Ball Of Radius Research Articles

Super eagle optimization algorithm based three-dimensional ball security corridor planning method for fixed-wing UAVs

In modern warfare, unmanned aerial vehicles (UAVs) have become a vital military force. Planning feasible paths for UAVs has become an essential part of forming autonomous attack systems. Aiming at the task requirements of fix-wing UAVs, this paper proposes a ball security corridor planning method by combining the ball Said-Ball curve and a novel super eagle optimization algorithm (SEOA). Firstly, the ball security corridor planning model is established by taking the center curve and radius function of the ball Said-Ball (BSB) curve as the flight path and security boundary. It not only considers the smoothness of the flight path, but also reserves enough security range to avoid cumulative errors. Then, a novel super eagle optimization algorithm is proposed to solve the established model with higher accuracy and efficiency. It simulates the hunting process of virtual super eagles with intelligent thinking. For super eagles, two patterns are designed for determining prey at different stages to avoid premature convergence. An information sharing strategy is also introduced to balance the exploitation and exploration abilities. For prey, they can choose orderly and emergency strategies based on emotional function to escape capture to speed up the convergence of SEOA. According to the results of solving the test function in CEC2017, the SEOA has smaller average rank compared to other flying animal-based algorithms and recent popular algorithms. For the solving successful rate, the SEOA reaches 82.7586% and 89.6552%, respectively. Additionally, the results of Wilcoxon rank-sum test statistically prove that the SEOA is a newly developed intelligent optimization method with significantly higher accuracy. Finally, the SEOA is used to solve the established model by optimizing the positions and radii of control balls of BSB curves. Experimental results in three complex simulation scenes show that the flight corridors obtained by SEOA outperform those generated by other algorithms in overall fitness values. This indicates a more rational optimization in flight altitude, distance, smoothness, and security range. Its faster convergence speed can ensure higher efficiency in planning a security corridor satisfying constraints and requirements.

Growth rate of Dehn twist lattice points in Teichmüller space

Athreya et al. [Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J. 161 (2012) 1055–1111] have shown the number of mapping class group lattice points intersecting a closed ball of radius [Formula: see text] in Teichmüller space is asymptotic to [Formula: see text], where [Formula: see text] is the dimension of the Teichmüller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius [Formula: see text] is coarsely asymptotic to [Formula: see text]. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius [Formula: see text] grows coarsely at least at the rate of [Formula: see text].

Influence of various balls and groove curvature radii on dynamic stiffness of ball bearings based on nonlinear dynamic model

In the context of this study, a novel dynamic model is integrated with the stiffness matrix model so that the dynamic forces effect the stiffness calculation real-timely to form the dynamic stiffness, which is a novel solution for the dynamic performance analysis of rolling bearings under different loads, assemblies and bearing structures. Based on this model, the influence of ball quantity and groove curvature radii on the dynamic stiffness of ball bearings is explored. The optimal ball's quantity and groove curvature radii combination were determined to attain the desired magnitude and fluctuation of bearing stiffness. The results show that favorable stiffness can be achieved when the number of balls is decreased by one or two, the radius of outer groove curvature approximates the ball radius, and the radius of inner groove curvature is appropriately larger than the ball radius. This configuration offers theoretical backing for optimizing the structural dimensions of ball bearings.

On Blaschke–Santaló-Type Inequalities for r-Ball Bodies

Let 𝔼𝑑 denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼𝑑 is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼𝑑 the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

Far field broadband approximate cloaking for the Helmholtz equation with a Drude-Lorentz refractive index

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in Rd for d=2 or d=3. Using ideas from transformation optics, we construct an approximate cloak by “blowing up” a small ball of radius ϵ>0 to one of radius 1. In the anisotropic cloaking layer resulting from the “blow-up” change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed ϵ) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as ϵ approaches 0, the L2-norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves.

A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut($$Fi_{22}$$) Which Has a Nontrivial Stabilizer of a Ball of Radius $$2$$

A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut($$Fi_{22}$$) Which Has a Nontrivial Stabilizer of a Ball of Radius $$2$$

Deterministic rendezvous in infinite trees

The rendezvous task calls for two mobile agents, starting from different nodes of a network modeled as a graph to meet at the same node. Agents have different labels which are integers from a set {1,…,L}. They wake up at possibly different times and move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous algorithms. The time of such an algorithm is the number of rounds since the wakeup of the earlier agent till the meeting. In most of the literature concerning rendezvous in graphs, the graph is finite and the time of rendezvous depends on its size. This approach is impractical for very large graphs and impossible for infinite graphs. For such graphs it is natural to design rendezvous algorithms whose time depends on the initial distance D between the agents. In this paper we adopt this approach and consider rendezvous in infinite trees. All our algorithms work in finite trees as well. Our main goal is to study the impact of orientation of a tree on the time of rendezvous.We first design a rendezvous algorithm working for unoriented regular trees, whose time is in O(z(D)log⁡L), where z(D) is the size of the ball of radius D, i.e., the number of nodes at distance at most D from a given node. The algorithm works for arbitrary delay between waking times of agents and does not require any initial information about parameters L or D. Its disadvantage is its complexity: z(D) is exponential in D for any degree d>2 of the tree. We prove that this high complexity is inevitable: Ω(z(D)) turns out to be a lower bound on rendezvous time in unoriented regular trees, even for simultaneous start and even when agents know L and D. Then we turn attention to oriented trees. While for arbitrary delay between waking times of agents the lower bound Ω(z(D)) still holds, for simultaneous start the time of rendezvous can be dramatically shortened. We show that if agents know either a polynomial upper bound on L or a linear upper bound on D, then rendezvous can be accomplished in oriented trees in time O(Dlog⁡L), which is optimal. If no such extra knowledge is available, a significant speedup is still possible: in this case we design an algorithm working in time O(D2+log2⁡L).

On edge chipping in molar teeth from blunt occlusal contact

Edge chipping is a leading failure mode in dental teeth. Virtually all chipping studies are limited to Vickers indentation on polished cusps of molar teeth. Such works are here extended to spherical contact. Occlusal loads are applied on the tooth's central fossa or a polished cusp using ball radii ranging from 0.4 to 5.16 mm. The chip dimensions are characterized by h/Dm and D/Dm, where h, D and Dm denote indent distance, chip size and tooth crown diameter. For the fossa loading, h/Dm, D/Dm and the least chipping force Pch are virtually independent of ball radius r for r < ≈ 4 mm. In this range, h/Dm and D/Dm lie between ≈0.30 to 0.36 and 0.51 to 0.69, respectively, while Pch equals ≈1330 N. For r > ≈ 4 mm, the failure occurs by debonding of enamel sectors from the dentin core. In the case of cusp loading, h/Dm < ≈ 0.3 while D/Dm and Pch vary with r. For relatively small h or large r, the failure occurs as soon as radial cracks initiate under the loading point. For a load applied near a cusp tip, the failure occurs by enamel debonding. Finally, the present work is easily extendable to fossil teeth of hominins and apes as well as prosthetic teeth. The morphological features obtained in such studies should provide quantitative means to assess the relationships between chip dimensions, chipping force and diet characteristics.

Homogenization in perforated domains at the critical scale

We describe the asymptotic behaviour of the minimal heterogeneous d-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set Ω⊆Rd, with d≥2. Two parameters are involved: ɛ, the radius of the ball, and δ, the length scale of the heterogeneity of the medium. We prove that this capacity behaves as C|logɛ|1−d, where C=C(λ) is an explicit constant depending on the parameter λ≔limɛ→0|logδ|/|logɛ|.We determine the Γ-limit of oscillating integral functionals subjected to Dirichlet boundary conditions on periodically perforated domains. Our first result is used to study the behaviour of the functionals near the perforations which, in this instance, are balls of radius ɛ. We prove that an additional strange term arises involving C(λ).

Monitoring method of cutting forces and vibrations by using frequency separation of acceleration sensor signals during milling process with small ball end mills

Recently, there has been a further demand for precise monitoring of milling process with a machine tool by a simple and cost-effective method. One of the ways for monitoring is to install acceleration sensors to the tool spindle and estimate cutting forces from the model of the tool spindle structure. This method generally allows stable monitoring of cutting forces for tools with large diameters. However, in case of using tools with small diameters, it is difficult to estimate the cutting force due to higher frequency vibrations generated near the tool center point. Therefore, to solve the problem, we propose a new monitoring method by signal analysis of acceleration sensors in this research. The problem is that the acceleration sensor signals contain two types of signals, such as ‘acceleration change due to mechanical displacement of a tool spindle generated by cutting load’ and ‘high frequency self-excited and forced vibration’. In our method, we separate these two signals by using an approximation of sequential quadratic regression. From the former, cutting forces are estimated from equation of motions which are obtained in advance from frequency responses of the tool spindle, and from the latter, intensities of vibrations due to milling are estimated. This method was tested several milling patterns such as large cutting loads, fluctuations of cutting loads, and cutting during abnormal vibrations. As a result, we have achieved in-process monitoring of milling process not only in the X and Y directions, but also in the Z direction with a small radius ball end mill.

Dynamic Response Study of Piezoresistive Ti3C2-MXene Sensor for Structural Impacts.

MXenes are a new family of two-dimensional (2D) nanomaterials. They are inorganic compounds of metal carbides/nitrides/carbonitrides. Titanium carbide MXene (Ti3C2-MXene) was the first 2D nanomaterial reported in the MXene family in 2011. Owing to the good physical properties of Ti3C2-MXenes (e.g., conductivity, hydrophilicity, film-forming ability, elasticity) various applications in wearable sensors, energy harvesters, supercapacitors, electronic devices, etc., have been demonstrated. This paper presents the development of a piezoresistive Ti3C2-MXene sensor followed by experimental investigations of its dynamic response behavior when subjected to structural impacts. For the experimental investigations, an inclined ball impact test setup is constructed. Stainless steel balls of different masses and radii are used to apply repeatable impacts on a vertical cantilever plate. The Ti3C2-MXene sensor is attached to this cantilever plate along with a commercial piezoceramic sensor, and their responses for the structural impacts are compared. It is observed from the experiments that the average response times of the Ti3C2-MXene sensor and piezoceramic sensor are 1.28±0.24μs and 31.19±24.61μs, respectively. The fast response time of the Ti3C2-MXene sensor makes it a promising candidate for monitoring structural impacts.

A Dynamic Model of Outer Race Defective Bearing Considering the Unbalanced Shaft-Bearing System With Experimental Simulation

Abstract In this study, the effects of the evolution of bearing outer race defect size and increase in speed on the vibration characteristics of a shaft-bearing system under unbalanced conditions are simulated and analyzed. A two degrees-of-freedom mathematical model is presented for a ball bearing used in an unbalanced shaft-bearing system. The contact stiffness between the races and the balls is considered as a series of springs is incorporated in the model. Hertzian contact deformation theory is used to obtain the contact stiffness. This model considers the contact deformation between the balls and the races, the additional displacement between the balls and the inner race due to radial clearance and due to defect geometry. The maximum possible radial displacement of the ball into the defect is calculated analytically using the groove radius, ball radius, and defect diameter. The rectangular function is used for modeling the defect. matlab codes are developed for modeling the bearing and for solving the differential equations of motion using the Runge–Kutta method. The vibration responses (peak and root-mean-square (RMS) values) obtained by modeling and by experimentation show similar vibration characteristics. The investigation shows that the values of statistical parameters initially increase with the increase in defect size and then decrease with a further increase in defect size. While peak and RMS increase with the increase in speed, crest factor and kurtosis decrease with the increase in speed. Peak is more sensitive for diagnosing spalls on outer race and its evolution. This study helps as an effective diagnosis of antifriction bearings having spalls on the outer race under unbalanced conditions.

Maximum entanglement of mixed symmetric states under unitary transformations

We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable (SAS) states, or absolutely classical for spin states. The results for the two-qubit system are deduced analytically. In particular, we determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that contains the set of SAS states. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin-1 system with a spin-squeezing Hamiltonian. For the symmetric three-qubit case, our results are mostly numerical, and we conjecture a 3-parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.

Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

Let {Zn}n≥0 be a d-dimensional supercritical branching random walk started from the origin. Write Zn(S) for the number of particles located in a set S⊂Rd at time n. Denote by Rn:=inf⁡{ρ≥0:Zi({|x|≥ρ})=0,∀0≤i≤n} the radius of the minimal ball (centered at the origin) containing the range of {Zi}i≥0 up to time n. In this work, we show that under some mild conditions Rn/n converges in probability to some positive constant x⁎ as n→∞. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates ofP(Rn≤xn)forx∈(0,x⁎);P(Rn≥xn)forx∈(x⁎,∞) as n→∞.

Intersections of Poisson [formula omitted]-flats in constant curvature spaces

Poisson processes in the space of k-dimensional totally geodesic subspaces (k-flats) in a d-dimensional standard space of constant curvature κ∈{−1,0,1} are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order m together with their (d−m(d−k))-dimensional Hausdorff measure within a geodesic ball of radius r. Asymptotic normality for fixed r is shown as the intensity of the underlying Poisson process tends to infinity for all m satisfying d−m(d−k)≥0. For κ∈{−1,0} the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2k≤d+1 a central limit theorem is shown for all possible values of m. However, while for κ=0 asymptotic normality still holds if 2k>d+1, we prove for κ=−1 convergence to a non-Gaussian infinitely divisible limit distribution in the special case m=1. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin–Stein method. We also show for general κ∈{−1,0,1} that, roughly speaking, the variances within a general observation window W are maximal if and only if W is a geodesic ball having the same volume as W. Along the way we derive a new integral-geometric formula of Blaschke–Petkantschin type in a standard space of constant curvature.

Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

A parabolic–elliptic Keller–Segel system ut=Δu−χ∇⋅(uf(|∇v|)∇v),0=Δv−M+u,with homogeneous Neumann boundary condition is considered in a radially symmetric domain Ω=BR(0)⊂RN(N≥3), where f(ξ)=(ξp−2(1+ξp)q−pp),ξ≥0,p≥2,1<q≤p<∞,and BR(0) is a N-dimensional ball of radius R>0. We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when NN−1<q<2.

Macroscopic scalar curvature and codimension 2 width

We show that a complete [Formula: see text]-dimensional Riemannian manifold [Formula: see text] with finitely generated first homology has macroscopic dimension [Formula: see text] if it satisfies the following “macroscopic curvature” assumptions: every ball of radius [Formula: see text] in [Formula: see text] has volume at most [Formula: see text], and every loop in every ball of radius [Formula: see text] in [Formula: see text] is null-homologous in the concentric ball of radius [Formula: see text].

Fatigue Analysis of Hip Prosthesis

The present work covers the analytical design process of three dimensional (3-D) hip joint prosthesis with numerical fatigue stress analysis. The analytical generation equations describing the different stem constructive parts (ball, neck, tour, cone, lower ball) have been presented to reform the stem model in a mathematical feature. The generated surface has been introduced to FE solver (Ansys version 11) in order to simulate the induced dynamic stresses and investigate the effect of every design parameter (ball radius, angle of neck, radius of neck, neck ratio, main tour radius, and outer tour radius) on the max. equivalent stresses for hip prosthesis made from titanium alloy. The dynamic loading case has been studied to a stumbling case. The load has been applied on the cap tip as a concentrated load distributed on the interface of ball and socket. The results show that the decreasing of max. Fatigue stress by (175) MPa could be obtained by increasing the outer tour radius from (10)mm to (15) mm and that will change the max. Fatigue zone location from the tour section to the neck. The ball radius and neck angle must be as lower as possible to decrease the fatigue stresses. The most dominate parameter to increase the safety factor is the radius of neck.

Growth of pseudo-Anosov conjugacy classes in Teichmüller space

Athreya, Bufetov, Eskin and Mirzakhani (2012) have shown that the number of mapping class group lattice points intersecting a closed ball of radius R in Teichmüller space is asymptotic to e^{hR} , where h is the dimension of the Teichmüller space. We show for any pseudo-Anosov mapping class f , there exists a power n , such that the number of lattice points of the f^n conjugacy class intersecting a closed ball of radius R is coarsely asymptotic to e^{\frac{h}{2}R} .

Local Hadwiger's Conjecture

We propose local versions of Hadwiger's Conjecture, where only balls of radius Ω(log⁡(v(G))) around each vertex are required to be Kt-minor-free. We ask: if a graph is locally-Kt-minor-free, is it t-colourable? We show that the answer is yes when t≤5, even in the stronger setting of list-colouring, and we complement this result with a O(log⁡v(G))-round distributed colouring algorithm in the LOCAL model. Further, we show that for large enough values of t, we can list-colour locally-Kt-minor-free graphs with 13⋅max⁡{h(t),⌈312(t−1)⌉}) colours, where h(t) is any value such that all Kt-minor-free graphs are h(t)-list-colourable. We again complement this with a O(log⁡v(G))-round distributed algorithm.