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The biological system has attracted many researchers with a lot of smart advances in the control system field. The control approaches have been effectively utilized in biological application to regulate the biological parameter functions based on the set point and threshold value. However, the controller's high disturbance has resulted in low gain and tuning performance. So, the current work has expected to plan a new African Buffalo based Recurrent Fractional-Order Sliding Mode (ABR-FOSM) for the biological system to tune the specific parameters like protein, pancreas, and Genetic Regulatory Network (GRN). Here, the incorporation of the buffalo algorithm has provided better tuning results. Moreover, the robustness of the designed controller has been validated by gaining the finest tuning performance and reduced Error and settling time. [ABSTRACT FROM AUTHOR]

In the geometric theory of functions of a complex variable, the well-known direction is related to the estimates of the products of the inner radii of pairwise non-overlapping domains. This direction is called extreme problems in classes of pairwise non-overlapping domains. One of the problems of this type is considered in the present work. [ABSTRACT FROM AUTHOR]

With the rapid advancement of drone technology and the growing applications in the field of drone engineering, the demand for precise and efficient path planning in complex and dynamic environments has become increasingly important. Traditional algorithms struggle with complex terrain, obstacles, and weather changes, often falling into local optima. This study introduces an Improved Crown Porcupine Optimizer (ICPO) for drone path planning, which enables drones to better avoid obstacles, optimize flight paths, and reduce energy consumption. Inspired by porcupines' defense mechanisms, a visuo-auditory synergy perspective is adopted, improving early convergence by balancing visual and auditory defenses. The study also employs a good point set population initialization strategy to enhance diversity and eliminates the traditional population reduction mechanism. To avoid local optima in later stages, a novel periodic retreat strategy inspired by porcupines' precise defenses is introduced for better position updates. Analysis on the IEEE CEC2022 test set shows that ICPO almost reaches the optimal value, demonstrating robustness and stability. In complex mountainous terrain, ICPO achieved optimal values of 778.1775 and 954.0118; in urban terrain, 366.2789 and 910.1682 and ranked first among the compared algorithms, proving its effectiveness and reliability in drone delivery path planning. Looking ahead, the ICPO will provide greater efficiency and safety for drone path planning in navigating complex environments. [ABSTRACT FROM AUTHOR]

Purpose: The present study aimed to compare the clinical outcomes and safety at a 1‐year follow‐up after 5 or 6 weeks of non‐weight bearing after a Talar OsteoPeriostic grafting from the Iliac Crest (TOPIC) for a medial osteochondral lesion of the talus (OLT). Methods: A retrospective comparative case‐control analysis of prospectively followed patients who underwent a TOPIC procedure with medial malleolus osteotomy was performed. Patients were matched in two groups with either 5 or 6 weeks of non‐weight bearing. Clinical outcomes were evaluated using the Numeric Rating Scale (NRS) during walking, rest, running, and stairclimbing. Additionally, the Foot and Ankle Outcome Score (FAOS) and American Orthopaedic Foot and Ankle Society (AOFAS) ankle‐hindfoot score were assessed. Moreover, radiology and complications were assessed. Results: Eleven patients were included in the 5‐week non‐weight bearing group and 22 in the 6‐week non‐weight bearing group. No significant differences were found in any of the baseline variables. The NRS during walking in the 5‐week group improved by 3.5 points and 4 points for the 6‐week group (p = 0.58 at 1‐year post‐operatively). In addition, all other NRS scores, FAOS subscales and the AOFAS scores improved (all n.s. at 1 year follow‐up). No significant differences in radiological (osteotomy union and cyst presence in the graft) were found. Moreover, no significant differences were found in terms of complications and reoperations. Conclusion: No statistical significant differences were found in terms of clinical, radiological and safety outcomes between 5 or 6 weeks of non‐weight bearing following a TOPIC for a medial OLT. Level of Evidence: Level III, Therapeutic. [ABSTRACT FROM AUTHOR]

Let X be an n-element point set in the k-dimensional unit cube [ 0 , 1 ] k where k ≥ 2 . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour) x 1 , x 2 , ... , x n through the n points, such that ∑ i = 1 n | x i - x i + 1 | k 1 / k ≤ c k , where | x - y | is the Euclidean distance between x and y, and c k is an absolute constant that depends only on k, where x n + 1 ≡ x 1 . From the other direction, for every k ≥ 2 and n ≥ 2 , there exist n points in [ 0 , 1 ] k , such that their shortest tour satisfies ∑ i = 1 n | x i - x i + 1 | k 1 / k = 2 1 / k · k . For the plane, the best constant is c 2 = 2 and this is the only exact value known. Bollobás and Meir showed that one can take c k = 9 2 3 1 / k · k for every k ≥ 3 and conjectured that the best constant is c k = 2 1 / k · k , for every k ≥ 2 . Here we significantly improve the upper bound and show that one can take c k = 3 5 2 3 1 / k · k or c k = 2.91 k (1 + o k (1)) . Our bounds are constructive. We also show that c 3 ≥ 2 7 / 6 , which disproves the conjecture for k = 3 . Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. [ABSTRACT FROM AUTHOR]

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. We do not assume that the points are in general position. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers. [ABSTRACT FROM AUTHOR]

Given a set of dot pattern point set P with the size n in the plane, we propose a Delaunay triangulation-based shape reconstruction algorithm, entitled prudent carving, that can reconstruct the inner boundaries (holes) and outer boundaries of P with the same approach. Our prudent carving algorithm is a parameter-free algorithm that has the ability to detect multiple components, sharp corners, and nested holes independent from the number of them in the total O (n log n) time. The qualitative and quantitative comparison of the results with the state-of-the-art algorithms shows that the prudent carving algorithm provides an enhancement over the previously best-known algorithms for reconstructing the boundaries of points. [ABSTRACT FROM AUTHOR]

We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. Under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method. [ABSTRACT FROM AUTHOR]

We consider the multilinear polytope defined as the convex hull of the set of binary points z, satisfying a collection of equations of the form z e = ∏ v ∈ e z v for all e ∈ E . The complexity of the facial structure of the multilinear polytope is closely related to the acyclicity degree of the underlying hypergraph. We obtain a polynomial-size extended formulation for the multilinear polytope of β -acyclic hypergraphs, hence characterizing the acyclic hypergraphs for which such a formulation can be constructed. [ABSTRACT FROM AUTHOR]

With the development of point cloud-based telemetry technology in recent years, the point cloud data of large field scenes acquired by various sensors have been applied to farmland boundary division, crop growth monitor, area surveying, etc. However, the large field point cloud will cost huge amounts of computational resources in the following transmission, storage and processing, which make it more important to simplify the field point cloud appropriately. In light of limitation of existing algorithms in point cloud simplification of large field scenes, we propose a novel feature-preserved simplification algorithm for large field point cloud data. By introducing the average local entropy as the threshold for area division, our algorithm effectively solves the problem of fuzzy boundary division, as well as preserving the field features and reducing the simplification errors. In view of the problem that the evaluation of current simplification algorithm is mainly focused on qualitative assessment, a quantitative evaluation index for the point cloud simplification is proposed by employing the point-average local entropy, which takes both model retention and simplification efficiency into account. Finally, comparable experiments are performed on four sets of point clouds. The results show that, compared with the statistics of six typical algorithms, the proposed algorithm increases the local entropy by 0.029%, 0.146% and 0.088% on our datasets, and increases 0.029% on the public dataset. The method accurately evaluates the simplification effect. Additionally, the surface area change rate is also used to further evaluate the performance of proposed algorithm, and the quantitative evaluation index is lower than others, which verify the advantages of proposed algorithm in feature protection and large field simplification. [ABSTRACT FROM AUTHOR]

Raw point cloud (PC) data acquired by 3D sensors or reconstruction algorithms inevitably contain noise and outliers, which can seriously impact downstream tasks, such as surface reconstruction and target detection. To address this problem, this paper proposes an innovative denoising method including the adaptive feature extraction (AFE) module, the gradient field estimation (GFE) module, and the further accelerated gradient ascent (FAGA) module. The method is based on considering the noisy PC as a convolutional distribution of clean PC and noise, and denoising is achieved by updating the positions of the points through gradient ascent iterations. Specifically, for a given noisy PC as input, we first extract global features in the AFE module, which are used as conditions to dynamically guide the extraction of local and non-local features to achieve adaptive feature acquisition. Next, these adaptive features are used as input to the GFE module to estimate the gradient field of the PCs and combined with our proposed FAGA module for denoising operations. Extensive qualitative and quantitative experiments are conducted on synthetic and natural PC datasets, and the results show that the proposed method exhibits superior performance relative to previous state-of-the-art methods. [ABSTRACT FROM AUTHOR]

We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct partitions of the set of lattice points in the plane. Our object is to analyze these trajectories and to point out a few special arithmetic properties of the integers they represent. We also introduce and study the parabolic-taxicab distance, which measures the fast traveling on the steps of the stairs defined by points on the parabolic trajectories whose coordinates are based on triangular numbers. [ABSTRACT FROM AUTHOR]

We revisit the following nonlinear Schrödinger equation - ε 2 Δ u + V (x) u = u p , u > 0 , u ∈ H 1 (R N) , where ε > 0 is a small parameter, N ≥ 2 and 1 < p < 2 ∗ - 1 . It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei's work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions. [ABSTRACT FROM AUTHOR]

Conformal predictions make it possible to define reliable and robust learning algorithms. But they are essentially a method for evaluating whether an algorithm is good enough to be used in practice. To define a reliable learning framework for classification from the very beginning of its design, the concept of scalable classifier was introduced to generalize the concept of classical classifier by linking it to statistical order theory and probabilistic learning theory. In this paper, we analyze the similarities between scalable classifiers and conformal predictions by introducing a new definition of a score function and defining a special set of input variables, the conformal safety set, which can identify patterns in the input space that satisfy the error coverage guarantee, i.e., that the probability of observing the wrong (possibly unsafe) label for points belonging to this set is bounded by a predefined ε error level. We demonstrate the practical implications of this framework through an application in cybersecurity for identifying DNS tunneling attacks. Our work contributes to the development of probabilistically robust and reliable machine learning models. [ABSTRACT FROM AUTHOR]

Focusing on the extreme points of the solution sets of matrix two-person games, we propose the notions of characteristic sets and characteristic numbers. The characteristic sets (resp., the characteristic numbers) are the sets (resp., the numbers) of the extreme points of the solution set of the game and the optimal solution sets of the players. These concepts allow us to measure the complexity of the game. The larger the characteristic numbers, the more complex the game is. Among other things, we obtain upper bounds for the characteristic numbers and give a novel geometric construction. By the construction, we get useful descriptions of the optimal strategy set of each player of a game given by any nonsingular square matrix. Namely, the investigation of the geometry the just-mentioned sets reduces to computing or studying certain simpler sets. We also formulate several open problems. [ABSTRACT FROM AUTHOR]

Imaging antiferromagnetic 180° domains with actively controlled visibility is vital for both fundamental science and sophisticated applications. While optical second-harmonic generation (SHG) is a well-known technique for distinguishing such domains in non-centrosymmetric antiferromagnets, a general material-based strategy to control domain contrast remains elusive. Using van der Waals antiferromagnet MnPS3 as a proof of concept, we demonstrate the tuning of nonreciprocity-induced domain contrast in SHG through applying an in-plane electric field that transforms the magnetic point group to its unitary subgroup. The interference among intrinsic electric-dipole, magnetic-dipole, and field-induced electric-dipole transitions, each carrying distinct characters under space-inversion (P ) and time-reversal (T ) operations, enables large tuning of domain contrast and nonreciprocity in a broad spectral range. This strategy, generically applicable to systems characterized by P T -symmetric magnetic groups with a polar unitary subgroup, offers a path to fast electrical modulation of nonlinear nonreciprocal photonic behaviors using antiferromagnets. Controlling properties associated with antiferromagnetic states could unlock numerous applications, though this has long seemed elusive. Here, Wang et al. demonstrate how electric fields can manipulate the coupling between light and antiferromagnetic domains in a nonlinear optical interaction. [ABSTRACT FROM AUTHOR]

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. The existence of such a partition for any number of regions remains an open question. In this paper, we address this issue by developing an algorithm to find such a convex fair partition without restrictions on the number of regions. Our approach utilizes the normal flow algorithm (a generalization of Newton's method) to find a zero for the excess areas and perimeters of the convex hulls of the regions. The initial partition is generated by applying Lloyd's algorithm to a randomly selected set of points within the polygon, after appropriate scaling. We performed extensive experimentation, and our algorithm can find a convex fair partition for 100% of the tested problem. Our findings support the conjecture that a convex fair partition always exists. [ABSTRACT FROM AUTHOR]

In this study, we propose an algorithmic framework for solving a class of optimal control problems. This class is associated with the minimum-time interception of moving target problems, where a plant with a given state equation must approach a moving target whose trajectory is known a priori. Our framework employs an analytical description of the distance from an arbitrary point to the reachable set of the plant. The proposed algorithm is always convergent and cannot be improved without losing the guarantee of its convergence to the correct solution for arbitrary Lipschitz continuous trajectories of the moving target. In practice, it is difficult to obtain an analytical description of the distance to the reachable set for the sophisticated state equation of the plant. Nevertheless, it was shown that the distance can be obtained for some widely used models, such as the Dubins car, in an explicit form. Finally, we illustrate the generality and effectiveness of the proposed framework for simple motions and the Dubins model. [ABSTRACT FROM AUTHOR]

Contrary to the second-order Phase field model (PFM) of fracture, fourth-order PFM provides a more precise representation of the crack surface by incorporating higher-order derivatives (curvature) of the phase-field order parameter in the so-called crack density functional. As a result, in a finite element setting, the weak form of the phase-field governing differential equation requires C 1 continuity in the basis function. C 0 Sibson interpolants or Natural element interpolants are obtained by the ratio of area traced by the second-order Voronoi cell over the first-order Voronoi cells, which is based on the natural neighbor of a nodal point set. C 1 Sibson interpolants are obtained by degree elevating the evaluated C 0 interpolants in the Bernstein-Bezier patch of a cubic simplex. For better computational efficiency while accounting only for the tensile part for driving fracture, a hybrid PFM is adopted. In this work, the numerical implementation of higher-order PFM with C 1 Sibson interpolants along with some benchmark examples are presented to showcase the performance of this method for simulating fracture in brittle materials. [ABSTRACT FROM AUTHOR]

When two tasks are presented simultaneously or in close succession, such as in the overlapping task paradigm of the psychological refractory period, dual-task performance on those tasks is usually impaired compared with separate single-task performance. Numerous theories explain these emerging dual-task costs in terms of the existence of capacity limitations in the constituent component tasks. The current paper proposes active dual-task coordination processes that work on the scheduling of these capacity-limited processes. Further, there are recent findings that point to a meta-cognitive control level in addition to these active coordination processes. This additional level's responsibility is to adjust the dual-task coordination of capacity-limited stages (i.e., coordination adjustment). I review evidence focusing on the existence of dual-task coordination processes and processes of coordination adjustment. The remainder of the paper elaborates on preliminary findings and points to the separability of these sets of processes, which is a key assumption of the framework of dual-task coordination adjustment. [ABSTRACT FROM AUTHOR]

In this study, we investigate the connection between the efficient frontier (EF) of a general multiobjective mixed integer linear optimization problem (MILP) and the so-called restricted value function (RVF) of a closely related single-objective MILP. In the first part of the paper, we detail the mathematical structure of the RVF, including characterizing the set of points at which it is differentiable, the gradients at such points, and the subdifferential at all nondifferentiable points. We then show that the EF of the multiobjective MILP is comprised of points on the boundary of the epigraph of the RVF and that any description of the EF suffices to describe the RVF and vice versa. Because of the close relationship of the RVF to the EF, we observe that methods for constructing the so-called value function (VF) of an MILP and methods for constructing the EF of a multiobjective optimization problem are effectively interchangeable. Exploiting this observation, we propose a generalized cutting-plane algorithm for constructing the EF of a multiobjective MILP that arises from an existing algorithm for constructing the classical MILP VF. The algorithm identifies the set of all integer parts of solutions on the EF. We prove that the algorithm converges finitely under a standard boundedness assumption and comes with a performance guarantee if terminated early. [ABSTRACT FROM AUTHOR]

We investigate a general boundary-value problem for nonuniformly 2 b → -parabolic equations with degeneration. The coefficients of parabolic equations and boundary conditions admit power singularities of any order in any variables on a certain set of points. By using a priori estimates and the Arzelà and Riesz theorems, we establish the existence and integral representation for the unique solution of the formulated boundary-value problems. The estimates of the solution of the general parabolic boundary- value problem and its derivatives in Hölder spaces with power weight are obtained. The order of the power weight is determined via the values of the orders of power singularities and degenerations of the coefficients of 2 b → -parabolic equations and boundary conditions. [ABSTRACT FROM AUTHOR]

When developing numerical methods for solving nonlinear minimax problems, the following auxiliary problem arose: in the convex hull of a certain finite set in Euclidean space, find a point that has the smallest norm. In 1971, B. Mitchell, V. Demyanov and V. Malozemov proposed a non-standard algorithm for solving this problem, which was later called the MDM algorithm (based on the first letters of the authors' last names). This article considers a specific minimax problem: finding the smallest volume ball containing a given finite set of points. It is called the Sylvester problem and is a special case of the problem about the Chebyshev center of a set. The Sylvester problem is associated with a convex quadratic programming problem with simplex constraints. To solve this problem, it is proposed to use a variant of the MDM algorithm. With its help, a minimizing sequence of feasible solutions is constructed such that two consecutive feasible solutions differ in only two components. The indices of these components are selected based on certain optimality conditions. We prove the weak convergence of the resulting sequence of feasible solutions that implies that the corresponding sequence of vectors converges in norm to a unique solution to the Sylvester problem. Four typical examples on a plane are given. [ABSTRACT FROM AUTHOR]

The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set P ~ of lifted points in the hyperbolic plane. Since P ~ is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than 12 g - 6 with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space. [ABSTRACT FROM AUTHOR]

The existence of nonzero fast points and linear structures reflects the properties of Boolean function's higher order derivatives, which is closely related to many cryptographic differential attacks. Rotation symmetric Boolean functions (RSBFs) is a super-class of symmetric functions, which are used widely in cryptography. We first obtain some existence results of nonzero linear structures of n-variable RSBFs with degree n - 2 . Moreover, we determine all the possible sets of fast points of n-variable RSBFs with degrees n - 3 and n - 4 based on integer partition. Finally, we investigate the existence of fast points of p-variable and 2p-variable RSBFs when p is an odd prime. [ABSTRACT FROM AUTHOR]

This paper develops a new method for analyzing the relationship between a set of points and another single point, the latter of which we call a reference point. This relationship has been discussed in various academic fields, such as geography, criminology, and epidemiology. Analytical methods, however, have not yet been fully developed, which has motivated this paper. Our method reveals how the number of points varies by the distance from a reference point and by direction. It visualizes the spatial pattern of points in relation to a reference point, describes the point pattern using mathematical models, and statistically evaluates the difference between two sets of points. We applied the proposed method to analyze the spatial pattern of the climbers of Mt. Azuma, Japan. The result gave us useful and interesting findings, indicating the method's soundness. [ABSTRACT FROM AUTHOR]

The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features. [ABSTRACT FROM AUTHOR]

As the world is more focused on developing products and services, which are sustainable, special attention is needed in the design and manufacture of machine tools. There is a multitude of elements that contribute to the sustainability of machine tools. Moreover, the elements are interdependent with each other. The presence and absence of each element can be measured by different indicators. This paper analyzes the influence of elements and their indicators on the sustainability of machine tools. Sustainability elements and their indicators have been identified first. Further, a new hybrid model was developed to find the interdependencies between the sustainability elements for machine tools and determine the priorities of the indicators. Two structural models, i.e., Decision-Making Trial and Evaluation Laboratory and Analytic Network Process, have been combined to establish levels and intensities of interdependence as well as the relative weight of selected indicators. The results classify the indicators into the receiver, dispatcher, and neutral groups and also point out the indicators with feedback dependencies, i.e., indicators that depend on themselves. The methodology can be applied for evaluating as well as comparing the sustainability of different machines. The observations would be helpful to the designers of machine tools to make strategies for improving the sustainability of their machines. [ABSTRACT FROM AUTHOR]

Apollonius defined the circle as the set of points that have a given ratio μ of distances from two given points, where the ratio is not equal to one. In a more general sense, consider two 0-symmetric, bounded, convex bodies K and K ′ , which define two norms. Their unit balls are K and K ′ . The surface of Apollonius is defined as the set of points equidistant from the centres of bodies K and K ′ with respect to the aforementioned norms. In this paper we demonstrate that the surface of Apollonius of two ellipsoids is a quadratic surface. We also examine the circumstances under which this surface becomes a sphere. [ABSTRACT FROM AUTHOR]

A two-way segmentation model is proposed in this article. The model is used to solve the problem that the objective contour can not be completely extracted from image due to occlusion between objects within similar image groups or image sequences. The proposed model first decomposes incomplete contours into sub-segments using local features identified by seed points set along each path. Then, locate the occluded part of the target object and reconstruct the target. Finally, a new vector field is generated based on the reconstructed object from the proposed model, followed by iterative evolution. The experimental results show that the proposed algorithm can better handle the problem of occlusion or misleading features of targets in composite images and medical images. Not only does it facilitate subsequent measurement and analysis, but it also preserves the original shape of the object during the segmentation process without prior information. It is worth noting that the accuracy of the proposed model is robust to our initialization strategy. [ABSTRACT FROM AUTHOR]

A number field K is primitive if K and Q are the only subextensions of K. Let C be a curve defined over Q . We call an algebraic point P ∈ C (Q ¯) primitive if the number field Q (P) is primitive. We present several sets of sufficient conditions for a curve C to have finitely many primitive points of a given degree d. For example, let C / Q be a hyperelliptic curve of genus g, and let 3 ≤ d ≤ g - 1 . Suppose that the Jacobian J of C is simple. We show that C has only finitely many primitive degree d points, and in particular it has only finitely many degree d points with Galois group S d or A d . However, for any even d ≥ 4 , a hyperelliptic curve C / Q has infinitely many imprimitive degree d points whose Galois group is a subgroup of S 2 ≀ S d / 2 . [ABSTRACT FROM AUTHOR]

We investigate how the complexity of Euclidean TSP for point sets P inside the strip (- ∞ , + ∞) × [ 0 , δ ] depends on the strip width δ . We obtain two main results. For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in O (n log 2 n) time using an existing algorithm) is guaranteed to be a shortest tour overall when δ ⩽ 2 2 , a bound which is best possible. We present an algorithm that is fixed-parameter tractable with respect to δ . Our algorithm has running time 2 O (δ) n + O (δ 2 n 2) for sparse point sets, where each 1 × δ rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle [ 0 , n ] × [ 0 , δ ] , it has an expected running time of 2 O (δ) n . These results generalise to point sets P inside a hypercylinder of width δ . In this case, the factors 2 O (δ) become 2 O (δ 1 - 1 / d) . [ABSTRACT FROM AUTHOR]

For any given full rank lattice Λ and natural number K, we consider the point set P (K) = Λ / K ∩ (0 , 1) 2 , with N = # P (K) ≈ K 2 , and bound the spherical cap discrepancy of the projection of these points under the Lambert map to the unit sphere. The bound is of order 1 / N , with leading coefficient given explicitly and depending on Λ only. The proof is established using a lemma that bounds the number of intersections of certain curves with fundamental domains that tile R 2 , and even allows for local perturbations of Λ without affecting the bound, proving to be suitable for numerical applications. A special case yields the smallest constant for the leading term of the cap discrepancy for deterministic algorithms up to date. [ABSTRACT FROM AUTHOR]

Currently, the digitization of environment objects (vegetation, terrain, architectural structures, etc.) in the form of point clouds is actively being developed. The integration of such digitized objects into virtual environment systems allows the quality of the modeled environment to be improved, but requires efficient methods and algorithms for real-time visualization of large point volumes. In this paper the solution of this task on modern multicore GPUs with support of hardware-accelerated ray tracing is researched. A modified method is proposed where the original unordered point cloud is split up into point groups whose visualization is effectively parallelized on ray tracing cores. The paper describes an algorithm for constructing such groups using swapping arrays of point indices, which works faster than alternative solutions based on linked lists, and also has lower memory overhead. The proposed method and algorithm were implemented in point cloud visualization software complex and approbated on a number of digitized environment objects. The results of the approbation confirmed the efficiency of proposed solutions as well as their applicability for virtual environment systems, video simulators, geoinformation systems, virtual laboratories, etc. [ABSTRACT FROM AUTHOR]

This text explores the relationship between mathematicians Georg Cantor and Richard Dedekind and their differing perspectives on set theory. Cantor saw sets as vast and infinite, while Dedekind viewed them as closed and finite. Cantor sought Dedekind's support for his ideas on inconsistent sets, but Dedekind was not interested. Cantor's mental health declined, and he sought a position outside of academia. Despite their differences, both mathematicians made important contributions to the field of set theory. [Extracted from the article]

Combining the Lagrange multiplier method, the Uzawa algorithm, and the least square progressive and iterative approximation (LSPIA), we proposed the constrained least square progressive and iterative approximation (CLSPIA) to solve the problem of B-spline curve and surface fitting with constraint on data interpolation, i.e., computing the control points of a B-spline curve or surface which interpolates one set of input points while approximating the other set of given points. Compared with the method of solving the linear system directly, CLSPIA has some advantages as it inherits all the nice properties of LSPIA. Because of the data reuse property of LSPIA, CLSPIA reduces a great amount of computation. Using the local property of LSPIA, we can get shape preserving fitting curves by CLSPIA. CLSPIA is efficient for fitting large-scale data sets due to the fact that its computational complexity is linear to the scale of the input data. The many numerical examples in this paper show the efficiency and effectiveness of CLSPIA. [ABSTRACT FROM AUTHOR]

Diffusion models have been successfully applied to point cloud generation tasks recently. The main notion is using a forward process to progressively add noises into point clouds and then use a reverse process to generate point clouds by denoising these noises. However, since point cloud data is high-dimensional and exhibits complex structures, it is challenging to adequately capture the surface distribution of point clouds. Moreover, point cloud generation methods often resort to sampling methods and local operations to extract features, which inevitably ignores the global structures and overall shapes of point clouds. To address these limitations, we propose a latent diffusion model based on Transformers for point cloud generation. Instead of directly building a diffusion process based on the points, we first propose a latent compressor to convert original point clouds into a set of latent tokens before feeding them into diffusion models. Converting point clouds as latent tokens not only improves expressiveness, but also exhibits better flexibility since they can adapt to various downstream tasks. We carefully design the latent compressor based on an attention-based auto-encoder architecture to capture global structures in point clouds. Then, we propose to use transformers as the backbones of the latent diffusion module to maintain global structures. The powerful feature extraction ability of transformers guarantees the high quality and smoothness of generated point clouds. Experiments show that our method achieves superior performance in both unconditional generation on ShapeNet and multi-modal point cloud completion on ShapeNet-ViPC. Our code and samples are publicly available at https://github.com/Negai-98/LDT. [ABSTRACT FROM AUTHOR]

Let S be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of S with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that S defines a pair of crossing edges of the same color is equal to 1 / 4 . This is connected to a recent result of Aichholzer et al. [1] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halved. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation 1 2 - 7 50 of the total number of crossings. [ABSTRACT FROM AUTHOR]

Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of Θ (n 3) triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them. For families of Θ (n 3 - α) triangles, with 0 ≤ α ≤ 1 , there might not be a point in more than Θ (n 3 - 2 α) of those triangles. An empty triangle of P is a triangle spanned by P not containing any point of P in its interior. Bárány conjectured that there exists an edge spanned by P that is incident to a super-constant number of empty triangles of P. The number of empty triangles of P might be as low as Θ (n 2) ; in such a case, on average, every edge spanned by P is incident to a constant number of empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number 0 ≤ α ≤ 1 there exists a point set of size n with Θ (n 3 - α) empty triangles such that any point of the plane is only in O (n 3 - 2 α) empty triangles. [ABSTRACT FROM AUTHOR]

This paper focuses on the minimization of a sum of a twice continuously differentiable function f and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of f involving the ϱ th power of the KKT residual. For ϱ = 0 , we justify the global convergence of the iterate sequence for the KL objective function and its R-linear convergence rate for the KL objective function of exponent 1/2. For ϱ ∈ (0 , 1) , by assuming that cluster points satisfy a locally Hölderian error bound of order q on a second-order stationary point set and a local error bound of order q > 1 + ϱ on the common stationary point set, respectively, we establish the global convergence of the iterate sequence and its superlinear convergence rate with order depending on q and ϱ . A dual semismooth Newton augmented Lagrangian method is also developed for seeking an inexact minimizer of subproblems. Numerical comparisons with two state-of-the-art methods on ℓ 1 -regularized Student's t-regressions, group penalized Student's t-regressions, and nonconvex image restoration confirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Point cloud completion is a challenging task that involves predicting missing parts in incomplete 3D shapes. While existing strategies have shown effectiveness on point cloud datasets with regular shapes and continuous surfaces, they struggled to manage the morphologically diverse structures commonly encountered in real-world scenarios. This research proposed a new point cloud completion method, called SegCompletion, to derive complete 3D geometries from a partial shape with different structures and discontinuous surfaces. To achieve this, morphological segmentation was introduced before point cloud completion by deep hierarchical feature learning on point sets, and thus, the complex morphological structure was segmented into regular shapes and continuous surfaces. Additionally, each instance of a point cloud that belonged to the same type of feature could also be effectively identified using HDBSCAN (Hierarchical Density-Based Spatial Clustering of Applications with Noise). Furthermore, the multiscale generative network achieved sophisticated patching of missing point clouds under the same geometric feature based on feature points. To compensate for the variance in the mean distances between the centers of the patches and their closest neighbors, a simple yet effective uniform loss was utilized. A number of experiments on ShapeNet and Pheno4D datasets have shown the performance of SegCompletion on public datasets. Moreover, the contribution of SegCompletion to our dataset (Cotton3D) was discussed. The experimental results demonstrated that SegCompletion performed better than existing methods reported in the literature. [ABSTRACT FROM AUTHOR]

High-speed railway bridges are subjected to normative limitations concerning maximum permissible deck accelerations. For the design of these structures, the European norm EN 1991-2 introduces the high-speed load model (HSLM)—a set of point loads intended to include the effects of existing high-speed trains. Yet, the evolution of current trains and the recent development of new load models motivate a discussion regarding the limits of validity of the HSLM. For this study, a large number of randomly generated load models of articulated, conventional, and regular trains are tested and compared with the envelope of HSLM effects. For each type of train, two sets of 100,000 load models are considered: one abiding by the limits of the EN 1991-2 and another considering wider limits. This comparison is achieved using both a bridge-independent metric (train signatures) and dynamic analyses on a case study bridge (the Canelas bridge of the Portuguese Railway Network). For the latter, a methodology to decrease the computational cost of moving loads analysis is introduced. Results show that some theoretical load models constructed within the stipulated limits of the norm can lead to effects not covered by the HSLM. This is especially noted in conventional trains, where there is a relation with larger distances between centres of adjacent vehicle bogies. [ABSTRACT FROM AUTHOR]

A new exact projective penalty method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing its value at the projection of an infeasible point on the feasible set with the distance to the projection. Beside Euclidean projections, also a pointed projection in the direction of some fixed internal feasible point can be used. The equivalence means that local and global minimums of the problems coincide. Nonconvex sets with multivalued Euclidean projections are admitted, and the objective function may be lower semicontinuous. The particular case of convex problems is included. The obtained unconstrained or box constrained problem is solved by a version of the branch and bound method combined with local optimization. In principle, any local optimizer can be used within the branch and bound scheme but in numerical experiments sequential quadratic programming method was successfully used. So the proposed exact penalty method does not assume the existence of the objective function outside the allowable area and does not require the selection of the penalty coefficient. [ABSTRACT FROM AUTHOR]

For a set Q of points in the plane and a real number δ ≥ 0 , let G δ (Q) be the graph defined on Q by connecting each pair of points at distance at most δ .We consider the connectivity of G δ (Q) in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set P of n - k points in the plane and a set S of k line segments in the plane, find the minimum δ ≥ 0 with the property that we can select one point p s ∈ s for each segment s ∈ S and the corresponding graph G δ (P ∪ { p s ∣ s ∈ S }) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in O (f (k) n log n) time, for a computable function f (·) . This implies that the problem is FPT when parameterized by k. The best previous algorithm uses O ((k !) k k k + 1 · n 2 k) time and computes the solution up to fixed precision. [ABSTRACT FROM AUTHOR]

This paper introduces a new way to extract a set of representative points from a continuous distribution, which focuses on a method where the selection of points is essentially deterministic, with an emphasis on achieving accurate approximation when the size of points is small. These points are generated by minimizing the Kullback–Leibler divergence, which is an information-based measure of the disparity between two probability distributions. We refer to these points as Kullback–Leibler points. Based on the link between the total variation and the Kullback–Leibler divergence, we prove that the empirical distribution of Kullback–Leibler points converges to the target distribution. Additionally, we illustrate that Kullback–Leibler points have advantages in simulations when compared with representative points generated by Monte Carlo or other representative points methods. In addition, to prevent the frequent evaluation of complex functions, a sequential version of Kullback–Leibler points is proposed, which adaptively updates the representative points by learning about the complex or unknown functions sequentially. Two potential applications of Kullback-Leibler points in simulation of complex probability densities and optimization of complex response surfaces are discussed and demonstrated with examples. [ABSTRACT FROM AUTHOR]

The recently proposed semantic-aligned matching detection transformer (SAM–DETR model) accelerates the convergence of the detection transformer (DETR) by mapping object queries into an identical embedding space as the encoder's output feature map. However, SAM–DETR model has the problem of low detection accuracy compared to other DETR variants. We observe that the lower detection accuracy of SAM–DETR model is caused by the insufficient number of sample points and the inaccurate localization of the sample points during re-sampling, which blurs the generated attention map. This paper proposes an object detector based on a feature correction and double sampling DETR (FCDS-DETR) to solve this problem. FCDS-DETR takes SAM–DETR model as a baseline and builds on it by adding a feature correction module and a double sampling mechanism to achieve further improvement in detection accuracy with a limited number of additional parameters without sacrificing convergence speed. Firstly, FCDS-DETR improves the sampling point localization accuracy by adding a feature correction module to model the inter-channel dependence of the feature maps to be sampled. Secondly, the number of sampled points is increased by the double sampling mechanism, and attention fusion is used to fuse the attention weight maps corresponding to the two sets of sampled points to improve the recognizability of the attention weight maps. The experimental results show that the average precision is improved by +0.7 on the COCO dataset compared with the SAM–DETR model, and the number of parameters is increased by only 10.34 % , which improves the detection performance of the model very well. [ABSTRACT FROM AUTHOR]

This paper introduces Dyck free groups that are geometric realizations of polygon cycles, the ratio of either two or three collinear points in the Desargues affine plane. A biproduct of this work is a concise presentation of Desargues affine planar cycles as a free group. In this paper, we include (1) the study of properties for the ratio of two and three points in a line in the Desargues affine plane. Also, we discuss the cases related to the "line-skew field" characteristic, when it is two and when it is different from two. (2) We have constructed the maps for ratio points-set for two and three points and have proved that these maps are bijections of the lines. (3) We observe that the set of ratio points (for two and for three points) with addition and multiplication of points forms a skew field and these skew fields are sub-skew fields of the "line-skew field" in a Desargues affine plane. (4) We also observe that every Dyck polygon cycle in the Desargues affine plane is the realization of a Dyck path cycle. (5) We prove that every Dyck polygon containing collinear ratio vertices has a free group presentation. [ABSTRACT FROM AUTHOR]

Understanding elevation changes on the Tibetan Plateau is crucial to comprehend the changes in topography, landscape, climate, environmental conditions, and water resources. However, some of the current products that track elevation changes only cover specific surface types or limited areas, and others have low spatial resolution. We propose an algorithm to extract ICESat-2 crossover points dataset for the Tibetan Plateau, and form a dataset. The crossover points dataset has a density of 2.015 groups/km², and each group of crossover points indicates the amount of change in elevation before and after a period of time over an area of approximately 17 meters in diameter. Comparing ICESat-2 crossover points data with existing studies on glaciers and lakes, we demonstrated the reliability of the derived elevation changes. The ICESat-2 crossover points provide a refined data source for understanding high-spatial-resolution elevation changes on the Tibetan Plateau. This dataset can provide validation data for various studies that require high-precision or high-resolution elevation change data on the Tibetan Plateau. [ABSTRACT FROM AUTHOR]

This paper attempts to prove the Lipschitz continuity of the resolvent operator associated with a (P , η) -accretive mapping and compute an estimate of its Lipschitz constant. This is done under some new appropriate conditions that are imposed on the parameter and mappings involved in it; with the goal of approximating a common element of the solution set of a system of generalized variational-like inclusions and the fixed point set of a total asymptotically nonexpansive mapping in the framework of real Banach spaces. A new iterative algorithm based on the resolvent operator technique is proposed. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the two sets mentioned above. The final section is dedicated to investigating and analyzing the notion of a generalized H(.,.)-accretive mapping introduced and studied by Kazmi et al. (Appl Math Comput 217:9679–9688, 2011). In this section, we provide some comments based on the relevant results presented in their work. [ABSTRACT FROM AUTHOR]

A method for detecting and approximating fault lines or surfaces, respectively, or decision curves in two and three dimensions with guaranteed accuracy is presented. Reformulated as a classification problem, our method starts from a set of scattered points along with the corresponding classification algorithm to construct a representation of a decision curve by points with prescribed maximal distance to the true decision curve. Hereby, our algorithm ensures that the representing point set covers the decision curve in its entire extent and features local refinement based on the geometric properties of the decision curve. We demonstrate applications of our method to problems related to the detection of faults, to multi-criteria decision aid and, in combination with Kirsch's factorization method, to solving an inverse acoustic scattering problem. In all applications we considered in this work, our method requires significantly less pointwise classifications than previously employed algorithms. [ABSTRACT FROM AUTHOR]