an efficient algorithm for decomposing a polygon into star shaped polygons abstract in this paper we show how a theorem in plane geometry can be converted into a o n log n algorithm for decomposing a polygon into star shaped subsets the computational efficiency of this new decomposition contrasts with the heavy computational burden of existing methods

discontinuous galerkin isogeometric analysis of elliptic diffusion problems on segmentations with gaps we propose a new discontinuous galerkin dg isogeometric analysis iga technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with nonmatching interfaces due to an incorrect segmentation procedure it may happen that the interfaces of adjacent subdomains do not coincide in this way gap regions which are not present in the original physical domain are created in this paper the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains we apply a multipatch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem the last formulation is further modified by deriving interface conditions without the presence of the solution in the gap finally the solution of this modified problem is approximated b

discontinuous galerkin isogeometric analysis of elliptic problems on segmentations with non matching interfaces abstract in this paper we develop a discontinuous galerkin isogeometric analysis method for solving elliptic problems on decompositions of the computational domain into volumetric patches with non matching parametrized interfaces we specially focus on high order numerical solutions for complex gap regions and extend ideas from our previous work on simple gap regions for the communication of the numerical solution between the subdomains which are separated by the gap region discontinuous galerkin numerical fluxes are constructed taking into account the diametrically opposite points on the boundary of the gap due to lack of information on the behavior of the solution in the gap region the fluxes coming from the interior of the gap are approximated by taylor expansions with respect to the adjacent subdomain solutions we follow the same ideas of our previous work and show a priori error estimates in the dg norm with respect to the mesh size and the gap distance numerical examples performed for two three and even four dimensional computational domains demonstrate the robustness of the proposed numerical method and validate the estimates predicted by the theory

some np hard polygon decomposition problems the inherent computational complexity of polygon decomposition problems is of theoretical interest to researchers in the field of computational geometry and of practical interest to those working in syntactic pattern recognition three polygon decomposition problems are shown to be np hard and thus unlikely to admit efficient algorithms the problems are to find minimum decompositions of a polygonal region into perhaps overlapping convex star shaped or spiral subsets we permit the polygonal region to contain holes the proofs are by transformation from boolean three satisfiability a known np complete problem several open problems are discussed

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

an algorithm for covering polygons with rectangles decomposing a polygon into simple shapes is a basic problem in computational geometry with applications in pattern recognition and integrated circuit manufacture here we examine the special case of covering a rectilinear polygon or polyomino with the minimum number of rectangles with overlapping allowed the problem is np hard however we give here an o v2 algorithm for constructing a minimum rectangle cover when the polygon is vertically convex here v is the number of vertices the problem is first reduced to a 1 dimensional interval basis problem in showing our algorithm produces an optimal cover we give a new proof of a minimum basis maximum independent set duality theorem first proved by e gyori j combin theory ser b 37 no 1 1 9

convex decomposition of polyhedra and robustness this paper presents a simple algorithm to compute a convex decomposition of a nonconvex polyhedron of arbitrary genus handles and shells internal voids for such a polyhedron s with n edges and rnotches features causing nonconvexity in polyhedra the algorithm produces a worst case optimal o r 2 number of convex polyhedra s i with u i 1 k s i s in o nr 2 r 7 2 time and o nr r 5 2 space recently chazelle and palios have given a fast o n r 2 log r time and o n r 2 space algorithm to tetrahedralize a nonconvex polyhedron their algorithm however works for a simple polyhedron of genus zero and with no shells internal voids the algorithm presented here is based on the simple cut and split paradigm of chazelle with the help of zone theorems on arrangements it is shown that this cut and split method is quite efficient the algorithm is extended to work for a certain class of nonmanifold polyhedra also presented is an algorithm for the same problem

on covering orthogonal polygons with star shaped polygons abstract in this paper we present an o n time algorithm for covering a horizontally convex orthogonal polygon with the minimum number of orthogonal star shaped polygons finding minimum covers by star shaped polygons is equivalent to finding the minimum number of guards needed so that every point in the polygon is visible to at least one guard this problem has important applications in robotics and pattern recognition

on a convex hull algorithm for polygons and its application to triangulation problems abstract a frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases due to its simplicity the algorithm is nevertheless attractive in this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons some application areas where such polygons occur are briefly discussed in addition it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0 n time

a domain decomposition method for isogeometric multi patch problems with inexact local solvers abstract in isogeometric analysis the computational domain is often described as multi patch where each patch is given by a tensor product spline nurbs parametrization in this work we propose a feti like solver where local inexact solvers exploit the tensor product structure at the patch level to this purpose we extend to the isogeometric framework the so called all floating variant of feti that allows us to use the fast diagonalization method at the patch level we construct then a preconditioner for the whole system and prove its quasi robustness with respect to the local mesh size h and patch size h precisely the condition number of the preconditioned system is bounded by the square of the logarithm of h h our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree p

explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements we derive new trace inequalities for nurbs mapped domains in addition to sobolev type inequalities we derive discrete trace inequalities for use in nurbs based isogeometric analysis all dependencies on shape size polynomial degree and the nurbs weighting function are precisely specified in our analysis and explicit values are provided for all bounding constants appearing in our estimates as hexahedral finite elements are special cases of nurbs our results specialize to parametric hexahedral finite elements and our analysis also generalizes to t spline based isogeometric analysis we compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants and we discuss application of our results to a laplace problem we finish this paper with a brief exploration of so called patch wise trace inequalities for isogeometric analysis

parallelization of continuous and discontinuous galerkin dual primal isogeometric tearing and interconnecting methods abstract in this paper we investigate the parallelization of dual primal isogeometric tearing and interconnecting ieti dp type methods for solving large scale continuous and discontinuous galerkin systems of equations arising from isogeometric analysis of elliptic boundary value problems these methods are extensions of the finite element tearing and interconnecting methods to isogeometric analysis the algorithms are implemented by means of energy minimizing primal subspaces we discuss how these methods can efficiently be parallelized in a distributed memory setting weak and strong scaling studies presented for two and three dimensional problems show an excellent parallel efficiency

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

ordered structural shape matching with primitive extraction by mathematical morphology abstract structural shape matching procedures begin by decomposing the shapes to be matched into primitive parts and extracting the properties of and relationships among these primitives the resultant structural descriptions are matched to structural models usually via a tree search whose time is exponential in the number of primitives in this paper we introduce a new fast method of structural shape matching called ordered structural matching which has polynomial time complexity instead of exponential the approach presented also uses mathematical morphology to rapidly decompose the shapes into primitives to be used in the matching the primitives so produced are pieces of the original shape but not necessarily those pieces that a human would select the method has been tested on the toy task of determining which plastic shapes match the shapes of the holes in a child s six sided toy box it is also directly applicable to the inspection and manipulation of certain airplane parts

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

isogeometric analysis and domain decomposition methods in the present work we use the mortar finite element method for the coupling of nonconforming discretized sub domains in the framework of nonlinear elasticity the mortar method has been shown to preserve optimal convergence rates see laursen 2002 25 for details and is variationally consistent we show that the method can be applied to isogeometric analysis with little effort once the framework of nurbs based shape functions has been implemented furthermore a specific coordinate augmentation technique allows the design of an energy momentum scheme for the constrained mechanical system under consideration the excellent performance of the redesigned mortar method as well as the energy momentum scheme is illustrated in representative numerical examples

a set covering based approach to find the reduct of variable precision rough set abstract attribute reduction is one of the core problems in rough set rs theory in the variable precision rough set vprs model attribute reduction faces two difficulties firstly in the vprs model a reduct anomaly problem may arise and it may cause an inconsistency of positive regions and decision rules after attribute reduction secondly the attribution reduction problem has been proved an np hard problem accordingly we would need to find a tradeoff between calculating the minimal reduct and reducing computing complexity to avoid the combinatorial explosion problem we propose a new approach to calculate the reduct in vprs model this new method focuses on calculating a distribution reduct while avoiding the anomaly problem in the vprs model the basic idea of the proposed approach is to convert the reduct problem into a set covering problem scp according to the positive regions in the vprs model and consequently a set covering heuristic function schf algorithm is applied to calculate the reduct after this conversion this approach keeps the positive regions consistent after the attribute reduction and moreover based on the scp the performance ratio of the proposed method to calculate the minimal reduct ranges between ln u ln ln u o 1 and 1 o 1 ln u with a computational complexity having an upper bound as o mn m n 2 finally we demonstrate the practical application of the vprs model using real case scenario from china s electricity power yield to verify the validity of our proposed approach we then apply statistical evaluation to explain the economic significance of the attributes

isogeometric schwarz preconditioners for linear elasticity systems isogeometric schwarz preconditioners are constructed and analyzed for both compressible elasticity in primal formulation and almost incompressible elasticity in mixed formulation these preconditioners require the solution of local elasticity problems on overlapping subdomains forming a decomposition of the problem domain and the solution of a coarse elasticity problem associated with the subdomain coarse mesh an h analysis of the preconditioner for the primal formulation of compressible elasticity yields an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes extensive numerical experiments in 2d and 3d confirm this theoretical bound and show that an analogous bound holds for the mixed formulation of almost incompressible elasticity the numerical tests also show the good preconditioner performance with respect to the polynomial degree p and regularity k of the isogeometric basis functions as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation

the tractability of segmentation and scene analysis one of the fundamental problems in computer vision is the segmentation of an image into semantically meaningful regions based only on image characteristics a single segmentation can be determined using a linear number of evaluations of a uniformity predicate however minimising the number of regions is shown to be an np complete problem we also show that the variational approach to segmentation based on minimising a criterion combining the overall variance of regions and the number of regions also gives rise to an np complete problem when a library of object models is available segmenting the image becomes a problem of scene analysis a sufficient condition for the reconstruction of a 3d scene from a 2d image to be solvable in polynomial time is that the scene contains no cycles of mutually occluding objects and that no range information can be deduced from the image it is known that relaxing the no cycles condition renders the problem np complete we show that relaxing the no range information condition also produces an np complete problem

the searchlight scheduling problem the problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered a searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon the direction of the ray can be changed continuously and a point is detected by a searchlight at a given time if and only if it is on the ray a robber is a point that can move continuously with unbounded speed first it is shown that the problem of obtaining a search schedule for an instance having at least one searchlight on the polygon boundary can be reduced to that for instances having no searchlight on the polygon boundary the reduction is achieved by a recursive search strategy called the one way sweep strategy then various sufficient conditions for the existence of a search schedule are presented by using the concept of a searchlight visibility graph finally a simple necessary and sufficient condition for the existence of a search schedule for instances having exactly two sea

a robust multigrid method for isogeometric analysis in two dimensions using boundary correction we consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations for classical finite elements such methods are well known to be fast solvers showing optimal convergence behavior however the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased recently a robust approximation error estimate and a corresponding inverse inequality for b splines of maximum smoothness have been shown both with constants independent of the spline degree we use these results to construct multigrid solvers for discretizations of two dimensional problems based on tensor product b splines with maximum smoothness which exhibit robust convergence rates

the internodes method for the treatment of non conforming multipatch geometries in isogeometric analysis abstract in this paper we apply the internodes method to solve second order elliptic problems discretized by isogeometric analysis methods on non conforming multiple patches in 2d and 3d geometries internodes is an interpolation based method that on each interface of the configuration exploits two independent interpolation operators to enforce the continuity of the traces and of the normal derivatives internodes easily handles both parametric and geometric nurbs non conformity we specify how to set up the interpolation matrices on non conforming interfaces how to enforce the continuity of the normal derivatives and we give special attention to implementation aspects the numerical results show that internodes exhibits optimal convergence rate with respect to the mesh size of the nurbs spaces and that it is robust with respect to jumping coefficients

coupling adaptively refined multi patch spline discretizations via boundary compatibility abstract the present paper studies adaptive refinement on multi patch domains in isogeometric analysis in particular we investigate the gluing construction for adaptively refined spline spaces to obtain discretizations that are c 0 smooth across interfaces we will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction given a spline basis or more generally a generating system if linear independence is not guaranteed on a d dimensional box domain there are two possibilities for constructing the spline basis on the domain boundary firstly one can simply restrict the basis functions to the boundary secondly one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh the two constructions do not necessarily produce the same set of functions if they do then the spline bases are said to be compatible we study this property for hierarchical hb and truncated hierarchical b splines thb splines and identify sufficient conditions these conditions are weaker for thb than for hb splines finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis in particular when considering multi patch domains

computing conforming partitions of orthogonal polygons with minimum stabbing number abstract let p be an orthogonal polygon with n vertices a partition of p into rectangles is called conforming if it results from cutting p along a set of interior disjoint line segments each having both endpoints on the boundary of p the stabbing number of a partition of p into rectangles is the maximum number of rectangles stabbed by any orthogonal line segment inside p in this paper we consider the problem of finding a conforming partition of p with minimum stabbing number we first give an o n log n time algorithm to solve the problem when p is a histogram for an arbitrary orthogonal polygon even with holes we give an integer programming formulation of the problem and show that a simple rounding results in a 2 approximation algorithm for the problem finally we show that the problem is np hard if p is allowed to have holes

the complexity of the residual node connectedness reliability problem this paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities the network is in an operational state if the surviving nodes induce a connected graph the residual node connectedness reliability r g of a network g is the probability that the graph induced by the surviving nodes is connected this reliability measure is very different from the widely studied k terminal network reliability measure it is proven that the problem of computing the residual connectedness reliability is np hard by showing that the problem of counting the number of node induced connected subgraphs of a given graph is bf p complete the problem remains bf p complete for split graphs as well as planar and bipartite graphs

overlapping schwarz methods for isogeometric analysis we construct and analyze an overlapping schwarz preconditioner for elliptic problems discretized with isogeometric analysis the preconditioner is based on partitioning the domain of the problem into overlapping subdomains solving local isogeometric problems on these subdomains and solving an additional coarse isogeometric problem associated with the subdomain mesh we develop an h analysis of the preconditioner showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and overlap sizes for fixed polynomial degree p and regularity k of the basis functions numerical results in two and three dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h p k number of subdomains n overlap size and also jumps in the coefficients of the elliptic operator

decomposing a polygon into simpler components the problem of decomposing a polygon into simpler components is of interest in fields such as computational geometry syntactic pattern recognition and graphics in this paper we consider decompositions which do not introduce steiner points the simpler components we consider are convex polygons spiral polygons star shaped polygons and monotone polygons we apply a technique for improving the efficiency of dynamic programming algorithms in order to achieve polynomial time algorithms for the problems of decomposing a simple polygon into the minimum number of each of the component types using the same technique we are able to exhibit polynomial time algorithms for the problems of decomposing a simple polygon into each of the component types while minimizing the length of the internal edges used to form the decomposition when the polygons are allowed to contain holes many of the problems become np hard

a feti dp preconditioner for a composite finite element and discontinuous galerkin method in this paper a nitsche type discretization based on a discontinuous galerkin dg method for an elliptic two dimensional problem with discontinuous coefficients is considered the problem is posed on a polygonal region omega which is a union of n disjoint polygonal subdomains omega i of diameter o h i the discontinuities of the coefficients possibly very large are assumed to occur only across the subdomain interfaces partial omega i inside each subdomain a conforming finite element space on a quasi uniform triangulation with mesh size o h i is introduced to handle the nonmatching meshes across the subdomain interfaces a dg discretization is applied only on the interfaces for solving the resulting discrete system a feti dp type method is proposed and analyzed it is established that the condition number of the preconditioned linear system is estimated by c 1 max i log h i h i 2 with a constant c independent of h i h i h j h i and the jumps of coefficients the

dual primal isogeometric tearing and interconnecting solvers for multipatch dg iga equations abstract in this paper we consider a new version of the dual primal isogeometric tearing and interconnecting ieti dp method for solving large scale linear systems of algebraic equations arising from discontinuous galerkin dg isogeometric analysis of diffusion problems on multipatch domains with non matching meshes the dg formulation is used to couple the local problems across patch interfaces the purpose of this paper is to present this new method and provide numerical examples indicating a polylogarithmic condition number bound for the preconditioned system and showing an incredible robustness with respect to large jumps in the diffusion coefficient across the interfaces

convex partitions of polyhedra a lower bound and worst case optimal algorithm the problem of partitioning a polyhedron into a minimum number of convex pieces is known to be np hard we establish here a quadratic lower bound on the complexity of this problem and we describe an algorithm that produces a number of convex parts within a constant factor of optimal in the worst case the algorithm is linear in the size of the polyhedron and cubic in the number of reflex angles since in most applications areas the former quantity greatly exceeds the latter the algorithm is viable in practice

approximation algorithms for pspace hard hierarchically and periodically specified problems we study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in t lengauer j assoc comput mach 36 1989 pp 474 509 or are specified by one dimensional finite narrow periodic specifications as in e wanke paths and cycles in finite periodic graphs in lecture notes in comp sci 711 springer verlag new york 1993 pp 751 760 we show that for most of the problems pi considered when specified using k level restricted hierarchical specifications or k narrow periodic specifications the following hold let rho be any performance guarantee of a polynomial time approximation algorithm for pi when instances are specified using standard specifications then forall epsilon 0 pi has a polynomial time approximation algorithm with performance guarantee 1 epsilon rho pi has a polynomial time approximation scheme when restricted to planar instances these are the first polynomial time approximation schemes for pspace hard hierarchically or periodically specified problems since several of the problems considered are pspace hard our results provide the first examples of natural pspace hard optimization problems that have polynomial time approximation schemes this answers an open question in condon et al chicago j theoret comput sci 1995 article 4

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

radiocoloring in planar graphs complexity and approximations the frequency assignment problem fap in radio networks is the problem of assigning frequencies to transmitters by exploiting frequency reuse while keeping signal interference to acceptable levels the fap is usually modelled by variations of the graph coloring problem a radiocoloring rc of a graph g v e is an assignment function v n such that u v 2 when u v are neighbors in g and u v 1 when the distance of u v in g is two the number of discrete frequencies and the range of frequencies used are called order and span respectively the optimization versions of the radiocoloring problem rcp are to minimize the span or the order in this paper we prove that the radiocoloring problem for general graphs is hard to approximate unless np zpp within a factor of n1 2 e for any e 0 where n is the number of vertices of the graph however when restricted to some special cases of graphs the problem becomes easier we prove that the min span rcp is np complete for planar graphs next we provide an o n time algorithm v n which obtains a radiocoloring of a planar graph g that approximates the minimum order within a ratio which tends to 2 where the maximum degree of g finally we provide a fully polynomial randomized approximation scheme fpras for the number of valid radiocolorings of a planar graph g with colors in the case where 4 50

optimal trace inequality constants for interior penalty discontinuous galerkin discretisations of elliptic operators using arbitrary elements with non constant jacobians abstract in this paper a new method to numerically calculate the trace inequality constants which arise in the calculation of penalty parameters for interior penalty discretisations of elliptic operators is presented these constants are provably optimal for the inequality of interest as their calculation is based on the solution of a generalised eigenvalue problem involving the volumetric and face stiffness matrices the method is applicable to any element type for which these matrices can be calculated including standard finite elements and the non uniform rational b splines of isogeometric analysis in particular the presented method does not require the jacobian of the element to be constant and so can be applied to a much wider variety of element shapes than are currently available in the literature numerical results are presented for a variety of finite element and isogeometric cases when the jacobian is constant it is demonstrated that the new method produces lower penalty parameters than existing methods in the literature in all cases which translates directly into savings in the solution time of the resulting linear system when the jacobian is not constant it is shown that the naive application of existing approaches can result in penalty parameters that do not guarantee coercivity of the bilinear form and by extension the stability of the solution the method of manufactured solutions is applied to a model reaction diffusion equation with a range of parameters and it is found that using penalty parameters based on the new trace inequality constants result in better conditioned linear systems which can be solved approximately 11 faster than those produced by the methods from the literature

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by b f svaiter abstract in this paper we give two explicit examples of unbounded linear maximal monotone operators the first unbounded linear maximal monotone operator s on l 2 is skew we show its domain is a proper subset of the domain of its adjoint s and s is not maximal monotone this gives a negative answer to a recent question posed by svaiter the second unbounded linear maximal monotone operator is the inverse volterra operator t on l 2 0 1 we compare the domain of t with the domain of its adjoint t and show that the skew part of t admits two distinct linear maximal monotone skew extensions these unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened and they are simpler than the example given by phelps simons interesting consequences on fitzpatrick functions for sums of two maximal monotone operators are also given

minimal antiderivatives and monotonicity abstract we consider settings in convex analysis which give rise to families of convex functions that contain their lower envelope given certain partial data regarding a subdifferential we consider the family of all convex antiderivatives that comply with the given data we prove that this family is not empty and in particular contains a minimal antiderivative under a fairly general assumption on the given data it turns out that the representation of monotone operators by convex functions fits naturally in these settings duality properties of representing functions are also captured by these settings and the gap between the fitzpatrick function and the fitzpatrick family is filled by this broader sense of minimality of the fitzpatrick function

following a curve with the discrete frechet distance finding the similarity between curves is an important problem that comes up in many areas such as 3d modeling gis applications ordering and reachability a related problem is to find one of the curves given a measure of similarity and another curve given a set of points s a polygonal curve p and an e 0 the discrete set chain matching problem is to find another polygonal curve q such that the nodes of q are points in s and d f p q e here d f is the discrete frechet distance between the two polygonal curves for the first time we study the set chain matching problem based on the discrete frechet distance rather than the continuous frechet distance we further extend the problem based on unique or non unique nodes and on limiting the number of points used we prove that three of the variations of the set chain matching problem are np complete for the version of the problem that is polynomial we give an o p s time greedy solution

a nearly optimal sensor placement algorithm for boundary coverage locating visual sensors in 2d can be often modeled as an art gallery problem tasks such as surveillance require observing or covering the interior of a polygon with a minimum number of sensors or guards for other tasks such as inspection and image based rendering observing the boundaries of the environment is sufficient as interior covering ic also edge covering ec is np hard and no finite algorithm is known for its exact solution approximate ec solutions are provided by many heuristic algorithms but their performances with respect to optimality minimum number of sensors is unknown in this paper we propose a new ec sensors location technique the algorithm is incremental and converges toward the optimal solution it refines an initial approximation provided by an integer covering algorithm iec where each edge is observed entirely by at least one sensor a lower bound for the number of sensors specific of the polygon considered is used at each step for evaluating the quality of the current solution and a set of rules are provided for performing a local refinement to reduce the computational burden the algorithm has been implemented and tests over hundreds of random polygons show that it supplies solutions very close to and often coincident with the lower bound and then suboptimal or optimal in addition the approximate starting solutions provided by the iec algorithms are on the average close to optimum the tight lower bound can also be used for testing other ec sensor location algorithms running times allow dealing with polygons with up to a few hundreds of edges which appears adequate for many practical cases an enhanced version of the algorithm also taking into account range and incidence constraints has also been implemented and tested

computational complexity of art gallery problems we study the computational complexity of the art gallery problem originally posed by klee and its variations specifically the problem of determining the minimum number of vertex guards that can see an n wall simply connected art gallery is shown to be np hard the proof can be modified to show that the problems of determining the minimum number of edge guards and the minimum number of point guards in a simply connected polygonal region are also np hard as a byproduct the problem of decomposing a simple polygon into a minimum number of star shaped polygons such that their union is the original polygon is also shown to be np hard

primal dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators a fundamental result due to rockafellar states that every cyclically monotone operator a admits an antiderivative f in the sense that the graph of a is contained in the graph of the subdifferential operator partial f given a method mathfrak m that assigns every finite cyclically monotone operator a some antiderivative mathfrak m a we say that the method is primal dual symmetric if mathfrak m applied to the inverse of a produces the fenchel conjugate of mathfrak m a rockafellar s antiderivatives do not possess this property utilizing fitzpatrick functions and the proximal average we present novel primal dual symmetric intrinsic methods the antiderivatives produced by these methods provide a solution to a problem posed by rockafellar in 2005 the results leading to this solution are illustrated by various examples