Search Results (161)

The importance of statistical distributions in representing real-world scenarios and aiding in decision-making is widely acknowledged. However, traditional models often face limitations in achieving optimal fits for certain datasets. Motivated by this challenge, this paper introduces a new probability distribution termed the weighted sine generalized Kumaraswamy (WSG-Kumaraswamy) distribution. This model is constructed by integrating the Kumaraswamy baseline distribution with the weighted sine-G family, which incorporates a trigonometric transformation to enhance flexibility without adding extra parameters. Various statistical properties of the WSG-Kumaraswamy distribution, including the quantile function, moments, moment-generating function, and probability-weighted moments, are derived. Maximum likelihood estimation is employed to obtain parameter estimates, and a comprehensive simulation study is performed to assess the finite-sample performance of the estimators, confirming their consistency and reliability. To illustrate the practical advantages of the proposed model, two real-world datasets from epidemiology and reliability engineering are analyzed. Comparative evaluations using goodness-of-fit criteria demonstrate that the WSG-Kumaraswamy distribution provides superior fits compared to established competitors. The results highlight the enhanced adaptability of the model for unit-interval data, positioning it as a valuable tool for statistical modeling in diverse applied fields. Full article

The generalized trigonometric functions called Ateb-functions are considered. On this basis, a generalization of the Fourier transform is constructed and called the Ateb-transform. From the operator theory point of view, the Ateb-transform is considered as a formalism of the convolution algebra in which multiplication is defined by means of hypergroups of the generalized shift operator. In this formal approach, the algebraic structure is presented, and its properties are developed. The eigenvalue problem for the differential equation of nonlinear oscillation is investigated. Some properties are illustrated numerically. The application of this approach for modeling vibration motion is considered. Full article

Path planning, as a key technology in unmanned aerial vehicle (UAV) systems, affects the overall efficiency of task completion and is often limited by energy consumption, obstacles, and maneuverability in complex application environments. Traditional algorithms have insufficient performance in nonlinear, multimodal, and multiconstraints problems. Based on this, this paper proposes an improved exponential-trigonometric optimization (ETO) to solve a 3D smooth path planning model based on a spherical Bezier curve. Firstly, a fixed arc length resampling strategy is proposed to address the issue of the insufficient adaptability of existing path smoothing methods to dynamic threats. Generate a uniformly distributed set of reference points along the Bezier curve and combine it with spherical projection to improve the safety and efficiency of the flight path. On this basis, establish a total cost function that includes four types of costs. Secondly, a new ETO variant called IETO is proposed by introducing the alpha evolution strategy, noise and physical attack strategy, and opposition-based cross teaching strategy into ETO. Then, the effectiveness of IETO for addressing various optimization problems is showcased through population diversity analysis, ablation analysis, and benchmark experiments. Finally, the results of the simulation experiment indicate that IETO stably provides shorter and smoother safe paths for UAVs in three elevation maps with different terrain features. Full article

We study the Sneddon $\mathcal{R}$-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel $xcos\left(xt\right)+hsin\left(xt\right)$, the $\mathcal{R}$-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted $L^p$-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite $\mathcal{R}$-transform. Full article

Background/Objectives: Rapid and accurate diagnosis of acute appendicitis is crucial for patient health and management, and the diagnostic process can be prolonged due to varying clinical symptoms and limitations of diagnostic tools. This study aims to shorten the timeframe for these vital processes and increase accuracy by developing a quantum-inspired hybrid model to identify appendicitis types. Methods: The developed model initially selects the two most performing architectures using four convolutional neural networks (CNNs) and two Transformers (ViTs). Feature extraction is then performed from these architectures. Phase-based trigonometric embedding, low-order interactions, and norm-preserving principles are used to generate a Quantum Feature Map (QFM) from these extracted features. The generated feature map is then passed to the Multiple Head Attention (MHA) layer after undergoing Hadamard fusion. At the end of this stage, classification is performed using a multilayer perceptron (MLP) with a ReLU activation function, which allows for the identification of acute appendicitis types. The developed quantum-inspired hybrid model is also compared with six different CNN and ViT architectures recognized in the literature. Results: The proposed quantum-inspired hybrid model outperformed the other models used in the study for acute appendicitis detection. The accuracy achieved in the proposed model was 97.96%. Conclusions: While the performance metrics obtained from the quantum-inspired model will form the basis of deep learning architectures for quantum technologies in the future, it is thought that if 6G technology is used in medical remote interventions, it will form the basis for real-time medical interventions by taking advantage of quantum speed. Full article

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering. Full article

We establish precise exponential tail estimates for lacunary trigonometric sums of the form $f_N\left(x\right)={\sum }_{k=1}^N{c}_k\cos \left(2\pi n_{k}x\right)$, under the Hadamard gap condition. Using cumulant expansions and moment-generating function techniques, we obtain non-asymptotic upper bounds for the tail probabilities, including third-order corrections that refine the classical central limit theorem estimates. Furthermore, several examples illustrate these bounds for various choices of coefficients, highlighting the transition from subgaussian to stretched-exponential tail behavior. Full article

Mathematics is a discipline that forms the foundation of many fields and should be learned gradually, starting from early childhood. However, some subjects can be difficult to learn due to their abstract nature, the need for attention and planning, and math anxiety. Therefore, in this study, a system that contributes to mathematics teaching using computer vision approaches has been developed. In the proposed system, users can write operations directly in their own handwriting on the system interface, learn their results, or test the accuracy of their answers. They can also test themselves with random questions generated by the system. In addition, visual graph generation has been added to the system, ensuring that education is supported with visuals and made enjoyable. Besides the character recognition test, which is applied on public datasets, the system was also tested with images obtained from 22 different users, and successful results were observed. The study utilizes CNN networks for handwritten character detection and self-created image processing algorithms to organize the obtained characters into equations. The system can work with equations that include single and multiple unknowns, trigonometric functions, derivatives, integrals, etc. Operations can be performed, and successful results can be achieved even for users who write in italicized handwriting. Furthermore, equations written within each closed figure on the same page are evaluated locally. This allows multiple problems to be solved on the same page, providing a user-friendly approach. The system can be an assistant for improving performance in mathematics education. Full article

This study employs the modified extended direct algebraic method (MEDAM) to investigate the generalized nonlinear fractional $(3+1)$-dimensional wave equation with gas bubbles. This advanced analytical framework is used to construct a comprehensive class of exact wave solutions and explore the associated dynamical characteristics of diverse wave structures. The analysis yields several categories of soliton solutions, including rational, hyperbolic (sech, tanh), and trigonometric (sec, tan) function forms. To the best of our knowledge, these soliton solutions have not been previously documented in the existing literature. By selecting appropriate standards for the permitted constraints, the qualitative behaviors of the derived solutions are illustrated using polar, contour, and two- and three-dimensional surface graphs. Furthermore, a stability analysis is performed on the obtained soliton solutions to ascertain their robustness and dynamical stability. The suggested analytical approach not only deepens the theoretical understanding of nonlinear wave phenomena but also demonstrates substantial applicability in various fields of applied sciences, particularly in engineering systems, mathematical physics, and fluid mechanics, including complex gas–liquid interactions. Full article

This paper investigates the reverse space-time nonlocal nonlinear Schrödinger (NNLS) equation, which arises in nonlinear optics, Bose–Einstein condensation, integrable systems, and plasma physics. Several classes of exact solutions are constructed using multiple analytical techniques. First, traveling wave solutions of Jacobi elliptic, hyperbolic, and trigonometric function types are ultimately obtained by employing a traveling wave transformation combined with a Weierstrass-type Riccati equation expansion method. Second, Lie symmetry analysis is applied to the NNLS equation, and the corresponding infinitesimal generators are determined. Using these generators, the original equation is reduced to local and nonlocal ordinary differential equations (ODEs), whose invariant solutions are subsequently obtained through integration. Finally, the NNLS equation is generalized to a multi-component system, for which the general form of the infinitesimal symmetries is derived. Symmetry reductions of the extended system yield further classes of reduced ODEs. In particular, the general form of the multi-component solutions is derived. Full article

Many authors analyze the prediction of chaos in a Josephson junction with quadratic damping by the Melnikov technique. Due to the lack of an explicit presentation of the Melnikov integral, the researchers apply numerical methods and illustrative examples to verify a good agreement between the numerical method and the analytical one. The reader has difficulty navigating and touching upon Melnikov’s elegant theory and, in particular, the formulation of the Melnikov criterion for the possible occurrence of chaos in a dynamical system, based solely on the provided illustrations of dependencies between the main parameters of the model under consideration. The statements in a number of publications devoted to this interesting topic, such as “It is easy to see that Melnikov’s integrals are finite and not zero. It is possible to see that the transverse zeros of the Melnikov function”, do not shed enough light on the origin of the “horseshoe”-type chaos. In this paper we will try to shed additional light on this important problem. A new planar system corresponding to the N-generalized Josephson junction with quadratic damping with many free parameters is considered, which may be of interest to specialists in the field of engineering sciences. Prediction of chaos in the proposed model by the Melnikov technique is closely related to the problem of approximately simultaneously finding all roots (simple or multiple) of generalized trigonometric polynomials. Several simulations are composed. We also demonstrate some specialized modules for investigating the dynamics of the model. One application about generating stochastic construction for possible control over oscillations is also discussed. Full article

The Lomb–Scargle method (LSM) constitutes a robust method for frequency and amplitude estimation in cases where data exhibit irregular or sparse sampling. Conventional spectral analysis techniques, such as the discrete Fourier transform (FT) and wavelet transform, rely on orthogonal mode decomposition and are inherently constrained when applied to non-equidistant or fragmented datasets, leading to significant estimation biases. The classical LSM, originally formulated for univariate time series, provides a statistical estimator that does not assume a Fourier series representation. In this work, we extend the LSM to multivariate datasets by redefining the shifting parameter $\tau$ to preserve the orthogonality of trigonometric basis functions in ${\mathbb{R}}^n$. This generalization enables simultaneous estimation of the frequency, phase, and amplitude vectors while maintaining the statistical advantages of the LSM, including consistency and noise robustness. We demonstrate its application to solar activity data, where sunspots serve as intrinsic markers of the solar dynamo process. These observations constitute a randomly sampled two-dimensional binary dataset, whose characteristic frequencies are identified and compared with the results of solar research. Additionally, the proposed method is applied to an ultrasound velocity profile measurement setup, yielding a three-dimensional velocity dataset with correlated missing values and significant temporal jitter. We derive confidence intervals for parameter estimation and conduct a comparative analysis with FT-based approaches. Full article

In this paper, we investigate the generalized coupled Zakharov system (GCZS), a fundamental model in plasma physics that describes the nonlinear interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves, including the influence of magnetic fields on weak ion-acoustic wave propagation. This research aims to achieve three main objectives. First, we uncover soliton solutions of the coupled system in hyperbolic, trigonometric, and rational forms, both in single and combined expressions. These results are obtained using the extended rational sinh-Gordon expansion method and the $\left({\displaystyle \frac{G^{\prime }}{G}},{\displaystyle \frac{1}{G}}\right)$-expansion method. Second, we analyze the dynamic characteristics of the model by performing bifurcation and sensitivity analyses and identifying the corresponding Hamiltonian function. To understand the mechanisms of intricate physical phenomena and dynamical processes, we plot 2D, 3D, and contour diagrams for appropriate parameter values. We also analyze the bifurcation of phase portraits of the ordinary differential equations corresponding to the investigated partial differential equation. The novelty of this study lies in the fact that the proposed model has not been previously explored using these advanced methods and comprehensive dynamical analyses. Full article

In this study, the extended (3 + 1)-dimensional Jimbo–Miwa equation, which has not been previously studied using Lie symmetry techniques, is the focus. We derive new symmetry reductions and exact invariant solutions, including lump and rogue wave structures. Additionally, precise solitary wave solutions of the extended (3 + 1)-dimensional Jimbo–Miwa equation using the multivariate generalized exponential rational integral function technique (MGERIF) are studied. The extended (3 + 1)-dimensional Jimbo–Miwa equation is crucial for studying nonlinear processes in optical communication, fluid dynamics, materials science, geophysics, and quantum mechanics. The multivariate generalized exponential rational integral function approach offers advantages in addressing challenges involving exponential, hyperbolic, and trigonometric functions formulated based on the generalized exponential rational function method. The solutions provided by MGERIF have numerous applications in various fields, including mathematical physics, condensed matter physics, nonlinear optics, plasma physics, and other nonlinear physical equations. The graphical features of the generated solutions are examined using 3D surface graphs and contour plots, with theoretical derivations. This visual technique enhances our understanding of the identified answers and facilitates a more profound discussion of their practical applications in real-world scenarios. We employ the MGERIF approach to develop a technique for addressing integrable systems, providing a valuable framework for examining nonlinear phenomena across various physical contexts. This study’s outcomes enhance both nonlinear dynamical processes and solitary wave theory. Full article

In this work, explicit Two-Derivative Runge–Kutta methods of the general case (that use several evaluations of the right-hand side function and its derivative per step) are considered. In order to address problems of orbital or oscillatory character, we develop methods with frequency-dependent coefficients. We construct three exponentially/trigonometrically fitted methods following two approaches suggested by Vanden Berghe and Simos. Also, we construct a phase-fitted and amplification-fitted method. The efficiency of the new modified methods is demonstrated by numerical examples. Full article