AppliedMath

An algebraic system is introduced which is very useful for performing scattering calculations in quantum field theory. It is the set of all real numbers greater than or equal to −m² with parity designation and a special rule for addition and subtraction, where m is the rest mass of the scattered particle. Full article

We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using the Wilks likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology was illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna, respectively, and also to the earlier (1998) Innsbruck experiment of Weihs et al. and the recent (2022) Munich experiment of Zhang et al., which investigates the use of a loophole-free Bell experiment as part of a protocol for device-independent quantum key distribution (DIQKD). Full article

Hepatocellular carcinoma (HCC) is the primary liver cancer that occurs the most frequently. The risk of developing HCC is highest in those with chronic liver diseases, such as cirrhosis brought on by hepatitis B or C infection and the most common type of liver cancer. Knowledge-based interpretations are essential for understanding the HCC microarray dataset due to its nature, which includes high dimensions and hidden biological information in genes. When analyzing gene expression data with many genes and few samples, the main problem is to separate disease-related information from a vast quantity of redundant gene expression data and their noise. Clinicians are interested in identifying the specific genes responsible for HCC in individual patients. These responsible genes may differ between patients, leading to variability in gene selection. Moreover, ML approaches, such as classification algorithms, are similar to black boxes, and it is important to interpret the ML model outcomes. In this paper, we use a reliable pipeline to determine important genes for discovering HCC from microarray analysis. We eliminate redundant and unnecessary genes through gene selection using principal component analysis (PCA). Moreover, we detect responsible genes with the random forest algorithm through variable importance ranking calculated from the Gini index. Classification algorithms, such as random forest (RF), naïve Bayes classifier (NBC), logistic regression, and k-nearest neighbor (kNN) are used to classify HCC from responsible genes. However, classification algorithms produce outcomes based on selected genes for a large group of patients rather than for specific patients. Thus, we apply the local interpretable model-agnostic explanations (LIME) method to uncover the AI-generated forecasts as well as recommendations for patient-specific responsible genes. Moreover, we show our pathway analysis and a dendrogram of the pathway through hierarchical clustering of the responsible genes. There are 16 responsible genes found using the Gini index, and CCT3 and KPNA2 show the highest mean decrease in Gini values. Among four classification algorithms, random forest showed $96.53\%$ accuracy with a precision of $97.30\%$. Five-fold cross-validation was used in order to collect multiple estimates and assess the variability for the RF model with a mean ROC of $0.95\pm 0.2$. LIME outcomes were interpreted for two random patients with positive and negative effects. Therefore, we identified 16 responsible genes that can be used to improve HCC diagnosis or treatment. The proposed framework using machine-learning-classification algorithms with the LIME method can be applied to find responsible genes to diagnose and treat HCC patients. Full article

In the first part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude. Full article

In this work, we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes a two-step iteration without surd or complex numbers. The computational test we performed reveals that such a generalization improves the accuracy in computation of the arctangent function by many orders of magnitude with increasing integer M, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of $\pi$ in the Machin-like formulas. Full article

Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of ${10}^{-16}$, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals. Full article

Our research involves analyzing the latest models used for electricity price forecasting, which include both traditional inferential statistical methods and newer deep learning techniques. Through our analysis of historical data and the use of multiple weekday dummies, we have proposed an innovative solution for forecasting electricity spot prices. This solution involves breaking down the spot price series into two components: a seasonal trend component and a stochastic component. By utilizing this approach, we are able to provide highly accurate predictions for all considered time frames. Full article

In interval-valued three-way decision, the reflection of decision-makers’ preference under the full consideration of interval-valued characteristics is particularly important. In this paper, we propose an interval-valued three-way decision model based on the cumulative prospect theory. First, by means of the interval distance measurement method, the loss function and the gain function are constructed to reflect the differences of interval radius and expectation simultaneously. Second, combined with the reference point, the prospect value function is utilized to reflect decision-makers’ different risk preferences for gains and losses. Third, the calculation method of cumulative prospect value for taking action is given through the transformation of the prospect value function and cumulative weight function. Then, the new decision rules are deduced based on the principle of maximizing the cumulative prospect value. Finally, in order to verify the effectiveness and feasibility of the algorithm, the prospect value for decision-making and threshold changes are analyzed under different risk attitudes and different radii of the interval-valued decision model. In addition, compared with the interval-valued decision rough set model, our method in this paper has better decision prospects. Full article

In this work, we studied convergence rates using quotient convergence factors and root convergence factors, as described by Ortega and Rheinboldt, for Hestenes’ Gram–Schmidt conjugate direction method without derivatives. We performed computations in order to make a comparison between this conjugate direction method, for minimizing a nonquadratic function f, and Newton’s method, for solving $\nabla f=0$. Our primary purpose was to implement Hestenes’ CGS method with no derivatives and determine convergence rates. Full article

Numerous studies have established a correlation between creativity and intrinsic motivation to learn, with creativity defined as the process of generating original and valuable ideas, often by integrating perspectives from different fields. The field of educational technology has shown a growing interest in leveraging technology to promote creativity in the classroom, with several studies demonstrating the positive impact of creativity on learning outcomes. However, mining creative thinking patterns from educational data remains a challenging task, even with the proliferation of research on adaptive technology for education. This paper presents an initial effort towards formalizing educational knowledge by developing a domain-specific Knowledge Base that identifies key concepts, facts, and assumptions essential for identifying creativity patterns. Our proposed pipeline involves modeling raw educational data, such as assessments and class activities, as a graph to facilitate the contextualization of knowledge. We then leverage a rule-based approach to enable the mining of creative thinking patterns from the contextualized data and knowledge graph. To validate our approach, we evaluate it on real-world datasets and demonstrate how the proposed pipeline can enable instructors to gain insights into students’ creative thinking patterns from their activities and assessment tasks. Full article

In this research work, a new three-parameter lifetime distribution is introduced and studied. It is called the Harris extended Bilal distribution due to its construction from a mixture of the famous Bilal and Harris distributions, resulting from a branching process. The basic properties, such as the moment generating function, moments, quantile function, and Rényi entropy, are discussed. We show that the hazard rate function has ideal features for modeling increasing, upside-down bathtub, and roller-coaster data sets. In a second part, the Harris extended Bilal model is investigated from a statistical viewpoint. The maximum likelihood estimation is used to estimate the parameters, and a simulation study is carried out. The flexibility of the proposed model in a hydrological data analysis scenario is demonstrated using two practical data sets and compared with important competing models. After that, we establish an acceptance sampling plan that takes advantage of all of the features of the Harris extended Bilal model. The operating characteristic values, the minimum sample size that corresponds to the maximum possible defects, and the minimum ratios of lifetime associated with the producer’s risk are discussed. Full article

The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure, and wobble pairs are known as non-Watson–Crick base pairs. We used topics from RNA combinatorics and Riordan array theory to establish connections among combinatorial objects related to linear trees, lattice walks, and RNA arrays. In this paper, we establish interesting new explicit bijections (one-to-one correspondences) involving certain subclasses of linear trees, lattice walks, and RNA secondary structures. We provide an interesting generalized lattice walk interpretation of RNA array I. In addition, we provide a combinatorial interpretation of RNA array II as RNA secondary structures with $n$ bases and $k$ base-point mutations where ω of the structures contain wobble base pairs. We also establish an explicit bijection between RNA structures with mutations and wobble bases and a certain subclass of lattice walks. Full article

This research studies the occurrence of data breaches in healthcare provider settings regarding patient data. Using visual analytics and data visualization tools, we study the distribution of healthcare breaches by state. We review the main causes and types of breaches, as well as their impact on both providers and patients. The research shows a range of data breach victims. Network servers are the most popular location for common breaches, such as hacking and information technology (IT) incidents, unauthorized access, theft, loss, and improper disposal. We offer proactive recommendations to prepare for a breach. These include, but are not limited to, regulatory compliance, implementing policies and procedures, and monitoring network servers. Unfortunately, the results indicate that the probability of data breaches will continue to rise. Full article

Copula analysis was created to explain the dependence of two or more quantitative variables. Due to the need for in-depth data analysis involving complex variable relationships, there is always a need for new copula models with original features. As a modern example, for the analysis of circular or periodic data types, trigonometric copulas are particularly attractive and recommended. This is, however, an underexploited topic. In this article, we propose a new collection of eight trigonometric and hyperbolic copulas, four based on the sine function and the others on the tangent function, all derived from the construction of the famous Farlie–Gumbel–Morgenstern copula. In addition to their original trigonometric and hyperbolic functionalities, the proposed copulas have the feature of depending on three parameters with complementary roles: one is a dependence parameter; one is a shape parameter; and the last can be viewed as an angle parameter. In our main findings, for each of the eight copulas, we determine a wide range of admissible values for these parameters. Subsequently, the capabilities, features, and functions of the new copulas are thoroughly examined. The shapes of the main functions of some copulas are illustrated graphically. Theoretically, symmetry in general, stochastic dominance, quadrant dependence, tail dependence, Archimedean nature, correlation measures, and inference on the parameters are investigated. Some copula shapes are illustrated with the help of figures. On the other hand, some two-dimensional inequalities are established and may be of separate interest. Full article

We show renormalization in Quantum Brain Dynamics (QBD) in $3+1$ dimensions, namely Quantum Electrodynamics with water rotational dipole fields. First, we introduce the Lagrangian density for QBD involving terms of water rotational dipole fields, photon fields and their interactions. Next, we show Feynman diagrams with 1-loop self-energy and vertex function in dipole coupling expansion in QBD. The counter-terms are derived from the coupling expansion of the water dipole moment. Our approach will be applied to numerical simulations of Kadanoff–Baym equations for water dipoles and photons to describe the breakdown of the rotational symmetry of dipoles, namely memory formation processes. It will also be extended to the renormalization group method for QBD with running parameters in multi-scales. Full article

A multilevel Monte Carlo (MLMC) method is applied to simulate a stochastic optimal problem based on the gradient projection method. In the numerical simulation of the stochastic optimal control problem, the approximation of expected value is involved, and the MLMC method is used to address it. The computational cost of the MLMC method and the convergence analysis of the MLMC gradient projection algorithm are presented. Two numerical examples are carried out to verify the effectiveness of our method. Full article

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we improve the theoretical results presented in a previous paper “A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices” and provide a fast algorithm to find the upper and lower bounds of the interval eigenvalues of a class of symmetric tridiagonal interval matrices. Full article