Search Results (90)

This paper introduces a new family of Szász–Mirakyan-type operators defined by a convex combination of two Poisson-type constructions. The operators preserve the constant function and provide a continuous transition between different exponential behaviors through a parameter sequence. Basic properties of the operators are studied, including the preservation of exponential test functions and the behavior of the first and second central moments. Voronovskaja-type asymptotic results are obtained, describing the effect of the parameter on the asymptotic structure. Moreover, a necessary condition for faster-than $1/n$ approximation is derived. The behavior of the operators is examined through computational evidence, which also confirms the theoretical findings. Full article

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane ${\mathbb{H}}^2$. Given an initial curve ${\gamma }_1$ and a target curve ${\gamma }_2$ of the same hyperbolic length, we evolve ${\gamma }_1$ by a normal speed given by the difference of the reciprocals of geodesic curvatures evaluated at points with the same outward unit normal, together with a time-dependent scalar term $\mathrm{\Gamma }\left(t\right)$ chosen to preserve the hyperbolic length. Using Leichtweiβ’s hyperbolic support function and Howe’s curvature formula, the flow is reformulated as a quasilinear uniformly parabolic equation on ${\mathbb{S}}^1$ with a nonlocal term $\mathrm{\Gamma }\left(t\right)$. We prove short-time existence, uniqueness, and preservation of strict horocyclic convexity. Linearizing the support function equation at the target support function yields a uniformly elliptic operator whose kernel contains the infinitesimal isometry directions. Under a spectral gap assumption on a normalized slice transverse to the isometry orbit, we prove global existence and exponential convergence for initial data sufficiently close to the target curve. In the last section, this assumption is verified explicitly when the target curve is a geodesic circle. Full article

We introduce the one-parameter bounded p-exponential distribution on $(0, p+1)$, which includes the uniform model as a special case and converges pointwise to the exponential law as $p\to \infty$. Closed-form expressions are derived for the CDF and PDF, the survival function, an explicit increasing-failure-rate hazard function, the quantile function (enabling inversion-based simulation), moments, and entropy, along with a constructive scaled beta or Kumaraswamy representation. We also establish stochastic ordering with respect to p in stop-loss and increasing convex order, formalizing how dispersion varies with the parameter while preserving the mean scale. Inference is discussed under parameter-dependent support, a non-regular setting, and we develop and compare several estimation procedures, including a likelihood-based boundary MLE, a variance-matching method-of-moments estimator, and Bayesian estimation under a gamma prior implemented via numerical quadrature or MCMC. Monte Carlo simulation studies evaluate finite-sample performance and interval behavior, and two real-world applications in survival and reliability analysis illustrate competitive goodness-of-fit relative to standard benchmark models. Full article

We study Maximum Entropy density estimation on continuous domains under finitely many moment constraints, formulated as the minimization of the Kullback–Leibler divergence with respect to a reference measure. To model uncertainty in empirical moments, constraints are relaxed through convex penalty functions, leading to an infinite-dimensional convex optimization problem over probability densities. The main contribution of this work is a rigorous convex-analytic treatment of such relaxed Maximum Entropy problems in a functional setting, without discretization or smoothness assumptions on the density. Using convex integral functionals and an extension of Fenchel duality, we show that, under mild and explicit qualification conditions, the infinite-dimensional primal problem admits a dual formulation involving only finitely many variables. This reduction can be interpreted as a continuous-domain instance of partially finite convex programming. The resulting dual problem yields explicit primal–dual optimality conditions and characterizes Maximum Entropy solutions in exponential form. The proposed framework unifies exact and relaxed moment constraints, including box and quadratic relaxations, within a single variational formulation, and provides a mathematically sound foundation for relaxed Maximum Entropy methods previously studied mainly in finite or discrete settings. A brief numerical illustration demonstrates the practical tractability of the approach. Full article

Strongly convex functions form a central subclass of convex functions and have gained considerable attention due to their structural advantages and broad applicability, particularly in optimization and information theory. In this paper, we investigate the class of strongly F-convex functions, which generalizes the classical notion of strong convexity by introducing an auxiliary convex control function F. We establish several fundamental structural characterizations of this class and provide a variety of nontrivial examples such as power, logarithmic, and exponential functions. In addition, we derive refined Jensen-type and Hermite–Hadamard-type inequalities adapted to the strongly F-convex concept, thereby extending and sharpening their classical forms. As applications, we obtain new analytical inequalities and improved error bounds for entropy-related quantities, including Shannon, Tsallis, and Rényi entropies, demonstrating that the concept of strong F-convexity naturally yields strengthened divergence and uncertainty estimates. Full article

This study looks at the stability and stabilization issues concerning the nonlinear time-delay systems specified by conformable derivatives. These requirements can be used for many useful applications. Through the construction of appropriate Lyapunov–Krasovskii functionals, we develop novel linear matrix inequality (LMI) conditions for the exponential stability of autonomous systems and practical exponential stability for systems subject to bounded perturbations. Furthermore, we propose state-feedback stabilization strategies that transform the controller design problem into a convex optimization framework solvable via efficient LMI techniques. The theoretical developments are comprehensively validated through numerical examples that demonstrate the effectiveness of the proposed stability and stabilization criteria. The results establish a rigorous framework for analyzing and controlling conformable fractional-order systems with time delays, bridging theoretical advances with practical implementation considerations. Full article

In this paper, we introduce a new class of analytic functions, denoted by $\mathfrak{S}(\nu ,{\phi }_{\vartheta ,e})$, and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter et al. in relation to the exponential function. A significant contribution of this work is the derivation of sharp bounds for various coefficient-related problems within this class. The computational challenges involved in deriving these bounds were effectively addressed using Mathematica^TM codes. Additionally, figures illustrating the geometric properties and essential computations have been incorporated into the paper. Full article

Background: The management of inventory under realistic supply chain disruptions, which are often non-exponential, challenges classical control theory. This study addresses the critical question of whether the optimality of simple base-stock policies holds under the combined influence of non-exponential disruptions and random yield. Methods: We model the system as a Piecewise Deterministic Markov Process (PDMP) with impulse control, using Phase-Type (PH) distributions to capture non-memoryless event timings. The analysis involves proving the existence of a solution to the Average Cost Optimality Equation (ACOE) via a vanishing discount approach, and the framework is validated with a numerical experiment. Results: Our primary finding is a rigorous proof that a state-dependent base-stock policy is optimal, a significant generalisation of classical theory. We establish this by demonstrating the value function’s convexity. The numerical experiment quantifies the significant cost penalties (over 12%) incurred by using simpler, memoryless models for supply disruptions. Conclusions: The study provides a crucial theoretical justification for the robustness of simple threshold-based control policies in complex, realistic settings. It highlights for managers the importance of modelling the variability of disruptions, not just their average duration, to avoid costly strategic errors. Full article

A Multi-Layer Perceptron (MLP), as the basic structure of neural networks, is an important component of various deep learning models such as CNNs, RNNs, and Transformers. Nevertheless, MLP training faces significant challenges, with a large number of saddle points and local minima in its non-convex optimization space, which can easily lead to gradient vanishing and premature convergence. Compared with traditional heuristic algorithms relying on a population-based parallel search, such as GA, GWO, DE, etc., the Besiege and Conquer Algorithm (BCA) employs a one-spot update strategy that provides a certain level of global optimization capability but exhibits clear limitations in search flexibility. Specifically, it lacks fast detection, fast adaptation, and fast convergence. First, the fixed sinusoidal amplitude limits the accuracy of fast detection in complex regions. Second, the combination of a random location and fixed perturbation range limits the fast adaptation of global convergence. Finally, the lack of a hierarchical adjustment under a single parameter (BCB) hinders the dynamic transition from exploration to exploitation, resulting in slow convergence. To address these limitations, this paper proposes a Flexible Besiege and Conquer Algorithm (FBCA), which improves search flexibility and convergence capability through three new mechanisms: (1) the sine-guided soft asymmetric Gaussian perturbation mechanism enhances local micro-exploration, thereby achieving a fast detection response near the global optimum; (2) the exponentially modulated spiral perturbation mechanism adopts an exponential spiral factor for fast adaptation of global convergence; and (3) the nonlinear cognitive coefficient-driven velocity update mechanism improves the convergence performance, realizing a more balanced exploration–exploitation process. In the IEEE CEC 2017 benchmark function test, FBCA ranked first in the comprehensive comparison with 12 state-of-the-art algorithms, with a win rate of 62% over BCA in 100-dimensional problems. It also achieved the best performance in six MLP optimization problems, showing excellent convergence accuracy and robustness, proving its excellent global optimization ability in complex nonlinear MLP optimization training. It demonstrates its application value and potential in optimizing neural networks and deep learning models. Full article

This study provides new equivalent descriptions of the Bounded Lower Oscillation ($\mathrm{BLO}$) space through Luxemburg-type $L^{\phi }$ integrability conditions, where $\phi$ is a nonnegative function with either convexity or concavity. The framework accommodates various representative forms of $\phi$, such as the power function $\phi \left(t\right)=t^p$, exponential-type functions $\phi \left(t\right)=e^{pt}-1$, and logarithmic functions $\phi \left(t\right)={log}_{+}^{k}t$, with parameters $p\in (0,\infty )$ and $k\in \mathbb{N}$. These results unify and extend existing characterizations of $\mathrm{BLO}$ by encompassing a broad class of generating functions. Full article

Correlations have been found for the base value of hardness (as the ratio between the heat of fusion and molar volume) and the softening temperature (as the ratio of heat of fusion and specific heat capacity). The relative change of bulk hardness as a function of temperature, H(T), is studied by three new parametric formulas beside the well-known exponential decay and Arrhenius-type expressions. Mathematically, two formulas can be considered as deriving from the exponential decay; the third one is a new rational fraction expression based on the power of normalized temperature. The normalizing temperature is taken as the softening temperature. In the Arrhenius expression, a temperature-dependent activation energy is introduced, which increases steadily with heating but never surpasses the value of self-diffusion. This rational fracture expression has been shown to be applicable to both pure metals and alloys with arbitrary H(T) curve shapes, from convex (pure metals) to concave (alloys). A detailed description of the fitting of these parametric formulas is given, applying the H(T) data from the literature and from our own measurements. Measuring our refractory high entropy alloy (RHEA) samples, an early softening temperature, smaller than the expected half of the melting point (Ts < Tm/2) was detected, signaling a phase instability in the case of Ti-, Zr- and Hf-containing alloys. Full article

Vertical structure monitoring of urban vegetation provides data support for urban green space planning and ecological management, playing a significant role in promoting sustainable urban ecological development. Three-dimensional green volume (3DGV) is a comprehensive index used to characterize the ecological benefit of urban vegetation. As a critical component of urban vegetation, street trees play a key role in urban ecological benefits evaluation, and the quantitative estimation of their 3DGV serves as the foundation for this assessment. However, current methods for measuring 3DGV based on point cloud data often suffer from issues of overestimation or underestimation. To improve the accuracy of the 3DGV for urban street trees, this study proposed a novel approach that used convex hull coupling k-means clustering convex hulls. A new method based on terrestrial laser scanning (TLS) data was proposed, referred to as the Convex Hull Coupling Method (CHCM). This method divides the tree crown into two parts in the vertical direction according to the point cloud density, which better adapts to the lower density of the upper layer of TLS data and obtains a more accurate 3DGV of individual trees. To validate the effectiveness of the CHCM method, 30 sycamore (Platanus × acerifolia (Aiton) Willd.) plants were used as research objects. We used the CHCM and five traditional 3DGV calculation methods (frustum method, convex hull method, k-means clustering convex hulls, alpha-shape algorithm, and voxel-based method) to calculate the 3DGV of individual trees. Additionally, the 3DGV was predicted and analyzed using five fitting models. The results show the following: (1) Compared with the traditional methods, the CHCM improves the estimation accuracy of the 3DGV of individual trees and shows a high consistency in the data verification, which indicates that the CHCM method is stable and reliable, and (2) the fitting results R² of the five models were all above 0.75, with the exponential function model showing the best fitting accuracy (R² = 0.89, RMSE = 74.85 m³). These results indicate that for TLS data, the CHCM can achieve more accurate 3DGV estimates for individual trees, outperforming traditional methods in both applicability and accuracy. The research results not only offer a novel technical approach for 3DGV calculation using TLS data but also establish a reliable quantitative foundation for the scientific assessment of the ecological benefits of urban street trees and green space planning. Full article

Maritime transportation is responsible for most global trade and is generally considered more environmentally efficient compared to other modes of transport, particularly for long-distance trade. With increasingly stringent emission regulations, however, accurately quantifying emissions and identifying their key determinants has become essential for effective environmental management. This study introduced a structured and comparative statistical modeling framework for ship-based emission modeling using gross tonnage (GT) as the primary predictor variable, due to its strong correlation with emission levels. Emissions for hydrocarbon (HC), carbon monoxide (CO), particulate matter with an aerodynamic diameter of less than 10 μm (PM10), carbon dioxide (CO₂), sulfur dioxide (SO₂), nitrogen oxides (NO_x), and volatile organic compounds (VOC) were estimated using a bottom-up approach based on emission factors and formulas defined by the U.S. Environmental Protection Agency (EPA), using data from 38,304 vessel movements through the Bosphorus in 2021. These EPA-estimated values served as dependent variables in the modeling process. The modeling framework followed a three-step strategy: (1) outlier detection using Rosner’s test to reduce the influence of outliers on model accuracy, (2) curve fitting with 12 regression models representing four curve types—polynomial (e.g., linear, quadratic), concave/convex (e.g., exponential, logarithmic), sigmoidal (e.g., logistic, Gompertz, Weibull), and spline-based (e.g., cubic spline, natural spline)—to capture diverse functional relationships between GT and emissions, and (3) model comparison using difference performance metrics to ensure a comprehensive assessment of predictive accuracy, consistency, and bias. The findings revealed that nonlinear models outperformed polynomial models, with spline-based models—particularly natural spline and cubic spline—providing superior accuracy for HC, PM₁₀, SO₂, and VOC, and the Weibull model showing strong predictive performance for CO and NO_x. These results underscore the necessity of using pollutant-specific and flexible modeling strategies to capture the intricacies of maritime emission dynamics. By demonstrating the advantages of flexible functional forms over standard regression techniques, this study highlights the need for tailored modeling strategies to better capture the complex relationships in maritime emission data and offers a scalable and transferable framework that can be extended to other vessel types, emission datasets, or maritime regions. Full article

This paper seeks to present the fundamental features of the category of conditional exponential convex functions (CECFs). Additionally, the study of continuous CECFs contributes to the characterization of convolution semigroups. In this context, we expand our focus to include a much broader class of Gaussian processes, where we define the generalized Fourier transform in a more straightforward manner. This approach is closely connected to the method by which we derived the Gaussian process, utilizing the framework of a Gelfand triple and the theorem of Bochner–Minlos. A part of this work involves constructing the reproducing kernel Hilbert spaces (RKHS) directly from CECFs. Full article

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space $\left({\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when $\beta =0$ and to classical Lebesgue spaces when $\mathtt{q}=\infty ,\beta =0$. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis. Full article