Search Results (159)

This paper is about a problem at the intersection of differential geometry, spectral analysis and the theory of manifolds. The study of finite-type subvarieties was initiated by Chen in the 1970s, with the aim of obtaining improved estimates for the mean total curvature of compact subvarieties in Euclidean space. The concept of a finite-type subvariety naturally extends that of a minimal subvariety or surface, the latter being closely related to variational calculus. In this work, we classify factorable surfaces in the Lorentz–Heisenberg space ${\mathbb{H}}_3$, equipped with a flat metric satisfying ${\Delta }^I{r}_i={\lambda }_i{r}_i$, which satisfies algebraic equations involving coordinate functions and the Laplacian operator with respect to the surface’s first fundamental form. Full article

This study explores the dynamics of particle motion in pseudo-Galilean $3-$space $G_3^1$ by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in $G_3^1$. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in $G_3^1$ and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications. Full article

Objectives: The purpose of the study is to evaluate the diagnostic accuracy of artificial intelligence (AI) software in detecting a common set of periodontal and restorative conditions, including marginal bone loss, dental caries, periapical lesions, calculus, endodontic treatment, crowns, restorations, and open crown margins, using intraoral periapical radiographs. Additionally, the study will assess how this AI software influences the diagnostic accuracy of dentists with varying levels of experience in identifying these conditions. Methods: A total of three hundred digital IOPARs representing 1030 teeth were selected based on predetermined selection criteria. The parameters assessed included (a) calculus, (b) periapical radiolucency, (c) caries, (d) marginal bone loss, (e) type of restorative (filling) material, (f) type of crown retainer material, and (g) detection of open crown margins. Two oral radiologists performed the initial diagnosis of the selected radiographs and independently labeled all the predefined parameters for the provided IOPARs under standardized conditions. This data served as reference data. A pre-trained AI-based computer-aided detection (“CADe”) software (Second Opinion^®, version 1.1) was used for the detection of the predefined features. The reports generated by the AI software were compared with the reference data to evaluate the diagnostic accuracy of the AI software. In the second phase of the study, thirty dental interns and thirty dental specialists were randomly selected. Each participant was randomly assigned five IOPARs and was asked to detect and diagnose the predefined conditions. Subsequently, all the participants were requested to reassess the IOPARs, this time with the assistance of the AI software. All the data was recorded using a self-designed Performa. Results: The sensitivity of the AI software in detecting caries, periapical lesions, crowns, open crown margins, restoration, endodontic treatment, calculus, and marginal bone loss was 91.0%, 86.6%, 97.1%, 82.6%, 89.3%, 93.4%, 80.2%, and 91.1%, respectively. The specificity of the AI software in detected caries, periapical lesions, crowns, open crown margins, restoration, endodontic treatment, calculus, and marginal bone loss was 87%, 98.3%, 99.6%, 91.9%, 96.4%, 99.3%, 97.8%, and 93.1%, respectively. The differences between the AI software and radiologist diagnoses of caries, periapical lesions, crowns, open crown margins, restoration, endodontic treatment, calculus, and marginal bone loss were statistically significant (all p values < 0.0001). The results showed that the diagnostic accuracy of operators (interns and specialists) with AI software revealed higher accuracy, sensitivity, and specificity in detecting caries, PA lesions, restoration, endodontic treatment, calculus, and marginal bone loss compared to that without using AI software. There were variations in the improvements in the diagnostic accuracy of interns and dental specialists. Conclusions: Within the limitations of the study, it can be concluded that the tested AI software has high accuracy in detecting the tested dental conditions in IOPARs. The use of AI software enhanced the diagnostic capabilities of dental operators. The present study used AI software to detect a clinically useful set of periodontal and restorative conditions, which can help dental operators in fast and accurate diagnosis and provide high-quality treatment to their patients. Full article

The application of fractional calculus in power electronics modeling provides an innovative method for improving inverter performance. This paper presents a three-phase inverter topology with fractional-order LC filter characteristics, analyzes its frequency response, and develops mathematical models in both stationary and rotating reference frames. Based on these models, a dual closed-loop decoupling control strategy for voltage and current is designed to enhance system stability and dynamic performance. In the power control loop, fractional-order virtual synchronous generator control (FOVSG) is employed. Observations show that increasing the fractional-order of the rotor leads to a higher transient frequency variation rate. To address this, an adaptive rotational inertia control scheme is integrated into the FOVSG structure (ADJ-FOVSG), enabling real-time adjustment of inertia to suppress transient frequency fluctuations. Experimental results demonstrate that when the reference active power changes, ADJ-FOVSG effectively suppresses power overshoot. Compared to traditional VSG, ADJ-FOVSG reduces the power regulation time by approximately 34.5% and decreases the peak frequency deviation by approximately 37.2%. Compared to the adaptive rotational inertia control in traditional VSG, ADJ-FOVSG improves regulation time by about 24% and reduces peak frequency deviation by roughly 24.4%. Full article

This research introduces an innovative fractal–fractional synergy framework for multiscale analysis of stress field dynamics in geo-energy systems. By integrating fractional calculus with multiscale fractal dimension analysis, we develop a coupled approach examining stress redistribution patterns across different geological scales. The methodology combines fractal characterization of rock mechanical parameters with fractional-order stress gradient modeling, validated through integrated analysis of core testing, well logging, and seismic inversion data. Our fractal–fractional operators enable simultaneous characterization of stress memory effects and scale-invariant fracture propagation patterns. Key insights reveal the following: (1) Non-monotonic variations in rock mechanical properties (fractal dimension D = 2.31–2.67) correlate with oil–water ratio changes, exhibiting fractional-order transitional behavior. (2) Critical stress thresholds (12.19–25 MPa) for fracture activation follow fractional power-law relationships with fracture orientation deviations. (3) Fracture network evolution demonstrates dual-scale dynamics—microscale tip propagation governed by fractional stress singularities (order α = 0.63–0.78) and macroscale expansion obeying fractal growth patterns (Hurst exponent H = 0.71 ± 0.05). (4) Multiscale modeling reveals anisotropic development with fractal dimension increasing by 18–22% during multi-well fracturing operations. The fractal–fractional formalism successfully resolves the stress-shadow paradox while quantifying water channeling risks through fractional connectivity metrics. This work establishes a novel paradigm for coupled geomechanical–fluid dynamics analysis in complex reservoir systems. Full article

This paper aims to develop a rheological model with fewer parameters that accurately describes the primary and secondary creep behavior of wood materials. The models studied are grounded in Riemann–Liouville fractional calculus theory. A comparison was conducted between the constant-order fractional Zener model and the variable-order fractional Maxwell model, with four parameters each. Using experimental creep data from four-point bending tests on two tropical wood species, along with an optimization algorithm, the variable-order fractional model demonstrated greater effectiveness. The selected fractional derivative order, modeled as a linearly increasing function of time, helped to elucidate the internal mechanisms in the wood structure during creep tests. Analyzing the parameters of this order function enabled an interpretation of their physical meanings, showing a direct link to the material’s mechanical properties. The Sobol indices have demonstrated that the slope of this function is the most influential factor in determining the model’s behavior. Furthermore, to enhance descriptive performance, this model was adjusted by incorporating stress non-linearity to account for the effects of the variation in constant loading level in wood. Consequently, this new formulation of rheological models, based on variable-order fractional derivatives, not only allows for a satisfactory simulation of the primary and secondary creep of wood but also provides deeper insights into the mechanisms driving the viscoelastic behavior of this material. Full article

The non-parametric version of Amari’s dually affine Information Geometry provides a practical calculus to perform computations of interest in statistical machine learning. The method uses the notion of a statistical bundle, a mathematical structure that includes both probability densities and random variables to capture the spirit of Fisherian statistics. We focus on computations involving a constrained minimization of the Kullback–Leibler divergence. We show how to obtain neat and principled versions of known computations in applications such as mean-field approximation, adversarial generative models, and variational Bayes. Full article

This paper extends the fractional calculus of variations to include generalized fractional derivatives with dependence on a given kernel, encompassing a wide range of fractional operators. We focus on variational problems involving the composition of functionals, deriving the Euler–Lagrange equation for this generalized case and providing optimality conditions for extremal curves. We explore problems with integral and holonomic constraints and consider higher-order derivatives, where the fractional orders are free. Full article

Generative diffusion models have achieved spectacular performance in many areas of machine learning and generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference, and stochastic calculus, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We show that these phase transitions are always in a mean-field universality class, as they are the result of a self-consistency condition in the generative dynamics. We argue that the critical instability arising from these phase transitions lies at the heart of their generative capabilities, which are characterized by a set of mean-field critical exponents. Finally, we show that the dynamic equation of the generative process can be interpreted as a stochastic adiabatic transformation that minimizes the free energy while keeping the system in thermal equilibrium. Full article

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs. Full article

This study examines the asymptotic stability of a continuous stirred tank reactor (CSTR) used for poly(methyl methacrylate) (PMMA) polymerisation, utilizing nonlinear fractional-order mathematical models. By applying Taylor series and Laplace transform techniques analytically and incorporating real plant data, we focus exclusively on the chemical reaction effects in the kinetic constants, disregarding mass transport phenomena. Our results confirm that fractional derivatives significantly enhance the stability and performance of dynamic models compared to traditional integer-order approaches. Specifically, we analyze the stability of a linearized fractional-order system at steady state, demonstrating that the system maintains asymptotic stability within feasible operational limits. Variations in the fractional order reveal distinct impacts on stability regions and system performance, with optimal values leading to improved monomer conversion, polymer concentration, and weight-average molecular weight. Comparative analyses between fractional- and integer-order models show that fractional-order operators broaden stability regions and enable precise tuning of process variables. These findings underscore the efficiency gains achievable through fractional differential equations in polymerisation reactors, positioning fractional calculus as a powerful tool for optimizing CSTR-based polymer production. Full article

The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Boole’s-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Boole’s-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Boole’s formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations. Full article

We consider the minimization of a functional of the calculus of variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the calculus of variations can be applied in a generalized Sobolev space, which is in turn diffeomorphism invariant. Under a suitable (invariant) assumption, the minima in this larger space belong to a usual Sobolev space and are bounded. Full article

This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings. Full article

Heat engines transform thermal energy into useful work, operating in a cyclic manner. For centuries, they have played a key role in industrial and technological development. Historically, only gases and liquids have been used as working substances, but the technical advances achieved in recent decades allow for expanding the experimental possibilities and designing engines operating with a single particle. In this case, the system of interest cannot be addressed at a macroscopic level and their study is framed in the field of stochastic thermodynamics. In the present work, we study mesoscopic heat engines built with a Brownian particle submitted to harmonic confinement and immersed in a fluid acting as a thermal bath. We design a Stirling-like heat engine, composed of two isothermal and two isochoric branches, by controlling both the stiffness of the harmonic trap and the temperature of the bath. Specifically, we focus on the irreversible, non-quasi-static case—whose finite duration enables the engine to deliver a non-zero output power. This is a crucial aspect, which enables the optimisation of the thermodynamic cycle by maximising the delivered power—thereby addressing a key goal at the practical level. The optimal driving protocols are obtained by using both variational calculus and optimal control theory tools. Furthermore, we numerically explore the dependence of the maximum output power and the corresponding efficiency on the system parameters. Full article