Ball Of Radius Research Articles

Cayley graphs with few automorphisms

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

The Exploration of the Possibility of Using Continuous-Cast 55 Grade Steel in the Manufacturing of 40-mm-in-Diameter Grinding Balls of IV and V Hardness Class

Results of a field–experimental study on the applicability of continuous-cast medium-carbon 55 grade steel sparingly alloyed with chromium to the manufacturing of hardness IV and V class grinding balls 40 mm in diameter are presented. The possibility of the stable production of hardness IV class grinding balls is shown since chromium added in small quantities facilitates the hardness stabilization over the half of the ball radius and increases the volumetric hardness by 2–3 HRC figures. The established increase in the volumetric hardness did not allow for the HRC 57 level to be reached to certify the balls as the hardness V class products. Rational process variables have been recommended to ensure the production of 40-mm-in-diameter grinding balls for the V hardness class of continuous-cast medium-carbon 55 grade steel on the ShPS 30–80 ball-rolling mill.

Aerobatics of Flying Saucers

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘$$G_2$$ mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated $$G_2$$ manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group $$G_2$$ as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $$G_2$$ geometries is used to construct a ‘$$G_2$$ joystick,’ consisting of two balls of radii in ratio $$1\!:\!3$$ that transforms the difficult $$G_2$$ manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The hbox {PPT}^2 conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert–Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.

Explicit isometric embeddings of collapsing dust ball

The work is devoted to the search for explicit isometric embeddings of a metric corresponding to the collapse of spherically symmetric matter with the formation of a black hole. Two approaches are considered: in the first, the embedding is constructed for the whole manifold at once; in the second, the idea of a junction of solutions, obtained separately for areas inside and outside the dust ball, is used. In the framework of the first approach, a global smooth embedding in 7D space with a signature (2 + 5) was constructed. It corresponds to the formation of the horizon as a result of matter falling from infinity. The second approach generally leads to an embedding in 7D space with the signature (1 + 6). This embedding corresponds to the case when matter flies out of a white hole with the disappearance of its horizon, after which the radius of the dust ball reaches its maximum, and then a collapse occurs with the formation of the horizon of a black hole. The embedding obtained is not smooth everywhere—it contains a kink on the edge of the dust ball, and also, it is not quite global. In the particular case, when the maximum radius of the dust ball coincides with the radius of the horizon, it is possible to construct a global smooth embedding in a flat 6D space with a signature (1 + 5).

Isoentangled Mutually Unbiased Bases, Symmetric Quantum Measurements, and Mixed-State Designs.

Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.

Local convergence of parameter based method with six and eighth order of convergence

This paper dealt with the local convergence study of the parameter based sixth and eighth order iterative method. This analysis discuss under assumption that the first order Frechet derivative satisfied the Lipschitz continuity condition. In this way, we also proposed the theoretical radius of convergence of these methods. Finally, some numerical examples demonstrate that our results apply to compute the radius of convergence ball of iterative method to solve nonlinear equations. We compare the results with the method in Kumar et al. (J Comput Appl Math 330:676–694, 2018) and observe that by our approach we get much larger balls as existing ones.

Microwave synthesis of histidine-functionalized graphene quantum dots/Ni-Co LDH with flower ball structure for supercapacitor

A flower ball-like histidine-functionalized graphene quantum dots/Ni-Co LDH (His-GQD/LDH) composite is synthesized via microwave method. The GQDs are uniformly interspersed on LDH surface and the radius of flower balls is approximately 200 nm. The synergistic effect of His-GQD and LDH can signally increase the specific surface areas and conductivity of the composite, thus endowing the composite high specific capacitance (1526 F g−1) and an admirable cycling stability (82.36% capacitance retention). Moreover, we have manufactured a supercapacitor employing His-GQD/LDH and active carbon (AC) as positive/negative electrodes. The device exhibits the maximum energy of 48.89 W h kg−1 at 0.80 kW kg−1, as well as prominent cycling stability (91.13% capacitance retention). This work provides a practicle experimental method of synthesizing hybridizing histidine-functionalized carbon derivatives with LDH by microwave synthesis. Meanwhile, we are optimistic to believe that the electrode material can be extensively applied for supercapacitor because of its splendid electrochemical properties and facile synthesis.

Performance Guarantees of Distributed Algorithms for QoS in Wireless Ad Hoc Networks

Consider a wireless network where each communication link has a minimum bandwidth quality-of-service requirement. Certain pairs of wireless links interfere with each other due to being in the same vicinity, and this interference is modeled by a conflict graph. Given the conflict graph and link bandwidth requirements, the objective is to determine, using only localized information, whether the demands of all the links can be satisfied. At one extreme, each node knows the demands of only its neighbors; at the other extreme, there exists an optimal, centralized scheduler that has global information. The present work interpolates between these two extremes by quantifying the tradeoff between the degree of decentralization and the performance of the distributed algorithm. This open problem is resolved for the primary interference model, and the following general result is obtained: if each node knows the demands of all links in a ball of radius $d$ centered at the node, then there is a distributed algorithm whose performance is away from that of an optimal, centralized algorithm by a factor of at most $(2d+3)/(2d+2)$. The tradeoff between performance and complexity of the distributed algorithm is also analyzed. It is shown that for line networks under the protocol interference model, the row constraints are a factor of at most $3$ away from optimal. Both bounds are best possible.

A novel energy-absorbing rock bolt with high constant working resistance and long elongation: Principle and static pull-out test

With mining at deep levels worldwide, surrounding rocks of roadways are prone to suffer from high stress fields, large deformation, and burst-potential. Therefore, it is critical to develop high performance energy-absorbing rock bolts to carry high load, accommodate large rock deformation, and energy absorbing. In this paper, a novel energy-absorbing rock bolt with high constant working resistance and long elongation is theoretically and experimentally investigated. The new bolt is composed of three main parts: a rebar, a sleeve tube with a partial slope in the inner surface, and a circle of steel balls. The working resistance is generated by plastic flow of the sleeve tube by contact force between the steel balls and the sleeve tube. A mechanical model is established based on Contact Mechanics to better understand the factors affecting the working resistance and to guide the static pull-out tests. Two batches of the new rock bolt with constant working resistances ranging from 60 kN to 145 kN were developed by changing the number and radius of the steel balls in static pull-out experiments. The load-displacement curves of the new rock bolt indicate the performance of the new rock bolt is characterized by high stiffness, high constant working resistance and adjustable supporting length. The constant working resistance corresponds positively to the gouging depth and the radial expansion of the sleeve tube, based on theoretical analysis verified by test results. A numerical simulation of deep soft roadway is applied to investigate the performance of new energy absorbing rock bolt.

Parking 3-sphere swimmer: II. The long-arm asymptotic regime

.The paper carries on our previous investigations on the complementary version of Purcell’s rotator (sPr3): a low-Reynolds-number swimmer composed of three balls of equal radii. In the asymptotic regime of very long arms, the Stokes-induced governing dynamics is derived, and then experimented in the context of energy-minimizing self-propulsion characterized in the first part of the paper.Graphical abstract

Some local–global phenomena in locally finite graphs

In this paper we present some results for a connected infinite graph G with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of G. (For a vertex w of a graph G the ball of radius r centered at w is the subgraph of G induced by the set Mr(w) of vertices whose distance from w does not exceed r). In particular, we prove that if every ball of radius 2 in G is 2-connected and G satisfies the condition dG(u)+dG(v)≥|M2(w)|−1 for each path uwv in G, where u and v are non-adjacent vertices, then G has a Hamiltonian curve, introduced by Kündgen et al. (2017). Furthermore, we prove that if every ball of radius 1 in G satisfies Ore’s condition (1960) then all balls of any radius in G are Hamiltonian.

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge ${\mathbf{c}}_{\mathrm M}\in(-\infty,1]$. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating $\gamma$ times a variant of the planar Gaussian free field (GFF), where $\gamma\in(0,2]$ satisfies $\mathbf c_{\mathrm M}=25-6(2/\gamma+\gamma/2)^2$. Physics considerations suggest that LQG should also make sense in the regime when $\mathbf c_{\mathrm M}>1$. However, the behavior in this regime is rather mysterious in part because the corresponding value of $\gamma$ is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of $\mathbf c_{\mathrm M}\in(-\infty,25)$. Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same "LQG size" with respect to the GFF. We prove that several formulas for dimension-related quantities are still valid for $\mathbf c_{\mathrm M}\in(1,25)$, with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for $\mathbf c_{\mathrm M}\in(1,25)$, which gives a finite quantum dimension iff the Euclidean dimension is at most $(25-\mathbf c_{\mathrm M})/12$. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius $r$ grows faster than any power of $r$ (which suggests that the Hausdorff dimension of LQG is infinite for $\mathbf c_{\mathrm M}\in(1,25)$). We include a substantial list of open problems.

Wasserstein distributionally robust shortest path problem

This paper proposes a data-driven distributionally robust shortest path (DRSP) model where the distribution of the travel time in the transportation network can only be partially observed through a finite number of samples. Specifically, we aim to find an optimal path to minimize the worst-case α-reliable mean-excess travel time (METT) over a Wasserstein ball, which is centered at the empirical distribution of the sample dataset and the ball radius quantifies the level of its confidence. In sharp contrast to the existing DRSP models, our model is equivalently reformulated as a tractable mixed 0–1 convex problem, e.g., 0–1 linear program or 0–1 second-order cone program. Moreover, we also explicitly derive the distribution achieving the worst-case METT by simply perturbing each sample. Experiments demonstrate the advantages of our DRSP model in terms of the out-of-sample performance and computational complexity. Finally, our DRSP model is easily extended to solve the distributionally robust bi-criteria shortest path problem and the minimum cost flow problem.

Online unit covering in Euclidean space

We revisit the online Unit Covering problem in higher dimensions: Given a set of n points in Rd, that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in Rd using the Euclidean distance.(I) We give an online deterministic algorithm with competitive ratio O(1.321d), thereby improving on the previous record, O(2ddlog⁡d), due to Charikar et al. (2004), by an exponential factor. In particular, the competitive ratios are 5 in the plane and 12 in 3-space (the previous ratios were 7 and 21, respectively). For d=3, the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual).(II) We show that the competitive ratio of every deterministic online algorithm for Unit Covering in Rd under the L2 norm is at least d+1 for every d≥1. This greatly improves upon the previous best lower bound, Ω(log⁡d/log⁡log⁡log⁡d), due to Charikar et al. (2004).(III) We generalize the above result to Unit Covering in Rd under the LC norm, where C is a centrally symmetric convex body, via the illumination number.(IV) We obtain lower bounds of 4 and 5 for the competitive ratio of any deterministic algorithm for online Unit Covering in R2 and R3, respectively; the previous best lower bounds were 3 for both cases.(V) When the input points are from the square or hexagonal lattice in R2, we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of 3. For the cubic lattice in R3, we give a deterministic online algorithm with a competitive ratio of 5.

Local convergence of fourth and fifth order parametric family of iterative methods in Banach spaces

This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Frechet derivative fulfills the Lipschitz continuity condition is needed. Under these conditions, a convergence theorem is established to study the existence and uniqueness regions for the solution for each method. The efficacy of our convergence study is shown solving various numerical examples as a nonlinear integral equation and calculating the radius of the convergence balls. We compare the radii of convergence balls and observe that by our approach, we get much larger balls as existing ones. In addition, we also include the real and complex dynamic study of one of the methods applied to a generic polynomial of order two.

The compared costs of domination, location-domination and identification

Let G = (V, E) be a finite graph and r ≥ 1 be an integer. For v ∈ V , let B r (v) = {x ∈ V : d(v, x) ≤ r} be the ball of radius r centered at v. A set C ⊆ V is an r-dominating code if for all v ∈ V , we have B r (v) ∩ C = ∅; it is an r-locating-dominating code if for all v ∈ V , we have B r (v) ∩ C = ∅, and for any two distinct non-codewords x ∈ V C, y ∈ V C, we have B r (x) ∩ C = B r (y) ∩ C; it is an r-identifying code if for all v ∈ V , we have B r (v) ∩ C = ∅, and for any two distinct vertices x ∈ V , y ∈ V , we have B r (x) ∩ C = B r (y) ∩ C. We denote by γ r (G) (respectively, ld r (G) and id r (G)) the smallest possible cardinality of an r-dominating code (respectively, an r-locating-dominating code and an r-identifying code). We study how small and how large the three differences id r (G)−ld r (G), id r (G)−γ r (G) and ld r (G) − γ r (G) can be.

Smoothness and monotonicity of the excursion set density of planar Gaussian fields

Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant as $R\to \infty $. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(\ell )$ and $c_{LS}(\ell )$ that encode the density of excursion/level set components at the level $\ell $. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the functionals from the decay of the probability of `four-arm events' for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which $c_{ES}(\ell )$ and $c_{LS}(\ell )$ are monotone.

Analytical investigation of an inertial propulsion system using rotating masses

The paper proposes an inertial system capable of generating a propulsive force from the rotation on an eccentric circular path of 8 steel balls placed between two rotating discs constructed with a radial slot system. In addition to a detailed description of the mechanical components of this proposed device, particular attention is paid to the theoretical treatment of the innovative principle upon which the device is based. The paper begins by detailing the equations of motion of the steel balls, from which the propulsion force of the system is computed. Furthermore, the influence of the ball radius on the propulsion force is investigated. The obtained findings support the capability of the proposed system to achieve a linear propulsive force.

Role of zero point energy in promoting ice formation in a spherical drop of water

We demonstrate that the Lifshitz interaction energy (excluding the self-energies of the inner and outer spherical regions) for three concentric spherical dielectric media can be evaluated easily using the immense computation power in recent processors relative to those of a few decades ago. As a prototype, we compute the Lifshitz interaction energy for a spherical shell of water immersed in water vapor of infinite extent while enclosing a spherical ball of ice inside the shell, such that two concentric spherical interfaces are formed: one between solid ice and liquid water and the other between liquid water and gaseous vapor. We evaluate the Lifshitz interaction energy for the above configuration at the triple point of water when the solid, liquid, and gaseous states of water coexist, and, thus, extend the analysis of Elbaum and Schick in Phys. Rev. Lett. 66 (1991) 1713 to spherical configurations. We find that, when the Lifshitz energy contributes dominantly to the total energy of this system, which is often the case when electrostatic interactions are absent, a drop of water surrounded by vapor of infinite extent is not stable at the triple point. This instability, that is a manifestation of the quantum fluctuations in the medium, will promote formation of ice in water, which will then grow in size indefinitely. This is a consequence of the finding here that the Lifshitz energy is minimized for large (micrometer size) radius of the ice ball and small (nanometer size) thickness of the water shell surrounding the ice. These results might be relevant to the formation of hail in thunderclouds. These results are tentative in that the self-energies are omitted; surface tension and nucleation energy are not considered.