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This article concerns with terminal recurrent neural network (RNN) models for time-variant computing, featuring finite-valued activation functions (AFs), and finite-time convergence of error variables. Terminal RNNs stand for specific models that admit terminal attractors, and the dynamics of each neuron retains finite-time convergence. The might-existing imperfection in solving time-variant problems, through theoretically examining the asymptotically convergent RNNs, is pointed out for which the finite-time-convergent models are most desirable. The existing AFs are summarized, and it is found that there is a lack of the AFs that take only finite values. A finitely valued terminal RNN, among others, is taken into account, which involves only basic algebraic operations and taking roots. The proposed terminal RNN model is used to solve the time-variant problems undertaken, including the time-variant quadratic programming and motion planning of redundant manipulators. The numerical results are presented to demonstrate effectiveness of the proposed neural network, by which the convergence rate is comparable with that of the existing power-rate RNN.

Although there have been many fuzzy or probabilistic relational database models proposed for representing and handling imprecise and uncertain information of objects in real-world applications, models combining the relevance and strength of both fuzzy set theory and probability theory appear sporadic. In this paper, we propose a new fuzzy and probabilistic relational database model where the imprecision of an attribute value is represented by a fuzzy set and the uncertainty of a relational tuple is represented by a probability interval. The mass assignment theory is employed to deal with the challenge of integration and computation of both fuzzy sets and probabilities in the same model. The conjunction and disjunction strategies to combine imprecise and uncertain information are introduced. Then the fundamental concepts of the classical relational database model are extended and generalized in this new model. The syntax and semantics of the selection operation are formally defined. Finally, the other important algebraic operations on imprecise attributes and uncertain tuples are developed.

This work focuses on the robust leader-follower synchronization navigation for the under-actuated vessels with only independent the surge control force and the yaw moment. The ocean disturbances and thruster saturation are simultaneously addressed. A virtual vessel placed at the relative distance with the moving main vessel alleviates the requirements on main vessel's velocity measurements. The disturbance estimation performance is achieved by utilizing the disturbance observer. The additional controls produce the filtered non-achievable command control signals due to the thrust saturation effects. The filtered signals correct feedback errors for avoiding disturbance estimation and rejection compromise. The dynamic surface control technique is employed to derive the under-actuated synchronization control laws. By employing first-order filters, the differential terms are replaced by the algebraic operations such that the control is convenient to implement. It is analyzed that the supply vessel control system is semi-globally stable and the synchronization navigation along with the main vessel is ensured with the small errors. Simulations in the different disturbance cases validate the synchronization scheme.

This paper proposes an optimization model for the joint planning of sectionalizing switches (SSs), including remote-controlled switches (RCSs) and manual switches (MSs), and tie lines among distribution feeders. Considering a flexible installation of SSs and tie lines (i.e., deployment of SSs on both sides of a line and construction of external and internal tie lines on of a feeder), this model is conducive to developing an economic investment scheme while preserving the expected energy not supplied (EENS) reliability criterion. First, the minimum total cost of investment, maintenance and penalty for EENS is chosen as the objective function. Next, a route analysis method is introduced to analyze particular performances of different deployment schemes for RCSs, MSs, external tie lines and internal tie lines. Then, the joint planning problem is formulated as a mixed-integer linear programming (MILP) model with the aid of algebraic operations to calculate the global optimal solution. The effectiveness of the proposed model is verified for several cases in a 54-node distribution system. Numerical results indicate that the proposed planning solution merits higher economy and lower EENS as compared to the schemes offered by other prevalent methods

In this study, trigonometric and hyperbolic type traveling wave solutions are produced by using the sub-equation analytical method by taking into account the Kerr Law properties of the equation defining nematic liquid crystals. The resulting traveling wave solutions of the equation play an important role in the energy transport in soliton molecules in liquid crystals. In addition, the solitary wave behaviors obtained for different values of the constants in the produced traveling wave solutions are discussed. The hyperbolic type traveling wave solutions of the equation defining the nematic liquid crystals incorporating Kerr Law property is represented as dark and singular solitons. Ready-made package programs are used for algebraic operations and graphic drawings. It is emphasized that the analytical method is effective, useful and valid.

The current remote sensing (RS) open data policy for multispectral (MS) missions such as Sentinel-2 and Landsat-8, together with the availability of free cloud distributed processing platforms such as Google Earth Engine, makes it possible the quick generation of burned area (BA) products even for nonexperts in the field. Indeed, fires and BAs can be detected using burn severity indices, which are usually obtained by simple band algebra operations. However, simple approaches can aid BA estimation only if typical error patterns are known and accounted for, especially when working at large (e.g., continental) scales. This article proposes an automatic BA detection system based on burn severity index thresholding, which integrates dedicated false and missed alarm mitigation strategies to improve the detection accuracy. The system is tested on Sentinel-2 and Landsat-8 data over ten different locations in Europe and spanning year 2018. Three known burn severity indices plus a custom one defined to improve the performance in the considered study area are under study. Results show that burned index thresholding is possible within accuracy bounds slightly larger than the state of the art, which is acceptable by considering the proposed simplified processing framework.

Pythagorean fuzzy soft set (PFSS) is the most powerful and effective extension of Pythagorean fuzzy sets (PFS) which deals with the parametrized values of the alternatives. It is also a generalization of intuitionistic fuzzy soft set (IFSS) which provides us better and precise information in the decision-making process comparative to IFSS. The core objective of this work is to construct some algebraic operations for PFSS such as OR-operation, AND-operation, and necessity and possibility operations. Furthermore, some fundamental properties have been established for PFSS utilizing the developed operations. Moreover, a decision-making technique has been offered for PFSS based on a score matrix. To demonstrate the validity of the proposed approach, a numerical example has been presented. Finally, to ensure the practicality of the established approach, a comprehensive comparative analysis has been presented. The obtained results show that our developed approach is most effective and delivers better information comparative to prevailing techniques.

This study aims to determine the effect of the Problem-Based Learning (PBL) learning model with algebraic box media on students' problem-solving abilities. This research is quantitative research conducted in Class VII at one of the Madrasa Tsanawiyah in Kediri. The research design used a one-group pretest-posttest design. The sample used was 20 students who were taken by a simple random sampling technique. Mathematical problem-solving ability tests in the form of essays are used in the pretest and posttest. The results showed that the pretest average value was 55.50 while the posttest score was 70.50. Inferential, the sig value is obtained. (2-tailed) is equal to 0.000 which is smaller than the significance value of 0.05, so it is concluded that the PBL learning model using algebraic box media affects student problem-solving abilities. Abstark: Penelitian ini bertujuan untuk mengetahui pengaruh model pembelajaran Problem-Based Learning (PBL) dengan media kotak aljabar terhadap kemampuan pemecahan masalah siswa. Penelitian ini adalah penelitian kuantitatif yang dilakukan di Kelas VII di salah satu Madrasa Tsanawiyah di Kota Kediri. Desain penelitian menggunakan one–group pretest-posttest design. Sampel yang digunakan sebanyak 20 siswa yang diambil dengan teknik simple random sampling. Tes kemampuan pemecahan masalah matematis berbentuk essay digunakan pada pretes dan posttes. Hasil penelitian menunjukkan bahwa rata-rata nilai pretest sebesar 55,50 sedangkan posttest sebesar 70,50. Secara inferensial, diperoleh nilai sig. (2-tailed) adalah 0,000 yang lebih kecil dari nilai signifikansi 0,05 sehingga disimpulkan model pembelajaran PBL menggunakan media kotak aljabar berpengaruh terhadap pemecahan masalah siswa.

It is shown that the set of finite multisets can be identified with the space of symmetric polynomials on a linear space of finite sequences whose completion in a suitable metric gives a topologization of the set of multisets as a subspace of the space of symmetric analytic functions on l1. As one of possible topologizations, we can mention the topology of Hilbert space. Moreover, we consider some algebraic operations on the set of multisets, in particular, the analogs of fractional mappings.

This paper presents a system of cyclic tensor algebra for operations involving fourth-order tensors. The advantages are that the system is objectively and consistently defined in three ways that each fall into one of three universal classes. Operators within a given class are called conjugate operators such that many familiar and fundamental identities of scalars and second-order tensors are maintained in fourth order; this provides greater insight along with anthropological and pedagogical advantages over current systems, while also revealing new identities and solutions. The relationship between operators of a different class is such that a property of cyclic symmetry arises whereby mixed-class product operators can be cycled around without invalidating an equation. In defining this system, we have considered the following: preservation from identities in zeroth- (scalar) and second-order tensor identities to fourth-order tensor identities; possible permutations of definitions and subsequent logical restrictions; the visual notational consistency throughout the system; and maintenance to legacy definitions and operator symbols. Additionally, we present many new and useful algebraic identities and provide a comparison to some selected contemporary systems used in the literature. We also provide, to complete at least a basic exposition of our proposed system, a set of identities for matrix-equivalent operations, which facilitate programming for numerical computing. This article is designed to be used as a reference work for anyone choosing to adopt this system of tensor operations in continuum mechanics theory involving fourth-order tensors.

In this paper, we propose six methods for binarization of the swallow swarm optimization (SSO) algorithm to solve the feature selection problem. The relevance of the selected feature subsets is estimated by two classifiers: a fuzzy rule-based classifier and a classifier based on k-nearest neighbors. To find an optimal subset of features, we take into account the number of features and classification accuracy. The developed algorithms are tested on datasets from the KEEL repository. For the statistical evaluation of the binarization methods, we use Friedman’s two-way analysis of variance by ranks for related samples. The best feature selection result is shown by a hybrid method based on modified algebraic operations and MERGE operation introduced by the authors of this paper. The best classification accuracy is achieved with a V-shaped transfer function.

Counting CSPd is the counting constraint satisfaction problem (# CSP in short) restricted to the instances where every variable occurs a multiple of d times. This paper revisits tractable structures in # CSP and gives a complexity classification theorem for # CSPd with algebraic complex weights. The result unifies affine functions (stabilizer states in quantum information theory) and related variants such as the local affine functions, the discovery of which leads to all the recent progress on the complexity of Holant problems. The Holant is a framework that generalizes counting CSP. In the literature on Holant problems, weighted constraints are often expressed as tensors (vectors) such that projections and linear transformations help analyze the structure. This paper gives an example showing that different classes of constraints distinguished by these algebraic operations may share the same closure property.

The t-norm fuzzy sets is a class of fuzzy sets equipped with a pair of algebraic operation, a notion of T-convexity and a metric, where all these notions are based on a strict triangular norm T. In this paper we introduce in this class a notion of T-convex hull of a fuzzy set. We prove theorem that binds the T-convex hull of an upper semicontinuous fuzzy set with the convex hull of a (crisp) set. We further show two applications of this result. First, we prove that the operation of forming T-convex hull behaves well with respect to algebraic operations. Second, we show an analogue of Shapley-Folkman theorem. Shapley-Folkman theorem is a well-known result that provides an upper bound on the distance between Minkowski sum of sets and the convex hull of this sum. In this paper we show that the distance between the sum of upper semicontinuous fuzzy subsets of a finite dimensional Euclidean space and the T-convex hull of this sum has an upper bound. As a consequence of this result, we present an iterative procedure for forming T-convex hull of an upper semicontinuous fuzzy set. As an application of the results regarding t-norm fuzzy sets we show an example from a game theoretic setting, where t-norm fuzzy sets were used to handle uncertainty in a repeated two player game.

A sheer number of techniques and web resources are available for software engineering practice and this number continues to grow. Discovering semantically similar or related technical terms and web resources offers the opportunity to design appealing services to facilitate information retrieval and information discovery. In this study, we extract technical terms and web resources from a community of question and answer (Q&A) discussions and propose an approach based on a neural language model to learn the semantic representations of technical terms and web resources in a joint low-dimensional vector space. Our approach maps technical terms and web resources to a semantic vector space based only on the surrounding technical terms and web resources of a technical term (or web resource) in a discussion thread, without the need for mining the text content of the discussion. We apply our approach to Stack Overflow data dump of March 2018. Through both quantitative and qualitative analyses in the clustering, search, and semantic reasoning tasks, we show that the learnt technical-term and web-resource vector representations can capture the semantic relatedness of technical terms and web resources, and they can be exploited to support various search and semantic reasoning tasks, by means of simple K-nearest neighbor search and simple algebraic operations on the learnt vector representations in the embedding space.

This research is a quantitative descriptive study that aims to get a picture of students' ability to change story problems into algebraic forms. This study uses mathematical modeling indicators to change story problems into algebraic forms.The subjects of this study were students of SMP Negeri 33 Palembang. The learning process is adapted to the principles and phases of problem based learning m0odel. The data collection technique used in this study was written test consisted of three essay question and interview needed to obtain supporting data. The results of this study are there are nine students who have a good category, eight students have a very good category, enough categories as many as three students, less categories are as many as two students and students who have very poor category abilities are four students. And also seen at the time of work on LKPD where the average ability of students is able to change story problems into algebraic operations well.

With the widespread application of cloud storage, ensuring the integrity of user outsourced data catches more and more attention. To remotely check the integrity of cloud storage, plenty of protocols have been proposed, implemented by checking the equation constructed by the aggregated blocks, tags, and indices. However, the verifier only has the knowledge of the indices of the audited blocks and tags, which thus requires the cloud to store both data blocks and tags for integrity verification. In this article, we present a compressive secure cloud storage protocol inspired by Goldreich-Goldwasser-Halevi (GGH) cryptosystem. Since the aggregated blocks can be reconstructed from the aggregated tags without the help of data indices, the cloud can only store data tags for providing the verifiable integrity proof. In this way, communication and storage costs can be hugely reduced and user private information can be hidden from the cloud. Furthermore, the proposed protocol only contains a few basic algebraic operations, making it highly efficient. We also provide formal security proof of the proposed protocol regarding forge, replay and replace attacks. In addition, we explore a new technique to support data dynamics. Furthermore, we establish a generic framework of compressive secure cloud storage protocols. Finally, we provide the theoretical analysis and experimental results, which further validate the effectiveness of the proposed protocol.

There has been a rise in the popularity of algebraic methods for graph algorithms given the development of the GraphBLAS library and other sparse matrix methods. An exemplar for these approaches is Breadth-First Search (BFS). The algebraic BFS algorithm is simply a recurrence of matrix-vector multiplications with the $n \times n$ adjacency matrix, but the many redundant operations over nonzeros ultimately lead to suboptimal performance. Therefore an optimal algebraic BFS should be of keen interest especially if it is easily integrated with existing matrix methods. Current methods, notably in the GraphBLAS, use a Sparse Matrix masked-Sparse Vector (SpMmSpV) multiplication in which the input vector is kept in a sparse representation in each step of the BFS, and nonzeros in the vector are masked in subsequent steps. This has been an area of recent research in GraphBLAS and other libraries. While in theory these masking methods are asymptotically optimal on sparse graphs, many add work that leads to suboptimal runtime. We give a new optimal, algebraic BFS for sparse graphs, thus closing a gap in the literature. Our method multiplies progressively smaller submatrices of the adjacency matrix at each step. Let $n$ and $m$ refer to the number of vertices and edges, respectively. On a sparse graph, our method takes $O(n)$ algebraic operations as opposed to $O(m)$ operations needed by theoretically optimal sparse matrix approaches. Thus for sparse graphs it matches the bounds of the best-known sequential algorithm and on a Parallel Random Access Machine (PRAM) it is work-optimal. Our result holds for both directed and undirected graphs. Compared to a leading GraphBLAS library our method achieves up to 24x faster sequential time and for parallel computation it can be 17x faster on large graphs and 12x faster on large-diameter graphs., Will appear in ACM Transactions on Knowledge Discovery from Data, Vol. 15, No. 5, 2021

The problem of general non-linear stochastic optimal control with small Wiener noise is studied. The problem is approximated by a Markov Decision Process. Bellman Equation is solved using Value Iteration (VI) algorithm in the low rank Tensor Train format (TT-VI). In this paper a modification of the TT-VI algorithm called TT-Q-Iteration (TT-QI) is proposed by authors. In it, the nonlinear Bellman Optimality Operator is iteratively applied to the solution as a composition of internal Tensor Train algebraic operations and TT-CROSS algorithm. We show that it has lower asymptotic complexity per iteration than the method existing in the literature, provided that TT-ranks of transition probabilities are small. In test examples of an underpowered inverted pendulum and Dubins cars our method shows up to 3–10 times faster convergence in terms of wall clock time compared with the original method.

This study proposes a problem-oriented methodology, using the algebraic operations proposed in the Quality Function Deployment's house-of-quality, to prioritise the integration of Industry 4.0 (I4....

This paper proposes a technique to identify nonlinear dynamical systems with time delay. The sparse optimization algorithm is extended to nonlinear systems with time delay. The proposed algorithm combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We further integrate the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. We use Taylor expansion to parameterize time delay. The proposed algorithm in this paper is computationally efficient and robust to noise. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type , while the underlying derivation’s weight gives a lower bound to the term’s expected number of steps to normal form. Noticeably, both approximations are tight—not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that.

We consider transformations of random variables on finite sets by algebraic operations. A system of operations is said to be approximation complete if any random variable may be approximated with arbitrary precision by applying the given operations to mutually independent identically distributed random variables whose distributions have no zero components. We establish some necessary conditions for a function system to be approximation complete and construct examples of approximation incomplete systems.