SMS-mötet i Göteborg 14-15 nov 2003: abstracts

Classification of the real flexible division algebras
Erik Darpö, Uppsala

An \emph{algebra} $A$ over a a field $k$ is a $k$-vectorspace endowed with a bilinear multiplication $A\times A\rightarrow A,\:(x,y)\mapsto xy$. If $0<\dim A<\infty$, and for all $z\neq 0$ the linear mappings $L_a:A\rightarrow A,\: x\mapsto ax$ and $R_a:A\rightarrow A,\: x\mapsto xa$ of left and right multiplication both are bijective, $A$ is called a \emph{division algebra}. Due to Bott and Milnor's theorem (1958), every real division algebra has dimension 1,2,4 or 8. The problem of classifying all real division algebras of dimension $2^b$ has a trivial solution for $b=0$, while it is still unsolved for $b\in\{1,2,3\}$. Theorems by Frobenius (1878) and Zorn (1931) state that the sets $\{\mathbb{R},\mathbb{C},\mathbb{H}\}$ and $\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ classify all associative respectively alternative real division algebras. Attempting to generalize these celebrated results, one is naturally led to study the class of all \emph{flexible} real division algebras, defined by the axiom $(xy)x=x(yx)$. Given any real algebra $A$, its \emph{algebra of derivations} $\mbox{Der}A=\{\delta\in\mathcal{L}(A)\mid\delta(xy)=\delta(x)y+x\delta(y)\}$ is a real Lie algebra, and $A$ is a module over $\mbox{Der}A$. For division algebras, the algebra of derivations has shown to be a useful invariant. We will give a complete classification of the real flexible division algebras, and show that the question of irredundancy could be reduced the normal form problem for a certain group action. The computation of the algebras of derivations will also be discussed.

Summation-by-parts operators with simultaneous approximation term methods for hyperbolic initial boundary value problems
Jing Gong, Uppsala

For wave propagating problems, the computational domain is often large compared with wavelengths, which means that waves have to trave relatively long distances during long times. In order to obtain an accurate solution, numerical methods with high accuracy in both time and space are needed to keep errors from dispersion low. High-order finite difference methods (HOFDM) fulfill all these requirements and can easily be parallelized to execute on high performance computers. The main difficulty with HOFDM is to show stablility for hyperolic initial boundary value problems (IBVPs). Kreiss and Scherer proved stability of difference approximations of hyperbolic IBVPs using explicit difference operators satisfying a summation by parts (SBP) methods. In this paper we will implement high-order finite difference operators which satisfy SBP properties to numerically solve hyperbolic IBVPs. This work is based on Nordström et al. By employing the SBP operators with simultaneous approximation term (SAT) methods high order accurate numerical solutions are obtained. Numerical experiments are performed and the simulation results are in good agreement with the practice results.

On the doubling of quadratic algebras
Lars Lindberg, Uppsala

A $k$-algebra is a vector space $\A$ over a field $k$ endowed with a $k$-bilinear multiplication $\A\times\A\rightarrow\A$, $(x,y)\mapsto xy$. The algebra $\A$ is said to be \emph{quadratic} if $0<\dim \A<\infty$, there exists an identity element $1\in\A$ and for every $x\in\A$ the vectors $1,~x,~x^{2}$ are linearly dependent. An algebra is said to be a \emph{division algebra} if $0<\dim\A<\infty$ and $xy=0$ implies $x=0$ or $y=0$. Considering a quadratic algebra $\A$, the vector space $\V(\A)=\A\times\A$ with multiplication $(w,x)(y,z)=(wy-\bar{z}x,x\bar{y}+zw)$ is again a quadratic algebra with $\dim\B=2\dim\A$. The construction $\A\mapsto\V(\A)$ is called the doubling of a quadratic algebra $\A$. A question which naturally arises is which properties of the algebra $\A$ are preserved under doubling and in particular whether doubling preserves the property of being a division algebra. In the case of $k=\R$, a famous theorem of Bott, Milnor and Kervaire (1958) asserts that every real division algebra has dimension 1,2,4 or 8. So in this case we see that if we take any 8-dimensional real quadratic division algebra $\A$, then $\V(\A)$ is not a division algebra. In this talk we will view the doubling as an endofunctor on the category of all quadratic $k$-algebras, where $k$ is a field of characteristic not two, and we will discuss some properties of this functor. The results give evidence to the conjecture that there is no 16-dimensional quadratic division algebra over $k$.

Garaded modules
Patrik Lundström, Troullhättan/Uddevalla

A mathematical object is classically called graded if it can be decomposed as a direct sum subject to certain multiplicative relations. In ring theory this grading is often defined by a group. Natural examples of such objects are given by the complex numbers, the quaternions, group rings, and more generally crossed product algebras and orders. Many properties of these rings are determined by the corresponding group and the principal component of the ring. A well known example of this kind is Mascke's theorem concerning the separability of group rings. We consider generalizations of classical separability results for graded rings and modules from the group graded case to the more general setting with gradings defined by groupoids, that is, categories with the property that all morphisms are isomorphisms. We then discuss applications of these results to category rings and crossed product algebras and orders defined by separable (not necessarily normal) Galois extensions. The talk ends with a short digression on some future ideas for research in this area.

Digitala räta linjer i Khalimskyplanet
Erik Melin, Uppsala

Digital geometri kan populärt uttryckt sägas vara datorskärmens geometri. Vi diskuterar detta område allmänt och introducerar Khalimskys topologi på \mathbb{Z}^n. På datorskärmen, eller mer allmänt, i det digitala planet \mathbb{Z}^2 är det av stor betydelse att kunna tala om digitala räta linjer. Vad man ska mena med en digital rät linje är dock a priori oklart -- förutom att den i någon mening bör approximera en Euklidisk linje. Azriel Rosenfeld preciserade år 1974 begreppet digital rät linje. Rosenfelds linjer respekterar dock ingen topologi i planet och två Rosenfeldlinjer kan korsa varandra utan att ha en gemensam skärningspunkt. Vi ger en ny, alternativ definition av digital linje sådan att den digitala linjen betraktad som ett delrum av Khalimskyplanet blir homeomorf med Khalimskylinjen. Speciellt följer att två korsande digitala linjer alltid har en gemensam skärningspunkt.

Radontransformer och tomografi
Hans Rullgård, Stockholms universitet

Låt $f$ vara en funktion med kompakt stöd definierad i planet. Radontransformen av $f$ är en funktion $Rf$ definierad på mängden av linjer i planet, vars värde på linjen $L$ är integralen av $f$ längs $L$. En viktad Radontransform $R_\rho f$ får man om $f$ multipliceras med en viktfunktion $\rho$ som beror på $L$ före integrationen. Det inversa problemet för viktade Radontransformer är att bestämma $f$ när $R_\rho f$ och $\rho$ är kända. Att lösa detta problem har betydelse för tillämpningar i datortomografi. För vissa klasser av viktfunktioner kan $f$ uttryckas med en explicit formel; i andra fall måste problemet lösas numeriskt. I båda fall är man intresserad av stabilitetsresultat, till exempel hur mycket den beräknade lösningen kan påverkas av mätfel. Jag kommer att redogöra för en del matematiska resultat inom detta område.

Harmonic analysis on the Lie ball
Henrik Seppänen, Chalmers

We will consider some geometrical properties of the Lie ball, such as different realizations (as a submanifold of $\mathbb{C}^n$, as a Grassmannian manifold and as a domain of real matrices), its group of automorphisms and generic submanifolds. We will also study Bergman spaces of holomorphic functions on the Lie ball.

An overview of phase transitions
Jeff Steif, Chalmers

I will discuss the notion of phase transitions that arise in the study of statistical mechanics (mathematically, this will simply be probability theory). I will discuss three models (percolation, the Ising model, and the Potts model) in which this phenomenon occurs. I will also discuss the important distinction between a 1st order and a 2nd order phase transition.

Algebraic classification of discrete kinetic models
Mirela Cristina Vinerean, Karlstad

The basic equation in kinetic theory is the Boltzmann equation for time-evolution of the particle density $f=f(x,t;v,\varepsilon ,...)$, where $% x,t,v$,$\varepsilon $ represent the position, the time, the velocity and the internal energy of the particle in the phase space. Discrete kinetic models (DKMs) or simply, discrete velocity models (DVMs) in the particular case when there exist no internal degrees of freedom, are models where all phase coordinates, except the space one, are discretized ( i.e. the velocities are assumed to be able to take a finite number of values). In this case, the Boltzmann equation is replaced by a system of differential equations easier to analyze from the mathematical or numerical point of view. In many interesting papers on DVMs, authors postulate from the beginning that the finite velocity space with ''good'' properties is given and only after this step, study the Boltzmann equation (system). Contrary to this approach, our aim is not to study the equations, but to discuss all possible choices of finite phase spaces (sets) satisfying this type of ''good restrictions''. Due to the velocity discretization is well-known that it is possible to have DVMs with ''spurious'' summational invariants (conservation laws which are not linear combination of physical invariants). Our purpose is to give a method (algorithm) for constructing normal models (without spurious invariants) and to classify all normal plane models with small number of velocities (which usually appear in applications). In the first step we describe DKMs as algebraic systems. We introduce for this an abstract discrete model (ADM) which is defined by the matrix of reactions (same as for the concrete model). This matrix contains as rows all vector of reactions, which can be written as $n$-dimensional vectors $% (k_{1},.,k_{n})$ with $k_{i}\in \mathbb{Z},$ describing the ''jump'' from a pre-reaction state to a new reaction state. The conservation laws corresponding to the many-particle system are uniquely determined by the ADM, or equivalently, by its corresponding matrix of reactions, and do not depend on the concrete realization. We find the restrictions on ADM such that it is a realization of some concrete DM and in the next step we give a general method of constructing normal models (using the results on ADMs). Having the general algorithm, we consider in more details, the particular cases of models with mass and momentum conservation (inelastic lattice gases with pair collisions) and models with mass, momentum and energy conservation (elastic lattice gases with pair collisions). In the first case , for ADM fulfilling the necessary conditions, the algorithm will always lead to a unique solution. This is not true in the second case, when we might have no solution. We made some improvements of the algorithm for this particular case, and classify all normal plane elastic models with small velocities. For model with less than $% 8$ velocities, this classification can be completely described using only analytical and geometrical arguments. It becomes a lot more difficult for bigger number of velocities $(9,10,..)$, when we need to use the help of computer. We hope to give also all complete classification of normal model with $9$ and $10$ velocities . We have already done a classification of these models, but only for the ones coming from 1-extensions.We try to find out, using computer, if there is possible other kind of extension.

Klassificering av automorfigrupper på trigonala Riemannytor
Daniel Ying, Linköping

Låt $X$ vara en sluten Riemannyta. Om $X$ kan ses som en 3-skiktad övertäckning av Riemannsfären $\widehat{\mathbb{C}}$ så att $f:X\stackrel{3:1}{\rightarrow} \widehat{\mathbb{C}}$ så sägs $X$ vara en trigonal Riemannyta och övertäckningen sägs vara en trigonal morfism. Om övertäckningen dessutom är cyklisk säger vi att $X$ är en cykliskt trigonal yta. Om nu $(X,f)$ är en cyklisk trigonal yta, så finns en automorfi, $\varphi:X\rightarrow X$, av ordning 3 sådan att $X/\langle\varphi\rangle\cong\widehat{\mathbb{C}}$ (med koniska punkter av ordnind 3) och $\varphi$ är en decktransformation av övertäckningen $f$. Om $X_{g}$ är en kompakt Riemannyta med genus $g\geq2$ så kan $X_{g}$ representeras som kvoten mellan det övre halvplanet, $\mathbb{H}$, och en fuchsisk ytgrupp, $\Gamma$, som verkar på $\mathbb{H}$, det vill säga $X_{g}=\mathbb{H}/\Gamma$. Låt $\Delta$ vara gruppen av automorfier på Riemannsfären med koniska punkter av ordning 3, det vill säga en grupp med signaturen $s(\Delta)=(0,+,[\underbrace{3,\ldots,3}_{\stackrel{\textrm{Antalet }} {\textrm{koniska punkter}}}])$. Då finns en NEC-grupp $\Lambda$ och en epimorfi $\theta:\Lambda \rightarrow\Delta$ med $ker(\theta)=\Gamma$. Om $G$ är en automorfigrupp till $X_{g}$, $g\geq5$, så verkar $\overline{G}=G/\langle\varphi\rangle$ på sfären. $X/G$ uniformeras av en NEC-grupp $\Lambda$ sådan att det finns en fuchsisk undergrupp $\Gamma\leqslant$ som uniformiserar $X$. Typerna av $\Lambda$ är redan kända men detta föredrag skall beskriva hur vi jobbar med att undersöka hur typerna av $G$ som en extension av $\langle\varphi\rangle$ med $\overline{G}$. Genom att skriva upp extensionsgrupperna på ett allmänt sätt och sedan använda GAP för att sortera ut de extensionerna som är verkliga extensioner av sfäriska grupper, kan vi sedan börja klassificera dem.