Lie algebra methods for the applications to the statistical theory of turbulence

Approximate Lie symmetries of the Navier-Stokes equations are used for the applications to scaling phenomenon arising in turbulence. In particular, we show that the Lie symmetries of the Euler equations are inherited by the Navier-Stokes equations in the form of approximate symmetries that allows to involve the Reynolds number dependence into scaling laws. Moreover, the optimal systems of all finite-dimensional Lie subalgebras of the approximate symmetry transformations of the Navier-Stokes are constructed. We show how the scaling groups obtained can be used to introduce the Reynolds number dependence into scaling laws explicitly for stationary parallel turbulent shear flows. This is demonstrated in the framework of a new approach to derive scaling laws based on symmetry analysis [11]-[13].

Hypercentral unit groups and the hyperbolicity of a modular group algebra

We classify groups G such that the unit group U-1 (ZG) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups U-1 (KG).

Nuclear elements of degree 6 in the free alternative algebra

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known clegree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.

CLASSIFICATION OF SUBALGEBRAS OF THE CAYLEY ALGEBRA OVER A FINITE FIELD

We classify all unital subalgebras of the Cayley algebra O(q) over the finite field F(q), q = p(n). We obtain the number of subalgebras of each type and prove that all isomorphic subalgebras are conjugate with respect to the automorphism group of O(q). We also determine the structure of the Moufang loops associated with each subalgebra of O(q).

Free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus (Omega(R)/dR)

In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus Omega(R)/dR where R = C[t. t(-1), u vertical bar u(2) = t(3) - 2bt(2) + t]. The first realization provides an analogue of Wakimoto`s construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new types of representations analogous to Imaginary Verma modules in the Affine setting. (c) 2009 Elsevier B.V. All rights reserved.

Förlorad i övergången från aritmetik till algebra : Hur gymnasieelever översätter aritmetik till algebra

The aim of this thesis is to look for signs of students’ understanding of algebra by studying how they make the transition from arithmetic to algebra. Students in an Upper Secondary class on the Natural Science program and Science and Technology program were given a questionnaire with a number of algebraic problems of different levels of difficulty. Especially important for the study was that students leave comments and explanations of how they solved the problems. According to earlier research, transitions are the most critical steps in problem solving. The Algebraic Cycle is a theoretical tool that can be used to make different phases in problem solving visible. To formulate and communicate how the solution was made may lead to students becoming more aware of their thought processes. This may contribute to students gaining more understanding of the different phases involved in mathematical problem solving, and to students becoming more successful in mathematics in general.The study showed that the students could solve mathematical problems correctly, but that they in just over 50% of the cases, did not give any explanations to their solutions.

Gymnasieelevers uppfattningar om algebra och problemlösning : En undersökning med utgångspunkt i elevernas kön, slutbetyg i grundskolan och val av gymnasieprogram

Syftet med den här uppsatsen är att undersöka elevers uppfattningar om algebra och problemlösning samt granska hur dessa uppfattningar påverkas beroende på elevernas val av gymnasieprogram, kön och slutbetyg i grundskolan. Syftet är vidare att ta reda på vilka eventuella hinder och svårigheter eleverna själva uppfattar då de använder algebra för att lösa matematiska problem. Som metod för att söka svar på syfte och frågeställningar har valts att genomföra en enkätundersökning med elever som går första året på gymnasiet och som läser antingen naturvetenskapsprogrammet eller bygg- och anläggningsprogrammet. Enkätundersökningen består av två delar, en del som undersöker elevers uppfattningar om matematik i allmänhet och algebra och problemlösning i synnerhet, samt en del som försöker reda ut vilka svårigheter eleverna uppfattar då de ska lösa matematiska problem med algebra. Svaren sammanställs genom en analys av vilka eventuella skillnader och likheter som finns beroende på elevernas val av gymnasieprogram, kön och betyg i grundskolan. Resultatet visar på att elever på naturvetenskapsprogrammet som hade MVG i betyg i grundskolan har en mer positiv inställning till algebra och problemlösning i jämförelse med elever från bygg- och anläggningsprogrammet som fått G i betyg. Vad gäller elevernas kön finns det inte några indikationer på att denna faktor har någon större påverkan på deras uppfattningar. Resultatet kan vara en indikation på att elevernas uppfattningar främst påverkas av deras förståelse för det algebraiska tankesättet. Det eleverna upplever som svårast när de ska lösa problem med hjälp av algebra är att översätta den skrivna texten till en algebraisk framställning. När eleverna löser matematiska problem indikerar även resultatet att de till stor del styrs av sina förväntningar och förutfattade föreställningar om uppgiften. Resultatet ger en indikation om att eleverna behöver arbeta mer med problemlösning i olika former för att genom det kunna träna upp sin resonemangsförmåga och sin förmåga att behärska alla de tre faserna, översättning, omskrivning och tolkning, i den algebraiska cykeln.

A comparative study of some computer algebra packages which determine the lie point symmetries of differential equations

The study seeks to determine which of five computer algebra packages is best at finding the Lie point symmetries of systems of partial differential equations with minimal user intervention. The chosen packages are LIEPDE and DIMSYM for REDUCE, LIE and BIGLIE for MUMATH, DESOLV for MAPLE, and MATHLIE for MATHEMATICA. A series of systems of partial differential equations are used in the study. The paper concludes that while all of the computer packages are useful, DESOLV appears to be the most successful system at determining the complete set of Lie point symmetries of systems of partial differential equations.

Algebra skills and traps and diagnostic teaching for the future

Explores on some research about teaching and learning algebra and related classroom issues. Diagnostic instruments that may be used by senior secondary teachers in teaching algebra to senior classes; Strategies for remediating algebraic difficulties and misconceptions; Impact of technology on the algebra curriculum; Usefulness of copying algebraic expressions while using Computer Algebra Systems or mathematics processing software in a calculus class.

Algebra concepts and skills survey - a draft

Presents questions adapted from the 'Group Review of Algebra Topics' (ACER 1991) that form a draft survey test of 24 items ranging across aspects of algebra and secondary algebra instruction.