A thesis submitted to the faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Doctor of Philosophy, / The inverse problem in the Calculus of Variations involves determining the Lagrangians, if any, associated with a given (system of) differential equation(s). One can classify Lagrangians according to the Lie algebra of symmetries of the Action integral (the Noether algebra). We give a complete classification of first-order Lagrangians defined on the line and produce results pertaining to the dimensionality of the algebra of Noether symmetries and compare and contrast these with similar results on the algebra of Lie symmetries of the corresponding Euler-Lagrange .equations. It is proved that the maximum dimension of the Noether point symmetry algebra of a particle Lagrangian. is five whereas it is known that the maximum dimension Qf the Lie algebra of the corresponding scalar second-order Euler-Lagrange equation is eight. Moreover, we show th'a.t a particle Lagrangian does not admit a maximal four-dimensional Noether point symmeiry algebra and consequently a particle Lagrangian admits the maximal r E {O, 1,2,3, 5}-dimensional Noether point symmetry algebra, It is well .known that an important means of analyzing differential equations lies in the knowledge of the first integrals of the equation. We deliver an algorithm for finding first integrals of partial differential equations and show how some of the symmetry properties of the first integrals help to 'further' reduce the order of the equations and sometimes completely solve the equations. Finally, we discuss some open questions. These include the inverse problem and classification of partial differential equations. ALo, there is the question of the extension of the results to 'higher' dimensions. / Andrew Chakane 2018

PhD., Faculty of Science, University of the Witwatersrand, 2011 / The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian (or via the recently developed concept of partial Lagrangians) is well known. The formulae to determine these for higher-order flows is somewhat cumbersome and becomes more so as the order increases. We carry out these for a class of fourth, fifth and sixth order PDEs. In the latter case, we involve the fifth-order KdV equation using the concept of ‘weak’ Lagrangians better known for the third-order KdV case. We then consider the case of a mixed ‘high-order’ equations working on the Shallow Water Wave and Regularized Long Wave equations. These mixed type equations have not been dealt with thus far using this technique. The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian is well known. In some of the examples, our focus is that the resultant conserved flows display some previously unknown interesting ‘divergence properties’ owing to the presence of the mixed derivatives. We then analyse the conserved flows of some multi-variable equations that arise in Relativity. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we also show the variations that occur for the unknown functions. We discuss symmetries of classes of wave equations that arise as a consequence of the Vaidya metric. The objective of this study is to show how the respective geometry is responsible for giving rise to a nonlinear inhomogeneous wave equation as an alternative to assuming the existence of nonlinearities in the wave equation due to physical considerations. We find Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical 4 conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations (on a ‘flat geometry’). Finally, we pursue the nature of the flow of a third grade fluid with regard to its underlying conservation laws. In particular, the fluid occupying the space over a wall is considered. At the surface of the wall, suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a class of PDEs. A complete class of conservation laws for the resulting equations are constructed and analysed using the invariance properties of the corresponding multipliers/characteristics.

In this paper, we study primary decomposition of any proper submodule N of a module M over a noetherian ring R. We start by briefly discussing basic facts about the very well known case where M is a finitely generated module over a Noetherian ring R, then we proceed to discuss the general case where M is any module over a Noetherian ring R. We put a lot of focus on the associated primes that occur with the primary decomposition, essentially studying their uniqueness and their relation to the associated primes of M/N.

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / Seja p um nÃmero primo Ãmpar. Uma famÃlia de reticulados (p-1)-dimensional produzindo novos empacotamentos pra vÃrios valores de p no intervalo [149... 3001] à apresentado. O resultado à obtido atravÃs da modificaÃÃo da construÃÃo de Craig e considerando conveniente escolhidos Z-submÃdulos de Q (ζ), onde ζ à raiz p-Ãsima primitiva da unidade. Para p ≥ 59, à mostrado que a densidade de centro do reticulado (p-1)-dimensional na nova famÃlia à pelo menos 2 vezes a densidade de centro do (p-1)-dimensional reticulado de Craig. / Let p an odd prime. A family of (p-1)-dimensional over-lattices yielding new record packings for several values of p in interval [149... 3001] is presented. The result is obtained by modifying Craig's construction and considering conveniently chosen Z-submodules of Q (ζ), where ζ is a primitive pth root of unity. For p ≥ 59, it is shown that the center density of the (p-1)-dimensional lattice in the new family is at least twice the center density of the (p-1)-dimensional Craig lattice.

The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings. / by Marcela Chiorescu. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one-dimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the Krull-Schmidt property for direct sums of torsion-free rank one modules for a reduced local commutative Noetherian one-dimensional ring R. / by Basak Ay. / Thesis (Ph.D.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness of the intersection algebra of principal ideals in a UFD. It concludes by discussing possible applications of this result to other areas of commutative algebra.

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness of the intersection algebra of principal ideals in a UFD. It concludes by discussing possible applications of this result to other areas of commutative algebra.

Thesis (Ph.D.)--McMaster University (Canada), 1990. / Source: Dissertation Abstracts International, Volume: 52-10, Section: B, page: 5315. Supervisor: Bruno J. Muller.

Οι αντιμεταθετικοί δακτύλιοι έχουν την προέλευσή τους από τη θεωρία αριθμών και από την αλγεβρική γεωμετρία στον 19ο αιώνα. Σήμερα είναι ιδιαίτερα σημαντικοί και έχουν ενδιαφέρουσα επίδραση στην αλγεβρική γεωμετρία και στην θεωρία αριθμών, χρησιμοποιώντας μεθόδους αντιμεταθετικής άλγεβρας. Εδώ περιγράφουμε τις βασικές μεθόδους και κάνουμε τα πρώτα βήματα σε αυτό το θέμα. Στο εξής όλοι οι δακτύλιοι θα είναι αντιμεταθετικοί, εκτός αν θεωρήσουμε κάτι άλλο. Το κεντρικό θέμα της αξιωματικής ανάπτυξης της γραμμικής άλγεβρας είναι ένας διανυσματικός χώρος επί ενός σώματος. Η αξιωματοποίηση της γραμμικής άλγεβρας, η οποία επιτεύχθηκε το 1920, μορφοποιήθηκε σε μια μεγάλη έκταση, από την επιθυμία να εισάγουμε γεωμετρικές έννοιες στη μελέτη συγκεκριμένων κλάσεων των συναρτήσεων στην ανάλυση. Κατ’ αρχάς, ασχοληθήκαμε αποκλειστικά με τους διανυσματικούς χώρους των πραγματικών αριθμών ή των μιγαδικών αριθμών. Η έννοια ενός module είναι μια άμεση γενίκευση ενός διανυσματικού χώρου. Η γενίκευση αυτή επιτυγχάνεται απλά αντικαθιστώντας το σώμα των συντελεστών διά ενός δακτυλίου. Ο ευκολότερος τρόπος για να ορίσουμε ένα module μπορούμε να πούμε ότι είναι ένα αλγεβρικό σύστημα το οποίο ικανοποιεί τα ίδια αξιώματα όπως ένας διανυσματικός χώρος εκτός του ότι οι συντελεστές ανήκουν σ’ ένα δακτύλιο R με μονάδα αντί ενός σώματος F. Αυτή η φαινομενικά σεμνή γενίκευση οδηγεί σε μια αλγεβρική δομή η οποία είναι μεγίστης σημασίας. Ιστορικά ο πρώτος δακτύλιος που μελετήθηκε ήταν ο δακτύλιος ℤ των ακεραίων, ο όρος “δακτύλιος” πρωτοχρησιμοποιήθηκε από τον Hilbert (1897) στο “Zahlbericht” του για έναν δακτύλιο αλγεβρικών ακεραίων. Στο ℤ κάθε δακτύλιος είναι κύριος. Στην πραγματικότητα τα ιδεώδη είχαν πρώτα εισαχθεί (από Kummer) ως “ιδεώδεις αριθμοί” στους δακτυλίους αλγεβρικών ακεραίων οι οποίοι εστερούντο μοναδικής παραγοντοντοποίησης (unique factorization). Στο ℤ μπορούμε από δύο αριθμούς a,b να ορίσουμε τον μέγιστο κοινό διαιρέτη (ΜΚΔ) αυτών, (a,b), το γινόμενό τους ab και το ελάχιστο κοινό πολλαπλάσιο (ΕΚΠ) αυτών, [a,b]. Αυτές οι πράξεις αντιστοιχούν σε πράξεις ιδεωδών σε κάθε δακτύλιο. / Commutative ring has its origins in number theory its origins in number theory and algebraic geometry in the 19th century. Today it is of particular importance in algebraic geometry, and there has been an interesting interaction of algebraic geometry and number theory, using the methods of commutative algebra. Here we can do no more than describe the basic techniques and take the first steps in the first steps in the subject. Throughout this chapter all rings will be commutative, unless otherwise stated. The central concept of the axiomatic development of linear algebra is that of a vector space over a field. The axiomatization of linear algebra, which was effected in the 1920’s, was motivated to a large extend by the desire to introduce geometric notions in the study of certain classes of functions in analysis. At first one dealt exclusively with vector spaces over the reals or the complexes. It soon became apparent that this restriction was rather artificial , since a large body of the results depended only on the solution of linear equations and thus were valid for arbitrary fields. This led to the study of vactor spaces over arbitrary fields and this is what presently constitutes linear algebra. The concept of a module is an immediate generalization of that of a vector space. One obtains the generalization by simply replacing the underlying field by any ring.In the first place, one learns from experience that the internal logical structure of mathematics strongly urges the pursuit of such ‘natural’ generalizations. These often result in an improved insight into the theory which led to them in the first place. The easiest way to define a module is to say that it is an algebraic system that satisfies the same axioms as a vector space except that the scalars come from a ring R with a 1 instead of from a field F. This seemingly modest generalization leads to an algebraic structure that is of the greatest importance. We use here the term R-module, it being understood that the scalars are written on the left. Historically the first ring to studied was the ring Z of integers, the term ‘ring’ was first used by Hilbert (1897) in his ‘Zahlbericht’ for a ring of algebraic integers. In Z every ideal is principal, in fact ideals were first introduced (by Kummer) as ‘ideal numbers’ in rings of algebraic integers which lacked unique factorization. In Z we can from any two numbers a,b form their highest common factor (HCF, also greatest common divisor, GCD) (a,b), their product ab and their least common multiple (LCM) [a,b]. These operations correspond to operations on ideals in any ring. Valuation theory may be described as the study of divisibility (in commutative rings) in its purest form, but that is only one aspect. The general formulation leads to the introduction of topological concepts like completion, which provides a powerful tool. It also emphasizes the parallel with the absolute value on the real and complex numbers. After the initial definitions we shall prove the essential uniqueness of the absolute value on R and C, and go on to describe the p-adic numbers, before looking at simple cases of the extension problem.