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Tensors in computations

Published online by Cambridge University Press:  04 August 2021

Lek-Heng Lim
Lek-Heng Lim*
Affiliation:
Computational and Applied Mathematics Initiative, University of Chicago, Chicago, IL 60637, USA E-mail: lekheng@uchicago.edu
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Abstract

The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.

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Research Article
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Acta Numerica , Volume 30 , May 2021 , pp. 555 - 764
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© The Author(s), 2021. Published by Cambridge University Press

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References

Abraham, R., Marsden, J. E. and Ratiu, T. (1988), Manifolds, Tensor Analysis, and Applications, Vol. 75 of Applied Mathematical Sciences, second edition, Springer.CrossRefGoogle Scholar
Aerts, D. and Daubechies, I. (1978), Physical justification for using the tensor product to describe two quantum systems as one joint system, Helv . Phys. Acta 51, 661–675.Google Scholar
Aerts, D. and Daubechies, I. (1979a), A characterization of subsystems in physics, Lett. Math. Phys. 3, 11–17.CrossRefGoogle Scholar
Aerts, D. and Daubechies, I. (1979b), A mathematical condition for a sublattice of a propositional system to represent a physical subsystem, with a physical interpretation, Lett. Math. Phys. 3, 19–27.CrossRefGoogle Scholar
Affleck, I., Kennedy, T., Lieb, E. H. and Tasaki, H. (1987), Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799–802.CrossRefGoogle ScholarPubMed
Akbarov, S. S. (2003), Pontryagin duality in the theory of topological vector spaces and in topological algebra, J. Math. Sci. 113, 179–349.CrossRefGoogle Scholar
Alman, J. and Williams, V. V. (2021), A refined laser method and faster matrix multiplication, in Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA) (Marx, D., ed.), Society for Industrial and Applied Mathematics (SIAM), pp. 522–539.CrossRefGoogle Scholar
Anderson, P. W. (1959), New approach to the theory of superexchange interactions, Phys. Rev. (2) 115, 2–13.CrossRefGoogle Scholar
Arias, A. and Farmer, J. D. (1996), On the structure of tensor products of l p -spaces, Pacific J. Math. 175, 13–37.CrossRefGoogle Scholar
Baldick, R. (1995), A unified approach to polynomially solvable cases of integer ‘non-separable’ quadratic optimization, Discrete Appl. Math. 61, 195–212.CrossRefGoogle Scholar
Barnes, J. and Hut, P. (1986), A hierarchical O(NlogN) force-calculation algorithm, Nature 324, 446–449.CrossRefGoogle Scholar
Bellman, R. (1997), Introduction to Matrix Analysis, Vol. 19 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM).Google Scholar
Bényi, A. and Torres, R. H. (2013), Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141, 3609–3621.CrossRefGoogle Scholar
Berberian, S. K. (2014), Linear Algebra, Dover Publications.Google Scholar
Beylkin, G. (1993), Wavelets and fast numerical algorithms, in Different Perspectives on Wavelets (San Antonio, TX, 1993), Vol. 47 of Proceedings of Symposia in Applied Mathematics, American Mathematical Society, pp. 89–117.CrossRefGoogle Scholar
Beylkin, G., Coifman, R. and Rokhlin, V. (1991), Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44, 141–183.CrossRefGoogle Scholar
Beylkin, G., Coifman, R. and Rokhlin, V. (1992), Wavelets in numerical analysis, in Wavelets and their Applications (Ruskai, M. B. et al., eds), Jones & Bartlett, pp. 181–210.Google Scholar
Bhatia, R. (1997), Matrix Analysis, Vol. 169 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Bini, D. (1980), Relations between exact and approximate bilinear algorithms: Applications, Calcolo 17, 87–97.CrossRefGoogle Scholar
Bini, D., Capovani, M., Romani, F. and Lotti, G. (1979), O(n 2.7799) complexity for n × n approximate matrix multiplication, Inform. Process. Lett. 8, 234–235.CrossRefGoogle Scholar
Bini, D., Lotti, G. and Romani, F. (1980), Approximate solutions for the bilinear form computational problem, SIAM J. Comput. 9, 692–697.CrossRefGoogle Scholar
Bishop, C. M. (2006), Pattern Recognition and Machine Learning, Information Science and Statistics, Springer.Google Scholar
Blass, A. (1984), Existence of bases implies the axiom of choice, in Axiomatic Set Theory (Boulder, CO, 1983), Vol. 31 of Contemporary Mathematics, American Mathematical Society, pp. 31–33.Google Scholar
Bleecker, D. (1981), Gauge Theory and Variational Principles, Vol. 1 of Global Analysis Pure and Applied Series A, Addison-Wesley.Google Scholar
Blum, K. (1996), Density Matrix Theory and Applications, Physics of Atoms and Molecules, second edition, Plenum Press.Google Scholar
Board, J. and Schulten, L. (2000), The fast multipole algorithm, Comput . Sci. Eng. 2, 76–79.Google Scholar
Bogatskiy, A., Anderson, B., Offermann, J., Roussi, M., Miller, D. and Kondor, R. (2020), Lorentz group equivariant neural network for particle physics, in Proceedings of the 37th International Conference on Machine Learning (ICML 2020) (H. Daumé III and A. Singh, eds), Vol. 119 of Proceedings of Machine Learning Research, PMLR, pp. 992–1002.Google Scholar
Boneh, D. and Silverberg, A. (2003), Applications of multilinear forms to cryptography, in Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis, MD, 2001), Vol. 324 of Contemporary Mathematics, American Mathematical Society, pp. 71–90.CrossRefGoogle Scholar
Boothby, W. M. (1986), An Introduction to Differentiable Manifolds and Riemannian Geometry, Vol. 120 of Pure and Applied Mathematics, second edition, Academic Press.Google Scholar
Borg, S. F. (1990), Matrix–Tensor Methods in Continuum Mechanics, second edition, World Scientific.CrossRefGoogle Scholar
Borisenko, A. I. and Tarapov, I. E. (1979), Vector and Tensor Analysis with Applications, Dover Publications.Google Scholar
Bourbaki, N. (1998), Algebra I, Chapters 1–3, Elements of Mathematics (Berlin), Springer.Google Scholar
Bourouihiya, A. (2008), The tensor product of frames, Sampl . Theory Signal Image Process. 7, 65–76.CrossRefGoogle Scholar
Boyd, S. and Vandenberghe, L. (2004), Convex Optimization, Cambridge University Press.CrossRefGoogle Scholar
Boyd, S. and Vandenberghe, L. (2018), Introduction to Applied Linear Algebra, Cambridge University Press.CrossRefGoogle Scholar
Bracewell, R. N. (1986), The Fourier Transform and its Applications, McGraw-Hill Series in Electrical Engineering, third edition, McGraw-Hill.Google Scholar
Brachat, J., Comon, P., Mourrain, B. and Tsigaridas, E. (2010), Symmetric tensor decomposition, Linear Algebra Appl. 433, 1851–1872.CrossRefGoogle Scholar
Brand, L. (1947), Vector and Tensor Analysis, Wiley.Google Scholar
Brass, H. and Petras, K. (2011), Quadrature Theory: The Theory of Numerical Integration on a Compact Interval, Vol. 178 of Mathematical Surveys and Monographs, American Mathematical Society.CrossRefGoogle Scholar
Brent, R. P. (1970), Algorithms for matrix multiplication. Report Stan-CS-70-157, Stanford University.CrossRefGoogle Scholar
Brent, R. P. and Zimmermann, P. (2011), Modern Computer Arithmetic, Vol. 18 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Bröcker, T. and Dieck, T. tom (1995), Representations of Compact Lie Groups, Vol. 98 of Graduate Texts in Mathematics, Springer.Google Scholar
Brualdi, R. A. (1992), The symbiotic relationship of combinatorics and matrix theory, Linear Algebra Appl. 162/164, 65–105.CrossRefGoogle Scholar
Buhmann, M. D. (2003), Radial Basis Functions: Theory and Implementations, Vol. 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Bürgisser, P. and Cucker, F. (2013), Condition: The Geometry of Numerical Algorithms, Vol. 349 of Grundlehren der Mathematischen Wissenschaften, Springer.CrossRefGoogle Scholar
Bürgisser, P., Clausen, M. and Shokrollahi, M. A. (1997), Algebraic Complexity Theory, Vol. 315 of Grundlehren der Mathematischen Wissenschaften, Springer.CrossRefGoogle Scholar
Callaway, E. (2020), ‘It will change everything’: Deepmind’s AI makes gigantic leap in solving protein structures, Nature 588 (7837), 203–204.CrossRefGoogle ScholarPubMed
Cayley, A. (1845), On the theory of linear transformations, Cambridge Math. J. 4, 193–209.Google Scholar
Chern, S. S., Chen, W. H. and Lam, K. S. (1999), Lectures on Differential Geometry, Vol. 1 of Series on University Mathematics, World Scientific.CrossRefGoogle Scholar
Choquet-Bruhat, Y., DeWitt-Morette, C. and Dillard-Bleick, M. (1982), Analysis, Manifolds and Physics, second edition, North-Holland.Google Scholar
Chou, P. C. and Pagano, N. J. (1992), Elasticity: Tensor, Dyadic and Engineering Approaches, Dover Publications.Google Scholar
Cobos, F., Kühn, T. and Peetre, J. (1992), Schatten–von Neumann classes of multilinear forms, Duke Math. J. 65, 121–156.CrossRefGoogle Scholar
Cobos, F., Kühn, T. and Peetre, J. (1999), On G p -classes of trilinear forms, J. London Math. Soc. (2) 59, 1003–1022.CrossRefGoogle Scholar
Cohen, A. and Daubechies, I. (1993), Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9, 51–137.CrossRefGoogle Scholar
Cohen, M. B., Lee, Y. T. and Song, Z. (2019), Solving linear programs in the current matrix multiplication time, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), ACM, pp. 938–942.Google Scholar
Cohen, T. and Welling, M. (2016), Group equivariant convolutional networks, in Proceedings of the 33rd International Conference on Machine Learning (ICML 2016) (M. F. Balcan and K. Q. Weinberger, eds), Vol. 48 of Proceedings of Machine Learning Research, PMLR, pp. 2990–2999.Google Scholar
Cohen-Tannoudji, C., Diu, B. and Laloë, F. (2020a), Quantum Mechanics 1: Basic Concepts, Tools, and Applications, second edition, Wiley-VCH.Google Scholar
Cohen-Tannoudji, C., Diu, B. and Laloë, F. (2020b), Quantum Mechanics 2: Angular Momentum, Spin, and Approximation Methods, second edition, Wiley-VCH.Google Scholar
Conrad, K. (2018), Tensor products. Available at https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf.Google Scholar
Conway, J. B. (1990), A Course in Functional Analysis, Vol. 96 of Graduate Texts in Mathematics, second edition, Springer.Google Scholar
Cook, S. A. and Aanderaa, S. O. (1969), On the minimum computation time of functions, Trans. Amer. Math. Soc. 142, 291–314.CrossRefGoogle Scholar
Cucker, F. and Smale, S. (2002), On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.) 39, 1–49.CrossRefGoogle Scholar
Davidson, E. R. (1976), Reduced Density Matrices in Quantum Chemistry, Vol. 6 of Theoretical Chemistry, Academic Press.Google Scholar
Davie, A. M. (1985), Matrix norms related to Grothendieck’s inequality, in Banach Spaces (Columbia, MO, 1984), Vol. 1166 of Lecture Notes in Mathematics, Springer, pp. 22–26.Google Scholar
Davis, P. J. and Rabinowitz, P. (2007), Methods of Numerical Integration, Dover Publications.Google Scholar
Concini, C. De and Procesi, C. (2017), The Invariant Theory of Matrices, Vol. 69 of University Lecture Series, American Mathematical Society.Google Scholar
de la Madrid, R. (2005), The role of the rigged Hilbert space in quantum mechanics, European J. Phys. 26, 287–312.CrossRefGoogle Scholar
De Lathauwer, L., De Moor, B. and Vandewalle, J. (2000), A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl. 21, 1253–1278.CrossRefGoogle Scholar
De Silva, V. and Lim, L.-H. (2008), Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl. 30, 1084–1127.CrossRefGoogle Scholar
De Terán, F. (2016), Canonical forms for congruence of matrices and T-palindromic matrix pencils: A tribute to Turnbull, H. W. and Aitken, A. C., SeMA J. 73, 7–16.CrossRefGoogle Scholar
Debnath, J. and Dahiya, R. S. (1989), Theorems on multidimensional Laplace transform for solution of boundary value problems, Comput. Math. Appl. 18, 1033–1056.CrossRefGoogle Scholar
Defant, A. and Floret, K. (1993), Tensor Norms and Operator Ideals, Vol. 176 of North-Holland Mathematics Studies, North-Holland.Google Scholar
Demaine, E. D., Demaine, M. L., Edelman, A., Leiserson, C. E. and Persson, P.-O. (2005), Building blocks and excluded sums, SIAM News 38, 1, 4, 6.Google Scholar
Demmel, J. W. (1997), Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Deng, L. (2012), The MNIST database of handwritten digit images for machine learning research, IEEE Signal Process. Mag. 29, 141–142.CrossRefGoogle Scholar
Dickinson, P. J. C. and Gijben, L. (2014), On the computational complexity of membership problems for the completely positive cone and its dual, Comput . Optim. Appl. 57, 403–415.CrossRefGoogle Scholar
Diestel, J., Fourie, J. H. and Swart, J. (2008), The Metric Theory of Tensor Products: Grothendieck’s Résumé Revisited, American Mathematical Society.CrossRefGoogle Scholar
Dodson, C. T. J. and Poston, T. (1991), Tensor Geometry: The Geometric Viewpoint and its Uses, Vol. 130 of Graduate Texts in Mathematics, second edition, Springer.CrossRefGoogle Scholar
Dongarra, J. and Sullivan, F. (2000), Guest editors’ introduction to the top 10 algorithms, Comput . Sci. Eng. 2, 22–23.Google Scholar
Dudgeon, D. E. and Mersereau, R. M. (1983), Multidimensional Digital Signal Processing, Prentice Hall.Google Scholar
Dummit, D. S. and Foote, R. M. (2004), Abstract Algebra, third edition, Wiley.Google Scholar
Dunford, N. and Schwartz, J. T. (1988), Linear Operators, Part I, Wiley Classics Library, Wiley.Google Scholar
Dunn, G. (1988), Tensor product of operads and iterated loop spaces, J. Pure Appl. Algebra 50, 237–258.CrossRefGoogle Scholar
Dym, H. (2013), Linear Algebra in Action, Vol. 78 of Graduate Studies in Mathematics, second edition, American Mathematical Society.CrossRefGoogle Scholar
Earman, J. and Glymour, C. (1978), Lost in the tensors: Einstein’s struggles with covariance principles 1912–1916, Stud. Hist. Philos. Sci. A 9, 251–278.CrossRefGoogle Scholar
Einstein, A. (2002), Fundamental ideas and methods of the theory of relativity, presented in its development, in The Collected Papers of Albert Einstein (Janssen, M. et al., eds), Vol. 7: The Berlin Years, 1918–1921, Princeton University Press, pp. 113–150.Google Scholar
Eisenhart, L. P. (1934), Separable systems of Stackel, Ann. of Math. (2) 35, 284–305.CrossRefGoogle Scholar
Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V. (2015), Tensor Categories, Vol. 205 of Mathematical Surveys and Monographs, American Mathematical Society.CrossRefGoogle Scholar
Faddeev, L. D. and Yakubovskiĭ, O. A. (2009), Lectures on Quantum Mechanics for Mathematics Students, Vol. 47 of Student Mathematical Library, American Mathematical Society.CrossRefGoogle Scholar
Fefferman, C. L. (2006), Existence and smoothness of the Navier–Stokes equation, in The Millennium Prize Problems (Carlson, J., Jaffe, A. and Wiles, A., eds), Clay Mathematics Institute, pp. 57–67.Google Scholar
Feichtinger, H. G. and Gröchenig, K. (1994), Theory and practice of irregular sampling, in Wavelets: Mathematics and Applications, CRC, pp. 305–363.Google Scholar
Feynman, R. P., Leighton, R. B. and Sands, M. (1963), The Feynman Lectures on Physics, Vol. 1: Mainly Mechanics, Radiation, and Heat, Addison-Wesley.Google Scholar
Fischer, C. F. (1977), The Hartree–Fock Method for Atoms, Wiley.Google Scholar
Fock, V. (1930), Näherungsmethode zur Lösung des quantenmechanischen Mehrkörper-problems, Z . Physik 61, 126–148.CrossRefGoogle Scholar
Fortnow, L. (2013), The Golden Ticket: P, NP, and the Search for the Impossible, Princeton University Press.CrossRefGoogle Scholar
Frank, A. and Tardos, E. (1987), An application of simultaneous Diophantine approximation in combinatorial optimization, Combinatorica 7, 49–65.CrossRefGoogle Scholar
Frazier, M. and Jawerth, B. (1990), A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93, 34–170.CrossRefGoogle Scholar
Friedberg, S. H., Insel, A. J. and Spence, L. E. (2003), Linear Algebra, fourth edition, Prentice Hall.Google Scholar
Friedland, S. (2013), On tensors of border rank l in Cm×n×l , Linear Algebra Appl. 438, 713–737.CrossRefGoogle Scholar
Friedland, S. and Gross, E. (2012), A proof of the set-theoretic version of the salmon conjecture, J. Algebra 356, 374–379.CrossRefGoogle Scholar
Friedland, S. and Lim, L.-H. (2018), Nuclear norm of higher-order tensors, Math. Comp. 87, 1255–1281.CrossRefGoogle Scholar
Friedland, S., Lim, L.-H. and Zhang, J. (2019), Grothendieck constant is norm of Strassen matrix multiplication tensor, Numer. Math. 143, 905–922.Google Scholar
Friedman, J. (1991), The spectra of infinite hypertrees, SIAM J. Comput. 20, 951–961.CrossRefGoogle Scholar
Friedman, J. and Wigderson, A. (1995), On the second eigenvalue of hypergraphs, Combinatorica 15, 43–65.CrossRefGoogle Scholar
Fuchs, F., Worrall, D. E., Fischer, V. and Welling, M. (2020), SE(3)-transformers: 3D roto-translation equivariant attention networks, in Advances in Neural Information Processing Systems 33 (NeurIPS 2020) (Larochelle, H. et al., eds), Curran Associates, pp. 1970–1981.Google Scholar
Fulton, W. and Harris, J. (1991), Representation Theory: A First Course, Vol. 129 of Graduate Texts in Mathematics, Springer.Google Scholar
Fürer, M. (2009), Faster integer multiplication, SIAM J. Comput. 39, 979–1005.CrossRefGoogle Scholar
García, A. A., Hehl, F. W., Heinicke, C. and Macías, A. (2004), The Cotton tensor in Riemannian spacetimes, Classical Quantum Gravity 21, 1099–1118.CrossRefGoogle Scholar
Garcke, J. (2013), Sparse grids in a nutshell, in Sparse Grids and Applications, Vol. 88 of Lecture Notes in Computational Science and Engineering, Springer, pp. 57–80.Google Scholar
Garey, M. R. and Johnson, D. S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman.Google Scholar
Garg, S., Gentry, C. and Halevi, S. (2013), Candidate multilinear maps from ideal lattices, in Advances in Cryptology (EUROCRYPT 2013), Vol. 7881 of Lecture Notes in Computer Science, Springer, pp. 1–17.Google Scholar
Garrett, P. (2010), Non-existence of tensor products of Hilbert spaces. Available at http://www-users.math.umn.edu/~garrett/m/v/nonexistence_tensors.pdf.Google Scholar
Gel′fand, I. M., Kapranov, M. M. and Zelevinsky, A. V. (1992), Hyperdeterminants, Adv . Math. 96, 226–263.Google Scholar
Gel′ fand, I. M., Kapranov, M. M. and Zelevinsky, A. V. (1994), Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser.CrossRefGoogle Scholar
Gentry, C., Gorbunov, S. and Halevi, S. (2015), Graph-induced multilinear maps from lattices, in Theory of Cryptography (TCC 2015), part II, Vol. 9015 of Lecture Notes in Computer Science, Springer, pp. 498–527.Google Scholar
Geroch, R. (1985), Mathematical Physics, Chicago Lectures in Physics, University of Chicago Press.Google Scholar
Gerstner, T. and Griebel, M. (1998), Numerical integration using sparse grids, Numer. Algorithms 18, 209–232.CrossRefGoogle Scholar
Golub, G. H. and Loan, C. F. Van (2013), Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, fourth edition, Johns Hopkins University Press.Google Scholar
Golub, G. H. and Welsch, J. H. (1969), Calculation of Gauss quadrature rules, Math. Comp. 23, 221–230, A1–A10.CrossRefGoogle Scholar
Golub, G. H. and Wilkinson, J. H. (1976), Ill-conditioned eigensystems and the computation of the Jordan canonical form, SIAM Rev. 18, 578–619.CrossRefGoogle Scholar
Goodman, R. and Wallach, N. R. (2009), Symmetry, Representations, and Invariants, Vol. 255 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Grafakos, L. (2014), Classical Fourier Analysis, Vol. 249 of Graduate Texts in Mathematics, third edition, Springer.CrossRefGoogle Scholar
Grafakos, L. and Torres, R. H. (2002a), Discrete decompositions for bilinear operators and almost diagonal conditions, Trans. Amer. Math. Soc. 354, 1153–1176.CrossRefGoogle Scholar
Grafakos, L. and Torres, R. H. (2002b), Multilinear Calderón–Zygmund theory, Adv . Math. 165, 124–164.Google Scholar
Greengard, L. and Rokhlin, V. (1987), A fast algorithm for particle simulations, J. Comput. Phys. 73, 325–348.CrossRefGoogle Scholar
Greub, W. (1978), Multilinear Algebra, Universitext, second edition, Springer.CrossRefGoogle Scholar
Griffiths, D. (2008), Introduction to Elementary Particles, second edition, Wiley-VCH.Google Scholar
Grothendieck, A. (1953), Résumé de la théorie métrique des produits tensoriels topologiques, Bol . Soc. Mat. São Paulo 8, 1–79.Google Scholar
Grothendieck, A. (1955), Produits Tensoriels Topologiques et Espaces Nucléaires, Vol. 16 of Memoirs of the American Mathematical Society, American Mathematical Society.CrossRefGoogle Scholar
Grover, L. K. (1996), A fast quantum mechanical algorithm for database search, in Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC 1996), ACM, pp. 212–219.Google Scholar
Hannabuss, K. (1997), An Introduction to Quantum Theory, Vol. 1 of Oxford Graduate Texts in Mathematics, The Clarendon Press, Oxford University Press.Google Scholar
Harris, J. (1995), Algebraic Geometry: A First Course, Vol. 133 of Graduate Texts in Mathematics, Springer.Google Scholar
Hartmann, E. (1984), An Introduction to Crystal Physics, Vol. 18 of Commission on Crystallographic Teaching: Second series pamphlets, International Union of Crystallography, University College Cardiff Press.Google Scholar
Hartree, D. R. (1928), The wave mechanics of an atom with a non-Coulomb central field, I: Theory and methods, Proc . Cambridge Philos. Soc. 24, 89–132.CrossRefGoogle Scholar
Hartshorne, R. (1977), Algebraic Geometry, Vol. 52 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Harvey, D. and van der Hoeven, J. (2021), Integer multiplication in time O(n log n), Ann. of Math. (2) 193, 563–617.CrossRefGoogle Scholar
Hassani, S. (1999), Mathematical Physics: A Modern Introduction to its Foundations, Springer.CrossRefGoogle Scholar
Hastie, T. J. and Tibshirani, R. J. (1990), Generalized Additive Models, Vol. 43 of Monographs on Statistics and Applied Probability, Chapman & Hall.Google Scholar
Hatcher, A. (2002), Algebraic Topology, Cambridge University Press.Google Scholar
Hauser, R. A. and Lim, Y. (2002), Self-scaled barriers for irreducible symmetric cones, SIAM J. Optim. 12, 715–723.CrossRefGoogle Scholar
Hay, G. E. (1954), Vector and Tensor Analysis, Dover Publications.Google Scholar
Heil, C. (2011), A Basis Theory Primer, Applied and Numerical Harmonic Analysis, expanded edition, Birkhäuser/Springer.Google Scholar
Heiss, F. and Winschel, V. (2008), Likelihood approximation by numerical integration on sparse grids, J. Econometrics 144, 62–80.CrossRefGoogle Scholar
Helgason, S. (1978), Differential Geometry, Lie Groups, and Symmetric Spaces, Vol. 80 of Pure and Applied Mathematics, Academic Press.Google Scholar
Helton, J. W. and Putinar, M. (2007), Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization, in Operator Theory, Structured Matrices, and Dilations, Vol. 7 of Theta Ser. Adv. Math., Theta, Bucharest, pp. 229–306.Google Scholar
Higham, N. J. (1992), Stability of a method for multiplying complex matrices with three real matrix multiplications, SIAM J. Matrix Anal. Appl. 13, 681–687.CrossRefGoogle Scholar
Higham, N. J. (2002), Accuracy and Stability of Numerical Algorithms, second edition, Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Hillar, C. J. and Lim, L.-H. (2013), Most tensor problems are NP-hard, J. Assoc. Comput. Mach. 60, 45.CrossRefGoogle Scholar
Hitchcock, F. L. (1927), The expression of a tensor or a polyadic as a sum of products, J. Math. Phys. Mass. Inst. Tech. 6, 164–189.Google Scholar
Hochbaum, D. S. and Shanthikumar, J. G. (1990), Convex separable optimization is not much harder than linear optimization, J. Assoc. Comput. Mach. 37, 843–862.CrossRefGoogle Scholar
Hofmann, T., Schölkopf, B. and Smola, A. J. (2008), Kernel methods in machine learning, Ann. Statist. 36, 1171–1220.CrossRefGoogle Scholar
Höllig, K. and Hörner, J. (2013), Approximation and Modeling with B-Splines, Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Holub, J. R. (1970), Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151, 563–579.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (1994), Topics in Matrix Analysis, Cambridge University Press.Google Scholar
Horn, R. A. and Johnson, C. R. (2013), Matrix Analysis, second edition, Cambridge University Press.Google Scholar
Irgens, F. (2019), Tensor Analysis, Springer.CrossRefGoogle Scholar
Jackson, J. D. (1999), Classical Electrodynamics, third edition, Wiley.Google Scholar
Jacobson, N. (1975), Lectures in Abstract Algebra, Vol II: Linear Algebra, Vol. 31 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Jaeger, J. C. (1940), The solution of boundary value problems by a double Laplace transformation, Bull. Amer. Math. Soc. 46, 687–693.CrossRefGoogle Scholar
Johnson, D. S. (2012), A brief history of NP-completeness, 1954–2012, Doc. Math. pp. 359–376.Google Scholar
Jordan, T. F. (2005), Quantum Mechanics in Simple Matrix Form, Dover Publications.Google Scholar
Joux, A. (2004), A one round protocol for tripartite Diffie–Hellman, J. Cryptology 17, 263–276.CrossRefGoogle Scholar
Kaarnioja, V. (2013), Smolyak quadrature. Master’s thesis, University of Helsinki, Finland.Google Scholar
Kalnins, E. G., Kress, J. M. and Miller, W. Jr. (2018), Separation of Variables and Superin-tegrability: The Symmetry of Solvable Systems , IOP Expanding Physics, IOP Publishing.Google Scholar
Kanwal, R. P. (1997), Linear Integral Equations, second edition, Birkhäuser.CrossRefGoogle Scholar
Karatsuba, A. A. and Ofman, Y. (1962), Multiplication of many-digital numbers by automatic computers, Dokl. Akad. Nauk SSSR 145, 293–294.Google Scholar
Karmarkar, N. (1984), A new polynomial-time algorithm for linear programming, Combinatorica 4, 373–395.CrossRefGoogle Scholar
Kearsley, E. A. and Fong, J. T. (1975), Linearly independent sets of isotropic Cartesian tensors of ranks up to eight, J. Res. Nat. Bur. Standards B 79, 49–58.CrossRefGoogle Scholar
Keshavarzzadeh, V., Kirby, R. M. and Narayan, A. (2018), Numerical integration in multiple dimensions with designed quadrature, SIAM J. Sci. Comput. 40, A2033–A2061.CrossRefGoogle Scholar
Khachiyan, L. (1979), A polynomial algorithm in linear programming, Soviet Math. Dokl. 20, 191–194.Google Scholar
Khachiyan, L. (1996), Diagonal matrix scaling is NP-hard, Linear Algebra Appl. 234, 173–179.CrossRefGoogle Scholar
Khanna, S., Linial, N. and Safra, S. (2000), On the hardness of approximating the chromatic number, Combinatorica 20, 393–415.CrossRefGoogle Scholar
Kleinjung, T. and Wesolowski, B. (2021), Discrete logarithms in quasi-polynomial time in finite fields of fixed characteristic, J. Amer. Math. Soc. doi:10.1090/jams/985.CrossRefGoogle Scholar
Kljuev, V. V. and Kokovkin-Ščerbak, N. I. (1965), On the minimization of the number of arithmetic operations for solving linear algebraic systems of equations, Ž. Vyčisl. Mat i Mat. Fiz. 5, 21–33.Google Scholar
Klyachko, A. A. (1998), Stable bundles, representation theory and Hermitian operators, Selecta Math . (N.S.) 4, 419–445.CrossRefGoogle Scholar
Knuth, D. E. (1998), The Art of Computer Programming , Vol. 2: Seminumerical Algorithms, third edition, Addison-Wesley.Google Scholar
Knutson, A. and Tao, T. (1999), The honeycomb model of GL n (C) tensor products I: Proof of the saturation conjecture, J. Amer. Math. Soc. 12, 1055–1090.CrossRefGoogle Scholar
Kondor, R. and Trivedi, S. (2018), On the generalization of equivariance and convolution in neural networks to the action of compact groups, in Proceedings of the 35th International Conference on Machine Learning (ICML 2018) (J. Dy and A. Krause, eds), Vol. 80 of Proceedings of Machine Learning Research, PMLR, pp. 2747–2755.Google Scholar
Kondor, R., Lin, Z. and Trivedi, S. (2018), Clebsch–Gordan nets: A fully Fourier space spherical convolutional neural network, in Advances in Neural Information Processing Systems 31 (NeurIPS 2018) (S. Bengio et al., eds), Curran Associates, pp. 10138–10147.Google Scholar
Koornwinder, T. H. (1980), A precise definition of separation of variables, in Geometrical Approaches to Differential Equations (Proc. Fourth Scheveningen Conference, 1979), Vol. 810 of Lecture Notes in Mathematics, Springer, pp. 240–263.Google Scholar
Kostoglou, M. (2005), On the analytical separation of variables solution for a class of partial integro-differential equations, Appl. Math. Lett. 18, 707–712.CrossRefGoogle Scholar
Kostrikin, A. I. and Manin, Y. I. (1997), Linear Algebra and Geometry, Vol. 1 of Algebra, Logic and Applications, Gordon & Breach Science Publishers.Google Scholar
Krantz, S. G. (2009), Explorations in Harmonic Analysis: With Applications to Complex Function Theory and the Heisenberg Group, Applied and Numerical Harmonic Analysis, Birkhäuser.CrossRefGoogle Scholar
Krishna, S. and Makam, V. (2018), On the tensor rank of 3 × 3 permanent and determinant. Available at arXiv:1801.00496.Google Scholar
Krivine, J.-L. (1979), Constantes de Grothendieck et fonctions de type positif sur les sphères, Adv . Math. 31, 16–30.Google Scholar
Krizhevsky, A., Sutskever, I. and Hinton, G. E. (2012), Imagenet classification with deep convolutional neural networks, in Advances in Neural Information Processing Systems 25 (NIPS 2012) (Pereira, F. et al., eds), Curran Associates, pp. 1097–1105.Google Scholar
Lacey, M. and Thiele, C. (1997), L p estimates on the bilinear Hilbert transform for 2 < p < ∞, Ann. of Math. (2) 146, 693–724.CrossRefGoogle Scholar
Lacey, M. and Thiele, C. (1999), On Calderón’s conjecture, Ann. of Math. (2) 149, 475–496.CrossRefGoogle Scholar
Landsberg, J. M. (2006), The border rank of the multiplication of 2 × 2 matrices is seven, J. Amer. Math. Soc. 19, 447–459.CrossRefGoogle Scholar
Landsberg, J. M. (2012), Tensors: Geometry and Applications, Vol. 128 of Graduate Studies in Mathematics, American Mathematical Society.CrossRefGoogle Scholar
Landsberg, J. M. (2017), Geometry and Complexity Theory, Vol. 169 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
Landsberg, J. M. (2019), Tensors: Asymptotic Geometry and Developments 2016–2018, Vol. 132 of CBMS Regional Conference Series in Mathematics, American Mathematical Society.CrossRefGoogle Scholar
Landsberg, J. M., Qi, Y. and Ye, K. (2012), On the geometry of tensor network states, Quantum Inform. Comput. 12, 346–354.CrossRefGoogle Scholar
Landsman, K. (2017), Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Vol. 188 of Fundamental Theories of Physics, Springer.CrossRefGoogle Scholar
Lang, S. (1993), Real and Functional Analysis, Vol. 142 of Graduate Texts in Mathematics, third edition, Springer.Google Scholar
Lang, S. (2002), Algebra , Vol. 211 of Graduate Texts in Mathematics, third edition, Springer.Google Scholar
Lax, P. D. (2007), Linear Algebra and its Applications, Pure and Applied Mathematics (Hoboken), second edition, Wiley-Interscience.Google Scholar
Lee, J. M. (2013), Introduction to Smooth Manifolds, Vol. 218 of Graduate Texts in Mathematics, second edition, Springer.CrossRefGoogle Scholar
Light, W. A. and Cheney, E. W. (1985), Approximation Theory in Tensor Product Spaces, Vol. 1169 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Lovelock, D. and Rund, H. (1975), Tensor, Differential Forms, and Variational Principles, Wiley-Interscience.Google Scholar
Lovett, S. (2019), The analytic rank of tensors and its applications, Discrete Anal. 2019, 7.Google Scholar
Mackey, G. W. (1978), Unitary Group Representations in Physics, Probability, and Number Theory, Vol. 55 of Mathematics Lecture Note Series, Benjamin / Cummings.Google Scholar
Mallat, S. (2009), A Wavelet Tour of Signal Processing, third edition, Elsevier / Academic Press.Google Scholar
Martin, D. (1991), Manifold Theory: An Introduction for Mathematical Physicists, Ellis Horwood Series in Mathematics and its Applications, Ellis Horwood.Google Scholar
McConnell, A. J. (1957), Application of Tensor Analysis, Dover Publications.Google Scholar
McCullagh, P. (1987), Tensor Methods in Statistics, Monographs on Statistics and Applied Probability, Chapman & Hall.Google Scholar
McCulloch, W. S. and Pitts, W. (1943), A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys. 5, 115–133.CrossRefGoogle Scholar
Mercer, J. (1909), Functions of positive and negative type and their connection with the theory of integral equations, Philos. Trans. Roy. Soc. London Ser. A 209, 415–446.Google Scholar
Meyer, Y. (1992), Wavelets and Operators, Vol. 37 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
Meyer, Y. and Coifman, R. (1997), Wavelets: Calderón–Zygmund and Multilinear Operators, Vol. 48 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
Michal, A. D. (1947), Matrix and Tensor Calculus with Applications to Mechanics, Elasticity, and Aeronautics, Wiley / Chapman & Hall.Google Scholar
Miller, W. Jr (1977), Symmetry and Separation of Variables, Vol. 4 of Encyclopedia of Mathematics and its Applications, Addison-Wesley.Google Scholar
Milnor, J. W. and Stasheff, J. D. (1974), Characteristic Classes, Vol. 76 of Annals of Mathematics Studies, Princeton University Press / University of Tokyo Press.CrossRefGoogle Scholar
Misner, C. W., Thorne, K. S. and Wheeler, J. A. (1973), Gravitation, W. H. Freeman.Google Scholar
Moon, P. and Spencer, D. E. (1961), Field Theory for Engineers, The Van Nostrand Series in Electronics and Communications, Van Nostrand.Google Scholar
Moon, P. and Spencer, D. E. (1988), Field Theory Handbook: Including Coordinate Systems , Differential Equations and Their Solutions, second edition, Springer.Google Scholar
Morse, P. M. and Feshbach, H. (1953), Methods of Theoretical Physics, McGraw-Hill.Google Scholar
Murty, K. G. and Kabadi, S. N. (1987), Some NP-complete problems in quadratic and nonlinear programming, Math. Program. 39, 117–129.CrossRefGoogle Scholar
Muscalu, C. and Schlag, W. (2013), Classical and Multilinear Harmonic Analysis II, Vol. 138 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
Nakahara, M. and Ohmi, T. (2008), Quantum Computing: From Linear Algebra to Physical Realizations, CRC Press.CrossRefGoogle Scholar
Narayanan, H. (2006), On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients, J. Algebraic Combin. 24, 347–354.CrossRefGoogle Scholar
Nesterov, Y. and Nemirovskii, A. (1994), Interior-Point Polynomial Algorithms in Convex Programming, Vol. 13 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Nie, J. (2017), Generating polynomials and symmetric tensor decompositions, Found . Comput. Math. 17, 423–465.Google Scholar
Niederreiter, H. (1992), Random Number Generation and Quasi-Monte Carlo Methods, Vol. 63 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM).Google Scholar
Nielsen, M. A. and Chuang, I. L. (2000), Quantum Computation and Quantum Information, Cambridge University Press.Google Scholar
Niyogi, P., Smale, S. and Weinberger, S. (2008), Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39, 419–441.CrossRefGoogle Scholar
O’Meara, K. C., Clark, J. and Vinsonhaler, C. I. (2011), Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form, Oxford University Press.Google Scholar
Orús, R. (2014), A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117–158.CrossRefGoogle Scholar
Osgood, B. G. (2019), Lectures on the Fourier Transform and its Applications, Vol. 33 of Pure and Applied Undergraduate Texts, American Mathematical Society.Google Scholar
Parlett, B. N. (2000), The QR algorithm, Comput . Sci. Eng. 2, 38–42.Google Scholar
Passman, D. S. (1991), A Course in Ring Theory, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software.Google Scholar
Pauli, W. (1980), General Principles of Quantum Mechanics, Springer.CrossRefGoogle Scholar
Pisier, G. (2012), Grothendieck’s theorem, past and present, Bull. Amer. Math. Soc. (N.S.) 49, 237–323.CrossRefGoogle Scholar
Plebański, J. and Krasiński, A. (2006), An Introduction to General Relativity and Cosmology, Cambridge University Press.CrossRefGoogle Scholar
Rao, K. R. and Yip, P. (1990), Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press.CrossRefGoogle Scholar
Reed, M. and Simon, B (1980), Functional Analysis, Vol. I of Methods of Modern Mathematical Physics, second edition, Academic Press.Google Scholar
Renegar, J. (2001), A Mathematical View of Interior-Point Methods in Convex Optimization, MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM) / Mathematical Programming Society (MPS).CrossRefGoogle Scholar
Reznick, B. (1992), Sums of Even Powers of Real Linear Forms, Vol. 96 of Memoirs of the American Mathematical Society, American Mathematical Society.CrossRefGoogle Scholar
Ricci, M. M. G. and Levi-Civita, T. (1900), Méthodes de calcul différentiel absolu et leurs applications, Math. Ann. 5, 125–201. Translation by R. Hermann: Ricci and Levi-Civita’s Tensor Analysis Paper (1975), Vol. II of Lie Groups: History, Frontiers and Applications, Math Sci Press.Google Scholar
Riehl, E. (2016), Category Theory in Context, Dover Publications.Google Scholar
Rindler, W. (2006), Relativity: Special, General, and Cosmological, second edition, Oxford University Press.Google Scholar
Roberts, M. D. (1995), The physical interpretation of the Lanczos tensor, Nuovo Cimento B (11) 110, 1165–1176.CrossRefGoogle Scholar
Rockmore, D. N. (2000), The FFT: An algorithm the whole family can use, Comput . Sci. Eng. 2, 30–34.Google Scholar
Rosenblatt, F. (1958), The perceptron: A probabilistic model for information storage and organization in the brain, Psychol. Rev. 65, 386–408.CrossRefGoogle Scholar
Ryan, R. A. (2002), Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer.CrossRefGoogle Scholar
Schaefer, H. H. and Wolff, M. P. (1999), Topological Vector Spaces, Vol. 3 of Graduate Texts in Mathematics, second edition, Springer.CrossRefGoogle Scholar
Schatten, R. (1950), A Theory of Cross-Spaces, Vol. 26 of Annals of Mathematics Studies, Princeton University Press.Google Scholar
Scheffé, H. (1970), A note on separation of variables, Technometrics 12, 388–393.Google Scholar
Scheffé, H. (1999), The Analysis of Variance, Wiley Classics Library, Wiley.Google Scholar
Schmüdgen, K. (2017), The Moment Problem, Vol. 277 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Schöbel, K. (2015), An Algebraic Geometric Approach to Separation of Variables, Springer Spektrum.CrossRefGoogle Scholar
Scholköpf, B. and Smola, A. J. (2002), Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press.Google Scholar
Schönhage, A. (1972/73), Unitäre Transformationen grosser Matrizen, Numer . Math. 20, 409–417.Google Scholar
Schönhage, A. and Strassen, V. (1971), Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7, 281–292.Google Scholar
Schouten, J. A. (1951), Tensor Analysis for Physicists, Clarendon Press.Google Scholar
Schrijver, A. (1986), Theory of Linear and Integer Programming, Wiley-Interscience Series in Discrete Mathematics, Wiley-Interscience.Google Scholar
R. F. Service (2020), ‘The game has changed’: AI triumphs at protein folding, Science 370 (6521), 1144–1145.CrossRefGoogle Scholar
Shi, Y. Y., Duan, L. M. and Vidal, G. (2006), Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A 74, 022320.CrossRefGoogle Scholar
Shor, P. W. (1994), Algorithms for quantum computation: Discrete logarithms and factoring, in 35th Annual Symposium on Foundations of Computer Science (1994), IEEE Computer Society Press, pp. 124–134.CrossRefGoogle Scholar
Sickel, W. and Ullrich, T. (2009), Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross, J. Approx. Theory 161, 748–786.CrossRefGoogle Scholar
Simmonds, J. G. (1994), A Brief on Tensor Analysis, Undergraduate Texts in Mathematics, second edition, Springer.CrossRefGoogle Scholar
Skiena, S. S. (2020), The Algorithm Design Manual, Texts in Computer Science, third edition, Springer.CrossRefGoogle Scholar
Smale, S. (1998), Mathematical problems for the next century, Math. Intelligencer 20, 7–15.CrossRefGoogle Scholar
Smolyak, S. A. (1963), Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148, 1042–1045.Google Scholar
Sokolov, N. P. (1960), Spatial Matrices and their Applications, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow. In Russian.Google Scholar
Sokolov, N. P. (1972), Introduction to the Theory of Multidimensional Matrices, Izdat. ‘Naukova Dumka’, Kiev. In Russian.Google Scholar
Spain, B. (1960), Tensor Calculus: A Concise Course, third edition, Oliver and Boyd / Interscience.Google Scholar
Stein, E. M. (1993), Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscil-latory Integrals, Vol. 43 of Princeton Mathematical Series, Princeton University Press.CrossRefGoogle Scholar
Steinwart, I. and Christmann, A. (2008), Support Vector Machines, Information Science and Statistics, Springer.Google Scholar
Stewart, G. W. (2000), The decompositional approach to matrix computation, Comput . Sci. Eng. 2, 50–59.Google Scholar
Strang, G. (1980), Linear Algebra and its Applications, second edition, Academic Press.Google Scholar
Strassen, V. (1969), Gaussian elimination is not optimal, Numer. Math. 13, 354–356.CrossRefGoogle Scholar
Strassen, V. (1973), Vermeidung von Divisionen, J. Reine Angew. Math. 264, 184–202.Google Scholar
Strassen, V. (1987), Relative bilinear complexity and matrix multiplication, J. Reine Angew. Math. 375/376, 406–443.Google Scholar
Strassen, V. (1990), Algebraic complexity theory, in Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity, Elsevier, pp. 633–672.Google Scholar
Stuart, A. M. and Humphries, A. R. (1996), Dynamical Systems and Numerical Analysis, Vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Sylvester, J. J. (1886), Sur une extension d’un théorème de Clebsch relatif aux courbes du quatrième degré, C. R. Math. Acad. Sci. Paris 102, 1532–1534.Google Scholar
Synge, J. L. and Schild, A. (1978), Tensor Calculus, Dover Publications.Google Scholar
Tai, C.-T. (1997), Generalized Vector and Dyadic Analysis, second edition, IEEE Press.CrossRefGoogle Scholar
Takesaki, M. (2002), Theory of Operator Algebras I, Vol. 124 of Encyclopaedia of Mathematical Sciences, Springer.CrossRefGoogle Scholar
Takhtajan, L. A. (2008), Quantum Mechanics for Mathematicians, Vol. 95 of Graduate Studies in Mathematics, American Mathematical Society.CrossRefGoogle Scholar
Tardos, E. (1986), A strongly polynomial algorithm to solve combinatorial linear programs, Oper. Res. 34, 250–256.CrossRefGoogle Scholar
Temple, G. (1960), Cartesian Tensors: An Introduction, Methuen’s Monographs on Physical Subjects, Methuen / Wiley.Google Scholar
Teschl, G. (2014), Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Vol. 157 of Graduate Studies in Mathematics, second edition, American Mathematical Society.Google Scholar
Thomas, J. W. (1995), Numerical Partial Differential Equations: Finite Difference Methods, Vol. 22 of Texts in Applied Mathematics, Springer.CrossRefGoogle Scholar
Thorne, K. S. and Blandford, R. D. (2017), Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, Princeton University Press.Google Scholar
Toom, A. L. (1963), The complexity of a scheme of functional elements simulating the multiplication of integers, Dokl . Akad. Nauk SSSR 150, 496–498.Google Scholar
Trefethen, L. N. and Bau, D. III (1997), Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Trèves, F. (2006), Topological Vector Spaces, Distributions and Kernels, Dover Publications.Google Scholar
Vaidya, P. M. (1990), An algorithm for linear programming which requires O(((m + n)n 2 + (m + n)1.5 n)L) arithmetic operations, Math. Program. 47, 175–201.CrossRefGoogle Scholar
Valiant, L. G. (1979), The complexity of computing the permanent, Theoret . Comput. Sci. 8, 189–201.Google Scholar
van der Vorst, H. A. (2000), Krylov subspace iteration, Comput . Sci. Eng. 2, 32–37.Google Scholar
Varopoulos, N. T. (1965), Sur les ensembles parfaits et les séries trigonométriques, C. R. Acad. Sci. Paris 260, 4668–4670, 5165–5168, 5997–6000.Google Scholar
Varopoulos, N. T. (1967), Tensor algebras and harmonic analysis, Acta Math. 119, 51–112.CrossRefGoogle Scholar
Vavasis, S. A. (1991), Nonlinear Optimization: Complexity Issues, Vol. 8 of International Series of Monographs on Computer Science, The Clarendon Press, Oxford University Press.Google Scholar
Verstraete, F. and Cirac, J. I. (2004), Renormalization algorithms for quantum-many body systems in two and higher dimensions. Available at arXiv:cond-mat/0407066.Google Scholar
Vinberg, E. B. (2003), A Course in Algebra, Vol. 56 of Graduate Studies in Mathematics, American Mathematical Society.CrossRefGoogle Scholar
Voigt, W. (1898), Die fundamentalen physikalischen Eigenschaften der Krystalle in ele-mentarer Darstellung , Von Veit.CrossRefGoogle Scholar
Wald, R. M. (1984), General Relativity, University of Chicago Press.CrossRefGoogle Scholar
Wallach, N. R. (2008), Quantum computing and entanglement for mathematicians, in Representation Theory and Complex Analysis, Vol. 1931 of Lecture Notes in Mathematics, Springer, pp. 345–376.Google Scholar
Walter, W. (1998), Ordinary Differential Equations, Vol. 182 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
Wazwaz, A.-M. (2011), Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press (Beijing) / Springer.CrossRefGoogle Scholar
Weinreich, G. (1998), Geometrical Vectors, Chicago Lectures in Physics, University of Chicago Press.CrossRefGoogle Scholar
Weyl, H. (1997), The Classical Groups: Their Invariants and Representations, Princeton Landmarks in Mathematics, Princeton University Press.Google Scholar
White, S. R. (1992), Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863–2866.CrossRefGoogle ScholarPubMed
White, S. R. and Huse, D. A. (1993), Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S = 1 Heisenberg chain, Phys. Rev. B 48, 3844–3853.CrossRefGoogle ScholarPubMed
Whitfield, J. D., Love, P. J. and Aspuru-Guzik, A. (2013), Computational complexity in electronic structure, Phys . Chem. Chem. Phys. 15, 397–411.CrossRefGoogle Scholar
Woodhouse, N. M. J. (2003), Special Relativity, Springer Undergraduate Mathematics Series, Springer.CrossRefGoogle Scholar
Wrede, R. C. (1963), Introduction to Vector and Tensor Analysis, Wiley.Google Scholar
Yau, S.-T. (2020), Shiing-Shen Chern: A great geometer of 20th century. https://cmsa.fas.harvard.edu/wp-content/uploads/2020/05/2020-04-22-essay-on-Chern-english-v1.pdf.CrossRefGoogle Scholar
Ye, K. and Lim, L.-H. (2018a), Fast structured matrix computations: Tensor rank and Cohn–Umans method, Found . Comput. Math. 18, 45–95.Google Scholar
Ye, K. and Lim, L.-H. (2018b), Tensor network ranks. Available at arXiv:1801.02662.Google Scholar