Library

Description: The paper's main result is a simple derivation rule for the Jacobi polynomials with respect to their parameters, i.e. for $\partial_{\alpha} P_n^{\alpha,\beta}$, and $\partial_{\beta} P_n^{\alpha,\beta}$. It is obtained via relations for the Gaussian hypergeometric function concerning parameter derivatives and integer shifts in the first two arguments. These have an interest on their own for further applications to continuous and discrete orthogonal polynomials. The study is motivated by a Galerkin method with moving weight, presents all proofs in detail, and terminates in a brief discussion of the generated polynomials.

Description: We derive a simple accuracy matching strategy for inexact Gauss Newton methods and apply it to the numerical solution of boundary value problems of ordinary differential equations by collocation. The matching strategy is based on an affine contravariant convergence theorem, i. e. , the characteristic constants are invariant under affine transformations of the domain. The inexact Gauss Newton method is applied to an integral formulation of the BVP. As discretization for the arising linear subproblems we employ adaptive collocation at Gaussian nodes with varying local orders and stepsizes. The grids are chosen via adaptive refinement and order selection.

Description: In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities like differential equations, power series representations, and hypergeometric representations can even dealt with algorithmically, i.\ e.\ they can be computed by the Computer Algebra system, rather than only verified. The types of functions that can be treated by the given technique cover the generalized hypergeometric functions, and therefore most of the special functions that can be found in mathematical dictionaries. The types of identities for which we present verification algorithms cover differential equations, power series representations, identities of the Rodrigues type, hypergeometric representations, and algorithms containing symbolic sums. The current implementations of special functions in existing Computer Algebra systems do not meet these high standards as we shall show in examples. They should be modified, and we show results of our implementations.

Description: {\newcommand{\R} {{\rm {\mbox{\protect\makebox[.15em][l]{I}R}}}} Given a list of $n$ numbers in $\R $, one wants to decide wether every number in the list occurs at least $k$ times. I will show that $(1-\epsilon)n\log_3(n/k)$ is a lower bound for the depth of a linear decision tree determining this problem. This is done by using the Björner-Lov\'asz method, which turns the problem into one of estimating the Möbius function for a certain partition lattice. I will also calculate the exponential generating function for the Möbius function of a partition poset with restricted block sizes in general.}

Description: Die Autoren sind sich dar{ü}ber im klaren, daß zahlreiche Imponderabilien die Ergebnisse der vorgelegten Modellrechnungen beeinflussen k{ö}nnen. Immerhin repräsentiert jedoch das vorgelegte mathematische Modell die Summe unserer derzeitigen Kenntnisse zur Fragestellung AIDS--Epidemie; neu hinzukommende Erkenntnisse lassen sich rasch einarbeiten und in ihren Konsequenzen überschauen. Die Entwicklung neuer effizienter numerischer Methoden spielte eine Schlüsselrolle bei der tatsächlichen Simulation dieses umfangreichen realitätsnahen mathematischen Modells.

Description: We present an adaptive Rothe method for two--dimensional problems combining an embedded Runge--Kutta scheme in time and a multilevel finite element discretization in space. The spatial discretization error is controlled by a posteriori error estimates based on interpolation techniques. A computational example for a thermodiffusive flame propagation model illustrates the high accuracy that is possible with the proposed method.

Description: The multicommodity linear formulation of the Fixed Charge Network Flow Design problem is known to have significantly sharp linear relaxation lower bounds. However the tradeoff is the introduction of a large amount of artificial variables. We exhibit a class of special instances for which the lower bound is tight. Further we completly describe the polyhedron in the space of the natural variables.

Description: Die von der DMV-Fachgruppe Scientific Computing in Kooperation mit dem gleichnamigen GAMM-Fachausschuss organisierte Tagung \glqq Scientific Computing in der Theoretischen Physik\grqq~fand vom 16. - 18. März 1994 am Fachbereich Mathematik und Informatik der Freien Universität Berlin statt. Ziel des Workshops war, die Kontakte zwischen den Fachleuten der Gebiete {\em Computational Physics} und {\em Scientific Computing} zu intensivieren. Schwerpunkte des Workshops waren numerische Simulation von Transportmodellen der Astrophysik und Halbleiterphysik, Multilevel-Methoden für partielle Differentialgleichungen sowie Monte-Carlo-Methoden und molekulardynamische Verfahren für Probleme der Statistischen Physik und Quantenfeldtheorie. Der Workshop fand ein äusserst positives Echo und führte über 70 Teilnehmer der verschiedenen Teilgebiete der Theoretischen Physik und ca. 50 Teilnehmer aus dem Bereich der Numerischen Mathematik zusammen. \originalTeX

Description: We give an overview of an approach on special functions due to Truesdell, and show how it can be used to develop certain type of identities for special functions. Once obtained, these identities may be verified by an independent algorithmic method for which we give some examples.

Description: {\newcommand{\N} {{\rm {\mbox{\protect\makebox[.15em][l]{I}N}}}} In several publications the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series $\sum\limits_{k=0}^{\infty}a_{k}z^{k}$ at the origin and vice versa, implementations of which exist in {\sc Mathematica}, {\sc Maple} and {\sc Reduce}. One main part of this procedure is an algorithm to derive a homogeneous linear differential equation with polynomial coefficients for the given function. We call this type of ordinary differential equations {\sl simple}. Whereas the opposite question to find functions satisfying given differential equations is studied in great detail, our question to find differential equations that are satisfied by given functions seems to be rarely posed. In this paper we consider the family $F$ of functions satisfying a simple differential equation generated by the rational, the algebraic, and certain transcendental functions. It turns out that $F$ forms a linear space of transcendental functions. % with polynomial function coefficients. Further $F$ is closed under multiplication and under the composition with rational functions and rational powers. These results had been published by Stanley who had proved them by theoretical algebraic considerations. In contrast our treatment is purely algorithmically oriented. We present algorithms that generate simple differential equation for $f+g$, $f\cdot g$, $f\circ r$ ($r$ rational), and $f\circ x^{p/q}$ ($p,q\in\N_0$), given simple differential equations for $f$, and $g$, and give a priori estimates for the order of the resulting differential equations. We show that all order estimates are sharp. After finishing this article we realized that in independent work Salvy and Zimmermann published similar algorithms. Our treatment gives a detailed description of those algorithms and their validity.}

Description: The power of the symbolic math system {\small REDUCE} for solving large and difficult problems in science and engineering is demonstrated by a set of model problems. These include algebraic equation solving, formal variable elimination, formal power series, symbolic treatment of differential equations and applications from theoretical physics.

Description: Results from finite-element-calculations are usually visualized by colored surface- and contour-line-plots or polygonal patches or simply displaced lines and grids. In computer graphics however more advanced techniques like texture-mapping and NURBS are well established and there exist efficient algorithms and implementations. We show that these techniques are not only easy to use, but form a very natural and far more efficient approach for visualization of higher order finite-element's solutions like in $p$- and $h$-$p$-version. Texture-mapping is useful for displaying vector-valued data, too.

Description: In Quantum Chemistry the field of Laser--Assisted Molecular Control'' has received a considerable amount of attention recently. One key problem in this new field is the simulation of the dynamical reaction of a molecule subjected to external radiation. This problem is described by the Schrödinger equation, which, after eigenfunction expansion, can be written in the form of a large system of ordinary differential equations, the solutions of which show a highly oscillatory behaviour. The oscillations with high frequencies and small amplitudes confine the stepsizes of any numerical integrator -- an effect, which, in turn, blows up the simulation time. Larger stepsizes can be expected by averaging these fast oscillations, thus smoothing the trajectories. Standard smoothing techniques (averaging, filtering) would kill the whole process and thus, lead to wrong numerical results. To avoid this unwanted effect and nevertheless speed up computations, this paper presents a quasiresonant smoothing algorithm (QRS). In QRS, a natural splitting parameter $\delta$ controls the smoothing properties. An adaptive QRS--version (AQRS) is presented which includes an error estimation scheme for choosing this parameter $\delta$ in order to meet a given accuracy requirement. In AQRS $\delta$ is permanently adapted to the solution properties for computing the chemically necessary information'' only. The performance of AQRS is demonstrated in several test problems from the field Laser--Assisted Selective Excitation of Molecules'' in which the external radiation is a picosecond laser pulse. In comparison with standard methods speedup factors of the order of $10^2$ are observed.

Description: This article describes the REDUCE package ZEILBERG implemented by Gregor Stölting and the author. The REDUCE package ZEILBERG is a careful implementation of the Gosper (The sum package contains also a partial implementation of the Gosper algorithm.) and Zeilberger algorithms for indefinite, and definite summation of hypergeometric terms, respectively. An expression $a_k$ is called a {\sl hypergeometric term} (or {\sl closed form}), if $a_{k}/a_{k-1}$ is a rational function with respect to $k$. Typical hypergeometric terms are ratios of products of powers, factorials, $\Gamma$ function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer-linear in their arguments.

Description: We test the performance of the multicanonical approach for biological molecules. The simulated molecules are frustrated systems with a complicated energy landscape. The resulting slowing down in simulations is alleviated by our ansatz. We perform a multicanonical simulation of nonpolar amino acids and study their $\alpha$-helix propensities. The results are shown to be in agreement with recent experimental results.

Description: In an array of coupled oscillators {\em synchronous chaos} may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries and this leads to the idea of {\em symmetry breaking bifurcations of chaotic attractors}. Finally the results and ideas are illustrated for several systems of coupled oscillators.

Description: We present new Monte Carlo results in non-compact lattice QED with staggered fermions down to $m_0 = 0.005$. This extends our previous investigations on the nature of the continuum limit of QED.

Description: The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $\Gamma$ function terms, binomial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments. We consider the more general case of ratios of products of exponentials, factorials, $\Gamma$ function terms, binomial coefficients, and Pochhammer symbols that are rational-linear with respect to $n$ and $k$ in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities. The given algorithms on definite summation apply to many cases in the literature to which neither the Zeilberger approach nor the Wilf-Zeilberger method is applicable. Examples of this type are given by theorems of Watson and Whipple, and a large list of identities (``Strange evaluations of hypergeometric series'') that were studied by Gessel and Stanton. It turns out that with our extended algorithms practically all hypergeometric identities in the literature can be verified. Finally we show how the algorithms can be used to generate new identities. REDUCE and MAPLE implementations of the given algorithms can be obtained from the author, many results of which are presented in the paper.

Description: We present a set of C++ classes that realize an abstract inexact Gauss Newton method in combination with a continuation process for the solution of parameter dependent nonlinear problems. The object oriented approach allows the continuation of different types of solutions within the same framework. \\{\bf Abstract: }We present a set of C++ classes that realize an abstract inexact Gauss Newton method in combination with a continuation process for the solution of parameter dependent nonlinear problems. The object oriented approach allows the continuation of different types of solutions within the same framework.

Description: We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we develop combinatorial models for the Klee-Minty cubes and similar linear programs with exponential decreasing paths. The analysis of two randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds conjectured in the literature. At the same time, we establish quadratic upper bounds for random pivots on the linear programs under investigation. This motivates the question whether some randomized pivot rules possibly have quadratic worst-case behavior on general linear programs.

Description: We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them {\em cascadic iterations}. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments that the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of $H^2$-regular two-dimensional problems with quasi-uniform triangulations. We show that his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two- and three-dimensional $H^{1+\alpha}$-regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method.

Description: The results from invariant theory and the results for semi-invariants and equivariants are summarized in a way suitable for the combination with Gröbner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincar\'{e} series is described. Secondly, an algorithm is given for the representation of an equivariant in terms of the fundamental equivariants. Several ways for the exact determination of zeros of equivariant systems are discussed

Description: We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.

Description: In this paper we consider the problem of $k$-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for so-called cycle, cycle with ears, knapsack tree and path-block-cycle inequalities. The separation heuristics, plus primal heuristics, have been implemented in a branch-and-cut routine using a formulation including the edges with nonzero costs and node variables. Results are presented for three classes of problems: equipartitioning problems arising in finite element methods and partitioning problems associated with electronic circuit layout and compiler design.

Description: {\def\N{{\mbox{{\rm I\kern-0.22emN}}}} Let a set $N$ of items, a capacity $F \in \N$ and weights $a_i \in \N$, $i \in N$ be given. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality $$\sum_{i \in N} a_i x_i \leq F.$$ In this paper we present a linear description of the 0/1 knapsack polytope for the special case where $a_i \in \{\mu,\lambda\}$ for all items $i \in N$ and $1 \leq \mu 〈 \lambda \leq b$ are two natural numbers. The inequalities needed for this description involve elements of the Hilbert basis of a certain cone. The principle of generating inequalities based on elements of a Hilbert basis suggests further extensions.}

Description: We compare a few variants of the recently proposed multicanonical method with the well known simulated annealing for the effectiveness in search of the energy global minimum of a biomolecular system. For this we study in detail Met-enkephalin, one of the simplest peptides. We show that the new method not only outperforms simulated annealing in the search of the energy groundstate but also provides more statistical-mechanical information about the system.

Description: We demonstrate the effectiveness of the multicanonical algorithm for the tertiary structure prediction of peptides and proteins. Unlike to simulated annealing the relationship to the canonical ensemble remains exactly controlled. Hence, the new method allows not only the prediction of the lowest-energy conformation, but also the calculation of thermodynamic quantities at various temperature from one run.

Description: Die enorme algorithmische Beschleunigung durch diskrete Galerkin-Methoden f{ü}r abz{ä}hlbare Differentialgleichungssysteme hat der Simulation von Polymerisationsprozessen neue, industriell relevante Problembereiche er{ö}ffnet, die mit den bis dahin verf{ü}gbaren Methoden nicht zug{ä}nglich waren.

Description: Large chemical computations show the need for full adaptivity supporting the development of robust and highly efficient programs. For solutions possessing sharp moving spatial transitions, as travelling wavefronts or emerging boundary and internal layers, an automatic adjustment of both the space and the time stepsize is generally accepted to be more successful in efficient resolving critical regions of high spatial and temporal activity. In contrast to the widespread discretization sequence first space then time the reversed sequence first time then space is employed. Full adaptivity of the proposed algorithm is realized by combining embedded time discretization and multilevel finite element space discretization. In this paper the algorithm is described for one--dimensional problems. The numerical results show the significantly new perspectives opened by this approach.

Description: An error controlled finite elemente method (FEM) for solving stationary Schrödinger equations in three space dimensions is proposed. The method is based on an adaptive space discretization into tetrahedra and local polynomial basis functions of order $p=1$--$5$ defined on these tetrahedra. According to a local error estimator the triangulation is automatically adapted to the solution. Numerical results for standard problems appearing in vibrational motion and molecular structure calculations are presented and discussed. Relative precisions better than 1e-8 are obtained. For equilateral H$_3^{++}$ the adaptive FEM turns out to be superior to global basis set expansions in the literature. Our precise FEM results exclude in a definite manner the stability or metastability of equilateral H$_3^{++}$ in its groundstate.

Description: {\def\NP{\hbox{$\cal N\kern-.1667em\cal P$}} The {\sl storage assignment problem} asks for the cost minimal assignment of containers with different sizes to storage locations with different capacities. Such problems arise, for instance, in the optimal control of automatic storage devices in flexible manufacturing systems. This problem is known to be $\NP$-hard in the strong sense. We show that the storage assignment problem is $\NP$-hard for {\sl bounded sizes and capacities}, even if the sizes have values $1$ and~$2$ and the capacities value~$2$ only, a case we encountered in practice. Moreover, we prove that no polynomial time $\epsilon$-approximation algorithm exists. This means that almost all storage assignment problems arising in practice are indeed hard.}

Description: We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step-size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.

Description: We investigate the generating sets (``Gröbner bases'') of integer lattices which correspond to the Gröbner bases of the associated binomial ideals. Extending results in Sturmfels and Thomas, preprint 1994, we obtain a geometric characterization of the universal Gröbner basis in terms of the vertices and edges of the associated corner polyhedra. We emphasize the special case where the lattice has finite index. In this case the corner polyhedra were studied by Gomory, and there is a close connection to the ``group problem in integer programming'' Schrijver, p.~363. We present exponential lower and upper bounds for the size of a reduced Gröbner basis. The initial complex of (the ideal of) a lattice is shown to be dual to the boundary of a certain simple polyhedron.

Description: We consider periodic orbits of autonomous parameter dependent ODE's. Starting from a shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincar\'e-section we develop a pathfollowing algorithm for periodic solutions based on a tangential continuation method with implicit reparametrization. For ODE's equivariant w.r.t. a finite group we show that spatial as well as spatio-temporal symmetries of periodic orbits can be exploited within the (multiple) shooting context. We describe how turning points, period doubling bifurcations and Hopf points along the branch of periodic solutions can be handled. Furthermore equivariant Hopf points and generic secondary bifurcations of periodic orbits with $ Z_m$-symmetry are treated. We tested the code with standard examples, e.g., the period doubling cascade in the Lorenz equations. To show the efficiency of the described methods we also used the program for an application from electronics, a ring oscillator with $n $ inverters. In this example the exploitation of symmetry reduces the amount of work for the continuation of periodic orbits from ${\cal O}(n^2)$ to ${\cal O}(n)$

Description: In this article algorithmic methods are presented that have been incorporated in computer algebra systems in the last five years. The methods can be used to identify transcendental functions. The essential idea due to Zeilberger is to represent the functions by systems of holonomic differential and recurrence equations, and the identification procedure utilizes algorithms of Stanley and Zeilberger. Also a noncommutative version of the Buchberger algorithm can be applied.

Description: In this paper we modify Buchberger's $S$-pair reduction algorithm for computing a Gröbner basis of a toric ideal so as to apply to an integer program in inequality form with fixed right hand sides and fixed upper bounds on the variables. We formulate the algorithm in the original space and interpret the reduction steps geometrically. In fact, three variants of this algorithm are presented and we give elementary proofs for their correctness. A relationship between these (exact) algorithms, iterative improvement heuristics and the Kernighan-Lin procedure is established.

Description: Adopting a statistical approach for the computation of turbulent combustion flows an approximation for the probability density function (PDF) of the composition variables is often required to treat the highly non-linear reaction term in a satisfactory way. One class of methods currently being used are the moment methods which employ transport equations for low order statistical moments and use a parametrized shape of the PDF. A second class solves a transport equation for the joint PDF by a Monte Carlo method. In the present paper we develop an intermediate algorithm based on a Galerkin method for the PDF transport equation. The solution is developed in terms of an orthogonal or bi-orthogonal basis of a suitable Hilbert space. The unconventional use of the related weight function as a prefactor (moving weight approach) permits adaptivity and results in a generalization of the $\beta-$closure for bounded scalar quantities. We present the approximation procedure in detail and apply it to the evolution of the composition in a homogeneous well-stirred reactor. The extension to non-homogeneous flow simulations is straightforward.

Description: Combinatorial optimization problems pervade many areas of human problem solving especially in the fields of business, economics and engineering. Intensive mathematical research and vast increases in raw computing power have advanced the state of the art in exact and heuristic problem solving at a pace that is unprecedented in human history. We survey here in layman's terms some of the fundamental concepts and principles that have led this progress. (This article will appear -- possibly in modified form -- in a popular science magazine.)

Description: {\def\N{{\mbox{{\rm I\kern-0.22emN}}}} Given a set $N$ of items and a capacity $b \in \N$, and let $N_j$ be the set of items with weight $j$, $1 \leq j \leq b$. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality $$\sum_{j=1}^b \sum_{i \in N_j} jx_i \leq b.$$ In this paper we first present a complete linear description of the 0/1 knapsack polytope for two special cases: (a) $N_j = \emptyset$ for all $1 〈 j \leq \lfloor {b \over 2} \rfloor$ and (b) $N_j = \emptyset$ for all $1 〈 j \leq \lfloor {b \over 3} \rfloor$ and $N_j = \emptyset$ for all $j \geq \lfloor {b \over 2} \rfloor + 1$. It turns out that the inequalities that are needed for the complete description of these special polytopes are derived by means of some ``reduction principle''. This principle is then generalized to yield valid and in many cases facet defining inequalities for the general 0/1 knapsack polytope. The separation problem for this class of inequalities can be solved in pseudo polynomial time via dynamic programming techniques.}

Description: In the field of deep regional hyperthermia, one of the most widely used devices is the BSD--2000 Hyperthermia System which employs the Sigma 60 applicator. The Sigma 60 consists of four independent sources, giving it the potential to control the energy pattern within the patient. The independent amplitudes and phases, as well as frequency selection and applicator position, present a large number of parameters for the operator to determine. Computer simulation has long been recognized as an attractive approach to optimizing these parameters. A treatment planning program was used in clinical practice at Stanford University Medical Center for two years. It demonstrated the feasibility of computer simulation for deep regional hyperthermia in a clinical situation. However, several parts of this system were written in a language specific to one workstation, which severely restricted the wider distribution of the program to other users of the Sigma 60. A new treatment planning system for the BSD 2000 has been developed and put into clinical practice at the Rudolf Virchow Clinic of the Free University of Berlin. The new method, which we will refer to as the Berlin system, has a simpler model construction program and a considerably better graphics capability. However, the most important feature is that all programs are written in FORTRAN, C, or the X Window graphics system. Therefore, the entire treatment planning system is completely portable to other workstations.

Description: In this paper we describe several versions of the routing problem arising in VLSI design and indicate how the Steiner tree packing problem can be used to model these problems mathematically. We focus on switchbox routing problems and provide integer programming formulations for routing in the knock-knee and in the Manhattan model. We give a brief sketch of cutting plane algorithms that we developed and implemented for these two models. We report on computational experiments using standard test instances. Our codes are able to determine optimum solutions in most cases, and in particular, we can show that some of the instances have no feasible solution if Manhattan routing is used instead of knock-knee routing.

Description: The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lifting Algorithm, proposed by Huber and Sturmfels, can be applied to symmetric Newton polytopes. This symmetric version of the Lifting Algorithm enables the efficient construction of the symmetric subdivision, giving rise to a symmetric homotopy, so that only the generating solutions have to be computed. Efficiency is obtained by combination with the product homotopy. Applications illustrate the practical significance of the presented approach.

Description: In this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible. The corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting.

Description: Die Kombination von elektronischer Fachinformation mit den neuen Mitteln der Kommunikation eröffnet neue Perspektiven für die Mathematik in Deutschland. Aufbauend auf das laufende, BMFT-geförderte Projekt ``Fachinformation'' der Deutschen Mathematiker-Vereinigung (DMV) plant die DMV eine Querschnittsaktivität für die Mathematik. Diese soll nicht nur die mathematischen Fachbereiche und Forschungsinstitute einbeziehen, sondern auch mathematische Forschungslabors aus der Industrie und die Informationsversorger der Mathematik, insbesondere die wissenschaftlichen Verlage, die TIB Hannover, die Bibliotheken der Universitäten und \mbox{-- nicht zuletzt --} das FIZ Karlsruhe mit dem Zentralblatt für Mathematik. \noindent Auf der technischen Grundlage des Internet und seiner weltweit verbreiteten Informationsdienste (Gopher, WAIS, WorldWideWeb, Hyper-G, ftp und E-mail) soll ein verteiltes, mathematisches Informationssystem geschaffen werden, bei dem die Partner des gemeinsamen Vorhabens ihre örtlichen Ressourcen untereinander und zugleich weltweit verfügbar machen. Diese Ziele sind durch konsequenten Einsatz von Software erreichbar, die nach dem Client-Server-Modell strukturiert und die im Internet verbreitet bzw.~anerkannt ist. \noindent Die Bereitstellung von Technik reicht keineswegs aus. Ohne die Schaffung einer personellen, organisatorischen und technischen Infrastruktur ist die Durchführung eines solchen Projekts nicht möglich. Als Massnahmen sind - für alle Partner - geplant: \hspace*{2mm} * Schaffung der Funktion des Informationsbeauftragten \hspace*{2mm} * Einrichtung eines gemeinsamen ``Forums für mathematische Information'' \hspace*{2mm} * Errichtung und Betreuung von mathematischen Informationsstationen \hspace*{2mm} * Teilnahme an nationalen und internationalen Standardisierungsmassnahmen \hspace*{2mm} * Workshops und regelmässige Schulungen, Öffentlichkeitsarbeit \noindent Als Aktivitäten - mit Focus auf mathematische Fachinformation - sind folgende Teilaufgaben und Pilotvorhaben bei allen Partnern geplant: \hspace*{2mm} * Einrichtung von elektronischen Informationssystemen \hspace*{2mm} * Verteiltes elektronisches Angebot von Preprints und Skripten (Volltexte) \hspace*{2mm} * Verteiltes elektronisches Angebot von Software- und Datensammlungen \hspace*{2mm} * Zugriff auf globale Informationssysteme in der Mathematik \hspace*{2mm} * Organisation eines Lebenden Mathematischen Museums \hspace*{2mm} * Zugriff auf elektronische Kataloge von Bibliotheken, insbesondere Fachbereichsbibliotheken \hspace*{2mm} * Schaffung eines organisatorischen Rahmens für elektronische mathematische Journale \hspace*{2mm} * Elektronisches Angebot von eingescannten historischen Dokumenten und Büchern \hspace*{2mm} * Schaffung einer Organisationsform für elektronisches Reviewing \hspace*{2mm} * Erprobung von Verfahren zur elektronischen Lieferung von Dokumenten \hspace*{2mm} * Elektronische Projektorganisation \noindent Das Vorhaben zielt ab auf die Schaffung einer informationstechnischen Infrastruktur für Datenbank-Retrieval, E-mail, ``Electronic Conferencing'' und fachspezifische Informationsnetze in exemplarischer Form für Wissenschaft und Industrie - auch für andere Fachgebiete.

Description: \begin{footnotesize} \noindent Combining electronic specialized information, such as electronic information retrieval from central databases, with the new means of communication opens up new perspectives for mathematics in Germany. Based on the current Fachinformationsprojekt (Specialized Information Project) run by the Deutsche Mathematiker-Vereinigung (DMV) and supported by the Federal Minister for Research and Technology (BMFT), the DMV is planning a new infrastructure activity for mathematics. This activity will not only include mathematical university departments and research institutes, but also partners from mathematical research laboratories in industry as well as suppliers of mathematical information, in particular, scientific publishing houses, the Technical Information Library (TIB) Hannover, university libraries and -- last but not least -- the Fachinformationszentrum (FIZ) Karlsruhe with the Zentralblatt für Mathematik. \noindent On the technical basis of the Internet and its worldwide information services (Gopher, WAIS, World Wide Web, Hyper-G, ftp and e-mail), a distributed mathematical information system is to be created, whose partners make their local resources available both to other partners and to the worldwide Internet community. These aims can be achieved by means of the consequent use of software that is structured according to the client-server model and distributed and accepted within the Internet. \noindent Providing the technical equipment, however, is by far not enough. Without the creation of a personal, technical and organizational infrastructure, realization of such a project will not be possible. For every partner the realization of the following measures is envisaged:\\[-7mm] \noindent \end{footnotesize} \begin{small} \begin{itemize} \item[*] Establishing an Information Coordinator\/ at every participating institution\\[-8mm] \item[*] Creation of a Forum for Mathematical Information\/\\[-8mm] \item[*] Installation and maintenance of Mathematical Information Stations\/\\[-8mm] \item[*] Participation in national and international Standardization Activities\/\\[-8mm] \item[*] Regular workshops and training courses, public relations\\[-7mm] \end{itemize} \noindent The following activities (carried out by all partners or within pilot projects by special groups) focusing on mathematical and mathematics-related information are planned:\\[-7mm] \noindent \begin{itemize} \item[*] Set up of Electronic Information Systems\/ by all partners \\[-8mm] \item[*] Distributed\/ electronic offer of Preprints\/ and scripts (full texts) by all partners\\[-8mm] \item[*] Distributed\/ electronic offer of Software and Data Collections\/ by all partners\\[-8mm] \item[*] Access to Global Information Systems in Mathematics\/\\[-8mm] \item[*] Organization of a Living Museum of Mathematics\/\\[-8mm] \item[*] Access to {\it Electronic Library Catalogs,}\/ esp.~to libraries of university departments\\[-8mm] \item[*] Electronic offer of Scanned Historical Books and Documents\/\\[-8mm] \item[*] Creation and management of Electronic Mathematical Journals\/\\[-8mm] \item[*] Creation of a framework for various kinds of Electronic Reviewing\/\\[-8mm] \item[*] Testing of new methods for Electronic Document Delivery\/\\[-8mm] \item[*] Electronic Project Organization\/\\[-7mm] \end{itemize} \noindent \end{small} \begin{footnotesize} The project aims, at its beginning, at the creation of an information infrastructure for database retrieval, e-mail, electronic conferencing and subject-specific information networks. Its focus is on mathematics in Germany. The general scope, however, is broader. The project is open for discussion, coordination, and cooperation with partners in other areas of science, industry or in other countries. It is also hoped that this project may form an example along which other models of electronic information and communication can be developed. Moreover, the project is also meant to enhance the offer from German mathematics to the worldwide "give and take" within the Internet community. \end{footnotesize}

Description: The present paper contains a generalization of a refinement of the Newton- Mysovskii theorem, recently obtained by the authors, to the case of Gauss-Newton procedures for solving nonlinear least-squares problems with full Jacobians. Invariant sufficient conditions are given that ensure the convergence of the Gauss-Newton iterates towards a solution of the problem, as well as the uniqueness of that solution in an explicitely defined neighborhood. It is shown by a counter- example that the results do not carry over to the rank deficient case.

Description: The breakdown voltage highly depends on the electric field in the depletion area whose computation is the most time consuming part of the simulation. We present a self adaptive Finite Element Method which reduces dramatically the required computation time compared to usual Finite Difference Methods. A numerical example illustrates the efficiency and reliability of the algorithm.

Description: Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30.

Description: Most nonlinear computations require the evaluation of first and higher derivatives of vector functions defined by computer programs. It is shown here how vectors of such partial derivatives can be obtained automatically and efficiently if the computer language allows overloading (as is or will be the case for C++, PASCAL-XSC, FORTRAN90, and other modern languages). Here, overloading facilitates the extension of arithmetic operations and univariate functions from real or complex arguments to truncated Taylor-series (or other user- defined types), and it generates instructions for the subsequent evaluation of adjoints. Similar effects can be achieved by precompilation of FORTRAN77 programs. The proposed differentiation algorithm yields gradients and higher derivatives at a small multiple of the run-time and RAM requirement of the original function evaluation program. {\bf Keywords:} Automatic Differentiation, Chain Rule, Overloading, Taylor Coefficients, Gradients, Hessians, Reverse Accumulation, Adjoint Equations. {\bf Abbreviated title:} Automatic Differentiation by Overloading.

Description: The description of chain length distributions in macromolecular reaction kinetics leads to so-called countable systems of differential equations. In particular, when the appearing reaction rate coefficients depend on the chain length of the reacting macromolecules itself, an efficient numerical treatment of these systems is very difficult. Then even the evaluation of the right-hand side of the system can become prohibitively expensive with respect to computing time. In this paper we show how the discrete Galerkin method can be applied to such problems. The existing algorithm CODEX is improved by use of a multiplicative error correction scheme for time discretization and a new type of numerical preprocessing by means of a Gauss summation. Both ideas are exemplary for a wide class of approximation types and are described very briefly here. The new numerical techniques are tested on an example from soot formation, where the coagulation of molecules is modeled in terms of reaction coefficients depending on the surface of the particles and their collision frequency.

Description: The numerical solution of optimal control problems by indirect methods (such as multiple shooting or collocation) requires a considerable amount of analytic calculation to establish a numerically tractable system. These analytic calculations, though being rather tedious in realistic examples, are nowadays mostly still done by hand-and thus prone to calculation errors. The paper aims at automating this analytic processing to a reasonable extent by means of a modern symbolic manipulation language (here: REDUCE). In its present stage of development the package OCCAL (mnemotechnically for \underline{O}ptimal \underline{C}ontrol \underline{CAL}culator) permits an interactive use, covering tasks like automatic determination of control and, in case of a singular control, of its order. In simpler problems, the present version of OCCAL automatically produces the full subroutine input for a MULtiple shooting code (MULCON) with adaptive numerical CONtinuation. In more complicate problems where singular sub-arcs may occur or where the sequence of sub-arcs of the optimal trajectory is unclear OCCAL is a significant help in reducing analytic pre-processing. Numerical examples illustrate the performance of OCCAL/MULCON.

Description: If $B$ is an arrangement of linear complex Hyperplanes in $C^d$, then the following can be constructed from knowledge of its intersection lattice: (a) the cohomology groups of the complement [Br], (b) the cohomology algebra of the complement [OS], (c) the fundamental group of the complement, if $d\le2$, (d) the singularity link up to homeomorphism, if $d\le3$, (e) the singularity link up to homotopy type [ZZ]. If $B'$ is, more generally, a 2-arrangement in $ R^{2d}$ (an arrangement of real subspaces of codimension 2 with even-dimensional intersections), then the intersection lattice still determines (a) the cohomology groups of the complement [GM] and (e) the homotopy type of the singularity link [ZZ]. We show, however, that for 2-arrangements the data (b), (c) and (d) are not determined by the intersection lattice. They require the knowledge of extra information on sign patterns, which can be computed as determinants of linear relations, or (equivalently) as linking coefficients in the sense of knot theory.

Description: We prove combinatorial formulas for the homotopy type of the union of the subspaces in an (affine, compactified affine, spherical or projective) subspace arrangement. From these formulas we derive results of Goresky & MacPherson on the homology of the arrangement and the cohomology of its complement. The union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset. A closely related space is obtained by taking the homotopy direct limit of this diagram. Our method consists in constructing a combinatorial model diagram over the same poset, whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces.

Description: A new adaptive multilevel approach for linear partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction. The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A Multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy.

Description: NUMSIM 91 fand vom 6.-08. Mai 1991 im Wissenschaftlichen Kommunikations- & Konferenz- Zentrum der Humboldt-Universität in Gosen bei Berlin statt. Im Mittelpunkt dieses Seminars standen Probleme der numerischen Simulation von Ladungstransport- und Technologieprozessen der Mikro- und Optoelektronik. Es führte Spezialisten der physikalischen Modellierung, der numerischen Mathematik und mathematischen Analysis sowie Nutzer von Simulationsprogrammen aus dem deutschsprachigen Raum zusammen. Ziel war die Vermittlung und der Austausch von Erfahrungen, die Diskussion gemeinsamer Aufgabenstellungen und Projekte und - in Anbetracht der deutschen Vereinigung nicht zuletzt - das gegenseitige Kennen- und Verstehenlernen.

Description: The rapidly increasing number of AIDS cases requires a realistic estimation of the future development of the HIV/AIDS disease. For that purpose we develop a large system of coupled nonlinear differential equations describing simultaneously the dynamics of the development of the disease, the population size, the gender and age structure. A set of 1650 coupled equations are linked by balanced parameters. The balancing procedure is described by a set of (formally) 2,178,000 additional algebraic conditions. As the considered system is stiff, it requires new special extrapolation methods combined with techniquees of dynamical sparsing for the solution of sparsely filled systems. According to our simulations we expect 19,0,000 deaths caused by AIDS in the Federal Republic of Germany (former territories) in the year 2000. Such an epidemical spread would tie up about 4-7 percent of the actual health care workers.

Description: In this paper we introduce a discontinuous finite element method. In our approach, it is possible to combine the advantages of finite element and finite difference methods. The main ingredients are numerical flux approximation and local orthogonal basis functions. The scheme is defined on arbitrary triangulations and can be easily extended to nonlinear problems. Two different error indicators are derived. Especially the second one is closely connected to our approach and able to handle arbitrary variing flow directions. Numerical results are given for boundary value problems in two dimensions. They demonstrate the performance of the scheme, combined with the two error indicators. {\bf Key words:} neutron transport equation, discontinuous finite element, adaptive grid refinement. {\bf Subject classifications:} AMS(MOS) 65N30, 65M15.

Description: In this paper it is shown that for highly nonuniformly refined triangulations the condition number of the BPX preconditioner for elliptic finite element problems grows at most linearly in the depth of refinement. This is achieved by viewing the computational available version of the BPX preconditioner as an abstract additive Schwarz method with exact solvers. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65N20, 65N30.

Description: We study the space of all extensions of a real hyperplane arrangement by a new pseudo- hyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the "Generalized Baues Problem" of Billera, Kapranov & Sturmfels, via the Bohne-Dress Theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not even known whether the extension space is connected. We show that the subspace of realizable extensions is always connected but not necessarily spherical.

Description: We consider such variational inequalities which either describe obstacle problems or result from an implicit time discretization of moving boundary problems of two phase Stefan type. Based on a discretization in space by means of continuous, piecewise linear finite elements with respect to a nested hierarchy of triangulations, in both cases we use iterative processes consisting of inner and outer iterations. The outer iterations are either active set strategies or generalized Newton methods while the inner iterations are preconditioned cg- iterations with multilevel preconditioners.

Description: One-step discretizations of order $p$ and step size $\varepsilon$ of ordinary differential equations can be viewed as time-$\varepsilon$ maps of \begin{displaymath} \dot{x} (t) = f(\lambda ,x(t)) + \varepsilon^p g(\varepsilon,\lambda,t/\varepsilon,x(t)), x \in R^N,\lambda \in R, \end{displaymath} where $g$ has period $\varepsilon$ in $t$. This is a rapidly forced nonautonomous system. We study the behavior of a homoclinit orbit $\Gamma$ for $\varepsilon = 0, \lambda =0$, under discretization. Under generic assumptions we show that $\Gamma$ becomes transverse for positive $\in$. The transversality effects are estimated from above to be exponentially small in $\in$. For example, the length $l(\varepsilon$) of the parameter interval of $\lambda$ for which $\Gamma$ persists can be estimated by \begin{displaymath} l(\varepsilon)\le Cexp(-2\pi\eta/\varepsilon), \end{displaymath} where $C,\eta$ are positive constants. The coefficient $\eta$ is related to the minimal distance from the real axis of the poles of $\Gamma(t)$ in the complex time domain. Likewise, the region where complicated, "chaotic" dynamics prevail is estimated to be exponentially small, provided $x \in R^2$ and the saddle quantity of the associated equilibrium is nonzero. Our results are visualized by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality becomes pratically invisible under normal circumstances, already for only moderately small step size. {\bf Keywords:} Homoclinic orbit, ordinary differential equations, discretization, transversality, averaging, exponential smallness, chaos. {\bf Subject Classifications:} (AMS): 34C15, 34C35, 58F14, 65L60

Description: We consider the discretization of obstacle problems for the Laplacian by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned c-g iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semi-local and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.

Description: Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and statistics. They can be considered as ordinary differential equations in sequence spaces. In this work, a fully adaptive algorithm for the computational treatment of such systems is developed. The method is based on a time discretization of an abstract Cauchy problem in Hilbert space and a discrete Galerkin approach for the discretization of the arising stationary subproblems. The Galerkin method uses orthogonal functions of a discrete variable, which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations are solved adaptively either by use of analytical properties of the orthogonal functions or by an appropriate numerical summation. The resulting algorithm CODEX is applied to examples of technological interest, in particular from polymer chemistry.

Description: The deformation of a hexagonal lattice dome under an external load is an example of a parameter dependent system which is equivariant under the symmetry group of a regular hexagon. In this paper the mixed symbolic-numerical algorithm SYMCON is applied to analyze its steady state solutions automatically showing their different symmetry and stability properties.

Description: We study the higher Bruhat orders $B(n,k)$ of Manin & Schechtman [MaS] and - characterize them in terms of inversion sets, - identify them with the posets $U(C^{n+1,r},n+1)$ of uniform extensions of the alternating oriented matroids $C^{n,r}$ for $r:=n-k$ (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane), - show that $B(n,k)$ is a lattice for $k =1$ and for $r\le 3$, but not in general, - show that $B(n,k)$ is ordered by inclusion of inversion sets for $k=1$ and for $r\le 4$. However, $B(8,3)$ is not ordered by inclusion. This implies that the partial order $B_\subseteq (n,k)$ defined by inclusion of inversion sets differs from $B(n,k)$ in general. We show that the proper part of $B_\subseteq (n,k)$ is homotopy equivalent to $S^{r-2}$. Consequently, - $B(n,k)\simeq S^{r-2}$ for $k=1$ and for $r\le 4$. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for $r=3$, and that the proper part is not always homotopy equivalent to $S^{r(M)-2}$.

Description: The C-implementation of KASKADE, an adaptive solver for linear elliptic differential equations in 2D, is object of a set of numerical experiments to analyze the use of resources (time and memory) with respect to numerical accuracy. We study the dependency of the reliability, robustness, and efficiency of the program from the parameters controlling the algorithm.

Description: A high resolution Godunov-type method is developed and applied to a two-dimensional version of the Burgers equation. On one-dimensional testproblems the scheme is second order accurate for smooth flow and resolves discontinuities sharply. On two-dimensional problems, however, large numerical oscillations are introduced when discontinuities traveling obliquely to the grid directions are present. These oscillations are caused by a Strang- type operator splitting, and cannot be eliminated using flux limiters or small timesteps. {\bf Keywords:} High resolution methods, Godunovs method, operator splitting, complex Burgers equation. {\bf AMS:} 35: A40, L45, L60, L65, L, 67. 65: M05, M10.

Description: For adaptive solution of convection- difussion problems with the streamline-diffusion finite element method, an error estimator based on interpolation techniques is developed. It can be shown that for correctness of this error estimator a restriction of the maximum angle is to be sufficient. Compared to usual methods, the adaptive process leads to more accurate solutions at much less computational cost. Numerical tests are enclosed. {\bf Keywords: } Adaptive finite elements, convection- diffusion equation, internal and boundary layers, streamline-diffusion. {\bf Subject Classifications:} AMS(MOS): 65N15, 65N30

Description: The success of the German Reunification requires a unique migration from the traditional Eastern Block system to a modern free society. The successful migration must include a shift from a low state of the art in technology to the high level of science and technology present in the Western Community. The role of supercomputing in this transition will include leading the process of organizational change in research and development. The paper contains a short overview on the current state of this evolution, on the currently used political and administrative tools to keep this evolution on track, and on the influence on the Konrad-Zuse-Zentrum, which is, as a Berlin institution, in the geographical and cultural center of this transformation.