Polynomial Function Research Articles

A framework for performance enhancement of classifiers in detection of prostate cancer from microarray gene

Prostate cancer is a major world health problem for men. This shows how important early detection and accurate diagnosis are for better treatment and patient outcomes. This study compares different ways to find Prostate Cancer (PCa) and label tumors as normal or abnormal, with the goal of speeding up current work in microarray gene data analysis. The study looks at how well several feature extraction methods work with three feature selection strategies: Harmonic Search (HS), Firefly Algorithm (FA), and Elephant Herding Optimization (EHO). The techniques tested are Expectation Maximization (EM), Nonlinear Regression (NLR), K-means, Principal Component Analysis (PCA), and Discrete Cosine Transform (DCT). Eight classifiers are used for the task of classification. These are Random Forest, Decision Tree, Adaboost, XGBoost, and Support Vector Machine (SVM) with linear, polynomial, and radial basis function kernels. This study looks at how well these classifiers work with and without feature selection methods. It finds that the SVM with radial basis function kernel, using DCT for feature extraction and EHO for feature selection, does the best of all of them, with an accuracy of 94.8 % and an error rate of 5.15 %.

Development and Application of an IoT-Based System for Soil Water Status Monitoring in a Soil Profile.

Soil water content (θ), matric potential (h) and hydraulic conductivity (K) are key parameters for hydrological and environmental processes. Several sensors have been developed for measuring soil θ-h-K relationships. The cost of such commercially available sensors may vary over several orders of magnitude. In recent years, some sensors have been designed in the framework of Internet of Things (i.e., IoT) systems to make remote real-time soil data acquisition more straightforward, enabling low-cost field-scale monitoring at high spatio-temporal scales. In this paper, we introduce a new multi-parameter sensor designed for the simultaneous estimation of θ and h at different soil depths and, due to the sensor's specific layout, the soil hydraulic conductivity function via the instantaneous profile method (IPM). Our findings indicate that a second-order polynomial function is the most suitable model (R2 = 0.99) for capturing the behavior of the capacitive-based sensor in estimating θ in the examined soil, which has a silty-loam texture. The effectiveness of low-cost capacitive sensors, coupled with the IPM method, was confirmed as a viable alternative to time domain reflectometry (TDR) probes. Notably, the layout of the sensor makes the IPM method less labor-intensive to implement. The proposed monitoring system consistently demonstrated robust performance throughout extended periods of data acquisition and is highly suitable for ongoing monitoring of soil water status.

Computational Construction of Weierstrass Sections for Semi-Invariant Polynomial Functions on Lie Algebras

In this paper, we present a computational approach to construct Weierstrass sections for semi-invariant polynomial functions on Lie algebras, extending the foundational work of Bourbaki and Popov. We focus on simple Lie algebras of type B, C, or D, and their associated parabolic subalgebras, particularly those with Levi factors composed of successive blocks of size two. Our method extends the notion of Weierstrass sections introduced by Popov, enabling us to explicitly construct these sections and establish their polynomiality. Furthermore, we demonstrate how these sections facilitate the linearization of semi-invariant generators. Central to our approach is the construction of an adapted pair, akin to a principal sl_2-triple in the non-reductive case. We provide computational algorithms and implementations for constructing these Weierstrass sections, offering a novel avenue for research in Lie algebra theory and algebraic geometry.

Experimental investigation of brain contusion characteristics and dynamic response in low-age children using an animal model

IntroductionBrain contusion is a prevalent traumatic brain injury (TBI) in low-age children, bearing the potential for coma and fatality. Hence, it is imperative to undertake comprehensive research in this field. MethodsThis study employed 4-week-old piglets as surrogates for children and introduced self-designed devices for both free-fall drop impact tests and drop-hammer impact tests. The study explored the characteristics of brain contusion and dynamic responses of brain under these distinct testing conditions. ResultsBrain contusions induced by free-fall and drop-hammer conditions both were categorized as the coup injury, except that slight difference in the contusion location was observed, with contusion occurring mainly in the surrounding regions beneath the impact location under free-fall condition and the region just right beneath the impact location under drop-hammer condition. Analysis of impact force and intracranial pressure (ICP) curves indicated similar trends in impact forces under both conditions, yet different trends in ICPs. Further examination of the peak impact forces and ICPs elucidated that, with increasing impact energy, the former followed a combined power and first-order polynomial function, while the latter adhered to a power function. The brain contusion was induced at the height (energy) of 2 m (17.2 J), but not at the heights of 0.4, 0.7, 1, 1.35 and 1.7 m, when the vertex of the piglet head collided with a rigid plate. In the case of a cylindrical rigid hammer (cross-sectional area constituting 40 % of the parietal bone) striking the head, the brain contusion was observed under the energy of 21.9 J, but not under energies of 8.1 J, 12.7 J and 20.3 J. Notably, the incidence of brain contusion was more pronounced under the free-fall condition. ConclusionsThese findings not only facilitate a comprehensive understanding of brain contusion dynamics in pediatric TBIs, but also contribute to the validation of theories and finite element models for piglet heads, which are commonly employed as surrogates for children.

Mechanical Characteristics and Macro–Microscopic Response Mechanisms of Cemented Paste Backfill under Different Curing Temperatures

Macroscopic and microscopic properties of cemented paste backfill (CPB) were studied through uniaxial compressive testing, acoustic emission (AE) monitoring, and microscopic feature analysis. The research shows that the uniaxial compressive strength (UCS) and elastic modulus have an exponential function type positive correlation with the increase in curing time and a polynomial function type with the rise of curing temperature; the mechanical parameters reach the maximum when the curing temperature is 40 °C. Increasing the curing time and curing temperature can promote the transition from shear crack to tensile crack. Increasing the curing time and raising the curing temperature both promote the transition of shear crack to tensile crack in the CPB. Overall, the crack mode is a combination of tensile and shear crack. At room temperature, the shear cracks dominates in the initial stage, but the proportion of the shear cracks decreases as the pressure increases in uniaxial compression test. At a curing temperature of 60 °C, the crack mode transitions to a tensile-shear mixed crack, with tension becoming the dominant crack mode. Microscopic analysis suggests an excellent linear correlation between the pore fractal dimension, UCS, and elastic modulus. When the pore fractal dimension decreases, the mechanical parameters also decrease. The pore fractal dimension can effectively characterize the macroscopic mechanical properties. Finally, the curing temperature is divided into two stages, with 40 °C as the dividing line for analysis. In the first stage, the increase in curing temperature effectively improved the mechanical parameters; in the second stage, the excessively high hydration reaction rate weakened the mechanical parameters.

New experimental measurements of X-ray emissions induced by protons of several tens of MeV on elements with Z=22 to Z=79: Toward quantitative high-energy PIXE

K-shell X-ray production cross-sections measurement campaigns were carried out at the ARRONAX cyclotron with titanium, chromium, copper, molybdenum, silver and gold targets for different energies from 15 to 68 MeV. These measurements, carried out using three detectors (SDD, CdTe and HPGe), complete the database with 66 new values. Our different measurements agree with previously published data and are consistent for all three detectors, confirming the absence of experimental bias. The universal ionisation cross-section representation allows an optimisation of the data using a 3rd-order polynomial function. This function can be used for HE-PIXE quantification for elements with atomic numbers between 22 (Ti) and 79 (Au) in the studied energy region.

Physical mechanism-corrected degradation trend prediction network under data missing

Accurate degradation trend prediction (DTP) is crucial for optimizing equipment operation and maintenance, thereby boosting production efficiency. This study introduces a novel Data Repair and Dual-data-stream LSTM (DR-DLSTM) network to tackle the challenge of missing data in equipment DTP. The proposed DR-DLSTM framework employs convex optimization to consider both the trend and periodic variations in the data, incorporating polynomial and trigonometric functions into the implicit feature matrix to construct latent vectors for missing data rectification. The network features a Dual-LSTM block with dual data streams to enhance feature extraction, with two gating update units correlating time series components and redistributing feature weights. The Dual-LSTM enables separate and accurate prediction of trend and periodic components, thereby enhancing the feature extraction capability of the prediction model. Additionally, the integration of physical rule information through Fourier and wavelet transform frequency correction modules allows for dynamic adjustments in prediction outcomes, from global trends to localized details. The DR-DLSTM's effectiveness is demonstrated through comprehensive comparisons with state-of-the-art models across multiple datasets, highlighting its superior performance. The results demonstrate the superiority of the proposed model. These algorithms were implemented in Python using Torch on a 2.9 GHz Intel I7 CPU and TITAN Xp GPU.

A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell’s equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.

An efficient polynomial-based verifiable computation scheme on multi-source outsourced data

With the development of cloud computing, users are more inclined to outsource complex computing tasks to cloud servers with strong computing capacity, and the cloud returns the final calculation results. However, the cloud is not completely trustworthy, which may leak the data of user and even return incorrect calculations on purpose. Therefore, it is important to verify the results of computing tasks without revealing the privacy of the users. Among all the computing tasks, the polynomial calculation is widely used in information security, linear algebra, signal processing and other fields. Most existing polynomial-based verifiable computation schemes require that the input of the polynomial function must come from a single data source, which means that the data must be signed by a single user. However, the input of the polynomial may come from multiple users in the practical application. In order to solve this problem, the researchers have proposed some schemes for multi-source outsourced data, but these schemes have the common problem of low efficiency. To improve the efficiency, this paper proposes an efficient polynomial-based verifiable computation scheme on multi-source outsourced data. We optimize the polynomials using Horner’s method to increase the speed of verification, in which the addition gate and the multiplication gate can be interleaved to represent the polynomial function. In order to adapt to this structure, we design the corresponding homomorphic verification tag, so that the input of the polynomial can come from multiple data sources. We prove the correctness and rationality of the scheme, and carry out numerical analysis and evaluation research to verify the efficiency of the scheme. The experimental indicate that data contributors can sign 1000 new data in merely 2 s, while the verification of a delegated polynomial function with a power of 100 requires only 18 ms. These results confirm that the proposed scheme is better than the existing scheme.

Algebraic properties of the maps .

The Boolean map defined by (where ) is used in various permutations that are part of cryptographic schemes, e.g., Keccak-f (the SHA-3-permutation), ASCON (the winner of the NIST Lightweight competition), Xoodoo, Rasta and Subterranean (2.0). In this paper, we study various algebraic properties of this map. We consider (through vectorial isomorphism) as a univariate polynomial. We show that it is a power function if and only if . We furthermore compute bounds on the sparsity and degree of these univariate polynomials, and the number of different univariate representations. Secondly, we compute the number of monomials of given degree in the inverse of (if it exists). This number coincides with binomial coefficients. Lastly, we consider as a polynomial map, to study whether the same rule ( ) gives a bijection on field extensions of . We show that this is not the case for extensions whose degree is divisible by two or three. Based on these results, we conjecture that this rule does not give a bijection on any extension field of .

A powered descent trajectory planning method with quantitative consideration of safe distance to obstacle

High scientific value areas on celestial bodies such as the Moon and Mars are often located in hazardous terrains. To achieve safe landing exploration, a novel planning method is proposed, which can ensure that the planned trajectory maintains a user-specified distance from obstacles, thus reducing potential collision risk induced by factors such as the body size and model uncertainties. Firstly, the basic model of the trajectory optimization problem and its convexification version is given. The obstacles are modeled as polynomial functions, based on which the “if-then” obstacle avoidance logic explicitly considering the safe distance, is described as a sum of squares constraint. This constraint formulation applies to any obstacle described by a finite number of polynomials, independent of the specific expression of the polynomials (reflecting the shape of the obstacle). Subsequently, the convexification process for the obstacle avoidance constraint is given. Finally, the sequential sum of squares programming problem for the obstacle avoidance trajectory is established, which boils down to a series of semidefinite programming problems. Simulation results show that the closest distance between the planned trajectory and obstacles strictly satisfies the specified distance constraint, and the trajectory could avoid non-convex obstacles. With the promising convergence properties of underlying convex optimization algorithms, advanced autonomous obstacle avoidance guidance schemes are expected to be formed based on the proposed trajectory planning method.

Improving cross-lingual low-resource speech recognition by Task-based Meta PolyLoss

Multilingual meta learning has emerged as a promising paradigm for transferring knowledge from source languages to facilitate the learning of low-resource target languages. Loss functions are a type of meta-knowledge that is crucial to the effective training of neural networks. However, the misalignment between the loss functions and the learning paradigms of meta learning degrades the network’s performance. To address this challenge, we propose a new method called Task-based Meta PolyLoss (TMPL) for meta learning. By regarding speech recognition tasks as normal samples and applying PolyLoss to the meta loss function, TMPL can be denoted as a linear combination of polynomial functions based on task query loss. Theoretical analysis shows that TMPL improves meta learning by enabling attention adjustment across different tasks, which can be tailored for different datasets. Experiments on three datasets demonstrated that gradient-based meta learning methods achieve superior performance with TMPL. Furthermore, our experiments validate that the task-based loss function effectively mitigates the misalignment issue.

Some criminological indicators of land mismanagement

Based on official statistical sources, this article presents a long-term criminological analysis of the situation with land mismanagement in Ukraine spanning the period from 2002 to 2023. The author determines absolute, average and relative indicators, in particular, the rate and dynamics of crimes in this category. The total number of recorded crimes in the category under study (1,240 crimes), their average annual occurrence (56 crimes), and the average growth rate (22.9 %) are calculated. Employing a 5th degree polynomial function, a trend depicting the dynamics of these crimes is constructed. The chosen model aptly captures the development trend of the process with high precision (coefficient of determination R2 = 0.916) and facilitates the identification and characterization of four distinct periods, each reflecting the main trends of land mismanagement in Ukraine.
 A paradoxical situation is revealed by the analysis of the practical application of the criminal law provisions which have supplemented Art. 254 of the Ukrainian Criminal Code « Land Mismanagement» in accordance with Law of Ukraine No. 402-IX of December 19, 2019 « On Amendments to Certain Legislative Acts of Ukraine Regarding Improvement of Legislation on Mining of Amber and Other Minerals». Notably, no crimes were recorded under Part 2 of Article 254 of the Criminal Code of Ukraine (the primary corpus delicti). Conversely, 15 crimes were recorded under Parts 3 and 4 of Article 254 (qualifying elements of the crime). The author concludes that this correlation would be impossible in the case of objective recording of crimes, and it also underscores a noteworthy latent aspect of land mismanagement, in particular, deliberate evasion of mandatory reclamation of lands damaged in the course of pilot commercial development of amber deposits or amber mining under a special subsoil use permit, resulting in considerable damage. The conducted analysis holds an intrinsic value both for scientific research and for law enforcement, ensuring an appropriate information support base for countering land mismanagement, firstly, by providing a clear understanding of the current situation with this phenomenon. Furthermore, it lays the groundwork for the development of the relevant system of criminal preventive measures tailored to address this specific issue.

TriKSV-LG: a robust approach to disease prediction in healthcare systems using AI and Levy Gazelle optimization

A seamless connection between the Internet and people is provided by the Internet of Things (IoT). Furthermore, lives are enhanced using the integration of the cloud layer. In the healthcare domain, a reactive healthcare strategy is turned into a proactive one using predictive analysis. The challenges faced by existing techniques are inaccurate prediction and a time-consuming process. This paper introduces an Artificial Intelligence (AI) and IoT-based disease prediction method, the TriKernel Support Vector-based Levy Gazelle (TriKSV-LG) Algorithm, which aims to improve accuracy, and reduce the time of predicting diseases (kidney and heart) in healthcare systems. The IoT sensors collect information about patients’ health conditions, and the AI employs the information in disease prediction. TriKSV utilizes multiple kernel functions, including linear, polynomial, and radial basis functions, to classify features more effectively. By learning from different representations of the data, TriKSV better handles variations and complexities within the dataset, leading to more robust disease prediction models. The Levy Flight strategy with Gazelle optimization algorithm tunes the hyperparameters and balances the exploration and exploitation for optimal hyperparameter configurations in predicting chronic kidney disease (CKD) and heart disease (HD). Furthermore, TriKSV's incorporation of multiple kernel functions, combined with the Gazelle optimization strategy, helps mitigate overfitting by providing a more comprehensive search space for optimal hyperparameter selection. The proposed TriKSV-LG method is applied to two different datasets, namely the CKD dataset and the HD dataset, and evaluated using performance measures such as AUC-ROC, specificity, F1-score, recall, precision, and accuracy. The results demonstrate that the proposed TriKSV-LG method achieved an accuracy of 98.56% in predicting kidney disease using the CKD dataset and 98.11% accuracy in predicting HD using the HD dataset.

A Novel Deep Convolutional Neural Network Approach using Jacobi Polynomial and Laplacian Function (JPLF) in Recognition of Plant Leaf Disease

Background/Objectives: Enhancing agricultural productivity is crucial for fostering economic growth. Plant diseases significantly threaten crops, necessitating timely detection to mitigate adverse impacts on quality, quantity, and overall productivity. This research addresses the importance of early disease detection in agriculture and proposes an innovative method utilizing Jacobian Polynomial and Laplacian Function for precise identification. Methods: Efficient monitoring of large-scale crop farms with minimal workforce is essential. To achieve this, an automatic method for plant disease detection is proposed. The method leverages Jacobian polynomials to expand input features, mitigating correlation issues among input vectors. The expanded Jacobi polynomial is the input vector for a backpropagation algorithm with a novel activation function based on the Laplacian function. Findings: The efficacy of the proposed JPLF model is demonstrated through the accurate identification of leaf diseases, achieving a high testing accuracy of 92.07%. Comparative analysis with existing models, such as CNN with MobileNet V2 (85.38%) and the IoU model (83.75%), highlights the superiority of the JPLF model in plant disease detection. Novelty: To overcome the limitations of existing approaches, the incorporation of Jacobian polynomials plays a pivotal role in expanding input features. This expansion aids in eliminating correlations among input vectors, enhancing the efficacy of disease detection. The proposed model, Jacobi Polynomial and Laplacian Function (JPLF) introduces a unique activation function based on the Laplacian function, improving accuracy. Keywords: Plant Disease Detection, Jacobi Polynomial, Laplacian Transform, Deep Learning Model, Feature Expansion

Separating invariants for two-dimensional orthogonal groups over finite fields

We described a minimal separating set for the algebra of O2+(Fq)-invariant polynomial functions of m-tuples of two-dimensional vectors over a finite field Fq.

Generation of initial debris cloud distributions for breakup events based on CARDC-SBM

With the dramatic increase in the number of resident space objects, the risk of collision between space objects and debris is growing. To protect spacecraft from such collisions, it is crucial to determine the generation patterns of space debris and analyze the evolution of debris clouds. In this paper, a continuous distribution formula is derived for each parameter of the breakup debris based on the China Aerodynamics Research and Development Center spacecraft breakup model (CARDC-SBM). Using CARDC-SBM, we theoretically establish a normal distribution for the separation velocity of the breakup debris in relation to the breakup mass center. Based on this foundation, the mean and standard deviation of the debris separation velocity distribution are empirically modeled using polynomial and exponential functions. Consequently, an empirical formula with a broad range of applicability is derived, which significantly reduces the computation time of debris cloud simulations. The initial velocity distribution given by this empirical formula is employed as the initial condition for the evolution of debris clouds from the perspective of a boundary value problem. Simulations using CARDC-SBM and subsequent analysis of the debris cloud evolution resulting from a breakup event reveal that the distribution of each parameter is determined solely by the relative collision velocity and the spacecraft's nominal density in the impact breakup scenario. Our results show that the empirical formula for the debris cloud distribution provides good agreement with Monte Carlo simulations.

Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves

The Riccati polynomial differential systems are differential systems of the form [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable [Formula: see text] defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.

Machine learning-based electricity theft detection using support vector machines

Electricity theft is a serious issue that many nations face, especially in developing areas where non-technical losses can make up a significant percentage of the overall losses sustained by utilities. Electricity theft detection (ETD) is a very challenging task because it frequently introduces irregularities in customer electricity consumption patterns. In recent times, machine learning (ML) techniques have been investigated as a potential solution for ETD. In this research, author propose electricity theft detection based on four kernel functions of support vector machines (SVM). The proposed method analyzes the electricity consumption patterns and then predicts the category of the user. The kernel functions utilized includes polynomial, sigmoid, radial basis function (RBF) and linear kernel function. For experimentation and model training, a dataset of Pakistani utility company is used, which contains the electricity consumption information. The results highlight SVM method works well for accurate ETD. The detection accuracy of the various kernel functions of SVM is 83%, 79%, 80%, and 76% for RBF, polynomial, sigmoid, and linear kernel functions, respectively, demonstrating the effectiveness of the proposed SVM-based method for theft detection. By leveraging these ML-based methods, utility companies can strengthen their ability to detect and prevent electricity theft, leading to improved revenue management and dependability of services.

Parabola or second-degree polynomial function exploration in initial teacher training: Integrating intuition and didactic situations

This work is the result of a master’s investigation in Brazil, which discusses the teaching of the parabola in the initial training of mathematics teachers. Our theoretical framework addresses the relationship between intuition and the dialectics of the theory of didactic situations, which supports the analysis of the results of this study Its objective was to identify and register categories of intuitive reasoning manifested by the subjects when solving a didactic situation involving the parabola with GeoGebra software contribution. The methodology adopted was didactic engineering, experimented with eight students in initial training at a Brazilian public university, among 6<sup>th</sup> and 9<sup>th</sup> undergraduate semesters. The posterior analysis and validation allowed us to verify the need to discuss the parabola, articulating its geometric, algebraic, and analytical views, as well as to reinforce the importance of its teaching with the use of technology.