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Let R be a real closed field and D ⊂ R an ordered domain. We describe an algorithm that given as input a polynomial P ∈ D [ X 1 , … , X k ] and a finite set, A = { p 1 , … , p m } , of points contained in V = Zer ( P , R k ) described by real univariate representations, computes a roadmap of V containing A . The complexity of the algorithm, measured by the number of arithmetic operations in D , is bounded by ( ∑ i = 1 m D i O ( log 2 ( k ) ) + 1 ) ( k log ( k ) d ) O ( k log 2 ( k ) ) , where d = deg ( P ) and D i is the degree of the real univariate representation describing the point p i . The best previous algorithm for this problem had complexity card ( A ) O ( 1 ) d O ( k 3 / 2 ) (Basu et al., ArXiv, 2012 ), where it is assumed that the degrees of the polynomials appearing in the representations of the points in A are bounded by d O ( k ) . As an application of our result we prove that for any real algebraic subset V of R k defined by a polynomial of degree d , any connected component C of V contained in the unit ball, and any two points of C , there exists a semi-algebraic path connecting them in C , of length at most ( k log ( k ) d ) O ( k log ( k ) ) , consisting of at most ( k log ( k ) d ) O ( k log ( k ) ) curve segments of degrees bounded by ( k log ( k ) d ) O ( k log ( k ) ) . While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965, 2006 ), that there always exists a path of length ( O ( d ) ) k - 1 connecting two such points, there was no upper bound on the complexity of such a path.

We consider a partially linear framework for modeling massive heterogeneous data. The major goal is to extract common features across all subpopulations while exploring heterogeneity of each subpopulation. In particular, we propose an aggregation type estimator for the commonality parameter that possesses the (nonasymptotic) minimax optimal bound and asymptotic distribution as if there were no heterogeneity. This oracle result holds when the number of subpopulations does not grow too fast. A plug-in estimator for the heterogeneity parameter is further constructed, and shown to possess the asymptotic distribution as if the commonality information were available. We also test the heterogeneity among a large number of subpopulations. All the above results require to regularize each subestimation as though it had the entire sample. Our general theory applies to the divide-and-conquer approach that is often used to deal with massive homogeneous data. A technical by-product of this paper is statistical inferences for general kernel ridge regression. Thorough numerical results are also provided to back up our theory.

Fiber optic gyroscope (FOG)-based north finding is extensively applied in navigation, positioning, and various fields. In dynamic north finding, an accelerated turntable speed shortens the time required for north finding, resulting in a rapid north-finding response. However, with an increase in turntable speed, the turntable’s jitter contributes to signal contamination in the FOG, leading to a deterioration in north-finding accuracy. This paper introduces a divide-and-conquer algorithm, the segmented cross-correlation algorithm, designed to mitigate the impact of turntable speed jitter. A model for north-finding error is established and analyzed, incorporating FOG’s self-noise and the turntable’s speed jitter. To validate the feasibility of our method, we implemented the algorithm on a FOG. The simulation and experimental results exhibited a strong concordance, affirming the validity of our proposed north-finding error model. The experimental findings indicate that, at a turntable speed of 180°/s, the north-finding bias error within a 360 s duration is 0.052°, representing a 64% improvement over the traditional algorithm. These results indicate the effectiveness of the proposed algorithm in mitigating the impact of unstable turntable speeds, offering a solution for north finding with both prompt response and enhanced accuracy.

Astochastic, mixed-integer program (MIP) involving joint chance constraints is developed that generates least-cost vehicle redistribution plans for shared-vehicle systems such that a proportion of all near-term demand scenarios are met. The model aims to correct short-term demand asymmetry in shared-vehicle systems, where flow from one station to another is seldom equal to the flow in the opposing direction. The model accounts for demand stochasticity and generates partial redistribution plans in circumstances when demand outstrips supply. This stochastic MIP has a nonconvex feasible region. A novel divide-and-conquer algorithm for generating p -efficient points, used to transform the problem into a set of disjunctive, convex MIPs and handle dual-bounded chance constraints, is proposed. Assuming independence of random demand across stations, a faster cone-generation method is also presented. In a real-world application for a system in Singapore, the potential of redistribution as a fleet management strategy and the value of accounting for inherent stochasticities are demonstrated.

ApoB-100 is a member of a large lipid transfer protein superfamily and is one of the main apolipoproteins found on low-density lipoprotein (LDL) and very low-density lipoprotein (VLDL) particles. Despite its clinical significance for the development of cardiovascular disease, there is limited information on apoB-100 structure. We have developed a novel method based on the “divide and conquer” algorithm, using PSIPRED software, by dividing apoB-100 into five subunits and 11 domains. Models of each domain were prepared using I-TASSER, DEMO, RoseTTAFold, Phyre2, and MODELLER. Subsequently, we used disuccinimidyl sulfoxide (DSSO), a new mass spectrometry cleavable cross-linker, and the known position of disulfide bonds to experimentally validate each model. We obtained 65 unique DSSO cross-links, of which 87.5% were within a 26 Å threshold in the final model. We also evaluated the positions of cysteine residues involved in the eight known disulfide bonds in apoB-100, and each pair was measured within the expected 5.6 Å constraint. Finally, multiple domains were combined by applying constraints based on detected long-range DSSO cross-links to generate five subunits, which were subsequently merged to achieve an uninterrupted architecture for apoB-100 around a lipoprotein particle. Moreover, the dynamics of apoB-100 during particle size transitions was examined by comparing VLDL and LDL computational models and using experimental cross-linking data. In addition, the proposed model of receptor ligand binding of apoB-100 provides new insights into some of its functions.

We develop a fast divide-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical scheme does not have a Toeplitz structure. In this paper, we derive a fast approximation of the coefficient matrix by the means of a finite sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O ( N log 2 N ) memory and O ( N log 3 N ) computational complexity with N being the numbers of unknowns. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

This case study tackles a real-world problem of a transportation company that is modeled as a scheduling optimization problem. The main goal of the considered problem is to schedule the maximum number of jobs that must be performed by vehicles over a specific planning horizon in order to minimize the total operational costs. Here, each customer request corresponds to a job composed of multiple operations, such as loading, unloading, and mandatory jobs, each associated with a specific location and time window. Once a job is allocated to a vehicle, all its operations must be executed by that same vehicle within their designated time constraints. Due to the imposed limitations, not every job can feasibly be scheduled. To address this challenge, two distinct methodologies are proposed. The first, a Holistic approach, solves the entire problem formulation using a black-box optimizer, serving as a comprehensive benchmark. The second, a Divide-and-Conquer approach, combines a heuristic greedy algorithm with a binary linear programming, decomposing the problem into sequential subproblems. Both approaches are implemented using the solver Hexaly. A comparative analysis is conducted under different scenarios and problem settings to highlight the advantages and drawbacks of each approach. The results show that the Divide-and-Conquer approach significantly improves computational efficiency, reducing time by up to 99% and vehicle usage by around 15–20% compared to the Holistic method. On the other hand, the Holistic method better ensures that mandatory jobs are completed, although at the cost of more resources.

There has been a growing attention to efficient simulations of multibody systems, which is apparently seen in many areas of computer-aided engineering and design both in academia and in industry. The need for efficient or real-time simulations requires high-fidelity techniques and formulations that should significantly minimize computational time. Parallel computing is one of the approaches to achieve this objective. This paper presents a novel index-3 divide-and-conquer algorithm for efficient multibody dynamics simulations that elegantly handles multibody systems in generalized topologies through the application of the augmented Lagrangian method. The proposed algorithm exploits a redundant set of absolute coordinates. The trapezoidal integration rule is embedded into the formulation and a set of nonlinear equations need to be solved every time instant. Consequently, the Newton–Raphson iterative scheme is applied to find the system coordinates and joint constraint loads in an efficient and highly parallelizable manner. Two divide-and-conquer-based mass-orthogonal projections are performed then to circumvent the effect of constraint violation errors at the velocity and acceleration level. Sample open- and closed-loop multibody system test cases are investigated in the paper to confirm the validity of the approach. Challenging simulations of multibody systems featuring long kinematic chains are also performed in the work to demonstrate the robustness of the algorithm. The details of OpenMP-based parallel implementation on an eight-core shared memory computer are presented in the text and the parallel performance results are extensively discussed. Significant speedups are obtained for the simulations of small- to large-scale multibody open-loop systems. The mentioned features make the proposed algorithm a good general purpose approach for high-fidelity, efficient or real-time multibody dynamics simulations.

In this paper, a new framework is presented for the dynamic modeling and control of fully actuated multibody systems with open and/or closed chains as well as disturbance in the position, velocity, acceleration, and control input of each joint. This approach benefits from the computed torque control method and embedded fractional algorithms to control the nonlinear behavior of a multibody system. The fractional Brunovsky canonical form of the tracking error is proposed for a generalized divide-and-conquer algorithm (GDCA) customized for having a shortened memory buffer and faster computational time. The suite of a GDCA is highly efficient. It lends itself easily to the parallel computing framework, that is used for the inverse and forward dynamic formulations. This technique can effectively address the issues corresponding to the inverse dynamics of fully actuated closed-chain systems. Eventually, a new stability criterion is proposed to obtain the optimal torque control using the new fractional Brunovsky canonical form. It is shown that fractional controllers can robustly stabilize the system dynamics with a smaller control effort and a better control performance compared to the traditional integer-order control laws.