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We introduce the concepts of multiple attribute decision-making (MADM) using square root neutrosophic normal interval-valued sets (SRNSNIVS). The square root neutrosophic (SRNS), interval-valued NS, and neutrosophic normal interval-valued (NSNIV) sets are extensions of SRNSNIVS. A historical analysis of several aggregating operations is presented in this article. In this article, we discuss a novel idea for the square root NSNIV weighted averaging (SRNSNIVWA), NSNIV weighted geometric (SRNSNIVWG), generalized SRNSNIV weighted averaging (GSRNSNIVWA), and generalized SRNSNIV weighted geometric (GSRNSNIVWG). Examples are provided for the use of Euclidean distances and Hamming distances. Various algebraic operations will be applied to these sets in this communication. This results in more accurate models and is closed to an integer$ \\Delta $ . A medical robotics system is described as combining computer science and machine tool technology. There are five types of robotics such as Pharma robotics, Robotic-assisted biopsy, Antibacterial nano-materials, AI diagnostics, and AI epidemiology. A robotics system should be selected based on four criteria, including robot controller features, affordable off-line programming software, safety codes, and the manufacturer's experience and reputation. Using expert judgments and criteria, we will be able to decide which options are the most appropriate. Several of the proposed and current models are also compared in order to demonstrate the reliability and usefulness of the models under study. Additionally, the findings of the study are fascinating and intriguing.

Estimating the state of a system by fusing sensor data is a major prerequisite in many applications. When the state is time-variant, derivatives of the Kalman filter are a popular choice for solving that task. Two variants are the square-root unscented Kalman filter (SRUKF) and the square-root cubature Kalman filter (SCKF). In contrast to the unscented Kalman filter (UKF) and the cubature Kalman filter (CKF), they do not operate on the covariance matrix but on its square root. In this work, we modify the SRUKF and the SCKF for use on manifolds. This is particularly relevant for many state estimation problems when, for example, an orientation is part of a state or a measurement. In contrast to other approaches, our solution is both generic and mathematically coherent. It has the same theoretical complexity as the UKF and CKF on manifolds, but we show that the practical implementation can be faster. Furthermore, it gains the improved numerical properties of the classical SRUKF and SCKF. We compare the SRUKF and the SCKF on manifolds to the UKF and the CKF on manifolds, using the example of odometry estimation for an autonomous car. It is demonstrated that all algorithms have the same localization performance, but our SRUKF and SCKF have lower computational demands.

Motivated by the problems of analyzing protein backbones, diffusion tensor magnetic resonance imaging (DT-MRI) fiber tracts in the human brain, and other problems involving curves, in this study we present some statistical models of parameterized curves, in , in terms of combinations of features such as shape, location, scale, and orientation. For each combination of interest, we identify a representation manifold, endow it with a Riemannian metric, and outline tools for computing sample statistics on these manifolds. An important characteristic of the chosen representations is that the ensuing comparison and modeling of curves is invariant to how the curves are parameterized. The nuisance variables, including parameterization, are removed by forming quotient spaces under appropriate group actions. In the case of shape analysis, the resulting spaces are quotient spaces of Hilbert spheres, and we derive certain wrapped truncated normal densities for capturing variability in observed curves. We demonstrate these models using both artificial data and real data involving DT-MRI fiber tracts from multiple subjects and protein backbones from the Shape Retrieval Contest of Non-rigid 3D Models (SHREC) 2010 database.

The square root is one of the most important basic operations in computers. Based on the parallel carry-free TW-MSD adder, the parallel R4-MSD square root algorithm is proposed and presented. The algorithm is designed and implemented on the protype SD16 of ternary optical computer, which is developed by Shanghai University in 2016. The processes of the R4-MSD square root algorithm is described in detail. The optical experiment of the square root algorithm is presented based on the protype SD16 of ternary optical computer. It is proved that there are at most two square roots for MSD number representations of a positive decimal number within a given precision scope. It takes clock cycles to complete the square root of a MSD number of length n . This algorithm also shows that it requires the same clock cycle for multiple MSD numbers as for a single MSD number with the equal length. It is better than the radix-2 square root for MSD numbers and other embedded systems. Theoretical proof and optical experiments show that the parallel R4-MSD square root algorithm is feasible. It makes full use of the advantages of the ternary optical processor such as huge digits, bit function reconfiguration, and bit-based allocation.

We consider the following Schrödinger equation with combined Hartree type and square-root nonlinearities - ▵ u = λ u + μ I α ∗ | u | p | u | p - 2 u + 1 - 1 1 + u 2 u , in R N , having prescribed mass ∫ R N | u | 2 = c , where N ≥ 2 and c > 0 is a given real number, λ appears as a Lagrange multiplier. Under the assumption of μ > 0 and N + α N < p < N + α + 2 N , we prove the existence of the normalized ground state by combining Concentration-compactness principle and estimate on the square-root nonlinearity. The main results extend and complement the earlier works.

•The problem of square-rooting in high-degree cubature Kalman filters is addressed for nonlinear continuous-discrete stochastic systems.•The overall SR solution in the realm of the unscented and cubature Kalman filters is devised by means of hyperbolic QR decompositions.•The new square-root fifth-degree cubature Kalman filter implementations are examined in estimating an aircraft executing a coordinated turn in the presence of Gaussian noise and ill-conditioned measurements. This paper addresses the problem of square-rooting in the Cubature Kalman Filtering (CKF) originated from Arasaratnam and Haykin in 2009. Presently, this technique has been accommodated to various cubature rules, including high-degree ones. Since its discovery the CKF has become one of the most powerful state estimation methods because of outstanding performance and robustness in numerous engineering applications. Its high-degree versions are shown to be accurate and even comparable to particle filters, which are considered to be among the most effective algorithms for treating nonlinear stochastic systems. However, the lack of square-root implementations within high-degree CKFs makes them vulnerable to round-off and other errors committed because of a potential covariance matrix positivity loss, which may encounter in practice. This shortcoming affects severely and fails high-degree CKFs since the Cholesky factorization of predicted and filtering covariances underlying the filters in use may not be fulfilled for indefinite matrices. Here, we resolve it by means of hyperbolic QR transforms applied for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined and compared numerically to the existing non-square-root CKF and some other available filters in a simulated flight control scenario, including that with ill-conditioned measurements.

It is imperative that quantum computing devices perform floating-point arithmetic operations. This paper presents a circuit design for floating-point square root operations designed using classical Babylonian algorithm. The proposed Babylonian square root, is accomplished using Clifford+T operations. This work focuses on realizing the square root circuit by employing the bit Restoring and bit Non-restoring division algorithms as two different approaches. The multiplier of the proposed circuit uses an improved structure of Toom-cook 2.5 multiplier by optimizing the T-gate count of the multiplier. It is determined from the analysis that the proposed square root circuit employing slow-division algorithms results in a T-count reduction of 80.51% and 72.65% over the existing work. The proposed circuit saves a significant number of ancillary qubits, resulting in a qubit cost savings of 61.67 % When compared to the existing work.

This paper presents a comprehensive review of digital floating-point arithmetic algorithms that utilize Taylor series expansion in combination with mantissa-region division techniques, and it further demonstrates their generalization and applicability based on the findings of our research. While the discussion is broad in scope, this paper consolidates and systematizes the authors’ method within a broader contextual discussion, rather than presenting a fully systematic review of the entire state of the art in floating-point arithmetic algorithms. In many scientific computing applications, compact and low-power hardware implementations are essential. To address these requirements, this review presents algorithms specifically designed to operate under such constraints. The focus is placed on efficient floating-point operations—including division, inverse square root, square root, exponentiation, and logarithmic functions—all realized through Taylor series expansion with mantissa region division techniques. Furthermore, the trade-offs are examined in detail, covering factors such as the required numbers of additions, subtractions, and multiplications, along with the look-up table (LUT) size. The study further identifies the environments and application domains where the Taylor series expansion method combined with mantissa-region division is most effective, based on comparisons with various other floating-point computation algorithms and their corresponding hardware implementations. Overall, the review underscores the value of this unified framework in enabling efficient and adaptable floating-point computation across a wide range of hardware-constrained environments.

The “Contact Mechanics Challenge” posed to the tribology community by Müser and Dapp in 2015 detailed a 100 µm × 100 µm randomly rough surface with a root-mean-square gradient of unity, g ¯ = 1 . Many surfaces, both natural and synthetic, can be described as randomly rough, but rarely with a root-mean-square gradient as steep as g ¯ = 1 . The selection of such a challenging surface parameter was intentional, but potentially limiting for broad comparisons across existing models and theories which may be limited by small-slope approximations. In this manuscript, the root-mean-square gradients ( g ¯ ) of the “Contact Mechanics Challenge” surface were produced on 1000 × scaled models such that there were three different surfaces for study with g ¯ = 0.2 , 0.5 , and 1. In situ measurements of the real area of contact and contact area distributions were performed using frustrated total internal reflectance along with surface deformation measurements performed using digital image correlation. These optical in situ experiments used the scaled 3D-printed rough surfaces that were loaded into contact with smooth, flat, and elastic samples that were made from unfilled PDMS: (10:1) E * = 2.1 MPa Δ γ  = 4 mJ/m 2 ; (20:1) E * = 0.75 MPa Δ γ  = 3 mJ/m 2 ; (30:1) E * = 0.24 MPa Δ γ  = 2 mJ/m 2 . All of the loading was performed using a uniaxial load frame under force control. A Green’s function molecular dynamics simulation assuming the small-slope approximation was compared to all experimental data. These measurements reveal that decreasing root-mean-square gradient noticeably increases real area of contact area under conditions of “equal” applied load, but variations in the root-mean-square gradient did not significantly alter the contact patch geometry under conditions of nearly equal real area of contact. Including g ¯ in the reduced pressure ( p = P / ( E ∗ g ¯ ) ) reduced the root-mean-square error between the simulation ( g ¯ = 1 ) and all experimental data for the relative area of contact as a function of reduced pressure over the entire range of surfaces, materials, and loads tested.

Purpose This research presents the incremental power-based seismic analysis (IPSA) technique by proposing formulations, and these formulations are implemented on the previous experimental steel and concrete tunnel-soil-pile interaction (TSPI) models. Methods For this reason, the square root sum of squares (SRSS) and root mean square (RMS) incremental momentum power (IMP) are evaluated by using proposed formulations and previous experimental data in the cases of the tunnel and pile. Results The differences of the SRSS and RMS responses of the steel and concrete tunnels and piles are calculated by using proposed formulations for Kobe seismic excitation with various peak ground accelerations of 0.05 g, 0.10 g, 0.15 g, and 0.20 g. The accuracy of the IPSA technique is evaluated for the SRSS and RMS responses of the tunnel and pile which is proved by using an example. The slight difference (< 2.5%) is found for the variations of the two locations (tunnel or pile) SRSS and RMS IMP. Conclusion The maximum fluctuation of the mass normalized incremental kinematic power (MN-IKP) is found within a range of (− 0.001 to 0.001) m 2 /s 3 . The accuracy of the IMP (98.7–100%) is higher than the moment (94.94–99.01%) in the cases of the tunnel and pile of the TSPI model.