Search Results (88)

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized $\delta$-Casorati curvatures ${\delta }_c(t;q_1+q_2-1)$ and ${\widehat{\delta }}_c(t;q_1+q_2-1)$ for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples. Full article

This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya–Santos–Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier–Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching $\mathcal{O}\left(n^{-1.5}\right)$ under Caputo derivatives, using parameters $\lambda =3.5$, $q=1.8$, and $n=100$. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion. Full article

In this work, we introduce a new subclass of bi-univalent functions using the $(p,q)$-derivative operator and the concept of subordination to generalized Laguerre polynomials ${\mathbb{L}}_t^{\varsigma }\left(k\right)$, which satisfy the differential equation $k{y}^{″}+(1+\varsigma -k)y^{\prime }+ty=0,$ with $1+\varsigma >0$, $k\in \mathbb{R}$, and $t\ge 0$. We focus on functions that blend the geometric features of starlike and convex mappings in a symmetric setting. The main goal is to estimate the initial coefficients of functions in this new class. Specifically, we obtain sharp upper bounds for $|a_2|$ and $|a_3|$ and for the Fekete–Szegö functional $|a_3-\eta a_2^2|$ for some real number $\eta$. In the final section, we explore several special cases that arise from our general results. These results contribute to the ongoing development of bi-univalent function theory in the context of $(p,q)$-calculus. Full article

In this study, we research the univariate quantitative symmetrized approximation of complex-valued continuous functions on a compact interval by complex-valued symmetrized and perturbed neural network operators. These approximations are derived by establishing Jackson-type inequalities involving the modulus of continuity of the used function’s high order derivatives. The kinds of our approximations are trigonometric and hyperbolic. Our symmetrized operators are defined by using a density function generated by a q-deformed and $\lambda$-parametrized hyperbolic tangent function, which is a sigmoid function. These accelerated approximations are pointwise and of the uniform norm. The related complex-valued feed-forward neural networks have one hidden layer. Full article

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple. Full article

Taylor expansion is a remarkable tool with broad applications in analysis, science, engineering, and mathematics. In this manuscript, we derive a proof of generalized Taylor expansion for polynomials and write its particular case in symmetric quantum calculus. In addition, we define a novel type of calculus that is called symmetric $(\mathit{p},\mathit{q})$- or dual basic symmetric quantum calculus. Moreover, we derive a symmetric $(\mathit{p},\mathit{q})$-Taylor expansion for polynomials based on this calculus. After that, we investigate Taylor’s formulae through an example. Furthermore, we define symmetric definite $(\mathit{p},\mathit{q})$-integral and derive a fundamental law of symmetric $(\mathit{p},\mathit{q})$-calculus. Finally, we derive the symmetric $(\mathit{p},\mathit{q})$-Cauchy formula for integrals that enables us to construct the fractional perspectives of $(\mathit{p},\mathit{q})$-symmetric integrals. Full article

In this study, a methodology using probabilistic distribution techniques to determine the parameters of the soil’s effective internal friction angle ($\phi ’$) was proposed. The method was grounded in quantitative survey information extracted from geotechnical reports. Extensive equivalent samples were estimated using Markov chain Monte Carlo (MCMC) simulations and probability density functions ($PDFs$). The effective internal friction angle ($\phi ’$) of silty clay layers was probabilistically characterized using the plasticity index ($P_I$), in situ static cone penetration test ($q_c$), and standard penetration test ($N_{SPT}$). A systematic quantitative analysis integrated prior information from different sources was systematically integrated with sampling data. By establishing a Bayesian framework that incorporated the regression relationship and uncertainties associated with the effective internal friction angle ($\phi ’$), the model ensured balance and symmetry in the treatment of prior information and observed data. The model was then transformed into equivalent sample values based on three models, reflecting the symmetrical consideration of different data sources. Further considerations involved correcting the three different analysis methods. A comparison of equivalent sample values with the mean values of the sampling data, along with the parameter optimization updates, was performed by combining the three models. Using three sets of sampling data, a linear relationship model for the new soil parameters was derived. The analysis results demonstrated that the proposed method could obtain equivalent samples for the effective internal friction angle. 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

The aim of this work is to demonstrate that all linear derivatives of the tensor algebra over a smooth manifold M can be viewed as specific cases of a broader concept—the operation of derivation. This approach reveals the universal role of differentiation, which simplifies and generalizes the study of tensor derivatives, making it a powerful tool in Differential Geometry and related fields. To perform this, the generic derivative is introduced, which is defined in terms of the quantities $Q_k^{\left(i\right)}\left(X\right)$. Subsequently, the transformation law of these quantities is determined by the requirement that the generic derivative of a tensor is a tensor. The quantities $Q_k^{\left(i\right)}\left(X\right)$ and their transformation law define a specific geometric object on M, and consequently, a geometric structure on M. Using the generic derivative, one defines the tensor fields of torsion and curvature and computes them for all linear derivatives in terms of the quantities $Q_k^{\left(i\right)}\left(X\right)$. The general model is applied to the cases of Lie derivative, covariant derivative, and Fermi derivative. It is shown that the Lie derivative has non-zero torsion and zero curvature due to the Jacobi identity. For the covariant derivative, the standard results follow without any further calculations. Concerning the Fermi derivative, this is defined in a new way, i.e., as a higher-order derivative defined in terms of two derivatives: a given derivative and the Lie derivative. Being linear derivative, it has torsion and curvature tensor. These fields are computed in a general affine space from the corresponding general expressions of the generic derivative. Applications of the above considerations are discussed in a number of cases. Concerning the Lie derivative, it is been shown that the Poisson bracket is in fact a Lie derivative. Concerning the Fermi derivative, two applications are considered: (a) the explicit computation of the Fermi derivative in a general affine space and (b) the consideration of Freedman–Robertson–Walker spacetime endowed with a scalar torsion field, which satisfies the Cosmological Principle and the computation of Fermi derivative of the spatial directions defining a spatial frame along the cosmological fluid of comoving observers. It is found that torsion, even in this highly symmetric case, induces a kinematic rotation of the space axes, questioning the interpretation of torsion as a spin. Finally it is shown that the Lie derivative of the dynamical equations of an autonomous conservative dynamical system is equivalent to the standard Lie symmetry method. Full article

This paper examines oscillations governed by the generic nonlinear differential equation ${u^{″}=\omega }_{p0}^2\left(1-u\mp {\displaystyle \frac{2{\beta }^2}{u^{\gamma }}}\right),$ where ${\omega }_{p0},\beta$ and $\gamma$ are positive constants. The aforementioned differential equation is of particular importance, as it describes electron plasma oscillations influenced by temperature effects and large oscillation amplitudes. Since no analytical solution exists for the oscillation period in terms of ${\omega }_{p0},\beta ,\gamma$ and the oscillation amplitude, accurate approximations are derived. A modified He’s approach is used to account for the non-symmetrical oscillation around the equilibrium position. The motion is divided into two parts: $u_{min}\le u<u_{eq}$ and $u_{eq}<u\le u_{max}$, where $u_{min}$ and $u_{max}$ are the minimum and maximum values of $u$, and $u_{eq}$ is its equilibrium value. The time intervals for each part are calculated and summed to find the oscillation period. The proposed method shows remarkable accuracy compared to numerical results. The most significant result of this paper is that He’s approach can be readily extended to strongly non-symmetrical nonlinear oscillations. It is also demonstrated that the same approach can be extended to any case where each segment of the function $f\left(u\right)$ in the differential equation $u^{″}+f\left(u\right)=0$ (for $u_{min}\le u<u_{eq}$ and for $u_{eq}<u\le u_{max}$) can be approximated by a fifth-degree polynomial containing only odd powers. Full article

Organic compounds with 1,3-diketone or 3-amino enone functional groups are extremely important as they can be converted into a plethora of carbo- or heterocyclic derivatives or can be used as ligands in the formation of metal complexes. Here, we have achieved the preparation of a series of non-symmetrical β-ketoenamines (O,N,N proligand) of the type (4-MeOC₆H₄)C(=O)CH=C(R)NH(Q) obtained through the Schiff base condensation of 1,3-diketones (1-anisoylacetone, 1-anisyl-3-(4-cyanophenyl)-1,3-propanedione, and 1-anisyl-3-(4,4,4-trifluorotolyl)-1,3-propanedione) functionalized with electron donor and electron-withdrawing substituents and 8-aminoquinoline (R = CH₃, 4-C₆H₄CN, 4-C₆H₄CF₃; Q = C₉H₇N). Schiff base ketoimines with a pendant quinolyl moiety were isolated as single regioisomers in yields of 22–56% and characterized with FT-IR, ¹H NMR, and UV-visible spectroscopy, as well as single-crystal X-ray crystallography, which allowed for the elucidation of the nature of the isolated regioisomers. The regioselectivity of the condensation of electronically unsymmetrical 1,3-diaryl-1,3-diketones with 8-aminoquinoline was studied by ¹H NMR, providing regioisomer ratios of ~3:1 and ~2:1 in the case of CN and CF₃ substituents, respectively. The electronic effects correlate well with the difference between the Hammett σ⁺ coefficients of the two para substituents on the aryl rings. Full article

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The $N_q$-transforms and ${}_{q}N$-transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters. Full article

This paper investigates the movement of a negligible mass body (third body) in the vicinity of the out-of-plane equilibrium points of the Hill three-body problem under the effect of radiation pressure of the primaries. We study the effect of the radiation parameters through the factors $q_i,i=1,2$ on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. These equilibrium positions are derived analytically under the action of radiation pressure exerted by the radiating primary bodies. We determined that these points emerge in symmetrical pairs, and based on the values of the radiation parameters, there may be two along the $Oz$ axis and either none or two on the $Oxz$ plane (outside the axes). A thorough numerical investigation found that both radiation factors have a strong influence on the position of the out-of-plane equilibrium points. Our results also reveal that the parameters have impact on the geometry of the zero-velocity curves. Furthermore, the stability of these points is examined in the linear sense. To do so, the spatial distribution of the eigenvalues on the complex plane of the linearized system is visualized for a wide range of radiation parameter combinations. By a numerical investigation, it is found that all equilibrium points are unstable in general. Full article

In this article, the $(p,q)$-analogs of the $\alpha$-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of $(p,q)$-generalized functions. Two spaces of infinitely $(p,q)$-differentiable functions are defined by introducing two $(p,q)$-differential symmetric operators. The $(p,q)$-analogs of the $\alpha$-th fractional Fourier transform are demonstrated to be continuous and linear between the spaces under discussion. Next, theorems pertaining to specific convolutions are established. This leads to the establishment of multiple symmetric identities, which in turn requires the construction of $(p,q)$-generalized spaces known as $(p,q)$-Boehmians. Finally, in addition to deriving the inversion formulas, the generalized $(p,q)$- analogs of the $\alpha$-th fractional Fourier transform are introduced, and their general properties are discussed. Full article

This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients of sequences, and establishes certain convolution theorems. The convolution theorems are utilized to accomplish q-equivalence classes of generalized distributions called q-Boehmians. Consequently, the q-gamma operators are therefore extended to the generalized spaces and performed to coincide with the classical integral operator. Further, the generalized q-gamma integral is shown to be linear, sequentially continuous and continuous with respect to some involved convergence equipped with the generalized spaces. Full article