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SC-94-06
{\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.}
SC-94-05
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.
SC-94-04
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.
SC-94-03
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.
SC-94-02
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.
SC-94-01
{\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.}
SC-93-35
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.
SC-93-34
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.)
SC-93-33
This paper throws light on the connection between the optimal condition number estimate for the BPX method and constructive approximation theory. We provide a machinery, which allows to understand the optimality as a consequence of an approximation property and an inverse inequality in $H^{1+\epsilon}$, $\epsilon > 0$. This machinery constructs so-called {\em approximation spaces}, which characterize a certain rate of approximation by finite elements and relates them with interpolation spaces, which characterize a certain smoothness.
SC-93-32
Adaptive numerical methods using the $h$-$p$-version of finite elements require special kinds of shape functions. Desirable properties of them are symmetry, hierarchy and simple coupling. In a first step it is demonstrated that for standard polynomial vector spaces not all of these features can be obtained simultaneously. However, this is possible if these spaces are extended. Thus a new class of polynomial shape functions is derived, which is well-suited for the $p$- and $h$-$p$-version of finite elements on unstructured simplices. The construction is completed by minimizing the condition numbers of the arising finite element matrices. The new shape functions are compared with standard functions widely used in the literature.