Hull Operation Research Articles

Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization

Via a family of monotone scalar functions, a preorder on a set is extended to its power set and then used to construct a hull operator and a corresponding complete lattice of sets. Functions mapping into the preordered set are extended to complete lattice-valued ones, and concepts for exact and approximate solutions for corresponding set optimization problems are introduced and existence results are given. Well-posedness for complete lattice-valued problems is introduced and characterized. The new approach is compared with existing ones in vector and set optimization. Its relevance is shown by means of many examples from multicriteria decision making, statistics, and mathematical economics and finance.

A classification of hull operators in archimedean lattice-ordered groups with unit

The category, or class of algebras, in the title is denoted by $bf W$. A hull operator (ho) in $bf W$ is a reflection in the category consisting of $bf W$ objects with only essential embeddings as morphisms. The proper class of all of these is $bf hoW$. The bounded monocoreflection in $bf W$ is denoted $B$. We classify the ho's by their interaction with $B$ as follows. A ``word'' is a function $w: {bf hoW} longrightarrow {bf W}^{bf W}$ obtained as a finite composition of $B$ and $x$ a variable ranging in $bf hoW$. The set of these,``Word'', is in a natural way a partially ordered semigroup of size $6$, order isomorphic to ${rm F}(2)$, the free $0-1$ distributive lattice on $2$ generators. Then, $bf hoW$ is partitioned into $6$ disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic order-preserving quotient of Word ($approx {rm F}(2)$). Of the $6$: $1$ is of size $geq 2$, $1$ is at least infinite, $2$ are each proper classes, and of these $4$, all quotients are chains; another $1$ is a proper class with unknown quotients; the remaining $1$ is not known to be nonempty and its quotients would not be chains.

Intuitionistic Fuzzy C Means Clustering for Lung Segmentation in Diffuse Lung Diseases

Lung segmentation is a challenging task when increased attenuation appears at the periphery of the lungs. In case of increased attenuation, the intensity of the lung tissue closely matches with surrounding non-lung tissues. This paper presents intuitionistic fuzzy c means clustering to segment the lungs with increased attenuation appearing at the border in diffuse lung diseases. In the proposed approach intuitionistic spatial kernel fuzzy c means is employed to obtain the initial lung segment and it is further refined using quick hull and morphological operations. The proposed approach is evaluated on ’TALISMAN’ diffuse lung diseases benchmark dataset. The dataset has 108 high resolution computed tomography scans with different types of diseases such as reticulation, fibrosis, ground glass opacity, consolidation, emphysema, micro and macro nodules. The evaluation results clearly indicate that the proposed approach achieves better segmentation accuracy on lung scans with increased attenuation appearing at the lung border.

Estimation of minimum volume of bounding box for geometrical metrology

This paper presents algorithms for estimating the minimum volume bounding box based on a three-dimensional point set measured by a coordinate measuring machine. A new algorithm, which calculates the minimum volume with high accuracy and reduced number of computations, is developed. The algorithm is based on the convex hull operation and established theories about a minimum bounding box circumscribing a convex polyhedron. The new algorithm includes a pre-processing operation that removes convex polyhedron faces located near the edges of the measured object. As showed in the paper, the solution of the minimum bonding box is not based on faces located near the edges; therefore, we can save computation time by excluding them from the convex polyhedron data set. The algorithms have been demonstrated on physical objects measured by a coordinate measuring machine, and on theoretical 3D models. The results show that the algorithm can be used when high accuracy is required, for example in calibration of reference standards.

Hull operators and interval operators in (L,M)-fuzzy convex spaces

Considering L being a continuous lattice and M being a completely distributive De Morgan algebra, several basic notions with respect to (L,M)-fuzzy convex structures in the sense of Shi and Xiu are introduced and their relationship with (L,M)-fuzzy convex structures are studied. Firstly, an equivalent form of (L,M)-fuzzy convex structures in the sense of Shi and Xiu is provided. Secondly, two types of fuzzy hull operators are introduced, which are called (L,M)-fuzzy hull operators and (L,M)-fuzzy restricted hull operators, respectively. It is shown that they can be used to characterize (L,M)-fuzzy convex structures. Finally, fuzzy counterparts of interval operators in the (L,M)-fuzzy case are proposed, which are called (L,M)-fuzzy interval operators. It is proved that there is a Galois correspondence between the category of (L,M)-fuzzy interval spaces and that of (L,M)-fuzzy convex spaces and further the category of arity 2 (L,M)-fuzzy convex spaces can be embedded in the category of (L,M)-fuzzy interval spaces as a fully reflective subcategory.

A categorical approach to abstract convex spaces and interval spaces

Abstract In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category CS of convex spaces is isomorphic to the category HS of hull spaces, and they are all topological over Set. Also, it is proved that there is an adjunction between the category IS of interval spaces and the category CS of convex spaces. In particular, the category CS(2) of arity 2 convex spaces can be embedded in IS as a reflective subcategory.

Representation of convex geometries by circles on the plane

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the result of Kashiwabara et al. (2005). Allowing circles rather than points, as was suggested by Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on the plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.

Detection of orchard citrus fruits using a monocular machine vision-based method for automatic fruit picking applications

Due to the variable illumination conditions and occlusion produced by neighbouring fruits and other background participants, vision systems are important in accurately and reliably detecting mature citrus in natural orchard environments for automatic fruit picking applications. A robust citrus fruit detection method based on a monocular vision system was proposed. An adaptive enhanced red and green chromatic map was generated from an illumination-compensated image, which was obtained using block-based local homomorphic filtering. Otsu thresholding, morphology operation, marker-controlled watershed transform and convex hull operation methods were then used in combination to locate potential citrus regions from the chromatic map. Local texture information was extracted from the potential regions using local binary patterns and fed to a histogram intersection kernel-based support vector machine to make the final decision. The performance of the proposed method was evaluated on 127 test images captured in two citrus orchards on both sunny and cloudy days. Under strict PASCAL criteria, the recall rate of correctly detected citrus was greater than 0.86, with 13 false detections.

Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces

For fuzzy counterparts of hull operators and interval operators, two types of fuzzy hull operators and one type of fuzzy interval operators are proposed. Firstly, the concept of L-hull operators is introduced and the resulting category is shown to be isomorphic to that of L-convex spaces. Secondly, considering the graded inclusions of L-subsets, the notion of L-ordered hull operators is presented, which is shown to be categorically isomorphic to strong L-convex structures. Finally, the concept of L-interval operators is introduced and it is shown that there is a Galois correspondence between the category of L-interval spaces and that of L-convex spaces. In particular, the category of arity 2 L-convex spaces can be reflectively embedded into that of L-interval spaces.

Some properties of M-fuzzifying convexities induced by M-orders

In this paper, the notions of geometric interval spaces, convex geometries, and base-point orders in the theory of convexity spaces are generalized to M-fuzzifying convexity spaces. Using M-fuzzifying restricted hull operators, the M-fuzzifying convexity spaces induced by M-orders are defined conveniently. Then it is shown that M-fuzzifying convexity spaces induced by M-orders are M-fuzzifying JHC and arity ⩽2, and if the M-order is strongly anti-symmetric, then the M-fuzzifying convexity space is an M-fuzzifying convex geometry and its segment operator is an M-fuzzifying geometric interval operator.

Strong inclusion orders between L-subsets and its applications in L-convex spaces

In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.

Guide on set invariance for delay difference equations

This paper addresses set invariance properties for linear time-delay systems. More precisely, the first goal of the article is to review known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by linear discrete time-delay difference equations (dDDEs).Secondly, we address the construction of invariant sets in the original state space (also called D-invariant sets) by exploiting the forward mappings. The notion of D-invariance is appealing since it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in some appropriate extended state space model.The present paper contains a sufficient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and sufficient condition for the existence of D-invariant sets in relation to linear time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance of linear dDDEs. In short, it is shown that the convex hull of the union of two or more D-invariant sets is not necessarily D-invariant, while the convex hull of a non-convex D-invariant set is D-invariant.

The restricted hull operator of M-fuzzifying convex structures1

In this paper, the notion of M-fuzzifying restricted hull operators is introduced and several equivalent characterizations are given. It is shown that there is a one-to-one correspondence between M-fuzzifying restricted hull operators and M-fuzzifying convex structures. As applications, some properties of the cut convex structures of an M-fuzzifying convex structure and of M-fuzzifying convexity preserving functions and of M-fuzzifying convex-to-convex functions are derived. In addition, using M-fuzzifying restricted hull operators, some M-fuzzifying convexities are naturally constructed from M-fuzzy quasi-orders.

A Rule-based Verification Strategy for Array Manipulating Programs

We present a method for verifying properties of imperative programs that manipulate integer arrays. Imperative programs and their properties are represented by using Constraint Logic Programs (CLP) over integer arrays. Our method is refutational. Given a Hoare triple {ϕ} prog {ψ} that defines a par tial correctness property of an imperative program prog, we encode the negation of the property as a predicate incorrect defined by a CLP program P, and we show that the property holds by proving that incorrect is not a consequence of P. Program verification is performed by applying a sequence of semantics preserving transformation rules and deriving a new CLP program T such that incorrect is a consequence of P iff it is a consequence of T. The rules are applied according to an automatic strategy whose objective is to derive a program T that satisfies one of the following properties: either (i) T is the empty set of clauses, hence proving that incorrect does not hold and prog is correct, or (ii) T contains the fact incorrect, hence proving that prog is incorrect. Our transformation strategy makes use of an axiomatization of the theory of arrays for the manipulation of array constraints, and also applies the widening and convex hull operators for the generalization of linear integer constraints. The strategy has been implemented in the VeriMAP transformation system and it has been shown to be quite effective and efficient on a set of benchmark array programs taken from the literature.

Set Invariance for Delay Difference Equations

This paper deals with set invariance for time delay systems. The first goal of the paper is to review the known necessary or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by discrete-time delay difference equations (dDDEs). Secondly, we address the construction of invariant sets in the original state space (also called D-invariant sets) by exploiting the forward mappings. As novelties, the present paper contains a sufficient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and sufficient condition for the existence of Dinvariant sets in relation to time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance. In short, tis shown that the convex hull of two D-invariant sets is not D-invariant but the convex hull of a non-convex D-invariant set is D-invariant.

On Borel Hull Operations

We show that some set-theoretic assumptions (for example Martin’s Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets (in \(\mathbb{R}^{n}\)). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the \(\sigma\)-algebra of sets with the property of Baire.

Program verification via iterated specialization

We present a method for verifying properties of imperative programs by using techniques based on the specialization of constraint logic programs (CLP). We consider a class of imperative programs with integer variables and we focus our attention on safety properties, stating that no error configuration can be reached from any initial configuration. We introduce a CLP program I that encodes the interpreter of the language and defines a predicate unsafe equivalent to the negation of the safety property to be verified. Then, we specialize the CLP program I with respect to the given imperative program and the given initial and error configurations, with the objective of deriving a new CLP program Isp that either contains the fact unsafe (and in this case the imperative program is proved unsafe) or contains no clauses with head unsafe (and in this case the imperative program is proved safe). If Isp enjoys neither of these properties, we iterate the specialization process with the objective of deriving a CLP program where we can prove unsafety or safety. During the various specializations we may apply different strategies for propagating information (either propagating forward from an initial configuration to an error configuration, or propagating backward from an error configuration to an initial configuration) and different operators (such as the widening and the convex hull operators) for generalizing predicate definitions. Each specialization step is guaranteed to terminate, but due to the undecidability of program safety, the iterated specialization process may not terminate. By an experimental evaluation carried out on a significant set of examples taken from the literature, we show that our method improves the precision of program verification with respect to state-of-the-art software model checkers.

Achievable and Crystallized Rate Regions of the Interference Channel with Interference as Noise

The interference channel achievable rate region is presented when the interference is treated as noise. The formulation starts with the 2-user channel, and then extends the results to the n-user case. The rate region is found to be the convex hull of the union of n power control rate regions, where each power control rate region is upperbounded by a (n-1)-dimensional hyper-surface characterized by having one of the transmitters transmitting at full power. The convex hull operation lends itself to a time-sharing operation depending on the convexity behavior of those hyper-surfaces. In order to know when to use time-sharing rather than power control, the paper studies the hyper-surfaces convexity behavior in details for the 2-user channel with specific results pertaining to the symmetric channel. It is observed that most of the achievable rate region can be covered by using simple On/Off binary power control in conjunction with time-sharing. The binary power control creates several corner points in the n-dimensional space. The crystallized rate region, named after its resulting crystal shape, is hence presented as the time-sharing convex hull imposed onto those corner points; thereby offering a viable new perspective of looking at the achievable rate region of the interference channel.

On hull classes of ℓ-groups and covering classes of spaces

Abstract W denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. Comp denotes the category of compact Hausdorff spaces with continuous maps. The Yosida functor is used to investigate the relationship between hull classes in W and covering classes in Comp. The central idea is that of a hull class whose hull operator preserves boundedness. We demonstrate how the Yosida functor may be used to identify hull classes in W and covering classes in Comp. In addition, we exhibit an array of order preserving bijections between certain families of hull classes and all covering classes, one of which was recently produced by Martínez. Lastly, we apply our results to answer a question of Knox and McGovern about the class of all feebly projectable ℓ-groups.

A New Approach to Random Access: Reliable Communication and Reliable Collision Detection

This paper applies Information Theoretic analysis to packet-based random multiple access communication systems. A new channel coding approach is proposed for coding within each data packet with built-in support for bursty traffic properties, such as message underflow, and for random access properties, such as packet collision detection. The coding approach does not require joint communication rate determination either among the transmitters or between the transmitters and the receiver. Its performance limitation is characterized by an achievable region defined in terms of communication rates, such that reliable packet recovery is supported for all rates inside the region and reliable collision detection is supported for all rates outside the region. For random access communication over a discrete-time memoryless channel, it is shown that the achievable rate region of the introduced coding approach equals the Shannon information rate region without a convex hull operation. Further connections between the achievable rate region and the Shannon information rate region are developed and explained.