Partial Order Relation Research Articles

Clusters of high-dimensional interval data and related Boolean functions of events in Euclidean space

Clustering interval data has been studied for decades. High-dimensional interval data can be expressed in terms of hyperrectangles in $$\mathbb {R}^d$$ (or d-orthotopes) in case of real-valued d-attributes data. This paper investigates such high-dimensional interval data: the Cartesian product of intervals, or a vector of interval. For the efficient computation of related Boolean functions, some interesting aspects have been discovered using vertices and edges of the graph, generated from given events. We also study the lower and upper-bounded orthants in $$\mathbb {R}^d$$ as events for which we show the existence of a polynomial-time algorithm to calculate the probability of the union of such events. This efficient algorithm has been discovered by constructing a suitable partial order relation based on a recursive projection onto lower-dimensional spaces. Illustrative real-life applications are presented.

On nonlinear fuzzy set-valued $ \Theta $-contractions with applications

<abstract><p>Among various improvements in fuzzy set theory, a progressive development has been in process to investigate fuzzy analogues of fixed point theorems of the classical fixed point results. In this direction, taking the ideas of $ \theta $-contractions as well as Feng-Liu's approach into account, some new fuzzy fixed point results for nonlinear fuzzy set-valued $ \theta $-contractions in the framework of metric-like spaces are introduced in this paper without using the usual Pompeiu-Hausorff distance function. Our established concepts complement, unify and generalize a few important fuzzy and classical fixed point theorems in the corresponding literature. A handful of these special cases of our notions are pointed and analyzed. Some of the main results herein are further applied to derive their analogues in metric-like spaces endowed with partial ordering and binary relations. Comparisons and nontrivial examples are given to authenticate the hypotheses and significance of the obtained ideas.</p></abstract>

Simulation of Sports Venue Based on Ant Colony Algorithm and Artificial Intelligence

In order to improve the congestion of the evacuation plan and further improve the evacuation efficiency, this paper proposes the priority Pareto partial order relation and the vector pheromone routing method based on the priority Pareto partial order relation. Numerical experiments show that compared with the hierarchical multiobjective evacuation path optimization algorithm based on the hierarchical network, the fragmented multiobjective evacuation path optimization algorithm proposed in this paper effectively improves the evacuation efficiency of the evacuation plan and the convergence of the noninferior plan set. However, the congestion condition of the noninferior evacuation plan obtained by the fragmented multiobjective evacuation route optimization algorithm is worse than the congestion condition of the noninferior evacuation plan obtained by the hierarchical multiobjective evacuation route optimization algorithm. The multiple factors that affect the routing process considered in the probability transfer function used in the traditional ant colony algorithm routing method must be independent of each other. However, in actual route selection, multiple factors that affect route selection are not necessarily independent of each other. In order to fully consider the various factors that affect the routing, this paper adopts the vector pheromone routing method based on the traditional Pareto partial order relationship instead of the traditional ant colony algorithm. The model mainly improves the original pheromone distribution and volatilization coefficient of the ant colony, speeds up the convergence speed and accuracy of the algorithm, and obtains ideal candidate solutions. The method is applied to the location of sports facilities and has achieved good results. The experimental results show that the improved ant colony algorithm model designed in this paper is suitable for solving the problem of urban sports facilities location in large‐scale space.

Learning Recommendation Algorithm Based on Improved BP Neural Network in Music Marketing Strategy.

The growth and popularity of streaming music have changed the way people consume music, and users can listen to online music anytime and anywhere. By integrating various recommendation algorithms/strategies (user profiling, collaborative filtering, content filtering, etc.), we capture users' interests and preferences and recommend the content of interest to them. To address the sparsity of behavioral data in digital music marketing, which leads to inadequate mining of user music preference features, a metric ranking learning recommendation algorithm with fused content representation is proposed. Relative partial order relations are constructed using observed and unobserved behavioral data to enable the model to be fully trained, while audio feature extraction submodels related to the recommendation task are constructed to further alleviate the data sparsity problem, and finally, the preference relationships between users and songs are mined through metric learning. Convolutional neural networks are used to extract the high-level semantic features of songs, and then the high-level semantic features of songs extracted from the previous layer are reformed into a session time sequence list according to the time sequence of user listening in order to build a bidirectional recurrent neural network model based on the attention mechanism so that it can reduce the influence of noisy data and learn the strong dependencies between songs.

Сравнительный логический анализ субстанциальной и процессуальной онтологий

В работах Аристотеля мы находим пример детально построенной субстанциальной онтологии, которой выпала роль философских оснований логики и всей западноевропейской науки. Исторически существовал и другой, процессуальный, взгляд на устройство окружающего мира, наиболее ярким представителем которого был Гераклит.
 В настоящей публикации проведен анализ субстанциальной онтологии и предложен подход к построению процессуальной. Основными характеристиками процессов являются их длительность и направленность. Введено отношение $ \textit{$\textbf{D}$}pxy $ деления по длительности процесса $ p $ на два последовательно связанных подпроцесса $ x $ и $ y $. Сформулирован ряд интуитивно очевидных аксиом, описывающих это отношение. С помощью данного отношения определено понятие локального времени процесса, обладающего свойствами плотного линейного порядка без первого и последнего элементов.
 Предметы в процессуальной онтологии понимаются как <<пучки>> подпроцессов. Для их характеризации вводится отношение $ x\leq y $, означающее, что процесс x является подпроцессом процесса y. Это отношение обладает свойствами частичного порядка с наибольшим элементом $\textit{$\textbf{W}$} $, который понимается как глобальный процесс, представляющий мир в целом. Показано, как с помощью данного отношения можно определить основные силлогистические константы для построения рассуждений о подпроцессах.
 Поскольку оба отношения описывают две стороны одной и той же онтологии, приняты дополнительные аксиомы связи между ними. Это позволило синхронизировать процессы, определив понятие глобального времени и понятие параллельных процессов.
 В завершение статьи показано, как в процессуальной онтологии можно определить понятия, соответствующие понятиям индивидов и свойства процессуальной и субстанциальной онтологий.

Predicting stock price trends based on financial news articles and using a novel twin support vector machine with fuzzy hyperplane

This study extracts the hidden topic model and emotional information from news articles. A novel fuzzy twin support vector machine is also developed to merge the large volume of information from on-line news, using this to predict stock price trends. Fuzzy set theory is very useful for this approach because the texts use fuzzy terminology, such as high/low and big/small, and the boundary between rising and falling categories is poorly defined. By using fuzzy partial ordering relation, the decision function of our approach is generalized such that the values assigned to the stocks fall within a specified range and indicate the membership grade of these stocks in the positive class (rising trends). The proposed approach is useful for dealing with outliers. Outliers only slightly increase the spread of the fuzzy decision boundary and the estimation of the mode of fuzzy hyperplane remains unchanged. Therefore, our model is more robust compared with other methods when the data contain some outliers. Besides, the membership grade estimated by the fuzzy decision boundary gives a better insight into the degree of confidence in the predicted outputs. The explicable characteristic for the degree of confidence is a vital aspect for decision-making applications.

Notes on Naimark’s dilation theorem

After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.

Fixed points of Kannan contractive mappings in relational metric spaces

In this paper, we prove an analogous version of fixed point theorem due to Kannan [Bull. Calcutta Math. Soc., 60, (1968), 71–76] via conditional contractive mappings and some other relational metrical notions. In the proof of present results, we utilize the notion of comparable mappings and some other well known classical fixed point theorems in the settings of relational and ordered metric spaces. We also, highlight the close connection of $$\alpha $$ -admissible mappings with the binary relation and partial order relation as well. Radically, these investigations open another new direction of metric fixed point theory for contractive type mappings. Moreover, non-trivial examples are given to demonstrate the importance and usefulness of such findings.

New fixed point theorems on b-metric spaces with applications to coupled fixed point theory

Let (X, d) be a complete b-metric space endowed with a partial order relation and $$f:X\rightarrow X$$ be a Ciric type operator. In this paper, an extended study of the fixed point equation $$x=f(x), \ x\in X$$ , is considered. As an application, coupled fixed point results are given in the same framework. Our results generalize some recent theorems in the literature.

Ordered models for concept representation

AbstractBasic notions linked with concept theory can be accounted for by partial order relations. These orders translate the fact that, for an agent, an object may be seen as a better or a more typical exemplar of a concept than anyother. They adequately model notions linked with categorial membership, typicality and resemblance, without any of the drawbacks that are classically encountered in conjunction theory. An interesting consequence of such a concept representation is the possibility of using the tools of non-monotonic logic to address some well-known problems of cognitive psychology. Thus, conceptual entailment and concept induction can be reexamined in the framework of preferential inference relations. This leads to a rigorous definition of the basic notions used in the study of category-based induction.

Optimization of fuzzy-valued functions using Zadeh's extension principle

This paper proposes a fuzzy optimization problem whose objective function is Zadeh's extension of a function with respect to a parameter and the independent variable. Making use of a partial order relation, the extension of a function mapping each parameter to the corresponding local minimizing point is proven to provide the local smallest fuzzy value for the extended function. Deploying the usual order relation in the literature, the same point is proven to provide the minimal value for the function. Examples illustrate the results.

A reduction theorem for the Galois–McKay conjecture

We introduce H {\mathcal {H}} -triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois–McKay conjecture to a question about simple groups.

Ranking via partial ordering for answer selection

Answer selection with deep neural networks has been studied extensively owing to its strong capability of encoding semantic features. Most previous work has treated the candidates as independent individuals for ranking but ignored the relations between these. In this study, we propose a ranking method via partial ordering for answer selection, which is applicable to various tasks, such as question answering and reading comprehension. First, we propose a comparative network, namely Candidate vs. Candidate, which aims to discover the possible partial order relations between (candidate, candidate) pairs. Thereafter, a multi-task learning framework is constructed for the answer selection problem, in which the main task is representing the relevance between (question, answer) pairs, while the auxiliary sub-task is learning the partial order relations between (answer, answer) pairs. By jointly training the networks with abundant supervision information, a reasonable relevance function and a comparison function can be approximated for these tasks. The experimental results on four benchmarks indicate that ranking candidate answers via partial ordering can significantly improve the answer selection performance.

Construction of three-way attribute partial order structure via cognitive science and granular computing

Partial order formal structure analysis (POFSA), as an emerging model of concept cognitive learning, has been extensively used in the field of knowledge processing. However, along with the development of information storage and network technology, the knowledge that people can master is growing dramatically, and it is difficult to effectively process the expanding knowledge just through one single theory. Therefore, this paper explores the construction method of a three-way attribute partial order structure (APOS) via multi granularity by incorporating the ideas of three-way decision (3WD) and granular computing (GrC) into the theory of POFSA. First, for a specific formal context, by taking object set as the whole domain and using attributes and their respective extensions to constitute granule, granular layers can be formed based on the binary relations of equivalence or compatibility between granules. Then, according to the partial order relations between the granules of different granular layers, the corresponding granular structure APOS can be generated. Finally, using the idea of 3WD to reasonably eliminate the cross connections between different branches of APOS, a three-way APOS via multi granularity can be constructed. In addition, based on the generation algorithm of the three-way APOS, the knowledge processing of several data sets from UCI have been conducted and discussed. Through discussion and experiment, it can be concluded that, the three-way APOS via multi granularity can not only improve the efficiency of knowledge processing, but also make the results of knowledge processing more reasonable.

Relationships between convexity and generalized convexity of comparable fuzzy functions

In this paper, we show some relations between the convexity and generalized convexity such as pseudo-convexity, strongly quasi-convexity of fuzzy mappings of several multivariables by using a partial order relation. Also we established a necessary and sufficient condition for convexity of fuzzy mapping. Finally we showed a necessary condition for global minimum of a fuzzy mapping.

A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions

In this study, we introduce and analyze the concepts of a fixed ordering structure and a variable ordering structure on intervals. The fixed ordering structures on intervals are defined with the help of a pointed convex cone of intervals. A variable ordering is defined by a set-valued map whose values are convex cones of intervals. In the sequel, a few properties of a cone of intervals are derived. It is shown that a binary relation, defined by a convex cone of intervals, is a partial order relation on intervals; further, the relation is antisymmetric if the convex cone of intervals is pointed. Several results under which a variable ordering map of intervals satisfies the conditions of a partial ordering relation of intervals are provided. The introduced fixed and variable ordering of intervals are applied to define and characterize optimal elements of an optimization problem with interval-valued functions. Finally, we propose a numerical technique and present its algorithmic implementation to obtain the set of optimal elements of an interval optimization problem. We also provide illustrative examples to support the study.

Concept Lattice Method for Spatial Association Discovery in the Urban Service Industry

A relative lag in research methods, technical means and research paradigms has restricted the rapid development of geography and urban computing. Hence, there is a certain gap between urban data and industry applications. In this paper, a spatial association discovery framework for the urban service industry based on a concept lattice is proposed. First, location data are used to form the formal context expressed by 0 and 1. Frequent closed itemsets and a concept lattice are computed on the basis of the formal context of the urban service industry. Frequent closed itemsets can filter out redundant information in frequent itemsets, uniquely determine the complete set of all frequent itemsets, and be orders of magnitude smaller than the latter. Second, spatial frequent closed itemsets and association rules discovery algorithms are designed and built based on the formal context. The inputs of the frequent closed itemsets discovery algorithms include the given formal context and frequent threshold value, while the outputs are all frequent closed itemsets and the partial order relationship between them. Newly added attributes create new concepts to guarantee the uniqueness of the new spatial association concept. The inputs of spatial association rules discovery algorithms include frequent closed itemsets and confidence threshold values, and a rule is confident when and only if its confidence degree is not less than the confidence threshold value. Third, the spatial association of the urban service industry in Nanning, China is taken as a case to verify the method. The results are basically consistent with the spatial distribution of the urban service industry in Nanning City. This study enriches the theories and methods of geography as well as urban computing, and these findings can provide guidance for location-based service planning and management of urban services.

Granularity and Entropy of Intuitionistic Fuzzy Information and Their Applications.

A granular structure of intuitionistic fuzzy (IF) information presents simultaneously the similarity and diversity of samples. However, this structural representation has rarely displayed its technical capability in data mining and information processing due to the lack of suitable constructive methods and semantic interpretation for IF information with regard to real data. To pursue better performance of the IF-based technique in real-world data mining, in this article, we examine information granularity, information entropy of IF granular structures, and their applications to data reduction of IF information systems. First, several types of partial-order relations at different hierarchical levels are defined to reveal the granularity of IF granular structures. Second, the granularity invariance between different IF granular structures is characterized by using relational mappings. Third, Shannon's entropies are generalized to IF entropies and their relationships with the partial-order relations are addressed. Based on the theoretical analysis above, the significance of intuitionistic attributes using the information measures is then introduced and the information-preserving algorithm for data reduction of IF information systems is constructed. Finally, by inducing substantial IF relations from public datasets that take both the similarity/diversity between the samples from the same/different classes into account, a collection of numerical experiments is conducted to confirm the performance of the proposed technique.

Optimality conditions in set-valued optimization problems with respect to a partial order relation by using subdifferentials

In this study, set-valued optimization problems are considered with respect to -order relation, which is a partial order relation on the family of nonempty and bounded sets. Four subdifferentials are defined for set-valued optimization maps. Some existence theorems and relations among these subdifferentials are obtained. By using them, necessary and sufficient optimality conditions are presented for set-valued optimization problems with respect to -order relation.

Continuity of Solution Mappings for Parametric Set Optimization Problems under Partial Order Relations

This paper mainly investigates the semicontinuity of solution mappings for set optimization problems under a partial order set relation instead of upper and lower set less order relations. To this end, we propose two types of monotonicity definition for the set-valued mapping introduced by two nonlinear scalarization functions which are presented by these partial order relations. Then, we give some sufficient conditions for the semicontinuity and closedness of solution mappings for parametric set optimization problems. The results presented in this paper are new and extend the main results given by some authors in the literature.