Mathematical Formula Research Articles

Validation of the practicability of logical assessment formula for evaluations with inaccurate ground-truth labels: An application study on tumour segmentation for breast cancer

Validation of the practicability of logical assessment formula for evaluations with inaccurate ground-truth labels: An application study on tumour segmentation for breast cancer

Effect of Interactive E-Book to Measure Mathematical Representations on Optics in High School

This study aims to measure the effect of using interactive e-books in enhancing students' mathematical representation skills on optics material in high school. Mathematical representation involves students' ability to use mathematical symbols, graphs, and formulas to solve physics problems, particularly in the concept of optics, which is often considered abstract. The research method used was a quasi-experiment with a pretest-posttest control group design. The sample consisted of two classes: an experimental class using interactive e-books as learning media and a control class using conventional methods. The results showed an improvement in mathematical representation skills in both classes, with the n-gain value of the experimental class being 0.604 and the control class being 0.557, both in the medium category. However, the experimental class demonstrated a more significant improvement compared to the control class, indicating that interactive e-books are more effective in helping students deeply understand optical concepts. The interactive features in the e-book, such as simulations and animations, provided a more engaging learning experience and helped students visualize abstract concepts. In conclusion, the use of interactive e-books has a positive impact on improving students' mathematical representation skills and can be considered an effective alternative learning medium, especially for science subjects that require better visualization and conceptual understanding.

Polynomial-time checking of generalized Sahlqvist syntactic shape

The best known modal logics are axiomatized by Sahlqvist axioms, i.e., axioms of a syntactic shape which guarantees these formulas to have such excellent properties as canonicity and elementarity. Recently, the definition of Sahlqvist formulas has been generalized and extended from formulas in classical modal logic to inequalities (sequents) in a wide family of logics known as LE-logics. We introduce an algorithm which checks if a given inequality is generalized Sahlqvist in polynomial time.

Enhanced framework for solving general energy equations based on metropolis-hasting Markov chain Monte Carlo

Due to the widespread presence of heat and mass transfer phenomena in industrial applications, numerous studies have been devoted to the accurate solution of energy equations, providing a foundation for the analysis of heat and mass transfer processes in practical applications. In this study, a Green's Function Markov Superposition Monte Carlo (GMSMC) for solving general energy equations has been developed based on probability and statistical principles owing to its advantageous features of insensitivity towards dimension and geometric complexity as well as the capability to handle multiple integrals in complex domains. The energy equation is first decomposed, and corresponding probability models are established for each component, considering their interrelationships. Subsequently, a solution framework for solving the general energy equation is constructed by integrating these probability models based on a Markov chain structure. The mathematical principles and formulae of the proposed method are derived in detail. The performance of the proposed method is validated by several heat transfer systems with different combinations of boundary conditions and features, which mainly include the distribution of the internal heat source and whether the convection or transient term is included. Results of the validation show that the temperatures obtained by the proposed method are in good agreement with the FEM based on a fine grid, no matter whether the calculation is for a single point or a distribution.

Generating Java code pairing with ChatGPT

The Large Language Models (LLMs) like ChatGPT 3.5 have created a new era of automatic code generation. However, the existing research primarily focuses on generating simple code based on datasets (such as HumanEval, etc.). Most of approaches pay less attention to complex and practical code generation. Therefore, in this paper, we propose a new approach called “Xd-CodeGen” which can be used to generate large scale Java code. This approach is composed of four phases: requirement analysis, modeling, code generation, and code verification. In the requirement analysis phase, ChatGPT 3.5 is utilized to decompose and restate user requirements. To do so, a knowledge graph is developed to describe entities and their relationship in detail. Further, Propositional Projection Temporal Logic (PPTL) formulas are employed to define the properties of requirements. In the modeling phase, we use knowledge graphs to enhance prompts and generate UML class and activity diagrams for each sub-requirement using ChatGPT 3.5. In the code generation phase, based on established UML models, we make use of prompt engineering and knowledge graph to generate Java code. In the code verification phase, a runtime verification at code level approach is employed to verify generated Java code. Finally, we apply the proposed approach to develop a practical Java web project.

Automated data-driven discovery of material models based on symbolic regression: A case study on the human brain cortex

We introduce a data-driven framework to automatically identify interpretable and physically meaningful hyperelastic constitutive models from sparse data. Leveraging symbolic regression, our approach generates elegant hyperelastic models that achieve accurate data fitting with parsimonious mathematic formulas, while strictly adhering to hyperelasticity constraints such as polyconvexity/ellipticity. Our investigation spans three distinct hyperelastic models—invariant-based, principal stretch-based, and normal strain-based—and highlights the versatility of symbolic regression. We validate our new approach using synthetic data from five classic hyperelastic models and experimental data from the human brain cortex to demonstrate algorithmic efficacy. Our results suggest that our symbolic regression algorithms robustly discover accurate models with succinct mathematic expressions in invariant-based, stretch-based, and strain-based scenarios. Strikingly, the strain-based model exhibits superior accuracy, while both stretch-based and strain-based models effectively capture the nonlinearity and tension-compression asymmetry inherent to the human brain tissue. Polyconvexity/ellipticity assessment affirm the rigorous adherence to convexity requirements both within and beyond the training regime. However, the stretch-based models raise concerns regarding potential convexity loss under large deformations. The evaluation of predictive capabilities demonstrates remarkable interpolation capabilities for all three models and acceptable extrapolation performance for stretch-based and strain-based models. Finally, robustness tests on noise-embedded data underscore the reliability of our symbolic regression algorithms. Our study confirms the applicability and accuracy of symbolic regression in the automated discovery of isotropic hyperelastic models for the human brain and gives rise to a wide variety of applications in other soft matter systems. Statement of significanceOur research introduces a pioneering data-driven framework that revolutionizes the automated identification of hyperelastic constitutive models, particularly in the context of soft matter systems such as the human brain. By harnessing the power of symbolic regression, we have unlocked the ability to distill intricate physical phenomena into elegant and interpretable mathematical expressions. Our approach not only ensures accurate fitting to sparse data but also upholds crucial hyperelasticity constraints, including polyconvexity, essential for maintaining physical relevance.

On the expressiveness and efficiency of guarded lists in Bach

Concurrency theory has received considerable attention, but mostly in the scope of synchronous process algebras such as CCS, CSP, and ACP. As another way of handling concurrency, data-based coordination languages aim to provide a clear separation between interaction and computation by synchronizing processes asynchronously by means of information being available or not on a shared space. Although these languages enjoy interesting properties, verifying program correctness remains challenging. In particular, model checking logic formulae is known to raise performance issues due to the state space explosion problem. In this paper, we propose a guarded list construct as a solution to address this problem. Beyond increasing performance, it also enriches the expressiveness of data-based coordination languages and allows for program transformations that further increase the speed of computations.

MATHEMATICAL APPROACH OF THE BACKPROPAGATION METHOD FOR CONSTRUCTING ARTIFICIAL NEURAL NETWORKS

Backpropagation isthe core part of a neural network. This method is used to efficiently train a network using a chain rule that allows differentiation of complex functions. In other words, after each pass through the network, the backpropagation method performs a backward pass to adjust the model parameters, such as weights and biases. This article highlights the importance of using the backpropagation method from the point of view of mathematical formulas for neural networks. The importance of using the backpropaga-tion learning algorithm to calculate the gradient (gradient descent) and the need to use the activation function to minimize the loss function is mathematically described and calculated by formulas, and also proven by calculating the matrix products of vectors for each layer of parameters - weights and biases and applying complex differential equations.

Interactive visual formula composition of multidimensional data classifiers

Understanding how a classification result is generated and what role individual features play in the classification is crucial in many applications and, in particular, in medical contexts such as the translation of diagnosis biomarkers into clinical practice. The goal is to find (ideally simple) relationships between the features in multi-dimensional data and the classification for an explanation of the underlying phenomenon. Mathematical formulas allow for the expression of these relationships and can serve as classifiers. However, there are infinitely many mathematical formulas for the given features and they bear an inherent trade-off between complexity and accuracy. We present an interactive visual approach that supports domain experts to mitigate the trade-off issue. Core to our approach is a novel feature selection method, from which formulas are composed using symbolic regression and where state-of-the-art classifiers serve as a reference. To evaluate our approach and compare the achieved classification performance to the performance achieved by other state-of-the-art feature selection techniques, we test our methods with well-known machine learning data sets. Our evaluation shows that our feature selection method performs better than randomly selecting features for data sets with many features or when a low number of generations in the symbolic regression is required. Moreover, it consistently matches or outperforms state-of-the-art methods. Moreover, we apply our approach in a case study to a hemodynamic cohort data set, where we report our findings and domain expert feedback. Our approach was able to find formulas containing features that are in agreement with literature. Also, we could find formulas that performed better in the micro-averaged F1 score when compared to established histological indices.

Advanced regression model fitting to experimental results in case of the effect of grinding conditions on a cutting force with the planer technical knives

The use of well-fit models enables users to make better decisions and draw accurate insights, so in this article a multi-step procedure for establishing regression functions for experimental data is presented. This was inspired by the need to determine well-fitted models for the results of the study of the relationship between the input factors of the grinding process of planer knives and the measured grinding power, as well as in relation to the knife cutting force. The sharpening process was carried out under various conditions, making it possible to assess their influence on the results and select the most favorable technological parameters to prepare the knives for operation. The equations adjusted by Excel or other basic methods require additional verification if the mathematical formula and values of the model coefficients correspond to the data and the physical meaning. For this purpose, one-, two- and three-factor analyses were performed to eliminate redundant factors and introduce significant interactions between input parameters. Additional analyses of variance and Tukey, Levene and Brown-Forsythe tests were used to determine the general form of the model. Detailed form of nonlinear models was worked out by comparing the distribution of residuals, looking for outliers and evidence of the presence or absence of collinearity, as well as by determining the values of additional model quality indicators, such as Akaike Information Criterion, Cook’s distance and Variance Inflation Factor. The models presented include a categorization variable (grinding type) and are presented graphically in the form of surface plots. With reference to the technological processes studied, a discussion of the physical significance of the developed models and the influence of the input factors on the model output value was conducted. The analyzes additionally include an evaluation of the model factors using prediction slice plots.

GDPR-anonymous population overlap estimation

The feasibility of many collaborative, data-driven research projects in Public Health and Social Sciences depends on the size of the overlap between the populations of contributing data partners, that is, the number of individuals for whom all parties can provide data. Where the identity of respective population members is subject to privacy protections, the overlap is traditionally estimated using de-identified cryptographic tokens to represent individuals. This does however not meet the strict standard of anonymity under GDPR and potentially runs counter to emerging privacy laws of US States, and may thus impose significant regulatory and governance burdens on research institutions, even at the early scoping stage. We present a novel method for overlap estimation, where two data partners construct a single, anonymous Bloom filter for their populations from which the overlap can be estimated. Two new techniques -- (a) the randomization of the contribution of individual population members to the Bloom filter and (b) an interactive bit-flipping protocol between data partners -- are presented which ensure that the Bloom filters are safe against dictionary attack by one party against the other party. We discuss the security aspects of this method, present the required mathematical formulas for calculating the overlap estimate, and report on the practical performance in terms of computational effort and resulting accuracy. The goal of the method is to facilitate new research by reducing the cost and complexity of initial planning and by making the overlap estimate GDPR-anonymous, and thus safer and subject to less onerous compliance requirements.

From formulae, via models to theories: Dynamic Energy Budget theory illustrates requirements

As sciences mature, they transition from observation and description to explanation and prediction. This transition is associated with qualitative changes in the way quantitative mathematical formulations are constructed and interpreted, resulting in a ‘theory’. Such transitions from phenomenology to theory are happening in biology but the heuristic framework involved is rarely articulated. We here describe how the use of models in research sets model requirements, using Dynamic Energy Budget (DEB) theory to illustrate the more elaborate ones. We first make the distinction between mathematical formulae and models based on their relation to the abstract and real worlds. We then explain how the transition from models to theory affects model construction and parameter estimation, and discuss the concept of parameter estimation-in-context using the database Add_my_Pet on animal energetics. The transition comes with the need to develop auxiliary and meta- theory and to work with multiple datasets, implying constraints for the loss function that is used for parameter estimation. Finally, we discuss the extra requirements for general explanatory models: they need to be explicit on relevant general principles and to be embedded in a wider scientific context. We also discuss how we see theory’s relationship to observation and prediction change in the future as we use it to deal with theoretical and applied problems in biology.

Spatio-Temporal Mechanisms of Rainfall in Kalaburagi District: An Analysis of Trends and Variability

The study used data from 11 taluks to analyse the changes and variability in annual and seasonal rainfall in the Kalaburagi district between 2016 and 2022. Sen Slope and Mann-Kendall analyses were used to evaluate trends, while the coefficient of variation was used to measure rainfall variability. The mean rainfall was calculated using simple mathematical formulas. The findings showed minimal seasonal and yearly variability, with the exception of the southwest monsoon season. The district as a whole as well as all 11 taluks consistently saw a rise in rainfall, according to annual Sen slope research. However, during the northeast monsoon season, negative trends were observed in Aland and Sedam Taluks. In the Mann-Kendall test, the district did not exhibit any general pattern; nevertheless, several taluks, such as Yadrami and Kamalapur, demonstrated remarkable trends in the southwest and northeast seasons, respectively. Due to geographical and climatic fluctuations, the district of Kalaburagi experienced an annual mean precipitation of 797 mm. Higher mean rainfall was recorded in Sedam, Kalagi, Chincholi, and Kamalapur, while lower mean rainfall was recorded in Jevaragi and Yadrami Taluks. In the water-scarce Kalaburagi district, this work is important for policymakers addressing climate change and water resource management.

Performance enhancement of NOMA multi-user networks with aerial reconfigurable intelligent surfaces and aerial relay

In this article, we propose a combination of multiple aerial reconfigurable intelligent surfaces (ARISs) with an aerial relay (AR) for improving the performance of multiple users adopting the non-orthogonal multiple access (NOMA) scheme. In particular, two groups of ARIS are utilized, one for aiding ground-to-air (G2A) communication and another one for aiding air-to-ground (A2G) communication. Mathematical formulas of outage probability (OP), throughput, and achievable capacity (AC) of every user in the proposed ARISs-AR-NOMA network are derived over the Nakagami-m channel. The propagation environment recommended for the fifth generation and beyond (5G/B5G) is applied to enhance the practical behavior of the proposed network. To confirm the performance enhancement, we compare the OP, throughput, and AC of the proposed system with the conventional AR-NOMA network (i.e., without ARISs). Numerical illustrations demonstrate the great benefits of the proposed ARISs-AR-NOMA network, its transmit power is dramatically lower than that of the conventional network. Moreover, the throughput and AC of the proposed network are considerably higher than the conventional network’s. Based on the effect of key parameters and network behaviors, various effective solutions are recommended to enhance the performance of the proposed ARISs-AR-NOMA network.

Effect of two-phase viscosity difference and natural fractures on the wormhole morphology formed by two-phase acidizing with self-diverting acid in carbonate rocks

Different types of acids are needed in the field to achieve various acidizing goals. Currently, there are no reliable acidizing models for multiphase flows and complex multiphysics coupling. This paper derives mathematical formulas for the oil–water acidizing process of a situ self-diverting acid combined with thermal–chemical–fracture interactions and discusses the influence of two-phase oil–water mixture and fractures on the wormhole morphology produced by self-diverting acid. The results show that the spent acid following the acid–rock reaction forms a high-viscosity sealing zone, causing the injected acid to be redirected. The self-diverting acid forms more numerous and longer branches than a conventional acid during single-phase acidizing. In the case of two-phase acidizing, the high viscosity difference produces distinct effects when using self-diverting acid compared with conventional acid. Specifically, the self-diverting acid extends the breakthrough time and forms a wormhole morphology with longer and more complex branches, whereas the conventional acid accelerates the breakthrough of the rock. As the viscosity difference decreases, the wormhole morphology of the self-diverting acid gradually approaches that of a single-phase acid. Large-aperture fractures completely determine the wormhole morphology, while smaller apertures determine the branch morphology of the wormhole. Fractures have a negative acidizing effect in the case of the self-diverting acid, unlike conventional acid. The proposed model accurately simulates the complex acidizing process of a self-diverting acid.

Analysis of chi angle distributions in free amino acids via multiplet fitting of proton scalar couplings

Abstract. Scalar couplings are a fundamental aspect of nuclear magnetic resonance (NMR) experiments and provide rich information about electron-mediated interactions between nuclei. 3J couplings are particularly useful for determining molecular structure through the Karplus relationship, a mathematical formula used for calculating 3J coupling constants from dihedral angles. In small molecules, scalar couplings are often determined through analysis of one-dimensional proton spectra. Larger proteins have typically required specialized multidimensional pulse programs designed to overcome spectral crowding and multiplet complexity. Here, we present a generalized framework for fitting scalar couplings with arbitrarily complex multiplet patterns using a weak-coupling model. The method is implemented in FitNMR and applicable to one-dimensional, two-dimensional, and three-dimensional NMR spectra. To gain insight into the proton–proton coupling patterns present in protein side chains, we analyze a set of free amino acid one-dimensional spectra. We show that the weak-coupling assumption is largely sufficient for fitting the majority of resonances, although there are notable exceptions. To enable structural interpretation of all couplings, we extend generalized and self-consistent Karplus equation parameterizations to all χ angles. An enhanced model of side-chain motion incorporating rotamer statistics from the Protein Data Bank (PDB) is developed. Even without stereospecific assignments of the beta hydrogens, we find that two couplings are sufficient to exclude a single-rotamer model for all amino acids except proline. While most free amino acids show rotameric populations consistent with crystal structure statistics, beta-branched valine and isoleucine deviate substantially.

Reverse degree-based topological indices study of molecular structure in triangular ϒ-graphyne and triangular ϒ-graphyne chain

Topological indices are mathematical descriptors of the structure of a molecule that can be used to predict its properties. They are derived from the graph theory, which describes the topology of a molecule and its connectivity. The main objective is mathematical modeling and topological properties of ϒ-graphyne. Current research focuses on two structures made from hexagonal honeycomb graphite lattices named triangular ϒ-graphyne and triangular ϒ-graphyne chains. The authors have simultaneously computed the first and second Reverse Zagreb indices, reverse hyper-Zagreb indices, and their polynomials. This research also derives mathematical closed-form formulas for some of its fundamental degree-based molecular descriptors. Researchers have been trying to synthesize a novel carbon form called Graphyne. For over a decade but with no success. Recently, some researchers have made a breakthrough in generating Carbons elusive allotrope and solved a long-standing problem in carbon materials. This wonder material is created to rival the conductivity of graphene but with control. These results opened new ways of research in the fields of semiconductors, electronics and optics. Furthermore, graphical and tabular results will help to investigate the structure-property relationships in γ-graphyne.

An efficient and cost-effective methods framework for performing biomechanical analysis on cadaveric skeletal muscle

IntroductionThe structure and arrangement of skeletal muscle fibers is the primary determinant of muscle function. While analysis of morphologic parameters to elucidate muscle structure-function relationships dates to the 17th century, considerable variability in methodology and technique to quantify these relationships exists within the literature. Additionally, applications to assess the impact of non-typical musculoskeletal morphologies on structure-function relationships are limited. This study aims to assemble and present a practical, step-by-step framework of combined methods and techniques for efficiently analyzing biomechanical impacts of cadaveric skeletal muscles. MethodsExisting skeletal muscle biomechanical formulas and experimentally determined parameters for typical fast-acting skeletal muscle were identified in the literature. The methods framework was assembled as a stepwise protocol that includes mathematical formulas, referenced accepted values, optional steps, section breaks, suggested techniques, important notes and footnotes, and materials needed for collecting necessary measurements. Proof of concept was achieved with primary histological data and imaging, along with references to examples where the methods have been successfully applied. ResultsThe assembled framework presents an ordered process for measuring skeletal muscle parameters, calculating the maximal isometric force of a skeletal muscle, and applying that data to understand musculoskeletal mechanics. Histology data, imaging, and a summary of studies that have successfully applied the framework methods provide visual aids and validation for the methods. ConclusionsThis study presents an efficient and convenient framework of combined methods and techniques for investigators to evaluate the biomechanical impact of skeletal muscles. The practical use of this framework should optimize project efficiency and spending, increase study rigor, and minimize procedural variation across different studies. This report may serve as a foundational resource for researchers studying cadaveric muscle biomechanics, and its use especially adds translational clinical value to case analyses of non-typical musculoskeletal morphologies.

Individualized generation of women's prototype based on the classification of body shape

The traditional prototype method only relies on the basic dimensions of the human body such as bust girth and back length to make the prototype pattern, ignoring the differences of human body details dimensions. In order to improve the fit of garments, the research on intelligent generation technology of individualized garment patterns has become one of the hot spots in the field of clothing. Based on the 3D body scanning technology, we proposed a method for generating prototype pattern based on body shape classification. The proposed method takes as inputs the body parameters (width, thickness and angle), while the output of the method is one of the four pattern generation rules-"Y" shape, "V" shape, "I" shape, or "X" shape. 207 subjects were divided into four categories based on the body angle parameters, namely "Y" shape, "V" shape, "I" shape and "X" shape, and a shape recognition model was built. The pattern database is required to obtain pattern generation rules by 3D point cloud reconstruction and flattening. Combined with the human body shape parameters, the mathematical formulas of the feature points on the pattern landmarks are built. The accuracy of body shape recognition model is 97.4%. The error analysis of the predicted pattern parameters shows that 90% of the pattern feature points have a goodness of fit above 0.7. In the 5 main landmarks, the proportion of pattern within the absolute error range is more than 80%, indicating that the prediction effect of the pattern is good. This method can be applied to the process of automatic pattern generation system based 2D measurement to improve work efficiency.

ИНФОРМАЦИЯЛЫК ТЕХНОЛОГИЯЛАРДЫ ОКУТУУДА МАТЕМАТИКАЛЫК МОДЕЛДИН ОРДУ

This paper discusses technologies for using mathematical modeling methods in teaching information technology. A method for selecting the types of problems to be solved using mathematical modeling and decision-making methods is proposed. Mathematical modeling makes it possible to present current information in a more compact, structured form, since mathematical formulas are designed to allow you to write the current problem in a more concise, understandable and memorable form. Another important area that can be used in educational projects is the assessment and calculation of alternative energy sources that are more efficient than the electricity used today. Of great interest is also the study of the dynamics of human reproduction in modern society. When implementing the above projects with the help of a computer, you can find a solution to the problem, create a simulation model and display a graphic image using software such as PowerSim.