The Department ofMATHEMATICS & STATISTICS Concordia University

Dr. Pawel Gora - List of Publications

Research articles published in reviewed journals or books: 

•  Ted Szylowiec and Pawe\l \ G\'ora, Memory maps with Elliptical Trajectories,  Int. Jour. Biff. Chaos. vol.33, No. 9, p.2330021 , 15 pages  (2023) DOI: https://doi.org/10.1142/S0218127423300215
• Pawe\l \ G\'ora and Ted Szylowiec, Islands Supporting ACIM for Two Dimensional map, Int. Jour. Biff. Chaos. Vol. 32, No. 8,  p. 2230017 (5 pages) (2022), DOI: https://doi.org/10.1142/S0218127422300178
•  Eyad AlFarajat and Pawe\l \ G\'ora,  On the absolutely continuous invariant measures for the random maps,  Int. Jour. Biff. Chaos. Vol. 32}, 03, 2250030 (2022),  https://doi.org/10.1142/S0218127422500304
•   Abraham Boyarsky and Pawe\l\ G\'ora, A Dynamical Systems Model for Face Perception, American Journal of Mathematical and Computer Modelling, Vol.6 (4), 76--80 (2021).    doi: https://doi.org/10.11648/j.ajmcm.20210604.13
• Abraham Boyarsky, Pawe\l \ G\'ora, Zhenyang Li. (2020) A Metastable Model for Quantum Super-position. Journal of Applied Mathematics and Computation, 4(4), 224--229 . DOI: https://doi.org/10.26855/jamc.2020.12.013
        
           Pawe\l\ G\'ora, Abraham Boyarsky and Christopher Keefe, "Absolutely Continuous Invariant measures for non-autonomous dynamical systems", Journal of Mathematical Analysis and Applications, 470  (2019), 159--168, DOI:                   https://doi.org/10.1016/j.jmaa.2018.09.060.
  
• Noah H. Rhee, Pawe\l\ G\'ora and Majid Bani-Yaghoub,  "Predicting and  estimating probability density functions of chaotic systems",  Discrete and Continuous  Dynamical System - B, DOI:10.3934/dcdsb.2017144
   Pawe{\l} G\'ora, Abraham Boyarsky, Zhenyang Li and Harald Proppe, "{Statistical and Deterministic Dynamics of Maps with Memory",  Discrete and Continuous  Dynamical System - A, 37 (8) (2017), 4347--4378, DOI: 10.3934/dcds.2017186,  preprint, http://arxiv.org/abs/1604.06991.
   Pawe{\l} G\'ora, Abraham Boyarsky and Zhenyang Li, "Singular SRB measures for a non 1--1 map of the unit square",    Journal of Stat. Physics 165 (2016), 409--433, DOI: 10.1007/s10955-016-1620-y, available at http://arxiv.org/abs/1607.01658, full-text view-only version: http://rdcu.be/kod0
   Abraham Boyarsky, Pawe\l\  G\'ora and Zhenyang Li,  "Optimal chaotic selectors", Dynamical Systems 30:3 (2015), 253-269, DOI: 10.1080/14689367.2015.1006585
   A. Boyarsky, P. Eslami, P. G\'ora,  Zh. Li,  J. Meddaugh and B. E. Raines,  "Chaos for successive maxima map implies chaos for the original map", Nonlinear Dynamics \textbf{79}, Issue 3 (2015),  2165--2175,     DOI 10.1007/s11071-014-1802-6
•   P. Gora, Zh. Li, A. Boyarsky and H. Proppe,    "Toward A Mathematical Holographic Principle", Journal of Statistical Physics \textbf{156}, No. 4 (2014), 775--799, DOI 10.1007/s10955-014-1029-4
•    A. Boyarsky, P. Gora and Zh. Li, "Selections and their Absolutely Continuous Invariant Measures", Journal of Mathematical Analysis and Applications \textbf{413}, No. 1 (2014),  100--113,   http://dx.doi.org/10.1016/j.jmaa.2013.11.043
•   Peyman Eslami and Pawel Gora, "Stronger Lasota-Yorke inequality for piecewise monotonic transformations", Proc. Amer. Math. Soc. \textbf{141}, No. 12 (2013), 4249--4260, DOI: S 0002-9939(2013)11676-X.
•   A. Boyarsky, Zh. Li, P. Gora, "A Framework for Interneural Dynamics in a Cerebral Cortex Center",   Int. Jour. Biff. Chaos \textbf{23},No. 9 (2013), p. 1350151 (8 pages),    DOI: 10.1141/S0218127413501514 .
  
•  Zh. Li, P. Gora, "Instability of Isolated Spectrum for W-shaped maps" , Ergodic Th. and Dyn. Syst. \textbf{33}, issue 4 (2013), 1052--59.
•   A. Boyarsky, P. Gora,  "Dynamical System Solutions for Homogeneous Linear Functional Equations of Higher Order",  Aequationes Math. \textbf{85} (2013), 221--230, DOI: 10.1007/s00010-013-0198-6 
•   Zhenyang Li, Pawel Gora, Abraham Boyarsky, Harald Proppe and Peyman Eslami,  "A Family of Piecewise Expanding Maps having Singular Measure as a limit  of ACIM's",  Ergodic Th. and Dyn. Syst. \textbf{33}, No. 1, (2013),  158--167, DOI:10.1017/S0143385711000836.
•  A. Boyarsky, P. Gora, "Stochastic Perturbations and Ulam's method for W-shaped Maps",  Discrete and Continuous  Dynamical System - A \textbf{33}, Number 5, (May 2013),  1937--1944, DOI:10.3934/dcds.2013.33.1937
•  P. Gora, A. Boyarsky and Peyman Eslami, "Metastable systems as random maps",   Int. Jour. Biff. Chaos \textbf{22}, No. 11, (2012),  1250279-1--1250279-11,  DOI: 10.1142/S0218127412502793
•   P. Gora, Zh. Li and  A. Boyarsky, "Harmonic average of slopes and the stability of  absolutely continuous invariant measure", J. Math. Anal. Appl.  396,    Issue 1 (2012), pages 1--6, DOI: http://10.1016/j.jmaa.2012.05.067

• (with Li, Zhenyang,  Boyarsky, Abraham and  Proppe, Harald), "Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval",  Journal of Statistical Physics \textbf{146} (2012), no. 4,  850-863, http://dx.doi.org/10.1007/s10955-012-0425-x .
• (with  Islam, Md Shafiqul), "Invariant measures of stochastic perturbations of dynamical systems using  Fourier approximations", Internat. J. Bifur. Chaos Appl. Sci. Engrg. \textbf{21} (2011), no. 1, 113--123.
• (with Peyman Eslami) "On Eventually Expanding Maps of the Interval", The American Mathematical Monthly, \textbf{118}, No. 7 (August-September 2011), pp. 629--635.
• (with R.J. Stern) "Subdifferential  analysis of the Van der Waerden function",  Journal of Convex Analysis vol. 18, No. 3 (2011),   699--705.
• (with W. Bahsoun) "SRB Measures for Certain Markov Processes",  DCDS-A 30, no. 1, (2011), 17--37.   (http://dx.doi.org/10.3934/dcds.2011.30.17)

• (with H. Doosti, M. S. Islam, Y. P. Chaubey) "Two-dimensional wavelets for nonlinear autoregressive models with an application in dynamical systems",  Italian Journal of Pure and Applied Mathematics, vol. 27 (2010), 39--62.
• (with M. S. Islam)  "Smoothness of density functions for random maps",  DCDIS, series A, 17 (2010), 249--262.
• (with A. Boyarsky) "Chaos of dynamical systems on general time domains", Int. Jour. Biff. Chaos  19, Issue: 11 (2009), 3829--3832, (http://dx.doi.org/10.1142/S0218127409025158)
• (with A. Boyarsky) "A Random Map Model for Quantum Interference",  Communications in Nonlinear Science and Numerical Simulations 15 (2010), 1974–-1979. (http://dx.doi.org/10.1016/j.cnsns.2009.08.018)

• (with A. Boyarsky and H. Proppe) "A model of holographic principle: Randomness and Additional Dimension" -  Physics Letters A 374 (2010), 435–-438.                                                        (http://dx.doi.org/10.1016/j.physleta.2009.11.005)
•  "Invariant densities for piecewise linear maps of interval",  Ergodic Th. and Dynamical Systems   29, Issue 05 (October 2009),  1549--1583.
•  (with A. Boyarsky) "An ergodic theory of consciousness", Int. Jour. Biff. Chaos  19, No. 4 (2009), 1397--1400.
•  (with A. Boyarsky) "A Definition of Time", International Journal of Theoretical Physics 48, Issue 6 (2009), 1589--1595.
• (with A. Boyarsky) "An Irreversible Process Represented by a Reversible One",  Int. Jour. Biff. Chaos 18, Issue 7 (2008), 2059--2061.
•  (with Erik Bollt,  Andrzej Ostruszka, and Karol Zyczkowski), "Basis Markov Partitions and Transition Matrices for Stochastic Systems",  SIAM Journal on Applied Dynamical Systems  7 (2008), Issue 2, 341--360.
•  "Invariant densities for generalized $\beta$-transformations", Ergodic Th. and Dynamical Systems 27, Issue 05  (2007), 1583--1598.
  
• (with Wael Bahsoun,  Silvia Mayoral, Manuel Morales)  "Random dynamics and finance: constructing implied binomial trees from a predetermined stationary density", Applied Stochastic Models in Business and Industry  23, Issue 3,  May/June 2007,  181--212.
  
• (with A. Boyarsky and Md Shafiqul Islam) "Invariant densities of random maps have lower bounds on their support.", Journal of Applied Mathematics and Stochastic Analysis 2006:1 (2006), 1--13.
  
• (with A. Boyarsky) "Invariant measures in brain dynamics",  Physics Letters A 358 (2006), 27--30.
  
• (with W. Bahsoun) "Random $\beta$-transformations",  Int. Jour. Biff. Chaos. 16, no.4 (2006), 1079--1081.

• (with W. Bahsoun)  "Invariant Densities for Position-Dependent  Random Maps on the Real Line: Existence, Approximation and Error Bounds",  Stochastics and Dynamics 6, no.2 (2006), 155--172.
• (with A. Boyarsky and W. Bahsoun) "Folding Maps  and Functional Equations",  Dynamical Systems, an International Journal 21, no. 2 (2006), 235-43.
• (with A. Boyarsky) "Strong chaotification of discrete time systems by small feedback control", Int. Jour.  Bif.  Chaos 16, No. 3 (2006), 715--719.
• (with W. Bahsoun) "Conditionally Invariant Measures for Position Dependent  Random Maps of an Interval With Holes",  Int. Jour. Bif. Chaos  16, No. 2 (2006), 437--444.
• (with A. Boyarsky,  Md Shafiqul Islam and Wael Bahsoun) "Absolutely Continuous Invariant Measures that Cannot be Observed Experimentally",  SIAM Journal on Applied Dynamical Systems  5, No. 1 (2006),  84-–90.
• (with A. Boyarsky) "Attainable densities for random maps"  J. Math. Anal. Appl.  317,  Issue 1, (2006),  257--270.
• (with A. Boyarsky and Md. Shafiqul Islam) "Approximation of acim's  for Markov switching position dependent random maps",  International Journal of Pure and Applied Mathematics 25, No. 1 (2005), 51--78.
• (with A. Boyarsky and Md. Sh. Islam) "Randomly chosen chaotic maps can give rise to nearly ordered behavior",   Physica D: Nonlinear Phenomena 210, issues 3-4 (2005),  284--294.
• (with A. Boyarsky and Md. Sh. Islam) "A generalization of Straube's theorem:  Existence of absolutely continuous invariant measures for random maps", Journal of Applied Mathematics and Stochastic Analysis 2005:2 (2005), 133–-141.
• (with A. Boyarsky and W. Bahsoun) "Markov switching for position dependent random maps with application to forecasting in financial markets",  SIAM Journal on Applied Dynamical Systems 4, No. 2 (2005), 391--406.
• (with W. Bahsoun) Position Dependent Random Maps in One and Higher  Dimensions,  Studia  Math. 166 (2005), 271--286.
• (with A. Boyarsky) "Nonobservable Space Dimensions and the Discreteness of Time", Chaos, Solitons and Fractals 24 (2005), 13--18.
• (with A. Boyarsky) "A Description of Stochastic Systems Using Chaotic Maps," J. Appl. Math. Stoch. Anal. 2004:2 (2004), 137--141.
• (with P. Bracken and A. Boyarsky) "Calculus of Variations for Functionals Containing Compositions", J. Math. Anal. Appl. 296 (2004), 658--664.
• (with W. Bahsoun) Dissipative and Exact Inner Functions, Complex Vraiables, Theory and Applications, an International Journal 49 (2004), 331--338.
• (with W. Bahsoun, A. Boyarsky) Small Stochastic Perturbations of Random Maps with Position Dependent Probabilities, J. Stoch. Anal. \& Appl. 22 (2004), 1121--1130.
• Paper (with A. Boyarsky) "Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $R^N$ ", Israel Jour. Math. 67 (1989), pp. 272--286, has been reprinted in a collection "The Theory of Chaotic Attractors", Editors: Hunt, B.R., Kennedy, J.A., Li, T-Y., Nusse, H.E., Springer, New York, 2004, pp. 349--363, as one of "the most influential papers in chaos theory from the past 40 years".
•  (with W. Bahsoun, A. Boyarsky) "Stochastic Perturbations of  Position Dependent Random Maps",   Stoch. & Dynam. 3  (2003), 545--557.
•  (with A. Boyarsky) "On the Significance of the Tent Map", International Journal of Bifurcation and Chaos 13, No. 5 (2003), 1299--1301.
• (with W. Bahsoun) "Weakly Convex and Concave Random Maps with Position Dependent Probabilities", Stoch. Anal. App. 21, no. 5 (2003), 983--994.
• (with W. Bahsoun, A. Boyarsky and M. Ebrahimi)" Filtering Entropy",  Physica D. 183, no. 3-4 (2003), 260--272.
• (with A. Boyarsky) "Absolutely continuous invariant measures for random maps with position dependent probabilities" , J. Math. Anal. Appl. 278 (2003), 225–-242.
•  (with P. Bracken and A. Boyarsky) "A minimum principle for chaotic dynamical systems", Physica D: Nonlinear Phenomena 166 (2002),  63--75.
• {with A. Boyarsky} "Chaotic maps derived from trajectory data", Chaos 12 (2002), no. 1, 42--48.
• (with A. Boyarsky) "A Dynamical System Interpretation of Irreducible Complexity", Discrete Dynamics in Nature and Society 7 (2002), 23--26.
• (with P. Bracken and A. Boyarsky) "Deriving chaotic dynamical systems from energy functionals",  Stochastics and Dynamics 1 (2001), 377--388.
•  (with A. Boyarsky) "Invariant Measures for Chebyshev Maps", J. Applied Math. Stoch. Analysis 14 (2001), 257--264.
•  (with A. Boyarsky) "Optimal Control of Chaotic Systems", International Journal of Bifurcation and Chaos 11, No. 7 (2001), 2007-2018. 
• (with A. Boyarsky) "Energy and Information of Chaotic Dynamical Systems", Chaos, Solitons & Fractals 12 (2001), 1611--1618.
•  (with A. Boyarsky and V. Lioubimov) "Snap-back repellers and scrambled sets in general topological spaces", Nonlinear Analysis 43 (2001), 591--604.
•  (with A. Boyarsky) "A Comparative Dynamical Analysis of Hebrew Texts", Discrete Dynamics in Nature and Society 4 (2000), 293--295. 

• (with A. Boyarsky and V. Lioubimov) "On the existence of ergodic continuous invariant measures for folding transformations", Erg. Th.\& Dyn. Sys. 20 (2000), 47--53.
• (with M. Jablo'nski) "Invariant measures and the compactness of the domain", Ann. Polon. Math. 69 (1998), 13--24.
• (with A. Boyarsky) "A new approach to controlling chaotic systems", Physica D 111 (1998), 1--15.
• (with A. Boyarsky and V. Lioubimov) "Dynamics on spaces of compact subsets with application to brain modeling", J. Math. Analysis and Appl. 216 (1997), 569--580.
• (with A. Boyarsky) "A lattice spacetime for the slit experiments of quantum mechanics", Phys. Lett. A 236 (1997), 263--269.
• (with A. Boyarsky and Y.S. Lou ) "Lyapunov exponents for higher dimensional random maps.", J. Appl. Math. & Stoch. Anal. 10 (1997), pages 209--218
• (with A. Boyarsky) "Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension", Birkhauser, New York, 1997
• (with A. Boyarsky) A model for the structure of spacetime, Chaos, Solitons & Fractals vol 7 pages 611--630 1996
• (with A. Boyarsky) Nelson's drift coefficient and the structure of quantum spacetime, Chaos, Solitons & Fractals vol 7 pages 939--954 1996
• (with M. Jablonski and A. Boyarsky) " A general existence theorem for absolutely continuous invariant measures on bounded and unbounded intervals.", Nonlinear World 3 (1996), 183--200
• (with A. Boyarsky) "An algorithm to control chaotic behavior in one-dimensional maps.", Computers \& Math. with Appl. 31 (1996), 13--22
• (with A. Boyarsky) "Iterated function systems and dynamical systems", Chaos 5 (1995), 634--639 .
• (with M. Jablonski and A. Boyarsky) "Invariant measures generated by sequences of approximating transformations.", Computers & Math. with Appl. 30 (1995), 75--91.
• "Properties of Invariant Measures for Piecewise Expanding One-Dimensional Transformations with Summable Oscillations of Derivative", Ergodic Th. & Dynam. Systems 14 (1994), 475-492.
• "Importance of Bounded Variation Condition for Stability of an Absolutely Continuous Invariant Measure.", J. Math. Analysis & Appl. 181 (1994), 422-428.
• (with A. Boyarsky and Y. S. Lou ) " Constructive approximations to the invariant densities of higher dimensional transformations." , J. Constructive Appr. 10 (1994), 1-13.
• (with A. Boyarsky) "A matrix solution to the inverse Perron-Frobenius problem", Proc. Amer. Math. Soc. 118 (1993), 409-414.
• "Graph theoretic bound on number of a.c.i.m. for random transformations" , Proc. Amer. Math. Soc.116 (1992), pp. 401-410.
• (with A. Boyarsky) "Modelling and simulating higher dimensional chaotic data", Computers Math. Applic. 24 (1992), pp. 101-105.
• (with A. Boyarsky) " A Dynamical Systems Model for Interference Effects and the Two-Slit Experiment of Quantum Physics", Physics Lett. A 168 (1992), p. 103-112.
• (with A. Boyarsky) "On Functions of Bounded Variation in Higher Dimensions" , Amer. Math. Monthly 99 (1992), 159-160.
• (with M. Pollak and A. Boyarsky) "A mathematical model describing consequences of abnormally high levels of epidermal growth factor receptor on the proliferation of neoplastic cells", Cancer Investigation 9 (No 5), (1991), 513-520.
• (with W. Byers and A. Boyarsky) "Measures on Periodic Orbits for Continuous Transformations on the Interval", Stoch. Anal. & Appl. 9 (1991), 263-270
• (with A. Boyarsky) "The Frobenius-Perron operator on spaces of curves ", Trans. Amer. Math. Soc. 324 (1991), 731-746.
• (with A. Boyarsky) "Computing the Topological Entropy of General One-Dimensional Maps", Trans. Amer. Math. Soc. 323 (1991), 39-49.
• (with A. Boyarsky and H. Proppe) "On the number of invariant measures for higher-dimensional chaotic transformations" , J. Stat. Phys. 62 (1991), 709-728.
• (with A. Boyarsky) "The pseudo-orbit shadowing property for Markov operators in the space of probability density functions" , Canadian J. Math. 42 (1990), 1000-1017.
• "Differential entropy : continuity properties and applications to nonsingular endomorphisms", Commentationes Math. 29 (1990), 169-182.
• (with W. Byers and A. Boyarsky) "Maximal Absolutely Continuous Measures for Piecewise Linear Markov Transformations" Erg. Th. & Dyn. Syst. 10 (1990), 645-656.
• (with A. Boyarsky and H. Proppe) "Inadequacy of the bounded Variation Technique in the Ergodic Theory of Higher Dimensional Transformations", Nonlinearity 3 (1990), 1081-1087.
• (with A. Boyarsky) "Absolutely Continuous Invariant Measures and Poincar\'e \ Sections of the Belousov-Zhabotinski Reaction", Dynamics and Stability of Systems 5 (1990), 65-81.
• (with A. Boyarsky) "Compactness of Eigenvectors for Families of Non-Negative Matrices", Jour. Math. Anal. Appl. 150 (1989), 425-438. (with A. Boyarsky) "Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $R^N$ ", Israel Jour. Math. 67 (1989), pp. 272-286.
• (with A. Boyarsky) "Higher Dimensional Point Transformations and Asymptotic Measures for Cellular Automata", Computers \& Math. Applic. 19 (1989), pp. 13-31.
• (with A. Boyarsky) "Approximating the Invariant Densities of Transformations with Infinitely Many Pieces on the Interval", Proceedings of American Math. Society 105 (1989), 922-928.
• (with A. Boyarsky) "Compactness of Invariant Densities for Families of Expanding Monotonic Transformations" , Can. Journal Math. 41 (1989), pp.855-869.
• (with B. Schmitt) "Un exemple de transformation dilatante et $C^1$ par morceaux de l'intervalle, sans probabilite absolument continue invariante", Erg. Th. \& Dyn. Syst. 9 (1989),101-113.
• "Countably piecewise expanding transformations without absolutely continuous invariant measure", Proceedings of Stefan Banach Center - Semester on Dynamical Systems and Ergodic Theory, Warsaw 1986 , PWN - Polish Scientific Publishers, 1989, pp.113-117.
• (with A. Boyarsky) "Why computers like Lebesgue measure", Computers and Mathematics with Applicatons 16, No. 4, 321-329, 1988.
• (with A. Boyarsky and H. Proppe) "Constructive approximations to Densities Invariant under Non-Expanding Transformations", Journal Stat. Phys. 51, No. 1/2, 179-194, 1988.
• "Random composing of mappings, small stochastic perturbations and attractors", Z. Wahrscheinlichkeitstheorie verw. Gebiete 69 (1985), 137-160 .
• "On small stochastic perturbations of mappings of the unit interval", Colloquium Math. 49 (1984), 73-85 .
• "On small stochastic perturbations of one-sided subshift of finite type", Bull. Acad. Polon. Sci. 27 (1979), 47-51 .

Other Publications:
 

• "Strange Attractors", Delta 7, 1982, 1-4 (in Polish).
• "Attractor, what is it ?", Wydawnictwo Specjalne Delty 1988 (in Polish).