Search Results (167)

In this paper, a new normalized integral operator is introduced, defined by an integral containing a power of the classical Bessel function of the first kind of order $\nu \in \mathbb{N}$ multiplied by $t^{\gamma -1}$. Sufficient conditions for this operator to be starlike and convex in the open unit disk are determined. To establish these geometric criteria, we apply the theory of differential subordination, initiated by S. S. Miller and P. T. Mocanu, also known as the method of admissible functions. Full article

In this paper, we introduce and investigate a normalized fractional Bessel–Maitland function generated through the Riemann–Liouville fractional derivative, thereby establishing a new connection between fractional calculus and geometric function theory. By combining coefficient estimates with suitable inequalities involving the Gamma and digamma functions and appropriate monotonicity properties, we derive sufficient conditions ensuring that the proposed function belongs to several important subclasses of analytic and univalent functions, including uniformly convex, starlike, convex, k-uniformly starlike, k-uniformly convex, exponential, and lemniscate classes. We further examine structural properties of the associated coefficient sequence and provide numerical illustrations depicting the theoretical results. The proposed fractional normalization extends the classical generalized Bessel–Maitland and Wright-type functions and offers a unified framework for investigating their geometric behaviour under fractional differential operators, opening new directions for the study of special functions in geometric function theory and fractional calculus. Full article

Motivated by the interplay between discrete orthogonal polynomials and geometric function theory, we introduce and investigate some new Ma-Minda-type subclasses of bi-univalent functions generated by the Poisson-Charlier polynomials through their following analytic generating function: ${\mathcal{E}}_a(x,z)={(1+z)}^x{e}^{-az}\left(\right|z|<1).$ Within the classical class $\Sigma$ of analytic and bi-univalent functions, we first define the Poisson-Charlier-generated bi-starlike and bi-convex families via the subordination relations involving ${\mathcal{E}}_a(x,·)$ for both a function f and its inverse $f^{-1}$. By combining the Carathéodory representation with the series expansion of ${\mathcal{E}}_a(x,z)$ and the Lagrange inversion formula for $f^{-1}$, we derive coefficient estimates for the initial Taylor-Maclaurin coefficients $a_2$ and $a_3$ of functions in the starlike class ${\Sigma }_{{𝒮}^{*}}\left(\mathcal{E}(a,x)\right)$ and in the convex class ${\Sigma }_{𝒦}\left(\mathcal{E}(a,x)\right)$. In addition, we obtain corresponding Fekete-Szegö type inequalities of the form $\left|a_3-\mu a_2^2\right|$ for a real parameter $\mu$ in both settings, which are expressed explicitly in terms of the parameters a and x of the Poisson-Charlier framework. To the best of our knowledge, these Poisson-Charlier-generated bi-starlike and bi-convex classes have not previously been investigated, so the resulting coefficient and Fekete-Szegö estimates constitute a distinct contribution rather than direct special cases of previously studied Ma-Minda families. Full article

Coefficient operators provide a natural way to reshape an analytic function before testing its geometry. This paper develops a two-parameter algebraically damped Hadamard-product operator and investigates the functions whose transformed images satisfy a disk-starlikeness condition. The construction retains a visible contribution from the first coefficients while allowing the higher-order tail to be attenuated at an adjustable algebraic rate. The geometric conclusion concerns the transformed function; it does not automatically imply that the original function is starlike or univalent. Sharp initial-coefficient estimates, a corrected Fekete–Szegö inequality, and corresponding extremal functions are obtained. The multiplier is compared quantitatively with several established coefficient operators, and the effects of its parameters on damping and coefficient flexibility are examined. A weighted-coefficient criterion yields transparent perturbation and convex-combination consequences, while explicit examples illustrate both membership and failure of the criterion. The practical profiles included in the paper are conceptual visualizations rather than empirically validated models. Full article

In this paper, motivated by the modified Tremblay fractional differential operator and the normalized generalized Bessel–Maitland function, we introduce the generalized Bessel–Maitland–Tremblay function via the Hadamard product transformation. This construction established an integrated framework that connects fractional calculus with geometric function theory. By utilizing estimates for the gamma and digamma functions together with monotonicity properties of the associated coefficient sequences, we establish sufficient conditions under which the proposed function belongs to various important subclasses of normalized analytic functions. In particular, criteria are obtained for uniform convexity, starlikeness and convexity of order $\delta$, $\nu$-uniform starlikeness, and $\nu$-uniform convexity, as well as exponential starlikeness, exponential convexity, and lemniscate-type conditions. Several special cases are shown to recover previously known results for the normalized generalized Bessel–Maitland function and the identity mapping. Graphical analysis and numerical examples are presented to show the geometric behavior of the proposed function and to verify the theoretical findings. Full article

Backpropagation is less a single algorithm than a pipeline of choices: how the error signal is propagated, how the weight gradient is assembled, and how the update is applied. This paper revisits three consecutive steps and proposes small, mathematically transparent modifications that improve gradient scaling and conditioning without changing the represented function class. The resulting method, Conditioned Backpropagation(CBP), combines (i) a layerwise gradient-norm equalization that counters the geometric depth dependence of the backpropagated error; (ii) an activation-centering reparameterization that removes the dominant rank-one mean term from the per-layer curvature; and (iii) a damped diagonal preconditioner that is positive-definite by construction. The composite operator is a bounded positive-definite preconditioner, so the method inherits standard nonconvex, Polyak–Łojasiewicz, and stochastic convergence guarantees at the per-step cost of ordinary backpropagation. No prior method composes these three repairs into one operator with a joint boundedness and positive-definiteness guarantee. Two further results, both new, concern equalization. On a block-structured strongly convex model, and for the curvature-equalizing target that the implemented gradient-energy equalizer approximates up to a quantified heterogeneity factor, equalization makes the convergence rate depth-uniform; the bounded-clip version that is actually run stays depth-uniform up to a clip-determined depth and retains a constant-factor improvement beyond it. Controlled experiments, run over ten or more seeds with paired significance tests, confirm the mechanisms: Equalization compresses an order-of-magnitude per-layer gradient disparity, centering cuts the top curvature eigenvalue about threefold and yields the lowest training loss, and the configurations combining centering with the damped preconditioner, including the full method, converge fastest. The effects persist on MNIST and on CIFAR-10 with a small residual convolutional network, at a measured per-iteration overhead below about twice that of Adam. Generalization is comparable across methods, and no end-to-end depth-scaling advantage is claimed, keeping the contribution focused on optimization geometry. Full article

In this article, we introduce and study two new subclasses of analytic univalent functions defined via the Ma–Minda function. Specifically, we consider the class ${\mathcal{C}}_{{\xi }_q}$ of q-convex functions involving a suitable Ma–Minda function ${\xi }_q\left(z\right),$ together with its classical counterpart ${\mathcal{C}}_{\xi }$ corresponding to $\xi \left(z\right),$ where $0<q<1.$ We determine bounds for the first few Taylor–Maclaurin coefficients and deduce Fekete–Szegö and Kruskal inequality. Moreover, we obtain the associated Toeplitz determinants related to this class. We highlight several new consequences of our results that are of independent interest in geometric function theory. Full article

Artificial intelligence applications in management, such as fraud detection and churn prediction, are frequently constrained by the class imbalance problem. Standard over-sampling methods, such as SMOTE, rely on local geometric interpolation, which assumes data convexity and struggles to model the disjoint structures typical of managerial datasets. We introduce Fourier-based Adaptive Spectral Synthesis (FASS), an over-sampling method that frames data generation as a signal reconstruction problem. By transforming the minority class data into the frequency domain via the empirical characteristic function, FASS isolates the global manifold structure from high-frequency sampling noise through an automated spectral filtering mechanism. We prove the $L^2$-consistency of the underlying estimator. Empirical evaluations on credit and marketing datasets demonstrate that FASS favorably shifts the precision–recall trade-off compared to geometric baselines, reducing false positives and providing a theoretically consistent, parameter-free approach to learning from imbalanced data. Full article

This paper establishes new Fejér and Hermite–Hadamard-type inequalities for functions of two variables whose mixed second-order partial derivatives satisfy coordinated GA-convexity or coordinated GA-quasi-convexity on a rectangle in the positive quadrant. Our main results are formulated for non-negative continuous weight functions that are not necessarily symmetric with respect to the geometric means of the interval endpoints, thereby extending the classical framework to genuinely asymmetric weights. However, to obtain explicit and sharp integral bounds in certain cases, we also employ a technical lemma that assumes a special symmetric setting where the weight function is symmetric on each coordinate with respect to $\sqrt{h_1{h}_2}$ and $\sqrt{k_1{k}_2}$. We clearly distinguish which theorems hold for general asymmetric weights and which depend on this symmetry condition. Our findings unify and extend numerous previously known results for both symmetric and non-symmetric weight functions. Full article

This study proposes a mathematical framework for predicting complete response-time distributions associated with parametric video-game configurations. Each configuration is encoded as a point in a normalized metric space, and the statistical descriptors of its response-time distribution (median, mean, selected quantiles, interquartile range, and Skewness) are treated as real-valued Lipschitz functions on that space. Predictions for unseen configurations are obtained through McShane–Whitney extension formulas, which provide geometrically controlled upper and lower bounds compatible with the empirical Lipschitz regularity of the observed data. To handle the sequential incorporation of new observations, a regularization mechanism is introduced that replaces raw descriptors violating Lipschitz continuity constraints with a convex combination of the observed value and a weighted historical estimate. In the extended version of the method, the coefficient of this combination is selected through a one-pass online Q-learning-inspired procedure that selects, for each descriptor and instability regime, a data-dependent trade-off between fidelity and geometric regularity. The final output is a continuous Log-Normal density fitted by nonlinear least squares to the predicted descriptors, together with a Wasserstein-type uncertainty band derived from the Lipschitz bounds. The framework is validated on a controlled experiment with 24 participants across 20 game levels. Results show that the predicted distributions shift systematically with the input configuration and that, in the illustrative comparison, the adaptive mechanism produces feature-specific smoothing decisions that differ from those obtained with a fixed coefficient. The method also provides interpretable predictions from small experimental datasets without requiring fully data-driven models. Full article

In this paper, we introduce the following quadratic functional equation: ${\displaystyle \sum _{i=1}^{m-1}}{\displaystyle \frac{i}{i+1}}f\left({\displaystyle \frac{{\sum }_{j=1}^i{x}_j}{i}}-x_{i+1}\right)+{\displaystyle \frac{1}{m}}f\left({\displaystyle \sum _{i=1}^m}{x}_i\right)={\displaystyle \sum _{i=1}^m}f\left(x_i\right)$ which is derived from a geometric median identity in the Euclidean plane. Then, we find a general solution to the quadratic functional equation and prove the stability results of the quadratic ∗-derivations in $\rho$-complete convex n-modular ∗-algebras. Full article

To address the pervasive engineering challenges of rigid interference and subpar machining efficiency encountered during the complex freeform surface polishing of P20 mold steel, this study proposes and fabricates a structurally designed, five-petal composite compliant polishing tool via fused granulation fabrication (FGF). The tool structurally integrates a passive thermoplastic polyurethane (TPU) compliant buffer layer with an active PA66/diamond micro-cutting functional layer, achieving monolithic precision assembly through dual-temperature-zone 3D printing. Tensile mechanical characterization (n = 6) reveals that the composite interface attains an average ultimate tensile strength (UTS) of 59.39 ± 15.41 MPa (with a peak of 78.90 MPa) and an average elongation at break of 27.42 ± 7.41%, demonstrating exceptional structural robustness and fracture toughness under heavy-load abrasive machining conditions. During adaptive polishing validations on complex convex topographies and deep concave mold cavities, the compliant tool effectively compensated for normal vector spatial errors intrinsic to three-axis CNC machining via passive geometric adaptation. Topographical evaluations suggest a ductile-regime, differential asperity planarization material removal paradigm, which is attributed to the macroscopic 3D elastic deformation of the tool synergized with the proposed compliance of the polymer matrix. Following high-intensity sequential polishing regimens, the original macroscopic milling striations were substantially reduced. Quantitative profilometric analysis reveals that the average surface roughness of the convex profiles decreased from an initial 13.33 µm to 7.42 µm, while that of the restrictive deep concave features was reduced from 10.84 µm to 4.11 µm. Ultimately, this technological framework circumvents the traditional reliance on capital-intensive, six-degree-of-freedom robotic platforms, providing a scalable automated polishing protocol compatible with standard CNC systems for the cost-effective surface planarization of precision molds. Full article

To accurately characterize the effects of polydisperse particle sizes, multimineral composition, and angular morphology on the packing structure of lunar regolith, a two-dimensional sequential packing method based on random convex octagons is proposed. The method establishes a particle parameter system using data from Chang’e-5 samples and generates polygonal particle models with controllable angular features through radial perturbation. On this basis, a sequential packing algorithm based on available arc analysis is developed. Non-overlapping particle insertion is achieved via geometric envelope constraints, and progressive filling is realized through effective arc sampling. Meanwhile, a packing control coefficient is introduced to enable continuous regulation of packing density. Results show that the proposed method can generate highly dense particle assemblies, with a maximum packing density of 0.8757 and an average coordination number of approximately 3.18, capturing the structural characteristics of “high compactness–low coordination number” in polydisperse angular particle systems. The algorithm exhibits a computational complexity of O(N^1.628), demonstrating high efficiency. Furthermore, contact area and contact strength are quantitatively characterized through contact contour extraction and an equivalent bow-shaped model. Radial distribution function and contact statistics indicate that the generated structures possess good randomness and physical consistency. The proposed method provides a high-fidelity mesoscopic structure generation approach for discrete element modeling (DEM) of lunar regolith and establishes a reliable foundation for analyzing the mechanical behavior of granular systems. Full article

This work presents an unsupervised machine learning approach for the synthesis of nonuniform antenna arrays with consideration of mutual coupling effects. By integrating physical array synthesis formulas into the loss function, the physics-informed generative adversarial network (PI-GAN) is adopted to generate candidate designs of nonuniform antenna arrays. The generator produces the array geometric layouts and complex excitation distributions, and the discriminator assesses the fidelity of the radiation pattern relative to the design target by utilizing adversarial training with physics-driven pattern matching losses. With this proposed PI-GAN architecture, a deep neural network (DNN)-based active element pattern (AEP) surrogate model is embedded as a differentiable physics layer to accurately characterize the element mutual coupling, replacing time-consuming full-wave simulations. This end-to-end optimization paradigm enables an efficient global search over the non-convex solution space while ensuring physical consistency of the synthesized array. The method is validated on a 16-element linear array and a 300-element planar array, achieving lower peak sidelobe levels (PSLL), respectively. A prototype of the 16-element linear array is fabricated and measured, and the experimental results closely match the simulations, further validating the practical feasibility of the proposed PI-GAN synthesis framework. Full article

This paper is devoted to defining and analyzing a new subclass $M(\nu ,\tau ,q)$ of q-valent harmonic mappings in the unit disk D, as well as investigating its connection with close-to-convex analytic functions. First, we prove that this newly defined class is non-empty and discuss its relationship with several known classes of harmonic mappings. Using arguments similar to those employed in the study of Mocanu-type harmonic mappings, we establish the close-to-convexity of functions belonging to this class. Necessary coefficient estimates for the analytic part are obtained, and auxiliary lemmas which play an essential role in the investigation of geometric properties of the class are derived. In particular, we establish distortion estimates for the derivative of the analytic part, which lead to a growth and distortion theorem for functions in the newly defined class $M(\nu ,\tau ,q)$. Furthermore, a covering theorem is obtained for these harmonic mappings. In addition, we also derive sharp bounds for the Fekete–Szegö-type functionals and several graphical examples are presented to analyze the geometric structure of the mappings in the class $M(\nu ,\tau ,q)$ that demonstrate how the parameters $q,\nu ,$ and $\tau ,$ affect the deformation of the unit disk. Full article