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Positive and unlabeled learning (PU learning) is a significant binary classification task in machine learning; it focuses on training accurate classifiers using positive data and unlabeled data. Most of the works in this area are based on a two-step strategy: the first step is to identify reliable negative examples from unlabeled examples, and the second step is to construct the classifiers based on the positive examples and the identified reliable negative examples using supervised learning methods. However, these methods always underutilize the remaining unlabeled data, which limits the performance of PU learning. Furthermore, many methods require the iterative solution of the formulated quadratic programming problems to obtain the final classifier, resulting in a large computational cost. In this paper, we propose a new method called the absolute value inequality support vector machine, which applies the concept of eccentricity to select reliable negative examples from unlabeled data and then constructs a classifier based on the positive examples, the selected negative examples, and the remaining unlabeled data. In addition, we apply a hyperparameter optimization technique to automatically search and select the optimal parameter values in the proposed algorithm. Numerical experimental results on ten real-world datasets demonstrate that our method is better than the other three benchmark algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p = − 1 , our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

A multivariate folded normal distribution is a distribution of the absolute value of a Gaussian random vector. In this paper, we provide the marginal and conditional distributions of the multivariate folded normal distribution, and also prove that independence and non-correlation are equivalent for it. In addition, we provide a numerical approach using the R language to fit a multivariate folded normal distribution. The accuracy of the estimated mean and variance parameters is then examined. Finally, a real data application to body mass index data are presented. [ABSTRACT FROM AUTHOR]

In this paper, the representability of the generalized Student's distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws. [ABSTRACT FROM AUTHOR]

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. [ABSTRACT FROM AUTHOR]

In this paper, the two-step method is considered with the generalized Newton method as a predictor step. The three-point Newton–Cotes formula is taken as a corrector step. The proposed method's convergence is discussed in detail. This method is very simple and therefore very effective for solving large systems. In numerical analysis, we consider a beam equation, transform it into a system of absolute value equations and then use the proposed method to solve it. Numerical experiments show that our method is very accurate and faster than already existing methods. [ABSTRACT FROM AUTHOR]

The paper describes PCDM and PCDM4MP as new tools and commands capable of exploring large datasets. They select variables based on identifying the absolute values of Pearson's pairwise correlation coefficients between a chosen response variable and any other existing in the dataset. In addition, for each pair, they also report the corresponding significance and the number of non-null intersecting observations, and all this reporting is performed in a record-oriented manner (both source and output). Optionally, using threshold values for these three as parameters of PCDM, any user can select the most correlated variables based on high magnitude, significance, and support criteria. The syntax is simple, and the tools show the exploration progress in real-time. In addition, PCDM4MP can trigger different instances of Stata, each using a distinct class of variables belonging to the same dataset and resulting after simple name filtering (first letter). Moreover, this multi-processing (MP) version overcomes the parallelization limitations of the existing parallel module, and this is accomplished by using vertical instead of horizontal partitions of large flat datasets, dynamic generation of the task pattern, tasks, and logs, all within a single execution of this second command, and the existing qsub module to automatically and continuously allocate the tasks to logical processors and thereby emulating with fewer resources a cluster environment. In addition, any user can perform further selections based on the results printed in the console. The paper contains examples of using these tools for large datasets such as the one belonging to the World Values Survey and based on a simple variable naming practice. This article includes many recorded simulations and presents performance results. They depend on different resources and hardware configurations used, including cloud vs. on-premises, large vs. small amounts of RAM and processing cores, and in-memory vs. traditional storage. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method was considered for solving the absolute value equation (AVE). We suggest a two-step method in which the well-known Gauss quadrature rule is the corrector step and the generalized Newton method is taken as the predictor step. The convergence of the proposed method is established under some acceptable conditions. Numerical examples prove the consistency and capability of this new method. [ABSTRACT FROM AUTHOR]

Overfitting often occurs in neural network training, and neural networks with higher generalization ability are less prone to this phenomenon. Aiming at the problem that the generalization ability of photovoltaic (PV) power prediction model is insufficient, a PV power time-sharing prediction (TSP) model combining variational mode decomposition (VMD) and Bayesian regularization neural network (BRNN) is proposed. Firstly, the meteorological sequences related to the output power are selected by mutual information (MI) analysis. Secondly, VMD processing is performed on the filtered sequences, which is aimed at reducing the non-stationarity of the data; then, normalized cross-correlation (NCC) and signal-to-noise ratio (SNR) between the components obtained by signal decomposition and the original data are calculated, after which the key influencing factors are screened out to eliminate the correlation and redundancy of the data. Finally, the filtered meteorological sequences are divided into two datasets based on whether the irradiance of the day is zero or not. Meanwhile, the predictions are performed using BRNN for each of the two datasets. Then, the results are reordered in chronological order, and the prediction of PV power is realized conclusively. It was experimentally verified that the mean absolute value error ( M A E ) of the method proposed in this paper is 0.1281 , which is reduced by 40.28 % compared with the back propagation neural network (BPNN) model on the same dataset, the mean squared error ( M S E ) is 0.0962 , and the coefficient of determination ( R 2 ) is 0.9907 . Other error indicators also confirm that VMD is of much significance and TSP is contributive. [ABSTRACT FROM AUTHOR]

For a simple graph with vertex set { v 1 , v 2 , ... , v n } with degree sequence d v i of vertex v i , i = 1 , 2 , ... , n , the inverse sum indeg matrix ( I S I -matrix) A I S I (G) = (a i j) n × n of G is defined by a i j = d v i d v j d v i + d v j , if v i is adjacent to v j , and zero, otherwise. The multiset of eigenvalues of A I S I (G) is the I S I -spectrum of G and the sum of their absolute values is the I S I -energy of G . In this paper, we modify the two results of (Li, Ye and Broersma, 2022), give the correct characterization of the extremal graphs and thereby obtain better bounds than the already known results. Moreover, we also discuss the QSPR analysis and carry the statistical modelling (linear, logarithmic and quadratic) of the physicochemical properties of anticancer drugs with the I S I -index (energy). [ABSTRACT FROM AUTHOR]

In this paper, a tensor splitting AOR iterative method for solving the H + -tensor absolute value equation is presented. Some sufficient conditions for the existence of the solution to the tensor absolute value equation are given. Under suitable conditions, the new method is proved to be convergent. Finally, some numerical examples demonstrate that our new method is effective. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given. [ABSTRACT FROM AUTHOR]

In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities. [ABSTRACT FROM AUTHOR]

Penetration and shared nodes between muscles, tendons and the plantar aponeurosis mesh elements in finite element models of the foot may cause inappropriate structural behavior of the tissues. Penetration between tissues caused using separate mesh without motion constraints or contacts can change the loading direction because of an inadequate mesh displacement. Shared nodes between mesh elements create bonded areas in the model, causing progressive or complete loss of load transmitted by tissue. This paper compares by the finite element method the structural behavior of the foot model in cases where a shared mesh has been used versus a separated mesh with sliding contacts between some important tissues. A very detailed finite element model of the foot and ankle that simulates the muscles, tendons and plantar aponeurosis with real geometry has been used for the research. The analysis showed that the use of a separate mesh with sliding contacts and a better characterization of the mechanical behavior of the soft tissues increased the mean of the absolute values of stress by 83.3% and displacement by 17.4% compared with a shared mesh. These increases mean an improvement of muscle and tendon behavior in the foot model. Additionally, a better quantitative and qualitative distribution of plantar pressure was also observed. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices and m edges, with maximum degree Δ (G) and minimum degree δ (G). Let A be the adjacency matrix of G, and let λ 1 ≥ λ 2 ≥ ... ≥ λ n be the eigenvalues of G. The energy of G, denoted by E (G) , is defined as the sum of the absolute values of the eigenvalues of G, that is E (G) = | λ 1 | + ... + | λ n | . The energy of G is known to be at least twice the minimum degree of G, E (G) ≥ 2 δ (G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E (G) ≥ Δ (G) + δ (G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E (G) . The results rely on elementary inequalities and their application. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n. If the maximal connected subgraph of G has no cut vertex then it is called a block. If each block of graph G is a clique then G is called clique tree. The distance energy E D (G) of graph G is the sum of the absolute values of the eigenvalues of the distance matrix D (G) . In this paper, we study the properties on the eigencomponents corresponding to the distance spectral radius of some special class of clique trees. Using this result we characterize a graph which gives the maximum distance spectral radius among all clique trees of order n with k cliques. From this result, we confirm a conjecture on the maximum distance energy, which was given in Lin et al. Linear Algebra Appl 467(2015) 29-39. [ABSTRACT FROM AUTHOR]

Variability is defined as the propensity at which a given signal is likely to change. There are many choices for measuring variability, and it is not generally known which ones offer better properties. This paper compares different variability metrics applied to irregularly (nonuniformly) sampled time series, which have important clinical applications, particularly in mental healthcare. Using both synthetic and real patient data, we identify the most robust and interpretable variability measures out of a set 21 candidates. Some of these candidates are also proposed in this work based on the absolute slopes of the time series. An additional synthetic data experiment shows that when the complete time series is unknown, as it happens with real data, a non-negligible bias that favors normalized and/or metrics based on the raw observations of the series appears. Therefore, only the results of the synthetic experiments, which have access to the full series, should be used to draw conclusions. Accordingly, the median absolute deviation of the absolute value of the successive slopes of the data is the best way of measuring variability for this kind of time series. [ABSTRACT FROM AUTHOR]

Due to the discreteness of integer data, there are a large number of gaps and continuous columns in the histogram based on integer data. Aiming at the characteristics, this paper presents a robust and reversible watermarking algorithm for a relational database based on continuous columns in histogram. Firstly, it groups the database tuples according to the watermark length and the grouping key. Secondly, it calculates the prediction errors and uses the absolute values of the prediction errors to construct the histogram. Thirdly, it traverses the histogram to find all the continuous columns and in turn, computes the sum of the height of each continuous column and selects the group of continuous columns that has the largest sum as the positions to embed the watermarks. FCTD (Forest cover type data set) is utilized for experimental verification. A large amount of experimental data shows that the method is effective and robust. Not only does the data distortion caused by shifting histogram columns not exist, but the robustness of the watermark is also greatly improved. [ABSTRACT FROM AUTHOR]

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model. [ABSTRACT FROM AUTHOR]

This paper provides results on the sign of the Green function (and its partial derivatives) of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed. [ABSTRACT FROM AUTHOR]

In this paper, we present the preliminaries of (p , q) -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over [ a , b ] × [ c , d ] by using the (p , q) -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the (p , q) -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the (p , q) -calculus of the functions of two variables. [ABSTRACT FROM AUTHOR]

In this paper, we introduced some new integral inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute values at certain powers are strongly η -convex functions via the Katugampola fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite–Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex. As an application, we derive certain inequalities involving the q-digamma and q-polygamma functions, respectively. As a consequence, new inequalities for the q-analogue of the harmonic numbers in terms of the q-polygamma functions are derived. Moreover, several inequalities for special means are also considered. [ABSTRACT FROM AUTHOR]