Dissertations / Theses: 'Differential and difference algebra'

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Hendriks, Peter Anne. "Algebraic aspects of linear differential and difference equations." [S.l. : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1996. http://irs.ub.rug.nl/ppn/153769580.

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Zhao, Xiangui. "Groebner-Shirshov bases in some noncommutative algebras." London Mathematical Society, 2014. http://hdl.handle.net/1993/24315.

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Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings. This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5). In Chapter 1, we introduce the background and the goals of the thesis. In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp. In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras. In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm. Finally, in Chapter 5, we summarize our results and propose possible future work.

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El-Nakla, Jehad A. H. "Finite difference methods for solving mildly nonlinear elliptic partial differential equations." Thesis, Loughborough University, 1987. https://dspace.lboro.ac.uk/2134/10417.

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This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.

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Chen, Shaoshi. "Quelques applications de l'algébre différentielle et aux différences pour le télescopage créatif." Phd thesis, Ecole Polytechnique X, 2011. http://pastel.archives-ouvertes.fr/pastel-00576861.

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Depuis les années 90, la méthode de création télescopique de Zeilberger a joué un rôle important dans la preuve automatique d'identités mettant en jeu des fonctions spéciales. L'objectif de long terme que nous attaquons dans ce travail est l'obtension d'algorithmes et d'implantations rapides pour l'intégration et la sommation définies dans le cadre de cette création télescopique. Nos contributions incluent de nouveaux algorithmes pratiques et des critères théoriques pour tester la terminaison d'algorithmes existants. Sur le plan pratique, nous nous focalisons sur la construction de télescopeurs minimaux pour les fonctions rationnelles en deux variables, laquelle a de nombreuses applications en lien avec les fonctions algébriques et les diagonales de séries génératrices rationnelles. En considérant cette classe d'entrées contraintes, nous parvenons à mâtiner la méthode générale de création télescopique avec réduction bien connue d'Hermite, issue de l'intégration symbolique. En outre, nous avons obtenu pour cette sous-classe quelques améliorations des algorithmes classiques d'Almkvist et Zeilberger. Nos résultats expérimentaux ont montré que les algorithmes à base de réduction d'Hermite battent tous les autres algorithmes connus, à la fois en ce qui concerne la complexité au pire et en ce qui concerne les mesures de temps sur nos implantations. Sur le plan théorique, notre premier résultat est motivé par la conjecture de Wilf et Zeilberger au sujet des fonctions hyperexponentielles-hypergéométriques holonomes. Nous présentons un théorème de structure pour les fonctions hyperexponentielles-hypergéométriques de plusieurs variables, indiquant qu'une telle fonction peut s'écrire comme le produit de fonctions usuelles. Ce théorème étend à la fois le théorème d'Ore et Sato pour les termes hypergéométriques en plusieurs variables et le résultat récent par Feng, Singer et Wu. Notre second résultat est relié au problème de l'existence de télescopeurs. Dans le cas discret à deux variables, Abramov a obtenu un critère qui indique quand un terme hypergéométrique a un télescopeur. Des résultats similaires ont été obtenus pour le $q$-décalage par Chen, Hou et Mu. Ces résultats sont fondamentaux pour la terminaison des algorithmes s'inspirant de celui de Zeilberger. Dans les autres cas mixtes continus/discrets, nous avons obtenu deux critères pour l'existence de télescopeurs pour des fonctions hyperexponentielles-hypergéométriques en deux variables. Nos critères s'appuient sur une représentation standard des fonctions hyperexponentielles-hypergéométriques en deux variables, sur sur deux décompositions additives.

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Masmali, Ibtisam Ali. "Hopf algebra and noncommutative differential structures." Thesis, Swansea University, 2010. https://cronfa.swan.ac.uk/Record/cronfa42676.

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In this thesis I will study noncommutative differential geometry, after the style of Connes and Woronowicz. In particular two examples of differential calculi on Hopf algebras are considered, and their associated covariant derivatives and Riemannian geometry. These are on the Heisenberg group, and on the finite group A4. I consider bimodule connections after the work of Madore. In the last chapter noncommutative fibrations are considerd, with an application to the Leray spectral sequence. NOTATION. In this thesis equations are numbered as round brackets (), where (a.b) denotes equation b in chapter a, and references are indicated by square brackets []. This thesis has been typeset using Latex, and some figures using the Visio program.

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Nagloo, Joel Chris Ronnie. "Model theory, algebra and differential equations." Thesis, University of Leeds, 2014. http://etheses.whiterose.ac.uk/6813/.

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In this thesis, we applied ideas and techniques from model theory, to study the structure of the sets of solutions XII - XV I , in a differentially closed field, of the Painlevé equations. First we show that the generic XII - XV I , that is those with parameters in general positions, are strongly minimal and geometrically trivial. Then, we prove that the generic XII , XIV and XV are strictly disintegrated and that the generic XIII and XV I are ω-categorical. These results, already known for XI , are the culmination of the work started by P. Painlevé (over 100 years ago), the Japanese school and many others on transcendence and the Painlevé equations. We also look at the non generic second Painlevé equations and show that all the strongly minimal ones are geometrically trivial.

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Khalid, Abdul Muqeet. "Hypergeometric equation and differential-difference bispectrality." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/21411/.

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The bispectral problem was posed by Duistermaat and Grünbaum in 1986. Since then, many interesting links of this problem with nonlinear integrable PDEs, algebraic geometry, orthogonal polynomials and special functions have been found. Bispectral operators of rank one are related to the KP equation and have been completely classified by G. Wilson. For rank greater than 1 some large families related to Bessel functions are known, although the classification problem remains open. If one generalises the bispectral problem by allowing difference operators in the spectral variable, then this has a clear parallel with the three-term recurrence relation in the theory of orthogonal polynomials. This differential-difference version of the bispectral problem has also been studied extensively, more recently in the context of the exceptional orthogonal polynomials. However, the associated special functions have not been treated in such a way, until now. In our work we make a step in that direction by constructing a large family of bispectral operators related to the hypergeometric equation. In this thesis, we will fully explain our construction.

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Khanizadeh, Farbod. "Symmetry structure for differential-difference equations." Thesis, University of Kent, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.655204.

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Having infinitely many generalised symmetries is one of the definition of integrability for non-linear differential-difference equations. Therefore, it is important to develop tools by which we can produce these quantities and guarantee the integrability. Two different methods of producing generalised symmetries are studied throughout this thesis, namely recursion operators and master symmetries. These are objects that enable one to obtain the hierarchy of symmetries by recursive action on a known symmetry of a given equation. Our first result contains new Hamiltonian, symplectic and recursion operators for several (1 + 1 )-dimensional differential-difference equations both scalar and multicomponent. In fact in chapter 5 we give the factorization of the new recursion operators into composition of compatible Hamiltonian and symplectic operators. For the list of integrable equations we shall also provide the inverse of recursion operators if it exists. As the second result, we have obtained the master symmetry of differentialdifference KP equation. Since for (2+1 )-dimensional differential-difference equations recursion operators take more complicated form, " master symmetries are alternative effective tools to produce infinitely many symmetries. The notion of master symmetry is thoroughly discussed in chapter 6 and as a result of this chapter we obtain the master symmetry for the differential-difference KP (DDKP) equation. Furthermore, we also produce time dependent symmetries through sl(2, C)-representation of the DDKP equation.

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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.

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10

Xue, Fei. "Asymptotic solutions of almost diagonal differential and difference systems." Morgantown, W. Va. : [West Virginia University Libraries], 2006. https://eidr.wvu.edu/etd/documentdata.eTD?documentid=4556.

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Kendal, Margaret. "Teaching and learning introductory differential calculus with a computer algebra system /." Connect to thesis, 2001. http://eprints.unimelb.edu.au/archive/00000299.

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Glencross, Alexander Iain. "Invariant differential operators on the representation space of a quiver." Thesis, University of Sheffield, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.366103.

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Voepel, Tammy. "Variable transformations for difference equations /." free to MU campus, to others for purchase, 1997. http://wwwlib.umi.com/cr/mo/fullcit?p9841344.

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Moroianu, Sergiu 1973. "Residue functionals on the algebra of adiabatic pseudo-differential operators." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85306.

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Bowen, Matthew K. "High-order finite difference methods for partial differential equations." Thesis, Loughborough University, 2005. https://dspace.lboro.ac.uk/2134/13492.

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General n-point formulae for difference operators and their errors are derived in terms of elementary symmetric functions. These are used to derive high-order, compact and parallelisable finite difference schemes for the decay-advection-diffusion and linear damped Korteweg-de Vnes equations. Stability calculations are presented and the speed and accuracy of the schemes is compared to that of other finite difference methods in common use. Appendices contain useful tables of difference operators and errors and present a stability proof for quadratic inequalities. For completeness, the appendices conclude with the standard Thomas method for solving tri-diagonal systems.

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Cheema, Tasleem Akhter. "Higher-order finite-difference methods for partial differential equations." Thesis, Brunel University, 1997. http://bura.brunel.ac.uk/handle/2438/7131.

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This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.

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Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.

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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.

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Sudhakar, Kulabalasingham. "Equivalence of second-order differential equations via Lagrangians using computer algebra." Thesis, Queen Mary, University of London, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397811.

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Panteki, Maria. "Relationships between algebra, differential equations and logic in England 1800-1860." Thesis, Middlesex University, 1991. http://eprints.mdx.ac.uk/6482/.

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This thesis surveys the links between mathematics and algebraic logic in England in the first half of the 19th century. In particular, we show the impact that De Morgan's work on the calculus of functions in 1836 had on the shaping of his logic of relations in 1860. Similarly we study Boole's background in D-operational methods and its impact on his calculus of logic in 1847. The starting point of the thesis is Lagrange's algebraic calculus and Laplace's analytical methods prominent in late 18th century French mathematics. Revival in mathematical research in early 19th century England was mainly effected through the diffusion of Lagrange's calculus of operations as further developed by Arbogast, Servois and others in the 1800's and of Laplace's theory of attractions. Lagrange's algebraic calculus and Laplace's methods in analysis – particularly on functional equations – were considerably developed by Herschel and Babbage during the period 1812-1820. Further research on the foundations of the calculus of operations and functions was provided by Murphy, De Morgan and Gregory in the late 1830's. Symbolic methods in analysis were further extended by Boole in 1844. Boole was followed by several analysts distinguished in their obsession in further vindicating these methods through applications on two differential equations which originally appeared in Laplace's planetary physics. We record the main issues of De Morgan's logic and their mathematical background. Special reference is given to his logic of relations and its connection with his foundational study of the calculus of functions. On similar lines we study Boole's algebraic cast of logic drawing consequently a comparison between his two major works on logic. Moreover we emphasise his epistemological views and his evaluation of symbolical methods within logic and analysis.

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Säckinger, Eduard. "Theory and monolithic CMOS integration of a differential difference amplifier /." Zürich, 1989. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=8854.

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Woods, Berietta F. "Numerical instabilities in finite-difference models of ordinary differential equations." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 1989. http://digitalcommons.auctr.edu/dissertations/389.

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We investigate various mechanisms for the occurrence of numerical instabilities in the discrete modeling of ordinary differential equations by finite-differences. The Logistic equation is used to illustrate the three types of numerical instabilities. Our results can be easily generalized to arbitrary first-order differential equations.

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Xu, Yuhua. "Disconjugacy and Oscillation Theory of Linear Differential and Difference Equations." DigitalCommons@USU, 1992. https://digitalcommons.usu.edu/etd/7130.

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This dissertation is both a literature survey and a presentation of new and independent results. The survey gives an overview of disconjugacy and oscillation theory for linear differential and difference equations with an emphasis on comparing the theory of difference equations to the theory of differential equations. higher order scalar equations. Second order scalar equations, matrix equations (systems) and Hamiltonian systems are discussed. A chapter on three-term recurrences of systems is also included. Both similarities and differences between differential and difference equations are described. The new and independent results are for Hamiltonian systems of difference equations. Those results include the representation of any solution in terms of an isotropic solution, necessary conditions for disconjugacy, the development of appropriate Riccati equations and the existence of principal solutions.

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Wu, Shuhui. "The asymptotic and oscillatory behaviour of difference and differential equations." Thesis, London Metropolitan University, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.507092.

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Lewis, Benjamin. "Primitive factor rings of p-adic completions of enveloping algebras as arithmetic differential operators." Thesis, Queen Mary, University of London, 2015. http://qmro.qmul.ac.uk/xmlui/handle/123456789/9549.

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We study the -adic completion dD[1] of Berthelot's differential operators of level one on the projective line over a complete discrete valuation ring of mixed characteristic (0; p). The global sections are shown to be isomorphic to a Morita context whose objects are certain fractional ideals of primitive factor rings of the -adic completion of the universal enveloping algebra of sl2(R). We produce a bijection between the coadmissibly primitive ideals of the Arens Michael envelope of a nilpotent finite dimensional Lie algebra and the classical universal enveloping algebra. We make limited progress towards characterizing the primitive ideals of certain a noid enveloping algebras of nilpotent Lie algebras under restrictive conditions on the Lie algebra. We produce an isomorphism between the primitive factor rings of these affinoid enveloping algebras and matrix rings over certain deformations of Berthelot's arithmetic differential operators over the a fine line.

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Kıran, Güçoğlu Arzu Tanoğlu Gamze. "The solution of some differential equations by nonstandard finite difference method/." [S.l.] : [s.n.], 2005. http://library.iyte.edu.tr/tezler/master/matematik/T000332.pdf.

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Thesis (Master)--İzmir Institute of Technology, İzmir, 2005
Keywords: Nonlinear differential equations, finite difference method, numeric simulation. Includes bibliographical references (leaves. 55-57).

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Peng, Linyu. "From differential to difference : the variational bicomplex and invariant Noether's theorems." Thesis, University of Surrey, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.600018.

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A solution of a differential system can be interpreted as a maximal submanifold determined by the Cartan distribution on jet bundle. By using an exact couple, we obtain the spectral sequence associated with the bigraded structure or equivalently the differential variational bicomplex on the jet bundle, that tremendously simplifies Vinogradov's approach. The spectral sequence consists of leaves of cohomology groups, which are closely related to conservation laws. Similarly to the jet bundle structure, we construct a space for difference systems, the prolongation bundle, on which the difference variational bicomplex is built accordingly. It allows us to describe discrete mechanics globally and is applied to find conservation laws for (difference) multisymplectic systems. The exact couple methodology is applied to the corresponding (difference) bigraded structure and shows a coherence between the differential and difference cases . When the difference system contributes to an empty constraint, we prove the exactness of an amended (difference) bicomplex. This thesis also focuses on the applications of the moving frame method to Noether's theorems. we generalize group actions on smooth manifolds to the prolongation bundle, which helps us to establish a discrete counterpart of a moving frame. This is used to illustrate a discrete version of Noether's first theorem by taking a relevant variational symmetry group action and its Adjoint action into consideration. We also consider Noether's second theorem dealing with invariant variational problems whose s:ymmetry characteristics are determined by some smooth functions. The Adjoint action is used to achieve the differential relationships among the invariant Euler-Lagrange equations. When symmetry characteristics depend on constrained smooth functions, conservation laws with respect to the related Euler-Lagrange equations can be constructed. The method is illustrated by some physical examples possessing (local) SU(n) gauge symmetries.

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Antar, Ezequiel. "Risk measures and financial innovation with backward stochastic difference/differential equations." Thesis, University of Cambridge, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.708320.

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Vidyarthi, Ananta. "Digital AM Radio Navigation using differential Time Difference of Arrival Principle." University of Cincinnati / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1336762773.

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Roustemoglou, Ilia. "On the classification of integrable differential/difference equations in three dimensions." Thesis, Loughborough University, 2015. https://dspace.lboro.ac.uk/2134/17086.

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Integrable systems arise in nonlinear processes and, both in their classical and quantum version, have many applications in various fields of mathematics and physics, which makes them a very active research area. In this thesis, the problem of integrability of multidimensional equations, especially in three dimensions (3D), is explored. We investigate systems of differential, differential-difference and discrete equations, which are studied via a novel approach that was developed over the last few years. This approach, is essentially a perturbation technique based on the so called method of dispersive deformations of hydrodynamic reductions . This method is used to classify a variety of differential equations, including soliton equations and scalar higher-order quasilinear PDEs. As part of this research, the method is extended to differential-difference equations and consequently to purely discrete equations. The passage to discrete equations is important, since, in the case of multidimensional systems, there exist very few integrability criteria. Complete lists of various classes of integrable equations in three dimensions are provided, as well as partial results related to the theory of dispersive shock waves. A new definition of integrability, based on hydrodynamic reductions, is used throughout, which is a natural analogue of the generalized hodograph transform in higher dimensions. The definition is also justified by the fact that Lax pairs the most well-known integrability criteria are given for all classification results obtained.

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Fialowski, Alice, Michael Penkava, and fialowsk@cs elte hu. "Deformation Theory of Infinity Algebras." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi906.ps.

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Fabiano, Richard H. "Approximation of integro-partial differential equations of hyperbolic type." Diss., Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/74733.

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A state space model is developed for a class of integro-partial differential equations of hyperbolic type which arise in viscoelasticity. An approximation scheme is developed based on a spline approximation in the spatial variable and an averaging approximation in the de1ay variable. Techniques from linear semigroup theory are used to discuss the well-posedness of the state space model and the convergence properties of the approximation scheme. We give numerical results for a sample problem to illustrate some properties of the approximation scheme.
Ph. D.

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Mensour, Boualem. "Dynamical invariants, multistability, controllability and synchronization in delay-differential and difference equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ28360.pdf.

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Alqahtani, Amani. "Comparative studies between difference and differential equations with emphasis on logistic model." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 2016. http://digitalcommons.auctr.edu/dissertations/3109.

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This study compares the behavior of differential equations and difference equations of various orders in order to predict the state of the systems at a given time by using initial information about the system. We have demonstrated that differential equations are used in a continuous domain whereas difference equations are employed for discrete dynamical systems. Furthermore, the difference between the two models is amplified in logistic models which both are known to give explicit solutions. However, the discrete logistic model is especially superior in exhibiting a chaotic behavior of the system which the differential equation is incapable of dealing with. The conclusions drawn from the findings conform the similarities and the stark differences between these two models.

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Medlock, Jan P. "Integro-differential-equation models in ecology and epidemiology /." Thesis, Connect to this title online; UW restricted, 2004. http://hdl.handle.net/1773/6790.

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Milici, Pietro. "A quest for exactness : machines, algebra and geometry for tractional constructions of differential equations." Thesis, Paris 1, 2015. http://www.theses.fr/2015PA010675.

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Dans La Géométrie de 1637, Descartes a trouvé un “équilibre” entre constructions géométriques et manipulation symbolique au moyen de l’introduction d’opportunes machines idéales. En particulier, les instruments de Descartes étaient l’algèbre polynomiale (analyse) et une classe de constructions diagrammatiques (synthèse). Cette approche implique une classification des courbes, suivant laquelle les courbes algébriques peuvent être considérées comme “purement géométriques”. Cette limite a été dépassée à l’aide d’une méthode générale par Newton et Leibniz, en introduisant l’infini dans la partie analytique, tandis que la perspective synthétique a graduellement et de plus en plus perdu de son importance par rapport à la perspective analytique (la géométrie devient un moyen de visualisation et cesse d’être un moyen de construction). L’approche fondationnelle de Descartes (analyse par éléments finis et synthèse par constructions diagrammatiques) a été tout de même étendue au-delà des limites des courbes algébriques, bien qu’en deux périodes distinctes. Vers la fin du XVII siècle la partie synthétique a été étendue avec le “mouvement tractionnel” (construction de courbes transcendantes à l’aide de machines idéalisées) et vers le début du XX siècle la partie analytique a été étendue avec l’“algèbre différentielle” (de nos jours, considérée comme une branche de l’algèbre computationnelle). L’objectif de cette thèse est de prouver comment il est possible d’obtenir un nouvel équilibre entre ces extensions (synthétique et analytique) des instruments cartésiens, un équilibre dépassant la limite des courbes algébriques et permettant de traiter une classe de problèmes transcendants. En d’autres termes, le but est de mettre en exergue comment une nouvelle convergence de machines, algèbre et géométrie est possible, permettant une fondation d’une partie de l’analyse infinitésimale sans exigence conceptuelle de l’infini. Ce travail se caractérise par l’attention qui est portée sur le rôle constructif de la géométrie (en tant qu’idéalisation du comportement de machines opportunes) à des fins fondationnelles. Cette approche, suite à la “dé-géométrisation” des mathématiques, se détache fortement du courant principal des discussions sur les mathématiques, notamment du point de vue fondationnel. Toutefois, même si aujourd’hui cette question est tombée dans l’oubli, le problème de définir des critères de constructions appropriés, très débattu à l’âge classique, a eu de profondes influences sur la façon dont les objets et les méthodes mathématiques de l’époque ont été définis. D’après la définition de Bos [2001], ce sont là les “problèmes d’exactitude” de la géométrie. Ces problèmes d’exactitude ont trait aux interprétations philosophiques et psychologiques, c’est pourquoi ils sont normalement considérés comme externes aux mathématiques. Toutefois, même si je ne vais pas apporter de réponse exhaustive, dans mes conclusions je propose une approche algorithmique (très primitive) pour cerner ces problèmes, que j’espère pouvoir approfondir dans des travaux à venir. Depuis la perspective des sciences cognitives, cette approche par rapport à l’analyse infinitésimale ne demande pas l’infini et, grâce aux machines idéalisées, peut être conçue au travers d’opportunes “métaphores fondatrices” (selon la terminologie de Lakoff and Núñez [2000]). Ce caractère concret peut avoir des retombées utiles dans la didactique des mathématiques, grâce à l’usage d’artefacts tant physiques que numériques (cette partie ne sera abordée que de façon marginale)
In La Géométrie, Descartes proposed a “balance” between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In particular, Cartesian tools were polynomial algebra (analysis) and a class of diagrammatic constructions (synthesis). This setting provided a classification of curves, according to which only the algebraic ones were considered “purely geometrical.” This limit was overcome with a general method by Newton and Leibniz introducing the infinity in the analytical part, whereas the synthetic perspective gradually lost importance with respect to the analytical one—geometry became a mean of visualization, no longer of construction. Descartes’s foundational approach (analysis without infinitary objects and synthesis with diagrammatic constructions) has, however, been extended beyond algebraic limits, albeit in two different periods. In the late 17th century, the synthetic aspect was extended by “tractional motion” (construction of transcendental curves with idealized machines). In the first half of the 20th century, the analytical part was extended by “differential algebra,” now a branch of computer algebra. This thesis seeks to prove that it is possible to obtain a new balance between these synthetic and analytical extensions of Cartesian tools for a class of transcendental problems. In other words, there is a possibility of a new convergence of machines, algebra, and geometry that gives scope for a foundation of (a part of) infinitesimal calculus without the conceptual need of infinity. The peculiarity of this work lies in the attention to the constructive role of geometry as idealization of machines for foundational purposes. This approach, after the “de-geometrization” of mathematics, is far removed from the mainstream discussions of mathematics, especially regarding foundations. However, though forgotten these days, the problem of defining appropriate canons of construction was very important in the early modern era, and had a lot of influence on the definition of mathematical objects and methods. According to the definition of Bos [2001], these are “exactness problems” for geometry. Such problems about exactness involve philosophical and psychological interpretations, which is why they are usually considered external to mathematics. However, even though lacking any final answer, I propose in conclusion a very primitive algorithmic approach to such problems, which I hope to explore further in future research. From a cognitive perspective, this approach to calculus does not require infinity and, thanks to idealized machines, can be set with suitable “grounding metaphors” (according to the terminology of Lakoff and Núñez [2000]). This concreteness can have useful fallouts for math education, thanks to the use of both physical and digital artifacts (this part will be treated only marginally)
Ne La Géométrie del 1637 Descartes ha proposto un “equilibrio” tra costruzioni geometriche e manipolazioni simboliche con l’introduzione di opportune macchine ideali. In particolare gli strumenti di Descartes erano l’algebra polinomiale (analisi) e una classe di costruzioni diagrammatiche (sintesi). Questa impostazione implica una classificazione delle curve, secondo cui solo quelle algebriche possono essere considerate “puramente geometriche”. Questo limite è stato superato con un metodo generale da Newton e Leibniz introducendo l’infinito nella parte analitica, mentre la prospettiva sintetica ha gradualmente sempre più perso importanza rispetto a quella analitica (la geometria diventa un mezzo di visualizzazione e non più di costruzione). L’approccio fondazionale di Descartes (analisi con oggetti finiti e sintesi con costruzioni diagrammatiche) è stato comunque esteso oltre i limiti delle curve algebriche, anche se in due periodi distinti. Nel tardo XVII secolo la parte sintetica è stata estesa con il “movimento trazionale” (costruzione di curve trascendenti con macchine idealizzate), e nella prima metà del XX secolo la parte analitica è stata estesa con la “algebra differenziale” (oggigiorno considerata una branca dell’algebra computazionale). L’obiettivo di questa tesi è di provare come sia possibile ottenere un nuovo equilibrio tra queste estensioni (sintentica e analitica) degli strumenti Cartesiani, un equilibrio che superi il limite delle curve algebriche e permetta di trattare una classe di problemi trascendenti. In altre parole, l’obiettivo è di evidenziare come sia possibile una nuova convergenza di macchine, algebra e geometria che permetta una fondazione di (parte della) analisi infinitesimale senza il bisogno concettuale dell’infinito. La caratteristica di questo lavoro è l’attenzione al ruolo costruttivo della geometria (come idealizzazione del comportamento di opportune macchine) per fini fondazionali. Questo approccio, dopo la “de-geometrizzazione” della matematica, è molto distante dal filone principale delle discussioni sulla matematica, specie dal punto di vista fondazionale. Comunque, anche se oggigiorno caduto in oblio, il problema di definire degli appropriati canoni di costruzioni era molto sentito nel periodo della prima età moderna, ed ha avuto profonde influenze sul modo in cui sono stati definiti gli oggetti e i metodi matematici dell’epoca. Secondo la definizione di Bos [2001], questi sono i “problemi di esattezza” per la geometria. Questi problemi di esattezza riguardano interpretazioni filosofiche e psicologiche, pertanto sono solitamente considerati esterni alla matematica. Comunque, anche se senza una risposta esaustiva, nelle conclusioni propongo un approccio algoritmico (molto primitivo) per inquadrare tali problemi, che spero di approfondire in lavori futuri

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Yillikci, Yildirim Kemal. "Finite difference techniques and rotor blade aeroelastic partial differential equations with quasisteady aerodynamics." Diss., Georgia Institute of Technology, 1988. http://hdl.handle.net/1853/11819.

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Eckel, Matthew G. "Difference Over Identity: Overturning the Paradigm of Identity with Gilles Deleuze's Differential Ontology." University of Toledo / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1396576703.

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Huber, Sarah. "Finite difference schemes for elliptic partial differential equations requiring a non-uniform mesh." Thesis, University of British Columbia, 2015. http://hdl.handle.net/2429/55607.

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A variety of finite difference schemes are explored for the numerical solution of elliptic partial differential equations, specifically the Poisson and convection-diffusion equations. Problems are investigated that require the use of a non-uniform or non-square mesh. This may be due to a non-square domain or a problem with a singularity. We explore the properties of the linear operators in the resulting systems of linear equations. In particular, we investigate the conditioning and eigenvalues of these operators, both numerically and in search of an approximation of these eigenvalues. We also investigate the choice of finite difference scheme with respect to accuracy and cost.
Science, Faculty of
Computer Science, Department of
Graduate

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Zeris, E. A. "Investigations of certain implicit finite difference schemes for integration of the long wave equations." Thesis, University of Strathclyde, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.381538.

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Veloso, Marcelo Oliveira. "Derivações localmente nilpotentes de certas k-algebras finitamente geradas." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306295.

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Orientador: Paulo Roberto Brumatti
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Este trabalho é dedicado ao estudo das derivações localmente nilpotentes de certas K-álgebras finitamente geradas, onde K é um corpo de característica zero. Estes domínios são generalizações de anéis bem conhecidos na literatura sendo um deles o anel de Fermat. Mais precisamente, caracterizamos o conjunto das derivações localmente nilpotentes destes domínios ou de um subconjunto deste conjunto. Também calculamos o ML invariante destes domínios e como aplicação direta destas informações encontramos um conjunto de geradores para o grupo dos automorfismos de um destes domínos. No caso do anel de Fermat mostramos que nem sempre temos um domíno rígido. Além disso, verificamos que a Conjectura de Nakai é verdadeira para o anel de Fermat.
Abstract: This work is dedicated to the study of locally nilpotent derivations of certain finitely generated K-algebras, where K is a field of zero characteristic. These domains are generalizations of the well-known rings in the literature. One of this is the Fermat ring. More precisely, we characterize the set of locally nilpotent derivations of these domains or some subsets of this set. We also calculate the ML invariant of these domains and as a direct application of these results we find a set of generators for the group of automorphisms of some of these domains. We show that the Fermat ring is not always a rigid domain. Furthermore, we prove that Nakai's conjecture is true for the ring Fermat.
Doutorado
Algebra Comutativa
Doutor em Matemática

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Svärd, Magnus. "Stable high-order finite difference methods for aerodynamics /." Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621.

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Cull, Laura Katherine. "Differential presence : Deleuze and performance." Thesis, University of Exeter, 2009. http://hdl.handle.net/10036/97094.

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This thesis argues that presence in the performing arts can be reconceived, via the philosophy of Gilles Deleuze, as an encounter with difference or ‘differential presence’ which is variously defined as immanence, destratification, affect/becoming, and duration. These definitions are developed through a series of four analyses of exemplary performance practices: 1) The Living Theatre; 2) Antonin Artaud; 3) Allan Kaprow and 4) Goat Island. Chapter One recuperates the Living Theatre from a dominant narrative of ‘failure’, aided by the Deleuzian concepts of ontological participation, immanence, production/creation and ‘the people to come’. Reframing the company as pioneers of methods such as audience participation and collective creation, the chapter argues that their theatrical ambition is irreducible to some simple pursuit of undifferentiated presence (as authenticity or communion). Chapter Two provides an exposition of three key concepts emerging in the encounter between Artaud and Deleuze: the body without organs, the theatre without organs, and the destratified voice. The chapter proposes that To have done with the judgment of god constitutes an instance of a theatre without organs that uses the destratified voice in a pursuit of differential presence – as a nonrepresentative encounter with difference that forces new thoughts upon us. Chapter Three defines differential presence in relation to Deleuze’s concepts of affect and becoming-imperceptible and Kaprow’s concepts of ‘experienced insight’, nonart, ‘becoming “the whole”’, and attention. The chapter argues that Kaprow and Deleuze share a concern to theorize the practice of participating in actuality beyond the subject/object distinction, in a manner that promotes an ethico-political sense of taking part in “the whole”. Finally, Chapter Four focuses on the temporal aspect of differential presence, arguing that through slowness, waiting, repetition and imitation, Goat Island’s performance work acknowledges and responds to ‘the need to open ourselves affectively to the actuality of others’ (Mullarkey 2003: 488).

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Hicks, Jesse W. "Classification of Spacetimes with Symmetry." DigitalCommons@USU, 2016. https://digitalcommons.usu.edu/etd/5054.

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Spacetimes with symmetry play a critical role in Einstein's Theory of General Relativity. Missing from the literature is a correct, usable, and computer accessible classification of such spacetimes. This dissertation fills this gap; specifically, we i) give a new and different approach to the classification of spacetimes with symmetry using modern methods and tools such as the Schmidt method and computer algebra systems, resulting in ninety-two spacetimes; ii) create digital databases of the classification for easy access and use for researchers; iii) create software to classify any spacetime metric with symmetry against the new database; iv) compare results of our classification with those of Petrov and find that Petrov missed six cases and incorrectly normalized a significant number of metrics; v) classify spacetimes with symmetry in the book Exact Solutions to Einstein’s Field Equations Second Edition by Stephani, Kramer, Macallum, Hoenselaers, and Herlt and in Komrakov’s paper Einstein-Maxwell equation on four-dimensional homogeneous spaces using the new software.

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Andersson, Tove. "Approaches to differential gene expression analysis in atherosclerosis." Doctoral thesis, KTH, Biotechnology, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3400.

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Todays rapid development of powerful tools for geneexpression analysis provides unprecedented resources forelucidating complex molecular events.

The objective of this workhas been to apply, combine andevaluate tools for analysis of differential gene expressionusing atherosclerosis as a model system. First, an optimisedsolid-phase protocol for representational difference analysis(RDA) was applied to twoin vitromodel systems. Initially, The RDA enrichmentprocedure was investigated by shotgun cloning and sequencing ofsuccessive difference products. In the subsequent steps,combinations of RDA and microarray analysis were used tocombine the selectivity and sensitivity of RDA with thehigh-throughput nature of microarrays. This was achieved byimmobilization of RDA clones onto microarrays dedicated forgene expression analysis in atherosclerosis as well ashybridisation of labelled RDA products onto global microarrayscontaining more than 32,000 human clones. Finally, RDA wasapplied for the investigation of the focal localisation ofatherosclerotic plaques in mice usingin vivotissue samples as starting material.

A large number of differentially expressed clones wereisolated and confirmed by real time PCR. A very diverse rangeof gene fragments was identified in the RDA products especiallywhen they were screened with global microarrays. However, themicroarray data also seem to contain some noise which is ageneral problem using microarrays and should be compensated forby careful verification of the results.

Quite a large number of candidate genes related to theatherosclerotic process were found by these studies. Inparticular several nuclear receptors with altered expression inresponse to oxidized LDL were identified and deserve furtherinvestigation. Extended functional annotation does not liewithin the scope of this thesis but raw data in the form ofnovel sequences and accession numbers of known sequences havebeen made publicly available in GenBank. Parts of the data arealso available for interactive exploration on-line through aninteractive software tool. The data generated thus constitute abase for new hypotheses to be tested in the field ofatherosclerosis.

Keywords:representational difference analysis, geneexpression profiling, microarray analysis, atherosclerosis,foam cell formation

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Kama, Phumezile. "Non-standard finite difference methods in dynamical systems." Thesis, Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-07132009-163422.

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Biânchi, Angelo Calil 1984. "Sobre a existencia de bases SAGBI finitas para o nucleo de k-derivações em k[x1,...,xn]." [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306306.

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Orientador: Paulo Roberto Brumatti
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica
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Resumo: O objetivo geral desse trabalho é entender a teoria das bases SAGBI num ponto de vista estrutural, buscando critérios para sua existência e resultados que comprovem sua eficácia para o estudo de certas k-subalgebras de k[x], bem como estudar a teoria geral das derivações sobre anéis de polinômios, suas localizações e quocientes, visando explorar as propriedades algébricas do núcleo destas derivações e as estruturas das k-subalgebras de k[x] que podem ser vistas como tais núcleos. O objetivo específico é estudar a teoria algébrico-geométrica para k-derivações em k[x], desenvolvida por Shigeru Kuroda, e utilizar dessa teoria para estabelecer uma condição para que o núcleo de uma tal derivação seja uma k-subalgebra finitamente gerada e outra para que este possua uma base SAGBI finita. Em cada momento ao longo do trabalho também é desejado enfatizar o comportamento das k-derivações que são localmente nilpotentes e obter uma forma algorítmica para determinar os geradores de seus núcleos, no caso particular da derivação ao possuir uma slice
Abstract: The general objective of this work is to understand the SAGBI bases theory from a structural point of view, seeking criterias for it¿s existence and results that prove it¿s effitiency in the study of certain subalgebras of k[x], as well as to study the general theory of derivations over polynomial rings, it¿s localizations and quotients, in order to explore the algebraic properties of the kernel of this derivations and the structures of the k-subalgebras of k[x] that may be seen as such kernels. The specific objective is to study the algebraic-geometric theory of k-derivations in k[x], developed by Shigeru Kuroda, and to use this theory to stabilish a condition for the kernel of one such derivation to be a finitely generated k-subalgebra and another condition for this derivation to have finite SAGBI base. Along this work we also want to emphasize the behavior of locally nilpotent k-derivations and to obtain an algorithmic way to determine the generators of it¿s kernels, in the particular case that the derivation has a slice
Mestrado
Matematica
Mestre em Matemática

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Neal, David R. "Finite difference approximations of advection-diffusion equations for modeling shark populations /." Electronic version (PDF), 2007. http://dl.uncw.edu/etd/2007-3/neald/davidneal.pdf.

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Pefferly, Robert J. "Finite difference approximations of second order quasi-linear elliptic and hyperbolic stochastic partial differential equations." Thesis, University of Edinburgh, 2001. http://hdl.handle.net/1842/11244.

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This thesis covers topics such as finite difference schemes, mean-square convergence, modelling, and numerical approximations of second order quasi-linear stochastic partial differential equations (SPDE) driven by white noise in less than three space dimensions. The motivation for discussing and expanding these topics lies in their implications in such physical phenomena as signal and information flow, gravitational and electromagnetic fields, large scale weather systems, and macro-computer networks. Chapter 2 delves into the hyperbolic SPDE in one space and one time dimension. This is an important equation to such fields as signal processing, communications, and information theory where singularities propagate throughout space as a function of time. Chapter 3 discusses some concepts and implications of elliptic SPDE's driven by additive noise. These systems are key for understanding steady state phenomena. Chapter 4 presents some numerical work regarding elliptic SPDE's driven by multiplicative and general noise. These SPDE's are open topics in the theoretical literature, hence numerical work provides significant insight into the nature of the process. Chapter 5 presents some numerical work regarding quasi-geostrophic geophysical fluid dynamics involving stochastic noise and demonstrates how these systems can be represented as a combination of elliptic and hyperbolic components.

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Niu, Li. "Studies of imbalance difference theory in modeling conversion between differential mode and common mode signals." Thesis, Clemson University, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3680740.

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This dissertation describes three related studies regarding the imbalance difference theory in modeling the conversion between differential mode and common mode/antenna mode signals. The topics covered are: rigorous derivation of imbalance difference theory for modeling radiated emission problems, modeling the conversion between differential mode and common mode propagation in transmission lines, and modeling the loading impedance on differential mode signals due to radiated emissions.

The imbalance difference theory describes a method for calculating the coupling between differential mode signals and common mode signals due to changes in electrical balance on a transmission line. It provides both physical insight and a simple technique for modeling the conversions between the two modes.

The first chapter presents a rigorous derivation of imbalance difference theory for modeling radiated emission problems. Although the theory has been successfully used to model a wide variety of important EMC problems over the past, it has not been rigorously derived. The derivation carefully defines the important quantities and demonstrates that imbalance difference calculations are exact provided that the differential-mode propagation is TEM and the current division factor, h, represents the actual ratio of currents on the two transmission line conductors excited by a common-mode source. This chapter also discusses the acquisition of the current division factor from 2D calculations of the cross-section of the transmission line.

The second chapter provides a rigorous development of the imbalance difference theory for three-conductor transmission lines where both the differential mode and common mode exhibit TEM propagation. It also derives expressions for the mode conversion impedances, which account for the energy converted from one mode to the other. They are essential for modeling the conversion between the two modes when they are strongly coupled.

The third chapter introduces conversion impedance to the existing imbalance difference theory model for modeling radiated emission problems, so that when the coupling between differential mode and antenna mode are strong, the imbalance difference theory can more accurately estimate the antenna mode current.

All three papers are about confirming, enriching and expanding the imbalance difference theory. The first chapter focuses on the rigorous derivation of theory for its most common application, radiated emission problems. The second chapter expands the theory to multi-conductor transmission line structure when the two modes are strongly coupled. The last chapter introduces conversion impedance to the theory in modeling radiated emission problems and improves the accuracy of the model at resonant frequencies.

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Tribastone, Mirco. "Scalable analysis of stochastic process algebra models." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4629.

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The performance modelling of large-scale systems using discrete-state approaches is fundamentally hampered by the well-known problem of state-space explosion, which causes exponential growth of the reachable state space as a function of the number of the components which constitute the model. Because they are mapped onto continuous-time Markov chains (CTMCs), models described in the stochastic process algebra PEPA are no exception. This thesis presents a deterministic continuous-state semantics of PEPA which employs ordinary differential equations (ODEs) as the underlying mathematics for the performance evaluation. This is suitable for models consisting of large numbers of replicated components, as the ODE problem size is insensitive to the actual population levels of the system under study. Furthermore, the ODE is given an interpretation as the fluid limit of a properly defined CTMC model when the initial population levels go to infinity. This framework allows the use of existing results which give error bounds to assess the quality of the differential approximation. The computation of performance indices such as throughput, utilisation, and average response time are interpreted deterministically as functions of the ODE solution and are related to corresponding reward structures in the Markovian setting. The differential interpretation of PEPA provides a framework that is conceptually analogous to established approximation methods in queueing networks based on meanvalue analysis, as both approaches aim at reducing the computational cost of the analysis by providing estimates for the expected values of the performance metrics of interest. The relationship between these two techniques is examined in more detail in a comparison between PEPA and the Layered Queueing Network (LQN) model. General patterns of translation of LQN elements into corresponding PEPA components are applied to a substantial case study of a distributed computer system. This model is analysed using stochastic simulation to gauge the soundness of the translation. Furthermore, it is subjected to a series of numerical tests to compare execution runtimes and accuracy of the PEPA differential analysis against the LQN mean-value approximation method. Finally, this thesis discusses the major elements concerning the development of a software toolkit, the PEPA Eclipse Plug-in, which offers a comprehensive modelling environment for PEPA, including modules for static analysis, explicit state-space exploration, numerical solution of the steady-state equilibrium of the Markov chain, stochastic simulation, the differential analysis approach herein presented, and a graphical framework for model editing and visualisation of performance evaluation results.

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Dissertations / Theses on the topic 'Differential and difference algebra'

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