http://hdl.handle.net/1842/113 2013-05-18T23:07:52Z http://hdl.handle.net/1842/3038 Title: Representations of algebras as universal localizations Authors: Ranicki, Andrew; Schofield, A.; Neeman, A. Abstract: Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra $S$ in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is $M_n(S)$ where $n$ is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra. 2004-01-01T00:00:00Z http://hdl.handle.net/1842/3010 Title: Frobenius n-homomorphisms, transfers and branched coverings Authors: Rees E.G.; Buchstaber V.M. Abstract: The main purpose is to characterise continuous maps that are n-branched coverings in terms of induced maps on the rings of functions. The special properties of Frobenius nhomomorphisms between two function spaces that correspond to n-branched coverings are determined completely. Several equivalent definitions of a Frobenius n-homomorphism are compared and some of their properties are proved. An axiomatic treatment of n-transfers is given in general and properties of n-branched coverings are studied and compared with those of regular coverings. 2008-01-01T00:00:00Z http://hdl.handle.net/1842/3009 Title: Averages in vector spaces over finite fields Authors: Wright J.; Carbery A.; Stones B. Abstract: We study the analogues of the problems of averages and maximal averages over a surface in R-n when the euclidean structure is replaced by that of a vector space over a finite field, and obtain optimal results in a number of model cases. 2008-01-01T00:00:00Z http://hdl.handle.net/1842/3008 Title: On the lines passing through two conjugates of a Salem number Authors: Smyth C.; Dubickas A. Abstract: We show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number alpha of degree d >= 3 is at most [d(2)/2] - d + 2, its conjugates being in general position if this number is attained. If, for instance, d >= 4 is even, then the conjugates of 01 G U of degree d are in general position if and only if a has 2 real conjugates, d - 2 complex conjugates, no three distinct conjugates of a lie on a line and any two lines that pass through two distinct conjugates of a are non-parallel, except for d/2 - 1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation alpha(1) + alpha(2) = alpha(3) + alpha(4) in distinct conjugates of a Pisot number. The Pisot number alpha(1) = (1 + root 3+2 root 5-)/2 shows that this equation has such a solution. 2008-01-01T00:00:00Z