A Knot Invariant Defined Based on the Skein Relation with Two Equations

In this paper the classification of n-dimensional knots in S"+2, bounding /--connected manifolds, where 3r>n + l>6, in terms of stable homotopy theory is suggested.The problem of isotopy classification is the fundamental one of knot theory.It was solved by A. Haefliger and J. Levine for knots of codimension > 3. The first step in reduction of a classification of knots of codimension two to a homotopy problem was made by R. Lashof and J. Shaneson [6].They showed that in the class of -dimensional knots with group Z for n > 5 there are at most two different knots having homotopy equivalent exteriors.In 1970 J. Levine [10] gave an algebraic classification of (2q -l)-dimensional knots in S2q+X bounding (q -1)connected manifolds.It turned out that the only invariant determining the isotopy type of such knots is the Seifert matrix considered to within 5-equivalence.Later Trotter [16] and C. Kearton [5] obtained a classification of knots, studied by J. Levine, in terms of Blanchfield pairing.C. Kearton has partially analyzed the more difficult problem of classification of 2^r-dimensional knots in S2q+2 bounding (q -l)-connected manifolds.In the present paper the classification of a wider class of higher-dimensional knots is obtained.It is the class of -dimensional knots in S"+2 bounding r-connected manifolds, where 3r > n + 1 > 6.The main result of this paper is the construction of a one-to-one correspondence between the set of isotopy types of such knots and some set described in purely homotopic terms.The plan of the paper is as follows:In 1 submanifolds of the sphere of codimension one are considered.Such a submanifold is assigned some pairing in the sense of the Spanier-Whitehead theory.This pairing describes the homotopy linking of the submanifold with its copy translated in the direction of the positive normal field.This pairing induces the usual Seifert pairing on middle dimensional homology groups and is called the homotopy Seifert pairing.The main result of 1 is the construction of a one-to-one correspondence between the set of isotopy types of such submanifolds and the set of homotopy pairings.

Using Yoshikawa's surface diagram, we constructed new invariants of ambient isotopy classes of smoothly embedded closed surfaces in ℝ4 via a state-sum model similar to the Kauffman's state-sum model for the Jones polynomial for classical knots and links in ℝ3. It is shown that the invariants can also be defined by skein relation and thus they are calculated from Yoshikawa's surface diagrams recurrently. Some of the properties of the invariants are given and explicit computations for several surfaces are included.

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander–Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n invariant.

Finite-type invariants of classical and virtual knots

Fuzzy classification is one of the most important applications of fuzzy logic. Its goal is to find a set of fuzzy rules which describe classification problems. Most of the existing fuzzy rule induction methods (e.g., the fuzzy decision trees induction method) focus on searching rules consisting of t-norms (i.e., AND) only, but not t-conorms (OR) explicitly. This may lead to the omission of generating important rules which involve t-conorms explicitly. This paper proposes a type of tree termed pattern trees which make use of different aggregations including both t-norms and t-conorms. Like decision trees, pattern trees are an effective machine learning tool for classification applications. This paper discusses the difference between decision trees and pattern trees, and also shows that the subsethood based method (SBM) and the weighted subsethood based method (WSBM) are two specific cases of pattern trees, with each having a fixed pattern tree structure.

Different from the Mittag–Leffler stability or asymptotic stability, the exponential stability issue, which provides faster and explicit convergence rate, is studied in this paper for fractional‐order uncertain systems with asynchronous switching and impulses, where the impulsive functions rely on not only switching modes but also impulsive time. Instead of using the inequality , by utilizing the theory of fractional‐order differential equations, the methods of Lyapunov function and mathematical induction, some novel and less‐conservative stability criteria are developed, respectively, for the case of switched stable subsystems or switched unstable subsystems. The obtained results build a tradeoff between impulsive function, impulsive interval, average dwell time and fractional order. In addition, our results with are also novel in contrast with the ones of integer‐order switched impulsive systems. Finally, five numerical examples are given to show the effectiveness of theoretical results.

Exponential synchronization of directed bipartite networks with node delays and hybrid coupling via impulsive pinning control

This paper deals with the stability problems of nonautonomous impulsive differential systems with time-delay. By utilizing the method of mathematic induction, proofs by contradiction and Lyapunov-Krasovskii functional, we obtain several uniformly stable criteria about the linear and nonlinear nonautonomous impulsive systems with time delay. One numerical example and its simulations are given to illustrate the effectiveness of the theoretical result.

In this paper, the stability of a class of impulsive control differential systems with variable delays is considered by using the method of mathematic induction, Lyapunov functions and Razumikhn-type techniques. Several criteria of uniform stability and global exponential stability are derived for linear and nonlinear differential systems, respectively. It is shown that the novelty stability conditions are the generalizations of some published results. Two numerical examples and their simulations are given to illustrate the effectiveness of the theoretical results.

Effective root cause analysis (RCA) of performance issues in modern cloud environ- ments remains a hard problem. Traditional RCA tracks complex issues by their signatures known as problem incidents. Common approaches to incident discovery rely mainly on expertise of users who define environment-specific set of alerts and >target detection of problems through their occurrence in the monitoring system. Adequately modeling of all possible problem patterns for nowadays extremely sophisticated data center applications is a very complex task. It may result in alert/event storms including large numbers of non-indicative precautions. Thus, the crucial task for the incident-based RCA is reduction of redundant recommendations by prioritizing those events subject to importance/impact criteria or by deriving their meaningful groupings into separable situations. In this paper, we consider automation of incident discovery based on rule induction algorithms that retrieve conditions directly from monitoring datasets without consuming the sys- tem events. Rule-learning algorithms are very flexible and powerful for many regression and classification problems, with high-level explainability. Since annotated or labeled data sets are mostly unavailable in this area of technology, we discuss data self-labelling principles which allow transforming originally unsupervised learning tasks into classification problems with further application of rule induction methods to incident detection.

Classification situations are ubiquitous and arise in a range of practical applications. Very often classification problems do not occur in isolation but in clusters where several classification problems need to be solved. Examples of these include employee contract renewal in large organizations, patient prognosis in hospitals, and crowdsourced problem solving. With any form of classification, however, there are invariably inaccuracies occurring in different forms, especially when multiple classification tasks need to be performed. Since typical classifiers are not free from errors, classification errors tend to accumulate, and having frequent misclassifications are often unacceptable. Moreover, in unsupervised learning situations, there are typically no pre-determined ground truth classes; in such a situation the ground truth class is determined by the view of the majority of classifiers. In this paper, we study the situation of multiple classification within the Naive Bayes framework, where the ground truth is determined by the decision of most classifiers, and where there is finite resources requiring decisions to be made within limited budget. Here, we represent the classification tasks as a random walk process, and perform a stochastic analysis of the situation and predict the occurrence of erroneous classification. We obtained closed-form solutions for the probability of classification inaccuracies and derive useful error bounds as a function of the budget. We find that by raising the budget, the probability of error in classification can be controlled, and the extent of the reduction can be quantified. Experimental measurements are also performed which show good agreement with the theoretical results.

In this article we develop a class of stochastic boosting (SB) algorithms, which build upon the work of Holmes and Pintore (Bayesian Stat. 8, Oxford University Press, Oxford, 2007). They introduce boosting algorithms which correspond to standard boosting (e.g. Buhlmann and Hothorn, Stat. Sci. 22:477---505, 2007) except that the optimization algorithms are randomized; this idea is placed within a Bayesian framework. We show that the inferential procedure in Holmes and Pintore (Bayesian Stat. 8, Oxford University Press, Oxford, 2007) is incorrect and further develop interpretational, computational and theoretical results which allow one to assess SB's potential for classification and regression problems. To use SB, sequential Monte Carlo (SMC) methods are applied. As a result, it is found that SB can provide better predictions for classification problems than the corresponding boosting algorithm. A theoretical result is also given, which shows that the predictions of SB are not significantly worse than boosting, when the latter provides the best prediction. We also investigate the method on a real case study from machine learning.

Time-domain electromagnetic induction method including on-time and off-time illumination is widely developed in unexploded ordnance (UXO) detection. The secondary field response detected in off-time is almost ineffective for targets with small size and great depth due to fast signal attenuation and low signal-to-noise ratio at late stage. In this paper, the on-time response for UXO detection with TEM system is discussed in detail. Firstly, the on-time response of a sphere is calculated theoretically and compared with the off-time response. Secondly, the transmitting circuit with four NMOS is designed to generate desired transmitting current to measure on-time response effectively. Finally, the on-time and off-time response of several targets are measured simultaneously. Results show that the late signal-to-noise ratio of on-time response is much higher and the attenuation of target with small size and iron pipe is slower than the off-time response, which is consistent with the theoretical results.

This dissertation explores the intersection of data fusion, federated learning, and Bayesian methods, with a focus on their applications in indoor localization, GNSS, and image processing. Data fusion involves integrating data and knowledge from multiple sources. It becomes essential when data is only available in a distributed fashion or when different sensors are used to infer a quantity of interest. Data fusion typically includes raw data fusion, feature fusion, and decision fusion. In this thesis, we will concentrate on feature fusion. Distributed data fusion involves merging sensor data from different sources to estimate an unknown process. Bayesian framework is often used because it can provide an optimal and explainable feature by preserving the full distribution of the unknown given the data, called posterior, over the estimated process at each agent. This allows for easy and recursive merging of sensor data with prior knowledge without the need for storage. However, there are challenges in data fusion, such as the multiple counting problem, which occurs when data is utilized numerous times without the user's knowledge. In a Bayesian setting, a priori information of the unknown quantities is available and can be present among different distributed estimators. When the local estimates are fused, the prior knowledge used to construct several local posteriors might be overused unless the fusion node accounts for this and corrects it. In this thesis, we analyze the effects of shared priors in Bayesian data fusion contexts. Our analysis helps to understand the performance behavior as a function of the number of collaborative agents and different types of priors, depending on different common fusion rules. To perform this analysis, we use two divergences that are common in Bayesian inference. The generality of the results allows us to analyze very generic distributions. These theoretical results are corroborated through experiments in a variety of estimation and classification problems, including linear and nonlinear models, and federated learning schemes in a Bayesian perspective. Federated Learning is a discipline of Distributed Data Fusion that can be used for model training on local data without exchanging the data itself. This technique is useful in scenarios like image classification, indoor positioning, and jammer signal classification, where privacy concerns make data crowdsourcing difficult. In this thesis, we explore FL in two specific applications: Jammer signal and indoor positioning. Jamming signals can disrupt the operation of GNSS receivers, making jamming mitigation and localization techniques essential. Jammer classification can help with these techniques. For indoor positioning, local data points are collected to create a local fingerprinting database. Data-driven machine learning methods are used to predict the location based on this model learned without data sharing. However, Federated Learning faces several challenges, including the non-IID (non-identically distributed) problem. This problem occurs when the data distribution varies across different clients, which can lead to a decrease in performance. In this thesis, we will discuss various FL frameworks and algorithms that can help address this problem. A FL method from Bayesian perspective are proposed to deal with the problem in indoor positioning problem. Additionally, we will delve into the concepts of personalized and clustered Federated Learning. The personalized approach tailors the learning process to individual clients, which enhances the system's performance by leveraging the unique data and characteristics of each client. We will discuss the application of this approach in indoor positioning and jammer signal classification, highlighting its potential improvements and challenges. Expanding on the personalized approach, we introduce and discuss clustered Federated Learning. This learning method groups clients with similar characteristics or data patterns. Each client is typically associated with one data distribution and participates in training a model for that distribution along with their cluster peers. We propose a Bayesian framework for clustered FL, which associates clients to clusters, along with several practical algorithms for efficient learning, considering the trade-offs between performance and computational complexity. In conclusion, this thesis provides a detailed investigation and evaluation of the integration of data fusion, federated learning, and Bayesian methods. The insights and conversations presented in this study are anticipated to enhance the progress and implementation of these technologies in various domains, particularly in indoor positioning, and GNSS.--Author's abstract

Multi-set query is a fundamental problem in computer systems and applications. Most traditional solutions for multi-set query are based on hash tables or bloom filters. However, when the sizes of multi-sets are large, these solutions cannot achieve small memory usage, fast query speed and high accuracy at the same time. In this work, we study the problem of using a learned oracle to improve the performance of traditional multi-set query processing empirically. The key idea is to train an oracle to predict which set contains a query item e as a classification problem. To ensure an exact query result, we combine the learned oracle with a standard bloom filter and an exact-match index to catch items that are not correctly identified by the oracle. When the oracle is both small and efficient, the whole query performance can be improved. In our framework, we treat the learned oracle as a complete black box, and is not dependent on its inner workings. Theoretical proofs and experimental results show that compared to the state-of-the-art, the error rate of our approach is 0% even with much less memory usage and a comparable speed.