Similarity classes of 3x3 matrices over a local principal ideal ring

In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.Comment: 14 pages, final version, to appear in Communications in Algebr

Growing a greater understanding of multiplication through lesson study: Mathematics teacher educators’ professional development

Research cites the need for developing teachers’ mathematical knowledge for teaching (MKT) as well as for developing mathematics teacher educators’ (MTEs) mathematical knowledge for teaching teachers (MKTT). Using the framework of lesson study: formulating goals and researching, planning, implementing and observing, and reflecting (Lewis & Hurd, 2011), a group of MTEs designed and analyzed a lesson on multiplication for prospective elementary teachers. A qualitative analysis of MTE journal reflections and prospective teacher work showed a greater understanding of MTEs’ MKTT related to multiplication after completion of the lesson study. The authors recommend MTEs conduct lesson studies for other mathematics topics to further understand what MKTT MTEs need to develop to best support prospective teachers’ MKT

Metastability in Two Dimensions and the Effective Potential

We study analytically and numerically the decay of a metastable phase in (2+1)-dimensional classical scalar field theory coupled to a heat bath, which is equivalent to two-dimensional Euclidean quantum field theory at zero temperature. By a numerical simulation we obtain the nucleation barrier as a function of the parameters of the potential, and compare it to the theoretical prediction from the bounce (critical bubble) calculation. We find the nucleation barrier to be accurately predicted by theory using the bounce configuration obtained from the tree-level (``classical'') effective action. Within the range of parameters probed, we found that using the bounce derived from the one-loop effective action requires an unnaturally large prefactor to match the lattice results. Deviations from the tree-level prediction are seen in the regime where loop corrections would be expected to become important.Comment: 13pp, LaTex with Postscript figs, CLNS 93/1202, DART-HEP-93/0

Understanding the small object argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.Comment: 42 pages; supersedes the earlier arXiv preprint math/0702290; v2: final journal version, minor corrections onl

Expanding Bubbles in a Thermal Background

Real scalar field models incorporating asymmetric double well potentials will decay to the state of lowest energy. While the eventual nature of the system can be discerned, the determination of the dynamics of the bubble wall provides many difficulties. In the present study we investigate numerically the evolution of spherically symmetric expanding bubbles coupled to a thermal bath in 3+1 dimensions. A Markovian Langevin equation is employed to describe the interaction between bubble and bath. We find the shape and velocity of the wall to be independent of temperature, yet extremely sensitive to both asymmetry and viscosity.Comment: RevTeX, 9 pages (multicols), 9 figures, submitted to Phys. Rev.

Coincident pairs of continuous sections in profinite groups

Given a profinite group G and a closed subgroup H, we show there exist continuous sections s : ( G / H ) 1 → G s:{(G/H)_1} \to G and s ′ : ( G / H ) r → G s’:{(G/H)_r} \to G of the projections G → ( G / H ) 1 G \to {(G/H)_1} and G → ( G / H ) r G \to {(G/H)_r} onto the left and right coset spaces, respectively, such that im ( s ) = im ( s ′ ) {\text {im}}(s) = {\text {im}}(s’) .</p

Periodic expansion of modules and its relation to units

AbstractA finitely generated full Z-module in an algebraic number field is seen to have a “graphically periodic” expansion, which, in turn, leads to the fundamental units of the coefficient ring. This generalizes the classical periodic simple continued fraction expansion of a real quadratic irrational

Similarity problem over 𝑆𝐿(𝑛,𝑍_{𝑝})

The question of conjugacy separability in S L ( n , Z ) SL(n,Z) leads to the conjugacy problem in S L ( n , Z p ) SL(n,{Z_p}) for various primes p p . We present a simple solution to the similarity problem over S L ( n , Z p ) SL(n,{Z_p}) , which solves the conjugacy problem as a special case.</p