Search Results (92)

Objective: Individuals with reactive hypoglycemia (RH) may be more likely to develop obesity and type 2 diabetes, but the ability to identify RH has been hampered by the lack of clear criteria. This study used calculus-based curve parameters from a mixed macronutrient liquid meal test (MMTT) to define RH in men and women with obesity. Methods: A total of 69 non-diabetic adults aged 35 ± 8.3 years with obesity (BMI 32.3 ± 4.2 kg/m²) underwent a 4 h MMTT to define RH, and an intravenous glucose tolerance test (IVGTT) to characterize RH (via insulin sensitivity, the acute insulin response to glucose (AIRg), insulin clearance, and the disposition index). Perceived hunger and fullness were assessed by visual analog scale. Results: RH was defined using curve properties of the MMTT. A total of 19 of the 69 participants had a reactive hypoglycemic response to the MMTT. Glucose AUC and nadir were lower, timing of glucose nadir was earlier, and insulin sensitivity was higher in RH compared to non-RH. Sex (female) and race (AA) were significant predictors of RH presence. Conclusions: Among individuals with obesity, RH is characterized by greater sensitivity to insulin and greater disposition index. We introduce a novel and reproducible method to define RH using curve-based criteria from a mixed meal test integrated with gold-standard IVGTT-derived outcomes. Full article

This study investigates the critical role of resonance capture dynamics in determining the energy dissipation performance of nonlinear energy sinks (NES). A fluid inerter combining mass amplification and damping characteristics is proposed as a core component, based on which two configurations of fractional-order NES (configured in series and parallel) are systematically constructed. The applicability of the complex averaging method in fractional-order systems has been addressed through fractional calculus (such as Leibniz properties), enabling it to be analyzed like integer-order systems. Employing the multi-scale perturbation method, the energy transfer mechanism between the primary oscillator and the NES is derived, leading to the analytical determination of optimal cubic stiffness and maximum energy transfer efficiency. Comparative simulation shows that the parameters of the inerter directly affect the magnitude of critical damping. The optimal cubic stiffness design method is more reliable than traditional methods and can ensure effective target energy transfer triggering. Further analysis of dissipation time shows that the performance of fractional-order NES is superior to integer-order NES; notably, the dissipation time of series fractional-order NES is significantly shorter than that of parallel and traditional NES. In summary, this study provides theoretical guidance for the design of lightweight and high-performance NES and will also promote the application of fractional calculus theory in the field of engineering vibration reduction. Full article

This article presents an improved formulation of Newton’s law of cooling using the conformable fractional derivative to model long-term thermal behavior more accurately. A key feature of our approach is the use of the fractional time variable $t^{\gamma }$, which introduces a simple scaling symmetry: the structure of the model remains unchanged even when time is proportionally stretched or compressed. This symmetry-based property provides additional flexibility compared to the classical formulation and enables the derivation of analytical solutions under both constant and non-constant ambient temperature. In particular, we incorporate sinusoidal models for ambient temperature to capture realistic environmental fluctuations over extended periods. Experimental measurements confirm that the conformable model achieves significantly better accuracy than traditional integer-order models. These results highlight the relevance of symmetry and fractional calculus in describing physical processes and demonstrate the potential of conformable methods for improving long-term thermal predictions. Full article

Quaternion-valued neural networks (QVNNs) that have multiple types of delays (leakage, time-varying, distributed, and neutral) and defined on time scales are discussed in this paper. Quaternions form a 4D normed division algebra and allow for a better representation of 3D and 4D data. QVNNs have been proposed and applications have appeared lately. Time-scale calculus was developed to allow the joint treatment of systems, or any hybrid mixing of them, and was also applied with success to the analysis of dynamic properties for neural networks (NNs). Because of its generality, encompassing the common properties of discrete-time (DT) and continuous-time (CT) NNs, time-scale NNs dynamics research does not benefit from a fully-developed Lyapunov theory. So, Halanay-type inequalities have to be used instead. To this end, we provide a novel generalization of inequalities of Halanay-type on time scales specifically suited for neutral systems, i.e., systems with neutral delays. Then, this new lemma is employed to obtain sufficient conditions presented both as linear matrix inequalities (LMIs) and as algebraic inequalities for the exponential stability and exponential synchronization of QVNNs on time scales with the mentioned delay types. The model put forward in this paper has a generality which is appealing for practical applications, in which both DT and CT dynamics are interesting, and all the discussed types of delays appear. For both the DT and CT scenarios, four numerical applications are used to illustrate the four theorems put forward in this research. Full article

This work presents refined and generalized forms of weighted Opial-type inequalities within the framework of time scale calculus. The proofs rely on several algebraic techniques, together with Hölder’s inequality and Keller’s chain rule. These results extend the classical Opial-type inequalities by embedding them into the time scale setting, which unifies both continuous and discrete analyses. Consequently, various integral and discrete inequalities emerge as particular cases of our main results, thereby broadening the applicability of Opial-type inequalities to dynamic systems and discrete models. Full article

This study introduces Itô-RMSProp, a novel extension of the RMSProp optimizer inspired by Itô stochastic calculus, which integrates adaptive Gaussian noise into the update rule to enhance exploration and mitigate overfitting during training. We embed this optimizer within Gated Recurrent Unit (GRU) networks for stock price forecasting, leveraging the GRU’s strength in modeling long-range temporal dependencies under nonstationary and noisy conditions. Extensive experiments on real-world financial datasets, including a detailed sensitivity analysis over a wide range of noise scaling parameters ($\epsilon$), reveal that Itô-RMSProp-GRU consistently achieves superior convergence stability and predictive accuracy compared to classical RMSProp. Notably, the optimizer demonstrates remarkable robustness across all tested configurations, maintaining stable performance even under volatile market dynamics. These findings suggest that the synergy between stochastic differential equation frameworks and gated architectures provides a powerful paradigm for financial time series modeling. The paper also presents theoretical justifications and implementation details to facilitate reproducibility and future extensions. Full article

Fractional-order chaotic systems have received increasing attention over the past few years due to their ability to effectively model memory and complexity in nonlinear dynamics. Nonetheless, most of the research conducted so far has been on constant-order formulations, which still have some limitations in terms of adaptability and reality. Thus, to evade these limitations, we present a recently designed four-dimensional hyperchaotic Chen system with variable-order fractional (VOF) derivatives in the Liouville–Caputo sense. In comparison with constant-order systems, the new system possesses excellent performance in numerous aspects. Firstly, with the use of variable-order derivatives, the system becomes more adaptive and flexible, allowing the chaotic dynamics of the system to evolve with changing fractional orders. Secondly, large-scale numerical simulations are conducted, where phase portrait orbits and time series for differences in VOF directly illustrate the effect of the order function on the system’s behavior. Thirdly, qualitative analysis is performed with the help of phase portraits, time series, and Lyapunov exponents to confirm the system’s hyperchaotic behavior and sensitivity to initial and control parameters. Finally, the model developed demonstrates a wide range of dynamic behaviors, which confirms the sufficient efficiency of VOF calculus for modeling complicated nonlinear processes. Numerous analyses indicate that this research not only shows meaningful findings but also provides thoughtful methodologies that might result in subsequent research on fractional-order chaotic systems. Full article

Brain tumors usually appear as masses formed by localized abnormal cell proliferation. Although complete removal of tumors is an ideal treatment goal, this process faces many challenges due to the aggressive nature of malignant tumors and the need to protect normal brain tissue. Therefore, early diagnosis is crucial to mitigate the harm posed by brain tumors. In this study, the classification accuracy is improved by improving the ResNet50 model. Specifically, the image is preprocessed and enhanced firstly, and the image is denoised by fractional calculus; then, transfer learning technology is adopted, the ECA attention mechanism is introduced, the convolutional layer in the residual block is optimized, and the multi-scale convolutional layer is fused. These optimization measures not only enhance the model’s ability to grasp the overall details but also improve its ability to recognize micro and macro features. This allows the model to understand data features more comprehensively and process image details more efficiently, thereby improving processing accuracy. In addition, the improved ResNet50 model is combined with EfficientNetB0 to further optimize performance and improve classification accuracy by utilizing EfficientNetB0’s efficient feature extraction capabilities through feature fusion. In this study, we used a brain tumor image dataset containing 5712 training images and 1311 validation images. The optimized ResNet50 model achieves a verification accuracy of 98.78%, which is 3.51% higher than the original model, and the Kappa value is also increased by 4.7%. At the same time, the lightweight design of the EfficientNetB0 improves performance while reducing uptime. These improvements can help diagnose brain tumors earlier and more accurately, thereby improving patient outcomes and survival rates. Full article

This work presents new results concerning weighted Hardy-type inequalities within the framework of delta conformable fractional integrals on arbitrary time scales. The proposed approach unifies the treatment of inequalities across continuous and discrete domains, enabling the derivation of original forms in both settings. The obtained results exhibit symmetry with classical inequalities, and several integral and discrete inequalities arise as special cases. These findings extend and generalize known results and enrich the theory of integral inequalities in fractional and dynamic calculus, providing a versatile platform for further developments in symmetric and weighted inequality analysis. Full article

In this work, Noether symmetries and conserved quantities of a non-standard Birkhoffian system based on the Caputo $\Delta$ Pfaff–Birkhoff principle on time scales are studied. Firstly, equations of motion for Caputo $\Delta$ non-standard Birkhoffian systems are set up from Caputo $\Delta$ variational principle. Secondly, invariance of Caputo non-standard Pfaff action on time scales is demonstrated, thus giving rise to Noether symmetry criterions which establish Noether’s theorems for the corresponding system. The validity of the methods and results presented in the paper is illustrated by means of examples provided at the end of the article. Full article

This article aims to clarify the applicability of He’s two-scale fractal dimension transform by replacing $t^{\mathsf{\alpha }}$ with $\mathsf{\tau }$. It demonstrates the potential to capture the influence of the fractal parameter on the system’s damping frequency, particularly when the viscoelastic term (damping) does not equal half of the fractional inertia force term. The analysis examines the elastomer materials’ dynamic fractal amplitude–time response, considering the viscohyperelastic effects related to the material’s energy dissipation capacity. To determine the amplitude of oscillations for the nonlinear equation of motion of a body supported by a viscohyperelastic elastomer subjected to uniaxial stretching, the harmonic balance perturbation method, combined with the two-scale fractal dimension transform and Ross’s formula, is employed. Numerical calculations demonstrate the effectiveness of He’s two-scale fractal transformation in capturing fractal phenomena associated with the fractional time derivative of deformation. This is due to a correlation between the fractional rate of viscoelasticity and the fractal structure of media in elastomer materials, which is reflected in the oscillation amplitude decay. Furthermore, the approach introduced by El-Dib to replace the original fractional equation of motion with an equivalent linear oscillator with integer derivatives is used to further assess the qualitative and quantitative performance of our derived solution. The proposed approach elucidates the applicability of He’s two-scale fractal calculus for determining the amplitude of oscillations in viscohyperelastic systems, where the fractal derivative order of the inertia and damping terms varies. Full article

To address the limitations of traditional tumor diagnostic methods in image feature extraction and model generalization, this study innovatively proposes a synergistic diagnostic model that integrates fractional Fourier transform (FrFT) and error back-propagation (BP) neural networks. The model leverages the time–frequency analysis capability of FrFT and incorporates the fractal characteristics observed during tumor proliferation, effectively enhancing multi-scale feature extraction and representation. Experimental results show that the proposed model achieves an accuracy of 93.177% in classifying benign and malignant tumors, outperforming the support vector machine (SVM) method. The integration of FrFT improves feature distinguishability and reduces dependence on manual extraction. This study not only represents a breakthrough in tumor diagnostic technology but also paves new avenues for the application of fractional calculus and fractal geometry in medical image analysis. The findings show great potential for clinical application and future development. Full article

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations. Full article

To enhance the operational efficiency of high-speed trains (HSTs), Train-to-Train (T2T) communication has received considerable attention. This paper introduces a T2T cooperative communication model that allows direct information exchange between HSTs, enhancing communication efficiency and system performance. The model incorporates a mix of dynamic and static nodes, and within this framework, we have developed a novel Dynamic Hierarchical Algorithm (DHA) to optimize communication paths. The DHA combines the stability of traditional algorithms with the flexibility of machine learning to adapt to changing network topologies. Furthermore, a communication link quality assessment function is proposed based on stochastic network calculus, which accounts for channel randomness, allowing for a more precise adaptation to the actual channel environment. Simulation results demonstrate that DHA has superior performance in terms of optimization time and effect, particularly in large-scale and highly dynamic network environments. The algorithm’s effectiveness is validated through comparative analysis with traditional and machine learning-based approaches, showing significant improvements in optimization efficiency as the network size and dynamics increase. Full article

The thermal impedance characteristics of insulated gate bipolar transistor (IGBT) modules are critical for the thermal management and design of electronic devices. This paper proposes a fractional-order equivalent thermal impedance model, which is inspired by the correlation between multi-time-scale dissipation characteristics of heat conduction processes and fractional calculus. The fractional-order equivalent thermal impedance model is derived based on the connection between fractional-order calculus and the Foster thermal network model in mathematical operations, with only two parameters to be identified: heat capacity $C$ and fractional order $\alpha$. Moreover, this paper provides a parameter identification method for the proposed fractional-order equivalent thermal impedance model based on the multi-objective particle swarm optimization (MOPSO) algorithm. In order to validate the effectiveness and superiority of this work, experiments and comparative works are provided in this paper. The results indicate that the fractional-order equivalent thermal impedance model can accurately describe the frequency domain characteristic curves of the thermal impedance of the Foster thermal network model for IGBT modules, with the difference between the amplitude frequency characteristics not exceeding 1 dB and the difference between the phase frequency characteristics not exceeding 1° within the operating frequency range of (1 kHz, 1 MHz). Full article