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An introduction to algebraic K-theory

Ausoni, Christian (2001)

Proceedings of the 20th Winter School "Geometry and Physics"

This paper gives an exposition of algebraic K-theory, which studies functors K n : Rings → Abelian Groups , n an integer. Classically n = 0 , 1 introduced by Bass in the mid 60’s (based on ideas of Grothendieck and others) and n = 2 introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite,...

An introduction to Cartan Geometries

Sharpe, Richard (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

A principal bundle with a Lie group H consists of a manifold P and a free proper smooth H -action P × H → P . There is a unique smooth manifold structure on the quotient space M = P / H such that the canonical map π : P → M is smooth. M is called a base manifold and H → P → M stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces H ⊂ G → G / H , where H is a closed subgroup of G . The pair ( 𝔤 , 𝔥 ) is a Klein pair. A model geometry consists of a Klein pair ( 𝔤 , 𝔥 ) and a Lie group H with Lie algebra 𝔥 . In this...

An Online Repository of Mathematical Samples

Baker, Josef B., Sexton, Alan P., Sorge, Volker (2009)

Towards a Digital Mathematics Library. Grand Bend, Ontario, Canada, July 8-9th, 2009

With a growing community of researchers working on the recognition, parsing and digital exploitation of mathematical formulae, a need has arisen for a set of samples or benchmarks which can be used to compare, evaluate and help to develop different implementations and algorithms. The benchmark set would have to cover a wide range of mathematics, contain enough information to be able to search for specific samples and be accessible to the whole community. In this paper, we propose an on-line system...

Analysis of the discontinuous Galerkin finite element method applied to a scalar nonlinear convection-diffusion equation

Hozman, Jiří, Dolejší, Vít (2008)

Programs and Algorithms of Numerical Mathematics

We deal with a scalar nonstationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. We present a discretization of this model equation by the discontinuous Galerkin finite element method. Moreover, under some assumptions on the nonlinear terms, domain partitions and the regularity of the exact solution,...

Analytical solution of rotationally symmetric Stokes flow near corners

Burda, Pavel, Novotný, Jaroslav, Šístek, Jakub (2013)

Applications of Mathematics 2013

We present analytical solution of the Stokes problem in rotationally symmetric domains. This is then used to find the asymptotic behaviour of the solution in the vicinity of corners, also for Navier-Stokes equations. We apply this to construct very precise numerical finite element solution.

Analytical solution of Stokes flow near corners and applications to numerical solution of Navier-Stokes equations with high precision

Burda, Pavel, Novotný, Jaroslav, Šístek, Jakub (2012)

Applications of Mathematics 2012

We present analytical solution of the Stokes problem in 2D domains. This is then used to find the asymptotic behavior of the solution in the vicinity of corners, also for Navier-Stokes equations in 2D. We apply this to construct very precise numerical finite element solution.

Application of Richardson extrapolation with the Crank-Nicolson scheme for multi-dimensional advection

Zlatev, Zahari, Dimov, Ivan, Faragó, István, Georgiev, Krassimir, Havasi, Ágnes, Ostromsky, Tzvetan (2013)

Applications of Mathematics 2013

Multi-dimensional advection terms are an important part of many large-scale mathematical models which arise in different fields of science and engineering. After applying some kind of splitting, these terms can be handled separately from the remaining part of the mathematical model under consideration. It is important to treat the multi-dimensional advection in a sufficiently accurate manner. It is shown in this paper that high order of accuracy can be achieved when the well-known Crank-Nicolson...

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