Relevant bibliographies by topics / Free Differential Algebras

Academic literature on the topic 'Free Differential Algebras'

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Published: 14 March 2023

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Journal articles on the topic "Free Differential Algebras"

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GAO, XING, LI GUO, and 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 (February 25, 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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QIU, JIANJUN, and YUQUN CHEN. "COMPOSITION-DIAMOND LEMMA FOR λ-DIFFERENTIAL ASSOCIATIVE ALGEBRAS WITH MULTIPLE OPERATORS." Journal of Algebra and Its Applications 09, no. 02 (April 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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Kadeishvili, T., and P. Real. "Free resolutions for differential modules over differential algebras." Journal of Mathematical Sciences 152, no. 3 (July 2008): 307–22. http://dx.doi.org/10.1007/s10958-008-9072-9.

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

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

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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 (June 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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SÁNCHEZ, OMAR LEÓN, and RAHIM MOOSA. "THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS." Journal of Symbolic Logic 81, no. 2 (June 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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BENKHALIFA, MAHMOUD. "WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA." International Journal of Mathematics 15, no. 10 (December 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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Salgado, P., and S. Salgado. "Extended gauge theory and gauged free differential algebras." Nuclear Physics B 926 (January 2018): 179–99. http://dx.doi.org/10.1016/j.nuclphysb.2017.10.026.

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

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Dissertations / Theses on the topic "Free Differential Algebras"

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Dabrowski, Yoann. "Free entropies, free Fisher information, free stochastic differential equations, with applications to Von Neumann algebras." Thesis, Paris Est, 2010. http://www.theses.fr/2010PEST1015.

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Ce travail étend nos connaissances des entropies libres et des équations différentielles stochastiques (EDS) libres dans trois directions. Dans un premier temps, nous montrons que l'algèbre de von Neumann engendrée par au moins deux autoadjoints ayant une information de Fisher finie n'a pas la propriété $Gamma$ de Murray et von Neumann. C'est un analogue d'un résultat de Voiculescu pour l'entropie microcanonique libre. Dans un second temps, nous étudions des EDS libres à coefficients opérateurs non-bornés (autrement dit des sortes d' EDP stochastiques libres ). Nous montrons la stationnarité des solutions dans des cas particuliers. Nous en déduisons un calcul de la dimension entropique libre microcanonique dans le cas d'une information de Fisher lipschitzienne. Dans un troisième et dernier temps, nous introduisons une méthode générale de résolutions d'EDS libres stationnaires, s'appuyant sur un analogue non-commutatif d'un espace de chemins. En définissant des états traciaux sur cet analogue, nous construisons des dilatations markoviennes de nombreux semigroupes complètement markoviens sur une algèbre de von Neumann finie, en particulier de tous les semigroupes symétriques. Pour des semigroupes particuliers, par exemple dès que le générateur s'écrit sous une forme divergence pour une dérivation à valeur dans la correspondance grossière, ces dilatations résolvent des EDS libres. Entre autres applications, nous en déduisons une inégalité de Talagrand pour l'entropie non-microcanonique libre (relative à une sous-algèbre et une application complètement positive). Nous utilisons aussi ces déformations dans le cadre des techniques de déformations/rigidité de Popa
This works extends our knowledge of free entropies, free Fisher information and free stochastic differential equations in three directions. First, we prove that if a $W^{*}$-probability space generated by more than 2 self-adjoints with finite non-microstates free Fisher information doesn't have property $Gamma$ of Murray and von Neumann (especially is not amenable). This is an analogue of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy. Second, we study a general free stochastic differential equation with unbounded coefficients (``stochastic PDE"), and prove stationarity of solutions in well-chosen cases. This leads to a computation of microstates free entropy dimension in case of Lipschitz conjugate variable. Finally, we introduce a non-commutative path space approach to solve general stationary free Stochastic differential equations. By defining tracial states on a non-commutative analogue of a path space, we construct Markov dilations for a class of conservative completely Markov semigroups on finite von Neumann algebras. This class includes all symmetric semigroups. For well chosen semigroups (for instance with generator any divergence form operator associated to a derivation valued in the coarse correspondence) those dilations give rise to stationary solutions of certain free SDEs. Among applications, we prove a non-commutative Talagrand inequality for non-microstate free entropy (relative to a subalgebra $B$ and a completely positive map $eta:Bto B$). We also use those new deformations in conjunction with Popa's deformation/rigidity techniques, to get absence of Cartan subalgebra results
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.

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The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.
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Rocha, Eugénio Alexandre Miguel. "Uma Abordagem Algébrica à Teoria de Controlo Não Linear." Doctoral thesis, Universidade de Aveiro, 2003. http://hdl.handle.net/10773/21444.

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Doutoramento em Matemática
Nesta tese de Doutoramento desenvolve-se principalmente uma abordagem algébrica à teoria de sistemas de controlo não lineares. No entanto, outros tópicos são também estudados. Os tópicos tratados são os seguidamente enunciados: fórmulas para sistemas de controlo sobre álgebras de Lie livres, estabilidade de um sistema de corpos rolantes, algoritmos para aritmética digital, e equações integrais de Fredholm não lineares. No primeiro e principal tópico estudam-se representações para as soluções de sistemas de controlo lineares no controlo. As suas trajetórias são representadas pelas chamadas séries de Chen. Estuda-se a representação formal destas séries através da introdução de várias álgebras não associativas e técnicas específicas de álgebras de Lie livres. Sistemas de coordenadas para estes sistemas são estudados, nomeadamente, coordenadas de primeiro tipo e de segundo tipo. Apresenta-se uma demonstração alternativa para as coordenadas de segundo tipo e obtêm-se expressões explícitas para as coordenadas de primeiro tipo. Estas últimas estão intimamente ligadas ao logaritmo da série de Chen que, por sua vez, tem fortes relações com uma fórmula designada na literatura por “continuous Baker-Campbell- Hausdorff formula”. São ainda apresentadas aplicações à teoria de funções simétricas não comutativas. É, por fim, caracterizado o mapa de monodromia de um campo de vectores não linear e periódico no tempo em relação a uma truncatura do logaritmo de Chen. No segundo tópico é estudada a estabilizabilidade de um sistema de quaisquer dois corpos que rolem um sobre o outro sem deslizar ou torcer. Constroem-se controlos fechados e dependentes do tempo que tornam a origem do sistema de dois corpos num sistema localmente assimptoticamente estável. Vários exemplos e algumas implementações em Maple°c são discutidos. No terceiro tópico, em apêndice, constroem-se algoritmos para calcular o valor de várias funções fundamentais na aritmética digital, sendo possível a sua implementação em microprocessadores. São também obtidos os seus domínios de convergência. No último tópico, também em apêndice, demonstra-se a existência e unicidade de solução para uma classe de equações integrais não lineares com atraso. O atraso tem um carácter funcional, mostrando-se ainda a diferenciabilidade no sentido de Fréchet da solução em relação à função de atraso.
In this PhD thesis several subjects are studied regarding the following topics: formulas for nonlinear control systems on free Lie algebras, stabilizability of nonlinear control systems, digital arithmetic algorithms, and nonlinear Fredholm integral equations with delay. The first and principal topic is mainly related with a problem known as the continuous Baker-Campbell-Hausdorff exponents. We propose a calculus to deal with formal nonautonomous ordinary differential equations evolving on the algebra of formal series defined on an alphabet. We introduce and connect several (non)associative algebras as Lie, shuffle, zinbiel, pre-zinbiel, chronological (pre-Lie), pre-chronological, dendriform, D-I, and I-D. Most of those notions were also introduced into the universal enveloping algebra of a free Lie algebra. We study Chen series and iterated integrals by relating them with nonlinear control systems linear in control. At the heart of all the theory of Chen series resides a zinbiel and shuffle homomorphism that allows us to construct a purely formal representation of Chen series on algebras of words. It is also given a pre-zinbiel representation of the chronological exponential, introduced by A.Agrachev and R.Gamkrelidze on the context of a tool to deal with nonlinear nonautonomous ordinary differential equations over a manifold, the so-called chronological calculus. An extensive description of that calculus is made, collecting some fragmented results on several publications. It is a fundamental tool of study along the thesis. We also present an alternative demonstration of the result of H.Sussmann about coordinates of second kind using the mentioned tools. This simple and comprehensive proof shows that coordinates of second kind are exactly the image of elements of the dual basis of a Hall basis, under the above discussed homomorphism. We obtain explicit expressions for the logarithm of Chen series and the respective coordinates of first kind, by defining several operations on a forest of leaf-labelled trees. It is the same as saying that we have an explicit formula for the functional coefficients of the Lie brackets on a continuous Baker-Campbell-Hausdorff-Dynkin formula when a Hall basis is used. We apply those formulas to relate some noncommutative symmetric functions, and we also connect the monodromy map of a time-periodic nonlinear vector field with a truncation of the Chen logarithm. On the second topic, we study any system of two bodies rolling one over the other without twisting or slipping. By using the Chen logarithm expressions, the monodromy map of a flow and Lyapunov functions, we construct time-variant controls that turn the origin of a control system linear in control into a locally asymptotically stable equilibrium point. Stabilizers for control systems whose vector fields generate a nilpotent Lie algebra with degree of nilpotency · 3 are also given. Some examples are presented and Maple°c were implemented. The third topic, on appendix, concerns the construction of efficient algorithms for Digital Arithmetic, potentially for the implementation in microprocessors. The algorithms are intended for the computation of several functions as the division, square root, sines, cosines, exponential, logarithm, etc. By using redundant number representations and methods of Lyapunov stability for discrete dynamical systems, we obtain several algorithms (that can be glued together into an algorithm for parallel execution) having the same core and selection scheme in each iteration. We also prove their domains of convergence and discuss possible extensions. The last topic, also on appendix, studies the set of solutions of a class of nonlinear Fredholm integral equations with general delay. The delay is of functional character modelled by a continuous lag function. We ensure existence and uniqueness of a continuous (positive) solution of such equation. Moreover, under additional conditions, it is obtained the Fr´echet differentiability of the solution with respect to the lag function.
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Pavarin, Alice. "Equivalences of additive categories." Doctoral thesis, Università degli studi di Padova, 2013. http://hdl.handle.net/11577/3422598.

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In the first part of the thesis, after an introduction of the concept of recollement and TTF triple in a triangulated category, we consider recollements of derived categories of differential graded algebras induced by self-orthogonal compact objects obtaining a generalization of Rickard's Theorem. Specializing to the case of partial tilting modules over a ring, we extend the results on triangle equivalences proved in [B] and [BMT]. After that we focus on the connection between recollements of derived categories of rings, bireflective subcategories and generalized universal localizations". In the second part of the thesis we give some results in the setting of monoidal categories and dual qausi-bialgebras. To every dual quasi-bialgebra H and every bialgebra R in the category of Yetter-Drinfeld modules over H, one can associate a dual quasi-bialgebra, called bosonization. In this thesis, using the fundamental theorem, we characterize as bosonizations the dual quasi-bialgebras with a projection onto a dual quasi-bialgebra with a preantipode. As an application we investigate the structure of the graded coalgebra grA associated to a dual quasibialgebra A with the dual Chevalley property (e.g. A is pointed).
Nella prima parte della tesi, dopo aver introdotto il concetto di incollamento e di triple TTF in una categoria triangolata, si considerano incollamenti di categorie derivate di algebre differenziali graduate indotti da oggetti compatti e auto ortogonali, ottenendo una generalizzazione del teorema di Rickard. Considerando il caso particolare del moduli partial tilting, estendiamo i risultati sulle equivalenze tra categorie triangolate ottenute in [B] e [BMT]. Segue una parte focalizzata sulla connessione tra incollamenti di categorie derivate di anelli, sottocategorie biriflessive e localizzazioni universali generalizzate. Nella seconda parte della tesi vengono dati alcuni risultati nell'ambito di categorie monoidali e dual quasi-bialgebre. Ad ogni dual quasi-bialgebra H e ad ogni bialgebra R nella categoria dei moduli di Yetter-Drinfeld su H, e possibile associare una dual quasi-bialgebra, chiamata bosonizzazione. In questa tesi, usando il teorema fondamentale, si caratterizza come bosonizzazione ogni dual quasi-bialgebra con proiezione su una dual quasi-bialgebra con preantipode. Come applicazione si studia la struttura della coalgebra graduata grA associata ad una dual quasi-bialgebra A con la proprieta di Chevalley duale (si vedra che A e puntata).
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Guibert, David. "Analyse de méthodes de résolution parallèles d’EDO/EDA raides." Thesis, Lyon 1, 2009. http://www.theses.fr/2009LYO10138/document.

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La simulation numérique de systèmes d’équations différentielles raides ordinaires ou algébriques est devenue partie intégrante dans le processus de conception des systèmes mécaniques à dynamiques complexes. L’objet de ce travail est de développer des méthodes numériques pour réduire les temps de calcul par le parallélisme en suivant deux axes : interne à l’intégrateur numérique, et au niveau de la décomposition de l’intervalle de temps. Nous montrons l’efficacité limitée au nombre d’étapes de la parallélisation à travers les méthodes de Runge-Kutta et DIMSIM. Nous développons alors une méthodologie pour appliquer le complément de Schur sur le système linéarisé intervenant dans les intégrateurs par l’introduction d’un masque de dépendance construit automatiquement lors de la mise en équations du modèle. Finalement, nous étendons le complément de Schur aux méthodes de type "Krylov Matrix Free". La décomposition en temps est d’abord vue par la résolution globale des pas de temps dont nous traitons la parallélisation du solveur non-linéaire (point fixe, Newton-Krylov et accélération de Steffensen). Nous introduisons les méthodes de tirs à deux niveaux, comme Parareal et Pita dont nous redéfinissons les finesses de grilles pour résoudre les problèmes raides pour lesquels leur efficacité parallèle est limitée. Les estimateurs de l’erreur globale, nous permettent de construire une extension parallèle de l’extrapolation de Richardson pour remplacer le premier niveau de calcul. Et nous proposons une parallélisation de la méthode de correction du résidu
This PhD Thesis deals with the development of parallel numerical methods for solving Ordinary and Algebraic Differential Equations. ODE and DAE are commonly arising when modeling complex dynamical phenomena. We first show that the parallelization across the method is limited by the number of stages of the RK method or DIMSIM. We introduce the Schur complement into the linearised linear system of time integrators. An automatic framework is given to build a mask defining the relationships between the variables. Then the Schur complement is coupled with Jacobian Free Newton-Krylov methods. As time decomposition, global time steps resolutions can be solved by parallel nonlinear solvers (such as fixed point, Newton and Steffensen acceleration). Two steps time decomposition (Parareal, Pita,...) are developed with a new definition of their grids to solved stiff problems. Global error estimates, especially the Richardson extrapolation, are used to compute a good approximation for the second grid. Finally we propose a parallel deferred correction
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(10724076), Daniel L. Bath. "Bernstein--Sato Ideals and the Logarithmic Data of a Divisor." Thesis, 2021.

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We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ f - f1···fr. We identify a large class of geometrically characterized germs so that the DX,x[s1,...,sr]-annihilator of fs11···fsrr admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain DX,x-map ∇A that is expected to characterize the roots of the Bernstein–Sato ideal.
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Books on the topic "Free Differential Algebras"

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A, Kaashoek M., Lancaster Peter, Langer Heinz, Lerer Leonid, and SpringerLink (Online service), eds. A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume. Basel: Springer Basel, 2012.

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I, Diaz J., and International Conference on "Free Boundary Problems: Theory and Applications" (1993 : Toledo, Spain), eds. Free boundary problems: Theory and applications. Harlow, Essex, England: Longman Scientific & Technical, 1995.

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Roe, John. Winding around: The winding number in topology, geometry, and analysis. Providence, Rhode Island: American Mathematical Society, 2015.

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1974-, Nelson Sam, ed. Quandles: An introduction to the algebra of knots. Providence, Rhode Island: American Mathematical Society, 2015.

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Heinz, Langer, Harry Dym, Marinus A. Kaashoek, Peter Lancaster, and Leonid Lerer. Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume. Springer Basel AG, 2014.

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Earl, Richard, and James Nicholson. The Concise Oxford Dictionary of Mathematics. 6th ed. Oxford University Press, 2021. http://dx.doi.org/10.1093/acref/9780198845355.001.0001.

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Over 4,000 entries This informative A to Z provides clear, jargon-free definitions of a wide variety of mathematical terms. Its articles cover both pure and applied mathematics and statistics, and include key theories, concepts, methods, programmes, people, and terminology. For this sixth edition, around 800 new terms have been defined, expanding on the dictionary’s coverage of algebra, differential geometry, algebraic geometry, representation theory, and statistics. Among this new material are articles such as cardinal arithmetic, first fundamental form, Lagrange’s theorem, Navier-Stokes equations, potential, and splitting field. The existing entries have also been revised and updated to account for developments in the field. Numerous supplementary features complement the text, including detailed appendices on basic algebra, areas and volumes, trigonometric formulae, and Roman numerals. Newly added to these sections is a historical timeline of significant mathematicians’ lives and the emergence of key theorems. There are also illustrations, graphs, and charts throughout the text, as well as useful web links to provide access to further reading.
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El Karoui, Noureddine. Algebraic geometry and matrix models. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.29.

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This article discusses the connection between the matrix models and algebraic geometry. In particular, it considers three specific applications of matrix models to algebraic geometry, namely: the Kontsevich matrix model that describes intersection indices on moduli spaces of curves with marked points; the Hermitian matrix model free energy at the leading expansion order as the prepotential of the Seiberg-Witten-Whitham-Krichever hierarchy; and the other orders of free energy and resolvent expansions as symplectic invariants and possibly amplitudes of open/closed strings. The article first describes the moduli space of algebraic curves and its parameterization via the Jenkins-Strebel differentials before analysing the relation between the so-called formal matrix models (solutions of the loop equation) and algebraic hierarchies of Dijkgraaf-Witten-Whitham-Krichever type. It also presents the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, along with higher expansion terms and symplectic invariants.
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Tretkoff, Paula. Algebraic Surfaces and the Miyaoka-Yau Inequality. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691144771.003.0005.

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This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.
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Vazquez, Juan Luis, J. I. Diaz, M. A. Herrero, and Amable Linan. Free Boundary Problems: Theory and Applications (Pitman Research Notes in Mathematics Series,). Chapman & Hall/CRC, 1995.

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Book chapters on the topic "Free Differential Algebras"

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Gao, Xing, and Li Guo. "Constructions of Free Commutative Integro-Differential Algebras." In Algebraic and Algorithmic Aspects of Differential and Integral Operators, 1–22. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-54479-8_1.

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Borowiec, A., V. K. Kharchenko, and Z. Oziewicz. "On Free Differentials on Associative Algebras." In Non-Associative Algebra and Its Applications, 46–53. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0990-1_8.

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Mikhalev, Alexander A., and Andrej A. Zolotykh. "Applications of Fox Differential Calculus to Free Lie Superalgebras." In Non-Associative Algebra and Its Applications, 285–90. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0990-1_47.

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Arias, Enrique, Vicente Hernández, and Jacinto-Javier Ibáñez. "High Performance Algorithms for Computing Nonsingular Jacobian-Free Piecewise Linearization of Differential Algebraic Equations." In Integral Methods in Science and Engineering, 7–12. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-0-8176-8184-5_2.

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Castellani, Leonardo, Riccardo D’ Auria, and Pietro Fré. "THE THEORY OF FREE DIFFERENTIAL ALGEBRAS AND SOME APPLICATIONS." In Supergravity and Superstrings: A Geometric Perspective, 794–831. WORLD SCIENTIFIC, 1991. http://dx.doi.org/10.1142/9789814542388_0023.

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"Differential idealizers and algebraic free divisors." In Commutative Algebra, 233–48. CRC Press, 2005. http://dx.doi.org/10.1201/9781420028324-20.

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Simis, Aron. "Differential idealizers and algebraic free divisors." In Commutative Algebra, 211–26. CRC Press, 2005. http://dx.doi.org/10.1201/9781420028324.ch15.

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Dawson, C. Bryan. "Alternate Representations: Parametric and Polar Curves." In Calculus Set Free, 1119–216. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780192895592.003.0009.

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“Alternate Representations: Parametric and Polar Curves” is an introduction to the algebra and differential calculus of these topics. Methods presented vary from the traditional, such as polar graph paper, to the use of technology, such as determining an interval for which a curve is traversed exactly once. Infinitesimals make only a brief appearance in the development of a formula for tangent lines. A review of conic sections precedes the presentation of conic sections in polar coordinates. Note that integration is not required for this chapter and, as a result, it can be covered earlier if desired.
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Aschenbrenner, Matthias, Lou van den Dries, and Joris van der Hoeven. "Newtonian Differential Fields." In Asymptotic Differential Algebra and Model Theory of Transseries. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691175423.003.0015.

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This chapter deals with Newtonian differential fields. Here K is an ungrounded H-asymptotic field with Γ‎ := v(Ksuperscript x ) not equal to {0}. So the subset ψ‎ of Γ‎ is nonempty and has no largest element, and thus K is pre-differential-valued by Corollary 10.1.3. An extension of K means an H-asymptotic field extension of K. The chapter first considers the relation of Newtonian differential fields to differential-henselianity before discussing weak forms of newtonianity and differential polynomials of low complexity. It then proves newtonian versions of d-henselian results in Chapter 7, leading to the following analogue of Theorem 7.0.1: If K is λ‎-free and asymptotically d-algebraically maximal, then K is ω‎-free and newtonian. Finally, it describes unravelers and newtonization.
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d’Inverno, Ray, and James Vickers. "Tensor algebra." In Introducing Einstein's Relativity, 65–84. 2nd ed. Oxford University PressOxford, 2022. http://dx.doi.org/10.1093/oso/9780198862024.003.0005.

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Abstract Chapter 5 introduces the mathematical tools needed for the study of relativity, namely, tensor analysis. To work effectively in Newtonian theory, one really needs the language of vectors, whereas, in relativity, one needs the language of tensor analysis. This part of the book is devoted to learning this formalism, which is a precondition for the rest of the book. The notion of a differentiable manifold is first introduced. The chapter then introduces tangents to curves and normals to surfaces as prototypes for vectors and co-vectors, respectively. It then goes on to introduce the elements of tensor algebra, including the tensor product and the notion of contraction. It ends by giving a coordinate-free interpretation of vector fields in terms of derivative operators.
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Conference papers on the topic "Free Differential Algebras"

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IZAURIETA, FERNANDO, EDUARDO RODRÍGUEZ, ALFREDO PÉREZ, and PATRICIO SALGADO. "EXPANDING LIE AND GAUGE FREE DIFFERENTIAL ALGEBRAS THROUGH ABELIAN SEMIGROUPS." In Proceedings of the MG12 Meeting on General Relativity. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814374552_0442.

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Rai, K. N., and D. C. Rai. "A Finite Element Method for the Solution of Free Boundary Problem." In ASME 2004 Heat Transfer/Fluids Engineering Summer Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/ht-fed2004-56777.

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A finite element method is presented for the solution of a free boundary problem which arises during planar melting of a semi-infinite medium initially at a temperature which is slightly below the melting temperature of the solid. The surface temperature is assumed to vary with time. Two different situations are considered (I) when thermal diffusivity is independent of temperature and (II) when thermal diffusivity varies linearly with temperature. The differential equation governing the process is converted to initial value problem of vector matrix form. The time function is approximated by Chebyshev series and the operational matrix of integration is applied, a linear differential equation can be represented by a set of linear algebraic equations and a nonlinear differential equation can be represented by a set of nonlinear algebraic equations. The solution of the problem is then found in terms of Chebyshev polynomial of second kind. The solution of this initial value problem is utilized iteratively in the interface heat flux equation to determine interface location as well as the temperature in two regions. The method appears to be accurate in cases for which closed form solutions are available, it agrees well with them. The effect of several parameters on the melting are analysed and discussed.
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Jackson, Dominic R., and S. Olutunde Oyadiji. "Free Vibration Analysis of Rotating Tapered Bresse-Rayleigh Beams Using the Differential Transformation Method." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-87843.

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The free vibration characteristics of a rotating tapered Rayleigh beam is analysed in this study. First, the strain-displacement relationship for the rotating beam is formulated and used to derive the kinetic and strain energies in explicit analytical form. Second, Hamilton’s variational principle is used to derive the governing differential equation of motion and the associated boundary conditions. Third, the Differential Transformation Method (DTM) is applied to reduce the governing differential equations of motion and the boundary conditions to a set of algebraic equations from which the frequency equation is derived. Next, a numerical algorithm implemented in the software package Mathematica is used to compute the natural frequencies of vibration for a few paired combinations of clamped, pinned and free end conditions of the beam. Also, the variation of the natural frequencies of vibration with respect to variations in the rotational speed, hub radius, taper ratio and the slenderness ratio is studied. The results obtained from the Bresse-Rayleigh theory are compared with results obtained from the Bernoulli-Euler and Timoshenko theories to demonstrate the accuracy and relevance of their application.
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Margetts, Rebecca, and Roger F. Ngwompo. "Comparison of Modeling Techniques for a Landing Gear." In ASME 2010 International Mechanical Engineering Congress and Exposition. ASMEDC, 2010. http://dx.doi.org/10.1115/imece2010-39722.

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A wide range of modeling techniques is available to the engineer. The objective of this paper is to compare some typical modeling techniques for the simulation of a multi-domain mechatronic system. Usual dynamic modeling methods, such as block diagrams and iconic diagrams, can cause problems for the engineer. Differential algebraic equations (DAEs) and algebraic loops can significantly increase simulation times and cause numeric errors. Bond graphs are less common in industry, and are presented here as a method which allows the engineer to easily identify causal loops and elements in differential causality. These can indicate DAEs in the underlying equations. An aircraft landing gear is given as an example of a multi-domain system, and is modeled as a block diagram, an iconic diagram and as a bond graph. The time to construct the model, time to solve and problems faced by the analyst are presented. Bond graphs offer distinct advantages in terms of the ease of implementing algebraic equations and visibility of causality. The time taken to model a system can be significantly reduced and the results appear free from computational errors. Bond graphs are therefore recommended for this type of multi-domain systems analysis.
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Bert, Charles W., and Huan Zeng. "Analysis of Axial Vibration of Compound Bars by Several Different Methods." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21564.

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Abstract Although solutions for the free vibration of dynamically slender prismatic or tapered bars in axial vibration are well known, apparently there have been only a few published analyses for the case of compound bars, i.e., bars of different cross section connected in mechanical series. The primary objective here is to analyze this problem by several exact methods: (1) classical solution of the set of governing differential equations, and (2) application of a relatively new technique originated in 1986 and known as the differential transformation (DT) method. The secondary objective is to present some simple approximate algebraic formulas for the fundamental frequency of an n-segment, axially vibrating bar for preliminary design use.
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Barhorst, Alan A., and Louis J. Everett. "Obtaining the Minimal Set of Hybrid Parameter Differential Equations for Mechanisms." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0395.

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Abstract Mechanisms are inherently constrained devices. Combining flexibility with mechanisms usually requires using Lagrange multipliers to handle the constraints. The added algebraic or numerical tedium, associated with the Lagrange multipliers, is well documented. Presented in this paper is a technique for obtaining the minimal set of hybrid parameter differential equations for a constrained device. That is, the set of equations that inherently incorporate the constraints. The technique illustrated in this paper is a recently developed hybrid parameter multiple body (HPMB) system modeling methodology. The variational nature of the methodology allows rigorous equation formulation providing not only the complete nonlinear, hybrid differential equations, but also the boundary conditions. The methodology is formulated in the constraint-free subspace of the system’s configuration space, thus Lagrange multipliers are not needed for constrained systems, regardless of the constraint type (holonomic or nonholonomic). To evince the utility of the method, a flexible four bar mechanism is modeled. Particularly, the inversion of the slider crank found in the quick return mechanism. A comparison of Hamilton’s principle and the described technique, as they are applied to the mechanism, is included. It is shown that the same equations result from either method, but the new technique is much more concise, more efficiently handles the constraints, and requires less algebraic tedium.
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Zhu, W. D., Y. G. Mao, and G. X. Ren. "Dynamic Modeling and Analysis of Three-Dimensional Slack Cables With Application to Elevator Traveling Cables." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-53691.

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This paper addresses three-dimensional dynamic modeling of a moving elevator traveling cable with bending and torsional stiffnesses and arbitrarily moving ends. An absolute nodal coordinate formulation based on Rayleigh beam theory is introduced to model the traveling cable. Dynamic equations of motion, which are presented as differential algebraic equations, are solved by the backward differentiation formula. Equilibria of a traveling cable with different cable parameters and car positions are first calculated. Motions of cable ends are prescribed next to simulate the free response of the traveling cable due to motion of the car. Finally, effects of different types of building sways on dynamic responses of the traveling cable are examined.
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Farid, Mehrdad, and Stanislaw A. Lukasiewicz. "On Dynamic Modeling of Multi-Link Spatial Manipulators With Flexible Links and Joints." In ASME 1997 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/imece1997-0536.

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Abstract A redundant Lagrangian/finite element approach is proposed to model the dynamics of lightweight spatial manipulators with both flexible links and joints. The links are assumed to be deformable due to bending and torsion. The elastic deformations of each link are expressed in its tangential (clamped free) local floating frame. The constraint equations due to the connectivity of the links are added to the equations of motion of the system by using Lagrange multipliers. The resulting mixed set of nonlinear differential equations and algebraic equations (DAEs) is solved numerically to predict the dynamic behavior of the system. The dynamic model derived here is free from the assumption of a nominal motion and takes into account not only the coupling effects between the rigid body motion and the elastic deformations of the links, but also the interaction between flexible links and actuated flexible joints.
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Terze, Zdravko, Milan Vrdoljak, and Dario Zlatar. "Numerical Flight Vehicle Forward Dynamics With State-Space Lie-Group Integration Scheme." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12319.

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Dynamic simulation procedures of flight vehicle maneuvers need robust and efficient integration methods in order to allow for reliable simulation missions. Derivation of such integration schemes in Lie-group settings is especially efficient since the coordinate-free Lie-group dynamical models operate directly on SO(3) rotational matrices and angular velocities, avoiding local rotation parameters and artificial algebraic constraints as well as kinematical differential equations. In the adopted modeling approach, a state-space of the flight vehicle (modeled as a multi-body system comprising rigid bodies) is modeled as a Lie-group. The numerical algorithm is demonstrated and tested within the framework of the characteristic case study of the aircraft 3D maneuver.
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Zhu, W. D., H. Ren, and C. Xiao. "A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application to Elevator Traveling and Compensation Cables." In ASME 2010 International Mechanical Engineering Congress and Exposition. ASMEDC, 2010. http://dx.doi.org/10.1115/imece2010-39219.

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A nonlinear, planar model of a slack cable with bending stiffness and arbitrarily moving ends is developed. The model uses the slope angle of the centroid line of the cable to describe the motion of the cable, and the resulting integro-partial differential equation with constraints is derived using Hamilton’s principle. A new method is developed to obtain the spatially discretized equations, and the Baumgarte stabilization procedure is used to solve the resulting differential-algebraic equations. The model can be used to calculate the equilibria and corresponding free vibration characteristics of the cable, as well as the dynamic response of the cable under arbitrarily moving ends. The results for an equilibrium and free vibration characteristics around the equilibrium are experimentally validated on a laboratory steel band. The methodology is applied to elevator traveling and compensation cables. It is found that a vertical motion of the car can introduce a horizontal vibration of a traveling or compensation cable. The results presented are verified by a commercial finite element software. The current method is shown to be more efficient than the finite element method as it uses a much smaller number of elements to reach the same accuracy. Some other interesting features include the condition for a traveling or compensation cable equilibrium to be closest to a natural loop and a direct proof that the catenary solution is unique.
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