Algorithmic operator algebras via normal forms in tensor rings

Poor, Jamal Hossein; Raab, Clemens G.; Regensburger, Georg

doi:10.1016/j.jsc.2017.07.011

Cited by 7 publications

(12 citation statements)

References 21 publications

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“…This extends the definition of integro-differential rings in [14] by dropping the additional requirement that the induced evaluation should be multiplicative. In the present paper, the notion of integro-differential rings always refers to Definition 2.2.…”

Section: Integro-differential Ringsmentioning

confidence: 90%

“…More general differential algebras with integration over rings were introduced in [11], see [12] for a unified presentation and comparison. In contrast, the integro-differential rings defined in [14,13] require the integration to be linear over all constants, but the construction of corresponding operators introduced there allows noncommutative coefficients and constants. Further references to the literature can be found in the respective sections of the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…We compute a complete set of rewrite rules to simplify such operators to normal form. A precise analysis of the uniqueness of these normal forms is presented in the appendix only, since it requires a refined construction relying on tensor rings (like in [14,13] for the multiplicative case). After a largely self-contained introduction to tensor reduction systems in the appendix, this is carried out in Section A.4 allowing also other functionals besides evaluation.…”

Section: Introductionmentioning

confidence: 99%

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The fundamental theorem of calculus in differential rings

Raab¹,

Regensburger²

2023

Preprint

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In this paper, we study the fundamental theorem of calculus and its consequences 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.

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Section: Integro-differential Ringsmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The fundamental theorem of calculus in differential rings

Raab¹,

Regensburger²

2023

Preprint

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“…However, termination of such rewriting relations is an issue. Finally, another topic for future research is to generalize the results of this paper to tensor reduction systems used for modelling linear operators as described in [12].…”

Section: Summary and Discussionmentioning

confidence: 99%

Compatible rewriting of noncommutative polynomials for proving operator identities

Chenavier¹,

Hofstadler²,

Raab³

et al. 2020

Preprint

Self Cite

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The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.

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“…Another approach consists in selecting the reducible monomials with more flexible orders than monomial ones, which may be used for proving Koszulness of algebras for which Gröbner bases give no result [17]. Finally, rewriting steps may be described in a functional manner [6,16,21], so that linear rewriting systems are represented by reduction operators. From this approach, the confluence property is characterised by means of lattice operations [10], which provides various applications to computer algebra and homological algebra: construction of Gröbner bases [11], computation of syzygies [12] or proofs of Koszulness [4,5,9,24].Rewriting methods based on monomial orders were also developed for formal power series, where the leading monomial of a series is the smallest monomial in its decomposition.…”

mentioning

confidence: 99%

Topological rewriting systems applied to standard bases and syntactic algebras

Chenavier

2020

Journal of Algebra

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We introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting systems with continuous reduction operators, we show that the topological confluence property is characterised by lattice operations. Using this characterisation, we show that standard bases induce topologically confluent rewriting systems on formal power series. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras. In particular, we use duality for proving that an algebra is syntactic or not.The binary relation → denotes to rewriting steps and * → is it reflexive transitive closure, so that it denotes to rewriting sequences. Under hypotheses of termination and confluence, computing normal forms, that are irreducible terms, provides algorithmic applications, for instance to the decision of the word problem for monoids or the ideal membership problem for polynomial algebras. It also provides effective methods for computing linear bases, Hilbert series, homotopy bases or free resolutions [1,18,23]. These methods induce constructive proofs of coherence theorems, which provide an explicit description of the action of a monoid on a category [15], or of homological properties, such as finite derivation type, finite homological type [19,30], or Koszulness [27].In the situation of rewriting systems on linear structures, a relation is usually oriented by rewriting one monomial into the linear combination of other monomials. There exist three main approaches for selecting the rewritten monomial, the most classical one consisting in using a monomial order. From this approach, Gröbner bases are characterised in terms of the confluence property: a set of polynomials is a Gröbner of the ideal it generates if and only if it induces a confluent rewriting system. As a consequence, effective confluence-based criteria were introduced for checking if a given set is a Gröbner basis or one of its numerous adaptations to (polynomial, free Lie, tensor) algebras [7,8,25,29], skew-polynomial rings [22] or free algebraic operads [13]. Another approach consists in selecting the reducible monomials with more flexible orders than monomial ones, which may be used for proving Koszulness of algebras for which Gröbner bases give no result [17]. Finally, rewriting steps may be described in a functional manner [6,16,21], so that linear rewriting systems are represented by reduction operators. From this approach, the confluence property is characterised by means of lattice operations [10], which provides various applications to computer algebra and homological algebra: construction of Gröbner bases [11], computation of syzygies [12] or proofs of Koszulness [4,5,9,24].Rewriting methods base...

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Algorithmic operator algebras via normal forms in tensor rings

Cited by 7 publications

References 21 publications

The fundamental theorem of calculus in differential rings

The fundamental theorem of calculus in differential rings

Compatible rewriting of noncommutative polynomials for proving operator identities

Topological rewriting systems applied to standard bases and syntactic algebras

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