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Facial approximation is a technique that involves constructing the facial muscles and applying a suitable facial soft tissue depth (FSTD) dataset. To date, several FSTD studies have been conducted for varying population groups. This study aims to establish a FSTD dataset of an adult Greek population sample for the first time. The facial depths of subjects were measured on 100 head CT scans of 50 male and 50 female subjects aged from 18 to 99. The 3D head and skull models of subjects were segmented in Amira 6.1 by using histogram method. FSTDs were measured at 22 cranial landmarks (5 mid-sagittal, 17 bilateral). The FSTD dataset was generated by considering the age and sex of subjects. The impact of age and sex on the FSTD was limited. Slight inter-population depth variations were reported. Facial asymmetry calculated between the bilateral landmarks was insignificant for both male and female subjects. [ABSTRACT FROM AUTHOR]

Although reality-based models are widely used to describe the geometric surfaces of an entity in a digital space, a systematic and universally recognised treatment of issues such as accuracy is lacking. The topic is certainly complex as this analysis should involve not only shape approximation but also other attributes (e.g., colour). Wanting to limit ourselves to geometry alone, this work proposes solutions for assessing the quality of photogrammetric models, differentiating them according to possible scenarios: sometimes, homologous models obtained using different techniques and technologies are available. In these cases, a comparison between digital reconstructions can serve to effectively quantify accuracy; more often, no terms of comparison are available, and one is forced to derive indicators from the same photogrammetric process to describe quality. We propose for this scenario a statistical analysis on the covariance matrix of the estimated coordinates for the tie points. The main goal is to provide a range of possible approaches to the conscious management of survey data. [ABSTRACT FROM AUTHOR]

This paper presents two new algorithms for solving multivalued variational inequality problems in a real Hilbert space. By combining the nonexpansiveness of proximal operators associated with the proper lower semicontinuous convex function of the problems and inertial techniques, we demonstrate the weak convergence of the iteration sequences generated by our first algorithm under monotone and Lipschitz continuous assumptions of the cost mappings. Next, we use Mann iteration technique to obtain the second algorithm and show its strong convergence. Finally, we give some numerical results for two proposed algorithms and comparison with some other known algorithms. [ABSTRACT FROM AUTHOR]

This paper aims to develop an efficient numerical scheme for simulating solid-state dewetting with strongly anisotropic surface energies in two dimensions. The governing equation is a sixth-order, highly nonlinear geometric partial differential equation, which makes it quite challenging to design an efficient numerical scheme. To tackle this problem, we first introduce an appropriate tangent velocity of the interface curve which could help mesh points equally distribute along the curve, then we reformulate the governing equation in terms of the tangent angle θ and the length L of the interface curve with a release of the stiffness brought by surface tension. To further reduce the numerical stability constraint from the high-order PDE, we propose a mixed finite element method for solving the reformulated θ -L equations. Numerical results are provided to demonstrate that the θ -L approach is not only efficient and accurate, but also has the mesh equidistribution property with an improved numerical stability. [ABSTRACT FROM AUTHOR]

Clinical simulation is vital in nursing education, providing realistic, hands-on experiences that prepare students for current challenges in healthcare settings. However, measuring the quality of its use requires the use of tools to deal with vague or inconsistent information associated with the subjective elements involved. The objective of this research was to evaluate the quality of the use of clinical simulation in nursing education by applying a set of qualitative indicators of a linguistic nature, expressed as single-valued neutrosophic sets. The combined use of the TOPSIS neutrosophic method with the calculation of the Hausdorff distance, allowed to evaluate a survey applied to professors of the UNIANDES university Nursing career. It was found that the best results were obtained with the realization of 3 or more clinical simulation sessions, mainly in the implementation and feedback offered by the students. [ABSTRACT FROM AUTHOR]

Existing line segment detectors break perceptually contiguous linear structures into multiple line segments. This can be offset by re-merging the segments, but existing merging algorithms over-merge and produce globally incorrect segments. Geometric cues are necessary but not sufficient for deciding whether to merge two segments or not. By restricting the result of any merging decision to have underlying image support, we reduce over-merging and globally incorrect segments. We propose a novel measure for evaluating merged segments based on line segment Hausdorff distance. On images from YorkUrbanDB, we show that our algorithm improves both qualitative and quantitative results obtained from four existing line segment detection methods and is better than two existing line segment merging methods. Our method does not suffer from inconsistent results produced by four recent deep learning-based models. The method is easily customisable to work for line drawings such as hand-drawn maps to obtain vectorized representations. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence of generalized pseudo-spectrum associated with bounded operators in a Hilbert space. We prove that the approximate generalized pseudo-spectrum converges to the exact set under norm convergence. To prove this result, we use the Hausdorff distance and the assumption that the generalized resolvent operator is not constant on any open subset of the generalized resolvent set. [ABSTRACT FROM AUTHOR]

The existing methods for determining adult age from human skeletons are mostly qualitative. However, a shift in quantifying age-related skeletal morphology on a quantitative scale is emerging. This study describes an intuitive variable extraction technique and quantifies skeletal morphology in continuous data to understand their aging pattern. A total of 200 postmortem CT images from the deceased aged 25–99 years (130 males, 70 females) who underwent forensic death investigations were used in the study. The 3D volume of the fourth lumbar vertebral body was segmented, smoothed, and post-processed using the open-source software ITK-SNAP and MeshLab, respectively. To measure the extent of 3D shape deformity due to aging, the Hausdorff distance (HD) analysis was performed. In our context, the maximum Hausdorff distance (maxHD) was chosen as a metric, which was subsequently studied for its correlation with age at death. A strong statistically significant positive correlation (P < 0.001) between maxHD and age at death was observed in both sexes (Spearman's rho = 0.742, male; Spearman's rho = 0.729, female). In simple linear regression analyses, the regression equations obtained yielded the standard error of estimates of 12.5 years and 13.1 years for males and females, respectively. Our study demonstrated that age-related vertebral morphology could be described using the HD method. Moreover, it encourages further studies with larger sample sizes and on other population backgrounds to validate the methodology. [ABSTRACT FROM AUTHOR]

Curve interpolation with B-spline is widely used in various areas. This problem is classic and recently raised in application scenario with new requirements such as path planning following the tangential vector field under certified error in CNC machining. This paper proposes an algorithm framework to solve Hausdorff distance certified cubic B-spline interpolation problem with or without tangential direction constraints. The algorithm has two stages: The first stage is to find the initial cubic B-spine fitting curve which satisfies the Hausdorff distance constraint; the second stage is to set up and solve the optimization models with certain constraints. Especially, the sufficient conditions of the global Hausdorff distance control for any error bound are discussed, which can be expressed as a series of linear and quadratic constraints. A simple numerical algorithm to compute the Hausdorff distance between a polyline and its B-spline interpolation curve is proposed to reduce our computation. Experimental results are presented to show the advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow. [ABSTRACT FROM AUTHOR]

The problem of constructing reachable and null-controllable sets for stationary linear discrete-time systems with a summary constraint on the scalar control is considered. For the case of quadratic constraints and a diagonalizable matrix of the system, these sets are built explicitly in the form of ellipsoids. In the general case, the limit reachable and null-controllable sets are represented as fixed points of a contraction mapping in the metric space of compact sets. On the basis of the method of simple iteration, a convergent procedure for constructing their external estimates with an indication of the a priori approximation error is proposed. Examples are given. [ABSTRACT FROM AUTHOR]

Image monitoring is a relatively new research area in statistics and machine learning that has wide applications in various fields including medical diagnostics and subsequently disease monitoring, satellite imaging, security systems, and so forth. In the literature, a vast majority of the methods use image intensity changes to detect out of control images. However, this approach is often unreasonable in many real-life applications where a change in contrast between the background and foreground of an image should not indicate an out of control image as long as the boundaries of the image objects remain unchanged. In this article, we propose a Shewhart-type control chart to monitor grayscale images using detected edges of the images. The central idea is to monitor the Hausdorff distance between the point-set of detected edge pixels in each image from the corresponding point-set of the estimated true in-control image. The proposed monitoring procedure should be easy to execute many real-life applications. Numerical studies show that it performs well in various types of situations in comparison with a number of competing methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to examine some trigonometric transformations of the Kies probability distribution. We consider the saturation in the Hausdorff sense which measures the speed of occurrence of a random event. Semi-closed form formulas for this saturation are derived. Also, some duality relations are obtained. Several numerical experiments are provided to validate the theoretical conclusions. The derived results can be applicable in many scientific and practical fields such as Biostatistics, Survival theory, Population dynamics, Lifetime analysis, Growth theory, Debugging and Test theory, Computer viruses propagation, Insurance mathematics, etc. [ABSTRACT FROM AUTHOR]

The study was aimed at developing a new automatic search technique for specific invariant patterns of movement-related brain potentials reflected in multidimensional electroencephalogram (EEG) signals. An adaptive band-pass filter with bandwidth closely matching the spectrum of the desired EEG pattern at the observed moment was synthesized based on the Singular Spectrum Analysis methodology. The preliminary filtering of the original EEG signals provides the required sensitivity for subsequent searching of time boundaries in patterns. The correctness of the developed method was confirmed with standard machine learning tools through the validation of the adaptive search method carried out on the general set of initial data. It is shown that the synthesized method has provided a reliable automatic search for induced pre-movement EEG patterns and the correct determination of their time boundaries (accuracy up 29% on average and reached maximum values to 100% for some individuals). The developed method expands the existing tools to improve the functionality and reliability of various Brain-computer interfaces for various purposes, including medical applications for paralyzed patients. [ABSTRACT FROM AUTHOR]

Affected by the angle of illumination, the images of small bodies taken by the spacecraft are incomplete, and it is difficult to reconstruct them with existing algorithms. To solve this problem, a 3D reconstruction algorithm for small bodies based on incomplete contours was proposed. Firstly, the image was subjected to threshold segmentation using the Otsu algorithm, and the extraction of incomplete contours was performed using the EDPF (edge drawing parameter free) algorithm. Subsequently, leveraging the solar azimuth information and combining it with the incomplete contours, the influence of shadows was taken into account. This process involved retaining both the illuminated regions and potential areas, resulting in the computation of expanded contours. Next, through the utilization of a set of expanded contours and projection matrices, the computation of the 3D bounding box was transformed into a linear inequality problem. Finally, voxel partitioning was performed to accomplish the 3D reconstruction of the small body. The quality of reconstruction was evaluated by normalizing the Hausdorff distance between the reconstructed results and reference models using the average radius. The simulation results indicate that the distribution probability of reconstruction errors being less than 4% is over 95%, and the algorithm also shows a certain level of robustness against changes in the angle of illumination. [ABSTRACT FROM AUTHOR]

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body , the areas of the maximum (resp. minimum) area convex -gons inscribed (resp. circumscribed) in is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex -gons by disk--gons, obtained as the intersection of closed Euclidean unit disks. It has been proved recently that if is the unit disk of a normed plane, then the same properties hold for the area of --gons circumscribed about a -convex disk and for the perimeters of --gons inscribed or circumscribed about a -convex disk , but for a typical origin-symmetric convex disk with respect to Hausdorff distance, there is a -convex disk such that the sequence of the areas of the maximum area --gons inscribed in is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of. [ABSTRACT FROM AUTHOR]

Abstract Repeatability involves the assessment of the agreement among repeated measurements from the same cluster of subjects, and this concept has been widely used in different scientific fields when data is structured in clusters. In the context of spatial trajectories, a degree of repeatability implies that individual trips can be distinguished from those of other individuals. Repeatability is usually assessed by the intraclass correlation coefficient (ICC), which is defined as the proportion of the total variance accounted for by among‐subject variability. However, where data are spatial trajectories the common approach to estimate the ICC does not apply because data involves sets of ordered spatial locations rather than single values. In this work, a novel approach based on spatial distances is proposed to estimate the ICC to assess the repeatability of spatial trajectories. The methodology is illustrated with a real case example of the flight trajectories of 36 Audouin's gulls (Ichthyaetus audouinii) moving through a heterogeneous landscape over a period of 18 days. Additionally, simulations were used to evaluate the performance of the approach under various scenarios. I demonstrate that ICC can be estimated on complex, spatially ordered data such as spatial trajectories, when using the appropriate spatial distance.

A hybrid UHVDC transmission system applying LCC as the rectifier and MMC as the inverter combines the advantages of both converter types, which makes this protection scheme more complicated. A new pilot protection scheme for a three-terminal hybrid DC transmission system applying energy functions is proposed. The energy function for LCC is applied to MMC to derive the energy level of the hybrid system. Furthermore, an improved Hausdorff distance (IHD) algorithm is proposed to detect the difference in energy levels between the normal and fault states. An abrupt change in energy level is characterized by IHD change rate. Time points at which the IHD change rate exceeds the threshold at converter stations are applied to determine the fault line and to estimate the fault section. The proposed protection scheme is then verified by a simulation model of the Wudongde ±800 kV three-terminal hybrid UHVDC transmission project. The appropriate sampling frequency is selected for a real-time calculation, and the threshold is selected considering the effect of noise. Results show the proposed scheme can identify and trip fault lines quickly and effectively, even for a 600 Ω grounding fault. Other waveshape similarity algorithms are compared and analyzed. Compared with existing protection schemes, the proposed scheme transmits less data to improve communication speed and reliability.

The similarity between arc fault current waveforms and nonlinear load currents can lead to misjudgments in arc fault identification methods that rely on arc current. In response to this issue, this paper establishes a unified fault criterion based on the conclusion that different types of loads that exhibit similar magnitudes of load side voltage sag during arc faults. Combined the voltage sag criterion with waveform criteria, this paper proposes a novel arc fault detection method. The method takes the load side voltage sag value exceeding the threshold as the detection start condition. After detection starts, the method identifies faults by calculating the degree of load side voltage waveform distortion using Hausdorff distance algorithm. Simulations are conducted on five different loads to determine the startup threshold and detection threshold. Test results show that the fault identification accuracy of the proposed method is 100% under the above thresholds.

Gene Expression Programming (GEP) is a powerful evolutionary algorithm with simple, linear and compact chromosomes, which has been applied in many fields to solve a large variety of complex problems such as logistic regression, function finding and time series prediction. Since online signature data are composed of discrete points, it is difficult to represent by functional forms, resulting in a limited amount of information used in calculating feature values. Hausdorff distance is utilized as a similarity measure to compute the maximum distance between two point sets, which reduces computational complexity compared with other distance measures. The main contributions of this work are: (1) In preprocessing stage, GEP is used to make signature curve continuous and control each parameter to obtain a fitting curve. Curve fitting is to find a suitable function that is the best fitting for a given set of data; (2) In feature extraction stage, curvature and torsion are utilized to construct eight feature sets for characterizing each user's signatures, and then Hausdorff distance is proposed to calculate the distances between feature sets of two signatures to form an eight-dimensional feature vector; (3) In verification stage, combined with Feed-Forward BP Neural Network classifier, distance matrices consisting of feature vectors are trained and tested many times. The best performances can be provided with false rejection rate, false acceptance rate, average error rate and standard deviation. The experimental results implemented on three available online signature databases MCYT-100, SVC2004 and SUSIG indicate the effectiveness and robustness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we proposed an approach for computing the Hausdorff distance between convex semialgebraic sets. We exploit the KKT conditions to rewrite the Hausdorff distance as polynomial maximization problems under some assumptions, in which polynomial and rational expressions of Lagrange multipliers are used. Then, polynomial maximization problems are solved by Lasserre's hierarchy of semi-definite relaxations. Finally, some numerical examples are reported. [ABSTRACT FROM AUTHOR]

We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀ ∃ < R . This implies that the problem is NP-, co-NP-, ∃ R -, and ∀ R -hard. [ABSTRACT FROM AUTHOR]

Repeatability involves the assessment of the agreement among repeated measurements from the same cluster of subjects, and this concept has been widely used in different scientific fields when data is structured in clusters. In the context of spatial trajectories, a degree of repeatability implies that individual trips can be distinguished from those of other individuals.Repeatability is usually assessed by the intraclass correlation coefficient (ICC), which is defined as the proportion of the total variance accounted for by among‐subject variability. However, where data are spatial trajectories the common approach to estimate the ICC does not apply because data involves sets of ordered spatial locations rather than single values.In this work, a novel approach based on spatial distances is proposed to estimate the ICC to assess the repeatability of spatial trajectories. The methodology is illustrated with a real case example of the flight trajectories of 36 Audouin's gulls (Ichthyaetus audouinii) moving through a heterogeneous landscape over a period of 18 days. Additionally, simulations were used to evaluate the performance of the approach under various scenarios.I demonstrate that ICC can be estimated on complex, spatially ordered data such as spatial trajectories, when using the appropriate spatial distance. [ABSTRACT FROM AUTHOR]

An implementation of a "rectilinear" geodesic lying in the Gromov–Hausdorff space is constructed in the form of the shortest geodesic with respect to the Hausdorff distance in some ambient metric space. [ABSTRACT FROM AUTHOR]

A nonlinear controlled system in a finite-dimensional Euclidean space and within a finite time interval is considered, the dynamics of which significantly changes over several small sections from a set time interval. The level of change in the reachable sets and integral funnels of the system under consideration is studied when it varies in these sections. The corresponding changes are estimated in the Hausdorff metric. [ABSTRACT FROM AUTHOR]

The mutual identification of spatial objects is a fundamental issue when updating geographic data with other data sets. Representations of spatial objects in different sources may not have the same identifiers, which would unambiguously assign them to each other. Intersections of spatial objects can be used for this purpose, but this does not allow for the detection of possible changes and their quantification. The aim of this paper is to propose a simple, applicable procedure for calculating the shape similarity measure, which should be able to efficiently identify different representations of spatial objects in two data sources, even though they may be changed or generalised. The main result is the aggregated index of shape similarity and instructions for its calculation and implementation. The shape similarity index is based on the calculation of the set similarity, the distance of the boundaries, and the differences in the area, perimeter, and number of the vertices of areal spatial objects. In the case study, the footprints of the building complexes in Dúbravka (part of the city of Bratislava, the capital of Slovakia) are compared using data from OpenStreetMap and INSPIRE (Infrastructure for Spatial Information in Europe) Buildings. A contribution to the quality check of the OpenStreetMap data is then a secondary result. The proposed method can be effectively used in the semi-automatic integration of heterogeneous data sources, updating the data source with other spatial data, or in their quality control. [ABSTRACT FROM AUTHOR]

The present paper is on fixed point theory in set valued analysis. Here putting several concepts together, we define a set valued contraction and establish that such contractions have fixed points in a complete metric space. We have two main theorems in one of which we use α -regularity condition of the space as a substitute of the continuity requirement on the multivalued contraction. There are some illustrative examples and corollaries. We also perform stability analysis of fixed point sets. We establish that the fixed point sets associated with the mappings we consider in our theorems are stable. In the last section we discuss some consequences of the main results of this paper. [ABSTRACT FROM AUTHOR]

Current differential protection (CDP) is widely used as main protection for transformers due to its high reliability and selectivity. Owing to non-fundamental frequency current, magnetizing inrush and current transformer (CT) saturation, CDP may undergo slow operating speed when the internal fault occurs. When the external fault occurs, CDP may false operate. Furthermore, given the disadvantage of Hausdorff distance being extremely sensitive to abnormal data, a new metric that reveals the dissimilarity degree of two waveforms is presented, namely waveform dissimilarity factor (WDF). WDF has inherent immunity to abnormal data. Subsequently, WDF-based protection for main transformer is proposed herein. In order to reduce the adverse effects of magnetizing inrush and CT saturation and accurately identify internal faults, the reward and punishment mechanism is designed. The protection principle is elaborated and its operating characteristics are analyzed under various conditions. Finally, by constructing the DFIG-based wind farm, PSCAD is used to verify the advantages of the novel protection algorithm based on WDF. Its performance is widely assessed in various scenarios in detail, including magnetizing inrushes, minor and server internal faults, inter-turn fault, turn-to-ground fault, internal fault with CT saturation and external fault with CT saturation, data loss, abnormal data, and CT error. The results show that the novel protection scheme has the characteristics of fast, secure, and robust.

Purpose – In this work, the authors are interested in the notion of vector valued and set valued Pettis integrable pramarts. The notion of pramart is more general than that of martingale. Every martingale is a pramart, but the converse is not generally true. Design/methodology/approach – In this work, the authors present several properties and convergence theorems for Pettis integrable pramarts with convex weakly compact values in a separable Banach space. Findings – The existence of the conditional expectation of Pettis integrable mutifunctions indexed by bounded stopping times is provided. The authors prove the almost sure convergence in Mosco and linear topologies of Pettis integrable pramarts with values in (cwk(E)) the family of convex weakly compact subsets of a separable Banach space. Originality/value – The purpose of the present paper is to present new properties and various new convergence results for convex weakly compact valued Pettis integrable pramarts in Banach space.

This paper proposes the general architecture of a multi-scale classification system based on the intelligent imputation method of multi-scale classification. SVM is used to transform the storage structure model of the object, obtain the trained support vector set with corresponding weights, measure the similarity between two data points based on the Hausdorff distance, realize the construction of a similarity matrix, adopt the idea of mean value, blur the data information, and improve the mechanism of on-scale imputation. The bicubic difference method is used as the theoretical basis of scale-down extrapolation, and the scale-down extrapolation algorithm DAMSC is established. Based on the quaternionic linkage number, the dataset data are preprocessed, and the data eigenvalue weights are calculated at the same time. The multi-scale classification validity index evaluates the model’s accuracy in classification and its performance in both upward and downward scale extrapolation. When the categorized data is 7, the MSCVI metric of the model attains a peak of 0.92254 in the segmentation dataset run results, which is more advantageous than other metrics. In the regression test of the model’s imputation results, the correlation coefficient of bus passengers is 0.99, the adjusted R-square is 0.97043, and the coefficient of the regression equation is 1.042, and the model in this paper is valid and reliable for the imputation of passenger drop-off points.

This paper discusses the Topp-Leone-G power series class of distributions. The greatest attention is paid to the investigation of intrinsic characteristic "saturation" to the horizontal asymptote in the Hausdorff sense. Some estimates for the value of the Hausdorff distance are obtained. We present a new family of recurrence generated adaptive functions with corresponding applications. The usefulness of the obtained results is demonstrated in a simulation study of some real data sets from the medical sector and insurance. Some suitable software modules within the programming environment CAS MATHEMATICA are proposed. [ABSTRACT FROM AUTHOR]

Let p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX) → (Y, dY) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) ≤ Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim FpX ≤ deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist → Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces. [ABSTRACT FROM AUTHOR]

The current best algorithms for the convex body chasing (CBC) problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm that always moves to the Steiner point of the request set is O(d) competitive for nested CBC, and this is optimal among memoryless algorithms [Bubeck et al.: Chasing nested convex bodies nearly optimally. In: 31st Annual ACM–SIAM Symposium on Discrete Algorithms (Salt Lake City 2020), pp. 1496–1508. SIAM, Philadelphia (2020)]. A memoryless algorithm coincides with the notion of a selector in functional analysis. The Steiner point is noted for being Lipschitz with respect to the Hausdorff metric, and for achieving the minimal Lipschitz constant possible. It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested CBC. We answer this question in the negative by exhibiting a selector that yields a non-competitive algorithm for nested CBC but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an L p -type analog of the Hausdorff distance is sufficient to guarantee competitiveness if and only if p = 1 . [ABSTRACT FROM AUTHOR]

We consider the problem of computing the (two-sided) Hausdorff distance between the unit ℓ p 1 and ℓ p 2 norm balls in finite dimensional Euclidean space for 1 ≤ p 1 < p 2 ≤ ∞ , and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the k 1 and k 2 unit D -norm balls, which are certain polyhedral norm balls in d dimensions for 1 ≤ k 1 < k 2 ≤ d . When two different ℓ p norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different ℓ p unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting. [ABSTRACT FROM AUTHOR]

In this paper we consider a circular arc approximation by hexic polynomial curves having 12 contacts with the circular arc. We present two methods for obtaining G k approximation curves, k = 3 , 4 , which interpolate at both endpoints and the midpoint of the circular arc. The approximation curves can be obtained by solving an equation of degree six. We show that the approximation orders of our methods are 12. We find the optimal approximation for each method and present numerical examples illustrating that the approximation orders are 12. [ABSTRACT FROM AUTHOR]