Partial Order Structure Research Articles

The First Mean Value Theorem of integral in Perplex Plane

Perplex numbers are an extension of the real numbers, consisting of pairs of real numbers that form a commutative ring. This article considers the partial order structure of perplex numbers and discusses the concept of perplex interval. Based on the definition of integration of perplex functions of perplex variable, this article we investigate the first mean value theorem of integral in perplex plane which can significantly enrich the mathematical toolbox by providing essential methods for estimating and approximating perplex number functions. It also simplifies the solution of complex integral problems involving perplex numbers and fosters theoretical extensions by exploring the behavior of perplex number functions under various conditions. This, in turn, promotes the development of perplex calculus, enhancing the overall theoretical and practical framework of perplex number analysis. In physics, particularly in relativity, studying the first mean value theorem for perplex numbers can provide more precise methods for describing and estimating physical quantities, enhancing the application of hyperbolic numbers in these fields.

Incremental concept cognitive learning in dynamic formal contexts based on attribute partial order structure diagram

Incremental concept cognitive learning in dynamic formal contexts based on attribute partial order structure diagram

Structural Order-Disorder Analysis of Thermally-Activated Water-Washed Kaolin Particles Using Fourier Transform Infrared Spectroscopy

During the calcination of kaolin particles, kaolinite is thermally activated at high temperatures, causing the crystal structure to collapse and yielding amorphous metakaolinite through dehydroxylation. This metakaolinite is used as a supplementary cementitious material, and one of the most important factors influencing the pozzolanic properties is calcination conditions. Fourier transform infrared spectroscopy (FTIR) has become useful in distinguishing and obtaining information about structural order-disorder and phase transformation following the calcination process. In this study, water-washed kaolin particles were thermally activated at elevated temperatures ranging from 600 to 800 °C for 3–4 h at a rate of 10 °C/min before being analyzed with FTIR to determine the optimum conditions for calcining kaolin particles by examining functional groups, and also to study structural order-disorder or crystallinity of calcined kaolin particles. The most reactive metakaolinite state of water-washed kaolin particles was achieved after 3 h of calcination at 800 °C. Using both empirical and numerical approaches, variations in the position and relative intensity of O-H stretching and deformation of hydroxyl groups in the infrared spectrum can be used to classify the degree of structural order of water-washed kaolin particles. By increasing the calcination temperatures and period, the well-ordered and partially-ordered structures of kaolin particles were transformed into well-ordered, partially-ordered, and poorly-ordered structures. These structural disorder and crystallinity have a significant impact on pozzolanic activity because well-ordered kaolinite can be transformed into less reactive metakaolinite, whereas poorly-ordered kaolinite with high defects can be transformed into more reactive metakaolinite. However, in this study, the structure of water-washed kaolin particles that achieved complete dehydroxylation was discovered to be partially-ordered to poorly-ordered and can be transformed into highly reactive pozzolans.

Model selection over partially ordered sets

In problems such as variable selection and graph estimation, models are characterized by Boolean logical structure such as the presence or absence of a variable or an edge. Consequently, false-positive error or false-negative error can be specified as the number of variables/edges that are incorrectly included or excluded in an estimated model. However, there are several other problems such as ranking, clustering, and causal inference in which the associated model classes do not admit transparent notions of false-positive and false-negative errors due to the lack of an underlying Boolean logical structure. In this paper, we present a generic approach to endow a collection of models with partial order structure, which leads to a hierarchical organization of model classes as well as natural analogs of false-positive and false-negative errors. We describe model selection procedures that provide false-positive error control in our general setting, and we illustrate their utility with numerical experiments.

Graph representation learning method based on three-way partial order structure

In the era of big data, handling massive datasets to extract valuable information has become increasingly critical. Knowledge representation emerges as a pivotal method to address this challenge. In the domain of knowledge representation, there exist two primary approaches: symbolic representation and vector representation. The integration of symbolic and vector representations to harness their respective strengths has become the cutting-edge approach to address challenges in the field of knowledge representation. This paper proposes a method that integrates a partial order formal structure analysis (POFSA) with graph representation learning. Specifically, we initially construct three-way partial order structure graphs, then create an attribute graph based on this structure, which can be processed by the graph representation learning methods. Finally, we utilize the graph representation learning to construct embeddings for three-way attribute partial order structure diagram (APOSD). We comprehensively assesse these embeddings across eight different datasets and present the results. The experiments indicate the feasibility of our proposed approach which is proven a novel approach of combining symbolic and vector representations for handling complex data and implicit knowledge.

Visualization and application method of acupuncture-moxibustion knowledge of senile dementia in ancient books based on partial order structure

Acupuncture-moxibustion has affirmative curative effect in the prevention and treatment of senile dementia. Starting from the literature research, a visualization and application method of acupuncture-moxibustion knowledge of senile dementia in ancient books based on partial order structure is proposed. This method could extract and integrate the acupuncture-moxibustion knowledge of senile dementia contained in ancient books of traditional Chinese medicine, and establish a standardized, structured and visual knowledge graph. Applying this method to knowledge visual analysis and clinical auxiliary guidance could provide reference for combing the knowledge of ancient books of traditional Chinese medicine and transforming the knowledge of ancient books into clinical application.

An exploration of new methods for metabolic syndrome examination by infrared thermography and knowledge mining

Metabolic syndrome (MS) is a clinical syndrome with multiple metabolic disorders. As the diagnostic criteria for MS still lacking of imaging laboratory method, this study aimed to explore the differences between healthy people and MS patients through infrared thermography (IRT). However, the observation region of the IRT image is uncertain, and the research tried to solve this problem with the help of knowledge mining technology. 43 MS participants were randomly included through a cross-sectional method, and 43 healthy participants were recruited through number matching. The IRT image of each participant was segmented into the region of interest (ROI) through the preprocessing method proposed in this research, and then the ROI features were granulated by the K-means algorithm to generate the formal background, and finally, the two formal background were separately built into a knowledge graph through the knowledge mining method based on the attribute partial order structure. The baseline data shows that there is no difference in age, gender, and height between the two groups (P > 0.05). The image preprocessing method can segment the IRT image into 18 ROI. Through the K-means method, each group of data can be separately established with a 43 × 36 formal background and generated a knowledge graph. It can be found through knowledge mining and independent-samples T test that the average temperature and maximum temperature difference between the chest and face of the two groups are statistically different (P < 0.01). IRT could reflect the difference between healthy people and MS people. The measurement regions were found by the method of knowledge mining on the premise of unknown. The method proposed in this paper may add a new imaging method for MS laboratory examinations, and at the same time, through knowledge mining, it can also expand a new idea for clinical research of IRT.

Rule acquisition based on attribute partial order structure diagram

Rule acquisition based on attribute partial order structure diagram

Rule acquisition based on attribute partial order structure diagram

Rule acquisition based on attribute partial order structure diagram

Arms Race in Adversarial Malware Detection: A Survey

Malicious software (malware) is a major cyber threat that has to be tackled with Machine Learning (ML) techniques because millions of new malware examples are injected into cyberspace on a daily basis. However, ML is vulnerable to attacks known as adversarial examples. In this article, we survey and systematize the field of Adversarial Malware Detection (AMD) through the lens of a unified conceptual framework of assumptions, attacks, defenses, and security properties. This not only leads us to map attacks and defenses to partial order structures, but also allows us to clearly describe the attack-defense arms race in the AMD context. We draw a number of insights, including: knowing the defender’s feature set is critical to the success of transfer attacks; the effectiveness of practical evasion attacks largely depends on the attacker’s freedom in conducting manipulations in the problem space; knowing the attacker’s manipulation set is critical to the defender’s success; and the effectiveness of adversarial training depends on the defender’s capability in identifying the most powerful attack. We also discuss a number of future research directions.

Conceptual Coverage Driven by Essential Concepts: A Formal Concept Analysis Approach

Formal concept analysis (FCA) is a mathematical theory that is typically used as a knowledge representation method. The approach starts with an input binary relation specifying a set of objects and attributes, finds the natural groupings (formal concepts) described in the data, and then organizes the concepts in a partial order structure or concept (Galois) lattice. Unfortunately, the total number of concepts in this structure tends to grow exponentially as the size of the data increases. Therefore, there are numerous approaches for selecting a subset of concepts to provide full or partial coverage. In this paper, we rely on the battery of mathematical models offered by FCA to introduce a new greedy algorithm, called Concise, to compute minimal and meaningful subsets of concepts. Thanks to its theoretical properties, the Concise algorithm is shown to avoid the sluggishness of its competitors while offering the ability to mine both partial and full conceptual coverage of formal contexts. Furthermore, experiments on massive datasets also underscore the preservation of the quality of the mined formal concepts through interestingness measures agreed upon by the community.

Infinite decreasing chains in the Mitchell order

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length omega _1.

Incremental concept cognitive learning based on three-way partial order structure

With the vigorous development of the information technology industry, the information data available to mankind has shown an explosive growth trend. Dynamic concept learning is an approach that can effectively process the acquired massive data and extract valuable information from them. Concept cognitive learning (CCL) is a very active research direction in the field of dynamic concept learning, while partial order formal structure analysis (POFSA) is a concrete and practical model of CCL. However, the existing CCL algorithms in POFSA face some challenges when processing constantly changing data. Therefore, this paper is devoted to explore an incremental CCL algorithm based on three-way object partial order structure diagram (OPOSD) in POFSA with the incorporation of the thoughts of incremental learning. The features of five object categories are considered, and their incremental influences on three-way OPOSD are analyzed and their incremental CCL algorithms in three-way OPOSD are established. Based on some real famous formal contexts, this paper conducts numerical experiments, and the results show that the incremental CCL algorithm based on three-way OPOSD is consistent with human cognitive principles, and can improve the CCL performance of POFSA as well.

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.

Stochastic order on metric spaces and the ordered Kantorovich monad

In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. Our proof of antisymmetry of the stochastic order on these spaces is more general than previously known results in this direction.The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances.We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone affine maps. Considering the category of L-ordered metric spaces as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively.In the unordered case, we had identified the Wasserstein space as the colimit of the spaces of empirical distributions of finite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.

Study on Medication Regularity of Sijunzi Decoction and Its Categorized Prescriptions Based on Attribute Partial Order Structure Diagram

Sijunzi decoction is a famous prescription in traditional Chinese medicine (TCM) for the treatment of spleen deficiency syndrome. Spleen deficiency syndrome is manifested in a variety of symptoms. Herbs can be added or replaced on the basis of Sijunzi decoction according to different symptoms, forming Sijunzi categorized prescriptions. In this paper, the medication regularity of Sijunzi decoction and its categorized prescriptions was studied based on attribute partial order structure diagram (APOSD). According to prescription-herb APOSD generated from Sijunzi decoction categorized prescriptions screened from comprehensive TCM database, it can be seen that three herbs are used in all prescriptions, i.e. Renshen, Baizhu and Fuling. Different forms of Sijunzi decoction mainly depend on whether the forth herb is Zhigancao, Chenpi-Gancao, or Gancao. The efficacy of common herbs in the prescriptions for different symptoms can be presented by prescription-herb APOSD and prescription-symptom APOSD generated from different prescription subgroups. The experimental results show that APOSD can effectively reveal the medication regularity of Sijunzi decoction and its categorized prescriptions, and it is an useful tool for theoretical study of TCM.

Research on the regular pattern of Professor Saimei Li using Chinese medicine alone in treating middle-aged type 2 diabetes mellitus on the basis of partial order structure theory

The method of knowledge discovery based on partial order structure theory has been applied to the study of Traditional Chinese Medicine. In this research, the information of symptoms, prescriptions, and herbs from 50 effective cases of middle-aged type 2 Diabetes Mellitus (T2DM) patients treated by Professor Saimei Li with Chinese Medicine was processed with formal contexts, and then the corresponding partially ordered diagrams were obtained for the exploration of the medication rules. The results showed that the common symptoms of the middle-aged patients are mainly of heat syndrome. In regard of the treatment, the main prescription formula is Lizhong Wan combined with Gegen Qin Lian Tang for middle-aged patients. The combination of warm and cold formula aims at warming the spleen and dispelling the pathogenic wind, clearing pathogenic heat and eliminating dampness. This research can promote the inheritance and innovative development of Chinese medicine in an effective way.

A novel classification tree based on local minimum Gini index and attribute partial order structure diagram

A novel classification tree based on local minimum Gini index and attribute partial order structure diagram

A novel classification tree based on local minimum Gini index and attribute partial order structure diagram

Decision tree is not only an important machine learning method, but also the basis of ensemble learning methods such as random forest and deep forest. Based on the theory of Formal Concept Analysis (FCA) and Attribute Partial Order Structure Diagram (APOSD), a new decision tree for classification is proposed in this paper. Firstly, the local minimum of Gini index is used to complete the data granulation, and the Formal Decision Mode Information Table (FDMIT) is constructed. Then, the Attribute Partial Order Classification Tree (APOCT) is generated based on APOSD to complete the pattern recognition and rule extraction. The method of APOCT separates the process of granulation and visualisation, and the granulation process is easy to parallelise and efficient. The experimental results show that APOCT is effective.

Knowledge discovery for spleen yang deficiency syndrome based on attribute partial order structure diagram

Syndromes and medications in Traditional Chinese Medicine (TCM) have been studied by advanced information technology in the modernisation of TCM, promoting the development of knowledge discovery in TCM. In this paper, based on the method of Attribute Partial Order Structure Diagram (APOSD), syndrome-symptom APOSD and prescription-herb APOSD are constructed for spleen yang deficiency syndrome, which is a common syndrome in TCM. The common symptoms and specific symptoms of spleen yang deficiency syndrome are extracted from the syndrome-symptom APOSD. The association rules among herbs in the prescriptions for treating spleen yang deficiency syndrome are visualised on the prescription-herb APOSD, and herb pairs and herb groups are extracted. Compared with Apriori algorithm, APOSD not only obtains important association rules, but also shows the compositions of prescriptions in the diagram. APOSD provides a new scientific research method for TCM.