spatiotemporal dynamics in nearly integrable regimes, as a warmup to turbulent dynamics...

	Mason A. Porter and Predrag Cvitanović A perturbative analysis of Modulated Amplitude Waves in Bose-Einstein Condensates CHAOS 14, 739 (2004) [ arXiv:nlin.CD/0308024 ]
	Mason A. Porter and Predrag Cvitanović Modulated Amplitude Waves in Bose-Einstein Condensates Phys. Rev. E 69, 047201 (2004) [ arXiv:nlin.CD/0307032 ]
	Yueheng Lan, Nicola Garnier and Predrag Cvitanović Modulated solutions of the complex Ginzburg-Landau equation Physica D 188 193, (2004) [ arXiv:nlin.PS/0208001 ]

A triptych of technical papers whose goal is to develop improved methods of computing higher order corrections to nontrivial saddles of path integrals:

	Predrag Cvitanović, C.P. Dettmann, R. Mainieri and G. Vattay Trace formulas for stochastic evolution operators: Weak noise perturbation theory J. Stat. Phys. 93, 981 (1998) [ ps.gz , chao-dyn/9807034 ]
	Predrag Cvitanović, C.P. Dettmann, R. Mainieri and G. Vattay Trace formulas for stochastic evolution operators: Smooth conjugation method Nonlinearity 12, 939 (1999) [ ps.gz published version , ps.gz , chao-dyn/9811003 ]
	Predrag Cvitanović, C.P. Dettmann, G. Palla, N. Søndergaard and G. Vattay Spectrum of stochastic evolution operators: Local matrix representation approach Phys. Rev. E 60, 3936 (1999) [ ps.gz , chao-dyn/9904027 ]
	Predrag Cvitanović Trace formulas for stochastic evolution operators (seminar abstract, 1999).
	Predrag Cvitanović Noisy Chaos a talk at Hans C. Fogedby 60 symposium.

A step toward generalizing Gutzwiller semiclassical theory to wave chaos in elastodynamical systems (professionals do not allow us to call this "acoustics")

P. Cvitanović, N. Søndergaard and A. Wirzba Wave Chaos in Elastodynamic Cavity Scattering Europhysics Letters 72, 534 (2005) Brief history: submitted to Phys Rev Letters Aug 28 2001; revised version submitted May 18 2004; referee offended by the "In elastodynamics, period two implies chaos" title - rejected; submitted to Europhysics Letters, May 2005, revised version with more resonances included Sep 23 2005, accepted Sep 26 2005. [ pdf , arXiv:nlin/0108053 ]

Related links: N. Søndergaard, P. Cvitanović, and A. Wirzba, Closed complex rays in scattering from elastic voids in B. Nilsson, ed., Mathematical Modelling of Wave Phenomena 2005, AIP Conference Proceedings (2006) N. Søndergaard, PhD thesis (Dec 2000) Andreas Wirzba, Georgia Tech seminar (Aug 2001)

A variational principle for robust periodic orbit and invariant tori searches:

	Yueheng Lan, Cristel Chandre, and Predrag Cvitanović Variational method for locating invariant tori Phys. Rev. E 74, 046206 (2006) [ arXiv:nlin.CD/0508026 ] (Aug 13, 2005)
	Predrag Cvitanović and Yueheng Lan Turbulent fields and their recurrences in N.~Antoniou, ed., Proceed. of 10. Intern. Workshop on Multiparticle Production: Correlations and Fluctuations in QCD (World Scientific, Singapore 2003); [ ps.gz , arXiv:nlin.CD/0308006 ]
	Yueheng Lan and Predrag Cvitanović Variational method for finding periodic orbits in a general flow Phys. Rev. E 69, 016217 (2004) [ arXiv:nlin.CD/0308008 , ps.gz ]

An early proposal on how to fish for periodic orbits by looking for near recurrences:

D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunaratne and I. Procaccia Exploring chaotic motion through periodic orbits Phys. Rev. Lett. 58, 2387-2389 (1987)

An attempt to demonstrate existence of unstable periodic orbits and chaos in a slices of rat brains, very noisy neurophysiology experimental data

	Marc W. Slutzky, Predrag Cvitanović and David J. Mogul Deterministic chaos and noise in three in vitro hippocampal models of epilepsy Annals of Biomedical Engineering 29, 607 (2001).
	Marc W. Slutzky, Predrag Cvitanović and David J. Mogul Manipulating epileptiform bursting in the rat hippocampus using chaos control and adaptive techniques IEEE Transactions on Biomedical Engineering 50, 559 (2001).
	Marc W. Slutzky, Predrag Cvitanović and David J. Mogul Identification of determinism in noisy neuronal systems J. Neuroscience Methods 118, 153 (2002)

Two papers demonstrating that periodic orbits must satisfy infinity many sum rules

	Predrag Cvitanović, Kim Hansen, Juri Rolf and Gabor Vattay Beyond periodic orbit theory Nonlinearity 11, 1209 (1998) [ ps.gz ]
	Sune F. Nielsen, Per Dahlqvist and Predrag Cvitanović Periodic orbit sum rules for billiards: Accelerating cycle expansions J. Phys. A 32, 6757 (1999) [ ps.gz , chao-dyn/9901001 , working notes ]

Non-hyperbolicity, intermittency, power-law correlations: convergence of cycle expansions, analyticity of dynamical zeta functions, approach to the border of order by renormalization methods.

	Roberto Artuso, Predrag Cvitanović and Gregor Tanner Cycle expansions for intermittent maps Proc. Theo. Phys. Supp. 150, 1 (2003) [ published version , arXiv:nlin.CD/0305008 , ps.gz ]
	Carl P. Dettmann and Predrag Cvitanović Cycle expansions for intermittent diffusion Phys. Rev. E 56, 6687 (1997)
	Roberto Artuso, Erik Aurell and Predrag Cvitanović Recycling of strange sets: II. applications Nonlinearity 3, 361 (1990)

Periodic orbit theory of deterministic diffusion

	Predrag Cvitanović, Jean-Pierre Eckmann and Pierre Gaspard Transport properties of the Lorentz gas in terms of periodic orbits Chaos, Solitons and Fractals 6, 113 (1995) - (61 kB) [ ps.gz ]
	Predrag Cvitanović, Pierre Gaspard and T. Schreiber) Investigation of the Lorentz Gas in terms of periodic orbits CHAOS 2, 85 (1992) [ ps.gz ]

Periodic orbit theory of diffusion extending to power spectra, with Pikovsky and Feigenbaum.
Both Mitchell's draft and Arkady's draft have interesting material not in the published, abreviated version:

Predrag Cvitanović and Arkady S. Pikovsky Cycle expansion for power spectrum Proc. SPIE - Int. Soc. Opt. Eng. (USA) 2038, 290 (1997)

How to implement the periodic orbit theory as a computational method

	Predrag Cvitanović Continuous symmetry reduced trace formulas (in preparation, July 2006)
	Predrag Cvitanović Trace formulas in classical dynamical systems in I.V. Lerner, J.P. Keating, D.E. Khmelnitskii, eds., Supersymmetry and Trace Formulae: Chaos and Disorder, (Plenum, New York 1998)
	G. Simon, Predrag Cvitanović, M.T. Levinsen, I. Csabai and Á. Horváth Periodic orbit theory applied to a chaotically oscillating gas bubble in water Nonlinearity 15, 25 (2002)
	Predrag Cvitanović, Gabor Vattay and Andreas Wirzba Quantum fluids and classical determinants in H. Friedrich and B. Gerhardt., eds., Classical, Semiclassical and Quantum Dynamics in Atoms - in Memory of Dieter Wintgen, Lecture Notes in Physics 485 (Springer, New York 1997) [ ps.gz , chao-dyn/9608012 ]
	Predrag Cvitanović Dynamical averaging in terms of periodic orbits Physica D 83, 109 (1995) [ps.gz]
	Predrag Cvitanović, B. Eckhardt, P.E. Rosenqvist, G. Russberg and P. Scherer Pinball scattering in G. Casati and B. Chirikov, eds., Quantum Chaos, (Cambridge University Press, Cambridge 1994) [ps.gz - sorry, no figures ]
	Neal J. Balmforth, Predrag Cvitanović, Glenn R. Ierley, Edward A. Spiegel and Gabor Vattay Advection of vector fields by chaotic flows in Stochastic Processes in Astrophysics, Annals of New York Academy of Sciences 706, 148 (1993) [ ps.gz - 69 Kb chao-dyn/9307011 - sorry, no figures ]
	Predrag Cvitanović and P.E. Rosenqvist A new determinant for quantum chaos in G.F. Dell'Antonio, S. Fantoni and V.R. Manfredi, eds., From Classical to Quantum Chaos, Soc. Italiana di Fisica Conf. Proceed. 41, pp. 57-64 (Ed. Compositori, Bologna 1993) [e-print not available]
	Predrag Cvitanović and G. Vattay Entire Fredholm determinants for evaluation of semi-classical and thermodynamical spectra Phys. Rev. Lett. 71, 4138 (1993) [ chao-dyn/930701 ]
	Predrag Cvitanović, P.E. Rosenqvist, G. Vattay, and H.H. Rugh A Fredholm determinant for semi-classical quantization CHAOS 3, 619 (1993) [ ps , chao-dyn/9307014 ]
	F. Christiansen and Predrag Cvitanović Periodic orbit quantization of the anisotropic Kepler problem CHAOS 2, 61 (1992)
	Periodic orbit theory in classical and quantum mechanics, CHAOS 2, 1 (1992)
	Kvantekaos, KVANT 5, 1 (1994) [e-print not available]
	Kvantes Lykkelige Dag (with Kenneth Krabat), Naturligvis 20 (1991) [e-print not available]
	Predrag Cvitanović and Bruno Eckhardt Periodic orbit expansions for classical smooth flows J. Phys A 24, L237 (1991) [ ps.gz ]
	Freddy Christiansen, Predrag Cvitanović and Hans Henrik Rugh The spectrum of the period-doubling operator in terms of cycles J. Phys A 23, L713 (1990)
	Roberto Artuso, Erik Aurell and Predrag Cvitanović Recycling of strange sets: I. cycle expansions Nonlinearity 3, 325 (1990)
	Predrag Cvitanović and Bruno Eckhardt Periodic orbit quantization of chaotic systems Phys. Rev. Lett. 63, 823 (1989)
	Predrag Cvitanović Invariant measurement of strange sets in terms of cycles Phys. Rev. Lett. 61, 2729 (1988)

How to partition the phase space of a chaotic dynamical system

	Mason A. Porter and Predrag Cvitanović Ground Control to Niels Bohr: Exploring Outer Space with Atomic Physics Notices Am. Math. Soc. 52, 1020 (2005) [ physics/0505085 , pdf ] Featured in: Science News 168 MSNotices200509.html (Sep 10 2005); American Mathematical Society Notices 52 (Oct 2005)
	Predrag Cvitanović Periodic orbits as the skeleton of classical and quantum chaos Physica D 51 138 (1991)
	Predrag Cvitanović, Gemunu H. Gunaratne and Itamar Procaccia Topological and metric properties of Hénon-type attractors Phys. Rev. A 38, 1503 (1988)
	Predrag Cvitanović and Kai T. Hansen Bifurcation structures in maps of Hénon type Nonlinearity 11, 1233 (1998)
	Kai T. Hansen and Predrag Cvitanović Symbolic Dynamics and Markov Partitions for the Stadium Billiard J. Stat. Phys. ? (20??) [ ps.gz , chao-dyn/9502005 ]
	The archived version was accepted for publication by J. Stat. Phys. in 1996, but then I had a brilliant idea how to make it better, and a revised version is still waiting to be resubmitted. If you can bring it back to a publishable state - current version is Notes for Kai (sept 95) - please do it, and join us as a co-author. The paper is as good as most stuff that gets published, but neither of us has the time to finish it.
	P. Cvitanović and Bruno Eckhardt Symmetry decomposition of chaotic dynamics Nonlinearity 6, 277 (1993) [ chao-dyn/9303016 ]

Transitions to chaos publications

Renormalization in chaos research overview

In 1976, M.J. Feigenbaum got me interested in his discovery of universality in one-dimensional iterative maps. Following his functional formulation of the problem, I derived the universal equation for period doubling, which has since played a key role in the theory of transitions to turbulence (see M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978) and 21, 669 (1979)). We have generalized the universal equations to period n-tuplings; since then we have constructed universal scaling functions for all winding numbers in circle maps, and established the universality of the Hausdorff dimension of the critical staircase.

	Predrag Cvitanović Universality in chaos (or, Feigenbaum for cyclists) lectures given at 1982 Zakopane School of Theoretical Physics, Acta Phys. Polonica A65, 203 (1984)
	Predrag Cvitanović and M.H. Jensen Universality in transitions to chaos in Chaos and universality (Nordita reprint selection, November 1981)
	Predrag Cvitanović and M.H. Jensen Universalitet i overgang til kaos Fysisk Tidsskrift 80, 82 (1982)

	Predrag Cvitanović Circle maps: irrationally winding, in C. Itzykson, P. Moussa and M. Waldschmidt, eds., Number Theory and Physics, Les Houches 1989 Spring School, (Springer, New York 1992) - sorry, no figures
	Predrag Cvitanović, Gemunu H. Gunaratne and M.J. Vinson On the mode-locking universality for critical circle maps Nonlinearity 3, 873 (1990) [ ps.gz - 108 Kb ]
	Predrag Cvitanović, M.H. Jensen, L.P. Kadanoff and I. Procaccia Renormalization, unstable manifolds and the fractal structure of mode locking Phys. Rev. Lett. 55, 343 (1985)
	Predrag Cvitanović Farey organization of the fractional Hall effect Phys. Scripta T9, 202 (1984)
	Predrag Cvitanović Universal scaling laws for maps on the interval and circle maps in R.W. Boyd, L.M. Narducci and M.G. Raymer, eds., Instabilities and Dynamics of Lasers and Nonlinear Optical Systems, (U. of Cambridge Press, Cambridge, 1985)
	Predrag Cvitanović, B. Shraiman and B. Söderberg Scaling laws for mode lockings in circle maps Phys. Scripta 32, 263 (1985)
	Predrag Cvitanović M.H. Jensen, L.P. Kadanoff and I. Procaccia Circle maps in the complex plane in L. Pietronero and E. Tosatti, eds., Fractals in Physics, Trieste, July 1985 (North Holland, New York, 1985)
	Predrag Cvitanović and T. Bohr Chaos is good news for physics Nature 329, 391-392 (1987)
	Predrag Cvitanović Chaos for cyclists in E. Moss, ed., Noise and Chaos in Nonlinear Dynamical Systems, (Cambridge Univ. Press, Cambridge 1989)
	Predrag Cvitanović Recycling chaos in A. Ferraz, F. Oliveira and R. Osorio, eds., Nonlinear Physical Phenomena, Brasilia 1989 Winter School, (World Scientific, Singapore 1990) - sorry, no figures
	Predrag Cvitanović The power of chaos in J.H. Kim and J. Stringer, eds., Applied Chaos, (John Wiley & Sons, New York 1992) - sorry, no figures

	Predrag Cvitanović and Jan Myrheim Complex universality Comm. Math. Phys. 121, 225 (1989)
	Predrag Cvitanović and Jan Myrheim Universality for period n-tuplings in complex mappings Phys. Lett. 94A, 329 (1983)
	Predrag Cvitanović Renormalization description of transitions to chaos in S. Lundquist, N.H. March and E. Tosatti, eds., Order and Chaos in Non-linear Physical Systems, pp. 73-97 (Plenum, New York 1988) - a subset of the "Complex Universality" paper
	Predrag Cvitanović, T. Bohr and M.H. Jensen Fractal aggregates in the complex plane Europhys. Lett. 6, 445 (1988)

The discovery of phase transitions on "strange sets" was followed up by many other authors; such transitions were subsequently found in a variety of dynamical systems.

	Predrag Cvitanović, A. Artuso and B. Kenny Phase transitions on strange irrational sets Phys. Rev. A 39, 268 (1989)
	Predrag Cvitanović Hausdorff dimension of irrational windings in R. Gilmore, ed., Proceedings of the XV International Colloquium on Group Theoretical Methods in Physics, pp. 184-198 (World Scientific, Singapore, 1987)
	Predrag Cvitanović Phase transitions on strange sets, in P. Zweifel, G. Gallavotti and M. Anile, eds., Non-linear Evolution and Chaotic Phenomena, pp. 349-361 (Plenum, New York 1988) - sorry, not scanned yet

Group theory publications

overview for my group theory publications click here Group theory webbook

Quantum field theory publications

Program whose goal is a non-perturbative theory of turbulent dynamics of classical, stochastic and quantum fields. Turbulent field theory is developed in two directions:
1) explorations of the feasibility of describing weak turbulence in terms of spatiotemporally recurrent patterns, and
2) new perturbation theory methods for computing corrections about nontrivial saddles of path integrals.

Read the sketch paper first, click through the trace formulas papers next.

One big calculation, the answer

	Predrag Cvitanovic´ and Toichiro Kinoshita Feynman-Dyson rules in parametric space Phys. Rev. D10, 3478 (1974) [ SPIRES citations ]
	A (hopefully) pedagogical overview of the Schwinger and Feynman parametric representation of Feynman integrals, should be useful for any QFT perturbative calculation. Some new results, for example the theorem derived in the appendix.
	Predrag Cvitanovic´ and Toichiro Kinoshita New approach to the separation of ultraviolet and infrared divergences of Feynman-parametric integrals Phys. Rev. D10, 3991 (1974) [ SPIRES citations ]
	A new method for dealing with Feynman diagram infrared divergences is introduced. The main result are the very ellegant and compact formulas (3.40) and (5.14) which extract the finite part from any general mass-shell Feynman diagram, removing both ultraviolet and infrared parts of the diagram and all of its counterterms. What remains after the projection is a pointwise-convergent integrand, well suited to numerical integrations.
	Predrag Cvitanovic´ and Toichiro Kinoshita Sixth-order radiative corrections to the electron magnetic moment Phys. Rev. Lett. 29, 1534 (1972)
	At the time perhaps the most demanding numerical computation in theoretical physics, it remained QED's most precise prediction for a number of years. A step in T. Kinoshita's heroic undertaking, but nothing you would want to read today.
	Predrag Cvitanovic´ and Toichiro Kinoshita Sixth order magnetic moment of the electron Phys. Rev. D10, 4007 (1974) [ SPIRES citations ]
	Most of this you can safely skip, unless you happen to be evaluating (g-2) to 3-loop level. However, the new formula (6.22) for the electron magnetic moment, Sect. VI, might be of interest: (g-2) is evaluated from a derivative of the 2-point electron self-energy, rather than a 3-point electron-photon vertex, with fewer Feynman graphs: (1) it enabled us to calculate the electron magnetic moment by two wholy independent methods.
	Predrag Cvitanovic´ Computer generation of integrands for Feynman parametric integrals Cornell preprint CLNS-234 (June, 1973) and Proc. 3rd Coll. on Advanced Comp. Meth. in Theoretical Physics (Marseille, 1973)
	One of the ur-algebraic symbol manipulation programs

For me the conceptually most striking lesson of these long QED calculations were the amazing cancellations induced by gauge invariance. The desire to understand and exploit gauge invariance more effectively has motivated much of my subsequent research. The most interesting results of this effort were the mass-shell QCD Ward identities and the construction of the QCD gauge sets. This work also motivated the formulation of planar field theory.

Other papers in this series are those on the QCD mass-shell infrared singularities and the diagram counting, both attempts to formulate gauge-invariant models computable to high orders, in order to investigate the nature of gauge-invariance induced cancellations.

Predrag Cvitanovic´ Asymptotic estimates and gauge invariance Nucl. Phys. B127, 176 (1977) [ SPIRES citations ]

On basis of very skimpy numerical evidence, I conjecture that the gauge invariance induced cancellations are so dramatic that the growth rate of high order perturbation theory corrections to mass-shell gauge-invariant quantities is slower than Dyson's asymptotic series n! estimate. In the case of QED vertex corrections, the smallest gauge invariant set contributing to (m+m'+k)th order consists of m photon "strands" attached to the incoming electron, m' photon "strands" attached to the outgoing electron, and k photon "strands" crossing the external photon vertex. Ignoring sets with electron loops and assuming that each gauge set gives a finite contribution leads to a guess that perturbation series for the electron magnetic moment sums up to approximately The story behind the conjecture.

The method that I have used to develop the planar field theory is somewhat different from what is in most field theory textbooks; consulting my Field Theory textbook might make this derivation more accessible.

	Predrag Cvitanovic´ Planar perturbation expansion Phys. Lett. 99B, 49 (1981) [ SPIRES citations ]
	Predrag Cvitanovic´ The planar sector of field theories (with P.G. Lauwers and P.N. Scharbach), Nucl. Phys. B203, 385 (1982) [ SPIRES citations ]
	Though the formalism is rather different, essentially the same theory was rederived in 1994 by M.A. Douglas, D.J. Gross, R. Gopakumar, D.V. Voiculescu and others.

	Predrag Cvitanovic´ Yang-Mills theory of the mass-shell Phys. Rev. Lett. 37, 1528 (1976) [ SPIRES citations ]
	Mass-shell amplitudes for both QED and QCD are defined via dimensional regularization, and shown to to be gauge invariant trough a cancellation between UL and IR singularities. Prior to this article, IR and UR were regularized by different methods which, when applied to QCD, violated gauge invariance.
	Predrag Cvitanovic´ Infra-red structure of Yang-Mills theories Phys. Lett. 65B, 272 (1976) [ SPIRES citations ]
	Predrag Cvitanovic´ Quantum Chromodynamics on the mass-shell Nucl. Phys. B130, 114 (1977) [ SPIRES citations ]
	Predrag Cvitanovic´, B. Lautrup and R.B. Pearson The number and weights of Feynman diagrams Phys. Rev. D18, 1939 (1978) [ SPIRES citations ]
	Predrag Cvitanovic´, J. Greensite and B. Lautrup The cross-over points in lattice gauge theories with continuous gauge groups Phys. Lett. 105B, 201 (1981)
	Predrag Cvitanovic´, P.G. Lauwers and P.N. Scharbach Gauge invariance structure of Quantum Chromodynamics Nucl. Phys. B186, 165 (1981) [ SPIRES citations ]

	Predrag Cvitanovic´ Wide-angle behavior of a double-scattering diagram Phys. Rev. D10, 338 (1974)
	Predrag Cvitanovic´, R.J. Gonsales and D.E. Neville Color charge algebras in Adler's Chromodynamics Phys. Rev. D18, 3881 (1978)
	Predrag Cvitanovic´, P. Hoyer and K. Konishi Partons and branching Phys. Lett. 85B 413 (1979)
	Predrag Cvitanovic´ and R. Horsley Exact solutions of the Altarelli-Parisi equations Nucl. Phys. B173, 229 (1980)
	Predrag Cvitanovic´, P. Hoyer and K. Zalewski Parton evolution as a branching process Nucl. Phys. B176, 429 (1980)

	Predrag Cvitanovic´ et al drafts in progress (of interest only to frustrated collaborators)
	Predrag Cvitanović Midnight rider Bike World (August 1974)
	Predrag Cvitanović and M.J. Feigenbaum, for D. Bensimon, T.C. Halsey, M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia, B.I. Shraiman and J. Stavans More on microcanonical paradigm (G\"oteborg, 3 a.m. of 17 Nov. 1985), rejected from every proceedings submitted to.