Tensor- and spinor-valued random fields with applications to continuum physics and cosmology. (English) Zbl 1502.60078

Summary: In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic.
Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic.
For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.


60G60 Random fields
74A40 Random materials and composite materials
74H50 Random vibrations in dynamical problems in solid mechanics
76F55 Statistical turbulence modeling
83F05 Relativistic cosmology


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[1] ADAMS, J. F. (1969). Lectures on Lie groups. W. A. Benjamin, Inc., New York-Amsterdam. · Zbl 0206.31604
[2] ARNOL′D, V. I. (1989). Mathematical methods of classical mechanics. Graduate Texts in Mathematics 60. Springer-Verlag, New York. Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, corrected reprint of the second edition. · Zbl 0692.70003
[3] ARNOL′D, V. I. (1998). On the teaching of mathematics. Uspekhi Mat. Nauk 53 229-234. · Zbl 1114.00300 · doi:10.1070/rm1998v053n01ABEH000005
[4] AUFFRAY, N., HE, Q.-C. and LE QUANG, H. (2019). Complete symmetry classification and compact matrix representations for 3D strain gradient elasticity. International Journal of Solids and Structures 159 197-210. · doi:10.1016/j.ijsolstr.2018.09.029
[5] AUFFRAY, N., KOLEV, B. and OLIVE, M. (2017). Handbook of bi-dimensional tensors: Part I: Harmonic decomposition and symmetry classes. Math. Mech. Solids 22 1847-1865. · Zbl 1391.74020 · doi:10.1177/1081286516649017
[6] BAEZ, J. C. (2012). Division algebras and quantum theory. Found. Phys. 42 819-855. · Zbl 1259.81023 · doi:10.1007/s10701-011-9566-z
[7] BALDI, P. and ROSSI, M. (2014). Representation of Gaussian isotropic spin random fields. Stochastic Process. Appl. 124 1910-1941. · Zbl 1319.60104 · doi:10.1016/j.spa.2014.01.007
[8] BARUT, A. O. and RĄCZKA, R. (1986). Theory of group representations and applications, Second ed. World Scientific Publishing Co., Singapore. · Zbl 0644.22011 · doi:10.1142/0352
[9] BEREZANS′KI˘I, YU. M. (1968). Expansions in eigenfunctions of selfadjoint operators. Translations of Mathematical Monographs 17. American Mathematical Society, Providence, R.I. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. · Zbl 0157.16601
[10] BOTT, R. H. (1965). The index theorem for homogeneous differential operators. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), (S. S. Cairns, ed.). Princeton Mathematical Series 27 167-186. Princeton Univ. Press, Princeton, N.J. · Zbl 0173.26001
[11] BOURBAKI, N. (1998). Algebra I. Chapters 1-3. Elements of Mathematics (Berlin). Springer-Verlag, Berlin Translated from the French, Reprint of the 1989 English translation [MR0979982 (90d:00002)].
[12] BOURBAKI, N. (2004). Integration. II. Chapters 7-9. Elements of Mathematics (Berlin). Springer-Verlag, Berlin Translated from the 1963 and 1969 French originals by Sterling K. Berberian. · Zbl 1095.28002
[13] BOURGUIGNON, J.-P., HIJAZI, O., MILHORAT, J.-L., MOROIANU, A. and MOROIANU, S. (2015). A spinorial approach to Riemannian and conformal geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich. · Zbl 1348.53001 · doi:10.4171/136
[14] BREDON, G. E. (1972). Introduction to compact transformation groups. Academic Press, New York-London Pure and Applied Mathematics, Vol. 46. · Zbl 0246.57017
[15] BRÖCKER, T. and TOM DIECK, T. (1995). Representations of compact Lie groups. Graduate Texts in Mathematics 98. Springer-Verlag, New York Translated from the German manuscript, Corrected reprint of the 1985 translation. · Zbl 0874.22001
[16] BUDINICH, P. and TRAUTMAN, A. (1988). The spinorial chessboard. Trieste Notes in Physics. Springer-Verlag, Berlin. · Zbl 0653.15022 · doi:10.1007/978-3-642-83407-3
[17] CLARKE, T. J., COPELAND, E. J. and MOSS, A. (2020). Constraints on primordial gravitational waves from the cosmic microwave background. J. Cosmol. Astropart. Phys. 2020 002, 31. · doi:10.1088/1475-7516/2020/10/002
[18] CURTIS, W. D. and LERNER, D. E. (1978). Complex line bundles in relativity. J. Mathematical Phys. 19 874-877. · Zbl 0402.53041 · doi:10.1063/1.523750
[19] DAI, L., KAMIONKOWSKI, M. and JEONG, D. (2012). Total angular momentum waves for scalar, vector, and tensor fields. Phys. Rev. D 86 125013. · doi:10.1103/PhysRevD.86.125013
[20] DENG, M. and DODSON, C. T. J. (1994). Paper: an engineered stochastic structure. Tappi Press, Atlanta, GA.
[21] DRAY, T. (1986). A unified treatment of Wigner \(D\) functions, spin-weighted spherical harmonics, and monopole harmonics. J. Math. Phys. 27 781-792. · Zbl 0605.33006 · doi:10.1063/1.527183
[22] DUISTERMAAT, J. J. and KOLK, J. A. C. (2000). Lie groups. Universitext. Springer-Verlag, Berlin. · Zbl 0955.22001 · doi:10.1007/978-3-642-56936-4
[23] DURRER, R. (2020). The Cosmic Microwave Background, 2 ed. Cambridge University Press. · doi:10.1017/9781316471524
[24] EASTWOOD, M. and TOD, P. (1982). Edth — a differential operator on the sphere. Math. Proc. Cambridge Philos. Soc. 92 317-330. · Zbl 0511.53026 · doi:10.1017/S0305004100059971
[25] ERDÉLYI, A., MAGNUS, W., OBERHETTINGER, F. and TRICOMI, F. G. (1981). Higher transcendental functions. Vol. II. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla. Based on notes left by Harry Bateman, Reprint of the 1953 original. · Zbl 0542.33001
[26] FAESSLER, A., HODÁK, R., KOVALENKO, S. and ŠIMKOVIC, F. (2017). Can one measure the Cosmic Neutrino Background? International Journal of Modern Physics E 26 1740008. · doi:10.1142/S0218301317400080
[27] FOLLAND, G. B. (2016). A course in abstract harmonic analysis, second ed. Textbooks in Mathematics. CRC Press, Boca Raton, FL. · Zbl 1342.43001
[28] FORTE, S. and VIANELLO, M. (1996). Symmetry classes for elasticity tensors. J. Elasticity 43 81-108. · Zbl 0876.73008 · doi:10.1007/BF00042505
[29] FROBENIUS, F. G. (1878). Über lineare Substitutionen und bilineare Formen. J. Reine Angew. Math. 84 1-63. · doi:10.1515/crelle-1878-18788403
[30] FULTON, W. and HARRIS, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics 129. Springer-Verlag, New York. · Zbl 0744.22001 · doi:10.1007/978-1-4612-0979-9
[31] GANCZARSKI, A. W., EGNER, H. and SKRZYPEK, J. J. (2015). Introduction to Mechanics of Anisotropic Materials. In Mechanics of Anisotropic Materials, (J. J. Skrzypek and A. W. Ganczarski, eds.). Engineering Materials 1-56. Springer International Publishing, Cham. · Zbl 1336.74005 · doi:10.1007/978-3-319-17160-9
[32] GAUNT, J. A. and FOWLER, R. H. (1929). The triplets of helium. Proc. Roy. Soc. A 122 513-532. · doi:10.1098/rspa.1929.0037
[33] GEL′FAND, I. M. and ŠAPIRO, Z. YA. (1952). Representations of the group of rotations in three-dimensional space and their applications. Uspehi Matem. Nauk (N.S.) 7 3-117. · Zbl 0049.15702
[34] GELLER, D. and MARINUCCI, D. (2010). Spin wavelets on the sphere. J. Fourier Anal. Appl. 16 840-884. · Zbl 1206.42039 · doi:10.1007/s00041-010-9128-3
[35] G¯IKHMAN, I. I. and SKOROKHOD, A. V. (2004). The theory of stochastic processes. I. Classics in Mathematics. Springer-Verlag, Berlin Translated from the Russian by S. Kotz, Reprint of the 1974 edition. · Zbl 1068.60004
[36] GODUNOV, S. K. and GORDIENKO, V. M. (2004). Clebsch-Gordan coefficients in the case of various choices of bases of unitary and orthogonal representations of the groups \[\text{SU}(2)\] and \[\text{SO}(3)\]. Sibirsk. Mat. Zh. 45 540-557. · Zbl 1048.22003 · doi:10.1023/B:SIMJ.0000028609.97557.b8
[37] GOLUBITSKY, M., STEWART, I. and SCHAEFFER, D. G. (1988). Singularities and groups in bifurcation theory. Vol. II. Applied Mathematical Sciences 69. Springer-Verlag, New York. · Zbl 0691.58003 · doi:10.1007/978-1-4612-4574-2
[38] GORDIENKO, V. M. (2002). Matrix elements of real representations of the groups \[\text{O}(3)\] and \[\text{SO}(3)\]. Sibirsk. Mat. Zh. 43 51-63, i. · Zbl 1008.43006 · doi:10.1023/A:1013816403253
[39] GORDIENKO, V. M. (2017). Matrices of Clebsch-Gordan coefficients. Sibirsk. Mat. Zh. 58 1276-1291. · Zbl 1398.22015 · doi:10.1134/s0037446617060088
[40] GREINER, W. (1998). Classical electrodynamics. Classical Theoretical Physics. Springer-Verlag, New York Translated from the 1991 German original, With a foreword by D. Allan Bromley. · doi:10.1007/978-1-4612-0587-6
[41] GRIGORIU, M. D. (2022). Finite dimensional models for random microstructures. Theory Probab. Math. Statist. 106 121-142. · Zbl 1495.60061 · doi:10.1090/tpms/1168
[42] GUILLEMINOT, J. (2020). Modeling non-Gaussian random fields of material properties in multiscale mechanics of materials. In Uncertainty Quantification in Multiscale Materials Modeling, (Y. Wang and D. L. McDowell, eds.). Elsevier Series in Mechanics of Advanced Materials 385-420. Woodhead Publishing. · doi:10.1016/B978-0-08-102941-1.00012-2
[43] GUILLEMINOT, J., NOSHADRAVAN, A., SOIZE, C. and GHANEM, R. G. (2011). A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Comput. Methods Appl. Mech. Engrg. 200 1637-1648. · Zbl 1228.74019 · doi:10.1016/j.cma.2011.01.016
[44] GUILLEMINOT, J. and SOIZE, C. (2011). Non-Gaussian positive-definite matrix-valued random fields with constrained eigenvalues: application to random elasticity tensors with uncertain material symmetries. Internat. J. Numer. Methods Engrg. 88 1128-1151. · Zbl 1242.74229 · doi:10.1002/nme.3212
[45] GUILLEMINOT, J. and SOIZE, C. (2013). On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties. J. Elasticity 111 109-130. · Zbl 1273.74007 · doi:10.1007/s10659-012-9396-z
[46] GUILLEMINOT, J. and SOIZE, C. (2013). Prior Representations of Random Fields for Stochastic Multiscale Modeling. Procedia IUTAM 6 44-49. IUTAM Symposium on Multiscale Problems in Stochastic Mechanics. · doi:10.1016/j.piutam.2013.01.005
[47] GUILLEMINOT, J. and SOIZE, C. (2013). Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media. Multiscale Model. Simul. 11 840-870. · Zbl 1355.74073 · doi:10.1137/120898346
[48] GUILLEMINOT, J. and SOIZE, C. (2014). Itô SDE-based generator for a class of non-Gaussian vector-valued random fields in uncertainty quantification. SIAM J. Sci. Comput. 36 A2763-A2786. · Zbl 1308.74166 · doi:10.1137/130948586
[49] GUILLEMINOT, J. and SOIZE, C. (2020). Non-Gaussian Random Fields in Multiscale Mechanics of Heterogeneous Materials. In Encyclopedia of Continuum Mechanics (H. Altenbach and A. Öchsner, eds.) 1826-1834. Springer Berlin Heidelberg, Berlin, Heidelberg. · doi:10.1007/978-3-662-55771-6_68
[50] HARVEY, F. R. (1990). Spinors and calibrations. Perspectives in Mathematics 9. Academic Press, Inc., Boston, MA. · Zbl 0694.53002
[51] HELD, A., NEWMAN, E. T. and POSADAS, R. (1970). The Lorentz group and the sphere. J. Mathematical Phys. 11 3145-3154. · Zbl 0202.27401 · doi:10.1063/1.1665105
[52] HOFMANN, K. H. and MORRIS, S. A. (2020). The structure of compact groups. A primer for the student—a handbook for the expert, fourth ed. De Gruyter Studies in Mathematics 25. De Gruyter, Berlin. · Zbl 1441.22001
[53] HOGER, A. (1994/95). Positive definiteness of the elasticity tensor of a residually stressed material. J. Elasticity 36 201-226. · Zbl 0824.73008 · doi:10.1007/BF00040848
[54] HUYBRECHTS, D. (2005). Complex geometry. An introduction. Universitext. Springer-Verlag, Berlin. · Zbl 1055.14001
[55] IGNACZAK, J. (1963). A completeness problem for stress equations of motion in the linear elasticity theory. Arch. Mech. Stos. 15 225-234. · Zbl 0117.18306
[56] INOMATA, K. and KAMIONKOWSKI, M. (2019). Circular polarization of the cosmic microwave background from vector and tensor perturbations. Phys. Rev. D 99 043501, 15. · doi:10.1103/physrevd.99.043501
[57] ITZKOWITZ, G., ROTHMAN, S. and STRASSBERG, H. (1991). A note on the real representations of \[\text{SU}(2,\mathbf{C})\]. J. Pure Appl. Algebra 69 285-294. · Zbl 0717.22011 · doi:10.1016/0022-4049(91)90023-U
[58] JARZYNSKI, C. (2011). Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale. Annual Review of Condensed Matter Physics 2 329-351. · doi:10.1146/annurev-conmatphys-062910-140506
[59] KAMIONKOWSKI, M., KOSOWSKY, A. and STEBBINS, A. (1997). Statistics of cosmic microwave background polarization. Phys. Rev. D 55 7368-7388. · doi:10.1103/PhysRevD.55.7368
[60] KARHUNEN, K. (1947). Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1947 79. · Zbl 0030.16502
[61] KARIMI, P., MALYARENKO, A., OSTOJA-STARZEWSKI, M. and ZHANG, X. (2020). RVE problem: mathematical aspects and related stochastic mechanics. Internat. J. Engrg. Sci. 146 103169, 16. · Zbl 1476.74004 · doi:10.1016/j.ijengsci.2019.103169
[62] KELLER, L. V. and FRIEDMANN, A. A. (1925). Differentialgleichungen für die turbulente Bewegung einer kompressibelen Flüssigkeit. In Proceedings of the first international congress for applied mechanics, Delft, 1924 (C. B. BIEZENO and J. M. BURGERS, eds.) 395-404. Technische Boekhandel en Drukkerij J. Waltman Jr., Delft.
[63] KLINE, M. (1990). Mathematical thought from ancient to modern times. Vol. 3, Second ed. The Clarendon Press, Oxford University Press, New York. · Zbl 0784.01048
[64] KRÖNER, E. (1958). Kontinuumstheorie der Versetzungen und Eigenspannungen. Ergebnisse der angewandten Mathematik. Bd. 5. Springer-Verlag, Berlin-Göttingen-Heidelberg. · Zbl 0084.40003
[65] LAWSON, H. B. JR. and MICHELSOHN, M.-L. (1989). Spin geometry. Princeton Mathematical Series 38. Princeton University Press, Princeton, NJ. · Zbl 0688.57001
[66] LERARIO, A., MARINUCCI, D., ROSSI, M. and STECCONI, M. (2022). Geometry and topology of spin random fields. Preprint arXiv: 2207.08413v1 [math.PR].
[67] LIM, L.-H. (2021). Tensors in computations. Acta Numer. 30 555-764. · Zbl 1512.65079 · doi:10.1017/S0962492921000076
[68] LOMAKIN, V. A. (1964). Statistical description of the stressed state of a body under deformation. Dokl. Akad. Nauk SSSR 155 1274-1277.
[69] LOMAKIN, V. A. (1965). Deformation of microscopically nonhomogeneous elastic bodies. J. Appl. Math. Mech. 29 1048-1054. · Zbl 0158.21304 · doi:10.1016/0021-8928(65)90125-5
[70] LUBARDA, V. A. and KRAJCINOVIC, D. (1993). Damage tensors and the crack density distribution. Internat. J. Solids Structures 30 2859-2877. · Zbl 0782.73058 · doi:10.1016/0020-7683(93)90158-4
[71] MA, Z.-Q. (2019). Group theory for physicists. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ. · Zbl 1402.20001 · doi:10.1142/11187
[72] MALYARENKO, A. (2011). Invariant random fields in vector bundles and application to cosmology. Ann. Inst. Henri Poincaré Probab. Stat. 47 1068-1095. · Zbl 1268.60072 · doi:10.1214/10-AIHP409
[73] MALYARENKO, A. (2013). Invariant random fields on spaces with a group action. Probability and its Applications (New York). Springer, Heidelberg With a foreword by Nikolai Leonenko. · Zbl 1268.60006 · doi:10.1007/978-3-642-33406-1
[74] MALYARENKO, A. (2017). Spectral expansions of random sections of homogeneous vector bundles. Teor. ˘ Imov¯ır. Mat. Stat. 97 142-156. · Zbl 1414.83106 · doi:10.1090/tpms/1054
[75] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2014). Statistically isotropic tensor random fields: correlation structures. Math. Mech. Complex Syst. 2 209-231. · Zbl 1317.60061 · doi:10.2140/memocs.2014.2.209
[76] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2016). Spectral expansions of homogeneous and isotropic tensor-valued random fields. Z. Angew. Math. Phys. 67 Art. 59, 20. · Zbl 1348.60077 · doi:10.1007/s00033-016-0657-8
[77] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2016). Spectral expansion of three-dimensional elasticity tensor random fields. In Engineering mathematics. I. Electromagnetics, fluid mechanics, material physics and financial engineering, (S. Silvestrov and M. Rančić, eds.). Springer Proc. Math. Stat. 178 281-300. Springer, Cham. · Zbl 1356.74022 · doi:10.1007/978-3-319-42082-0_16
[78] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2016). A random field formulation of Hooke’s law in all elasticity classes. Preprint arXiv: 1602.09066v2 [math-ph].
[79] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2017). A random field formulation of Hooke’s law in all elasticity classes. J. Elasticity 127 269-302. · Zbl 1418.60046 · doi:10.1007/s10659-016-9613-2
[80] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2017). Fractal planetary rings: energy inequalities and random field model. Internat. J. Modern Phys. B 31 1750236, 14. · Zbl 1434.85010 · doi:10.1142/S0217979217502368
[81] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2018). Random fields related to the symmetry classes of second-order symmetric tensors. In Stochastic processes and applications, (S. Silvestrov, A. Malyarenko and M. Rančić, eds.). Springer Proc. Math. Stat. 271 173-185. Springer, Cham SPAS2017, Västerås and Stockholm, Sweden, October 4-6, 2017, Based on the International Conference “Stochastic processes and algebraic structures—from theory towards applications”. · Zbl 1423.60084 · doi:10.1007/978-3-030-02825-1_10
[82] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2019). Tensor-valued random fields for continuum physics. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. · Zbl 1402.60002
[83] MALYARENKO, A. and OSTOJA-STARZEWSKI, M. (2022). Polyadic random fields. Z. Angew. Math. Phys. 73 Paper No. 204. · Zbl 1505.60056 · doi:10.1007/s00033-022-01842-5
[84] MALYARENKO, A., OSTOJA-STARZEWSKI, M. and AMIRI-HEZAVEH, A. (2020). Random fields of piezoelectricity and piezomagnetism. Correlation structures. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. · Zbl 1451.74001 · doi:10.1007/978-3-030-60064-8
[85] MARINUCCI, D. and PECCATI, G. (2011). Random fields on the sphere. Representation, limit theorems and cosmological applications. London Mathematical Society Lecture Note Series 389. Cambridge University Press, Cambridge. · Zbl 1260.60004 · doi:10.1017/CBO9780511751677
[86] MARINUCCI, D. and PECCATI, G. (2013). Mean-square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18 no. 37, 10. · Zbl 1329.60139 · doi:10.1214/ECP.v18-2400
[87] MATHEWS, J. (1962). Gravitational multipole radiation. J. Soc. Indust. Appl. Math. 10 768-780. · Zbl 0114.21201
[88] MIHAI, L. A. (2022). Stochastic Elasticity. A Nondeterministic Approach to the Nonlinear Field Theory. Interdisciplinary Applied Mathematics 55. Springer International Publishing AG. · Zbl 07555481
[89] MISNER, C. W., THORNE, K. S. and WHEELER, J. A. (1973). Gravitation. W. H. Freeman and Co., San Francisco, Calif.
[90] MONIN, A. S. and YAGLOM, A. M. (2007). Statistical fluid mechanics: mechanics of turbulence. Vol. I. Dover Publications, Inc., Mineola, NY Translated from the 1965 Russian original, Edited and with a preface by John L. Lumley, English edition updated, augmented and revised by the authors, Reprinted from the 1971 edition. · Zbl 1140.76003
[91] MONIN, A. S. and YAGLOM, A. M. (2007). Statistical fluid mechanics: mechanics of turbulence. Vol. II. Dover Publications, Inc., Mineola, NY Translated from the 1965 Russian original, Edited and with a preface by John L. Lumley, English edition updated, augmented and revised by the authors, Reprinted from the 1975 edition. · Zbl 1140.76003
[92] MONTGOMERY, D. and ZIPPIN, L. (1974). Topological transformation groups. Robert E. Krieger Publishing Co., Huntington, N.Y. Reprint of the 1955 original. · Zbl 0323.57023
[93] MOYAL, J. E. (1952). The spectra of turbulence in a compressible fluid; eddy turbulence and random noise. Proc. Cambridge Philos. Soc. 48 329-344. · Zbl 0046.42401 · doi:10.1017/s0305004100027675
[94] MUNTEANU, G. (2004). Complex spaces in Finsler, Lagrange and Hamilton geometries. Fundamental Theories of Physics 141. Kluwer Academic Publishers, Dordrecht. · Zbl 1064.53047 · doi:10.1007/978-1-4020-2206-7
[95] MURAKAMI, S. (2012). Continuum damage mechanics. A continuum mechanics approach to the analysis of damage and fracture. Solid Mechanics, Applicat. 185. Springer, Dordrecht. · doi:10.1007/978-94-007-2666-6
[96] NA˘IMARK, M. A. and ŠTERN, A. I. (1982). Theory of group representations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 246. Springer-Verlag, New York Translated from the Russian by Elizabeth Hewitt, Translation edited by Edwin Hewitt. · Zbl 0484.22018 · doi:10.1007/978-1-4613-8142-6
[97] NEWMAN, E. T. and PENROSE, R. (1966). Note on the Bondi-Metzner-Sachs group. J. Mathematical Phys. 7 863-870. · doi:10.1063/1.1931221
[98] NISHAWALA, V. V., OSTOJA-STARZEWSKI, M., LEAMY, M. J. and PORCU, E. (2016). Lamb’s problem on random mass density fields with fractal and Hurst effects. Proc. A. 472 20160638, 14. · Zbl 1371.74151 · doi:10.1098/rspa.2016.0638
[99] NOSHADRAVAN, A., GHANEM, R., GUILLEMINOT, J., ATODARIA, I. and PERALTA, P. (2013). Validation of a probabilistic model for mesoscale elasticity tensor of random polycrystals. Int. J. Uncertain. Quantif. 3 73-100. · Zbl 1513.74064 · doi:10.1615/Int.J.UncertaintyQuantification.2012003901
[100] OBUKHOV, A. M. (1947). Statistically homogeneous random fields on a sphere. Uspehi Mat. Nauk 2 196-198.
[101] OGDEN, R. W. (1974). On isotropic tensors and elastic moduli. Proc. Cambridge Philos. Soc. 75 427-436. · Zbl 0327.73007 · doi:10.1017/s0305004100048635
[102] OLIVE, M. (2019). Effective computation of \[\text{SO}(\text{3})\] and \[\text{O}(\text{3})\] linear representation symmetry classes. Math. Mech. Complex Syst. 7 203-237. · doi:10.2140/memocs.2019.7.203
[103] OLIVE, M. and AUFFRAY, N. (2013). Symmetry classes for even-order tensors. Math. Mech. Complex Syst. 1 177-210. · Zbl 1391.15089 · doi:10.2140/memocs.2013.1.177
[104] OLIVE, M. and AUFFRAY, N. (2014). Symmetry classes for odd-order tensors. Z. Angew. Math. Mech. 94 421-447. · Zbl 1302.15030 · doi:10.1002/zamm.201200225
[105] OLVER, F. W. J., LOZIER, D. W., BOISVERT, R. F. and CLARK, C. W., eds. (2010). NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge With 1 CD-ROM (Windows, Macintosh and UNIX). · Zbl 1198.00002
[106] OSTOJA-STARZEWSKI, M. (2008). Microstructural randomness and scaling in mechanics of materials. CRC Series: Modern Mechanics and Mathematics. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1148.74002
[107] OSTOJA-STARZEWSKI, M. (2019). Ignaczak equation of elastodynamics. Math. Mech. Solids 24 3674-3713. · Zbl 07273388 · doi:10.1177/1081286518757284
[108] OSTOJA-STARZEWSKI, M., KALE, S., KARIMI, P., MALYARENKO, A., RAGHAVAN, B. V., RANGANATHAN, S. I. and ZHANG, J. (2016). Scaling to RVE in Random Media. In Advances in Applied Mechanics, vol. 49 (S. P. A. Bordas and D. S. Balint, eds.) 111-211. Elsevier, Burlington. · doi:10.1016/bs.aams.2016.07.001
[109] OSTOJA-STARZEWSKI, M. and LAUDANI, R. (2020). Violations of the Clausius-Duhem inequality in Couette flows of granular media. Proc. R Soc. A 476 20200207, 17. · Zbl 1472.74047 · doi:10.1098/rspa.2020.0207
[110] OSTOJA-STARZEWSKI, M. and MALYARENKO, A. (2014). Continuum mechanics beyond the second law of thermodynamics. Proc. R Soc. A 470 20140531. · doi:10.1098/rspa.2014.0531
[111] OSTOJA-STARZEWSKI, M., SHEN, L. and MALYARENKO, A. (2015). Tensor random fields in conductivity and classical or microcontinuum theories. Math. Mech. Solids 20 418-432. · Zbl 1327.74017 · doi:10.1177/1081286513498524
[112] PENROSE, R. and RINDLER, W. (1987). Spinors and space-time. Vol. 1. Two-spinor calculus and relativistic fields. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. · Zbl 0663.53013
[113] PENZIAS, A. A. and WILSON, R. W. (1965). A Measurement of Excess Antenna Temperature at 4080 Mc/s. Astrophys. J. Lett. 142 419-421. · doi:10.1086/148307
[114] PERRIN, G. and SOIZE, C. (2020). Adaptive method for indirect identification of the statistical properties of random fields in a Bayesian framework. Comput. Statist. 35 111-133. · Zbl 1505.62316 · doi:10.1007/s00180-019-00936-5
[115] POINCARÉ, H. (1895). Analysis situs. J. de l’École Polytechnique (2) 1 1-123.
[116] PONTRYAGIN, L. S. (1966). Topological groups. Translated from the second Russian edition by Arlen Brown. Gordon and Breach Science Publishers, Inc., New York-London-Paris.
[117] RICCI-CURBASTRO, G. (1892). Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique. Bull. des Sci. Math. (II) 16 167-189.
[118] RICCI-CURBASTRO, G. and LEVI-CIVITA, T. (1900). Méthodes de calcul différentiel absolu et leurs applications. Math. Ann. 54 125-201. · doi:10.1007/BF01454201
[119] ROBERTSON, H. P. (1940). The invariant theory of isotropic turbulence. Proc. Cambridge Philos. Soc. 36 209-223. · Zbl 0023.42604
[120] RUIZ-MEDINA, M. D., PORCU, E. and FERNANDEZ-PASCUAL, R. (2011). The Dagum and auxiliary covariance families: Towards reconciling two-parameter models that separate fractal dimension and the Hurst effect. Probab. Engng. Mech. 26 259-268. · doi:10.1016/j.probengmech.2010.08.002
[121] SCHOENBERG, I. J. (1938). Metric spaces and completely monotone functions. Ann. of Math. (2) 39 811-841. · doi:10.2307/1968466
[122] SCORPAN, A. (2005). The wild world of 4-manifolds. American Mathematical Society, Providence, RI. · Zbl 1075.57001
[123] SELIVANOVA, S. (2014). Computing Clebsch-Gordan matrices with applications in elasticity theory. In Logic, computation, hierarchies, (V. Brattka, H. Diener and D. Spreen, eds.). Ontos Math. Log. 4 273-295. De Gruyter, Berlin Festschrift for Victor Selivanov. · Zbl 1311.74022
[124] SENA, M. P., OSTOJA-STARZEWSKI, M. and COSTA, L. (2013). Stiffness tensor random fields through upscaling of planar random materials. Probab. Engng. Mech. 34 131-156. · doi:10.1016/j.probengmech.2013.08.008
[125] SHERMERGOR, T. D. (1971). Relations between the components of the correlation functions of an elastic field. Journal of Applied Mathematics and Mechanics 35 392-397. · Zbl 0332.73008 · doi:10.1016/0021-8928(71)90007-4
[126] SOIZE, C. (2006). Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput. Methods Appl. Mech. Engrg. 195 26-64. · Zbl 1093.74065 · doi:10.1016/j.cma.2004.12.014
[127] SOIZE, C. (2008). Construction of probability distributions in high dimension using the maximum entropy principle: applications to stochastic processes, random fields and random matrices. Internat. J. Numer. Methods Engrg. 76 1583-1611. · Zbl 1195.74311 · doi:10.1002/nme.2385
[128] SOIZE, C. (2008). Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probabilistic Engineering Mechanics 23 307-323. 5th International Conference on Computational Stochastic Mechanics. · doi:10.1016/j.probengmech.2007.12.019
[129] SOIZE, C. (2010). Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data. Comput. Methods Appl. Mech. Engrg. 199 2150-2164. · Zbl 1231.74501 · doi:10.1016/j.cma.2010.03.013
[130] SOIZE, C. (2011). A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension. Comput. Methods Appl. Mech. Engrg. 200 3083-3099. · Zbl 1230.74241 · doi:10.1016/j.cma.2011.07.005
[131] SOIZE, C. (2021). Computational stochastic homogenization of heterogeneous media from an elasticity random field having an uncertain spectral measure. Comput. Mech. 68 1003-1021. · Zbl 1478.74017 · doi:10.1007/s00466-021-02056-8
[132] SOIZE, C. (2021). Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum. Theory Probab. Math. Statist. 105 113-136. · Zbl 1485.60051 · doi:10.1090/tpms/1159
[133] SOIZE, C. and DESCELIERS, C. (2010). Computational aspects for constructing realizations of polynomial chaos in high dimension. SIAM J. Sci. Comput. 32 2820-2831. · Zbl 1225.60118 · doi:10.1137/100787830
[134] SPENCER, A. J. M. (1971). Part III. Theory of invariants. In Continuum physics. Vol. I (A. C. Eringen, ed.) 239-355. Academic Press, New York-London.
[135] STABER, B. and GUILLEMINOT, J. (2018). A random field model for anisotropic strain energy functions and its application for uncertainty quantification in vascular mechanics. Comput. Methods Appl. Mech. Engrg. 333 94-113. · Zbl 1440.74021 · doi:10.1016/j.cma.2018.01.001
[136] STECCONI, M. (2021). Isotropic random spin weighted functions on \[{S^2}\] vs isotropic random fields on \[{S^3}\]. Theory Probab. Math. Statist. In press, preprint arXiv: 2108.00736v1 [math.PR].
[137] TAYLOR, G. I. (1935). Statistical theory of turbulence. Proc. Roy. Soc. A. 151 421-444. · doi:10.1098/rspa.1935.0158
[138] TEMPLE, G. F. J. (2004). Cartesian tensors. An introduction. Dover Publications, Inc., Mineola, NY Reprint of the 1960 original. · Zbl 0091.33804
[139] THORNE, K. S. (1980). Multipole expansions of gravitational radiation. Rev. Modern Phys. 52 299-339. · doi:10.1103/RevModPhys.52.299
[140] TRAUTMAN, A. (1997). Clifford and the “square root” ideas. In Geometry and nature (Madeira, 1995), (H. Nencka and J.-P. Bourguignon, eds.). Contemp. Math. 203 3-24. Amer. Math. Soc., Providence, RI In memory of W. K. Clifford, Papers from the Conference on New Trends in Geometrical and Topological Methods held in Madeira, July 30-August 5, 1995. · Zbl 0872.15021 · doi:10.1090/conm/203/02577
[141] TRAUTMAN, A. (2006). Clifford Algebras and Their Representations. In Encyclopedia of Mathematical Physics (J.-P. Françoise, G. L. Naber and T. S. Tsun, eds.) 518-530. Academic Press, Oxford. · doi:10.1016/B0-12-512666-2/00016-X
[142] TRAUTMAN, A. (2008). Connections and the Dirac operator on spinor bundles. J. Geom. Phys. 58 238-252. · Zbl 1140.53025 · doi:10.1016/j.geomphys.2007.11.001
[143] VANMARCKE, E. (2010). Random fields. Analysis and synthesis, new ed. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ. · Zbl 1220.60030 · doi:10.1142/5807
[144] VILENKIN, N. YA. and KLIMYK, A. U. (1991). Representation of Lie groups and special functions. Vol. 1. Mathematics and its Applications (Soviet Series) 72. Kluwer Academic Publishers Group, Dordrecht Simplest Lie groups, special functions and integral transforms, Translated from the Russian by V. A. Groza and A. A. Groza. · Zbl 0742.22001 · doi:10.1007/978-94-011-3538-2
[145] VILENKIN, N. YA. and KLIMYK, A. U. (1993). Representation of Lie groups and special functions. Vol. 2. Class I representations, special functions, and integral transforms. Mathematics and its Applications (Soviet Series) 74. Kluwer Academic Publishers Group, Dordrecht Translated from the Russian by V. A. Groza and A. A. Groza. · Zbl 0809.22001 · doi:10.1007/978-94-017-2883-6
[146] VON KÁRMÁN, T. (1937). On the statistical theory of turbulence. Proc. Nat. Acad. Sci. 23 98-105.
[147] VON KÁRMÁN, T. (1937). The fundamentals of the statistical theory of turbulence. J. Aeron. Sci. 4 131-138.
[148] VON KÁRMÁN, T. (1948). Sur la théorie statistique de la turbulence. C. R. Acad. Sci. Paris 226 2108-2111. · Zbl 0038.38304
[149] VON KÁRMÁN, T. (1948). Progress in the statistical theory of turbulence. Proc. Nat. Acad. Sci. U.S.A. 34 530-539. · Zbl 0032.22601 · doi:10.1073/pnas.34.11.530
[150] VON KÁRMÁN, T. and HOWARTH, L. (1938). On the statistical theory of isotropic turbulence. Proc. Roy. Soc. 164 192-215.
[151] VON KÁRMÁN, T. and LIN, C. C. On the statistical theory of isotropic turbulence. In Advances in Applied Mechanics, vol. 2 (R. von Mises and T. von Kármán, eds.) 1-19. · Zbl 0044.40603
[152] WALD, R. M. (1984). General relativity. University of Chicago Press, Chicago, IL. · Zbl 0549.53001 · doi:10.7208/chicago/9780226870373.001.0001
[153] WALLACH, N. R. (1973). Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics 19. Marcel Dekker, Inc., New York. · Zbl 0265.22022
[154] WEYL, H. (1997). Die Idee der Riemannschen Fläche. Teubner-Archiv zur Mathematik. Supplement [Teubner Archive on Mathematics. Supplement] 5. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart Reprint of the 1913 German original, With essays by Reinhold Remmert, Michael Schneider, Stefan Hildebrandt, Klaus Hulek and Samuel Patterson, Edited and with a preface and a biography of Weyl by Remmert. · doi:10.1007/978-3-663-07819-7
[155] WEYL, H. (1997). The classical groups, their invariants and representations. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ Fifteenth printing, Princeton Paperbacks. · Zbl 1024.20501
[156] WHITNEY, H. (1936). Differentiable manifolds. Ann. of Math. (2) 37 645-680. · Zbl 0015.32001 · doi:10.2307/1968482
[157] WINEMAN, A. S. and PIPKIN, A. C. (1964). Material symmetry restrictions on constitutive equations. Arch. Rational Mech. Anal. 17 184-214. · Zbl 0126.40604 · doi:10.1007/BF00282437
[158] YADRENKO, M. ˘I. (1983). Spectral theory of random fields. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York Translated from the Russian. · Zbl 0539.60048
[159] YAGLOM, A. M. (1948). Homogeneous and isotropic turbulence in a viscous compressible fluid. Izvestiya Akad. Nauk SSSR. Ser. Geograf. Geofiz. 12 501-522.
[160] YAGLOM, A. M. (1957). Certain types of random fields in \(n\)-dimensional space similar to stationary stochastic processes. Teor. Veroyatnost. i Primenen 2 292-338. · Zbl 0084.12804
[161] YAGLOM, A. M. (1961). Second-order homogeneous random fields. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II. Contributions to Probability Theory (J. NEYMAN, ed.) 593-622. Univ. California Press, Berkeley, Calif. · Zbl 0123.35001
[162] YAGLOM, A. M. (1987). Correlation theory of stationary and related random functions. Vol. I. Basic results. Springer Series in Statistics. Springer-Verlag, New York. · Zbl 0685.62077
[163] YAGLOM, A. M. (1987). Correlation theory of stationary and related random functions. Vol. II. Supplementary notes and references. Springer Series in Statistics. Springer-Verlag, New York. · Zbl 0685.62078
[164] ZALDARRIAGA, M. and SELJAK, U. (1997). An all sky analysis of polarization in the microwave background. Phys. Rev. D 55 1830-1840. · doi:10.1103/PhysRevD.55.1830
[165] ZERILLI, F. J. (1970). Tensor harmonics in canonical form for gravitational radiation and other applications. J. Mathematical Phys. 11 2203-2208. · doi:10.1063/1.1665380
[166] ZERILLI, F. J. (1970). Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics. Phys. Rev. D (3) 2 2141-2160. · Zbl 1227.83025 · doi:10.1103/PhysRevD.2.2141
[167] ZHANG, X., MALYARENKO, A., PORCU, E. and OSTOJA-STARZEWSKI, M. (2022). Elastodynamic problem on tensor random fields with fractal and Hurst effects. Meccanica 57 957-970. · Zbl 1534.74036 · doi:10.1007/s11012-021-01424-1
[168] ZHENG, Q. S. and BOEHLER, J.-P. (1994). The description, classification, and reality of material and physical symmetries. Acta Mech. 102 73-89. · Zbl 0811.73002 · doi:10.1007/BF01178519
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