Search Results (55)

This study proposes an intelligent prediction framework integrating native sparse attention (NSA) with the Chen-Guan (CHG) algorithm to optimize tunnel boring machine (TBM) operations in heterogeneous geological environments. The framework resolves critical limitations of conventional experience-driven approaches that inadequately address the nonlinear coupling between the spatial heterogeneity of rock mass parameters and mechanical system responses. Three principal innovations are introduced: (1) a hardware-compatible sparse attention architecture achieving O(n) computational complexity while preserving high-fidelity geological feature extraction capabilities; (2) an adaptive kernel function optimization mechanism that reduces confidence interval width by 41.3% through synergistic integration of boundary likelihood-driven kernel selection with Chebyshev inequality-based posterior estimation; and (3) a physics-enhanced modelling methodology combining non-Hertzian contact mechanics with eddy field evolution equations. Validation experiments employing field data from the Pujiang Town Plot 125-2 Tunnel Project demonstrated superior performance metrics, including 92.4% ± 1.8% warning accuracy for fractured zones, ≤28 ms optimization response time, and ≤4.7% relative error in energy dissipation analysis. Comparative analysis revealed a 32.7% reduction in root mean square error (p < 0.01) and 4.8-fold inference speed acceleration relative to conventional methods, establishing a novel data–physics fusion paradigm for TBM control with substantial implications for intelligent tunnelling in complex geological formations. Full article

In the modern era of informatics, where data are very important, efficient management of data is necessary and critical. Two of the most important data management techniques are searching and data ordering (technically sorting). Traditional sorting algorithms work in quadratic time $\left(\mathcal{O}\left(x^2\right)\right)$, and in the optimized cases, they take linearithmic time $\left(\mathcal{O}\left(x·\log x\right)\right)$, with no existing method surpass this lower bound, given arbitrary data, i.e., ordering a list of cardinality x in $\left(\mathcal{O}\left(x·\log x\right)-ϵ\left(x\right)\right)\forall ϵ\left(x\right)>0$. This research proposes Search-o-Sort, which reinterprets sorting in terms of searching, thereby offering a new framework for ordering arbitrary data. The framework is applied to classical search algorithms,–Linear Search, Binary Search (in general, k-ary Search), and extended to more optimized methods such as Interpolation and Jump Search. The analysis suggests theoretical pathways to reduce the computational complexity of sorting algorithms, thus enabling algorithmic development based on the proposed viewpoint. Full article

On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article. Full article

In this article, we give a brief review of a well-known integral inequality that gives information about the integral of the product of two functions using synchronous functions, the Chebyshev inequality. We have compiled the most relevant information about fractional and generalized integrals, which are one of the most dynamic topics in today’s mathematical sciences. After presenting the classical formulation of the inequality using Lebesgue integrable functions, the most general results known from the literature are collected in an attempt to present the reader with a current overview of this research topic. Full article

The core idea of prediction-based anomaly detection is to identify anomalies by constructing a prediction model and comparing predicted and observed values. However, most existing traffic flow prediction models primarily focus on spatio-temporal features, neglecting comprehensive frequency-domain feature learning. Additionally, anomaly detection accuracy is often limited by insufficient prediction error analysis. To address this limitation, this paper proposes a prediction-based anomaly detection method for traffic flow data with multi-domain feature extraction. The prediction model is built as follows: first, Bidirectional Long Short-Term Memory network (Bi-LSTM) and a Graph Attention Network (GAT) extract temporal and spatial features, respectively. Then, Fast Fourier Transform (FFT) converts time-domain signals into the frequency domain, where Transformer learns magnitude and phase features. Finally, a prediction model is constructed using the extracted time-domain and frequency-domain features. For error analysis, this paper innovatively applies Chebyshev’s inequality to determine the error threshold, identifying anomalies based on whether errors exceed this threshold. Experimental results show that integrating multi-domain features can more comprehensively capture data characteristics and improve model prediction accuracy. In the anomaly detection experiment, it was verified that constructing a high-accuracy prediction model and conducting reasonable error analysis can effectively enable anomaly detection in the data. Full article

Outliers are typically identified using frequentist methods. The data are classified as “outliers” or “not outliers” based on a test statistic that measures the magnitude of the difference between a value and the majority part of the data. The threshold for a data value to be an outlier is typically defined by the user. However, a subjective choice of the threshold increases the uncertainty associated with outlier status for each data value. A cellwise outlier detection algorithm named FuzzyHRT is used to automate the editing process in repeated surveys. This algorithm uses Bienaymé–Chebyshev’s inequality and fuzzy logic to detect four different types of outliers resulting from format inconsistencies, historical, tail, and relational anomalies. However, fuzzy logic is not suited for probabilistic reasoning behind the identification of anomalous cells. Bayesian methods are well suited for quantifying the uncertainty associated with the identification of outliers. Although, as suggested by the literature, there exist well-developed Bayesian methods for record-level outlier detection, Bayesian methods for identifying outliers within individual records (i.e., at the cell level) remain unexplored. This paper presents two approaches from the Bayesian perspective to study the uncertainty associated with identifying outliers. A Bayesian bootstrap approach is explored to study the uncertainty associated with the output scores from the FuzzyHRT algorithm. Empirical likelihoods in a Bayesian setting are also considered for probabilistic reasoning behind the identification of anomalous cells. NASS survey data for livestock and major crop yield (such as corn) are considered for comparing the performances of the two proposed approaches with recent cellwise outlier methods. Full article

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for $\mathrm{p}=2$ to results in the ${\mathbb{L}}^{\mathrm{p}}$ space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. Full article

By means of the functional analysis theory, reorder method, mathematical induction and the dimension reduction method, the Chebyshev-Jensen-type inequalities involving the $\chi$-products ${⟨·⟩}_{\chi }$ and ${[·]}_{\chi }$ are established, and we proved that our main results are the generalizations of the classical Chebyshev inequalities. As applications in probability theory, the discrete with continuous probability inequalities are obtained. Full article

This study aims to improve the quality of the clustering results of the density peak clustering (DPC) algorithm and address the privacy protection problem in the clustering analysis process. To achieve this, a DPC algorithm based on Chebyshev inequality and differential privacy (DP-CDPC) is proposed. Firstly, the distance matrix is calculated using cosine distance instead of Euclidean distance when dealing with high-dimensional datasets, and the truncation distance is automatically calculated using the dichotomy method. Secondly, to solve the difficulty in selecting suitable clustering centers in the DPC algorithm, statistical constraints are constructed from the perspective of the decision graph using Chebyshev inequality, and the selection of clustering centers is achieved by adjusting the constraint parameters. Finally, to address the privacy leakage problem in the cluster analysis, the Laplace mechanism is applied to introduce noise to the local density in the process of cluster analysis, enabling the privacy protection of the algorithm. The experimental results demonstrate that the DP-CDPC algorithm can effectively select the clustering centers, improve the quality of clustering results, and provide good privacy protection performance. Full article

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented. Full article

Sea-level rise, population growth, and changing land-use patterns will further constrain Florida’s already scarce groundwater and surface water supplies in the coming decades. Significant investments in water supply and water demand management are needed to ensure sufficient water availability for human and natural systems. Section 403.928 (1) (b) of the Florida Statutes requires estimating the expenditures needed to meet the future water demand and avoid the adverse effects of competition for water supplies to 2040. This study considers the 2020–2040 planning period and projects (1) future water demand and supplies; and (2) the total expenditures (capital costs) necessary to meet the future water demand in Florida, USA. The uniqueness of this study compared with the previous studies is the introduction of a probabilistic-based approach to quantify the uncertainty of the investment costs to meet future water demand. We compile data from the U.S. Geological Survey, Florida’s Department of Agriculture & Consumer Services, Florida’s Water Management Districts, and the Florida Department of Environmental Protection to project the future water demand and supplies, and the expenditures needed to meet the demand considering uncertainty in the costs of alternative water supply options. The results show that the total annual water demand is projected to increase by 1405 million cubic meters (+15.9%) by 2040, driven primarily by urbanization. Using the median capital costs of alternative water supply projects, cumulative expenditures for the additional water supplies are estimated between USD 1.11–1.87 billion. However, when uncertainty in the project costs is accounted for, the projected expenditure range shifts to USD 1.65 and USD 3.21 billion. In addition, we illustrate how using Modern Portfolio Theory (MPT) can increase the efficacy of investment planning to develop alternative water supply options. The results indicate that using MPT in selecting the share of each project type in developing water supply options can reduce the standard deviation of capital costs per one unit of capacity by 74% compared to the equal share allocation. This study highlights the need for developing more flexible funding strategies on local, regional, and state levels to finance additional water supply infrastructure, and more cost-effective combinations of demand management strategies and alternative water supply options to meet the water needed for the state in the future. Full article

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work. Full article

Integral inequalities make up a comprehensive and prolific field of research within the field of mathematical interpretations. Integral inequalities in association with convexity have a strong relationship with symmetry. Different disciplines of mathematics and applied sciences have taken a new path as a result of the development of new fractional operators. Different new fractional operators have been used to improve some mathematical inequalities and to bring new ideas in recent years. To take steps forward, we prove various Grüss-type and Chebyshev-type inequalities for integrable functions in the frame of non-conformable fractional integral operators. The key results are proven using definitions of the fractional integrals, well-known classical inequalities, and classical relations. Full article

The main objective of this paper is to use the generalized proportional Hadamard fractional integral operator to establish some new fractional integral inequalities for extended Chebyshev functionals. In addition, we investigate some fractional integral inequalities for positive continuous functions by employing a generalized proportional Hadamard fractional integral operator. The findings of this study are theoretical but have the potential to help solve additional practical problems in mathematical physics, statistics, and approximation theory. Full article