on the weak computability of a four dimensional orthogonal packing and time scheduling problem this paper proposes a four dimensional orthogonal packing and time scheduling problem the problem differs from the classical packing problems in that the position and orientation of each item in the container can be changed over time in this way the four dimensional space time problem better uses the container time also we consider a general case that all parameters are real numbers which makes the problems more difficult to solve this paper proposes an algorithm and proves that the algorithm could solve the problem optimally by a finite number of operations we say this problem is weak computational meaning that if there exists a universal machine that could represent real numbers and could do unit arithmetic or logical operation on real numbers in finite time then the algorithm could find optimal solutions in finite time this paper also presents a proof of the weak computability over a general case of the three dimensional orthogonal packing problem where all parameters are positive real numbers

on approximate and algebraic computability over the real numbers abstract we consider algebraic and approximate computations of partial real functions r d r algebraic computability is defined by means of parameter free finite algorithmic procedures the notion of approximate computability is a straightforward generalization of the ko friedman approach based on oracle turing machines to functions with not necessarily recursively open domains the main results of the paper give characterizations of approximate computability by means of the passing sets of finite algorithmic procedures i e characterizations from the algebraic point of view some consequences and also modifications of the concepts are discussed finally two variants of arithmetical hierarchies over the reals are considered and used to classify and mutually compare the domains graphs and ranges of algebraically resp approximately computable real functions

computability structure of the sobolev spaces and its applications in this paper we present a definition of s computability for functions in sobolev spaces hs rd s r two applications of s computability are then discussed one is to address several open problems regarding classical computable functions the other is to conduct computable analysis of several important classes of partial differential equations

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

curves that must be retraced we exhibit a polynomial time computable plane curve c that has finite length does not intersect itself and is smooth except at one endpoint but has the following property for every computable parametrization f of c and every positive integer m there is some positive length subcurve of c that f retraces at least m times in contrast every computable curve of finite length that does not intersect itself has a constant speed hence non retracing parametrization that is computable relative to the halting problem

a polynomial time computable curve whose interior has a nonrecursive measure abstract a polynomial time computable simple curve is constructed such that its measure in the two dimensional plane is positive this construction is applied to prove the following two results 1 1 there exists a polynomial time computable simple closed curve in the two dimensional plane such that the measure of its interior region is a nonrecursive real number 2 2 there exists a polynomial time computable simple curve in the two dimensional plane such that its length is finite but is a nonrecursive real number

the computational complexity of maximization and integration on montre que ces deux problemes de la theorie de la complexite de calcul aussi bien que les autres resolvant la maximisation et l integration sont equivalents aux problemes de complexite de calcul discret naturel dont la plupart ont deja beaucoup ete etudies

an optimal algorithm for euclidean shortest paths in the plane we propose an optimal time algorithm for a classical problem in plane computational geometry computing a shortest path between two points in the presence of polygonal obstacles our algorithm runs in worst case time o n log n and requires o n log n space where n is the total number of vertices in the obstacle polygons the algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane the map can be used to answer single source shortest path queries in o log n time the time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem finally we also discuss extensions to more general shortest path problems involving nonpoint and multiple sources

theory of representations abstract an approach for a simple general and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented a standard theory of effectivity on f f n n has been developed in a previous paper by representations f m this theory is extended to other sets m topological and recursion theoretical properties of representations are studied where the final topology of a representation plays an essential role it is shown that for any separable t0 space an up to equivalence unique admissible representation can be defined which reflects the topological properties correctly

computational complexity of two dimensional regions the computational complexity of bounded sets of the two dimensional plane is studied in the discrete computational model we introduce four notions of polynomial time computable sets in r 2 and study their relationship the computational complexity of the winding number problem membership problem distance problem and area problem is characterized by the relations between discrete complexity classes of the np theory

computability on computable metric spaces in previous papers e g weihrauch 1987 type 2 theory of effectivity tte has been shown to be a very general and powerful computer oriented theory for studying effectivity in mathematics in this contribution computability on certain computable separable metric spaces is studied in detail by applying the framework of tte computationally admissible representations of metric spaces are introduced after showing that four different effective representations are computationally equivalent for extending computability to the set of continuous functions several effective namings which are related to definitions of continuity are introduced and compared the definitions are used to prove effective g extension theorems for continuous functions and an effective g characterization of the domains of strongly continous functions which generalize the known properties of real functions kreitz 1984

on the computational complexity of ordinary differential equations the computational complexity of the solution y of the differential equation y x f x y x with the initial value y 0 0 relative to the computational complexity of the function f is investigated the lipschitz condition on the function f is shown to play an important role in this problem on the one hand examples are given in which f is polynomial time computable but none of the solutions y is computable on the other hand if f is polynomial time computable and if f satisfies a weak form of the lipschitz condition then the unique solution y is polynomial space computable furthermore there exists a polynomial time computable function f which satisfies this weak lipschitz condition such that the unique solution y is not polynomial time computable unless p pspace

a combinatorial formula for kazhdan lusztig polynomials in their fundamental paper 14 kazhdan and lusztig defined for every coxeter group w a family of polynomials indexed by pairs of elements of w which have become known as the kazhdan lusztig polynomials of w see e g 12 chap 7 these polynomials are intimately related to the bruhat order of w and to the algebraic geometry of schubert varieties and have proven to be of fundamental importance in representation theory our aim in this work is to give a simple nonrecursive combinatorial formula for any kazhdan lusztig polynomial of any coxeter group theorem 4 1 and to study some consequences of it the main idea involved in the proof and statement of this formula is that of extending the concept of the rpolynomial see e g 12 w to any finite multichain of w so that for multichains of length 1 one obtains apart from sign the usual r polynomials once this has been done then the kazhdan lusztig polynomial of a pair u v turns out to be just the sum over all multichains from u to v of the corresponding generalized r polynomials the r polynomial of a multichain can be readily defined and computed from the ordinary r polynomials see 9 10 and 11 since several combinatorial formulas and interpretations are known for these polynomials see e g 5 10 and for the case of symmetric groups 3 and simple recurrences exist for them we feel that this formula is a significant step forward in the understanding of the kazhdanlusztig polynomials though combinatorial formulas for kazhdan lusztig polynomials have appeared before in the literature see e g 16 21 8 6 none of them hold in complete generality

a heuristic for quadratic boolean programs with applications to quadratic assignment problems abstract a heuristic for quadratic boolean programs is presented computational tests with quadratic assignment problems qap showed that it finds very good suboptimal solutions in moderate time and behaves computationally stable in the appendix a fortran program for qap is listed which improves an earlier code published by burkard and derigs

jordan curves with polynomial inverse moduli of continuity computational complexity of two dimensional domains whose boundaries are polynomial time computable jordan curves with polynomial inverse moduli of continuity is studied it is shown that the membership problem of such a domain can be solved in p n p i e in polynomial time relative to an oracle in np in contrast to the higher upper bound p m p for domains without the property of polynomial inverse modulus of continuity on the other hand the lower bound of up for the membership problem still holds for domains with polynomial inverse moduli of continuity it is also shown that the shortest path problem of such a domain can be solved in pspace close to its known lower bound while no fixed upper bound was known for domains without this property

computability on subsets of euclidean space i closed and compact subsets in this paper we introduce and compare computability concepts on the set of closed subsets of euclidean space we use the language and framework of type 2 theory of effectivity tte which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems in particular type 2 theory of effectivity allows to separate topological from computational aspects of effectivity we consider three different computability concepts on the set of closed subsets each of which is characterized by several representations which are proved to be equivalent the three induced types of computable closed sets have already been considered by many authors however under different and partly inconsistent names our characterizations show that they can be regarded as straightforward generalizations of the r e co r e and recursive subsets of natural numbers therefore we suggest to call them the recursively enumerable the co recursively enumerable and the recursive closed subsets of euclidean space open subsets obtain the dual names we extend the investigation by introducing several natural representations of the compact subsets of euclidean space and proving equivalences the paper extends and generalizes earlier definitions adds new ones and compares them in a single framework the resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets

a heuristic method for the multi story layout problem in this paper we present a heuristic procedure designed expressly for solving a large layout problem in a multi story setting where the objective is to minimize total fixed and interaction costs this is achieved by decomposing the original facilities layout problem into several similar but smaller problems thus enabling solution of problems with as many as 150 facilities in reasonable time some of the novel features of the procedure described are the use of a heuristic k median subroutine to obtain groupings of facilities and a simple and fast exchange improvement method computational results for randomly generated problems compare the effectiveness of this method with the space planning heuristic method of liggett and mitchell

combining path relinking and genetic algorithms for the multiple level warehouse layout problem path relinking is a method to generate new solution by exploring trajectories that connect high quality solutions in this paper a class of new hybrid heuristics are proposed by combining a genetic algorithm and path relinking and applying these to a multiple level warehouse layout problem parallel and series combinations to integrate crossover and mutation operations of a genetic algorithm with path relinking are investigated we proposed position and sequence based path relinking methods to connect two solutions which are either elites or ones selected randomly extensive experiments are carried out to compare the performance of the new heuristics