Ball Of Radius Research Articles

On existence of perfect bitrades in Hamming graphs

A pair (T0,T1) of disjoint sets of vertices of a graph G is called a perfect bitrade in G if any ball of radius 1 in G contains exactly one vertex in T0 and T1 or none simultaneously. The volume of a perfect bitrade (T0,T1) is the size of T0. If C0 and C1 are distinct perfect codes with minimum distance 3 in G then (C0∖C1,C1∖C0) is a perfect bitrade. For any q≥3, r≥1 we construct perfect bitrades of volume (q!)r in the Hamming graph H(qr+1,q) and show that for r=1 their volume is minimum.

A gap theorem for complete submanifolds with parallel mean curvature in the hyperbolic space

Let M be an n(≥7)-dimensional complete submanifold with parallel mean curvature in the hyperbolic space Hn+p, whose mean curvature satisfies H2−1≤0. Denote by A˚ and BR(q) the trace free second fundamental form of M and the geodesic ball of radius R centered at q∈M, respectively. We prove that if lim supR→∞∫BR(q)|A˚|2dMR2=0, and if (∫M|A˚|ndM)2/n+2n(n−2)3n(n−1)H(∫M|A˚|n/2dM)2/n≤C(n), then M is congruent to an n-dimensional hyperbolic space or the Euclidean space Rn. Here C(n) is an explicit positive constant depending only on n. We also obtain a similar gap theorem in the case where n=5,6.

Constitutive Model of Thin Metal Sheet in Bulging Test Considering Strain Gradient Hardening

The study of the size effect was one of the most important subjects in the field of micro-forming. To investigate the stress of the thin sheet in the bulging test with the second order size effect, a constitutive equation considering the strain gradient hardening was proposed. Based on the equation, the stress of the thin sheet during the bulging test was calculated by the finite element method. The bulging tests with various thicknesses of brass sheets and radiuses of punching balls were performed to verify the proposed equation. The results showed that the constitutive equation could capture the stress variations, while the simulation using the constitutive equation from the conventional theory of plasticity showed the results with large deviation from those of the experiment. It was found that the stress was sensitive to the thickness of the sheet and the radius of the punching ball in bulging test of thin brass sheet. The bulging of the thin brass sheet with a thickness below ten times of its material intrinsic length would cause the generation of the geometrically necessary dislocations, which induced the strain gradient hardening. Besides that, the decrement of the punching ball radius would also increase the inhomogeneous deformation and enhance the strain gradient hardening during the thin sheet bulging process. The strain gradient hardening during the thin sheet bulging test was related to the strain of the sheet. The hardening effect of the strain gradient was obvious when the strain was small. The strain gradient hardening should be considered in the thin sheet bulging test with the second order size effect.

Local Universality for Zeros and Critical Points of Monochromatic Random Waves

This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves $$\phi _\lambda $$ of frequency $$\lambda $$ on a compact, smooth, Riemannian manifold (M, g) as $$\lambda \rightarrow \infty .$$ We prove global variance estimates for the measures of integration over the zeros and critical points of $$\phi _\lambda .$$ These global estimates hold for a wide class of manifolds—for example when (M, g) has no conjugate points—and rely on new local variance estimates on zeros and critical points of $$\phi _\lambda $$ in balls of radius $$\approx \lambda ^{-1}$$ around a fixed point. Our local results hold under conditions about the structure of geodesics that are generic in the space of all metrics on M.

Dynamics of ball-like flames in extremely low-speed counterflow field in near-lean limit low-Lewis number mixture

For the understanding of ball-like flame behavior in counterflow field, transient three-dimensional computations with thermal-diffusion model were conducted for a low-Lewis number mixture near lean limit. Three types of flame behaviors were confirmed: stable spherical ball-like flame (spherical BLF) in A ≤ 0.010, stable non-spherical ball-like flame (non-spherical BLF) in 0.01 < A < 0.089 and splitting ball-like flame (splitting BLF) in A ≥ 0.089, where A is ordinary stretch rate normalized with laminar burning velocity SL and thermal diffusivity α. Analysis of flame structure for non-spherical BLF located its center at the stagnation point showed that the maximum temperature on the stagnation plane was higher than that on the counterflow axis because of the small difference between the flame curvatures on the stagnation plane and that on the counterflow axis. With the increase of stretch rate, the maximum temperature of the non-spherical BLF on the stagnation plane increased and the position of maximum temperature got away from the stagnation point. The maximum temperature on the counterflow axis decreased and the position of maximum temperature got closer to the stagnation point. Existence of unburned fuel was also confirmed near the stagnation point at A = 0.085. Thus, net fuel velocity was newly introduced to evaluate the effect of the unburned fuel diffusion. The profile of the net fuel velocity revealed two peaks in the case of A < 0.050 and four peaks in the case of A > 0.050. In the case A > 0.050, the inner two peaks were found to be due to the diffusion of unburned fuel to the outward direction. The analyses on the peak positions showed that the flame splitting occurs when the positions of the inner two peaks of the net fuel velocity are located outside of the reference flame ball radius.

An upper bound for the first nonzero Neumann eigenvalue

Let M denote a complete, simply connected n-dimensional Riemannian manifold with sectional curvature KM≤k and Ricci curvature RicM≥(n−1)K, where k,K∈R. Then for a bounded domain Ω⊂M with smooth boundary, we prove that the first nonzero Neumann eigenvalue μ1(Ω)≤Cμ1(Bk(R)). Here Bk(R) is a geodesic ball of radius R>0 in the simply connected space form Mk such that vol(Ω) = vol(Bk(R)), and C is a constant which depends on the volume, diameter of Ω and the dimension of M.

Occupation measure of random walks and wired spanning forests in balls of Cayley graphs

We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius r is O(r 5/2 ). We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside any given ball of radius r is O(r 11/2 ).

On isomorphic embeddings of c into L1-preduals and some applications

First we study the distortion ‖T‖‖T−1‖ of an isomorphic embedding T of the space c of convergent sequences into an infinite-dimensional L1-predual X such that (extBX⁎)′⊂rBX⁎ for some r∈(0,1), that is, the set of all weak⁎ cluster points of the set of all extreme points of the closed unit ball of the dual space X⁎ is contained in the closed ball of radius r∈(0,1). We prove that ‖T‖‖T−1‖≥(3−r)/(1+r). Moreover, we give some examples showing that for every r∈(0,1) our bound is optimal. Then, we apply our theorem to provide stability results for polyhedrality and extendability of compact operators in the setting of L1-preduals. To achieve the latter goal, we additionally give a new characterization of infinite-dimensional Banach spaces having the compact norm-preserving extension property for compact operators. Furthermore, in the framework of ℓ1-preduals such that the standard basis in ℓ1 is weak⁎ convergent, we provide precise values of stability constants for the weak⁎ fixed point property.

On Cayley graphs of {\bb Z}^4.

The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components -1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

Embedding the Heisenberg group into a bounded-dimensional Euclidean space with optimal distortion

Let H := \Big(\begin{smallmatrix} 1 &amp; \mathbb{R} &amp; \mathbb{R} \\ 0 &amp; 1 &amp; \mathbb{R} \\ 0 &amp; 0 &amp; 1 \end{smallmatrix}\Big) denote the Heisenberg group with the usual Carnot–Carathéodory metric d . It is known (since the work of Pansu and Semmes) that the metric space (H,d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 &lt; \varepsilon \leq 1/2 , the snowflaked metric space (H,d^{1-\varepsilon}) embeds into an infinite-dimensional Hilbert space with distortion O( \varepsilon^{-1/2} ) . This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H . Assouad's argument allows \ell^2 to be replaced by \mathbb{R}^{D(\varepsilon)} for some dimension D(\varepsilon) dependent on \varepsilon . Naor and Neiman showed that D could be taken independent of \varepsilon , at the cost of worsening the bound on the distortion to O( \varepsilon^{-1-c_D} ) , where c_D \to 0 as D \to \infty . In this paper we show that one can in fact retain the optimal distortion bound O( \varepsilon^{-1/2} ) and still embed into a bounded-dimensional space \mathbb{R}^D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius R \geq 2 in \Gamma := \Big(\begin{smallmatrix} 1 &amp; \mathbb{Z} &amp; \mathbb{Z} \\ 0 &amp; 1 &amp; \mathbb{Z} \\ 0 &amp; 0 &amp; 1 \end{smallmatrix}\Big) can be embedded into a bounded-dimensional space \mathbb{R}^D with the optimal distortion bound of O(\log^{1/2} R) . The construction is iterative, and is inspired by the Nash–Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

The Non-Uniform k -Center Problem

In this article, we introduce and study the Non-Uniform k -Center (NUkC) problem. Given a finite metric space ( X , d ) and a collection of balls of radii { r 1 ≥ … ≥ r k }, the NUkC problem is to find a placement of their centers in the metric space and find the minimum dilation α, such that the union of balls of radius α ⋅ r i around the i th center covers all the points in X . This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds. The NUkC problem generalizes the classic k -center problem, wherein all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k -center with outliers (kCwO for short) problem, in which there are k balls of radius 1 and ℓ (number of outliers) balls of radius 0. Before this work, there was a 2-approximation and 3-approximation algorithm known for these problems, respectively; the former is best possible unless P=NP. We first observe that no O (1)-approximation to the optimal dilation is possible unless P=NP, implying that the NUkC problem is harder than the above two problems. Our main algorithmic result is an ( O (1), O (1))- bi-criteria approximation result: We give an O (1)-approximation to the optimal dilation; however, we may open Θ(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criterion), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor approximation for this problem. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees that have been studied recently in the TCS community. We show NUkC is at least as hard as the firefighter problem. While we do nt know whether the converse is true, we are able to adapt ideas from recent works [1, 3] in non-trivial ways to obtain our constant factor bi-criteria approximation.

Study the Effect of Oil Viscosity on Friction Coefficient at Point Contact Elasto-Hydrodynamic Lubrication Based on Experimental Analysis

This study presents an experimental work for elasto-hydrodynamic lubrication (EHL) in point contact case. In this study, experimental studies have been conducted to show how friction varies over a wide range of running conditions. The effect of the change in parameters such as lubricant’s viscosity, base oil type, and load on power loss is investigated. The oils have used (5W-20, 10W-30, 10W-40 and 20W-50) which have a different grade of viscosity. The test is conducted using a ball on disc test rig. Different loads have been used (30, 40, 50, and 60 N). The boundary conditions are: the oil’s temperature is (25 oC temperature room); the radius of ball is (0.018 m). The coefficient of friction of oils for each load has been calculated. At the same load and same initial conditions the results shows, the increase in friction coefficient with the increase of viscosity is almost linear with an average slope of 1.4 (Pa.s)-1. Also, the increase in load results in linear increase in friction coefficient by an average slope of 0.63 N-1

Domains Without Dense Steklov Nodal Sets

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem -Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\Delta \\phi _{\\sigma _j}=0,\\quad \\hbox { on }\\,\\,\\Omega ,\\quad \\partial _\\nu \\phi _{\\sigma _j}=\\sigma _j \\phi _{\\sigma _j}\\quad \\hbox { on }\\,\\,\\partial \\Omega \\end{aligned}$$\\end{document}in two-dimensional domains Omega . In particular, this paper presents a dense family mathcal {A} of simply-connected two-dimensional domains with analytic boundaries such that, for each Omega in mathcal {A}, the nodal set of the eigenfunction phi _{sigma _j} “is not dense at scale sigma _j^{-1}”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains Omega in mathcal {A}, the nodal sets of the eigenfunctions phi _{sigma _j} associated with the eigenvalue sigma _j have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each Omega in mathcal {A} there is a value r_1>0 such that for each j there is x_jin Omega such that phi _{sigma _j} does not vanish on the ball of radius r_1 around x_j.

New inequalities for Holmes–Thompson and Busemann measures

Some sharp bounds for the inner radius and the outer radius of the unit ball of a (normed or) Minkowski space with respect to its isoperimetrix are known. To find more such bounds is a challenging problem. Related to this motivation, we derive new sharp inequalities between inner and outer radii for the Holmes–Thompson and Busemann measures. Cross-section measures as well as the Blaschke–Santalo inequality will be used to obtain these new inequalities.

Extending the applicability of a third-order scheme with Lipschitz and H\xf6lder continuous derivative in Banach spaces

We extend the applicability of a cubically convergent nonlinear system solver using Lipschitz continuous first-order Fréchet derivative in Banach spaces. This analysis avoids the usual application of Taylor expansion in convergence analysis and extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, our study provides the radius of convergence ball and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is provided. Various numerical tests confirm that our analysis produces better results and it is useful in solving such problems where previous methods can not be implemented.

Local Convergence and Complex Dynamics of a Uni-parametric Family of Iterative Schemes

Using the Lipschitz continuous derivative, we study the local convergence analysis of a third and fourth-order convergent uni-parametric class of iterative schemes. This technique avoids the usual practice of Taylor expansion in convergence analysis and extends the applicability of the family by using the assumption based on the first-order derivative only. Our study provides the radii of balls of convergence and computable error bounds along with the uniqueness of the solution. Also, complex dynamical properties of the family are discussed to select good schemes in the view of numerical stability. Numerical illustrations show that our analysis is useful in solving such problems where previous studies fail to solve.

Evaluation of the measurement precision and accuracy in the dental CAD/CAM system.

The purpose of this study is to investigate the precision and trueness of the occlusal contact record obtained with a dental CAD/CAM device (ARCTICA). Sandblasted spherical steel balls with diameters of 10-20 mm were measured using ARCTICA and a three-dimensional measurement device (FN503). The radius of each steel ball was calculated from the measured value and the difference between the measured and nominal values was obtained. Upper/lower dental arch casts were measured and processed for occlusal contact image creation using both devices. The graphical images of the occlusal contacts at the intercuspal position obtained from both systems and an occlusal analysis device (BE-1) were compared with the occlusal contact area. Excellent correlation was observed between the measurement results of ARCTICA and FN503 (p<0.001, R2: 0.99). The occlusal contact areas were 186.0 mm2 (ARCTICA), 192.8 mm2 (FN503), and 196.1 mm2 (BE-1). This study showed that ARCTICA is capable of conducting highly accurate measurements and generating a graphical image of occlusal contacts with good reliability.

A localization method in Hamiltonian graph theory

The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius r centered at a vertex u is a subgraph of G induced by the set of vertices whose distances from u do not exceed r.) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs.In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of Bauer et al. (1989), Bondy (1980), Häggkvist and Nicoghossian (1981), and Moon and Moser (1963). Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by Kündgen, Li and Thomassen (2017).

Mass functions of a compact manifold

Let M be a compact manifold of dimension n. In this paper, we introduce the Mass Functiona≥0↦X+M(a) (resp. a≥0↦X−M(a)) which is defined as the supremum (resp. infimum) of the masses of all metrics on M whose Yamabe constant is larger than a and which are flat on a ball of radius 1 and centered at a point p∈M. Here, the mass of a metric flat around p is the constant term in the expansion of the Green function of the conformal Laplacian at p. We show that these functions are well defined and have many properties which allow to obtain applications to the Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all metrics on M).

Screening Effect in Isotropic Gaussian Processes

The screening effect is the phenomenon that optimal linear prediction of a spatial process on an unobserved location mostly depends on nearby observations. That is, the optimal predictor based on just nearby observations yields a good approximation of which based on the whole large dataset. However, the approximation does not always perform well since the screening effect may not hold in all situations. To determine when the screening effect holds is an important issue in spatial statistics. This paper provides some sufficient conditions for ensuring an asymptotic screening effect in ℝd based on the spectral density of the underlying isotropic Gaussian process and the geometries of nearby observations. These results apply to isotropic processes with an arbitrary degree of differentiability. Assuming we are predicting at origin, the conditions are (1) the spectral density is nearly a constant in balls of finite radius far from the origin, (2) the positions of nearby observations do not fall on a curve with non-zero intercept whose degree is less or equal to the order of mean square differentiability of the process. These conditions are easy to verify in practice. Convergence rates of the asymptotic screening effect are also obtained. These rates depend on the rate of decrease of the spectral density. Simulation studies on the screening effect for finite samples are also provided.