The fundamental theorem of calculus in differential rings.(English)Zbl 1548.12007
Summary: In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such integro-differential rings and discuss many examples. We also construct corresponding integro-differential operators and provide normal forms via rewrite rules. They are then used to derive several identities and properties in a purely algebraic way, generalizing well-known results from analysis. In identities like shuffle relations for nested integrals and the Taylor formula, additional terms are obtained that take singularities into account. Another focus lies on treating basics of linear ODEs in this framework of integro-differential operators. These operators can have matrix coefficients, which allow to treat systems of arbitrary size in a unified way. In the appendix, using tensor reduction systems, we give the technical details of normal forms and prove them for operators including other functionals besides evaluation.
MSC:
| 12H05 | Differential algebra |
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 34A30 | Linear ordinary differential equations and systems |
| 47E05 | General theory of ordinary differential operators |
Keywords:
integro-differential rings;integro-differential operators;normal forms;generalized shuffle relations;generalized Taylor formulaSoftware:
TenResCite
References:
| [1] | Aschenbrenner, M.; van den Dries, L.; van der Hoeven, J., Asymptotic Differential Algebra and Model Theory of Transseries, Annals of Mathematics Studies, vol. 195, 2017, Princeton University Press: Princeton University Press Princeton, NJ ·Zbl 1430.12002 |
| [2] | Bavula, V. V., The algebra of integro-differential operators on a polynomial algebra, J. Lond. Math. Soc. (2), 83, 517-543, 2011 ·Zbl 1225.16010 |
| [3] | Bergman, G. M., The diamond lemma for ring theory, Adv. Math., 29, 178-218, 1978 ·Zbl 0326.16019 |
| [4] | Buchberger, B., An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal (German), 1965, University of Innsbruck, Ph.D. thesis ·Zbl 1245.13020 |
| [5] | Cluzeau, T.; Hossein Poor, J.; Quadrat, A.; Raab, C. G.; Regensburger, G., Symbolic computation for integro-differential-time-delay operators with matrix coefficients, IFAC-PapersOnLine, 51, 14, 153-158, 2018 |
| [6] | Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations, 1955, McGraw-Hill Book Co., Inc.: McGraw-Hill Book Co., Inc. New York-Toronto-London ·Zbl 0064.33002 |
| [7] | Cohn, P. M., Further Algebra and Applications, 2003, Springer-Verlag London, Ltd.: Springer-Verlag London, Ltd. London ·Zbl 1006.00001 |
| [8] | Edgar, G. A., Transseries for beginners, Real Anal. Exch., 35, 2, 253-309, 2010 ·Zbl 1218.41019 |
| [9] | Gao, X.; Guo, L.; Rosenkranz, M., On rings of differential Rota-Baxter operators, Int. J. Algebra Comput., 28, 1, 1-36, 2018 ·Zbl 1416.16025 |
| [10] | Guo, L., An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, vol. 4, 2012, International Press/Higher Education Press: International Press/Higher Education Press Somerville, MA/Beijing ·Zbl 1271.16001 |
| [11] | Guo, L.; Keigher, W., On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212, 522-540, 2008 ·Zbl 1185.16038 |
| [12] | Guo, L.; Regensburger, G.; Rosenkranz, M., On integro-differential algebras, J. Pure Appl. Algebra, 218, 456-473, 2014 ·Zbl 1310.16019 |
| [13] | Hossein Poor, J., Tensor reduction systems for rings of linear operators, 2018, Johannes Kepler University Linz: Johannes Kepler University Linz Austria, Ph.D. thesis |
| [14] | Hossein Poor, J.; Raab, C. G.; Regensburger, G., Algorithmic operator algebras via normal forms in tensor rings, J. Symb. Comput., 85, 247-274, 2018 ·Zbl 06789139 |
| [15] | Kaplansky, I., An Introduction to Differential Algebra, Publ. Inst. Math. Univ. Nancago, vol. 5, 1957, Hermann: Hermann Paris ·Zbl 0083.03301 |
| [16] | Keigher, W. F., Adjunctions and comonads in differential algebra, Pac. J. Math., 59, 1, 99-112, 1975 ·Zbl 0327.12104 |
| [17] | Keigher, W. F., On the ring of Hurwitz series, Commun. Algebra, 25, 6, 1845-1859, 1997 ·Zbl 0884.13013 |
| [18] | Keigher, W. F.; Pritchard, F. L., Hurwitz series as formal functions, J. Pure Appl. Algebra, 146, 3, 291-304, 2000 ·Zbl 0978.12007 |
| [19] | Knuth, D. E.; Bendix, P. B., Simple word problems in universal algebras, (Computational Problems in Abstract Algebra, 1970, Pergamon: Pergamon Oxford), 263-297 ·Zbl 0188.04902 |
| [20] | Kummer, E. E., Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen, J. Reine Angew. Math., 21, 74-90, 1840, 193-225, and 328-371 ·ERAM 021.0659cj |
| [21] | Lappo-Danilevski, I. A., Résolution algorithmique des problèmes réguliers de Poincaré et de Riemann (Mémoire premier: Le problème de Poincaré, concernant de la construction d’un groupe de monodromie d’un système donné d’équations différentielles linéaires aux intégrales régulières), J. Soc. Phys.-Math. Léningrade, 2, 1, 94-120, 1928 |
| [22] | Mora, T., An introduction to commutative and noncommutative Gröbner bases, Theor. Comput. Sci., 134, 1, 131-173, 1994 ·Zbl 0824.68056 |
| [23] | Przeworska-Rolewicz, D., Algebraic Analysis, 1988, PWN—Polish Scientific Publishers/D. Reidel Publishing Co.: PWN—Polish Scientific Publishers/D. Reidel Publishing Co. Warsaw/Dordrecht ·Zbl 0696.47002 |
| [24] | Przeworska-Rolewicz, D., Two centuries of the term “algebraic analysis”, (Algebraic Analysis and Related Topics. Algebraic Analysis and Related Topics, Warsaw, 1999. Algebraic Analysis and Related Topics. Algebraic Analysis and Related Topics, Warsaw, 1999, Banach Center Publ., vol. 53, 2000, Polish Acad. Sci. Inst. Math: Polish Acad. Sci. Inst. Math Warsaw), 47-70 ·Zbl 1074.01516 |
| [25] | Quadrat, A., A constructive algebraic analysis approach to Artstein’s reduction of linear time-delay systems, IFAC-PapersOnLine, 48, 12, 209-214, 2015 |
| [26] | Quadrat, A.; Regensburger, G., Computing polynomial solutions and annihilators of integro-differential operators with polynomial coefficients, (Algebraic and Symbolic Computation Methods in Dynamical Systems. Algebraic and Symbolic Computation Methods in Dynamical Systems, Adv. Delays Dyn., vol. 9, 2020, Springer: Springer Cham), 87-114 ·Zbl 1540.47112 |
| [27] | C.G. Raab, G. Regensburger, The free commutative integro-differential ring, in preparation. |
| [28] | Ree, R., Lie elements and an algebra associated with shuffles, Ann. Math. (2), 68, 210-220, 1958 ·Zbl 0083.25401 |
| [29] | Regensburger, G.; Rosenkranz, M.; Middeke, J., A skew polynomial approach to integro-differential operators, (Proceedings of ISSAC ’09. Proceedings of ISSAC ’09, New York, NY, USA, 2009, ACM), 287-294 ·Zbl 1237.16023 |
| [30] | Rosenkranz, M., A new symbolic method for solving linear two-point boundary value problems on the level of operators, J. Symb. Comput., 39, 171-199, 2005 ·Zbl 1126.68104 |
| [31] | Rosenkranz, M.; Regensburger, G., Solving and factoring boundary problems for linear ordinary differential equations in differential algebras, J. Symb. Comput., 43, 515-544, 2008 ·Zbl 1151.34008 |
| [32] | Rosenkranz, M.; Regensburger, G.; Tec, L.; Buchberger, B., Symbolic analysis for boundary problems: from rewriting to parametrized Gröbner bases, (Langer, U.; Paule, P., Numerical and Symbolic Scientific Computing: Progress and Prospects, 2012, Springer Vienna: Springer Vienna Vienna), 273-331 ·Zbl 1250.65104 |
| [33] | Rosenkranz, M.; Serwa, N., An integro-differential structure for Dirac distributions, J. Symb. Comput., 92, 156-189, 2019 ·Zbl 1428.12008 |
| [34] | Rota, G.-C., Twelve problems in probability no one likes to bring up, (Algebraic Combinatorics and Computer Science, 2001, Springer Italia: Springer Italia Milan), 57-93 ·Zbl 0984.60007 |
| [35] | Rowen, L. H., Ring Theory, 1991, Academic Press, Inc.: Academic Press, Inc. Boston, MA ·Zbl 0743.16001 |
| [36] | Stanley, R. P., Differentiably finite power series, Eur. J. Comb., 1, 2, 175-188, 1980 ·Zbl 0445.05012 |
| [37] | van der Hoeven, J., Transseries and Real Differential Algebra, Lecture Notes in Mathematics, vol. 1888, 2006, Springer-Verlag: Springer-Verlag Berlin ·Zbl 1128.12008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
