Ball Of Radius Research Articles

On the number of lattice points in a ball

We prove a fairly general inequality that estimates the number of lattice points in a ball of positive radius in general position in a Euclidean space. The bound is uniform over lattices induced by a matrix having a bounded operator norm.

Bootstrap percolation in random geometric graphs

AbstractFollowing Bradonjić and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is $a\log n$ for some fixed $a>1$ . Each vertex is added with probability p to a set $A_0$ of initially infected vertices. Vertices subsequently become infected if they have at least $ \theta a \log n $ infected neighbours. Here $p, \theta \in [0,1]$ are taken to be fixed constants.We show that if $\theta < (1+p)/2$ , then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for $ \theta > (1+p)/2$ , even if one adversarially infects every vertex inside a ball of radius $O(\sqrt{\log n} )$ , with high probability the infection will spread to only o(n) vertices beyond those that were initially infected.In addition we give some bounds on the $(a, p, \theta)$ regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an ‘almost no percolation or almost full percolation’ dichotomy which may be of independent interest.

Distribution and moments of the error term in the lattice point counting problem for three-dimensional Cygan–Korányi balls

We study fluctuations of the error term for the number of integer lattice points lying inside a three-dimensional Cygan–Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.

Mean first exit times of Ornstein–Uhlenbeck processes in high-dimensional spaces

The d-dimensional Ornstein–Uhlenbeck process (OUP) describes the trajectory of a particle in a d-dimensional, spherically symmetric, quadratic potential. The OUP is composed of a drift term weighted by a constant and a diffusion coefficient weighted by σ > 0. In the absence of drift (i.e. θ = 0), the OUP simply becomes a standard Brownian motion (BM). This paper is concerned with estimating the mean first-exit time (MFET) of the OUP from a ball of finite radius L for large . We prove that, asymptotically for , the OUP takes (on average) no longer to exit than BM. In other words, the mean-reverting drift of the OUP (scaled by ) has asymptotically no effect on its MFET. This finding might be surprising because, for small , the OUP exit time is significantly larger than BM by a margin that depends on θ. As it allows for the drift to be ignored, it might simplify the analysis of high-dimensional exit-time problems in numerous areas.Finally, our short proof for the non-asymptotic MFET of OUP, using the Andronov–Vitt–Pontryagin formula, might be of independent interest.

Large nearest neighbour balls in hyperbolic stochastic geometry

Consider a stationary Poisson process in a d-dimensional hyperbolic space. For R>0 define the point process xi _R^{(k)} of exceedance heights over a suitable threshold of the hyperbolic volumes of kth nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius R centred at a fixed point. The point process xi _R^{(k)} is compared to an inhomogeneous Poisson process on the real line with intensity function e^{-u} and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum kth nearest neighbour ball with a limiting Gumbel distribution is derived.

On a Steklov-Robin eigenvalue problem

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider Ω=Ω0∖B‾r, where Br is the ball centered at the origin with radius r>0 and Ω0⊂Rn, n⩾2, is an open, bounded set with Lipschitz boundary, such that B‾r⊂Ω0. We impose a Steklov condition on the outer boundary and a Robin condition involving a positive L∞ function β(x) on the inner boundary. Then, we study the first eigenvalue σβ(Ω) and its main properties. In particular, we investigate the behavior of σβ(Ω) when we let vary the L1-norm of β and the radius of the inner ball. Furthermore, we study the asymptotic behavior of the corresponding eigenfunctions when β is a positive parameter that goes to infinity.

Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

The Weyl law of the Laplacian on the flat torus $${\mathbb {T}}^n$$ is concerning the number of eigenvalues $$\le \lambda ^2$$ , which is equivalent to counting the lattice points inside the ball of radius $$\lambda $$ in $${\mathbb {R}}^n$$ . The leading term in the Weyl law is $$c_n\lambda ^n$$ , while the sharp error term $$O(\lambda ^{n-2})$$ is only known in dimension $$n\ge 5$$ . Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators $$H_V=-\Delta _{g}+V$$ in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531 )) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284 ) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau’s heat kernel estimates, Blair–Sire–Sogge’s eigenfunction estimates, and Duhamel’s principle for the wave equation.

Do additional target points speed up evolutionary algorithms?

Most runtime analyses of randomised search heuristics focus on the expected number of function evaluations to find a unique global optimum. We ask a fundamental question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set OPT∪S where S is a set of non-optimal search points and OPT is the set of optima and compare it to the expected hitting time of OPT.We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions with polynomial expected optimisation times we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering Hamming balls around all global optima gives an easier target for some algorithms and functions and can shift the phase transition with respect to offspring population sizes in the (1,λ) EA on OneMax. However, for the one-dimensional Ising model the time to reach Hamming balls of radius (1/2−ε)n around optima does not reduce the asymptotic expected optimisation time in the worst case. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time.

Random Euclidean coverage from within

Let X_1,X_2, ldots be independent random uniform points in a bounded domain A subset mathbb {R}^d with smooth boundary. Define the coverage thresholdR_n to be the smallest r such that A is covered by the balls of radius r centred on X_1,ldots ,X_n. We obtain the limiting distribution of R_n and also a strong law of large numbers for R_n in the large-n limit. For example, if A has volume 1 and perimeter |partial A|, if d=3 then mathbb {P}[npi R_n^3 - log n - 2 log (log n) le x] converges to exp (-2^{-4}pi ^{5/3} |partial A| e^{-2 x/3}) and (n pi R_n^3)/(log n) rightarrow 1 almost surely, and if d=2 then mathbb {P}[n pi R_n^2 - log n - log (log n) le x] converges to exp (- e^{-x}- |partial A|pi ^{-1/2} e^{-x/2}). We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

Revisit of multi-dimensional Bohr radius

In this article, we obtain a new lower bound of the Bohr radius of the unit ball in ℓpn space, for 1≤p<∞ and n∈N, n≥2. Moreover, we observe that this new lower bound is sharper than previously all known lower bounds.

A minimization problem related to the principal frequency of the p -Bilaplacian with coupled Dirichlet–Neumann boundary conditions

For each fixed integer N ≥ 2 let Ω ⊂ R N be an open, bounded and convex set with smooth boundary. For each real number p ∈ ( 1 , ∞ ) define M ( p ; Ω ) = inf u ∈ W C 2 , ∞ ( Ω ) ∖ { 0 } ∫ Ω ( exp ⁡ ( | Δ u | p ) − 1 ) d x ∫ Ω ( exp ⁡ ( | u | p ) − 1 ) d x , where W C 2 , ∞ ( Ω ) := ∩ 1 &lt; p &lt; ∞ { u ∈ W 0 2 , p ( Ω ) : Δ u ∈ L ∞ ( Ω ) } . We show that if the radius of the largest ball which can be inscribed in Ω is strictly larger than a constant which depends on N then M ( p ; Ω ) vanishes while if the radius of the largest ball which can be inscribed in Ω is strictly less than 1 then M ( p ; Ω ) is a positive real number. Moreover, in the latter case when p is large enough we can identify the value of M ( p ; Ω ) as being the principal frequency of the p -Bilaplacian on Ω with coupled Dirichlet–Neumann boundary conditions.

Complete study of local convergence and basin of attraction of sixth-order iterative method

&lt;abstract&gt;&lt;p&gt;The local convergence analysis of the parameter based sixth-order iterative method is the primary focus of this article. This investigation was conducted based on the Fréchet derivative of the first order that satisfies the Lipschitz continuity condition. In addition, we developed a conceptual radius of convergence for these methods. Also, we discussed the solution behavior of complex polynomials with the basin of attraction. Finally, some numerical examples are provided to illustrate how the conclusions we got can be employed to determine the iterative approach's radius of convergence ball in the context of solving nonlinear equations. We compare the numerical results with our method and the existing sixth order methods proposed by Argyros et al. We observe that using our method yields significantly larger balls than those that already exist.&lt;/p&gt;&lt;/abstract&gt;

Application and Research of the White Light Interferometer

Surface topography measurement is the micro topography measurement of samples to be tested, which plays an important role in measuring white light interferometer. Taking the B3-shaped penetration test block for nondestructive testing and the profilometer standard ball for length measurement as the main research objects. The depth and width of crack defects in the penetration test block are measured and analyzed, the radius of the spherical groove on the reverse side of the penetration test block is measured and analyzed, and the spherical radius of the profilometer standard ball is measured and analyzed. The measurement results show that there is little difference between the radius measurement of the penetration test block and the theoretical value. The measurement of crack width and depth will contribute to the qualitative analysis of the crack, and the measurement of the profilometer standard ball provides a non-contact measurement means.

Mechanistic model for quantifying the effect of impact force on mechanochemical reactivity.

Conventional mechanochemical synthetic tools, such as ball mills, offer no methodology to quantitatively link macroscale reaction parameters, such as shaking frequency or milling ball radius, to fundamental drivers of reactivity, namely the force vectors applied to the reactive molecules. As a result, although mechanochemistry has proven to be a valuable method to make a wide variety of products, the results are seldom reproduceable between reactors, difficult to rationally optimize, and hard to ascribe to a specific reaction pathway. Here we have developed a controlled force reactor, which is a mechanochemical ball mill reactor with integrated force measurement and control during each impact. We relate two macroscale reactor parameters-impact force and impact time-to thermodynamic and kinetic transition state theories of mechanochemistry utilizing continuum contact mechanics principles. We demonstrate force controlled particle fracture of NaCl to characterize particle size evolution during reactions, and force controlled reaction between anhydrous copper(II) chloride and (1, 10) phenanthroline. During the fracture of NaCl, we monitor the evolution of particle size as a function of impact force and find that particles quickly reach a particle size of ∼100 μm largely independent of impact force, and reach steady state 10-100× faster than reaction kinetics of typical mechanochemical reactions. We monitor the copper(II) chloride reactivity by measuring color change during reaction. Applying our transition state theory developed here to the reaction curves of copper(II) chloride and (1, 10) phenanthroline at multiple impact forces results in an activation energy barrier of 0.61 ± 0.07 eV, distinctly higher than barriers for hydrated metal salts and organic ligands and distinctly lower than the direct cleavage of the CuCl bond, indicating that the reaction may be mediated by the higher affinity of Fe in the stainless steel vessel to Cl. We further show that the results in the controlled force reactor match rudimentary estimations of impact force within a commercial ball mill reactor Retsch MM400. These results demonstrate the ability to quantitatively link macroscale reactor parameters to reaction properties, motivating further work to make mechanochemical synthesis quantitative, predictable, and fundamentally insightful.

Linear bounds for constants in Gromov’s systolic inequality and related results

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Gromov that this result implies two by now famous Gromov's inequalities: $Fill Rad(M^n)\leq c(n)vol(M^n)^{1\over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)\leq 6c(n)vol(M^n)^{1\over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible closed curve in $M^n$. We prove that these results hold with $c(n)=({n!\over 2})^{1\over n}\leq {n\over 2}$. We demonstrate that for essential Riemannian manifolds $sys_1(M^n) \leq n\ vol^{1\over n}(M^n)$. All previously known upper bounds for $c(n)$ were exponential in $n$. Moreover, we present a qualitative improvement: In Guth's theorem the assumption that the volume of every metric ball of radius $r$ is less than $({r\over c(n)})^n$ can be replaced by a weaker assumption that for every point $x\in M^n$ there exists a positive $\rho(x)\leq r$ such that the volume of the metric ball of radius $\rho(x)$ centered at $x$ is less than $({\rho(x)\over c(n)})^n$ (for $c(n)=({n!\over 2})^{1\over n}$). Also, if $X$ is a boundedly compact metric space such that for some $r>0$ and an integer $n\geq 1$ the $n$-dimensional Hausdorff content of each metric ball of radius $r$ in $X$ is less than $({r\over 4n})^n$, then there exists a continuous map from $X$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$.

Golf swing in response to anteroposterior ball position

Studies for effective golf coaching such as providing a causal relationship between the address posture and golf-swing behaviors are limited. This study investigated the effects of ball position on golf-swing behaviors. Twenty professional golfers performed golf swing at five different anteroposterior (forward/backward) ball positions: reference ball position (R0) and ±2.14 cm (golf ball radius) and ±4.17 cm (golf ball diameter) to the R0. Their swings were captured using a motion capture system with two force platforms. Statistical parametric mapping was used to compare the ground reaction forces and swing kinematics for different ball positions. The backward (closer to the golfer) ball positions were associated with more extended trunk, knee, and ankle angles in the sagittal plane during the entire golf swing (from address to impact) compared to that of the R0. The backward ball positions were generally associated with decreased horizontal ground reaction forces and pelvis and X-factor angles in the transverse plane during the downswing compared to that of the R0. Whereas, these patterns were reversed for the forward ball positions. These results demonstrate the value of ball position for coaching methods including the importance of a consistent ball position and reveal a consistent human control system for professionals which can provide valuable information for modeling the features of a digitized coaching tool.

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Abstract We consider a finite, connected and simple graph Γ that admits a vertex-transitive group of automorphisms 𝐺. Under the assumption that, for all x ∈ V ⁢ ( Γ ) x\in V(\Gamma) , the local action G x Γ ⁢ ( x ) G_{x}^{\Gamma(x)} is the action of Sym ⁡ ( n ) \operatorname{Sym}(n) on ordered pairs, we show that the group G x [ 3 ] G_{x}^{[3]} , the point-wise stabiliser of a ball of radius three around 𝑥, is trivial.

Numerical Simulation of Bridging Ball Plugging Mechanism in Fractured-Vuggy Carbonate Reservoirs

Pores, fractures, caves, and other storage spaces are commonly distributed in fractured-vuggy carbonate reservoirs. During the drilling process, more than half of all drill-in fluid loss issues are caused by developed caves. Cave scales range from centimeters to meters, making leak prevention increasingly difficult through the use of traditional technologies. Currently, there is still high demand for the understanding of feasible loss control techniques, especially in fractured-vuggy carbonate reservoirs. Multistage Bridge Plugging (MBP) technology has facilitated pioneering experiments in many oilfields, but the success rate of plugging is less than 50%, and the effects of plugging are uncontrollable and difficult to predict. This is due to a lack of clarity regarding the plugging mechanism and the key controlling factors. In this study, we used the Discrete Element Method (DEM) simulation method to investigate the controlling factors of MBP technology, and analyzed its applicable conditions. We found that the prerequisite for the success of MBP is the presence of a constricted throat near the wellbore when drilling the well hole; the first-stage bridging ball is the key to the success of MBP. Larger ball radius, cave inclination and initial flow rate, and lower ball velocity are beneficial to the first-stage bridging. All discussion in this research is based on the ideal situation. However, the cave pattern is difficult to describe using several models, let alone by one ideal model. With the progress of seismic fine description technology and mud logging, more accurate characterization of caves in carbonate reservoirs will help to accurately formulate the plugging scheme and greatly improve the success rate of plugging technology. Additionally, the engineering risks of this technology, such as plugging the coiled tubing, need to be further studied.

Wave scattering by many small impedance particles and applications

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is a << d << λ , where d is the minimal distance between neighboring particles, λ is the wavelength, and the particles can be impedance balls B ( x m , a ) with centers x m located on a grid. Equations for the limiting effective (self-consistent) field in the medium are derived. It is proved that one can create material with a desired refraction coefficient by embedding in a free space many small balls of radius a with prescribed boundary impedances. The small balls can be centered at the points located on a grid. A recipe for creating materials with a desired refraction coefficient is formulated. It is proved that materials with a desired radiation pattern, for example, wave-focusing materials, can be created.

Computer Vision of Optimal Geometric Concentration Ratio for the Solar Ball Lens

This research evaluates the optical properties of an inhomogeneous and non-paraxial system using a solar ball lens (SBL) as a new thermal solar concentrated collector. This evaluation is based on detecting a diacaustic curve in a straightforward and accurate manner, with the diagnostic relying on image processing as a computational tool using the MATLAB program rather than a complicated numerical analytic procedure. The circle of least confusion (CLC) of the (SBL), (Fluorinated ethylene propylene (FEP) polymer – water core), was calculated. Furthermore, the study evaluated the maximum geometrical concentration ratio (G C) of refracted solar radiation that can be captured by a receiver of the (SBL). Without energy losses due to spherical aberration, the (G C) ratio was (14.10) at (h/r) ratio, circular aperture radius to ball radius, around (0.6). The investigation also revealed an aplanatic point for preventing the development of optical envelope inside the solar ball lens at (h/r) equal to 0.8638, as well as a maximum (h/r) value of 0.7351 for collecting solar radiation.