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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 25, 2015

The Algebraic Formulation: Why and How to Use it

Elena Ferretti

Abstract

Finite Element, Boundary Element, Finite Volume, and Finite Difference Analysis are all commonly used in nearly all engineering disciplines to simplify complex problems of geometry and change, but they all tend to oversimplify. This paper shows a more recent computational approach developed initially for problems in solid mechanics and electro-magnetic field analysis. It is an algebraic approach, and it offers a more accurate representation of geometric and topological features.

References

[1] Newton I., Philosophiae Naturalis Principia Mathematica, 1687.10.5479/sil.52126.39088015628399Search in Google Scholar

[2] Ferretti E., The Cell Method as a Case of Bialgebra, Mathematics and Computers in Science and Engineering Series Nº 34 - Recent Advances in Applied Mathematics, Modelling and Simulation: Proceedings of the 8th International Conference on Applied Mathematics, Simulation, Modelling (ASM ‘14), WSEAS Press, Athens (Greece), 322-331, 2014.Search in Google Scholar

[3] Ferretti E., Similarities between Cell Method and Non-Standard Calculus, Mathematics and Computers in Science and Engineering Series Nº 39 - Recent Advances in Computational Mathematics: Proceedings of the 3rd International Conference on Applied and Computational Mathematics (ICACM ’14), WSEAS Press, Athens (Greece), 110-115, 2014.Search in Google Scholar

[4] Bellina F., Bettini P., Tonti E., Trevisan F., Finite Formulation for the Solution of a 2D Eddy-Current Problem, IEEE Transaction on Magnetics, 2002, 38(2), 561-564.10.1109/20.996147Search in Google Scholar

[5] Ferretti E., Modellazione del Comportamento del Cilindro Fasciato in Compressione, Ph.D. Thesis (in Italian), University of Lecce, Lecce (Italy), 2001.Search in Google Scholar

[6] Ferretti E., Crack Propagation Modeling by Remeshing using the Cell Method (CM), CMES: Comput. Model. Eng. Sci., 2003, 4(1), 51-72.Search in Google Scholar

[7] Ferretti E., Crack-Path Analysis for Brittle and Non-Brittle Cracks: A Cell Method Approach, CMES: Comput. Model. Eng. Sci., 2004, 6(3), 227-244.Search in Google Scholar

[8] Ferretti E., A Cell Method (CM) Code for Modeling the Pullout Test Step-Wise, CMES: Comput. Model. Eng. Sci., 2004, 6(5), 453-476.Search in Google Scholar

[9] Ferretti E., A Discrete Nonlocal Formulation using Local Constitutive Laws, Int. J. Fracture, 2004, 130(3), L175-L182.10.1007/s10704-004-2588-1Search in Google Scholar

[10] Ferretti E., Modeling of the Pullout Test through the Cell Method, In G. C. Sih, L. Nobile, RRRTEA - International Conference of Restoration, Recycling and Rejuvenation Technology for Engineering and Architecture Application, Aracne, 180–192, 2004.Search in Google Scholar

[11] Ferretti E., A Local Strictly Nondecreasing Material Law for Modeling Softening and Size-Effect: A Discrete Approach, CMES: Comput. Model. Eng. Sci., 2005, 9(1), 19-48.Search in Google Scholar

[12] Ferretti E., On nonlocality and locality: Differential and discrete formulations, 11th International Conference on Fracture - ICF11, 2005, 3, 1728-1733.Search in Google Scholar

[13] Ferretti E., Cell Method Analysis of Crack Propagation in Tensioned Concrete Plates, CMES: Comput. Model. Eng. Sci., 2009, 54(3), 253-282.Search in Google Scholar

[14] Ferretti E., The Cell Method: An Enriched Description of Physics Starting from the Algebraic Formulation, CMC: Comput. Mater. Con., 2013, 36(1), 49-72.Search in Google Scholar

[15] Ferretti E., A Cell Method Stress Analysis in Thin Floor Tiles Subjected to Temperature Variation, CMC: Comput. Mater. Con., 2013, 36(3), 293-322.Search in Google Scholar

[16] Ferretti E., The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering. Momentum Press, New York, 2014.Search in Google Scholar

[17] Ferretti E., The Assembly Process for Enforcing Equilibriumand Compatibilitywith the CM: a Coboundary Process, CMES: Comput. Model. Eng. Sci., (in press).Search in Google Scholar

[18] Ferretti E., The Mathematical Foundations of the Cell Method, International Journal of Mathematical Models and Methods in Applied Sciences, (submitted).Search in Google Scholar

[19] Ferretti E., Some new Findings on the Mathematical Structure of the Cell Method, International Journal of Mathematical Models and Methods in Applied Sciences, (submitted).Search in Google Scholar

[20] Ferretti E., The Cell Method: An Overview on the Main Features, Curved and Layer. Struct., 2015, 2, 194-243.10.1515/cls-2015-0011Search in Google Scholar

[21] Ferretti E., Casadio E., Di Leo A., Masonry Walls under Shear Test: A CM Modeling, CMES: Comput. Model. Eng. Sci., 2008, 30(3), 163-190.Search in Google Scholar

[22] Ferretti E., Di Leo A., Modelling of Compressive Tests on FRP Wrapped Concrete Cylinders through a Novel Triaxial Concrete Constitutive Law, SITA: Scientific Israel – Technological Advantages, 2003, 5, 20-43.Search in Google Scholar

[23] Ferretti E., Di Leo A., Viola E., Computational Aspects and Numerical Simulations in the Elastic Constants Identification, CISM Courses and Lectures Nº 471 - Problems in Structural Identification and Diagnostic: General Aspects and Applications, Springer, Wien – New York, 133–147, 2003.10.1007/978-3-7091-2536-6_10Search in Google Scholar

[24] Freschi F., Giaccone L., Repetto M., Educational value of the algebraic numerical methods in electromagnetism, COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2008, 27(6), 1343-1357.10.1108/03321640810905828Search in Google Scholar

[25] Marrone M., Rodrìguez-Esquerre V.F., Hernàndez-Figueroa H.E., Novel Numerical Method for the Analysis of 2D Photonic Crystals: the Cell Method, Opt. Express, 2002, 10(22), 1299-1304.10.1364/OE.10.001299Search in Google Scholar

[26] Matoušek J., Lectures in Discrete Geometry, Graduate Texts in Mathematics 212, Springer, 2002.10.1007/978-1-4613-0039-7Search in Google Scholar

[27] Mattiussi C., An Analysis of Finite Volume, Finite Element, and Finite Difference Methods using some Concepts from Algebraic Topology, J. Comput. Phys., 1997, 133, 289-309.10.1006/jcph.1997.5656Search in Google Scholar

[28] Mattiussi C., The Finite Volume, Finite Difference, and Finite Elements Methods as Numerical Methods for Physical Field Problems, In: P. Hawkes (Ed.), Advances in Imaging and Electron Physics, 113, 1-146, 2000.10.1016/S1076-5670(00)80012-9Search in Google Scholar

[29] Nappi A., Rajgelj S., Zaccaria D., Application of the Cell Method to the Elastic-Plastic Analysis, Proc. Plasticity ‘97, 1997, 14-18.Search in Google Scholar

[30] Nappi A., Rajgelj S., Zaccaria D., A Discrete Formulation Applied to Crack Growth Problem, In: G. G. Sih (Ed.), Mesomechanics 2000, Tsinghua University Press, Beijing, P. R. China, 395-406, 2000.Search in Google Scholar

[31] Nappi A., Tin-Loi F., A discrete formulation for the numerical analysis of masonry structures, In: Wang C.M., Lee K.H., Ang K.K. (Eds.), Computational Mechanics for the Next Millennium, Elsevier, Singapore, 81-86, 1999.Search in Google Scholar

[32] Nappi A., Tin-Loi F., A Numerical Model for Masonry Implemented in the Framework of a Discrete Formulation, Struct. Eng. Mech., 2001, 11(2), 171-184.10.12989/sem.2001.11.2.171Search in Google Scholar

[33] Pani M., Taddei F., The Cell Method: Quadratic Interpolation with Tetraedra for 3D Scalar Fields, CMES: Comput. Model. Eng. Sci., 2013, 94(4), 279-300.Search in Google Scholar

[34] Tonti E., On the Mathematical Structure of a Large Class of Physical Theories, Rend. Acc. Lincei, 1972, 52, 48-56.Search in Google Scholar

[35] Tonti E., The Algebraic - Topological Structure of Physical Theories, Conference on Symmetry, Similarity and Group Theoretic Methods in Mechanics, Calgary (Canada), 1974, 441-467.Search in Google Scholar

[36] Tonti E., On the formal structure of physical theories, Monograph of the Italian National Research Council, 1975.Search in Google Scholar

[37] Tonti E., The Reason for Analogies between Physical Theories, Appl. Math. Modelling, 1976, 1, 37-50.10.1016/0307-904X(76)90023-8Search in Google Scholar

[38] Tonti E., On the Geometrical Structure of the Electromagnetism, In: G. Ferrarese (Ed.), Gravitation, Electromagnetism and Geometrical Structures, for the 80th birthday of A. Lichnerowicz, Pitagora, Bologna, 281-308, 1995.Search in Google Scholar

[39] Tonti E., Algebraic Topology and Computational Electromagnetism, Fourth International Worksop on the Electric and Magnetic Field: from Numerical Models to industrial Applications, Marseille, 1998, 284-294.Search in Google Scholar

[40] Tonti E., A Direct Discrete Formulation of Field Laws: the Cell Method, CMES: Comput. Model. Eng. Sci., 2001, 2(2), 237-258.Search in Google Scholar

[41] Tonti E., A Direct Discrete Formulation for the Wave Equation, J. Comput. Acoust., 2001, 9(4), 1355-1382.10.1142/S0218396X01001455Search in Google Scholar

[42] Tonti E., Finite Formulation of the Electromagnetic Field, Progress in Electromagnetics Research, PIER 32 (Special Volume on Geometrical Methods for Comp. Electromagnetics), 2001, 1-44.10.2528/PIER00080101Search in Google Scholar

[43] Tonti E., Finite Formulation of the Electromagnetic Field, International COMPUMAG Society Newsletter, 2001, 8(1), 5-11.10.2528/PIER00080101Search in Google Scholar

[44] Tonti E., Finite Formulation of the Electromagnetic Field, IEE Transactions on Magnetics, 2002, 38(2), 333-336.10.1109/20.996090Search in Google Scholar

[45] Tonti E., The Mathematical Structure of Classical and Relativistic Physics, Birkhäuser, 2013.10.1007/978-1-4614-7422-7Search in Google Scholar

[46] Tonti E., Zarantonello F., Algebraic Formulation of Elastostatics: the Cell Method, CMES: Comput. Model. Eng. Sci., 2009, 39(3), 201-236.Search in Google Scholar

[47] Tonti E., Zarantonello F., Algebraic Formulation of Elastodynamics: the Cell Method, CMES: Comput. Model. Eng. Sci., 2010, 64(1), 37-70.Search in Google Scholar

[48] Viola E., Tornabene F., Ferretti E., Fantuzzi N., Soft Core Plane State Structures Under Static Loads Using GDQFEM and Cell Method, CMES: Comput. Model. Eng. Sci., 2013, 94(4), 301-329.Search in Google Scholar

[49] Viola E., Tornabene F., Ferretti E., Fantuzzi N., GDQFEM Numerical Simulations of Continuous Mediawith Cracks and Discontinuities, CMES: Comput. Model. Eng. Sci., 2013, 94(4), 331-369.Search in Google Scholar

[50] Viola E., Tornabene F., Ferretti E., Fantuzzi N., On Static Analysis of Composite Plane State Structures via GDQFEM and Cell Method, CMES: Comput. Model. Eng. Sci., 2013, 94(5), 421-458.Search in Google Scholar

[51] Bott R., Tu L.W., Differential Forms in Algebraic Topology, Springer-Verlag, Berlin, New York, 1982.10.1007/978-1-4757-3951-0Search in Google Scholar

[52] Grothendieck A., Topological vector spaces, Gordon and Breach Science Publishers, New York, 1973.Search in Google Scholar

[53] Köthe G., Topological vector spaces, Grundlehren der mathematischen Wissenschaften 159, Springer-Verlag, New York, 1969.Search in Google Scholar

[54] McNulty G.F., Shallon C.R., Inherently nonfinitely based finite algebras, Universal algebra and lattice theory (Puebla, 1982), Lecture Notes in Math. 1004, Springer-Verlag, Berlin, New York, 206-231, 1983.Search in Google Scholar

[55] Robertson A.P., Robertson W.J., Topological vector spaces, Cambridge Tracts in Mathematics 53, Cambridge University Press, 1964.Search in Google Scholar

[56] Schaefer H.H., Topological vector spaces, GTM 3, Springer-Verlag, New York, 1971.10.1007/978-1-4684-9928-5Search in Google Scholar

[57] Trèves F., Topological Vector Spaces, Distributions, and Kernels, Academic Press, 1967.Search in Google Scholar

[58] Ayres F., Mandelson E., Calculus (Schaum's Outlines Series), 5th ed., Mc Graw Hill, 2009.Search in Google Scholar

[59] Barut A.O., Electrodynamics and Classical Theory of Fields and Particles, Courier Dover Publications, 1964.Search in Google Scholar

[60] Bishop R., Goldberg S.I., Tensor analysis on manifolds, Courier Dover Publications, 1980.Search in Google Scholar

[61] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Springer-Verlag, 1991.10.1007/978-1-4613-9714-4Search in Google Scholar

[62] Fenner R.T., Finite Element Methods for Engineers, Imperial College Press, London, 1996.10.1142/p014Search in Google Scholar

[63] Flanders H., Differential forms with applications to the physical sciences, Dover Publications, 1989.Search in Google Scholar

[64] Fleming W., Functions of Several Variables, 3rd ed., Springer-Verlag, New York, 1987.Search in Google Scholar

[65] Hairer E., Lubich C., Wanner G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, New York, 2006.Search in Google Scholar

[66] Huebner K.H., The Finite Element Method for Engineers, Wiley, 1975.Search in Google Scholar

[67] Lam T.Y., Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, 2005.10.1090/gsm/067Search in Google Scholar

[68] Lang S., Differential manifolds, Reading, Mass.–London–Don Mills, Ont., Addison-Wesley Publishing Co., Inc., 1972.Search in Google Scholar

[69] Larson R., Hosteler R.P., Edwards B.H., Calculus, 8th ed., Houghton Mifflin, 2005.Search in Google Scholar

[70] Latorre D.R., Kenelly J.W., Reed I.B., Biggers S., Calculus Concepts: An Applied Approach to the Mathematics of Change, Cengage Learning, 2007.Search in Google Scholar

[71] Livesley R.K., Finite Elements, an Introduction for Engineers, Cambridge University Press, 1983.Search in Google Scholar

[72] Lovelock D., Hanno R., Tensors, Differential Forms, and Variational Principles, Dover Publications, 1989 [1975].Search in Google Scholar

[73] Okada S., Onodera R., Algebraification of Field Laws of Physics by Poincaré Process, Bull. of Yamagata University – Natural Sciences, 1951, 1(4), 79-86.Search in Google Scholar

[74] Rudin W., Functional Analysis, McGraw-Hill Science/Engineering/Math, 1991.Search in Google Scholar

[75] Schouten J.A., Tensor Calculus for Physicists, Clarendon Press, Oxford, 1951.Search in Google Scholar

[76] Truesdell C., Noll W., The non-linear field theories of mechanics: Third edition, Springer, 2004.10.1007/978-3-662-10388-3Search in Google Scholar

[77] Veblen O., Whitehead J.H.C., The Foundations of Differential Geometry, Cambr. Tracts, no. 29, 1932.Search in Google Scholar

[78] Zill D.G., Wright S., Wright W.S., Calculus: Early Transcendentals, 3rd ed., Jones & Bartlett Learning, 2009.Search in Google Scholar

[79] Artin M., Algebra, Prentice Hall, 1991.Search in Google Scholar

[80] Bourbaki N., Algebra, Springer-Verlag, Berlin, New York, 1988.Search in Google Scholar

[81] Bourbaki N., Elements of mathematics, Algebra I, Springer-Verlag, 1989.Search in Google Scholar

[82] Curtis C.W., Linear Algebra, Allyn & Bacon, Boston, 1968.Search in Google Scholar

[83] Dummit D.S., Foote R.M., Abstract Algebra, 3rd ed., Wiley, 2003.Search in Google Scholar

[84] Frescura F.A.M., Hiley B.J., The implicate order, algebras, and the spinor, Foundations of Physics, 1980, 10(1-2), 7-31.10.1007/BF00709014Search in Google Scholar

[85] Frescura F.A.M., Hiley B.J., Algebras, quantum theory and prespace, Revista Brasileira de Fisica, Volume Especial, Los 70 anos de Mario Schonberg, 1984, 49-86.Search in Google Scholar

[86] Hazewinkel M., Contravariant tensor, Encyclopedia of Mathematics, Springer, ed. 2001.Search in Google Scholar

[87] Hazewinkel, M., Covariant tensor, Encyclopedia of Mathematics, Springer, ed. 2001.Search in Google Scholar

[88] Herstein I.N., Abstract Algebra, 3rd ed., Wiley, 1996.Search in Google Scholar

[89] Hestenes D., Space-time Algebra, Gordon and Breach, New York, 1966.Search in Google Scholar

[90] Lang S., Linear algebra, Springer-Verlag, Berlin, New York, 1987.Search in Google Scholar

[91] Lang S., Algebra, Graduate Texts in Mathematics 211, revised 3rd ed., Springer-Verlag, New York, 2002.10.1007/978-1-4613-0041-0_1Search in Google Scholar

[92] Lax P., Linear algebra, Wiley-Interscience, 1996.Search in Google Scholar

[93] Mac Lane S., Birkhoff G, Algebra, AMS Chelsea, 1999.Search in Google Scholar

[94] Oates-Williams S., On the variety generated by Murskiĭ's algebra, Algebra Universalis, 1984, 18(2), 175-177.10.1007/BF01198526Search in Google Scholar

[95] Roman S., Advanced Linear Algebra, Graduate Texts in Mathematics 135, 2nd ed., Springer-Verlag, Berlin, New York, 2005.Search in Google Scholar

[96] Wilder R.L., Introduction to Foundations of Mathematics, John Wiley and Sons, 1965.Search in Google Scholar

[97] Baylis W.E., Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, 1996.Search in Google Scholar

[98] Baylis W.E., Electrodynamics: A Modern Geometric Approach, 2nd ed., Birkhäuser, 2002.Search in Google Scholar

[99] Bohm D., Hiley B.J., Stuart A., On a New Mode of Description in Physics, Int. J. Theor. Phys. 1970, 3(3), 171-183.Search in Google Scholar

[100] Doran C., Lasenby A., Geometric algebra for physicists, University Press, Cambridge 2003.10.1017/CBO9780511807497Search in Google Scholar

[101] Dorst L., The inner products of geometric algebra, MA: Birkhäuser Boston, Boston, 2002.10.1007/978-1-4612-0089-5_2Search in Google Scholar

[102] Dorst L., Fontijne D., Mann S., Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, Elsevier/Morgan Kaufmann, Amsterdam, 2007.10.1016/B978-012369465-2/50004-9Search in Google Scholar

[103] Frescura F.A.M., Hiley B.J., Geometric interpretation of the Pauli spinor, American Journal of Physics, 1981, 49(2), 152.10.1119/1.12548Search in Google Scholar

[104] Hestenes D., New foundations for classical mechanics: Fundamental Theories of Physics, 2nd ed., Springer, 1999.Search in Google Scholar

[105] Jost, J., Riemannian Geometry and Geometric Analysis, Springer-Verlag, Berlin, 2002.10.1007/978-3-662-04672-2Search in Google Scholar

[106] Lasenby J., Lasenby A.N., Doran C.J.L., A Unified Mathematical Language for Physics and Engineering in the 21st Century, Philosophical Transactions of the Royal Society of London, 2000, A 358, 1-18.10.1098/rsta.2000.0517Search in Google Scholar

[107] Lounesto P., Clifford algebras and spinors, Cambridge University Press, Cambridge, 2001.10.1017/CBO9780511526022Search in Google Scholar

[108] Macdonald A., Linear and Geometric Algebra, CreateSpace, Charleston, 2011.Search in Google Scholar

[109] Micali A., Boudet R., Helmstetter J., Clifford Algebras and their Applications in Mathematical Physics, Workshop Proceedings: 2nd (Fundamental Theories of Physics), Kluwer, 1989.Search in Google Scholar

[110] Porteous I.R., Clifford algebras and the classical groups, Cambridge University Press, Cambridge, 1995.10.1017/CBO9780511470912Search in Google Scholar

[111] Schutz B., Geometrical methods of mathematical physics, Cambridge University Press, 1980.10.1017/CBO9781139171540Search in Google Scholar

[112] Snygg J., A New Approach to Differential Geometry Using Clifford's Geometric Algebra, Birkhäuser, 2012.10.1007/978-0-8176-8283-5Search in Google Scholar

[113] van Dantzing D., On the Relation Between Geometry and Physics and the Concept of Space-Time, Helv. Phys. Acta, 1956, Suppl. IV, 48-53.Search in Google Scholar

[114] Branin F.H.Jr., The Algebraic Topological Basis for Network Analogies and the Vector Calculus, Proc. Symp. on Generalized Networks, Brooklyn Polit., 1966, 453-487.Search in Google Scholar

[115] Frescura F.A.M., Hiley B.J., The algebraization of quantum mechanics and the implicate order, Foundations of Physics, 1980, 10(9-10), 705-722.10.1007/BF00708417Search in Google Scholar

[116] Twiss R.J., Moores E.M., §2.1 The orientation of structures, In: Structural geology, 2nd ed., Macmillan, 1992.Search in Google Scholar

[117] Morton K.W., Stringer S.M., Finite Volume Methods for Inviscid and Viscous Flows, Steady and Unsteady, Lecture, Series 1995-02, Computational Fluid Dynamics, Von Karman Institute of Fluid Dynamics, 1995.Search in Google Scholar

[118] Burke W.L., Applied Differential Geometry, Cambridge University Press, Cambridge, 1985.10.1017/CBO9781139171786Search in Google Scholar

[119] Sternberg S., Lectures on Differential Geometry, Prentice Hall, 1964.Search in Google Scholar

[120] Gardner J.W., Wiegandt R., Radical Theory of Rings, Chapman & Hall/CRC Pure and Applied Mathematics, 2003.10.1201/9780203913352Search in Google Scholar

[121] Halmos P., Finite dimensional vector spaces, Springer, 1974.10.1007/978-1-4612-6387-6Search in Google Scholar

[122] Coxeter H.S.M., Regular Polytopes, Dover Publications, Inc., New York, 1973.Search in Google Scholar

[123] Gosset T., On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900.Search in Google Scholar

[124] Grünbaum B., Convex Polytopes, Graduate Texts in Mathematics 221, 2nd ed., Springer, 2003.10.1007/978-1-4613-0019-9Search in Google Scholar

[125] Hilbert D., Grundlagen der Geometrie, 10th ed., Teubner, Stuttgart, 1968.10.1007/978-3-322-92726-2Search in Google Scholar

[126] Johnson N.W., The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966.Search in Google Scholar

[127] Ziegler G.M., Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, 1995.10.1007/978-1-4613-8431-1Search in Google Scholar

[128] Cromwell P.R., Polyhedra, Cambridge University Press, 1999.Search in Google Scholar

[129] Davey B.A., Idziak P.M., Lampe W.A., McNulty G.F., Dualizability and graph algebras, Discrete Mathematics, 2000, 214(1), 145-172.10.1016/S0012-365X(99)00225-3Search in Google Scholar

[130] Delić D., Finite bases for flat graph algebras, Journal of Algebra, 2001, 246(1), 453-469.10.1006/jabr.2001.8947Search in Google Scholar

[131] Kelarev A.V., Graph Algebras and Automata, Marcel Dekker, New York, 2003.10.1201/9781482276367Search in Google Scholar

[132] Kelarev A.V., Miller M., Sokratova O.V., Languages recognized by two-sided automata of graphs, Proc. Estonian Akademy of Science, 2005, 54(1), 46-54.10.3176/phys.math.2005.1.03Search in Google Scholar

[133] Kelarev A.V., Sokratova O.V., Directed graphs and syntactic algebras of tree languages, J. Automata, Languages & Combinatorics, 2001, 6(3), 305-311.Search in Google Scholar

[134] Kelarev A.V., Sokratova O.V., On congruences of automata defined by directed graphs, Theoretical Computer Science, 2003, 301(1-3), 31-43.10.1016/S0304-3975(02)00544-3Search in Google Scholar

[135] Kiss E.W., Pöschel R., Pröhle P., Subvarieties of varieties generated by graph algebras, Acta Sci. Math. (Szeged), 1990, 54(1-2), 57-75.Search in Google Scholar

[136] Lee S.-M., Graph algebras which admit only discrete topologies, Congr. Numer., 1988, 64, 147-156.Search in Google Scholar

[137] Lee S.-M., Simple graph algebras and simple rings, Southeast Asian Bull. Math., 1991, 15(2), 117-121.Search in Google Scholar

[138] Pöschel R., The equational logic for graph algebras, Z. Math. Logik Grundlag. Math., 1989, 35(3), 273-282.10.1002/malq.19890350311Search in Google Scholar

[139] Bain J., Spacetime structuralism: §5 Manifolds vs. geometric algebra, In: Dennis Dieks, The ontology of spacetime, Elsevier, 2006.Search in Google Scholar

[140] Catoni F., Boccaletti D., Cannata R., Mathematics of Minkowski Space, Birkhäuser Verlag, Basel, 2008.Search in Google Scholar

[141] Naber G.L., The Geometry of Minkowski Spacetime, Springer-Verlag, New York, 1992.10.1007/978-1-4757-4326-5Search in Google Scholar

Received: 2014-7-16
Accepted: 2015-1-12
Published Online: 2015-3-25
Published in Print: 2015-1-1

© Elena Ferretti

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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