Ellipsoid Method Research Articles

Combination of Boundary Elements and the Ellipsoid Method for Optimizing the Electromagnetic Fields of Overhead Power Lines

ABSTRACTThis study proposes a numerical approach aimed at mitigating electromagnetic field intensities at ground level generated by overhead power lines. The methodology integrates the boundary element method for field evaluation with the ellipsoid method for field optimization, while adhering to critical design and safety constraints. The optimized power line configurations, derived from this integrated approach, achieved significant reductions in both electric and magnetic field magnitudes at ground level without compromising any specified constraints. Moreover, the research findings demonstrate the robustness, efficiency, and practical applicability of the proposed methodology in the design of optimized power lines, offering potential for increased power transfer capability, and decreased environmental and health impacts.

Vulnerability Assessment for Naval Ships against Air-Explosive Impulses: Modified Damage-Extent Method Incorporating Structural Capacity

Abstract This study introduces an enhanced damage-extent method that incorporates the structural load-bearing capacity of the hull to assess the vulnerability of naval ships to explosive loads. This vulnerability assessment predicts the area of damage to the hull structure and calculates the probability of onboard equipment experiencing functional losses due to the explosive load, thus allowing various design alternatives to be evaluated. The proposed methodology improves upon traditional damage-volume-based approaches, such as damage-radius and ellipsoid methods, by considering the hull's structural stiffness and intrinsic damage resistance. It integrates the hull's structural resistance to the load, enhancing the damage assessment process for both the hull and equipment. This approach facilitates damage prediction for different hull designs by comparing the allowable impulse with the explosive pressure. In assessing the functionality loss and vulnerability of the equipment within the damaged hull, the network of equipment functions is considered. An anti-ship cruise missile with a sea-skimming trajectory is investigated as the explosive charge, with procedures established to simulate its trajectory and impact location on the hull. Hundreds of potential internal and external explosion points are generated, predicting the explosive pressure at each location. The shock wave, including incident overpressure, reflection pressure, and quasi-static gas pressure, is converted into impulses, taking into account the configuration of hull compartments to accurately predict these pressures and equivalent impulse. The resulting impulse is compared with the intrinsic damage capacity of each compartment's structure to assess potential damage. System network and fault tree analysis evaluate the loss of function and vulnerability of equipment within the damaged hull. Finally, the proposed capacity-based damage extent method demonstrates more accurate damage assessment compared to traditional methods, overcoming the limitations of damage-radius and ellipsoid approaches by considering hull strength

Kidney growth progression patterns in autosomal dominant polycystic kidney disease

BackgroundPrognosis for autosomal dominant polycystic kidney disease (ADPKD), the main inherited cause of kidney failure, relies on estimating cystic growth using linear formulas derived from height-adjusted total kidney volume (Ht-TKV). However, nonlinear renal growth patterns may occur in typical ADPKD. AimsTo determine kidney outcomes of subjects diagnosed with typical ADPKD exhibiting nonlinear, and unpredictable cystic growth during follow-up. MethodsRetrospective cohort study. We categorized TKV changes in individuals with typical ADPKD according to observed kidney growth trajectories. Ht-TKV was calculated from consecutive CT or MRI using the ellipsoid method. We compared estimated glomerular filtration rate (eGFR) trajectories with linear mixed models. ResultsWe included 83 individuals with ADPKD (67% women; age 47 ± 12 years; follow-up 5.2 years [IQR 2.8–9.0]). Three kidney growth patterns were observed: slow progression (24%, <3%/year linear increase), fast progression (39%, ≥3%/year linear increase), and atypical progression (37%, nonlinear growth). Adjusted ht-TKV change in mL/m/year was +1.4 (IQR –4.5 to +10.0), +40.3 (+16.9 to +89.3), and +32.8 (+15.9 to +85.9) for slow, fast, and atypical progressors, respectively (p <0.001). Atypical progressors exhibited a significantly greater decline in eGFR in mL/min/m²/year (–7.9, 95% CI –6.5, –3.9) compared to slow (–0.5, 95% CI –3.1 to +0.5) and fast progressors (–3.4, 95% CI –7.9, –2.0; between-group p <0.001). Atypical progressors had a higher proportion of acute complications, including hemorrhages, infections, and urolithiasis (84%), compared to slow (20%) and fast progressors (31%) (p <0.001). ConclusionIn typical ADPKD, nonlinear, abrupt, and unpredictable cyst growth occurs frequently, leading to a higher risk of acute complications and kidney function decline.

Three Ellipsoids for External Approximation of the Half-Ball

Introduction. Ellipsoids that approximate the half-ball in Rn (n≥2) and their volume is smaller than the volume of the ball can be used to construct algorithms for solving the problem of minimization of a convex (smooth or non-smooth) function, the problem of minimization of a convex function on a sphere, a general problem convex programming, saddle point problems of convex-concave functions and others. The speed of convergence of such algorithms will be determined by the ratio of the volume of the approximating ellipsoid to the volume of the ball. The purpose of the work is to describe properties of three ellipsoids for approximation of a n-dimensional half-ball, the volume reduction factor of which depends only on n - dimension of space. The first is the well-known ellipsoid of minimum volume, which is used in the classical Yudin-Nemirovsky-Shor ellipsoid method. It provides a minimum volume reduction factor and can be used if n≥2. If n=1, then the classical ellipsoid method replaces the dichotomy method. The other two ellipsoids are close to the minimum volume ellipsoid and at large n guarantee volume reduction coefficients close to the minimum. The results. It is shown that when n=1 the first approximate ellipsoid provides a coefficient of reduction of the segment length equal to 2–√2 ≈ 0.5858. The second approximate ellipsoid is new and the volume reduction factor for it is slightly larger than the factor for the first approximate ellipsoid. If n=1, then it provides a coefficient of reduction of the segment length equal to (√5–1)/2 ≈ 0.6180. For three ellipsoids, comparative results are given in terms of volume reduction coefficients (n=1/20) and the number of iterations for solving problems with relative accuracy 1e-10 (n=1/30). Conclusions. A new ellipsoid has been constructed to approximate the n-dimensional half-ball, which is close in volume to the minimal ellipsoid. The volume reduction factor of the proposed ellipsoid is approximated by an asymptotic formula 1–1/(2n)+1/n², which differs little from the formula 1–1/(2n) for an ellipsoid of minimum volume.

Fundamentals of Determination of the Biological Tissue Refractive Index by Ellipsoidal Reflector Method

This paper presents the theoretical fundamentals, prerequisites for creation, and peculiarities of modeling a new method for determining the refractive index of biological tissues. The method uses a mirror ellipsoid of revolution as an optical element to ensure total internal reflection phenomena. This paper thoroughly analyzes the differences in the refractive index of healthy and pathological tissues on a biometric diagnostic basis. The analysis is used to model the measurement setup’s parameters. This paper also considers various methods of determining the refractive index of biological tissues based on different principles of physical optics, such as interferometry, refractometry, ellipsometry, and goniophotometry. It systematizes typical optical elements of total internal reflection that can be used in goniophotometry. It justifies the selection of the element base for the goniometric installation based on the ellipsoidal reflector method. A simulation of the installation operation was carried out for various parameters of the ellipsoidal reflector, ensuring the measurement of the biological tissue refractive index from 1.33 to 1.7. This paper also proposes a constructive solution for manufacturing an ellipsoidal reflector of the required configuration.

Development of a true density-based automated quality grading device for unboiled arecanut kernels

The automated sorting of arecanut kernels is a significant challenge that has not been effectively addressed thus far. Scientific grading techniques are necessary given the paradigm change toward investigating alternative uses for arecanuts in industry and the medical field. This research work emphasizes the relatively unexplored aspect of the post-harvest process; quality grading of kernels based on physical properties. It aimed to develop a novel approach for classifying unboiled (Chali) arecanut kernels cultivated in Goa, India based on their true density, using a combination of mechanical and visual techniques. The study explored the potential of true density as a quality indicator for real-time grading of the kernels. To achieve this, automated grading equipment was devised, utilizing a load cell to measure the kernel's mass and the ellipsoid approximation method to estimate its volume. A machine vision system captured the top and side images of the kernels to measure their volume. Python programs were created to enable image acquisition, processing, object detection, measurement and kernel segregation. Real-time kernel classification was accomplished by establishing serial data communication between the Python code and the Arduino board. The kernel segregation process was facilitated by servomechanism and a stepper motor. The kernels were classified into acceptable and non-acceptable categories based on a threshold value of true density. The research successfully established a method that utilizes the physical attributes of arecanut kernels as parameters for quality grading. However, the study encountered challenges with the density measurements, as the paired t-test results revealed significant differences between the kernel true density measured by the device and the true density estimated using the weighing scale-water displacement method, indicating a percentage error of 13.2%. Addressing these challenges would lead to more accurate density calculations, thereby enhancing the overall effectiveness of the kernel classification process. Furthermore, the technique allowed for the offline estimation of the kernels' porosity, which was found to be 45.3%. In future research, the integration of density and porosity measurement systems could be explored for real-time quality evaluation based on porosity, offering potential opportunities for further enhancement and optimization of the grading process. The technology could be further applied to other types of nuts and agricultural products, thereby overcoming the limitations of color-based sorting using image processing. Key words: Quality grading, True density, Machine vision, Arecanut kernel, Python

Revolutionizing Bladder Health: Artificial-Intelligence-Powered Automatic Measurement of Bladder Volume Using Two-Dimensional Ultrasound.

Measuring elevated post-void residual volume is important for diagnosing urinary outflow tract obstruction and cauda equina syndrome. Catheter placement is exact but painful, invasive, and may cause infection, whereas an ultrasound is accurate, painless, and safe. The purpose of this single-center study is to evaluate the accuracy of a module for artificial-intelligence (AI)-based fully automated bladder volume (BV) prospective measurement using two-dimensional ultrasound images, as compared with manual measurement by expert sonographers. Pairs of transverse and longitudinal bladder images were obtained from patients evaluated in an urgent care clinic. The scans were prospectively analyzed by the automated module using the prolate ellipsoid method. The same examinations were manually measured by a blinded expert sonographer. The two methods were compared using the Pearson correlation, kappa coefficients, and the Bland-Altman method. A total of 111 pairs of transverse and longitudinal views were included. A very strong correlation was found between the manual BV measurements and the AI-based module with r = 0.97 [95% CI: 0.96-0.98]. The specificity and sensitivity for the diagnosis of an elevated post-void residual volume using a threshold ≥200 mL were 1.00 and 0.82, respectively. An almost-perfect agreement between manual and automated methods was obtained (kappa = 0.85). Perfect reproducibility was found for both inter- and intra-observer agreements. This AI-based module provides an accurate automated measurement of the BV based on ultrasound images. This novel method demonstrates a very strong correlation with the gold standard, making it a potentially valuable decision-support tool for non-experts in acute settings.

Uncertainty Optimization of Vibration Characteristics of Automotive Micro-Motors Based on Pareto Elliptic Algorithm

The NVH (Noise, Vibration, and Harshness) characteristics of micro-motors used in vehicles directly affect the comfort of drivers and passengers. However, various factors influence the motor’s structural parameters, leading to uncertainties in its NVH performance. To improve the motor’s NVH characteristics, we propose a method for optimizing the structural parameters of automotive micro-motors under uncertain conditions. This method uses the motor’s maximum magnetic flux as a constraint and aims to reduce vibration at the commutation frequency. Firstly, we introduce the Pareto ellipsoid parameter method, which converts the uncertainty problem into a deterministic one, enabling the use of traditional optimization methods. To increase efficiency and reduce computational cost, we employed a data-driven method that uses the one-dimensional Inception module as the foundational model, replacing both numerical models and physical experiments. Simultaneously, the module’s underlying architecture was improved, increasing the surrogate model’s accuracy. Additionally, we propose an improved NSGA-III (Non-dominated Sorting Genetic Algorithm III) method that utilizes adaptive reference point updating, dividing the optimization process into exploration and refinement phases based on population matching error. Comparative experiments with traditional models demonstrate that this method enhances the overall quality of the solution set, effectively addresses parameter uncertainties in practical engineering scenarios, and significantly improves the vibration characteristics of the motor.

Automatic reorientation to generate short-axis myocardial PET images

BackgroundAccurately redirecting reconstructed Positron emission tomography (PET) images into short-axis (SA) images shows great significance for subsequent clinical diagnosis. We developed a system for automatic redirection and quantitative analysis of myocardial PET images.MethodsA total of 128 patients were enrolled for 18 F-FDG PET/CT myocardial metabolic images (MMIs), including 3 image classifications: without defects, with defects, and excess uptake. The automatic reorientation system includes five modules: regional division, myocardial segmentation, ellipsoid fitting, image rotation and quantitative analysis. First, the left ventricular geometry-based canny edge detection (LVG-CED) was developed and compared with the other 5 common region segmentation algorithms, the optimized partitioning was determined based on partition success rate. Then, 9 myocardial segmentation methods and 4 ellipsoid fitting methods were combined to derive 36 cross combinations for diagnostic performance in terms of Pearson correlation coefficient (PCC), Kendall correlation coefficient (KCC), Spearman correlation coefficient (SCC), and determination coefficient. Finally, the deflection angles were computed by ellipsoid fitting and the SA images were derived by affine transformation. Furthermore, the polar maps were used for quantitative analysis of SA images, and the redirection effects of 3 different image classifications were analyzed using correlation coefficients.ResultsOn the dataset, LVG-CED outperformed other methods in the regional division module with a 100% success rate. In 36 cross combinations, PSO-FCM and LLS-SVD performed the best in terms of correlation coefficient. The linear results indicate that our algorithm (LVG-CED, PSO-FCM, and LLS-SVD) has good consistency with the reference manual method. In quantitative analysis, the similarities between our method and the reference manual method were higher than 96% at 17 segments. Moreover, our method demonstrated excellent performance in all 3 image classifications.ConclusionOur algorithm system could realize accurate automatic reorientation and quantitative analysis of PET MMIs, which is also effective for images suffering from interference.

Optimal rejection of bounded perturbations in linear leader-following consensus protocol: invariant ellipsoid method

Optimal rejection of bounded perturbations in linear leader-following consensus protocol: invariant ellipsoid method

Exploration of morphological coherence in open clusters with a “core-shell” structure

Context. The morphology of open clusters plays a major role in the study of their dynamic evolution. The study of their morphological coherence, namely, the three-dimensional (3D) difference between the inner and outer morphologies of open clusters, allows us to obtain a better understanding of the morphological evolution of open clusters. Aims. We aim to investigate the morphological coherence of 132 open clusters with up to 1 kpc from the Sun in the three-dimensional (3D) space within the heliocentric cartesian coordinate frame. The 132 open clusters have a 3D core-shell structure and conform to the ellipsoidal model, with all of them coming from a catalog of publicly available clusters in the literature. Methods. We employed the ellipsoid fitting method to delineate the 3D spatial structure of the sample clusters, while using the morphological dislocation (MD) defined in our previous work and the ellipticity ratio (ER) of the clusters’ inner and outer structures to characterize the morphological coherence of the sample clusters. Results. The results show an inverse correlation between the ER of the sample clusters and the number of their members, indicating that sample clusters with a much more elliptical external morphology than internal shape generally tend to host a large number of members. Meanwhile, a slight shrinking of the MD of the sample clusters with their members’ number may shed light on the significant role of the gravitational binding of the sample clusters in maintaining their morphological stability. Moreover, there are no correlations between the MD and ER of the sample clusters and their age. They are also not significantly correlated with the X-axis, the Y-axis, their orbital eccentricities, and the radial and vertical forces on them. However, the ER of the sample clusters displays some fluctuations in the distributions between it and the above covariates, implying that the morphologies of the sample clusters are sensitive to the external environment if sample effects are not taken into account. Finally, the analysis of the 3D spatial shapes of sample clusters with a small ER or a large ER demonstrates that the number of members lays an important foundation for forming a dense internal system for sample clusters. At the same time, the MD of the sample clusters can serve well as an indicator of their morphological stability, which is built upon a certain amount of member stars. Conclusions. We present a new insight into the morphological coherence of open clusters, attributed to the combination of their gravitational binding capacity and external environmental perturbations.

Attitude Control for a satellite with inertial wheels based on Attractive Ellipsoids Method

Abstract This article presents a robust control law based on the Attractive Ellipsoids Method applied to perform attitude control of a satellite that has inertial wheels and relatively small dimensions such as microsatellites or other similar architectures. The novelty lies in the fact that under certain characterized external and parametric perturbations it is possible to track the desired angular trajectory with an optimization method to determine the computed torque. Additionally, a demonstration of this methodology is carried out with a numerical example.

Decentralized leader-following consensus control design for discrete-time multi-agent systems with switching topology

The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS as a whole. In contrast to the traditional approach, which uses the concept of an extended dynamic multi-agent system model and communication topology graph Laplacian, this paper proposes a decomposition approach, which provides a separate design of local controllers. The control law is chosen in the form of distributed feedback with discrete PID controllers. The problem of local controllers’ design is reduced to a set of semidefinite programming problems using the method of invariant ellipsoids. Sufficient conditions for agents’ stabilization and global consensus condition fulfillment are obtained using the linear matrix inequality technique. The availability of information about a finite set of possible configurations between agents allows us to design local controllers offline at the design stage. A numerical example demonstrates the effectiveness of the proposed approach.

Structure of compound and component communities of fleas parasitic on small mammals in six different regions as revealed by environmental‐based co‐occurrence geometry analyses

AbstractWe inferred the patterns of co‐occurrence of flea species in compound (across all host species) and component (across conspecific hosts) communities from six regions of the world (Mongolia, Northwest Argentina, Argentinian Patagonia, West Siberia, Slovakia, and South Africa) using the novel eigenvector ellipsoid method. This method allows us to infer structural community patterns by comparing species’ environmental requirements with the pattern of their co‐occurrences. We asked whether: (a) communities are characterized by species segregation, nestedness, or modularity; (b) patterns detected by the novel method conform to the patterns identified by traditional methods that search for non‐randomness in community structure; and (c) the pattern of flea species co‐occurrences in component communities is associated with host species traits. The results of the application of the eigenvector ellipsoid method suggested that the co‐occurrence of flea species was random in all compound communities except in South Africa, where this community demonstrated a tendency to be nested. Flea species co‐occurrences were random in many component communities. Species segregation was detected in the flea community of one host, whereas the flea communities of 14 hosts from different regions appeared to be nested. No indication of a modular structure in any community was found. The nestedness of flea component communities was mainly characteristic of hosts with a low relative brain mass. We concluded that the application of this novel method that combines data on species distribution and their environmental requirements allows better identification of the community structural patterns and produces more reliable results as compared with traditional methods.

A phase 2 safety and efficacy study of neoadjuvant/adjuvant darovasertib for localized ocular melanoma.

9510 Background: Darovasertib is a protein kinase C (PKC) inhibitor with meaningful activity in metastatic uveal melanoma (UM) due to its effect on PKC delta downstream of canonical GNAQ/GNA11 mutations. To date, its clinical activity in patients with localized primary disease has not been assessed in either neoadjuvant or adjuvant settings. Methods: Patients planned for enucleation with localized UM were treated in an initial safety cohort with darovasertib 300mg BID for 1 month (n=3 patients), and then following DSMB agreement in an expansion cohort for up to 6 months (n=12 patients) as neoadjuvant treatment prior to definitive management (enucleation, plaque brachytherapy or EBRT) across 3 Australian centers. All patients were eligible to receive up to 6 months of adjuvant treatment with darovasertib at investigator discretion after definitive management of their primary tumour. Tumour volume was calculated by the rotational ellipsoid method. Results: 15 patients (male n=7, female n=8; median age 62 years (range, 33-76 years)) were enrolled. At baseline, AJCC tumor stages were T3a (n=5), T3b (n=4), T4a (n=4), T4b (n=2), and the median tumor size (maximum thickness/diameter/volume) was 9.7mm/ 15.6 mm/ 2463 mm3. At datalock; 11/15 patients had completed primary treatment, 4/15 remained on neoadjuvant treatment, 6 patients received adjuvant darovasertib after primary treatment of their UM with 3 patients completing the planned 6-months . Median tumor shrinkage (maximum height/base/volume change) was 11.2%/ 7.6%/ 22.7% after 1 month of treatment and 31.7%/ 11.9% /45.3% after 6 months. At datalock, 6/9 (66%) currently completed neoadjuvant patients were converted to plaque brachytherapy (n=5) or EBRT (n=1) with 3 ongoing. One patient with high-risk cytogenetic features had relapsed with metastatic disease despite receiving 6-months of neoadjuvant darovasertib and another 6-months of adjuvant treatment. Treatment emergent adverse events included postural hypotension (Gr1/2 – 13/13%), syncope (Gr3 – 13%), rash (Gr1/2 – 33/5%), pruritis (Grr1 – 13%), dizziness (Gr1 – 27%), fatigue (Gr1/2 – 30/5%), nausea (Gr1/2 – 73/6%), vomiting (Gr1 – 40%), and diarrhea (Gr1 – 60%). Updated results, histopathological and genomic outcomes will be presented. Conclusions: NADOM provides the first evidence that a globe-salvage neoadjuvant treatment strategy in UM is feasible, safe, and efficacious. The results suggest that PKC inhibition with darovasertib can induce clinically meaningful tumor shrinkage in patients with primary UM patients who otherwise require enucleation. Larger trials are in now progress (NCT05907954) to further quantify visual and oncological outcomes. Clinical trial information: 05187884.

Identification and Control of Flexible Joint Robots Based on a Composite-Learning Optimal Bounded Ellipsoid Algorithm and Prescribe Performance Control Technique

This paper presents an indirect adaptive neural network (NN) control algorithm tailored for flexible joint robots (FJRs), aimed at achieving desired transient and steady-state performance. To simplify the controller design process, the original higher-order system is decomposed into two lower-order subsystems using the singular perturbation technique (SPT). NNs are then employed to reconstruct the aggregated uncertainties. An adaptive prescribed performance control (PPC) strategy and a continuous terminal sliding mode control strategy are introduced for the reduced slow subsystem and fast subsystem, respectively, to guarantee a specified convergence speed and steady-state accuracy for the closed-loop system. Additionally, a composite-learning optimal bounded ellipsoid algorithm (OBE)-based identification scheme is proposed to update the NN weights, where the tracking errors of the reduced slow and fast subsystems are integrated into the learning algorithm to enhance the identification and tracking performance. The stability of the closed-loop system is rigorously established using the Lyapunov approach. Simulations demonstrate the effectiveness of the proposed identification and control schemes.

Optimal demand response aggregation in wholesale electricity markets: Comparative analysis of polyhedral; ellipsoidal and box methods for modeling uncertainties

This paper presents an optimal framework for demand response aggregators in wholesale electricity markets. Demand response aggregators provide customers with flexible contracts in the proposed model. These contracts allow for hourly load reductions through load curtailment, load shifting, onsite generation utilization, and energy storage systems. The study suggests the price-based self-scheduling model for demand response aggregators, which seeks to maximize the aggregator's payoff for participation in day-ahead energy markets. The proposed model is implemented on a sample demand response aggregator with uncertainty and without uncertainty. Three methods, box, polyhedral, and ellipsoidal, based on robust optimization, are proposed for uncertainty modeling in this work, and their effectiveness is compared with each other. The results show that if the uncertainty is modeled using the polyhedral method, the value of the objective function will deviate by only 22.72 % compared to the case without uncertainty. However, this deviation in box and ellipsoidal methods is 118.48 % and 49.20 %, respectively, which shows the superiority of the polyhedral method compared to the oval and box methods.

New method for calculating the windward area of irregular fragments

Average windward area is an important index for calculating the trajectory, velocity attenuation and terminal effect of explosive fragments. In order to solve the problems that existing theoretical method cannot calculate windward area of irregular fragment and experiment method is not convenient for automatic calculation and has low accuracy, a Monte Carlo subdivision projection simulation algorithm is proposed. The average windward area of arbitrary shaped fragments can be obtained with coordinate translation, random rotation, plane projection, convex-hull triangulation, concave boundary searching and sorting with maximum edge length constraint, subdivision area calculation, and averaging by thousands of cycles. Results show that projection area obtained by the subdivision projection algorithm is basically the same as that obtained by software method of computer aided design. Moreover, the maximum calculation error of the algorithm is less than 7%, and its accuracy is much higher than that of the equivalent ellipsoid method. The average windward area calculated by the Monte Carlo subdivision projection simulation algorithm is consistent with theoretical formula for prefabricated fragments, and the error is less than 3%. The convergence and accuracy of the Monte Carlo subdivision projection algorithm are better than those of the icosahedral uniform orientation method.

Development and Application of a High-Precision Portable Digital Compass System for Improving Combined Navigation Performance.

There are not many high-precision, portable digital compass solutions available right now that can enhance combined navigation systems' overall functionality. Additionally, there is a dearth of writing about these products. This is why a tunnel magnetoresistance (TMR) sensor-based high-precision portable digital compass system is designed. First, the least-squares method is used to compensate for compass inaccuracy once the ellipsoid fitting method has corrected manufacturing and installation errors in the digital compass system. Second, the digital compass's direction angle data is utilized to offset the combined navigation system's mistake. The final objective is to create a high-performing portable TMR digital compass system that will enhance the accuracy and stability of the combined navigation system (abbreviated as CNS). According to the experimental results, the digital compass's azimuth accuracy was 4.1824° before error compensation and 0.4580° after it was applied. The combined navigation system's path is now more accurate overall and is closer to the reference route than it was before the digital compass was added. Furthermore, compared to the combined navigation route without the digital compass, the combined navigation route with the digital compass included is more stable while traveling through the tunnel. It is evident that the digital compass system's design can raise the integrated navigation system's accuracy and stability. The integrated navigation system's overall performance may be somewhat enhanced by this approach.

Using the Ellipsoid Method for Sylvester's Problem and its Generalization

Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The generalized problem consists in finding the center of a sphere in an n-dimensional space that has a minimal radius and contains a finite set of n-dimensional spheres given by their centers and radii. The article consists of 3 sections. Section 1 describes the emshor algorithm – the algorithm of the ellipsoid method for problem of minimization of an arbitrary convex function, proves its convergence theorems, gives a geometric interpretation of the algorithm, which is based on the use of a minimum volume ellipsoid. Octave implementation of the emshor algorithm is presented here, which can be successfully applied for non-smooth convex functions minimization if the number of variables is n = 2 - 30. In Section 2, the sylvester1 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimizing a convex piecewise quadratic function, which is equivalent to the problem of finding a sphere of minimum radius for a finite set of points. In Section 3, the sylvester2 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimization of a convex function, which is equivalent to the generalized problem of finding a sphere of minimum radius for a finite set of spheres with given centers and radii. The results of testing the sylvester1 and sylvester2 algorithms demonstrate high working speed for modern computers and high accuracy in terms of optimal value of the objective function when solving problems in n-dimensional spaces for small values n = 2 - 30. Keywords: ellipsoid method, convex function, Sylvester's problem, piecewise smooth function, minimax problem.