Basic Algebraic Operations Research Articles

A Hybrid Fuzzy Extension and Its Application in Multi-Attribute Decision Making

The hypersoft set theory is an extension of soft set theory. The complex non-linear diophantine fuzzy set is a hybrid fuzzy extension that serves as a generalization of the q-rung linear diophantine fuzzy set and the complex linear diophantine fuzzy set. In this paper, to tackle multi-sub-attributed real-world situations under complex non-linear diophantine fuzzy ambiance, the concept of complex q-rung linear diophantine fuzzy hypersoft set is proposed along with its score and accuracy function. Also, the idea of lattice ordered complex q-rung linear diophantine fuzzy hypersoft set is proposed in this paper, along with some of its basic algebraic operations. Furthermore, a highly effective algorithm using lattice ordered complex q-rung linear diophantine fuzzy hypersoft set is provided for handling multi-attributed decision-making issues exquisitely, along with an illustrative example in the field of vertical farming. Then, a comparative analysis between the proposed and current notions is provided to demonstrate the superiority and benefits of the suggested concepts over the current ones.

Interval-valued bipolar complex fuzzy soft sets and their applications in decision making

In this manuscript we demonstrate interval-valued bipolar complex fuzzy set (IVBCFS) and then interval-valued bipolar complex fuzzy soft set (IVBCFSS), as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set and soft set. We also initiate operational laws and basic results and properties for IVBCFS and IVBCFSS. Further explanation is given for the basic algebraic operations like complement, extended union, extended intersection, restricted union, and restricted intersection, AND product and OR product for IVBCFSS. Moreover, we demonstrate some fundamental aggregation operators like IVBCFS average aggregation, IVBCFS geometric and as well as their properties. To emphasize the usefulness and application of the system, we also develop the decision-making method and joint instances of the IVBCFSS (set-ups 1 and 2). In order to describe the effectiveness and influence of the approaching novel work, this study uses a comparative analysis of the new creating concept with prevailing ideas.

Algeblocks as a playful technique for the development of algebraic operations: a pedagogical experience with high school students

This article presents the Algeblocks technique as a playful tool that uses colored blocks to represent variables and algebraic operations. This technique is widely used to teach basic algebraic operations, such as factoring, remarkable products, and first-degree equations. The population consisted of 150 students, with a sample of 50 high school students. A pre-test was conducted as a diagnostic evaluation at the beginning of the year, and a post-test was applied as a summative evaluation at the end of the program. The results showed an increase in the student's grades but also revealed that the strategy caused motivation, enthusiasm, and high collaboration in the study group, confirming that learning was successfully achieved.

EXPLORING CHANGE AND ALIGNMENT; AN ACADEMIC AUDIT OF PRIMARY-LEVEL MATHEMATICS IN PUBLIC SCHOOLS OF PUNJAB SINCE 2000

Primary-level Mathematics curriculum experienced five revisions since 2000 that could not break status qou in porrly performance of the students in basic algebraic operations on numbers. Alignment among curriculum, textbook, and assessment leads enhancing performance of the students. The instant study examines the change in the curriculum and determines the alignment among curricula, textbooks, and assessments proposed therein restricted to the basic algebraic operations on numbers since 2000.
 This investigation explores cosmetic quantitative changes in the focused curricula but no qualitative change because the ultimate scope remained unaltered. However, curricula, textbooks, and assessments proposed therein are found strictly aligned by employing Surveys Enacted Modal devised by Porter. The cosmetic change transformed the curriculum into more structured form that facilitates the textbook developers, whereas; reduces the scope of multiple textbooks. Moreover, the curriculum is a document for the textbook developers and is reviseable subject to the paradigm shift in the discipline.

A Low-Overhead Auditing Protocol for Dynamic Cloud Storage Based on Algebra

With the widespread adoption of cloud storage, ensuring the integrity of outsourced data has become increasingly important. Various cloud storage auditing protocols based on public key cryptography have been proposed. However, all of them require complex cryptographic operations and incur significant storage and communication costs. To address the issues of significant storage overhead for data tags, high computational complexity of cryptographic algorithms, and limited efficiency of dynamic data algorithms in signature algorithm-based cloud storage outsourcing data integrity verification protocols, we propose a dynamic auditing protocol called AB-DPDP, which is based on algebra. Our protocol reduces the computational complexity of tag generation by utilizing basic algebraic operations instead of the traditional cryptographic method used in most current auditing protocols. To reduce storage overhead and protect private data, our protocol stores only tags, allowing for data to be restored through these tags, as opposed to storing both tags and data on the cloud server. To accommodate for more frequent and efficient data dynamics, we propose the dynamic index skip table data structure. Furthermore, the security of our proposed protocol is thoroughly proven based on the security definition of secure cloud storage. Finally, through theoretical analysis and experimental evaluation, we demonstrate the advantages of our scheme in terms of data privacy, storage overhead, communication overhead, computation overhead, and data dynamic efficiency.

Phase Demodulation Strategy Based on Kalman Filter for Sinusoidal Encoders

Phase demodulation strategy based on Kalman filter for sinusoidal encoders is proposed in this paper. For the state equation and observation equation in the constructed phase demodulation model, the phase angle, the phase angular speed, and the speed fluctuation are the system states, and the sine and cosine signals output by the encoder are the system observations. In addition, due to the nonlinear relationship between the observations and the system states, the observation function is linearized by the extended Kalman approach. In order to implement the Kalman filter on the hardware platform with a field programmable gate array (FPGA), the matrix model of the Kalman filter is further derived into an algebraic model, which only contains sine, cosine, and basic algebraic operations. Compared to the arctangent method, the proposed method has the noise suppression capability, and compared to the conventional phase-locked loop (PLL) method, the proposed method has the better dynamic performance without the acceleration-associated phase error. Both simulation and experimental results can prove the effectiveness of the proposed method.

On a Finitely Activated Terminal RNN Approach to Time-Variant Problem Solving.

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.

The Cost of Compositionality: A High-Performance Implementation of String Diagram Composition

String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less attention has been given to their algorithmic properties, and efficient implementations of diagrammatic reasoning are almost an unexplored subject. This work intends to be a contribution in such a direction. We introduce a data structure representing string diagrams in terms of adjacency matrices. This encoding has the key advantage of providing simple and efficient algorithms for composition and tensor product of diagrams. We demonstrate its effectiveness by showing that the complexity of the two operations is linear in the size of string diagrams. Also, as our approach is based on basic linear algebraic operations, we can take advantage of heavily optimised implementations, which we use to measure performances of string diagrammatic operations via several benchmarks.

Formula omitted]-Laplace transforms and [formula omitted]-differential equations of the arbitrary-order, theory and applications

This paper provides a framework to study a class of arbitrary-order uncertain differential equations known as arbitrary-order Z+-differential equations. For this purpose, we first present the parametric form of Z+-numbers. Then, we introduce the basic algebraic operations on Z+-numbers, including addition, scalar multiplication, and Hukuhara difference. Definitely, these operations lead us to define the Z+-valued function. Afterward, the limit and continuity concepts of a Z+-valued function are provided under the definition of a metric on the space of Z+-numbers. Furthermore, the concepts of Z+-differentiability, Z+-integral, and Z+-Laplace transform with the convergence theorem for the Z+-valued function and its nth-order derivatives are introduced in detail. Considering all these concepts, a Z+-differential equation (Z+DE) can be expressed in the form of a bimodal differential equation combining a fuzzy initial value problem (FIVP) and a random differential equation (RDE). To this end, we use a combination of “FIVP under strongly generalized Hukuhara differentiability (SGH-differentiability)” and “random differential equation under mean-square differentiability (ms-differentiability)” to define the nth-order differential equations with Z+-number initial values. Further, the existence and uniqueness of the Z+-differential equations are examined by presenting several theorems. Finally, the effectiveness of the approaches is illustrated by solving two examples.

An Efficient Identity-Based Provable Data Possession Protocol With Compressed Cloud Storage

Cloud storage is more and more prevalent in practice, and thus how to check its integrity becomes increasingly essential. A classical solution is identity-based (ID-based) provable data possession (PDP), which supports certificateless cloud storage auditing without entire user data. However, existing ID-PDP protocols always require that cloud users outsource data blocks, authenticators and a small-sized file tag to the cloud, and make use of the heavy elliptic curve cryptography over bilinear pairing. These disadvantages would result in vast storage, communication, and computation costs, which is unexpected, especially for resource-limited cloud users. To improve the performance, this paper proposes a novel cryptographic primitive: ID-based PDP with compressed cloud storage. In this model, cloud storage auditing can be achieved by using only encrypted data blocks in a self-verified way, and original data blocks can be reconstructed from the outsourced data. Thus, data owners no longer need to store original data blocks on the cloud. We also use some basic algebraic operations to realize a concrete ID-based PDP protocol with compressed cloud storage, which is quite efficient due to no heavy cryptographic operations involved. The proposed protocol can easily be extended to support the other practical functions by using the primitive replacement technique. The proposed protocol is strictly proven to have the properties of correctness, privacy, unforgeability and detectability. Finally, we give plenty of theoretical analysis and experimental results to validate the efficiency of the proposed protocol.

Faster generation of holographic video of 3-D scenes with a Fourier spectrum-based NLUT method.

In this article, a new type of Fourier spectrum-based novel look-up table (FS-NLUT) method is proposed for the faster generation of holographic video of three-dimensional (3-D) scenes. This proposed FS-NLUT method consists of principal frequency spectrums (PFSs) which are much smaller in size than the principal fringe patterns (PFPs) found in the conventional NLUT-based methods. This difference in size allows for the number of basic algebraic operations in the hologram generation process to be reduced significantly. In addition, the fully one-dimensional (1-D) calculation framework of the proposed method also allows for a significant reduction of overall hologram calculation time. In the experiments, the total number of basic algebraic operations needed for the proposed FS-NLUT method were found to be reduced by 81.23% when compared with that of the conventional 1-D NLUT method. In addition, the hologram calculation times of the proposed method, when implemented in the CPU and the GPU, were also found to be 60% and 66% faster than that of the conventional 1-D NLUT method, respectively. It was also confirmed that the proposed method implemented with two GPUs can generate a holographic video of a test 3-D scene in real-time (>24f/s).

On Neutrosophic Crisp Sets and Neutrosophic Crisp Mathematical Morphology

In this paper, we analyze the basic algebraic operations of neutrosophic crisp sets andtheir properties, show that some operations of neutrosophic crisp sets with type 2 and type 3 arenot closed by counter examples, and give some new operations. Then, discuss some morphologicaloperators in neutrosophic crisp mathematical morphology, and study the application to edgeextraction of color images, and give some Python programs and related experimental results.

A Compressive Integrity Auditing Protocol for Secure Cloud Storage

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.

Incorporating Variability in Lean Manufacturing: A Fuzzy Value Stream Mapping Approach

In this paper, value stream mapping (VSM) is integrated with fuzzy set theory to incorporate variability and uncertainty in the lean production system. VSM is one of the primary analytical tools for identifying waste and optimizing a production line. However, the standard VSM fails to consider the variability in manufacturing environments, which is, in fact, one of the root causes of waste. Therefore, this article proposes fuzzy VSM to overcome this weakness. Two alternative forms of fuzzy numbers, triangular fuzzy numbers (TFNs) and normal fuzzy numbers (NFNs), are applied, respectively, to depict time intervals, inventories, and other operating variables in VSM. An industrial case for assessing the validity of the proposed approaches is presented. Both approaches make it possible to incorporate and analyze variability in VSM and can be easily applied to industrial cases, as they only require basic algebraic operations. The obtained results are compared and the choice between TFNs and NFNs is discussed accordingly. A triangular fuzzy VSM tends to overestimate the variability of the process in complex production environment with complicated operational processes. However, it permits a more accurate description of variation in the environment where the optimistic and pessimistic values have very different variations from the core.

The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

Method with horizontal fuzzy numbers for solving real fuzzy linear systems

The paper presents a method for solving real fuzzy systems of linear equations with the usage of horizontal fuzzy numbers (HFNs). Based on the multidimensional RDM interval arithmetic and a fuzzy number in a parametric form, the definition of a horizontal fuzzy number with nonlinear left and right borders was given. Additionally, the paper presents the properties of basic algebraic operations on these numbers. The usage of horizontal fuzzy numbers with linear and nonlinear borders was illustrated in the examples with n times n fuzzy linear systems. The obtained results are multidimensional and satisfy the fuzzy linear systems. Calculated solutions of fuzzy linear systems were compared with the results of other methods. The solution obtained with horizontal fuzzy numbers satisfies any equivalent form of fuzzy linear system, whereas the results of existing methods do not satisfy the equivalent forms of the system. The presented examples also show that the method with HFN delivers a full multidimensional solution (direct solution). The analyzed results of standard methods, which are only a part of span (indicator, indirect solution) of a full solution and/or include values that do not satisfy the fuzzy linear systems, are underestimated or overestimated. The proposed method gives a possibility to obtain a crisp solution together with a crisp system of equations; other methods do not possess these properties.

Portable implementation model for CFD simulations. Application to hybrid CPU/GPU supercomputers

ABSTRACTNowadays, high performance computing (HPC) systems experience a disruptive moment with a variety of novel architectures and frameworks, without any clarity of which one is going to prevail. In this context, the portability of codes across different architectures is of major importance. This paper presents a portable implementation model based on an algebraic operational approach for direct numerical simulation (DNS) and large eddy simulation (LES) of incompressible turbulent flows using unstructured hybrid meshes. The strategy proposed consists in representing the whole time-integration algorithm using only three basic algebraic operations: sparse matrix–vector product, a linear combination of vectors and dot product. The main idea is based on decomposing the nonlinear operators into a concatenation of two SpMV operations. This provides high modularity and portability. An exhaustive analysis of the proposed implementation for hybrid CPU/GPU supercomputers has been conducted with tests using up to 128 GPUs. The main objective consists in understanding the challenges of implementing CFD codes on new architectures.

A Type-2 Fuzzy Relational Database Model

This paper introduces a type-2 fuzzy relational database model (T-2FRDB) as an extension of type-1 fuzzy relational database models with a full set of basic fuzzy relational algebraic operations that can represent and query uncertain and imprecise information in real world applications. In this model, the membership degree of tuples in a fuzzy relation is represented by fuzzy numbers on [0,1], and fuzzy relational algebraic operations are defined by using the extension principle for computing minimum and maximum values of such fuzzy numbers. Some properties of the type-2 fuzzy relational algebraic operations in T-2FRDB are also formulated and proven as extensions of their counterpart in the type-1 fuzzy relational database models. DOI: 10.32913/rd-ict.vol3.no14.352

A Flexible Data Structure for Heterogeneous Information Integration

A uniform mechanism to express and operate various data is essential for the development of big data and internet of things, and the common data model provides fundamental supports for the integration of heterogeneous data. By comparing with the advantages and disadvantages of several primary common data models, this paper proposes a new common data model called GriDoc Data Model (GDM). Through the introduction of concepts, such as Paragraph, Section, GriData and GriDoc, the model establishes a flexible mechanism to express different kinds of heterogeneous data. On the basis of the GDM model concept, this paper defines the basic algebraic operations and presents the process of integrating and querying GDM object data.

One or two circular shapes? A binary detection for electrical capacitance tomography sensors

AbstractImage reconstruction using engine electrical capacitance tomography sensors rises several drawbacks. Recently, the authors proposed a novel methodology dedicated to a fast detection of the configuration: size and position, of a single circular form using basic algebraic operations. However, this technique is efficient when a single form is present in the sensor cross‐section under interest. The present work proposes 2 different techniques that determine if the sensor cross‐section contains 1 or 2 circular shapes. Both methods are applied to simulated measurements. They operate much differently, and each comes with its own pros and cons. The first method considers that the sensor signal presents an axis of symmetry if a single circular object lies in the sensor. Otherwise, the symmetry is broken. It achieves a recognition rate of 73.9%. In the second method, machine‐learning techniques are used to perform a binary supervised classification. Promising rates of recognition over 99% are obtained. The present study also reveals that a 4‐electrode sensor leads to the best recognition rates with both methods. This result was also established by the authors in a different framework when dealing with physical limitations on spatial resolution in electrical capacitance tomography sensors.