Ball Of Radius Research Articles

No-Dimensional Tverberg Theorems and Algorithms

Tverberg’s theorem states that for any kge 2 and any set Psubset {mathbb {R}}^d of at least (d+1)(k-1)+1 points in d dimensions, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class text {CLS} = text {PPAD} cap text {PLS}, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa (SODA 2019) gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set Psubset {mathbb {R}}^d and any kin {2,dots ,n} in O(ndlceil {log k}rceil ) time a k-partition of P such that there is a ball of radius O((k/sqrt{n}){text {diam}}(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.

Robust k-center with two types of radii

In the non-uniform k-center problem, the objective is to cover points in a metric space with specified number of balls of different radii. Chakrabarty, Goyal, and Krishnaswamy [ICALP 2016, Trans. on Algs. 2020] (CGK, henceforth) give a constant factor approximation when there are two types of radii. In this paper, we give a constant factor approximation for the two radii case in the presence of outliers. To achieve this, we need to bypass the technical barrier of bad integrality gaps in the CGK approach. We do so using “the ellipsoid method inside the ellipsoid method”: use an outer layer of the ellipsoid method to reduce to stylized instances and use an inner layer of the ellipsoid method to solve these specialized instances. This idea is of independent interest and could be applicable to other problems.

Statistics of the maximum and the convex hull of a Brownian motion in confined geometries

We consider a Brownian particle with diffusion coefficient D in a d-dimensional ball of radius R with reflecting boundaries. We study the maximum M x (t) of the trajectory of the particle along the x-direction at time t. In the long time limit, the maximum converges to the radius of the ball M x (t) → R for t → ∞. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations Δ(t) = [R − M x (t)]/R in the limit of large t in all dimensions. We find that the distribution of Δ(t) exhibits a rich variety of behaviors depending on the dimension d. These results are obtained by establishing a connection between this problem and the narrow escape time problem. We apply our results in d = 2 to study the convex hull of the trajectory of the particle in a disk of radius R with reflecting boundaries. We find that the mean perimeter ⟨L(t)⟩ of the convex hull exhibits a slow convergence towards the perimeter of the circle 2πR with a stretched exponential decay . Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries. Our results are corroborated by thorough numerical simulations.

Guarantees for Existence of a Best Canonical Polyadic Approximation of a Noisy Low-Rank Tensor

The canonical polyadic decomposition (CPD) of a low-rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery of underlying factors. However, it is well known that the low-rank CPD approximation problem is ill-posed. That is, a tensor may fail to have a best rank $R$ CPD approximation when $R>1$. This article gives deterministic bounds for the existence of best low-rank tensor approximations over ${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{K}}={\mathbb{C}}$. More precisely, given a tensor ${\mathcal T} \in {\mathbb{K}}^{I \times I \times I}$ of rank $R \leq I$, we compute the radius of a Frobenius norm ball centered at ${\mathcal T}$ in which best ${\mathbb{K}}$-rank $R$ approximations are guaranteed to exist. In addition we show that every ${\mathbb{K}}$-rank $R$ tensor inside of this ball has a unique canonical polyadic decomposition. This neighborhood may be interpreted as a neighborhood of “mathematical truth" in which CPD approximation and computation are well-posed. In pursuit of these bounds, we describe low-rank tensor decomposition as a “joint generalized eigenvalue" problem. Using this framework, we show that, under mild assumptions, a low-rank tensor which has rank strictly greater than border rank is defective in the sense of algebraic and geometric multiplicities for joint generalized eigenvalues. Bounds for existence of best low-rank approximations are then obtained by establishing perturbation theoretic results for the joint generalized eigenvalue problem. In this way we establish a connection between existence of best low-rank approximations and the tensor spectral norm. In addition we solve a “tensor Procrustes problem" which examines orthogonal compressions for pairs of tensors. The main results of the article are illustrated by a variety of numerical experiments.

Effects of ball size and microstructures on indentation curves of Fe9Cr model alloys and tempered martensitic steel Eurofer97

Nano-indentation with spherical indenters is a test technique largely used to determine the elastic–plastic behavior of a very small volume of material. In this work, the indentation size effects under spherical indentation were investigated for Fe9Cr model alloys and reduced activation tempered martensitic steel Eurofer97. Experimental tests and finite element method (FEM) simulations were performed. Results and analysis of the experimental results and FEM simulations revealed that the indentation size effects depend on both the spherical indenter radius and the microstructures characterized by the initial dislocation density and the grain size of the tested materials. The observed difference in the indentation stress–strain curves was qualitatively discussed in terms of the relative effects associated with the ball radius and the microstructural length scale.

Bounds for List-Decoding and List-Recovery of Random Linear Codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input $\ell$ if for every sufficiently large subset of codewords (of $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the list size in either case. The capacity, i.e., the largest possible rate for these notions as the $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet of the code family. In this work, we study the of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

Basketball self training shooting posture recognition and trajectory estimation using computer vision and Kalman filter

Abstract Self-shooting training is one of the fundamental criteria for success in basketball. Particularly, young players increase their performance with regular training. However, the training process becomes painful and time-consuming without a coach since the incorrect shooting posture causes missing shots, leading to reluctance. In this research, a self-shooting posture algorithm is developed to track the movement of basketball players and give them feedback about their position, angle, and basketball projectile trajectory information. The proposed algorithm uses computer vision techniques and Kalman filter to detect the best projectile trajectory using initial conditions such as acceleration due to gravity the initial velocity at the angle of launch having certain horizontal distance to the rim and the rim distance from the ground The acceleration of both gravity and air drag are altered by predefined parameters, including the drag coefficient basketball mass ball radius and silhouette area The proposed algorithm provides the shooting angle in real-time by placing the projectile angle on to the cropped image of the player posture and draws the projectile trajectory towards the basketball hoop According to the results, the players having a specified height can achieve the best shooting at the angle with air drag force. On the other hand, if there is no air resistance, the best shooting angle is deviated significantly. The other stats that are a total time of travel, maximum horizontal distance, maximum height and the time until the top are also given along with the results.

Knot graphs and Gromov hyperbolicity

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any n, there exists a knot K such that the ball of radius n in the Gordian graph centered at K contains no connected sum of torus knots.

ABELIAN DIFFERENCE SETS AS LATTICE COVERINGS AND LATTICE TILINGS

AbstractWe demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $ . This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

Real-time interface-tracking framework for the evolution of the phases during the quenching of the steel balls

The locus of the micro-structural phase boundaries and their respective volumetric fraction during the quench of the high-temperature austenite critically determines the resulted physical and mechanical properties. We have developed a new numerically dynamic framework and we have established analytical relationships, which combined with the experimental measurements, could predict the real-time formation of the equivalent phase borders and their marginal evolution within the steel ball throughout the quenching process. In this regard, the transient behavior of the temperature is computed upon quenching where the threshold of phase transformation, obtained from our experimental data, is tracked as the locus of the earliest formation of phase boundaries. Our parametric analysis predicts the role of the scale (i.e. radius) of the steel ball and the initial and quench (i.e. final) temperatures and additionally anticipates the onsets of parameters, where one and two of the phases cease to exist. The model could get utilized for the design of the quench parameters to obtain the desired mechanical properties from the original austenite state.

Existence and nonexistence of sign-changing solutions for singular semilinear equations on exterior domains

In this paper we study solutions of Δu+K(|x|)f(u)=0 in the exterior of the ball of radius R>0 in RN where f grows sublinearly at infinity and is singular at 0 with f(u)∼−1|u|q−1u, 0<q<1, for small u. We assume K(|x|)∼|x|−α for large |x| and establish existence of positive and sign-changing solutions when α>2(N−1) and R>0 is small. We prove nonexistence when R>0 is large and we also prove nonexistence for 2<α<N+q(N−2) and all R>0.

Determination of the optimum ball radius for researching materials using ball indenting

The article discusses the study of the effect of a change in the radius of the ball in the injecting of the sample on the curve in the coordinates «load – indentation depth», the deviation of the indentation depth during elastoplastic indentation from the indentation depth with the elastic indentation and the amount of the axial deformation of the ball. The study was conducted using the Ansys Mechanical APDL program implementing the fenite element method. In the process of the study, it was found that with a change in the radius of the ball, there is no obvious change in the behavior of the sample material, and the deviation of the indentation depth during the elastoplastic indulgence from the indentation depth during the elastic indentation is not dependent on the size of the ball radius. There was also an effect of changing the radius of the ball on the size of the axial deformation of the ball and proposed a formula for determining the size of the axial deformation of the ball for the ball of any diameter, which will determine the actual depth of the ball into the ball when using the balls of different radius.

On the structure of random graphs with constant $r$-balls

We continue the study of the properties of graphs in which the ball of radius r around each vertex induces a graph isomorphic to the ball of radius r in some fixed vertex-transitive graph F , for various choices of F and r . This is a natural extension of the study of regular graphs. More precisely, if F is a vertex-transitive graph and r \in \mathbb{N} , we say a graph G is r -locally F if the ball of radius r around each vertex of G induces a graph isomorphic to the graph induced by the ball of radius r around any vertex of F . We consider the following random graph model: for each n \in \mathbb{N} , we let G_n = G_n(F,r) be a graph chosen uniformly at random from the set of all unlabelled, n -vertex graphs that are r -locally F . We investigate the properties possessed by the random graph G_n with high probability, i.e. with probability tending to 1 as n \to \infty , for various natural choices of F and r . We prove that if F is a Cayley graph of a torsion-free group of polynomial growth, and r is sufficiently large depending on F , then the random graph G_n = G_n(F,r) has largest component of order at most n^{5/6} with high probability, and has at least \exp(n^{\delta}) automorphisms with high probability, where \delta&gt;0 depends upon F alone. Both properties are in stark contrast to random d -regular graphs, which correspond to the case where F is the infinite d -regular tree. We also show that, under the same hypotheses, the number of unlabelled, n -vertex graphs that are r -locally F grows like a stretched exponential in n , again in contrast with d -regular graphs. In the case where F is the standard Cayley graph of \mathbb{Z}^d , we obtain a much more precise enumeration result, and more precise results on the properties of the random graph G_n(F,r) . Our proofs use a mixture of results and techniques from geometry, group theory and combinatorics. We make several conjectures regarding what happens for Cayley graphs of other finitely generated groups.

A characterization of the Mathieu–Conway–Monster series of locally projective graphs

Let I=(V,L) be a vertex-line incidence system which is said to be thin or thick if every line contains two or three vertices, respectively. In either case any two lines can intersect in at most one vertex. Let Γ be the collinearity graph of I and let G be a group of automorphisms of I. Then Γ is said to be locally projective in dimension n with respect to G if (a) G is flag-transitive on I; (b) the stabilizer G(x) of a vertex x∈V in G induces on the set of lines containing x, the group Ln(2) for some n≥2 in its natural doubly transitive action of degree 2n−1. Thus a locally projective graph in dimension n has valency 2n−1 or 2(2n−1) depending on whether it is thin or thick. The G(x)-invariant structure of a projective GF(2)-space of linear dimension n on the set of lines containing x, will be denoted by Πx. In the thin case we further assume that (c) the action of G on Γ is 2- but not 3-arc transitive and (d) the action is of collineation type, meaning that for an edge {x,y} of Γ an element in G which swaps x and y can be chosen to centralize G(x)∩G(y) modulo O2(G(x)∩G(y)). The first examples of locally projective graphs come from affine (thin) and projective (thick) GF(2)-spaces. Less trivial classical examples are the dual polar GF(2)-spaces of orthogonal (thin) and symplectic (thick) types. Further examples coming from Petersen and Tilde geometries are associated with some sporadic simple groups. The fundamental problem is to establish an upper bound on |G(x)| in terms of the dimension n, by finding an absolute bound d such that the vertex-wise stabilizer in G of the ball of radius d in Γ centred at a vertex x is always trivial (independently of the graph): Gd(x)=1. In the classical examples G2(x)=1. The first examples of Petersen and Tilde geometries were constructed in the early 1980s to disprove the belief that the classical bound holds in general. In fact, for the vertex-transitive graph of the Petersen geometry of the Baby Monster group the smallest d with Gd(x)=1 is 5. In 1990s V.I. Trofimov in a brilliant series of papers proved that G6(x)=1 in the thin case without assuming (c) and (d). In 2002 it has been proved by S.V. Shpectorov and the author that in the thin case under consideration all the examples with the critical parameter d above the classical bound indeed come from Petersen geometries (one way or another). An ongoing project is to extend the classification of locally projective amalgams to the thick case, which is considerably more complicated. Until recently only a characterization of an important class of 3-dimensional locally projective graphs [3] was available. In a recent paper [6] the author has shown that if the standard geometrization of a thick locally projective graph contains planes locally isomorphic to the generalized quadrangle of order (2,2), then Γ contains a family of thin locally projective subgraphs which are densely embedded in Γ in the sense that the stabilizer in G(x) of such a subgraph containing x has index 2n. This result places in a general setting the famous embeddingsG(M22)↪G(M24),G(Co2)↪G(Co1),G(BM)↪G(M) among Petersen and Tilde geometries. In the present paper we characterize the graphs of the geometries on the right hand side of these embeddings in the class of thick locally projective graphs with planes of the above specified type and with critical parameter d above the classical bound.

Frequency domain of weakly translation invariant frame MRAs

Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions [Formula: see text] with frequency domain [Formula: see text]. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function [Formula: see text] is convex and completely symmetric about the origin, then it must be weakly invariant and [Formula: see text]. If [Formula: see text] has a ball-shaped frequency domain, the ball radius must be bounded by [Formula: see text]. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions.

Recursive gain controller for strict feedback systems

This paper presents a new solution to the ‘explosion of complexity’ problem arising from the repeated derivations of the virtual controls in Standard Backstepping approach. It is inspired by the principle of separation and is based on the use of ‘fictitious’ signals. This solution avoids the use of filters to approximate the virtual controls first derivatives as it is done both in Command Filtered Backstepping and Dynamic Surface Control and ensures global and exponential convergence of the tracking errors to zero within balls of radii that can be made as small as desired. This proposed solution, referred to as Recursive Gain Controller, is applicable to more wider class of strict feedback systems as it requires less conservative assumptions. Simulation results are given to illustrate the effectiveness of the proposed approach. It is shown that the Recursive Gain Controller has better performance than the Standard Backstepping approach.

Calibration of wet sand and gravel particles based on JKR contact model

The parameters of wet sand and gravel particles with different shapes are calibrated based on JKR contact model. The results reveal that surface energy and rolling friction coefficient are the significant factors affecting the tangent of repose angle, indicating the great influence on the simulation accuracy of wet sand and gravel particles during vibration screening. In addition, the tangent of repose angle and error between simulation and experiment both increase with the filling ball radius increasing, and the corresponding simulation accuracy decreases. Then the screening efficiency is further investigated to verify the accuracy of the calibration parameters. The parameters calibrated can well simulate the screening process. And it is found that non-spherical model instead of spherical model can better simulate the screening process of sand and gravel particles. Meanwhile, JKR model is also verified to better simulate the screening process than Hertz-Mindlin model. Moreover, as the filling ball radius increases, the screening efficiency increases while the simulation time and file size are greatly reduced. However, the corresponding error between simulation and experiment increases. Considering the error and screening time, the optimum filling ball radii for flat, triangular cone and ellipsoid particles are 2 mm, 1.5 mm and 1 mm, respectively.

Thermal elastohydrodynamic lubrication characteristics and optimisation of the ball-type tripod universal joint

Purpose This paper aims to study the influence of three-column groove shell radius, ball radius, lubricating oil viscosity and elastic modulus on the thermal elastohydrodynamic lubrication (TEHL) characteristics and optimisation of the ball-type tripod universal joint. Design/methodology/approach The point contact TEHL model of the joint was developed, and the multi-grid method was used to solve it. The influence of three-column groove shell radius, ball radius, lubricating oil viscosity and elastic modulus on the lubrication characteristics was analysed. Further, the optimisation of the joint TEHL performance was carried out by the Kriging approximation model combined with the multi-objective particle swarm optimisation (MOPSO) algorithm. Findings The research results show that increasing groove shell radius and ball radius can effectively increase the oil film thickness, and decrease the oil film pressure, as well as the temperature rise. Decreasing elastic modulus can reduce the oil film temperature rise and pressure, and increasing viscosity can effectively increase the oil film thickness. The optimised minimum oil film thickness increases by 33.23% and the optimised maximum oil film pressure and maximum temperature rise decrease by 11.92% and 28.87%, respectively. Furthermore, the relative error of each response output is less than 10%. Originality/value This study applies TEHL theory to the tribological research of the ball-type tripod universal joint, and the joint’s lubrication performance is improved greatly by the Kriging model and MOPSO algorithm, which provides an effective measure to raise the joint’s working efficiency.

Asymptotic mean value property for eigenfunctions of the Laplace–Beltrami operator on Damek–Ricci spaces

Let S be a Damek–Ricci space equipped with the Laplace–Beltrami operator \(\Delta \). In this paper, we characterize all eigenfunctions of \(\Delta \) through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to infinity.

Ideal structure of rings of analytic functions with non-Archimedean metrics

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].