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The resolution of the acceleration and jerk vectors of a particle moving on a space curve in the Euclidean 3-space is considered. By applying this resolution and Siacci's theorem, alternative resolutions of acceleration and jerk vectors are derived based on the quasi-frame. In the osculating plane, the acceleration vector is resolved as the sum of its tangential and radial components. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions. The maximum permissible speed on a space curve at all trajectory points is established via the jerk vector formula. Finally, some examples are presented to illustrate how the results work. [ABSTRACT FROM AUTHOR]

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example. [ABSTRACT FROM AUTHOR]

Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results by removing some restrictive conditions. Finally, an example is provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Dynamic systems of linear and nonlinear differential equations with pure delay are considered in this study. As an application, the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions is used to obtain the controllability and Hyers–Ulam stability results. By introducing a delay Gramian matrix, we establish some sufficient and necessary conditions for the controllability of linear delay differential systems. In addition, by applying Krasnoselskii's fixed point theorem, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear delay differential systems. Our results improve, extend, and complement some existing ones. Finally, two examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Nonhomogeneous systems of linear fractional equations with pure delay and multiple delays with linear parts given by permutable or nonpermutable matrices are considered. Representations of their solutions are obtained by using delayed Mittag–Leffler type matrix functions and Laplace transform. Our results are new and extend previous results. [ABSTRACT FROM AUTHOR]

• Nonhomogeneous linear systems of second order differential equations with pure delay and multiple delays are considered. • Representations of their solutions without a commutativity assumption on the matrix coefficients are obtained by using delayed matrix functions and Laplace transform. • Our results improve and extend previous results. Nonhomogeneous linear systems of second order differential equations with pure delay and multiple delays are considered. Representations of their solutions without a commutativity assumption on the matrix coefficients are obtained by using delayed matrix functions and Laplace transform. Our results improve and extend previous results. [ABSTRACT FROM AUTHOR]

In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear stochastic delay systems. In addition, by employing Krasnoselskii's fixed point theorem, we present some necessary criteria for the controllability of nonlinear stochastic delay systems. Our results improve and extend some existing ones. Finally, an example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii's fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle. [ABSTRACT FROM AUTHOR]

Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall's inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings. [ABSTRACT FROM AUTHOR]

Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the oscillatory properties of solutions to a class of delay differential equations of even order. We focus on criteria that exclude decreasing positive solutions. As in this paper, this type of solution emerges when considering the noncanonical case of even equations. By finding a better estimate of the ratio between the Kneser solution with and without delay, we obtain new constraints that ensure that all solutions to the considered equation oscillate. The new findings improve some previous findings in the literature. [ABSTRACT FROM AUTHOR]

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control system, we first use the characteristics of the modified Clarke sub-differential and a fixed point approach for multivalued functions. Then, we show that there are optimal state-control sets that are driven by Sobolev-type HF neutral stochastic evolution HVIs utilizing constrained Lagrange optimal systems. The optimal control (OC) results are created without taking the uniqueness of the control system solutions into account. Finally, the main finding is shown by an example. [ABSTRACT FROM AUTHOR]

The influence of the throughflow and gravity fluctuation on thermosolutal convection in an anisotropic porous bed with the Darcy–Brinkman effect is considered numerically. The critical Rayleigh numbers for the onset of stationary and oscillatory modes have been found via linear instability analysis. The impact of various gravitational functions in the presence of throughflow on stability is studied. The analysis has been carried out for decreasing and increasing gravity fluctuations. The convective problem has been numerically analyzed using a single-term Galerkin approach. The results show that the mechanical anisotropy parameter and Lewis number have a destabilizing effect, while the thermal anisotropy parameter, Darcy number, solutal Rayleigh number, throughflow parameter, and gravity parameter have a stabilizing effect on stationary and oscillatory convection. It is clear that the system changes in a way that makes it more stable for case (iii) gravity fluctuation and more unstable for case (iv) gravity fluctuation. [ABSTRACT FROM AUTHOR]

The effect of a heat source and temperature gradient on Brinkman–Bènard Triple-Diffusive magneto-Marangoni (BBTDMM) convection in a two-layer system is investigated. The two-layer system is horizontally infinite and is surrounded on all sides by adiabatic boundaries. It is exposed to basic uniform and non-uniform temperature profiles and heat sources. The appropriate eigenvalues and thermal Marangoni numbers (TMNs), which depend on temperature and concentration, are obtained for the temperature profiles (TPs) for lower rigid and higher free boundaries with surface tension. The transformed system of ordinary differential equations is solved by using an exact technique. For all three TPs, the impact of significant relevant parameters on these eigenvalues, and hence on BBTDMM convection, are investigated versus the thermal ratio. It is observed that, by increasing the values of the modified internal Rayleigh number for the fluid layer and the solute Marangoni numbers, the Darcy number, and the viscosity ratio for the set of physical parameters chosen in the study, one can postpone BBTDMM convection. Higher values of the modified internal Rayleigh numbers for the porous layer augment BBTDMM convection. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we first demonstrate adequate requirements for the existence of mild solutions to the concerned control system. Then, using limited Lagrange optimal systems, we demonstrate the existence of optimal state-control pairs that are regulated by an HFNSEHVI with a non-local condition. In order to demonstrate the existence of fixed points, the symmetric structure of the spaces and operators that we create is essential. Without considering the uniqueness of the control system's solutions, the best control results are established. Lastly, an illustration is used to demonstrate the major result. [ABSTRACT FROM AUTHOR]

In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the M o ¨ n c h fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application. [ABSTRACT FROM AUTHOR]

DBS entails the insertion of an electrode into the patient brain, enabling Subthalamic nucleus (STN) stimulation. Accurate delineation of STN borders is a critical but time-consuming task, traditionally reliant on the neurosurgeon experience in deciphering the intricacies of microelectrode recording (MER). While clinical outcomes of MER have been satisfactory, they involve certain risks to patient safety. Recently, there has been a growing interest in exploring the potential of local field potentials (LFP) due to their correlation with the STN motor territory. A novel STN detection system, integrating LFP and wavelet packet transform (WPT) with stacking ensemble learning, is developed. Initial steps involve the inclusion of soft thresholding to increase robustness to LFP variability. Subsequently, non-linear WPT features are extracted. Finally, a unique ensemble model, comprising a dual-layer structure, is developed for STN localization. We harnessed the capabilities of support vector machine, Decision tree and k-Nearest Neighbor in conjunction with long short-term memory (LSTM) network. LSTM is pivotal for assigning adequate weights to every base model. Results reveal that the proposed model achieved a remarkable accuracy and F1-score of 89.49% and 91.63%. Ensemble model demonstrated superior performance when compared to standalone base models and existing meta techniques. This framework is envisioned to enhance the efficiency of DBS surgery and reduce the reliance on clinician experience for precise STN detection. This achievement is strategically significant to serve as an invaluable tool for refining the electrode trajectory, potentially replacing the current methodology based on MER. [Display omitted] • LFP could give an insight into the electrophysiological neural rhythms of STN. • A unique LSTM based stacking ensemble learning model is designed for STN detection. • Soft thresholding offered noise reduction and robustness to signal variability. • Time–frequency information is extracted from LFP data using WPT. • LSTM helped to focus on relevant regions or patterns within the data. [ABSTRACT FROM AUTHOR]