2010Zeit-Experimente zur Faktorisierung

ein Beitrag zur Didaktik der KryptologieSchulz, Ralph-Hardo; Witten, Helmut

ein Beitrag zur Didaktik der KryptologieSchulz, Ralph-Hardo; Witten, Helmut

We report on experiments on the time of factorization of semiprimes (i.e., products of two primes) using the systems ''Sage'' and ''CrypTool''. With some exceptions the time grows exponentially with the length of the semiprimes - as expected. Using the quadratic sieve implemented in CrypTool 2, we could factorize the number RSA-100, a 100-decimal-digits semiprime, on our laptop in less than eight and a half hours.

2009Substructuring of a Signorini-type problem and Robin's method for the Richards
equation in heterogeneous soilBerninger, Heiko; Sander, Oliver

We prove a substructuring result for a variational inequality concerning - but
not restricted to - the Richards equation in homogeneous soil and including
boundary conditions of Signorini's type. This generalizes existing results for
the linear case and leads to interface conditions known from linear
variational equalities: continuity of Dirichlet and flux values in a weak
sense. In case of the Richards equation these ... View more

We prove a substructuring result for a variational inequality concerning - but not restricted to - the Richards equation in homogeneous soil and including boundary conditions of Signorini's type. This generalizes existing results for the linear case and leads to interface conditions known from linear variational equalities: continuity of Dirichlet and flux values in a weak sense. In case of the Richards equation these are the continuity of the physical pressure and of the water flux, which is hydrologically reasonable. Therefore, we also apply these interface conditions in the heterogeneous case of piecewise constant soil parameters, which we address by the Robin method. We prove that, for a certain time discretization, the homogeneous problems in the subdomains including Robin and Signorini-type boundary conditions can be solved by convex minimization. As a consequence we are able to apply monotone multigrid in the discrete setting as an efficient and robust solver for the local problems. Numerical results demonstrate the applicability of our approach.

View less2009Connectedness, disconnectedness, and light factorization structures in a fuzzy settingPreuß, Gerhard

Connectedness, disconnectedness, and light factorization structures are
studied in the realm of the topological constructs \textbf{FPUConv} and
\textbf{FSUConv} of fuzzy preuniform convergence spaces and fuzzy semiuniform
convergence spaces respectively which have been introduced by the author in
\cite{23} using fuzzy filters in the sense of Eklund and Gähler \cite{7}. The
presented theory profits from the fact ... View more

Connectedness, disconnectedness, and light factorization structures are studied in the realm of the topological constructs \textbf{FPUConv} and \textbf{FSUConv} of fuzzy preuniform convergence spaces and fuzzy semiuniform convergence spaces respectively which have been introduced by the author in \cite{23} using fuzzy filters in the sense of Eklund and Gähler \cite{7}. The presented theory profits from the fact that both constructs have hereditary quotients. Additionally, there are special features, e.g. a product theorem for the investigated connectedness concept and the existene of a proper class of light factorization structures on FPUConv as well as on FSUConv.

View less2013Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equationsGräser, Carsten; Kornhuber, Ralf; Sack, Uli

We present globally convergent nonsmooth Schur-Newton methods for the solution
of discrete vector-valued Cahn-Hilliard equations with logarithmic and
obstacle potentials. The method solves the nonlinear set-valued saddle-point
problems as arising from discretization by implicit Euler methods in time and
first order finite elements in space without regularization. Efficiency and
robustness of the convergence speed for ... View more

We present globally convergent nonsmooth Schur-Newton methods for the solution of discrete vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems as arising from discretization by implicit Euler methods in time and first order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments.

View less2010Rainwater-Simons-type convergence theorems for generalized convergence methodsHardtke, Jan-David

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for boundaries but for the more general notion of (I)-generating sets introduced by Fonf and Lindenstrauss.

2010A polynomial chaos approach to stochastic variational inequalitiesForster, Ralf; Kornhuber, Ralf

We consider stochastic elliptic variational inequalities of the second kind
involving a bilinear form with stochastic diffusion coefficient. We prove
existence and uniqueness of weak solutions, propose a stochastic Galerkin
approximation of an equivalent parametric reformulation, and show equivalence
to a related collocation method. Numerical experiments illustrate the
efficiency of our approach and suggest similar ... View more

We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show equivalence to a related collocation method. Numerical experiments illustrate the efficiency of our approach and suggest similar error estimates as for linear elliptic problems.

View less2010The geometry of Lp-spaces over atomless measure spaces and the Daugavet
propertySanchez Perez, Enrique A.; Werner, Dirk

We show that Lp-spaces over atomless measure spaces can be characterized in terms of a p-concavity type geometric property that is related with the Daugavet property.

2011Nigel Kalton's work in isometrical banach space theoryWerner, Dirk

This survey paper describes some of the work that the late Nigel Kalton did in the field of isometrical Banach space theory.

2010Walk the dog, or: products of open balls in the space of continuous functionsBehrends, Ehrhard

Let C[0,1] be the Banach algebra of real valued continuous functions on [0,1],
provided with the supremum norm. For f,g\in C[0,1] and balls B_{f}, B_{g} with
center f and g, respectively, it is not necessarily true that f\cdot g is in
the interior of B_{f}\cdot B_{g}. In the present paper we characterize those
pairs f, g where this is the case. The problem is illustrated by using a
suitable translation. One studies ... View more

Let C[0,1] be the Banach algebra of real valued continuous functions on [0,1], provided with the supremum norm. For f,g\in C[0,1] and balls B_{f}, B_{g} with center f and g, respectively, it is not necessarily true that f\cdot g is in the interior of B_{f}\cdot B_{g}. In the present paper we characterize those pairs f, g where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way.

View less2013Variational formulation of rate- and state-dependent friction problemsPipping, Elias; Sander, Oliver; Kornhuber, Ralf

We propose a variational formulation of rate- and state-dependent models for
the dynamic sliding of a linearly elastic block on a rigid surface in terms of
two coupled variational inequalities. Classical Dieterich-Ruina models are
covered as special cases. We show existence and uniqueness of solutions for
the two spatial subproblems arising from time discretisation. Existence of
solutions to the coupled spatial problems ... View more

We propose a variational formulation of rate- and state-dependent models for the dynamic sliding of a linearly elastic block on a rigid surface in terms of two coupled variational inequalities. Classical Dieterich-Ruina models are covered as special cases. We show existence and uniqueness of solutions for the two spatial subproblems arising from time discretisation. Existence of solutions to the coupled spatial problems is established for Dieterich's state equation through a fixed point argument.We conclude with some numerical experiments that suggest mesh independent convergence of the underlying fixed point iteration, and illustrate quasiperiodic occurrence of stick/slip events.

View less2010L- and M-structure in lush spacesPipping, Elias

Let X be a Banach space which is lush. It is shown that if a subspace of X is either an L-summand or an M-ideal then it is also lush.

2011Faktorisieren mit dem Quadratischen Sieb

ein Beitrag zur Didaktik der Algebra und KryptologieSchulz, Ralph-Hardo; Witten, Helmut

ein Beitrag zur Didaktik der Algebra und KryptologieSchulz, Ralph-Hardo; Witten, Helmut

Eines der zur Zeit schnellsten Verfahren zur Faktorisierung ganzer Zahlen ist
das ``Quadratische Sieb'' (engl. ``quadratic sieve factorization method''),
das 1981 von Carl Pomerance entwickelt wurde. Wir beschreiben im Folgenden die
Basisversion des Quadratischen Siebs sowie die Variante des Quadratischen
Siebs mit mehrfachen Polynomen, das sogenannte ``Multiple Polynomial Quadratic
Sieve'' MPQS, das unabhängig von J. ... View more

Eines der zur Zeit schnellsten Verfahren zur Faktorisierung ganzer Zahlen ist das ``Quadratische Sieb'' (engl. ``quadratic sieve factorization method''), das 1981 von Carl Pomerance entwickelt wurde. Wir beschreiben im Folgenden die Basisversion des Quadratischen Siebs sowie die Variante des Quadratischen Siebs mit mehrfachen Polynomen, das sogenannte ``Multiple Polynomial Quadratic Sieve'' MPQS, das unabhängig von J. Davis und D. Holdridge bzw. P. Montgomery gefunden wurde. Bei der Darstellung der Verfahren orientieren wir uns an [Buchmann 2010], [Crandall & Pomerance 2005], [Esslinger et al. 2011], [Pomerance 1996], 'Quadratisches Sieb' in [Wikipedia de] und 'quadratic sieve' in [Wikipedia en].

View less2013A multidomain discretization of the Richards equation in layered soilBerninger, Heiko; Kornhuber, Ralf; Sander, Oliver

We consider the Richards equation on a domain that is decomposed into
nonoverlapping layers, i.e., the decomposition has no cross points. We assume
that the saturation and permeability functions are space-independent on each
subdomain. Kirchhoff transformation of each subdomain problem separately then
leads to a set of semi-linear equations, which can each be solved efficiently
using monotone multigrid. The transformed ... View more

We consider the Richards equation on a domain that is decomposed into nonoverlapping layers, i.e., the decomposition has no cross points. We assume that the saturation and permeability functions are space-independent on each subdomain. Kirchhoff transformation of each subdomain problem separately then leads to a set of semi-linear equations, which can each be solved efficiently using monotone multigrid. The transformed subdomain problems are coupled by nonlinear continuity and flux conditions. This nonlinear coupled problem can be solved using substructuring methods like the Dirichlet-Neumann or Robin iteration. We give several numerical examples showing the discretization error, the solver robustness under variations of the soil parameters and a hydrological example with four soil layers and surface water.

View less2010The p-daugavet property for function spacesSanchez Perez, Enrique A.; Werner, Dirk

A natural extension of the Daugavet property for p-convex Banach function spaces and related classes is analysed. As an application, we extend the arguments given in the setting of the Daugavet property to show that no reflexive space falls into this class.

2010Subspaces of almost Daugavet spacesLücking, Simon

We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\ell_1$, then $Z$ has the almost Daugavet property, too.

2010Fast and robust numerical solution of the Richards equation in homogeneous
soilBerninger, Heiko; Kornhuber, Ralf; Sander, Oliver

We derive and analyse a solver-friendly finite element discretiza- tion of a
time discrete Richards equation based on Kirchhoff transformation. It can be
interpreted as a classical finite element discretization in physical variables
with non-standard quadrature points. Our approach allows for non- linear
outflow or seepage boundary conditions of Signorini type. We show convergence
of the saturation and, in the ... View more

We derive and analyse a solver-friendly finite element discretiza- tion of a time discrete Richards equation based on Kirchhoff transformation. It can be interpreted as a classical finite element discretization in physical variables with non-standard quadrature points. Our approach allows for non- linear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the non-degenerate case, of the discrete physical pressure. The associated discrete algebraic problems can be formu- lated as discrete convex minimization problems and, therefore, can be solved efficiently by monotone multigrid methods. In numerical examples for two and three space dimensions we observe L2-convergence rates of order O(h2) and H1-convergence rates of order O(h) as well as robust convergence behaviour of the multigrid method with respect to extreme choices of soil parameters.

View less2010Adaptive modelling of coupled hydrological processes with application in water
managementBastian, Peter; Berninger, Heiko; Dedner, Andreas; Engwer, Christian; Henning, Patrick; Kornhuber, Ralf; Kröner, Dietmar; Ohlberger, Mario; Sander, Oliver; Schiffler, Gerd; Shokina, Nina; Smetana, Kathrin

This paper presents recent results of a network project aiming at the
modelling and simulation of coupled surface and subsurface flows. In
particular, a discontinuous Galerkin method for the shallow water equations
has been developed which includes a special treatment of wetting and drying. A
robust solver for saturated-unsaturated groundwater flow in homogeneous soil
is at hand, which, by domain decomposition techniques, ... View more

This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated-unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet-Neumann method, using the dune-grid-glue module in the numerics software Dune.

View less2012An adaptive Newton multigrid method for a model of marine ice sheetsJouvet, Guillaume; Gräser, Carsten

In this paper, we consider a model for the time evolution of marine ice
sheets. This model combines the Shallow Ice Approximation (SIA) for the ice
deformation, the Shallow Shelf Approximation (SSA) for the basal sliding and
the mass conservation principle. At each time step, we solve a generalized
p-Laplace minimization-type problem with obstacle (SIA), a vectorial p-Laplace
minimization-type problem (SSA) and a ... View more

In this paper, we consider a model for the time evolution of marine ice sheets. This model combines the Shallow Ice Approximation (SIA) for the ice deformation, the Shallow Shelf Approximation (SSA) for the basal sliding and the mass conservation principle. At each time step, we solve a generalized p-Laplace minimization-type problem with obstacle (SIA), a vectorial p-Laplace minimization-type problem (SSA) and a transport equation (mass conservation). The two minimization problems are solved using a truncated nonsmooth Newton multigrid method while the transport equation is solved using a vertex-centred finite volume method. Our approach is combined to a mesh adaptive refinement procedure to face the large gradients of the solution that are expected close to the grounding line which separates the ice sheet and the ice shelf. As applications, we present some simulations of the marine ice sheet model inter- comparison project MISMIP in two and three space dimensions. In particular, we test the ability of our model to reproduce a reversible grounding line after being perturbed in model parameters.

View less2010Order convergence and convergence almost everywhere revisitedPreuß, Gerhard

In Analysis two modes of non-topological convergence are interesting: order
convergence and convergence almost everywhere. It is proved here that oder
convergence of sequences can be induced by a limit structure, even a finest
one, whenever it is considered in sigma-distributive lattices. Since
convergence almost everywhere can be regarded as order convergence in a
certain sigma-distributive lattice, this result can ... View more

In Analysis two modes of non-topological convergence are interesting: order convergence and convergence almost everywhere. It is proved here that oder convergence of sequences can be induced by a limit structure, even a finest one, whenever it is considered in sigma-distributive lattices. Since convergence almost everywhere can be regarded as order convergence in a certain sigma-distributive lattice, this result can be applied to convergence of sequences almost everywhere and thus generalizing a former result of U. Höhle obtained in a more indirect way by using fuzzy topologies.

View less2008Thickness of the unit sphere, l 1-types, and the almost Daugavet propertyKadets, Vladimir; Shepelska, Vavara; Werner, Dirk