Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 p-Adic Analysis: Essential Ideas and Results
- 2 Ultrametric Geometry: Cluster Networks and Buildings
- 3 p-Adic Wavelets
- 4 Ultrametricity in the Theory of Complex Systems
- 5 Some Applications of Wavelets and Integral Operators
- 6 p-Adic and Ultrametric Models in Geophysics
- 7 Recent Development of the Theory of p-Adic Dynamical Systems
- 8 Parabolic-Type Equations, Markov Processes, and Models of Complex Hierarchical Systems
- 9 Stochastic Heat Equation Driven by Gaussian Noise
- 10 Sobolev-Type Spaces and Pseudodifferential Operators
- 11 Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields
- 12 Heat Traces and Spectral Zeta Functions for p-Adic Laplacians
- References
- Index
- References
References
Published online by Cambridge University Press: 10 May 2018
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 p-Adic Analysis: Essential Ideas and Results
- 2 Ultrametric Geometry: Cluster Networks and Buildings
- 3 p-Adic Wavelets
- 4 Ultrametricity in the Theory of Complex Systems
- 5 Some Applications of Wavelets and Integral Operators
- 6 p-Adic and Ultrametric Models in Geophysics
- 7 Recent Development of the Theory of p-Adic Dynamical Systems
- 8 Parabolic-Type Equations, Markov Processes, and Models of Complex Hierarchical Systems
- 9 Stochastic Heat Equation Driven by Gaussian Noise
- 10 Sobolev-Type Spaces and Pseudodifferential Operators
- 11 Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields
- 12 Heat Traces and Spectral Zeta Functions for p-Adic Laplacians
- References
- Index
- References
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Ultrametric Pseudodifferential Equations and Applications , pp. 214 - 235Publisher: Cambridge University PressPrint publication year: 2018
References
[1] and , Anon-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies, Eur. J. Appl. Math. 9(3), 261–284 (1998).Google Scholar
[2] and , A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann. 310(3), 527–560 (1998).Google Scholar
[3] and , Stochastic processes in Qp associated with systems of nonlinear PIDEs, p-Adic Numbers Ultrametric Anal. Appl. 1(2), 105–117 (2009).Google Scholar
[4] , , and , p-Adic multiresolution analysis and wavelet frames, J. Fourier Anal. Appl. 16(5), 693–714 (2010) [arXiv:0802. 1079].
[5] , , and , p-Adic non-orthogonal wavelet bases, Proc. Steklov Inst. Math. 265, 1–12 (2009).Google Scholar
[6] , , and , p-Adic multiresolution analyses (2008) [arXiv:0810. 1147].
[7] , , and , Convoluted generalized white noise, Schwinger functions and their analytic continuation toWightman functions, Rev. Math. Phys. 8(6), 763–817 (1996).Google Scholar
[8] , , and , Euclidean rand om fields, pseudodifferential operators, and Wightman functions, in Stochastic Analysis and Applications (Powys, 1995), River Edge, NJ: World Scientific, 1996, pp. 20–37.Google Scholar
[9] , , and , SPDEs leading to local, relativistic quantum vector fields with indefinite metric and nontrivial S-matrix, in Stochastic Partial Differential Equations and Applications (Trento, 2002), Lecture Notes in Pure and Applied Mathematics 227, New York: Dekker, 2002, pp. 21–38.Google Scholar
[10] and , Diffusion in p-adic numbers, in and (eds.), Gaussian Rand om Fields, Singapore: World Scientific, 1991, pp. 86–99.Google Scholar
[11] and , A rand om walk on p-adics: the generator and its spectrum, Stochastic Process. Appl. 53, 1–22 (1994).Google Scholar
[12] and , Jump processes on leaves of multibranching trees, J. Math. Phys. 49(9), 093503, 20 pp. (2008).Google Scholar
[14] , , and , Memory retrieval as a p-adic dynamical system, BioSystems 49, 105–115 (1999).Google Scholar
[15] , , and , Harmonic analysis in the p-adic Lizorkin spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems, J. Fourier Anal. Appl. 12(4), 393–425 (2006).Google Scholar
[16] , , and , Pseudo-differential operators in the p-adic Lizorkin space, in p-Adic Mathematical Physics, AIP Conference Proceedings 826, New York: AIP, 2006, pp. 195–205.Google Scholar
[17] , , and , p-Adic semilinear evolutionary pseudodifferential equations in Lizorkin spaces, Dokl. Akad. Nauk 415(3), 295–299 (2007) [Dokl. Math. 76(1), 539–543 (2007)].
[18] , , and , Theory of p-Adic Distributions: Linear and Nonlinear Models, Cambridge: Cambridge University Press, 2010.Google Scholar
[19] , , and , The Cauchy problems for evolutionary pseudo-differential equations over p-adic field and thewavelet theory, J. Math. Anal. Appl. 375, 82–98 (2011).Google Scholar
[21] and , Coincidence of the continuous and discrete p-adic wavelet transforms (2007) [arXiv:math-ph/0702010].Google Scholar
[22] and , Frames of p-adicwavelets and orbits of the affine group, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 18–33 (2009) [arXiv:0801. 4713].Google Scholar
[23] and , Multidimensional basis of p-adic wavelets and representation theory, p-Adic Numbers Ultrametric Anal. Appl. 1(3), 181–189 (2009) [arXiv:0903. 0461].Google Scholar
[24] and , Multidimensional ultrametric pseudodifferential equations, Proc. SteklovMath. Inst. 265, 19–35 (2009) [arXiv:0708. 2074].Google Scholar
[25] and , Multidimensional p-adic wavelets for the deformed metric, p-Adic Numbers Ultrametric Anal. Appl. 2(4), 265–277 (2010) [arXiv:1105. 1524].
[26] and , Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function, p-Adic Numbers Ultrametric Anal. Appl. 2(1), 21–34 (2010) [arXiv:1105. 1506].Google Scholar
[27] and , Clustering by hypergraphs and dimensionality of cluster systems, p-Adic Numbers Ultrametric Anal. Appl. 4(3), 167–178 (2012) [arXiv:1204. 5952].Google Scholar
[28] , , and , p-Adic Schrodinger-type operator with point interactions, J. Math. Anal. Appl. 338, 1267–1281 (2008).Google Scholar
[29] , , and , Analytic and probabilistic aspects of Levy processes and fields in quantum theory, in Levy Processes, Boston, MA: Birkhauser Boston, 2001, pp. 187–224.Google Scholar
[30] , and , Euclidean rand om fields obtained by convolution from generalized white noise, J. Math. Phys. 36(10), 5217–5245 (1995).Google Scholar
[31] , Uniformly distributed sequences of p-adic integers, Math. Notes 55, 109–133 (1994).Google Scholar
[32] , Uniformly distributed sequences of p-adic integers, II, Discrete Math. Appl. 12(6), 527–590 (2002).Google Scholar
[33] , Ergodic transformations in the space of p-adic integers, in p-Adic Mathematical Physics. 2nd International Conference (Belgrade, Serbia and Montenegro, 21 September 2005), AIP Conference Proceedings 826, New York: AIP, 2006, pp. 3–24.Google Scholar
[34] , Non-Archimidean theory of T-functions, in Proceedings of Advanced Study Institute Boolean Functions in Cryptology and Information Security, Amsterdam: IOS Press, 2008, pp. 33–57.Google Scholar
[35] and , Applied Algebraic Dynamics, de Gruyter Expositions in Mathematics 49, Berlin: Walter de Gruyter, 2009.Google Scholar
[36] , , and , Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis, Dokl. Math. 86, 306–308 (2011).Google Scholar
[37] , , and , Ergodicity of dynamical systems on 2-adic spheres, Dokl. Math. 86, 843–845 (2012).Google Scholar
[38] , , and , Ergodicity criteria for non-expand ing transformations of 2-adic spheres, Discrete and Continuous Dynamical Systems, 34(2), 367–377 (2013).Google Scholar
[39] , , and , T-functions revisited: new criteria for bijectivity/ transitivity, Designs, Codes Cryptogr. 71(3), 383–407 (2014).Google Scholar
[40] , , , and , Nonlocal Diffusion Problems, Mathematical Surveys and Monographs 165, Providence, RI: American Mathematical Society; and Madrid: Real Sociedad Matematica Espanola, 2010.Google Scholar
[41] , , , , , , , and , Protein states and proteinquakes, Proc. Natl. Acad. Sci. USA 82, 5000–5004 (1985).Google Scholar
[42] , , , and , Wave function of the universe and p-adic gravity, Int. J. Mod. Phys. A 6, 4341–4358 (1991).Google Scholar
[43] , , and , On the p-adic summability of the anharmonic oscillator, Phys. Lett. B 200, 512–514 (1988).Google Scholar
[45] and , Beyond Planck energy to nonarchimedean geometry, Mod. Phys. Lett. A 6, 313–316 (1991).Google Scholar
[46] and , Strings, gravity and p-adic space-time, in , , and , Quantum Gravity. Proceedings of the Fourth Seminar on Quantum Gravity, Held May 25–29, 1987, in Moscow, USSR, Singapore: World Scientific, 1988, p. 409.Google Scholar
[47] , , , and , Vector-Valued Laplace Transforms and Cauchy Problems, New York: Birkhauser-Springer, 2011.Google Scholar
[48] , , , and , Weyl's law: spectral properties of the Laplacian in mathematical physics, in and (eds.), Mathematical Analysis of Evolution, Information, and Complexity, Weinheim: Wiley-VCH, 2009, pp. 1–71.
[49] and , Some p-adic representations of the Smale horseshoe, Phys. Lett. A. 176, 292–294 (1993).Google Scholar
[50] , Resolution of singularities and division of distributions, Commun. Pure Appl. Math. 23, 145–150 (1970).Google Scholar
[51] and , Protein ultrametricity and spectral diffusion in deeply frozen proteins, Biophys. Rev. Lett. 3(3), 387 (2008) [arXiv:0804. 4551].Google Scholar
[52] and , On the ultrametricity of the fluctuation dynamic mobility of protein molecules, Proc. Steklov Inst. Math. 265(1), 75–81 (2009).Google Scholar
[53] , , and , Application of p-adic analysis to models of breaking of replica symmetry, J. Phys. A 32(50), 8785–8791 (1999).Google Scholar
[54] , , and , Description of logarithmic relaxation by a model of a hierarchical rand om walk [in Russian], Dokl. Akad. Nauk 368(2), 164–167 (1999).Google Scholar
[55] , , and , p-Adic models of ultrametric diffusion in the conformational dynamics of macromolecules, Proc. Steklov Inst. Math. 245 (2), 48–57 (2004).Google Scholar
[56] , , and , p-Adic description of characteristic relaxation in complex systems, J. Phys. A 36(15), 4239–4246 (2003).Google Scholar
[57] , , and , and , p-Adic models of ultrametric diffusion constrained by hierarchical energy land scapes, J. Phys. A 35(2), 177–189 (2002).Google Scholar
[58] , , and , First passage time distribution and the number of returns for ultrametric rand om walks, J. Phys. A 42(8), 085003, 18 pp. (2009) [arXiv:0808. 3066].Google Scholar
[59] , , and , Mathematical modeling of molecular nanomachines [in Russian], Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz. -Mat. Nauki 1(22), 9–15 (2011).Google Scholar
[60] , , and , Ultrametric rand omwalk and dynamics of protein molecules, Proc. Steklov Inst. Math. 285, 3–25 (2014).Google Scholar
[61] , , and , Native ultrametricity of sparse rand om ensembles, J. Phys. A 49(3), 035101 (2016).Google Scholar
[62] , , , and , Fractal globule as a molecular machine, JETP Lett. 98(4), 242–246 (2013).Google Scholar
[64] , Nonlinear Differential Equations of Monotone Types in Banach Spaces, New York: Springer, 2010.Google Scholar
[65] and , Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. 354(7), 2933–2953 (2002).Google Scholar
[66] and , An integrodifferential model for phase transitions: stationary solutions in higher space dimensions, J. Statist. Phys. 95(5–6), 1119–1139 (1999).Google Scholar
[67] , , , and , Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138(2), 105–136 (1997).Google Scholar
[68] and , An empirical comparison of voting classification algorithms: bagging, boosting, and variants, Mach. Learn. 36, 105–139 (1999).Google Scholar
[69] , , , and , Ultraproducts and approximations in local rings. I, Invent. Math. 51(2), 189–203 (1979).Google Scholar
[70] and , The topology of multidimensional protein energy surfaces: theory and application to peptide structure and kinetics, J. Chem. Phys. 106, 1495–1517 (1997).Google Scholar
[71] , Feynman formulas for an infinite-dimensional p-adic heat type equation, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(1), 137–148 (2011).Google Scholar
[72] and , A wavelet theory for local fields and related groups, J. Geom. Anal. 3, 423–456 (2004).Google Scholar
[73] , p-Adic dynamics and Sullivan's no wand ering domain theorem, Compos. Math. 122, 281–298 (2000).Google Scholar
[75] , Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. 84, 231–256 (2002).Google Scholar
[76] , Examples of wavelets for local fields, in Wavelets, Frames, and Operator Theory (College Park, MD, 2003), Providence, RI: AmericanMathematical Society, 2004, pp. 27–47.Google Scholar
[78] , and , A semilinear equation in L1(Rn), Ann. Scuola Norm. Super. Pisa Cl. Sci. 2(4), 523–555 (1975).Google Scholar
[79] and , Potential Theory on Locally Compact Abelian Groups, New York: Springer-Verlag, 1975.Google Scholar
[80] , Modules over the ring of differential operators; the study of fundamental solutions of equations with constant coefficients, Functional Anal. Appl. 5(2), 1–16 (1972).Google Scholar
[81] , Stochastic equations of mathematical physics over the field of p-adic numbers, Theor. Math. Phys. 119(2), 594–604 (1999).Google Scholar
[83] and , Spin glasses: experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801–976 (1986).Google Scholar
[84] , The Witten complex and the degenerate Morse inequalities, J. Differential Geom. 23(3), 207–240 (1986).Google Scholar
[86] , , and , Rand om walks on ultrametric spaces: low temperature patterns, J. Phys. A 19, L861 (1986).Google Scholar
[87] and , Organization and function of the 3D genome, Nature Rev. Genet. 17, 661–678 (2016).Google Scholar
[88] , On p-adic classification, p-Adic Numbers Ultrametric Anal. Appl. 1(4), 271–283 (2009).Google Scholar
[89] , Ultrametricity indices for the Euclidean and Boolean hypercubes, p-Adic Numbers Ultrametric Anal. Appl. 8(4), 298–311 (2016).Google Scholar
[91] , , , and , Nonarchimedean string dynamics, Nucl. Phys. B 302(3), 365–402 (1988).Google Scholar
[92] , , and , Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks, Proteins: Struct. Funct. Genet. 41, 271–287 (2000).Google Scholar
[94] , Distributions sur un groupe localement compact et applications a l'etude des representations des groupes p-adiques, Bull. Soc. Math. France 89, 43–75 (1961).Google Scholar
[95] and , Groupes reductifs sur un corps local, I. Donnees radicielles valuees, Publ. Math. IHES 41, 5–251 (1972).Google Scholar
[96] , Scattering on the p-adic field and a trace formula, Int. Math. Res. Notices 2000, no. 2, 57–70.Google Scholar
[97] and , On the replica Fourier transform, Compt. Rendus Ac. Sci. Ser. IIB 325, 527 (1997) [arXiv:cond-mat/9709200].Google Scholar
[98] , Harmonic analysis on trees, in Harmonic Analysis on Homogeneous Spaces, Providence, RI: American Mathematical Society, 1973, pp. 419–424.
[99] , Representations of p-adic groups: a survey, in Automorphic Forms, Representations and L-Functions, Part 1, Providence, RI: American Mathematical society, 1979, pp. 111–155.Google Scholar
[100] and , Riesz kernels and pseudodifferential operators attached to quadratic forms over p-adic fields, p-Adic Numbers Ultrametric Anal. Appl. 5(3), 177–193 (2013).Google Scholar
[101] and , p-Adic elliptic quadratic forms, parabolic-type pseudodifferential equations with variable coefficients and Markov processes, p-Adic Numbers Ultrametric Anal. Appl. 6(1), 1–20 (2014).Google Scholar
[102] and , An Introduction to Semilinear Evolution Equations, Oxford: Oxford University Press, 1998.Google Scholar
[103] and , Nonlocal operators, parabolic-type equations, and ultrametric rand om walks, J. Math. Phys. 54, 113503 (2013); Erratum 55, 109901 (2014).Google Scholar
[104] and , Non-local operators, non-Archimedean parabolic-type equations with variable coefficients and Markov processes. Publ. Res. Inst. Math. Sci. 51(2), 289–317 (2015).Google Scholar
[105] , and , Heat traces and spectral zeta functions for p-Adic Laplacians, Algebra Analiz (St. Petersburg Math. J.) 29(3), 162–184 (2017).Google Scholar
[106] and , Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math. 2(3), 351–429 (2010).Google Scholar
[108] , Liquid and Vapour Flows in Porous Bodies: Surface Phenomena, Boca Raton FL: CRC Press, 2000.Google Scholar
[110] and , Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers, in Topology, Ergodic Theory, Real Algebraic Geometry, Providence, RI: American Mathematical Society, 2001, pp. 51–70.Google Scholar
[111] , Measurable choice of limit points and the existence of separable and measurable processes, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 22, 161–165 (1972).Google Scholar
[112] , Trace formula in non-commutative geometry and the zeros of the Riemann zeta function, Selecta Math (N. S.) 5, 29–106 (1999).Google Scholar
[113] and , Regularizing effects for ut + Aψ(u) = 0 in L1, J. Funct. Anal. 45, 194–212 (1982).Google Scholar
[114] , Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E.'s, Electron. J. Probab. 4(6), 29 pp. (1999).Google Scholar
[115] and , Stochastic integrals for SPDE's: a comparison, Expo. Math. 29(1), 67–109 (2011).Google Scholar
[116] and , Stochastic Equations in Infinite Dimensions, Cambridge: Cambridge University Press, 1992.Google Scholar
[117] , Ten Lectures on Wavelets, CBMS-NSR Series in Applied Mathematics, Philadelphia, PA: SIAM, 1992.Google Scholar
[118] , , and , Replica Fourier transforms on ultrametric trees, and block-diagonalizing multi-replicamatrices, J. Physique I (France) 7, 105–115 (1997) [arXiv:cond-mat/9703132].Google Scholar
[119] , , and , Exploring the three-dimensional organization of genomes, Nature Genetics 14, 390–403 (2013).Google Scholar
[120] , Report on Igusa's local zeta function, Seminaire Bourbaki 43 (1990–1991), exp. 741; Astérisque 201–202–203, 359–386 (1991). Available at www. wis. kuleuven. ac. be/algebra/denef. html.Google Scholar
[121] , On the nature of the “explicit formulas” in analytic number theory – a simple example, in Number Theoretic Methods (Iizuka, 2001), Dordrecht: Kluwer, 2002, pp. 97–118.Google Scholar
[122] and , A p-adic behaviour of dynamical systems, Rev. Mat. Complut. 12, 301–323 (1999).Google Scholar
[123] , , , , and , On nonlocal modified gravity and its cosmological solutions, Springer Proc. Math. Stat. 191, 35–51 (2016) [arXiv:1701. 02090[hep-th]].Google Scholar
[124] , Quantum Mechanics and Quantum Field Theory. A Mathematical Primer, Cambridge: Cambridge University Press, 2011.Google Scholar
[125] , , , and , p-adic and adelic minisuperspace quantum cosmology, Int. J. Mod. Phys. A 17(10), 1413–1433 (2002) [arXiv:gr-qc/0105050].Google Scholar
[126] , Eigenvalues of the Laplacian and the heat equation, Amer. Math. Monthly 88(9), 686–695 (1981).Google Scholar
[127] , On signature change in p-adic space-times, Mod. Phys. Lett. A 6, 2301–2307 (1991).Google Scholar
[128] , p-Adic perturbation series and adelic summability, Phys. Lett. B 256 (3–4), 392–396 (1991).Google Scholar
[129] , p-Adic and adelic quantum mechanics, Proc. Steklov Inst. Math. 245, 72–85 (2004) [arXiv:hep-th/0312046].Google Scholar
[130] , p-Adic structure of the genetic code (2012) [arXiv:1202. 2353[qbio. OT]].
[131] , On ultrametricity in bioinformation systems, in Conference Proceedings Theoretical Approaches to Bioinformation Systems, Belgrade: Institute of Physics, 2014, pp. 57–64.Google Scholar
[132] , Genetic code and number theory, Facta Universitatis: Phys. Chem. Techn. 14(3), 225–241 (2016) [arXiv:0911. 4014[q-bio. OT]].Google Scholar
[133] and , A p-adic model of DNA sequence and genetic code, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 34–41 (2009) [arXiv:qbio. GN/0607018].Google Scholar
[134] and , p-Adic modelling of the genome and the genetic code, Comp. J. 53, 432–442 (2010) [arXiv:0707. 3043, doi:10. 1093/comjnl/bxm083].Google Scholar
[135] , , , and , On p-adic mathematical physics, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 1–17 (2009) [arXiv:0904. 4205].Google Scholar
[136] , , and , Ultrametrics in the genetic code and the genome, Appl. Math. Comput. 309, 350–358 (2017).Google Scholar
[138] , , and , Generalized functions on adeles, J. Math. Sci. (N. Y.) 142(3), 2105–2112 (2007).Google Scholar
[139] , , , , and , Basic Phylogenetic Combinatorics, Cambridge: Cambridge University Press, 2012.
[140] , , and , Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ. 3(3), 499–521 (2003).Google Scholar
[141] , , , and , Attractors of rand om dynamical systems over p-adic numbers and a model of noisy cognitive processes. Physica D, 130, 1–12 (1999).Google Scholar
[142] and , Minimal polynomial dynamics on the set of 3-adic integers, Bull. London Math Soc. 41(2), 302–314 (2009).Google Scholar
[144] and , One-Parameter Semigroups for Linear Evolution Equations, Berlin: Springer, 2000.Google Scholar
[145] , Local properties of Levy processes on a totally disconnected group, J. Theor. Probab. 2(2), 209–259 (1989).Google Scholar
[147] , p-Adicwhite noise, chaos expansions, and stochastic integration, in Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994), River Edge, NJ: World Scientific, pp. 102–115, 1995.Google Scholar
[148] , Local fields, Gaussian measures, and Brownian motions, in (ed.), Topics in LieGroups and Probability: Boundary Theory, CRMProceedings and Lecture Notes 28, Providence, RI: American Mathematical Society, 2002, pp. 11–50.Google Scholar
[149] , , , and , p-Adic affine dynamical systems and applications, C. R. Acad. Sci. Paris Ser. I 342, 129–134 (2006).Google Scholar
[150] , , , and , Strict ergodicity of affine p-adic dynamical systems, Adv. Math. 214, 666–700 (2007).Google Scholar
[151] , , , and , p-Adic repellers in Qp are subshifts of finite type, C. R. Math. Acad. Sci. Paris. 344, 219–224 (2007).Google Scholar
[152] , , and , Modular forms in the spectral action of Bianchi IX gravitational instantons (2015) [arXiv:1511. 05321].
[153] , Orthogonal wavelets with compact support on locally compact Abelian groups, Izv. Ross. Akad. Nauk, Ser. Mat. 69(3), 193–220 (2005) [Izv. Math. 69, 623–650 (2005)].Google Scholar
[154] , Multiresolution analysis and wavelets on Vilenkin groups,Facta Universitatis (NI?S). Ser. : Elec. Energ. 21(3), 309–325 (2008).Google Scholar
[155] , Biorthogonal wavelets on Vilenkin groups, Tr. Mat. Inst. Im. V. A. Steklova, Ross. Akad. Nauk 265, 110–124 (2009) [Proc. Steklov Inst. Math. 265, 101–114 (2009)].Google Scholar
[156] , On wavelets related to the Walsh series, J. Approximation Theor. 161, 259–279 (2009).Google Scholar
[157] , Wavelets and frames based on Walsh–Dirichlet type kernels, Commun. Math. Appl. 1, 27–46 (2010).Google Scholar
[158] and , Theoreme d'equidistribution de Brolin en dynamique p-adique, C. R. Math. Acad. Sci. Paris. 339, 271–276 (2004).Google Scholar
[159] , , and , Anomalous Knudsen diffusion and reactions in disordered porous media. Center for Turbulence Research Annual Research Briefs 323 (2007).
[160] , Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics 28 Berlin: Springer, 1979.Google Scholar
[161] and , p-Adic models of turbulence, in p-Adic Mathematical Physics, AIP Conference Proceedings 286, Melville, NY: AIP, 2006, pp. 174–191.Google Scholar
[163] and , Effective scalar field theory of p-adic string, Phys. Rev. D 37, 3077–3084 (1988).Google Scholar
[165] , , and , Myoglobin: the hydrogen atom of biology and paradigm of complexity, Proc. Nat. Acid. Sci. 100(15), 8615–8617 (2003).Google Scholar
[166] , , and , The energy land scapes and motions of proteins, Science 254(5038), 1598–1603 (1991).Google Scholar
[167] , Langland s Correspondence for Loop Groups. Cambridge Studies in Advanced Mathematics 103, Cambridge: Cambridge University Press, Cambridge, 2007.Google Scholar
[169] , , , and , High order chromatin architecture shapes the land scape of chromosomal alterations in cancer, Nature Biotech. 29, 1109–1113 (2011).Google Scholar
[170] , , and , The and erson localization transition and eigenfunction multifractality in an ensemble of ultrametric rand om matrices, J. Stat. Mech. : Theory Exp. 12, L12001 (2009).Google Scholar
[171] , Notes on 2D conformal field theory and string theory, in et al. (eds.) Quantum Fields and Strings: A Course for Mathematicians, Volume 2 (Princeton, NJ, 1996/1997), Providence, RI: American Mathematical Society, 1999, pp. 1017–1089.Google Scholar
[172] and , Pseudo-differential operators with semi-quasielliptic symbols over p-adic fields, J. Math. Anal. Appl. 386(1), 32–49 (2012).Google Scholar
[174] , , and , Generalized functions. Volume 6. Representation Theory and Automorphic Functions Providence, RI: AMS Chelsea Publishing, 2010.Google Scholar
[176] and , Generalized Functions. Volume 2. Spaces of Fundamental and Generalized Functions. Providence, RI: AMS Chelsea Publishing, 2010.Google Scholar
[177] and , Generalized Functions. Volume 4. Applications of Harmonic Analysis. Providence, RI: AMS Chelsea Publishing, 2010.Google Scholar
[178] , Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem. Second edition. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press, 1995.Google Scholar
[179] and , Quantum Physics. A Functional Integral Point of View, second edition, New York: Springer-Verlag, 1987.Google Scholar
[180] , A modified strong dyadic integral and derivative, Mat. Sbornik. 193(4), 37–60 (2002) [Sb. Math. 193, 507–529 (2002)].Google Scholar
[181] , A dyadic analogue of Wiener's Tauberian theorem and some related questions, Izv. Ross. Akad. Nauk, Ser. Mat. 67(1), 33–58 (2003) [Izv. Math. 67, 29–53 (2003)].Google Scholar
[182] , Fractional modified dyadic integral and derivative on R+, Funkts. Anal. Prilozh. 39(2), 64–70 (2005) [Funct. Anal. Appl. 39, 135–139 (2005)].Google Scholar
[183] , Modified dyadic integral and fractional derivative on R+, Mat. Zametki 79(2), 213–233 (2006) [Math. Notes 79, 196–214 (2006)].Google Scholar
[184] , , and , Walsh Series and Transforms: Theory and Applications, second edition, Moscow: Izdatelstvo LKI, 2008; English translation of the first edition: Walsh Series and Transforms: Theory and Applications, Dordrecht: Kluwer, 1991.Google Scholar
[185] , Rational points in henselian discrete valuation rings, Publ. Math. IHÉS, 31, 59–64 (1966).Google Scholar
[186] , , and , Twisted spectral triples and quantum statistical mechanical systems, p-Adic Numbers Ultrametric Anal. Appl. 6(2), 81–104 (2014).Google Scholar
[187] , Patterns and Waves. The Theory and Applications of Reaction–Diffusion Equations, Oxford Applied Mathematics and Computing Science Series, New York: Clarendon Press, 1991.Google Scholar
[188] and , Multiresolution analysis, Haar bases, and self-similar tilings of Rn, IEEE Trans. Information Theor. 38(2), 556–568 (1992).Google Scholar
[189] , , and , The role of topological constraints in the kinetics of collapse of macromolecules, J. Physique 49, 2095–2100 (1988).Google Scholar
[190] , , , , , , , and , Edge length dynamics on graphs with applications to p-adic AdS/CFT (2016) [arXiv:1612. 09580].
[191] , , , , and , p-adic AdS/CFT, Commun. Math. Phys. 352(3), 1019–1059 (2017) [arXiv:1605. 01061].
[192] , , and , On ergodic behaviour of p-adic dynamical systems. Infinite Dimensional Analysis, Quantum Prob. Related Fields, 4 (4), 569–577 (2001).Google Scholar
[193] , , and , Topological transitivity for p-adic dynamical systems, in p-Adic Functional Analysis, Lecture Notes in Pure and Applied Mathematics 222, New York: Dekker, 2011, 127–132.Google Scholar
[194] , Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69, 331–371 (1910).Google Scholar
[195] , , , and , From a melt of rings to chromosome territories: the role of topological constraints in genome folding, Rep. Prog. Phys. 77, 022601, 24 pp. (2014).Google Scholar
[197] , , and , Anomalous diffusion with linear reaction dynamics, Phys. Rev. E 74, 031116 (2006).Google Scholar
[198] , , , and , Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence (2016) [arXiv:1605. 07639].
[199] , , , and , White Noise. An Infinite-Dimensional Calculus, Dordrecht: Kluwer, 1993.Google Scholar
[200] and , Reducing the dimensionality of data with neural networks, Science 313, 504–507 (2006).Google Scholar
[204] , Some aspects of the arithmetic theory of polynomials, in Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984), Progress in Mathematics 67, Boston, MA: Birkhauser, 1987, pp. 20–47.Google Scholar
[205] , An Introduction to the Theory of Local Zeta Functions, AMS/IP Studies in Advanced Mathematics 14, Providence, RI: American Mathematical Society, 2000.Google Scholar
[206] , , , and , Effects of topological constraints on globular polymers, Soft Matter 11, 665–671 (2015) [arXiv:1404. 0763].Google Scholar
[207] , Fizika pochvy. Vodno-fizicheskie svoystva i rezhimy pochv [Soil Physics. Hydrophysical Properties and Regime of Soils], Moscow: Vysshaia Shkola, 1970.Google Scholar
[209] , Symmetric stochastic integrals with respect to p-adic Brownian motion, Stochastics 79(6), 523–538 (2007).Google Scholar
[210] , p-adic Brownian motion over Qp, Proc. Steklov Inst. Math. 265(1), 115–130 (2009).Google Scholar
[211] , Fractal theoretic aspects of local field, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 51–57 (2009).Google Scholar
[212] and , Weak solutions of stochastic differential equations over the field of p-adic numbers, Tohoku Math. J. 59, 547–564 (2007).Google Scholar
[213] and , Electric–magnetic duality and the geometric Langland s program, Commun. Number Theory Phys. 1(1), 1–236 (2007).Google Scholar
[214] , Diffusion processes with ultrametric jumps, Rep. Math. Phys. 60(2), 221–235 (2007).Google Scholar
[215] and , Hierarchical structures and asymmetric stochastic processes on p-adics and adeles, J. Math. Phys. 35(9), 4637–4650 (1994).Google Scholar
[216] and , Dirichlet forms for diffusion in R2 and jumps on fractals: the regularity problem, p-Adic Numbers Ultrametric Anal. Appl. 2(4), 341–359 (2010).Google Scholar
[217] and , Orthogonal Series, second edition, Moscow: AFTs, Moscow, 1999; English translation of the first edition, Orthogonal Series, Translated Mathematical Monograph 75, Providence, RI: American Mathematical Society, 1989.
[218] , Fundamental solutions over the field of p-adic numbers, Algebra Anal. 4(3), 248–266 (1992) [St. Petersburg Math. J. 4, 613–628 (1993)].Google Scholar
[219] , Generalized functions and Gaussian path integrals over non- Archimedean function spaces, USSR-Izv. 39(1), 761–794 (1992).Google Scholar
[220] , Generalized functions on the non-Archimedean superspace, USSRIzv. 39(3), 1209–1238 (1992).
[222] , Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Dordrecht: Kluwer, 1997.Google Scholar
[223] , Human subconscious as the p-adic dynamical system. J. Theor. Biol. 193, 179–196 (1998).Google Scholar
[224] , Small denominators in complex p-adic dynamics. Indag. Mathem. 12(2), 177–189 (2001).Google Scholar
[225] , Non-Archimedean Analysis and Its Applications Moscow: Fizmatlit, 2003 [in Russian].Google Scholar
[226] , p-Adic information space and gene expression, in , , and (eds.) Integrative Approaches to Brain Complexity, London: Wellcome Trust Publications, 2006, p. 14.Google Scholar
[227] , Gene expression from 2-adic dynamical systems, Proc. Steklov Inst. Math. 265(1), 131–139 (2009).Google Scholar
[228] and , Generalized functionals of p-adic white noise, Dokl. Akad. Nauk 344(1), 23–26 (1995).Google Scholar
[229] and , p-Adic analogue of the porous medium equation [arXiv:1611. 08863 [math. AP]].
[230] , , and , Wavelet analysis on adeles and pseudo-differential operators, J. Fourier Anal. Appl. 18(6), 1215–1264 (2012).Google Scholar
[231] and , p-Adic pseudodifferential operators and analytic continuation of replica matrices, Theor. Math. Phys. 144(2), 1166–1170 (2005).Google Scholar
[232] and , Pseudodifferential operators on ultrametric spaces, and ultrametric wavelets, Izv. Math. 69(5), 989–1003 (2005).Google Scholar
[233] and , Wavelets on ultrametric spaces, Appl. Comp. Harmonic Analysis 19, 61–76 (2005).Google Scholar
[234] and , Replica symmetry breaking related to a general ultrametric space I: replica matrices and functionals, Physica A: Stat. Mech. Appl. 359, 222–240 (2006) [arXiv:cond-mat/0603685].Google Scholar
[235] and , Replica symmetry breaking related to a general ultrametric space II: RSB solutions and the n → 0 limit, Physica A: Stat. Mech. Appl. 359, 241–266 (2006) [arXiv:cond-mat/0603687].Google Scholar
[236] and , Ultrametric rand om field, Infin. Dim. Anal. Quantum. Prob. Related Topics 9(2), 199–213 (2006) [arXiv:math/0603584].Google Scholar
[237] and , Wavelets and the Cauchy problem for the Schrodinger equation on analytic ultrametric space, in and (eds.), Proceedings of the 2nd Conference on Mathematical Modelling of Wave Phenomena 2005 (14–19 August 2005, Växjö, Sweden), AIP Conference Proceedings 834, New York: Melville, 2006, pp. 344–350.Google Scholar
[238] and , Genetic code on the dyadic plane, Physica A: Stat. Mech. Appl. 381, 265–272 (2007) [arXiv:q-bio. QM/0701007].Google Scholar
[239] and , Replica symmetry breaking related to a general ultrametric space III: the case of general measure, Physica A: Stat. Mech. Appl. 378(2), 283–298 (2007) [arXiv:cond-mat/0603694].Google Scholar
[240] and , 2-Adic clustering of the PAM matrix, J. Theor. Biol. 261, 396–406 (2009) [arXiv:0903. 0137].Google Scholar
[241] and , 2-Adic degeneration of the genetic code and energy of binding of codons, in , , and (eds.) Quantum Bio-Informatics III, Singapore: World Scientific, 2010, pp. 193–204.Google Scholar
[242] and , Genetic code and deformation of the 2- dimensional 2-adic metric, p-Adic Numbers Ultrametric Anal. Appl. 3(2), 165–168 (2011).Google Scholar
[243] and , Replica procedure for probabilistic algorithms as a model of gene duplication, Dokl. Math. 84(2), 726–729 (2011) [arXiv:1105. 2893].Google Scholar
[244] , , and , Hierarchical model of the actomyosin molecular motor based on ultrametric diffusion with drift, Infin. Dim. Anal. Quantum Prob. Related Topics 18(2), 1550013, 16 pp. (2015) [arXiv:1312. 7528].Google Scholar
[245] , , , , and , Application of p-adic analysis to time series, Infin. Dim. Anal. Quantum Prob. Related Topics 16(4), 1350030, 15 pp. (2013).Google Scholar
[246] , , and , On p-adic Gibbs measures of the countable state Potts model on the Cayley tree, Nonlinearity 20, 2923–2937 (2007).Google Scholar
[247] and , Behaviour of Hensel perturbations of p-adic monomial dynamical systems. Anal. Math., 29, 107–133 (2003).Google Scholar
[249] , , and , Non-Archimedean dynamics, Bull. Belg. Math. Soc. Simon Stevin 9(suppl.), 141–147 (2002).Google Scholar
[250] , , and , The asymptotic number of periodic points of discrete polynomial p-adic dynamical systems. Contemp. Math. 319, 159–166 (2003).Google Scholar
[251] , , and , Applications of p-adic numbers: from physics to geology, Contemp. Math. 665, 121–131 (2016).Google Scholar
[252] , , and , Application of p-adic wavelets to model reaction–diffusion dynamics in rand om porous media, J. Fourier Anal. Appl. 22, 809–822 (2016).Google Scholar
[253] , , and , Modeling fluid's dynamics with master equations in ultrametric spaces representing the treelike structure of capillary networks, Entropy. 18, 249 (2016).Google Scholar
[254] and , On adelic analogue of Laplacian, Proc. Jangjeon Math. Soc. 6(1), 1–18 (2003).Google Scholar
[255] and , Distributional asymptotics and p-adic Tauberian and Shannon–Kotelnikov theorems, Asympt. Anal. 46(2), 163–187 (2006).Google Scholar
[256] and , p-Adic multidimensional wavelets and their application to p-adic pseudo-differential operators (2006) [arXiv:math-ph/0612049].
[257] and , Non-Haar p-adic wavelets and pseudodifferential operators, Dokl. Akad. Nauk 418(2), 167–170 (2008) [Dokl. Math. 77(1), 42–45 (2008)].
[258] and , An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators, p-Adic Numbers Ultrametric Anal. Appl. 1(3), 204–216 (2009).Google Scholar
[259] and , Non-Haar p-adic wavelets and their application to pseudodifferential operators and equations, Appl. Comput. Harm. Anal. 28(1), 1–23 (2010) [arXiv:0808. 3338v1].Google Scholar
[260] , , and , Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators, J. Fourier Anal. Appl. 19(6), 1323–1358 (2013).Google Scholar
[261] , , and , p-Adic orthogonal wavelet bases, p-Adic Numbers Ultrametric Anal. Appl. 1(2), 145–156 (2009).Google Scholar
[262] , , and , p-Adic refinable functions and MRA-based wavelets, J. Approx. Theory 161, 226–238 (2009) [arXiv:0711. 2820].Google Scholar
[263] , , and , Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators, J. Fourier Anal. Appl. 19, 1323–1358 (2013).Google Scholar
[264] and , Attracting points of polynomial dynamical systems in fields of p-adic numbers. Izv. Math. 71, 753–764 (2007).Google Scholar
[265] and , Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis. J. Number Theory, 133(2), 484–491 (2013).Google Scholar
[266] and , Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions, Chaos, Solitons & Fractals. 60, 11–30 (2014).Google Scholar
[267] and , Quincunx multiresolution analysis for L2(Q22), p-Adic Numbers Ultrametric Anal. Appl. 2(3), 222–231 (2010).Google Scholar
[269] , Schrodinger-type operator over p-adic number field, Teor. Mat. Fiz. 86 (3), 323–333 (1991) [Theor. Math. Phys. 86, 221–228 (1991)].Google Scholar
[270] , Parabolic equations over the field of p-adic numbers, Math. USSR Izv. 39, 1263–1280 (1992).Google Scholar
[271] , A Schrodinger-type equation over the field of p-adic numbers, J. Math. Phys. 34(8), 3420–3428 (1993).Google Scholar
[272] , The differentiation operator on subsets of the field of p-adic numbers, Izv. Ross. Akad. Nauk., Ser. Mat. 56(5), 1021–1039 (1992) [Russ. Acad. Sci., Izv. Math. 41(2), 289–305 (1993)].Google Scholar
[273] , Additive and multiplicative fractional differentiations over the field of p-adic numbers, in et al. (eds.), p-Adic Functional Analysis, Lecture Notes in Pure and AppliedMathematics 192, New York: Marcel Dekker, 1997, pp. 275–280.Google Scholar
[274] , Fundamental solutions of pseudodifferential equations connected with p-adic quadratic forms, Izv. Ross. Akad. Nauk., Ser. Mat. 62(6), 103–124 (1998) [Izv. Math. 62, 1169–1188 (1998)].Google Scholar
[275] , Pseudo-differential Equations and Stochastics over Non-Archimedean Fields, New York: Marcel Dekker, Inc., 2001.Google Scholar
[276] , Hausdorff measure for a stable-like process over an infinite extension of a local field, J. Theor. Probab. 15, 951–972 (2002).Google Scholar
[278] , Analysis in Positive Characteristic, Cambridge: Cambridge University Press, 2009.Google Scholar
[279] , Radial solutions of non-Archimedean pseudodifferential equations, Pacific J. Math. 269(2), 355–369 (2014).Google Scholar
[280] and , Interaction measures on the space of distributions over the field of p-adic numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6(3), 389–411 (2003).Google Scholar
[281] and , Molecular motors: a theorist's perspective, Ann. Rev. Phys. Chem. 58, 675–695 (2007).Google Scholar
[282] and , Character Sums with Exponential Functions and Their Applications, Cambridge: Cambridge University Press, 1999.Google Scholar
[283] , The Logic of Chance: The Nature and Origin of Biological Evolution, Upper Saddle River, NJ: FT Press Science, 2012.Google Scholar
[284] , , and , Pseudodifferential operators on adele rings and wavelet bases, Dokl. Math. 85, 358–362 (2012).Google Scholar
[286] , Wavelet theory as p-adic spectral analysis, Izv. Math. 66(2), 367–376 (2002) [arXiv:math-ph/0012019].Google Scholar
[287] , p-Adic pseudodifferential operators and p-adic wavelets, Theor. Math. Phys. 138, 322–332 (2004) [arXiv:math-ph/0303045].Google Scholar
[288] , p-Adic pseudodifferential operators: methods and applications, Proc. Steklov Inst. Math. 245, 143–153 (2004).Google Scholar
[289] , Ultrametric analysis and interbasin kinetics, AIP Conf. Proc. 826, 121–128 (2006).Google Scholar
[290] , Wavelets and spectral analysis of ultrametric pseudodifferential operators, Mat. Sbornik 198(1), 103–126 (2007) [Sb. Math. 198, 97–116 (2007)] [arXiv:math-ph/0412082].Google Scholar
[291] , Toward an ultrametric theory of turbulence, Theor. Math. Phys. 157(3), 1711–1720 (2008) [arXiv:0803. 2719].Google Scholar
[292] , Dynamics on rugged land scapes of energy and ultrametric diffusion, p-Adic Numbers Ultrametric Anal. Appl. 2(2), 122–132 (2010).Google Scholar
[293] , Multidimensional clustering and hypergraphs, Theor. Math. Phys. 164(3), 1163–1168 (2010).Google Scholar
[294] , Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics, Proc. Steklov Inst. Math. 274suppl.(1), 1–84 (2011).Google Scholar
[295] , Cluster networks and Bruhat–Tits buildings, Theor. Math. Phys. 180 (2), 958–966 (2014) [arXiv:1404. 6960].Google Scholar
[296] , Ultrametricity in the theory of complex systems, Theor. Math. Phys. 185 (2), 46–360 (2015).Google Scholar
[297] , Model of protein fragments and statistical potentials, p-Adic Numbers Ultrametric Anal. Appl. 8(4), 325–337 (2016) [arXiv:1504. 03940].Google Scholar
[298] and , Pseudodifferential operators on ultrametric spaces and ultrametric wavelets, Izv. Ross. Akad. Nauk., Ser. Mat. 69(5), 133–148 (2005) [Izv. Math. 69, 989–1003 (2005)] [arXiv:math-ph/0412062].Google Scholar
[299] and , Localization in space for a free particle in ultrametric quantum mechanics, Dokl. Akad. Nauk 411(3), 319–322 (2006) [Dokl. Math. 74(3), 906–909 (2006)].Google Scholar
[300] and , 2-Adic numbers in genetics and Rumer's symmetry, Dokl. Math. 81(1), 128–130 (2010).Google Scholar
[301] and , p-Adic integral operators in wavelet bases, Dokl. Akad. Nauk 437(4), 457–461 (2011) [Dokl. Math. 83(2), 209–212 (2011)].Google Scholar
[302] , , and , p-Adic wavelets and their applications, Proc. Steklov Inst. Math. 285, 157–196 (2014).Google Scholar
[303] , , and , Nondegenerate ultrametric diffusion, J. Math. Phys. 46, 063302–063317 (2005).Google Scholar
[304] and , The Arrhenius formula in kinetic theory and Witten's spectral asymptotics, J. Phys. A: Math. Theor. 44(21), 215202 (10 pp.) (2011) [arXiv:1008. 4487].Google Scholar
[305] and , p-Adic fractional differential operator with point interactions, Meth. Funct. Anal. Topol. 13(2), 169–180 (2007).Google Scholar
[306] , Orthogonal wavelets on the Cantor dyadic group, SIAM J. Math. Anal. 27, 305–312 (1996).Google Scholar
[307] , Wavelet analysis on the Cantor dyadic group, Houston J. Math. 24, 533–544 (1998).Google Scholar
[308] and , Scattering Theory for Automorphic Functions. Annals of Mathematics Studies 87, Princeton, NJ: Princeton University Press, 1976.Google Scholar
[309] , On the analogy between arithmetic geometry and foliated spaces, Rend. Mat. Appl. (7) 28(2), 163–188 (2008).Google Scholar
[310] , Finite ultrametric spaces and computer science, in and (eds.), Categorical Perspectives, Trends in Mathematics, Berlin: Springer, 2001, pp. 219–241.Google Scholar
[311] and , p-Adic conformal invariance and the Bruhat–Tits tree, Lett. Math. Phys. 22(2), 123–129 (1991).Google Scholar
[312] , On Siegel disk linearization theorem for fields of prime characteristic, Nonlinearity 17, 745–763 (2004).Google Scholar
[314] , Generalized Liouville differentiation and function spaces Lr p(En). Embedding theorems, Mat. Sbornik 60(3), 325–353 (1963).Google Scholar
[315] , Operators connected with fractional differentiation, and classes of differentiable functions, Tr. Mat. Inst. im. V. A. Steklova, Akad. Nauk SSSR 117, 212–243 (1972) [Proc. Steklov Inst. Math. 117, 251–286 (1974)].Google Scholar
[317] , p-Adic Numbers and Their Functions, Cambridge: Cambridge University Press, 1981.Google Scholar
[318] and , Knudsen self- and Fickian diffusion in rough nanoporous media, J. Chem. Phys. 119, 2808 (2003).Google Scholar
[319] and , Anomalous Knudsen diffusion in simple pore models, Diffusion Fundamentals 2, 14. 1–14. 2 (2005).Google Scholar
[320] , An efficient image representation for multiscale analysis, in Topical Meeting on Machine Vision, Washington, D. C.: Optical Society of America, 1980, pp. 172–175.Google Scholar
[321] , Multiresolution representation and wavelets, Ph. D. Thesis, University of Pennsylvania, Philadelphia, PA, 1988.Google Scholar
[322] , Reflections on arithmetical physics, in P. Diţǎ and (eds.), Conformal Invariance and String Theory, Boston, MA: Academic Press, 1989, pp. 293–303.Google Scholar
[323] , Numbers as functions, p-Adic Numbers Ultrametric Anal. Appl. 5(4), 313–325 (2013).Google Scholar
[324] , Painleve VI equations in p-adic time, p-Adic Numbers Ultrametric Anal. Appl. 8(3), 217–224 (2016).Google Scholar
[325] and , Big Bang, blowup, and modular curves: algebraic geometry in cosmology, SIGMA 10, 073 (2014) [arXiv:1402. 2158].Google Scholar
[326] and , Symbolic dynamics, modular curves, and Bianchi IX cosmologies (2015) [arXiv:1504. 04005].
[327] and , KMS weights on higher rank buildings, p-Adic Numbers Ultrametric Anal. Appl. 8(1), 45–67 (2016).Google Scholar
[328] , Cyclotomy and endomotives, p-Adic Numbers Ultrametric Anal. Appl. 1 (3), 217–263 (2009).Google Scholar
[329] and , Multifractals, Mumford curves and eternal inflation, p-Adic Numbers Ultrametric Anal. Appl. 6(2), 135–154 (2014).Google Scholar
[330] and , The rand om walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000).Google Scholar
[331] and , The restaurant at the end of the rand om walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen. 37, 161–208 (2004).Google Scholar
[332] , Ondelettes et fonctions splines, in Séminaire sur les équations aux dérivées partielles 1986–1987, Exp. No. VI, paris: Ecole Polytechnique Palaiseau, 1987, 18 pp.Google Scholar
[333] , Principe d'incertitude, bases hilbertiennes et algebres d'operateurs, Seminair N. Bourbaki, 1985–1986, Asterisque 145–146(4), pp. 209–223 (1987).Google Scholar
[337] , , and , Recurrent oligomers in proteins: an optimal scheme reconciling accurate and concise backbone representations in automated folding and design studies, Proteins: Struct. Funct. Genet. 40, 662–674 (2000).Google Scholar
[338] , Applied Functional Analysis and Partial Differential Equations, River Edge, NJ: World Scientific, 1998.Google Scholar
[339] and , A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27(2), 803–844 (1999).
[340] , Ageneralization of Epstein zeta functions. With a supplementary note by Hermann Weyl, Canadian J. Math. 1, 320–327 (1949).Google Scholar
[341] , Eigenfunctions on Riemannian manifolds, J. Indian Math. Soc. (N. S.) 17(4), 159–165 (1953).Google Scholar
[342] and , Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math. 1, 242–256 (1949).Google Scholar
[343] , Generalized rand om processes and their extension in measure Trudy Moskov. Mat. Obšč. 8, 497–518 (1959) [in Russian]; English translation in Selected Translations in Mathematical Statistics and Probability, Vol. 3, Providence, RI: American Mathematical Society, 1963, pp. 291–313.Google Scholar
[344] , Rand om fields on the adele ring and Wilson's renormalization group, Ann. Inst. Henri Poincaré (A): Phys. Théor. 50(3), 357–367 (1989).Google Scholar
[345] , p-Adic ?4-theory as a functional equation problem, Lett. Math. Phys. 39(3), 253–260 (1997).Google Scholar
[346] , p-Adic renormalization group solutions and the euclidean renormalization group conjectures p-Adic Numbers Ultrametric Anal. Appl. 4(2), 109–114 (2012).
[347] and , Asymptotic properties of combinatorial optimization problems in p-adic space, p-Adic Numbers Ultrametric Anal. Appl. 3(2), 114–128 (2011).Google Scholar
[349] , Are there p-adic knot invariants?, Theor. Math. Phys. 187(1), 447–454 (2016) [arXiv:1509. 04928].Google Scholar
[350] , On the existence of generalized Gibbs measures for the onedimensional p-adic countable state Potts model, Proc. Steklov Inst. Math. 265(1), 165–176 (2009).Google Scholar
[351] , A dynamical system approach to phase transitions for p-adic Potts model on the Cayley tree of order two, Rep. Math. Phys. 70(3), 385–406 (2012).Google Scholar
[352] and , On inhomogeneous p-adic Potts model on a Cayley tree, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8(2), 277–290 (2005).Google Scholar
[354] , From data to the p-adic or ultrametric model, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 58–68 (2009).Google Scholar
[355] , Sparse p-adic data coding for computationally efficient and effective big data analytics, p-Adic Numbers Ultrametric Anal. Appl. 8(3), 236–247 (2016).Google Scholar
[356] and , Algorithms for hierarchical clustering: an overview, Wiley Interdisci. Reviews: Data Mining Knowledge Discovery 2(1), 86–97 (2012).Google Scholar
[358] and , Ward's hierarchical agglomerative clustering method: which algorithms implement Ward's criterion?, J. Classif. 31(3), 274–295 (2014).Google Scholar
[359] and , On the metric structure of ultrametric spaces, Proc. Steklov Inst. Math. 245, 169–188 (2004) [arXiv:cond-mat/0310079].Google Scholar
[360] , Analysis of the information structure of protein sequences: a new method for analyzing the domain organization of proteins, J. Biomol. Struct. Dyn. 21 (5), 615–624 (2004).Google Scholar
[361] , , and , A new paradigm of protein structural organization, in Proceedings of the 2nd International Conference “Theoretical Approaches to Bioinformatic Systems” (TABIS. 2013), 2013, pp. 1–23.
[362] , On combinatorial analogs of the group of diffeomorphisms of the circle, Izv. Ross. Akad. Nauk., Ser. Mat. 56(5), 1072–1085 (1992) [Russ. Acad. Sci., Izv. Math. 41(2), 337–349 (1993)].Google Scholar
[363] , , and , Wavelet Theory, Moscow: Fizmatlit, 2005; English translation Providence, RI: American Mathematical Society, 2011.Google Scholar
[364] and , Why are Haar bases in various structures the same?, Mat. Zametki 91(6), 950–953 (2012).
[365] , White Noise Calculus and Fock Space, Lecture Notes in Mathematics 1577, Berlin: Springer, 1994.Google Scholar
[367] and , Applications of p-adics to geophysics: linear and quasilinear diffusion ofwater-in-oil and oil-in-water emulsions, Theor. Math. Phys. 190, 154–163 (2017).Google Scholar
[368] , , , , , , and , Fractal radar scattering from soil. Phys. Rev. E 67, 041403 (2003).Google Scholar
[369] , , , , , and , Weathering: toward a fractal quantifying. Math. Geol. 36, 607–627 (2004).Google Scholar
[370] , Classification of irreducible representations of groups of automorphisms of Bruhat–Tits trees, Funkts. Anal. Prilozh. 11(1), 32–42 (1977) [Funct. Anal. Appl. 11, 26–34 (1977)].Google Scholar
[371] and , Scattering theory and automorphic functions, in Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 6, Zap. Nau?cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 27, 161–193 (1972) [in Russian].Google Scholar
[372] , Infinite number of order parameters for spin-glasses, Phys. Rev. Lett. 43, 1754 (1979).Google Scholar
[373] and , p-Adic numbers and replica symmetry breaking, Eur. Phys. J. B 14, 535–542 (2000) [arXiv:cond-mat/9906095].Google Scholar
[374] , Generalized Coherent States and Their Applications, Moscow: Nauka, 1987; English translation Berlin: Springer, 1986.Google Scholar
[375] and , Dyadic wavelets and refinable functions on a halfline, Mat. Sbornik 197(10), 129–160 (2006) [Sb. Math. 197, 1529–1558 (2006)].Google Scholar
[377] and , Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-adjointness, Boston, MA: Academic Press, 1975.Google Scholar
[378] and , Methods of Modern Mathematical Physics: Functional Analysis I, Boston, MA: Academic Press, 1980.Google Scholar
[379] , Capillary conduction of liquids through porous mediums, Physics 1(5), 318–333 (1931).Google Scholar
[380] , The Soil As a Reactor (Modelling Processes in the Soil), Reiskirchen: Catena, 1987.Google Scholar
[382] , Dynamique des fonctions rationelles sur des corps locaux, PhD thesis, Orsay, 2000.
[383] , Dynamique des fonctions rationelles sur des corps locaux, Astérisque 147, 147–230 (2003).Google Scholar
[384] , Espace hyperbolique p-adique et dynamique des fonctions rationelles, Compos. Math. 138, 199–231 (2003).Google Scholar
[385] and , Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type, Mat. Zametki 86(3), 429–444 (2009) [Math. Notes 86, 407–421 (2009)].Google Scholar
[386] and , Taibleson operators, p-Adic parabolic equations and ultrametric diffusion, Pacific J. Math. 237(2), 327–347 (2008).Google Scholar
[387] and , Elliptic pseudodifferential equations and Sobolev spaces over p-adic fields, Pacific J. Math. 246(2), 407–420 (2010).Google Scholar
[388] , Hypersingular Integrals and Their Applications, Basingstoke: Taylor & Francis, 2002.Google Scholar
[389] , , and , Fractional Integrals and Derivatives and Some of Their Applications, Yverdon: Gordon and Breach Science Publishers, 1993.Google Scholar
[390] , Ultrametric Calculus. An Introduction to p-Adic Analysis, Cambridge: Cambridge University Press, 1984.Google Scholar
[391] , Arbres, amalgames, SL2, Asterisque 46, Paris: Societe Mathematique de France, 1977.Google Scholar
[393] and , p-Adic Haar multiresolution analysis and pseudodifferential operators, J. Fourier Anal. Appl. 15(3), 366–393 (2009).Google Scholar
[395] , The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics 241, Berlin: Springer, 2007.Google Scholar
[396] , Calculation of general p-adic Feynman amplitude, Commun. Math. Phys. 149(3), 623–636 (1992).Google Scholar
[398] , , and , Sparse Image and Signal Processing: Wavelets and Related Geometric Multiscale Analysis, Cambridge: Cambridge University Press, 2015.Google Scholar
[400] , , and , Cluster ensembles – a knowledge reuse framework for combining multiple partitions, J. Mach. Learn. Res. 3, 583–617 (2002).Google Scholar
[401] , Harmonic analysis on n-dimensional vector spaces over local fields. I. Basic results on fractional integration, Math. Annalen 176, 191–207 (1968).Google Scholar
[402] , Fourier Analysis on Local Fields, Princeton, MA: Princeton University Press, 1975.Google Scholar
[403] , Spin Glasses, a Challenge for Mathematicians, Berlin: Springer, 2003.
[404] , , , and , Temperedness of measures defined by polynomial equations over local fields (2016) [arXiv:1610. 08444].
[405] , and , Parabolic type equations and Markov stochastic processes on adeles, J. Fourier Anal. Appl. 19(4), 792–835 (2013).Google Scholar
[406] and , Ultrametric diffusion, exponential land scapes, and the first passage time problem (2016) [arXiv:1511. 08757].
[407] , Topological Vector Spaces, Distributions and Kernels, New York: Academic Press, 1967.Google Scholar
[408] , , , and , A 3D building blocks approach to analyzing and predicting structure of proteins, Proteins: Struct. Funct. Genet. 5, 355–373 (1989).Google Scholar
[409] , Algebres de fonctions continues p-adiques. I, Nederl. Akad. Wetensch. Proc. Ser. A 30, 401–411 (1968) and Algebres de fonctions continues p-adiques. II, Nederl. Akad. Wetensch. Proc. Ser. A 30, 412–420 (1968).
[410] , A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. of America J. 44(5), 892–898 (1980).Google Scholar
[411] , Path integrals for a class of p-adic Schrodinger equations, Lett. Math. Phys. 39(2), 97–106 (1997).Google Scholar
[412] , Arithmetic quantum physics: why, what, and whither, in Selected Topics of p-Adic Mathematical Physics and Analysis, Proc. V. A. Steklov Inst. Math. 245, 273–280 (2005).Google Scholar
[413] , Reflections on Quanta, Symmetries, and Supersymmetries, NewYork: Springer, 2011.Google Scholar
[414] , The Porous Medium Equation. Mathematical Theory, Oxford: Clarendon Press, 2007.Google Scholar
[415] and , Zeta functions for analytic mappings, logprincipalization of ideals, and Newton polyhedra, Trans. Amer. Math. Soc. 360, 2205–2227 (2008).Google Scholar
[416] , The arithmetic of discretized rotations, in , , and (eds.), p-Adic Mathematical Physics, AIP Conference Proceedings, Melville, New York, 826, New York: AIP, 2006, pp. 162–173.Google Scholar
[417] , Algebraic and arithmetic dynamics, www. maths. qmul. ac. uk/fv/database/ algdyn. pdf.
[418] and , Pseudo-rand omness of round-off errors in discretized linear maps on the plane, Int. J. Bifurcations Chaos. 13, 3373–3393 (2003).Google Scholar
[419] , Generalized functions over the field of p-adic numbers, Russ. Math. Surv. 43, 19–64 (1988).Google Scholar
[420] , On the spectrum of some pseudodifferential operators over the field of p-Adic numbers, St. Petersburg Math. J. 2(6), 1261–1278 (1991).Google Scholar
[421] , Derivation of Freund–Witten adelic formula for four-point Veneziano amplitudes, Theor. Math. Phys. 94(3), 251–259 (1993).Google Scholar
[422] , Freund–Witten adelic formulae for Veneziano and Virasoro–Shapiro amplitudes, Russ. Math. Surv. 48(6), 1–39 (1993).Google Scholar
[423] , On the Freund–Witten adelic formula for Veneziano amplitudes, Lett. Math. Phys. 27(2), 123–131 (1993).Google Scholar
[424] , Adelic formulas for four-particle string and superstring tree amplitudes in one-class quadratic fields, Proc. Steklov Inst. Math. 245, 3–21 (2004).Google Scholar
[425] , On the non-linear equation of a p-adic open string for a scalar field, Russ. Math. Surv. 60(6), 1077–1092 (2005).Google Scholar
[426] , Nonlinear equations for p-adic open, closed, and open-closed strings, Theor. Math. Phys. 149(3), 1604–1616 (2006).Google Scholar
[427] , On the equations for p-adic closed and open strings, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 79–87 (2009).Google Scholar
[428] , Solutions of p-adic string equations, Theor. Math. Phys. 167(2), 539–546 (2011).Google Scholar
[429] , Nonexistence of solutions of the p-adic strings, Theor. Math. Phys. 174(2), 178–185 (2013).Google Scholar
[431] and , Nonlinear dynamics equation in p-adic string theory, Theor. Math. Phys. 138, 297–307 (2004) [arXiv:math-ph/0306018].Google Scholar
[432] , , and , The spectral theory in the p-adic quantum mechanics, Izv. Akad. Nauk SSSR, Ser. Mat. 54, 275–302 (1990).Google Scholar
[433] , , and , Spectral theory in p-adic quantum mechanics, and representation theory, Math. USSR-Izv. 36(2), 281–309 (1991).Google Scholar
[434] , , and , p-Adic Analysis and Mathematical Physics, Singapore: World Scientific, 1994.Google Scholar
[435] , p-Adic space–time and string theory, Teor. Mat. Fiz. 71(3), 337–340 (1987) [Theor. Math. Phys. 71, 574–576 (1987)].Google Scholar
[437] , From p-adic strings to etale strings, Proc. Steklov Math. Inst. 203, 41–48 (1994).Google Scholar
[438] , Number theory as the ultimate physical theory, p-Adic Numbers Ultrametric Anal. Appl. 2(1), 77–87 (2010). This paper corresponds to the preprint CERNTH. 4781/87, Geneva, July 1987.Google Scholar
[439] , Spectral zeta functions, in and (eds.), Zeta Functions in Geometry, Advanced Studies in Pure Mathematics 21, Tokyo: Kinokuniya, 1992, pp. 327–358.Google Scholar
[441] , An introduction to stochastic partial differential equations, in (ed.), École d'été de probabilités de Saint-Flour, XIV–1984, Lecture Notes in Mathematics 1180, Berlin: Springer, 1986, pp. 265–439.Google Scholar
[444] , Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (3), 351–399 (1989).Google Scholar
[446] , Additive processes on local fields, J. Math. Sci. Univ. Tokyo 3, 629–654 (1996).Google Scholar
[447] , Extension of measures to infinite dimensional spaces over p-adic field, Osaka J. Math. 37, 967–985 (2000).Google Scholar
[448] , Trace formula on the p-adic upper half-plane, J. Funct. Anal. 216(2), 422–454 (2004).Google Scholar
[449] , Markov processes on the adeles and representations of Euler products, J. Theor. Probab. 23(3), 748–769 (2010).Google Scholar
[450] , Limit theorems for p-adic valued asymmetric semistable laws and processes, p-Adic Numbers Ultrametric Anal. Appl. 9(1), 62–77 (2017).Google Scholar
[451] , Functional Analysis, Die Grundlehren der Mathematischen Wissenschaften 123, Berlin: Springer, 1965.Google Scholar
[452] , Van der Put basis and p-adic dynamics, p-Adic Numbers, Ultrametric Anal. Appl. 2(2), 175–178 (2010).Google Scholar
[453] , Onmeasure-preserving functions over Z3. Ultrametric Anal. Appl. 4, 326–335 (2012).Google Scholar
[454] , On ergodicity of p-adic dynamical systems for arbitrary prime p, p-Adic Numbers, Ultrametric Anal. Appl. 5, 239–241 (2013).Google Scholar
[455] , , and , Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w. r. t. Haar measure, in , , and (eds.), Advances in non-Archimedean Analysis, Contemporary Mathematics 551, Providence, RI: American Mathematical Society, 2011, pp. 33–38.Google Scholar
[456] , A p-adic infinite-dimensional symplectic group, Russ. Acad. Sci. Izv. Math. 43(3), 421–441 (1994).Google Scholar
[457] , Quantum approximation theorem, p-Adic Numbers Ultrametric Anal. Appl. 1(1), 88–90 (2009).Google Scholar
[458] , Adelic decoherence, p-Adic Numbers Ultrametric Anal. Appl. 4(1), 84–87 (2012).Google Scholar
[461] , On stochastic generation of ultrametrics in high-dimensional Euclidean spaces, p-Adic Numbers Ultrametric Anal. Appl. 6(2), 155–165 (2014).Google Scholar
[462] , Fundamental solutions of pseudo-differential operators over padic fields, Rend. Sem. Mat. Univ. Padova 109(2003), 241–245.
[463] , Pseudo-differential equations connected with p-adic forms and local zeta functions, Bull. Austral. Math. Soc. 70(1), 73–86 (2004).Google Scholar
[464] , Parabolic equations and Markov processes over p-adic fields, Potential Anal. 28(2), 185–200 (2008).Google Scholar
[465] , Local zeta functions supported on analytic submanifolds and Newton polyhedra, Int. Math. Res. Not. 2009(15), 2855–2898 (2009).Google Scholar
[466] , Local zeta functions and fundamental solutions for pseudodifferential operators over p-adic fields, p-Adic Numbers, Ultrametric Anal. Appl. 3 (4), 344–358 (2011).Google Scholar
[467] , The Cauchy problem for non-Archimedean pseudodifferential equations of Klein–Gordon type, J. Math. Anal. Appl. 420(2), 1033–1050 (2014).Google Scholar
[468] , The non-Archimedean stochastic heat equation driven by Gaussian noise, J. Fourier Anal. Appl. 21(3), 600–627 (2015).Google Scholar
[469] , Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems (2016) [arXiv:1604. 06471]. To appear in Nonlinearity.
[470] , Pseudodifferential Equations over Non-Archimedean Spaces, Lecture Notes in Mathematics 2174, Berlin: Springer, 2016.Google Scholar
[471] , Local zeta functions, pseudodifferential operators, and Sobolevtype spaces over non-Archimedean local fields, p-Adic Numbers Ultrametric Anal. Appl. 9(4), 313–335 (2017).Google Scholar
[472] , Non-Archimedean white noise, pseudodifferential stochastic equations, and massive Euclidean fields, J. Fourier Anal. Appl. 23(2), 288–323 (2017).Google Scholar
[473] , Sobolev-type spaces and pseudodifferential operators, unpublished manuscript (2017).
[474] , Simple model of protein folding kinetics, Proc. Nat. Acad. Sci. U. S. A. 92 (21), 9801–9804 (1995).Google Scholar