No description available.

Ph.D. (Mathematics) / This thesis reports on an interdisciplinary research programme: an investigation of power structures, and their applications in various fields. A power construction is an attempt to lift whatever structure may exist between the elements of a set to subsets of that set. The notions of structure considered here are algebraic, relational and topological. It is shown how power constructions are useful in a number of contexts in Mathematics, in Logic, in Computer Science and in the Philosophy of Science. The thesis is therefore an exercise in what may be called lateral research, where the aim is to look horizontally across disciplinary boundaries, identify common basic concepts, and use these to fertilise each field with results from the others. This differs from the more common vertical research method, the two manifestations of which (in Mathematics, in particular) are specialisation and generalisation. To specialise means to narrow down the field of investigation, as with a group theorist studying specifically Abelian groups. Generalisation moves in the opposite direction - one may attempt, for example, to generalise a result first proved for Abelian groups to the case of arbitrary groups. But, whether narrowing down or opening up, in vertical research mode it is the concept alone which is under investigation - its own nature, rather than its relationships to other concepts. In particular, vertical research pays little attention to the occurrence and application of the concept under investigation in other disciplines. It is sad that 'research', in Mathematics, is often unthinkingly equated with 'vertical research'. This is detrimental to scholarship in at least two ways. One is the training of new scientists - more particularly, of new PhD's. It is ironic that though the requirement for a PhD is almost universally held to be 'original research', or 'a contribution to knowledge', few things are in fact more orthodox and conformist than a PhD thesis. Here I refer not just to presentation (uniformity of which may be beneficial), but to methodology: few PhD candidates would dare to prejudice their chances with unpredictable examiners by venturing outside the paradigm of vertical research. A second effect (which is also a cause) of equating 'research' with 'vertical research' is the allocation of research funding. Project proposals and grant applications must be evaluated; this is usually done by peer review, and it seems clear that referees' reports emanating from a smallish fraternity of specialists will have a more enthusiastic ring than...

Matematik är ett ämne som elever över hela världen behöver lära sig. I de senare årskurserna bygger ämnet på att elever ska kunna behärska algebra. Vi har valt att fokusera på mönster vilket många anser vara inkörsporten till algebra. För att lärare ska kunna erbjuda elever en rättvis utbildning bör de ha kunskap om elevers svårigheter och lärande. Vår litteraturstudie har riktat in sig på att försöka svara på vad forskning säger om elevers lärande till växande geometriska mönster och explicit formel. I de elva studier som valts ut har vi kommit fram till att några gemensamma slutsatser finns men även sådana som skiljer sig åt. Att elever behöver utveckla sitt språk och få förståelse för innebörden av bokstavssymboler i matematiken är en viktig slutsats för lärande relaterad till mönster. Studierna har dessutom visat en stark relevans i att samordna olika delar till en helhet. Svenska elevers resultat i internationella undersökningar problematiseras och vad som kan tänkas vara orsaken till att de presterar sämre i mönsterrelaterade uppgifter än vad de gör i övriga delar i matematiken. Vårt resultat är till stor del överensstämmande med hur de svenska styrdokumenten förhåller sig till ämnet. Trots det har svenska elever svårigheter med att uttrycka en explicit formel till ett växande geometriskt mönster. Just den explicita formeln är viktig för elevers helhetsförståelse av mönster och tidig algebra.

No description available.

No description available.

No description available.

No description available.

No description available.

The primary objective of this thesis is to present a unified account of the various generalizations of the concept of ring of quotients given by K, Asano (1949), R. E. Johnson (1951), Y. Utumi (1956), G. D. Findlay and J. Lambek (1958). A secondary objective is to investigate how far the commutative localization can be carried over to the noncommutative case. We begin with a formulation of the notion of D-system of right ideals of a ring R. The investigation of the D-systems was motivated by the fact that each maximal right quotient ring of R consists precisely of semi R-homomorphisms into R with domains in a specific D-system of right ideals of R or of R¹, the ring obtained from R by adjoining identity. A nonempty family X of right ideals of R is called a D-system provided the following three conditions holds: D1. Every right ideal of R containing some member of X is in X. D2. For any two right ideals A and B of R belonging to X, φ⁻¹B belongs to X for each R-homomorphism φ of A into R. D3. If A belongs to X and if for each a in A there exists Ba in X, then the ideal sum of aBa (a in A) is in X, Each D-system X of right ideals of R induces a modular closure operation on the lattice L(M) of all submodules of an R-module M and hence gives rise to a set Lx(M) of closed submodules of M. We are able to set up an isomorphism between the lattice of all modular closure operations on L(R) and the lattice of all D-systems of right ideals of R and characterize the D-systems X used in Asano's, Johnson's and Uturai's constructions of«quotient rings in terms of properties of Lx(R). In view of the intimate relation between the rings of quotients of a ring R and the extensions of R-modules, we generalize the concepts of infective R-module, rational and essential extensions of an R-module corresponding to a D-system T of right ideals of R¹. The existence and uniqueness of the maximal Y-essential extension, minimal Y-injective extension and maximal Y-rational extension of an R-module and their mutual relations are established. Finally, we come to the actual constructions of various extensions of rings and modules. The discussions center around the centralizer of a ring over a module, the maximal essential and rational extensions and the different types of rings of right quotients. We include here also a partial, though not satisfactory, solution of the noncommutative localization problem. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.