spacefilling curves and the planar travelling salesman problem to construct a short tour through points in the plane the points are sequenced as they appear along a spacefilling curve this heuristic consists essentially of sorting so it is easily coded and requires only o n memory and o n log n operations its performance is competitive with that of other fast methods

color level sets a multi phase method for structural topology optimization with multiple materials in this paper we address the problem of structural shape and topology optimization in a multi material domain a level set method is employed as an alternative approach to the popular homogenization based methods of rule of mixtures for multi material modeling a multi phase level set model is adapted for material and topology representation this model eliminates the need for a material interpolation or phase mixing scheme it only requires m level set functions to represent a structure of n 2m different material phases in a principle similar to combining colors from the three primary colors therefore this multi phase model may be referred to as a color level set representation which has its unique benefits it is flexible to handle complex topologies it substantially reduces the number of model functions when n 3 it automatically avoids the problem of overlap between material phases of a conventional partitioning approach we describe numerical techniques for efficient and robust implementation of the method by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain this would separate the issues of accuracy in numerical calculations of the physical equation and in the level set model propagation a gradient projection method is described for incorporating multiple constraints in the problem finally the benefits and the advantages of the developed method are illustrated with several 2d examples of mean compliance minimization of multi material structures

constructing competitive tours from local information abstract we consider the problem of a searcher exploring an initially unknown weighted planar graph g when the searcher visits a vertex v it learns of each edge incident to v the searcher s goal is to visit each vertex of g incurring as little cost as possible we present a constant competitive algorithm for this problem

submodularity and the traveling salesman problem in this paper we investigate the relationship between traveling salesman tour lengths and submodular functions this work is motivated by the one warehouse multi retailer inventory distribution problem with traveling salesman tour vehicle routing costs our goal is to find a submodular function whose values are close to those of optimal tour lengths through a central warehouse and a group of retailers our work shows that a submodular approximation to traveling salesman tour lengths whose error is bounded by a constant does not exist however we present heuristics that have errors which grow slowly with the number of retailers for the traveling salesman problem in the euclidean plane furthermore we perform computational tests that show that our submodular approximations of traveling salesman tour lengths have smaller errors than our theoretical worst case analysis would lead us to believe

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

competitive online routing in geometric graphs we consider online routing algorithms for finding paths between the vertices of plane graphs although it has been shown in bose et al internat j comput geom 12 4 2002 283 that there exists no competitive routing scheme that works on all triangulations we show that there exists a simple online o 1 memory c competitive routing strategy that approximates the shortest path in triangulations possessing the diamond property i e the total distance travelled by the algorithm to route a message between two vertices is at most a constant c times the shortest path our results imply a competitive routing strategy for certain classical triangulations such as the delaunay greedy or minimum weight triangulation since they all possess the diamond property we then generalize our results to show that the o 1 memory c competitive routing strategy works for all plane graphs possessing both the diamond property and the good convex polygon property

calabi yau manifolds motivations and constructions the possible ways of compacitification of thee8 e8 superstring theory to four dimensions are reviewed the phenomenological need forn 1 supersymmetry is argued on quite general grounds to favour the choice of a calabi yau manifold for the compact internal manifold the massless spectrum after compactification is derived in full detail revealing beside the usual particles others that may have great phenomenological impact the technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given a big family of such constructions giving rise to many new calabi yau manifolds is presented and its relevance to the search of a phenomenologically acceptable solution is discussed

lower and upper competitive bounds for online directed graph exploration abstract we study the problem of exploring all nodes of an unknown directed graph a searcher has to construct a tour that visits all nodes but only has information about the parts of the graph it already visited analogously to the traveling salesman problem the goal is to minimize the cost of such a tour in this article we present upper and lower bounds for the competitive ratio of both the deterministic and the randomized online version of exploring all nodes of directed graphs our bounds are sharp or sharp up to a small constant depending on the specific model as it turns out restricting the diameter the incoming outgoing degree or randomly choosing a starting point does not improve lower bounds beyond a small constant factor even supplying the searcher in a planar euclidean graph with the nodes coordinates does not help essentially exploring a directed graph has a multiplicative overhead linear in the number of nodes furthermore if one wants to search for a specific node in unweighted directed graphs a greedy algorithm with quadratic multiplicative overhead can only be improved by a small constant factor as well

calabi yau hypersurfaces in products of semi ample surfaces we study calabi yau manifolds that are embedded as hypersurfaces in products of semi ample complex surfaces we classify the deformation classes of the latter and thereby achieve a classification of the calabi yau manifolds that are constructed in this way complementing the results in the existing literature we obtain the complete hodge diamond for all calabi yau hypersurfaces in products of semi ample surfaces

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

a parametric simplex algorithm for linear vector optimization problems in this paper a parametric simplex algorithm for solving linear vector optimization problems lvops is presented this algorithm can be seen as a variant of the multi objective simplex the evans steuer algorithm math program 5 1 54 72 1973 different from it the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions instead it finds a solution in the sense of lohne vector optimization with infimum and supremum springer berlin 2011 that is it finds a subset of efficient solutions that allows to generate the whole efficient frontier in that sense it can also be seen as a generalization of the parametric self dual simplex algorithm which originally is designed for solving single objective linear optimization problems and is modified to solve two objective bounded lvops with the positive orthant as the ordering cone in ruszczynski and vanderbei econometrica 71 4 1287 1297 2003 the algorithm proposed here works for any dimension any solid pointed polyhedral ordering cone c and for bounded as well as unbounded problems numerical results are provided to compare the proposed algorithm with an objective space based lvop algorithm benson s algorithm in hamel et al j global optim 59 4 811 836 2014 that also provides a solution in the sense of lohne 2011 and with the evans steuer algorithm 1973 the results show that for non degenerate problems the proposed algorithm outperforms benson s algorithm and is on par with the evans steuer algorithm for highly degenerate problems benson s algorithm hamel et al 2014 outperforms the simplex type algorithms however the parametric simplex algorithm is for these problems computationally much more efficient than the evans steuer algorithm

interval analysis method for damage identification of structures description was adopted for the measured natural frequencies in this paper via the first order taylor series expansion the interval bounds of the elemental stiffness parameters of both undamaged and damaged structures were derived by using the model updating based on the initial analytical finite element model the damage can be identified by comparing the differences between the two models where the quantitative measure of the possibility of damage existence in the elements is introduced a larger value of possibility of damage existence implied a higher possibility of damage occurrence the damage identifications for a steel cantilever beam and a steel cantilever plate were performed by the presented method which is validated by monte carlo simulation moreover the case of the multidamage identification the number of the used natural frequencies and the effects of damage level and uncertainty level on the damage detection were studied as well the numerical results proved the validity and applicability of the presented interval analysis method

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

a distribution free tsp tour length estimation model for random graphs abstract traveling salesman problem tsp tour length estimations can be used when it is not necessary to know an exact tour e g when using certain heuristics to solve location routing problems the best estimation models in the tsp literature focus on random instances where the node dispersion is known those that do not require knowledge of node dispersion are either less accurate or slower in this paper we develop a new regression based tour length estimation model that is distribution free accurate and fast with a small standard deviation of the estimation errors when the distribution of the node coordinates is known it provides a close estimate of the well known asymptotic tour length estimation formula of beardwood et al 1959 more importantly when the distribution is unknown or non integrable so beardwood et al s estimation cannot be used our model still provides good fast tour length estimates

a new methodology of simulated annealing for the optimisation problems abstract complex optimisation problems with many degrees of freedom are often characterised by the enormously large configuration space typically o en or o n the idea of simulated annealing sa proposed by kirkpatrick has been applied to the complex optimisation problems which can be treated as annealing a statistical mechanical system from high temperature to low temperature however the sa is terribly slow for large problem sizes in typically o n3lnn time we discover the hybrid algorithm ha which is based on a hybrid mechanism which combines conventional heuristics with low temperature simulated annealing ltsa which could be parallelised easily the ha is a new approach of resolving optimisation problems with o n complexity where information propagation can be inhibited by restraining the range of searches in the configuration space we use the ha to resolve several famous combinatorial optimisation problems including the travelling salesman problem tsp of large sizes up to 1 000 000 cities within 3 to 5 percent of the optimal value in linear time and other nonuniformly distributed tsps as well we shall also discuss the applicability of the ha to the optimisation problems in general

an exact generalised function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping abstract this paper concerns the frequency response analysis of beams and plane frames with an arbitrary number of kelvin voigt viscoelastic dampers typical external and internal dampers are considered as grounded translational tuned mass rotational and axial dampers for bending and axial vibrations respectively using the theory of generalised functions within a 1d formulation of equations of motion exact closed form expressions are derived for beam dynamic green s functions and frequency response functions under arbitrary polynomial load for any number of dampers for a plane frame exact global frequency response matrix and load vector are built with size depending only on the number of beam to column nodes for any number of dampers and point polynomial loads along the frame members from the nodal displacement solution the exact frequency response in all frame members is also obtained in closed analytical form numerical applications show many of the advantages of the proposed method

polynomial deformations and cohomology of calabi yau manifolds we study the method of polynomial deformations that is used in the physics literature to determine the hodge numbers of calabi yau manifolds as well as the related yukawa couplings we show that the argument generally presented in the literature in support of these computations is seriously misleading give a correct proof which applies to all the cases we found in the literature and present examples which show that the method is not universally valid we present a general analysis which applies to all calabi yau manifolds embedded as complete intersections in products of complex projective spaces yields sufficient conditions for the validity of the polynomial deformation method and provides an alternative computation of all the hodge numbers in many cases in which the polynomial method fails

index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints multidimensional optimization problems where the objective function and the constraints are multiextremal non differentiable lipschitz functions with unknown lipschitz constants and the feasible region is a finite collection of robust nonconvex subregions are considered both the objective function and the constraints may be partially defined to solve such problems an algorithm is proposed that uses peano space filling curves and the index scheme to reduce the original problem to a holder one dimensional one local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search the method neither uses penalty coefficients nor additional variables convergence conditions are established numerical experiments confirm the good performance of the technique

routing heuristics for automated pick and place machines we examine the problem of sequencing the placements of multiple part types for a computer controlled placement machine the problem is modeled as a collection of interdependent traveling salesman problems a heuristic based on a space filling curve is shown to be easy to compute and quite effective numerical experiments show that on problems of realistic size our heuristics show very little divergence from optimality the probabilistic analysis of the proposed heuristic indicates that the proposed heuristics are asymptotically optimal finally we use our heuristic to perform the sequencing operation on real placement machines the results of this experiment are in accord with our numerical simulations the main ideas of this study have become part of the control system currently in use in a large electronic card assembly facility that produces approximately 12 000 boards per day improvements in throughput of between 4 and 9 have been reported

a level set approach for damage identification of continuum structures based on dynamic responses abstract the present paper aims to propose a novel approach for damage identification of continuum structures based on their dynamic performances the main idea is resorting to the level set model which is used to describe the shape and topology of the damage regions implicitly the original natural frequency based inverse problem is thus transferred into a generalized shape optimization problem which can be tackled by solving an evolution type hamilton jacobi equation compared to traditional approaches the distinctive advantage of the proposed approach is that it can deal with damage regions of complex shapes in a convenient way numerical examples demonstrate the effectiveness of the proposed approach

stress related topology optimization of continuum structures involving multi phase materials considering stress constraints in structural topology optimization is very important from both theoretical and application perspectives most of the corresponding studies in literature however are only focused on the problems involving single phase homogenous materials in the present paper stress constrained topology optimization of continuum structures involving multi phase heterogeneous materials is investigated a level set based variational consistent solution framework is developed numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

complete convergence of short paths and karp s algorithm for the tsp let xi 1 i 0 sum infty n 1 p left t n sqrt n beta gt epsilon right lt infty this result is essential in justifying karp s algorithm for the traveling salesman problem under the independent model and it settles a question posed by b w weide

pattern recognition algorithms based on space filling curves and orthogonal expansions the classical problem of constructing a multidimensional pattern classifier in the bayesian framework is considered preprocessing of the learning sequence by a quasi inverse of a space filling curve is proposed and properties of space filling curves which are necessary to obtain bayes risk consistency are indicated the learning sequence transformed into the unit interval is used to estimate the coefficients in an orthogonal expansion of the bayes decision rule to transform a new observation into the unit interval requires o d elementary operations where d is the dimension of the observation space strong bayes risk consistency of the classifiers is proved when distributions of the random pair of the observation vector and its class are absolutely continuous with respect to the lebesgue measure attainable convergence rate of the bayes risk is discussed details of the classification algorithm based on the haar series and its properties are presented distribution free consistency of the classifier is established the performance of such a classifier is tested both on simulated data and on the standard benchmarks providing misclassification errors comparable to or even better than the k nearest neighbors k nn method

calabi yau manifolds as complete intersections in products of complex projective spaces we consider constructions of manifolds withsu 3 holonomy as embedded in products of complex projective spaces by imposing certain homogeneous holomorphic constraints we prove that every such construction leads to one deformation class of manifolds withsu 3 holonomy for a subset of these manifolds we prove simple connectedness address the problem of calculating the second betti number and explicitly calculate it for a class of constructions this establishes a very wide class of manifolds withsu 3 holonomy that can give rise to yet many more constructions via dividing out the action of suitably chosen discrete groups some of the examples studied may yield phenomenologically acceptable models

tight bounds for undirected graph exploration with pebbles and multiple agents we study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents the vertices of the graph are unlabeled and cannot be distinguished by the agents but the edges incident to a vertex have locally distinct labels the graph is explored when all vertices have been visited by at least one agent in this setting it is known that for a single agent without pebbles log n bits of memory are necessary and sufficient to explore any graph with at most n vertices we are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph we give tight results for both questions showing that for a single agent with constant memory log log n pebbles are necessary and sufficient for exploration we further prove that using collaborating agents instead of pebbles does not help as log log n agents with constant memory each are necessary and sufficient for exploration for the upper bounds we devise an algorithm for a single agent with constant memory that explores any n vertex graph using o log log n pebbles even when n is not known a priori the algorithm terminates after polynomial time and returns to the starting vertex we further show that the algorithm can be realized with additional constant memory agents rather than pebbles implying that o log log n agents with constant memory can explore any n vertex graph for the lower bound we show that the number of agents needed for exploring any graph with at most n vertices is already log log n when we allow each agent to have at most o log n 1 e bits of memory for any e 0 our argument also implies that a single agent with sublogarithmic memory needs log log n pebbles to explore any n vertex graph

heuristics based on spacefilling curves for combinatorial problems in euclidean space we describe a family of heuristics to solve combinatorial problems such as routing and partitioning these heuristics exploit geometry but ignore specific distance measures consequently they are simple and fast but nonetheless fairly accurate and so seem well suited to operational problems where time or computing resources are limited we survey promising new application areas and show how procedures may be customized to reflect the structure of particular applications

online routing in triangulations we consider online routing algorithms for routing between the vertices of embedded planar straight line graphs our results include 1 two deterministic memoryless routing algorithms one that works for all delaunay triangulations and the other that works for all regular triangulations 2 a randomized memoryless algorithm that works for all triangulations 3 an o 1 memory algorithm that works for all convex subdivisions 4 an o 1 memory algorithm that approximates the shortest path in delaunay triangulations and 5 theoretical and experimental results on the competitiveness of these algorithms