Logarithmic Operators Research Articles

Prospects for the Use of Quasi-Mersen Numbers in the Design of Parallel-Serial Processors

It is shown that a serial-parallel processor, comparable in bit capacity to a 16-bit binary processor, can be implemented based on an algorithm built on the residue number system, a distinctive feature of which is the use of the first four quasi-Mersenne numbers, i.e., prime numbers representable as pk=2k+1, k=1,2,3,4. Such a set of prime numbers satisfies the criterion 2p1p2p3p4+1=P, where P is also a prime number. Fulfillment of this criterion ensures the possibility of convenient use of the considered RNS for calculating partial convolutions developed for the convenience of using convolutional neural networks. It is shown that the processor of the proposed type can be based on the use of a set of adders modulo a quasi-Mersenne number, each of which operates independently. A circuit of a modulo 2k+1 adder is proposed, which can be called a trigger circuit, since its peculiarity is the existence (at certain values of the summed quantities) of two stable states. The advantage of such a circuit, compared to known analogs, is the simplicity of the design. Possibilities for further development of the proposed approach related to the use of the digital logarithm operation, which allows reducing the operations of multiplication modulo 2k+1 to addition operations, are discussed.

Different algebraic structures and their properties setting logarithm operator applied to extended neutrosophic interval-valued set

We present the neutrosophic interval-valued set applied to the q-rung logarithmic operator (q-RLOANIVS). One might develop a q-rung neutrosophic interval-valued set by extending the Pythagorean interval-valued fuzzy set (PIVFS) and neutrosophic set (NS). We discuss the q-Rlogarithimic operator applied neutrosophic interval-valued weighted averaging (q-RLOANIVWA), q-Rlogarithimic operator applied neutrosophic intervalvalued weighted geometric (q-RLOANIVWG), extended q-Rlogarithimic operator applied neutrosophic intervalvalued weighted averaging (q-RELOANIVWA) and extended q-Rlogarithimic operator applied neutrosophic interval-valued weighted geometric (q-RELOANIVWG). Several algebraic attributes have been established, including distributivity, idempotency, and associativity of q-RLOANIVSs.

Rate Optimization of Intelligent Reflecting Surface-Assisted Coal Mine Wireless Communication Systems.

This paper proposes a three-step joint rate optimization method for intelligent reflecting surface (IRS)-assisted coal mine wireless communication systems. Different from terrestrial IRS-assisted communication scenarios, in coal mines, IRSs can be installed flexibly on the tops of rectangular tunnels to address the issues of signals being blocked and interfered with by mining equipment. Therefore, it is necessary to optimize the IRS deployment position, the transmit power and IRS phase shifts to achieve the maximum effective achievable rate at user stations equipped with the proposed system. However, due to the complex channel models of coal mines, the optimization problem of IRS deployment position is non-convex. To solve this problem, two auxiliary variables along with logarithmic operations and Taylor approximation are introduced. On this basis, a three-step joint rate optimization involving the transmit power, IRS phase shifts and IRS deployment position is proposed to maximize the effective achievable rates at the user station. The simulation results show that compared with other rate optimization schemes, the effective achievable rates at the user station using the proposed joint rate optimization scheme can be improved by approximately 12.32% to 54.17% for different parameter configurations. It is also pointed out that the deployment position of the IRS can converge to the same optimal position independent of the initial deployment position. Moreover, we investigate the effects of the roughness of the tunnel walls in a coal mine on the effective achievable rates at the user station, and the simulation results indicate that the proposed three-step joint rate optimization scheme performs better in the coal mine scenario regardless of the roughness.

Self-supervised image change detection method based on lightweight capsule network

In response to the significant impact of speckle noise on the accuracy of SAR image change detection, the high complexity of existing capsule network based image change detection methods, and the loss of a large amount of original image information in training samples, this paper proposes a self supervised image change detection method based on the Light Capsule Network (SCapsNet). First, the log ratio operator difference graph is generated, and the "pseudo label" of training samples with high confidence is obtained through the maximum inter class variance method and fuzzy C-means clustering method, which lays the foundation for self-supervised learning; Secondly, construct a three channel training sample based on the difference map of two temporal SAR images and logarithmic ratio operators to maximize the preservation of sample information; Then, SCapsNet is designed to extract training sample features through single scale convolution, and a single scale capsule network is used to mine spatial relationships between features; Finally, comparative experiments and ablation experiments were set up and tested on 5 real SAR Datasets. The experimental results show that the The advantage of this method is to improve the operational efficiency of the method while reducing model complexity, obtain stronger robust features, suppress the adverse impact of speckle noise on change detection performance, and improve change detection performance.

New algebraic structures such as q-rung interval-valued intuitionistic fuzzy set applied to logarithm operator and its extension

We introduce the interval-valued intuitionistic fuzzy set used with the q-rung logarithimic operator (q-LIVIFS). A q-rung interval-valued intuitionistic fuzzy set may be developed by expanding the Pythagorean interval-valued fuzzy set (PIVFS). We discuss the q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted averaging (q-LIVIFWA), interval-valued intuitionistic fuzzy weighted geometric (q-LIVIFWG), extended q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted averaging (q-ELIVIFWA), and extended q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted geometric (q-ELIVIFWG). Several algebraic properties of q-LIVIFSs, such as distributivity, idempotency, and associativity, have been identified.

Multi-criteria group decision-making problem under intuitionistic fuzzy logarithmic aggregation operators based on T-norm and T-conorm

Multi-criteria group decision-making problem under intuitionistic fuzzy logarithmic aggregation operators based on T-norm and T-conorm

Multi-Objective Optimization based decision-making process using Dombi Logarithmic aggregation operator in CNN environment and its application to optimally select suitable greenhouse site for tomato crops

In this research article, we have defined Dombi Logarithmic aggregation operational laws in the framework of Cylindrical Neutrosophic Numbers (CNN) and utilized these laws to establish a new aggregation operator namely Cylindrical Neutrosophic Dombi Weighted Logarithmic Aggregation Operator(C N DW L A ). The said aggregation operational laws & aggregation operator have been used to present a new and novel decision making process where Full Consistency Method (FUCOM) and Multi-Objective Optimization(MOO) have been inte- grated and embedded fruitfully. Here the objective functions were formulated using the concept of a single-layer neural network and the FUCOM method to assess criterion weights. We have resolved the most favorable Pareto optimal solution derived from Multi-Objective Optimization by employing simulation and the Technique for Or- der of Preference by Similarity to Ideal Solution (TOPSIS) approach. Finally, we have applied our proposed decision making method along with MARCOS and MOORA techniques to determine the best greenhouse site for cultivating tomato crops. An exhaustive sensitivity and comparative analysis has been conducted to assess the robustness and stability of our Multi Criteria Group Decision Making (MCGDM) techniques. It is observed that our proposed method has some advantages in cylindrical neutrosophic environment and it produces stable robust results with the variation of underlying model parameters.

Evaluation of Renewable Energy Sources for a Sustainable Future: A Multi-Criteria Decision-Making Approach

The urgent global challenges of climate change, energy security, and environmental degradation highlight the need for sustainable energy solutions. Renewable energy sources (RES) present a viable pathway towards sustainability by mitigating greenhouse gas emissions, reducing reliance on fossil fuels, and fostering economic resilience. Purpose: The purpose of this paper is to propose an advanced Multi-Criteria Decision-Making (MCDM) approach to evaluate various RES by integrating environmental, economic, technological, social acceptance, and resource availability criteria, to identify the most suitable RES for sustainable energy solutions. Methodology: The study employs a hybrid method combining Type-2 Neutrosophic Numbers (T2NN) with LOPCOW (Logarithmic Percentage Change Operator Weighting) and MAIRCA (Multi-Attributive Ideal-Real Comparative Assessment) to rank the suitability of different RES, including solar, wind, hydropower, and geothermal energy. Findings: The case study results reveal wind energy as the top-ranked alternative, supported by consistent findings across comparative methods such as COPRAS, MABAC, EDAS, and TOPSIS. Sensitivity analysis further confirms the stability of the proposed model under various scenarios. Originality: The originality of this study lies in the integration of T2NN, LOPCOW, and MAIRCA to address the limitations of traditional MCDM approaches in handling uncertainty and imprecision in data. The study demonstrates the efficacy of the proposed framework in providing a robust evaluation of RES, and its value lies in its potential to inform decision-making in the field of sustainable energy solutions.

Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.

Multilayer Fused Correntropy Reprsenstation for Fault Diagnosis of Mechanical Equipment.

Fault diagnosis is vital for improving the reliability and safety of mechanical equipment. Existing fault diagnosis methods require a large number of samples for model training. However, in real-world environments, mechanical equipment usually operates under healthy conditions during most of its service life, resulting in a scarcity of fault samples. To solve this problem, a novel multilayer fusion correntropy representation method combined with a support vector machine is proposed for the fault diagnosis of mechanical equipment. First, the monitoring signal is expanded into multilayer signal components using wavelet packet decomposition. Then, the correlation between the signal components of each layer is expressed by correntropy, and the corresponding correntropy matrix is constructed. After performing the matrix logarithm operator, all correntropy matrices composed of correntropy values are fused into a vector, which is viewed as a feature of the signal. Finally, a support vector machine is established using small samples to realize fault classification. The effectiveness of the proposed method is validated on four public datasets. The results indicate that compared with other methods, the proposed method has advantages in terms of diagnosis accuracy and noise immunity ability.

Prescribed performance dynamic surface control based on dual extended state observer for 2-dof hydraulic cutting arm

In tunnel section forming operations, the boom-type roadheader tracking target trajectory with high precision is greatly significant in avoiding over and under excavation and improving excavation efficiency. However, there exist complex cutting loads, measurement noise, and model uncertainties, seriously degrading the tracking performance of traditional nominal model-based controllers. Hence, this study first fully analyzes the kinematics of all members of the cutting mechanism and establishes its complete multi-body dynamic model using the Lagrange method. Furthermore, a dual extended state observer is designed to estimate the mechanical system’s angular velocity and unmodeled disturbances and actuators’ uncertain nonlinearities. In particular, introducing a nonlinear filter replaces the traditional first-order filter in dynamic surface technology, overcoming the “explosion of complexity” while attenuating the conservatism of gains tuning. Then, a dual extended state observer-based prescribed performance dynamic surface controller is developed for roadheaders for the first time. Simultaneously, integrating an improved error transformation function into controller design effectively avoids the online computational burden caused by traditional logarithmic operations. Utilizing Lyapunov theory, the cutting system’s prescribed transient response and steady-state performance are guaranteed. Finally, the proposed controller’s effectiveness is verified by comparative experiments on the roadheader.

Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator

Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator

EMAKAS: An efficient three-factor mutual authentication and key-agreement scheme for IoT environment

The adoption of IoT in healthcare revolutionizes remote patient monitoring and healthcare efficiency. Yet, it brings notable security and privacy challenges, particularly in resource-constrained environment. We propose a secure and efficient three-factor lightweight mutual authentication and key agreement scheme, designed for IoT-based smart healthcare systems, addressing these critical concerns. The scheme employs a fuzzy extractor, a one-way hash function, Elliptic Curve Discrete Logarithm and XOR operations for efficient cryptographic transformations, creating a robust framework for secure data handling. The scheme's design focuses on security and privacy while minimizing computational demands, making it ideal for resource-constrained IoT devices. We utilized both informal and formal security analyses to validate our scheme, employing the Random Oracle Model (ROM), Scyther tool and Burrows-Abadi-Needham (BAN) logic. The security and performance analysis showed that our scheme offers more security features across 15 defined criteria with minimal communication and computational costs compared to other related schemes. The scheme is not only robust against security threats but also practical for implementation in IoT healthcare environment, offering a solution for secure IoT communication by achieving mutual authentication and key agreement with minimized computational requirements.

Riemannian Geodesic Discriminant Analysis–Minimum Riemannian Mean Distance: A Robust and Effective Method Leveraging a Symmetric Positive Definite Manifold and Discriminant Algorithm for Image Set Classification

This study introduces a novel method for classifying sets of images, called Riemannian geodesic discriminant analysis–minimum Riemannian mean distance (RGDA-MRMD). This method first converts image data into symmetric positive definite (SPD) matrices, which capture important features related to the variability within the data. These SPD matrices are then mapped onto simpler, flat spaces (tangent spaces) using a mathematical tool called the logarithm operator, which helps to reduce their complexity and dimensionality. Subsequently, regularized local Fisher discriminant analysis (RLFDA) is employed to refine these simplified data points on the tangent plane, focusing on local data structures to optimize the distances between the points and prevent overfitting. The optimized points are then transformed back into a complex, curved space (SPD manifold) using the exponential operator to enhance robustness. Finally, classification is performed using the minimum Riemannian mean distance (MRMD) algorithm, which assigns each data point to the class with the closest mean in the Riemannian space. Through experiments on the ETH-80 (Eidgenössische Technische Hochschule Zürich-80 object category), AFEW (acted facial expressions in the wild), and FPHA (first-person hand action) datasets, the proposed method demonstrates superior performance, with accuracy scores of 97.50%, 37.27%, and 88.47%, respectively. It outperforms all the comparison methods, effectively preserving the unique topological structure of the SPD matrices and significantly boosting image set classification accuracy.

Quantum $\operatorname{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge

We provide a ribbon tensor equivalence between the representation category of small quantum \operatorname{SL}(2) , at parameter q=e^{\pi i/p} , and the representation category of the triplet vertex operator algebra at integral parameter p>1 . We provide similar quantum group equivalences for representation categories associated to the Virasoro and singlet vertex operator algebras at central charge c=1-6(p-1)^{2}/p . These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A_{1} .

Characters of logarithmic vertex operator algebras and coloured invariants of torus links

We show that the characters of s l r \mathfrak {sl}_r versions of the ( 1 , p ) (1,p) singlet and the ( 1 , p ) (1,p) triplet vertex operator algebras arise as limits of appropriately coloured s l r \mathfrak {sl}_r Jones invariants of certain torus links.

STaRNet: A spatio-temporal and Riemannian network for high-performance motor imagery decoding

Brain–computer interfaces (BCIs), representing a transformative form of human–computer interaction, empower users to interact directly with external environments through brain signals. In response to the demands for high accuracy, robustness, and end-to-end capabilities within BCIs based on motor imagery (MI), this paper introduces STaRNet, a novel model that integrates multi-scale spatio-temporal convolutional neural networks (CNNs) with Riemannian geometry. Initially, STaRNet integrates a multi-scale spatio-temporal feature extraction module that captures both global and local features, facilitating the construction of Riemannian manifolds from these comprehensive spatio-temporal features. Subsequently, a matrix logarithm operation transforms the manifold-based features into the tangent space, followed by a dense layer for classification. Without preprocessing, STaRNet surpasses state-of-the-art (SOTA) models by achieving an average decoding accuracy of 83.29% and a kappa value of 0.777 on the BCI Competition IV 2a dataset, and 95.45% accuracy with a kappa value of 0.939 on the High Gamma Dataset. Additionally, a comparative analysis between STaRNet and several SOTA models, focusing on the most challenging subjects from both datasets, highlights exceptional robustness of STaRNet. Finally, the visualizations of learned frequency bands demonstrate that temporal convolutions have learned MI-related frequency bands, and the t-SNE analyses of features across multiple layers of STaRNet exhibit strong feature extraction capabilities. We believe that the accurate, robust, and end-to-end capabilities of the STaRNet will facilitate the advancement of BCIs.

The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution

Let LΔ be the logarithmic Laplacian operator with Fourier symbol 2ln⁡|ζ|, we study the expression of the diffusion kernel which is associated to the equation∂tu+LΔu=0in(0,N2)×RN,u(0,⋅)=0inRN∖{0}. We apply our results to give a classification of the solutions of{∂tu+LΔu=0in(0,T)×RN,u(0,⋅)=finRN and obtain an expression of the fundamental solution of the associated stationary equation in RN, and of the fundamental solution u in a bounded domain, i.e. LΔu=kδ0 in the sense of distributions in Ω, such that u=0 in RN∖Ω.

Lost circulation detection method based on cepstrum analysis of transient pressure waves

Encountering natural fractures or unbalanced formation pressure during oil drilling can result in economic loss and environmental pollution due to well leakage. Existing detection methods encounter challenges such as high costs, complex downhole environments, and difficult data acquisition. To address these issues, we propose a well leakage detection method using cepstrum for analyzing transient pressure waves. Cepstrum is a signal Fourier transform after logarithmic operation and then Fourier inverse spectrum obtained. By studying the propagation of transient pressure waves in the wellbore, we identify drilling fluid leakage location and amount based on time-dependent and amplitude changes of pressure wave signal characteristic peaks. To handle noise in the pressure wave signal, we employ adaptive noise-complete ensemble empirical modal decomposition (CEEMDAN) and wavelet threshold (WT) joint denoising. Correlation coefficient (CCF) with the Hilbert joint spectrum (HJS) is used to extract main frequency components, achieving denoising. Experimental results confirm: ① Noise interference in transient pressure waves is effectively suppressed using the CEEMDAN-WT-CCF-HJS denoising method. ② Cepstrum analysis of the pressure wave signal during wellbore annulus system leakage reveals distinct reflected wave characteristic peaks, aiding in locating different leakage points, with the amplitude of these peaks reflecting the size of the leakage. ③ This method efficiently utilizes time-frequency information from the excitation pressure wave signal, offering advantages over traditional time-domain and frequency-domain analysis. Experiments covering various leakage scenarios, amounts, and borehole sizes yielded controlled experimental errors (2.25%–9.10%), within a reasonable range. The method's validity and reliability were confirmed, providing theoretical support and technical guidance for well leakage detection in oil drilling.

Selection of Software Development Methodology by Employing a Multi-Criteria Decision-Making Approach Based on Logarithmic Bipolar Complex Fuzzy Aggregation Operators

Selection of Software Development Methodology by Employing a Multi-Criteria Decision-Making Approach Based on Logarithmic Bipolar Complex Fuzzy Aggregation Operators