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Articles de revues sur le sujet « Free Differential Algebras »

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Auteur : Grafiati
Publié le 14 mars 2023

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1

GAO, XING, LI GUO et SHANGHUA ZHENG. « CONSTRUCTION OF FREE COMMUTATIVE INTEGRO-DIFFERENTIAL ALGEBRAS BY THE METHOD OF GRÖBNER–SHIRSHOV BASES ». Journal of Algebra and Its Applications 13, no 05 (25 février 2014) : 1350160. http://dx.doi.org/10.1142/s0219498813501600.

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In this paper, we construct free commutative integro-differential algebras by applying the method of Gröbner–Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota–Baxter (DRB) algebras of order n. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner–Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional monomials to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.
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2

QIU, JIANJUN, et YUQUN CHEN. « COMPOSITION-DIAMOND LEMMA FOR λ-DIFFERENTIAL ASSOCIATIVE ALGEBRAS WITH MULTIPLE OPERATORS ». Journal of Algebra and Its Applications 09, no 02 (avril 2010) : 223–39. http://dx.doi.org/10.1142/s0219498810003859.

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In this paper, we establish the Composition-Diamond lemma for λ-differential associative algebras over a field K with multiple operators. As applications, we obtain Gröbner–Shirshov bases of free λ-differential Rota–Baxter algebras. In particular, linear bases of free λ-differential Rota–Baxter algebras are obtained and consequently, the free λ-differential Rota–Baxter algebras are constructed by words.
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3

Kadeishvili, T., et P. Real. « Free resolutions for differential modules over differential algebras ». Journal of Mathematical Sciences 152, no 3 (juillet 2008) : 307–22. http://dx.doi.org/10.1007/s10958-008-9072-9.

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4

Iyer, Uma N., et Timothy C. McCune. « Differential operators on the free algebras ». Selecta Mathematica 18, no 2 (1 novembre 2011) : 329–55. http://dx.doi.org/10.1007/s00029-011-0076-9.

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5

Frønsdal, C., et A. Galindo. « The Ideals of Free Differential Algebras ». Journal of Algebra 222, no 2 (décembre 1999) : 708–46. http://dx.doi.org/10.1006/jabr.1999.8076.

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6

Qiu, Jianjun. « Gröbner–Shirshov bases for commutative algebras with multiple operators and free commutative Rota–Baxter algebras ». Asian-European Journal of Mathematics 07, no 02 (juin 2014) : 1450033. http://dx.doi.org/10.1142/s1793557114500338.

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In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gröbner–Shirshov bases and linear bases of free commutative Rota–Baxter algebra, free commutative λ-differential algebra and free commutative λ-differential Rota–Baxter algebra are given, respectively. Consequently, these three free algebras are constructed directly by commutative Ω-words.
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7

SÁNCHEZ, OMAR LEÓN, et RAHIM MOOSA. « THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS ». Journal of Symbolic Logic 81, no 2 (juin 2016) : 493–509. http://dx.doi.org/10.1017/jsl.2015.76.

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AbstractA model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.
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8

BENKHALIFA, MAHMOUD. « WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA ». International Journal of Mathematics 15, no 10 (décembre 2004) : 987–1005. http://dx.doi.org/10.1142/s0129167x04002673.

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Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.
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9

Salgado, P., et S. Salgado. « Extended gauge theory and gauged free differential algebras ». Nuclear Physics B 926 (janvier 2018) : 179–99. http://dx.doi.org/10.1016/j.nuclphysb.2017.10.026.

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10

Fré, Pietro, et Pietro Antonio Grassi. « Pure spinors, free differential algebras, and the supermembrane ». Nuclear Physics B 763, no 1-2 (février 2007) : 1–34. http://dx.doi.org/10.1016/j.nuclphysb.2006.10.026.

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11

Mikhailov, A. V., et V. V. Sokolov. « Integrable ordinary differential equations on free associative algebras ». Theoretical and Mathematical Physics 122, no 1 (janvier 2000) : 72–83. http://dx.doi.org/10.1007/bf02551171.

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12

Gao, Xing, Li Guo et Markus Rosenkranz. « Free integro-differential algebras and Gröbner–Shirshov bases ». Journal of Algebra 442 (novembre 2015) : 354–96. http://dx.doi.org/10.1016/j.jalgebra.2014.10.016.

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13

Qiu, Jianjun, et Yuqun Chen. « Free Lie differential Rota–Baxter algebras and Gröbner–Shirshov bases ». International Journal of Algebra and Computation 27, no 08 (décembre 2017) : 1041–60. http://dx.doi.org/10.1142/s0218196717500485.

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14

Castellani, L. « Quantum groups and free differential algebras in field theory ». Nuclear Physics B - Proceedings Supplements 56, no 3 (juillet 1997) : 170–82. http://dx.doi.org/10.1016/s0920-5632(97)00324-1.

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15

Castellani, L. « Infinite-dimensional free differential algebras and string field theory ». Nuclear Physics B 317, no 1 (avril 1989) : 46–62. http://dx.doi.org/10.1016/0550-3213(89)90560-9.

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16

Huebschmann, J. « Minimal Free Multi-Models for Chain Algebras ». gmj 11, no 4 (décembre 2004) : 733–52. http://dx.doi.org/10.1515/gmj.2004.733.

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Abstract Let 𝑅 be a local ring and 𝐴 a connected differential graded algebra over 𝑅 which is free as a graded 𝑅-module. Using homological perturbation theory techniques, we construct a minimal free multi-model for 𝐴 having properties similar to those of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi’ refers to the category of multicomplexes.
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17

Hartglass, Michael. « Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops ». Canadian Journal of Mathematics 69, no 3 (1 juin 2017) : 548–78. http://dx.doi.org/10.4153/cjm-2016-022-6.

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AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.
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18

Li, Yunnan, et Li Guo. « Construction of free differential algebras by extending Gröbner-Shirshov bases ». Journal of Symbolic Computation 107 (novembre 2021) : 167–89. http://dx.doi.org/10.1016/j.jsc.2021.03.002.

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19

Dall'Agata, Gianguido, Riccardo D'Auria et Sergio Ferrara. « Compactifications on twisted tori with fluxes and free differential algebras ». Physics Letters B 619, no 1-2 (juillet 2005) : 149–54. http://dx.doi.org/10.1016/j.physletb.2005.04.005.

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20

Thibault de Chanvalon, Manon. « Classification of Bicovariant Differential Calculi over free Orthogonal Hopf Algebras ». Algebras and Representation Theory 18, no 3 (13 février 2015) : 831–47. http://dx.doi.org/10.1007/s10468-015-9518-y.

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21

Shirvani, M., et J. Z. Gonçalves. « Free Group Algebras in the Field of Fractions of Differential Polynomial Rings and Enveloping Algebras ». Journal of Algebra 204, no 2 (juin 1998) : 372–85. http://dx.doi.org/10.1006/jabr.1997.7283.

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22

Dilxat, Munayim, Shoulan Gao, Dong Liu et Limeng Xia. « U(h)-Free Modules over the Lie Algebras of Differential Operators ». Mathematics 10, no 10 (18 mai 2022) : 1728. http://dx.doi.org/10.3390/math10101728.

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This paper mainly considers a class of non-weight modules over the Lie algebra of the Weyl type. First, we construct the U(h)-free modules of rank one over the differential operator algebra. Then, we characterize the tensor products of these kind of modules and the quasi-finite highest weight modules. Finally, we undertake such research for the differential operator algebra of multi-variables.
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23

Gao, Mingchu. « Free Markov processes and stochastic differential equations in von Neumann algebras ». Illinois Journal of Mathematics 52, no 1 (2008) : 153–80. http://dx.doi.org/10.1215/ijm/1242414126.

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24

Castellani, Leonardo, et Alberto Perotto. « Free differential algebras : Their use in field theory and dual formulation ». Letters in Mathematical Physics 38, no 3 (novembre 1996) : 321–30. http://dx.doi.org/10.1007/bf00398356.

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25

MIKHALEV, ALEXANDER A., et ANDREJ A. ZOLOTYKH. « RANK AND PRIMITIVITY OF ELEMENTS OF FREE COLOUR LIE (p-)SUPERALGEBRAS ». International Journal of Algebra and Computation 04, no 04 (décembre 1994) : 617–55. http://dx.doi.org/10.1142/s021819679400018x.

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Using Fox differential calculus we study characteristics of orbits of elements of free colour Lie (p-)superalgebras under action of the automorphism groups of these algebras. In particular, an effective criterion for an element to be primitive and an algorithm for finding the rank of an element are obtained.
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26

Karaçuha, Serkan, et Christian Lomp. « Integral calculus on quantum exterior algebras ». International Journal of Geometric Methods in Modern Physics 11, no 04 (avril 2014) : 1450026. http://dx.doi.org/10.1142/s0219887814500261.

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Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.
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27

MORENO, GIOVANNI. « ON FAMILIES IN DIFFERENTIAL GEOMETRY ». International Journal of Geometric Methods in Modern Physics 10, no 09 (30 août 2013) : 1350042. http://dx.doi.org/10.1142/s0219887813500424.

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Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.
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28

Poinsot, Laurent. « Wronskian Envelope of a Lie Algebra ». Algebra 2013 (29 mai 2013) : 1–8. http://dx.doi.org/10.1155/2013/341631.

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The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.
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29

PARK, EFTON. « THE INDEX OF TOEPLITZ OPERATORS ON FREE TRANSFORMATION GROUP C*-ALGEBRAS ». Bulletin of the London Mathematical Society 34, no 1 (janvier 2002) : 84–90. http://dx.doi.org/10.1112/s0024609301008554.

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Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.
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30

LERDA, A., J. E. NELSON et T. REGGE. « COVARIANT CANONICAL FORMALISM FOR POLYNOMIAL SUPERGRAVITY IN ANY DIMENSION ». International Journal of Modern Physics A 02, no 05 (octobre 1987) : 1643–54. http://dx.doi.org/10.1142/s0217751x87000855.

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We develop a covariant canonical formalism for supergravity theories and consider Lagrangian densities which are polynomial in the curvatures in any space-time dimension. Our formalism is particularly suited for generic free differential algebras.
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31

PILITOWSKA, A., A. B. ROMANOWSKA et D. STANOVSKÝ. « VARIETIES OF DIFFERENTIAL MODES EMBEDDABLE INTO SEMIMODULES ». International Journal of Algebra and Computation 19, no 05 (août 2009) : 669–80. http://dx.doi.org/10.1142/s0218196709005305.

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Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based.
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32

Rubenthaler, Hubert. « Algebras of invariant differential operators on a class of multiplicity free spaces ». Comptes Rendus Mathematique 347, no 23-24 (décembre 2009) : 1343–46. http://dx.doi.org/10.1016/j.crma.2009.10.015.

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33

Foussats, A., R. Laura et O. Zandron. « Factorisation and gauge transformations in supergravity theories constructed on free differential algebras ». Classical and Quantum Gravity 3, no 5 (1 septembre 1986) : 871–79. http://dx.doi.org/10.1088/0264-9381/3/5/016.

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34

Yang, Wen-Li, Yao-Zhong Zhang et Samuel Kault. « Differential operator realizations of superalgebras and free field representations of corresponding current algebras ». Nuclear Physics B 823, no 3 (décembre 2009) : 372–402. http://dx.doi.org/10.1016/j.nuclphysb.2009.06.029.

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35

RIZELL, GEORGIOS DIMITROGLOU. « Nontriviality results for the characteristic algebra of a DGA ». Mathematical Proceedings of the Cambridge Philosophical Society 162, no 3 (28 juillet 2016) : 419–33. http://dx.doi.org/10.1017/s0305004116000645.

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AbstractAssume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.
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36

Fiorani, Emanuele, Sandra Germani et Andrea Spiro. « Lie algebras of conservation laws of variational partial differential equations ». Advances in Geometry 18, no 2 (25 avril 2018) : 207–28. http://dx.doi.org/10.1515/advgeom-2018-0004.

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Abstract We establish a version of Noether’s first Theorem according to which the (equivalence classes of) conserved quantities of given Euler–Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action from which the Euler–Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such a bijective correspondence Φ͠ between equivalence classes comes from an explicit (non-bijective) linear map Φ from vector fields into conserved differential operators, and not from a map into divergences of conserved operators as it occurs in other proofs of Noether’s Theorem. Where possible, claims are given a coordinate-free formulation and all proofs rely just on basic differential geometric properties of finite-dimensional manifolds.
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37

Reid, Gregory J. « Finding abstract Lie symmetry algebras of differential equations without integrating determining equations ». European Journal of Applied Mathematics 2, no 4 (décembre 1991) : 319–40. http://dx.doi.org/10.1017/s0956792500000589.

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There are symbolic programs based on heuristics that sometimes, but not always, explicitly integrate the determining equations for the infinitesimal Lie symmetries admitted by systems of differential equations. We present a heuristic-free algorithm ‘Structure constant’, which can always determine whether the Lie symmetry group of a given system of PDEs is finite- or infinite-dimensional. If the group is finite-dimensional then ‘Structure constant’ can determine the dimension and structure constants of its associated Lie algebra without the heuristics of integration involved in other methods. If the group is infinite-dimensional, then ‘Structure constant’ computes the number of arbitrary functions which determine the infinite-dimensional component of its Lie symmetry algebra and also calculates the dimension and structure of its associated finite-dimensional subalgebra. ‘Structure constant’ employs the algorithms ‘Standard form’ and ‘Taylor’, described elsewhere. ‘Standard form’ is a heuristic-free algorithm which brings any system of determining equations to a standard form by including all integrability conditions in the system. ‘Taylor’ uses the standard form of a system of differential equations to calculate its Taylor series solution. These algorithms have been implemented in the symbolic language MAPLE. ‘Structure constant’ can also automatically determine the dimension and structure constants of the Lie symmetry algebras of entire classes of differential equations dependent on variable coefficients. In particular, we obtain new group classification results for some physically interesting classes of nonlinear telegraph equations depending on two variable coefficients, one representing a nonlinear wave speed and the other representing a nonlinear dispersion.
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38

Kratsios, Anastasis. « Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras ». Mathematics 9, no 3 (27 janvier 2021) : 251. http://dx.doi.org/10.3390/math9030251.

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The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.
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39

ALBEVERIO, S., YU G. KONDRATIEV et M. RÖCKNER. « DIFFEOMORPHISM GROUPS AND CURRENT ALGEBRAS : CONFIGURATION SPACE ANALYSIS IN QUANTUM THEORY ». Reviews in Mathematical Physics 11, no 01 (janvier 1999) : 1–23. http://dx.doi.org/10.1142/s0129055x99000027.

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The constuction of models of non-relativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space Γ of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on Γ, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on ℝd.
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40

Tcheka, Calvin. « On an Algebra Structure on the Hochschild Homology of Simply Connected Topological Space ». Algebra Colloquium 26, no 03 (12 août 2019) : 425–36. http://dx.doi.org/10.1142/s1005386719000312.

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In this note, we use the pairing induced by the interchange map in conjunction with the strongly homotopy commutative algebra structure to define products on the Eilenberg–Moore differential Tor and give a simplified proof of an improved outcome of Jones’s result due to Ndombol and Thomas. As a result, we establish an isomorphism of graded algebras between the Hochschild homology and the free loop space cohomology of a simply connected topological space.
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41

Iserles, Arieh, et Antonella Zanna. « On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations ». LMS Journal of Computation and Mathematics 3 (2000) : 44–75. http://dx.doi.org/10.1112/s1461157000000206.

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AbstractMany discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.
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42

Eager, Richard, et Ingmar Saberi. « Holomorphic field theories and Calabi–Yau algebras ». International Journal of Modern Physics A 34, no 16 (10 juin 2019) : 1950071. http://dx.doi.org/10.1142/s0217751x19500714.

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We consider the holomorphic twist of the worldvolume theory of flat D[Formula: see text]-branes transversely probing a Calabi–Yau manifold. A chain complex, constructed using the BV formalism, computes the local observables in the holomorphically twisted theory. Generalizing earlier work in the case [Formula: see text], we find that this complex can be identified with the Ginzburg dg algebra associated to the Calabi–Yau. However, the identification is subtle; the complex is the space of fields contributing to the holomorphic twist of the free theory, and its differential arises from interactions. For [Formula: see text], this holomorphically twisted theory is related to the elliptic genus. We give a general description for D1-branes probing a Calabi–Yau fourfold singularity, and for [Formula: see text] quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch [Formula: see text] genus of large-[Formula: see text] symmetric products and cyclic homology.
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43

de Azcárraga, J. A., J. M. Izquierdo, M. Picón et O. Varela. « Generating Lie and gauge free differential (super)algebras by expanding Maurer–Cartan forms and Chern–Simons supergravity ». Nuclear Physics B 662, no 1-2 (juillet 2003) : 185–219. http://dx.doi.org/10.1016/s0550-3213(03)00342-0.

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Naz, R., et F. M. Mahomed. « Dynamic Euler-Bernoulli Beam Equation : Classification and Reductions ». Mathematical Problems in Engineering 2015 (2015) : 1–7. http://dx.doi.org/10.1155/2015/520491.

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We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass densityg(x), and the applied load denoted byf(u), a function of transverse displacementu(t,x). The complete Lie group classification is obtained for different forms of the variable lineal mass densityg(x)and applied loadf(u). The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms ofg(x). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature wheng(x)is constant with variable applied loadf(u). For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.
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Boukraa, S. « The BRS algebra of a free minimal differential algebra ». Nuclear Physics B 303, no 2 (juin 1988) : 237–59. http://dx.doi.org/10.1016/0550-3213(88)90180-0.

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Hubert, Evelyne. « Factorization-free Decomposition Algorithms in Differential Algebra ». Journal of Symbolic Computation 29, no 4-5 (mai 2000) : 641–62. http://dx.doi.org/10.1006/jsco.1999.0344.

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Tambara, Daisuke. « A note on free differential graded algebra resolutions ». Tsukuba Journal of Mathematics 20, no 2 (décembre 1996) : 399–411. http://dx.doi.org/10.21099/tkbjm/1496163090.

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Vaulà, Silvia. « On the underlyingE11symmetry of theD= 11 free differential algebra ». Journal of High Energy Physics 2007, no 03 (2 mars 2007) : 010. http://dx.doi.org/10.1088/1126-6708/2007/03/010.

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Harrington, Heather A., Kenneth L. Ho et Nicolette Meshkat. « A Parameter-Free Model Comparison Test Using Differential Algebra ». Complexity 2019 (17 février 2019) : 1–15. http://dx.doi.org/10.1155/2019/6041981.

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We present a method for rejecting competing models from noisy time-course data that does not rely on parameter inference. First we characterize ordinary differential equation models in only measurable variables using differential-algebra elimination. This procedure gives input-output equations, which serve as invariants for time series data. We develop a model comparison test using linear algebra and statistics to reject incorrect models from their invariants. This algorithm exploits the dynamic properties that are encoded in the structure of the model equations without recourse to parameter values, and, in this sense, the approach is parameter-free. We demonstrate this method by discriminating between different models from mathematical biology.
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D’AURIA, RICCARDO, PIETRO FRE’, GUIDO DE MATTEIS et IGOR PESANDO. « SUPERSPACE CONSTRAINTS AND CHERN-SIMONS COHOMOLOGY IN D=4 SUPERSTRING EFFECTIVE THEORIES ». International Journal of Modern Physics A 04, no 14 (20 août 1989) : 3577–613. http://dx.doi.org/10.1142/s0217751x89001412.

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The linear multiplet, composed of the dilaton ϕ, of an antisymmetric gauge field Bμν and of a spinor χ is always present in any superstring induced N=1D=4 supergravity model. We consider its coupling to supergravity using only superspace Bianchi identities and the rheonomy approach. In this way, our results are fully general and independent from the choice of any Lagrangian, a concept which is never mentioned in this paper. We consider two situations corresponding to two different free differential algebras: (1) the case where there are no Chern-Simons terms in the Bμν field strength Hμνρ and (2) the case where such terms are included in Hμνρ. Case (2) is obviously the one chosen by string theory on the ground of anomaly cancellation. In both cases, we must solve the H-Bianchi identity using a solution of the super Poincare’ Bianchi identities as a background. Such a solution, besides the physical fields displays a certain number of auxiliary fields. The most general solution of the super Poincare’ Bianchis we have to consider corresponds to 16⊕16 off-shell multiplet which, by suitable choices can be reduced either to the so-called old minimal or to the new minimal 12⊕12 multiplet. We give the general solution of the H-Bianchi within the 16⊕16 formulation both with and without Chern-Simons terms. This is done through the D=4 analogue of Bonora-Pasti-Tonin theorem of the 10D anomaly free supergravity. By specializing our parameters, we obtain the form of the coupling in the new minimal model retrieving in this case the results of Cecotti, Ferrara and Villasante. In addition we clarify the geometrical meaning of R-symmetry showing that in the absence of Chern-Simons forms, the condition for the embedding of the linear multiplet into the Kaehler manifold [Formula: see text] spanned by the chiral multiplets (existence on [Formula: see text] of a U(1) Killing vector) is the same condition which guarantees the existence of a local Weyl transformation by means of which the 16⊕16 curvatures can be reduced to the new minimal form and the scalar complex scalar auxiliary field S can be set to zero. Finally, we discuss the arbitrariness contained in the solution of the H-Bianchi identities at the level of the (0, 3) superspace sector. We derive the D=4 analogue of the superspace cocycle which is responsible for the Grisaru-Zanon R4-terms in the D=10 case.
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