多复变数全纯映射精细的Bohr定理

本文首先建立不依赖自同构从复Banach空间平衡域到Cn单位多圆柱上一定限制条件下全纯映射精细的范数型Bohr定理及复Banach空间X上单位球到复Banach空间Y上单位球全纯映射精细的泛函型Bohr定理.其次,给出有界对称域上全纯映射精细的Bohr定理.最后,得到J*代数单位球上全纯映射精细的Bohr定理.所得结果将一维的Bohr定理推广至高维.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

On final motions of a Chaplygin ball on a rough plane

A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball.     

When to rush a ‘behind’ in Australian rules football: a dynamic programming approach

In Australian rules football, points are scored when the ball passes over the goal line. Six points are awarded for a goal when the ball passes between the two centre posts, and one point for a ‘behind’, when the ball passes between a centre post and an adjacent outer post. After a behind, the defending team has a free kick from the goal line. It may be worthwhile, particularly in the closing stages of a game, for a defending team voluntarily to concede a behind, by themselves passing the ball between the two outer posts, either to avert the possibility of an imminent goal or to increase the probability of scoring a goal themselves. A dynamic programming model is used to analyse this situation.     

Representation of the Green’s function of the exterior Neumann problem for the Laplace operator

We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.     

Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

Extension of concepts and techniques of linear spaces for the Riemannian setting has been frequently attempted. One reason for the extension of such techniques is the possibility to transform some Euclidean non-convex or quasi-convex problems into Riemannian convex problems. In this paper, a version of Kantorovich’s theorem on Newton’s method for finding a singularity of differentiable vector fields defined on a complete Riemannian manifold is presented. In the presented analysis, the classical Lipschitz condition is relaxed using a general majorant function, which enables us to not only establish the existence and uniqueness of the solution but also unify earlier results related to Newton’s method. Moreover, a ball is prescribed around the points satisfying Kantorovich’s assumptions and convergence of the method is ensured for any starting point within this ball. In addition, some bounds for the Q-quadratic convergence of the method, which depends on the majorant function, are obtained.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

Combinatorial Generalizations of Jung’s Theorem

We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.     

A Nonholonomic Model of the Paul Trap

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.     

A Direct Connection Between the Bergman and Szegő Projections

We use Stokes’s theorem to establish an explicit and concrete connection between the Bergman and Szeg? projections on the disc, the ball, and on strongly pseudoconvex domains.     

Proper Holomorphic Self-Maps of Quasi-balanced Domains

We show that proper holomorphic self-maps of smoothly bounded pseudoconvex quasi-balanced domains of finite type are automorphisms. This generalizes the classical Alexander’s theorem on proper holomorphic self-maps of the unit ball.     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

From approximate balls to approximate ellipses

A ball spans a set of n points when none of the points lie outside it. In Zarrabi-Zadeh and Chan (Proceedings of the 18th Canadian conference on computational geometry (CCCG’06), pp 139–142, 2006) proposed an algorithm to compute an approximate spanning ball in the streaming model of computation, and showed that the radius of the approximate ball is within 3/2 of the minimum. Spurred by this, in this paper we consider the 2-dimensional extension of this result: computation of spanning ellipses. The ball algorithm is simple to the point of being trivial, but the extension of the algorithm to ellipses is non-trivial. Surprisingly, the area of the approximate ellipse computed by this approach is not within a constant factor of the minimum and we provide an elegant proof of this. We have implemented this algorithm, and experiments with a variety of inputs, except for a very pathological one, show that it can nevertheless serve as a good heuristic for computing an approximate ellipse.     

On Lattice Coverings of Nil Space by Congruent Geodesic Balls

Nil geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from W. Heisenberg’s famous real matrix group. The aim of this paper is to study lattice-like ball coverings in Nil space. We introduce the notion of the density of the considered coverings and give upper and lower estimates to it, moreover in Section 3, we formulate a conjecture for the ball arrangement of the least dense latticelike geodesic ball covering and give its covering density ${\triangle \approx 1.42900615}$ . The homogeneous 3-spaces have a unified interpretation in the projective 3-sphere and in our work we will use this projective model.     

An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping

Set-Valued and Variational Analysis - The ‘heavy ball with friction’ dynamical system x + γx + ?f(x)=0 is a nonlinear oscillator with damping (γ&;gt;0). It has been...     

The Bloch theorem in several complex variables

The background theory for the Bloch theorem is generalized to several complex variables. This work involves study of the Bergman kernel functions in order to extend work of Landau and Bonk. The main conclusion is an estimate for Bloch’s constant for mappings of domains of the first classical type. In the special case of then-dimensional ball, the estimate of Bloch’s constant coincides with that of Liu.     

Krasnoselskii's iteration process in hyperbolic space

A well-known iteration scheme due to Krasnoselskii for approximation of fixed points of nonexpansive mappings in Banach spaces is extended to a wider class of spaces. This class includes convex metric spaces of ‘hyperbolic’ type, and the results apply to the study of holomorphic self-mappings of the unit ball in complex Hilbert space.     

The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom

The relativistically admissible velocities of Einstein’s special theory of relativity are regulated by the Beltrami–Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. It is shown in this expository article that the Einstein velocity addition law of relativistically admissible velocities enables Cartesian coordinates to be introduced into hyperbolic geometry, resulting in the Cartesian–Beltrami-Klein ball model of hyperbolic geometry. Suggestively, the latter is increasingly becoming known as the Einstein Relativistic Velocity Model of hyperbolic geometry. Möbius addition is a transformation of the ball linked to Clifford algebra. Einstein addition and Möbius addition in the ball of the Euclidean n-space are isomorphic to each other, and they share remarkable analogies with vector addition. Thus, in particular, Einstein (Möbius) addition admits scalar multiplication, giving rise to gyrovector spaces, just as vector addition admits scalar multiplication, giving rise to vector spaces. Moreover, the resulting Einstein (Möbius) gyrovector spaces form the algebraic setting for the Beltrami-Klein (Poincaré) ball model of n-dimensional hyperbolic geometry, just as vector spaces form the algebraic setting for the standard Cartesian model of n-dimensional Euclidean geometry. As an illustrative novel example special attention is paid to the study of the plane separation axiom (PSA) in Euclidean and hyperbolic geometry.     

Exponential Convergence of Time-Delay Systems in the Presence of Bounded Disturbances

In this paper, the problem of control design for exponential convergence of state/input delay systems with bounded disturbances is considered. Based on the Lyapunov–Krasovskii method combining with the delay-decomposition technique, a new sufficient condition is proposed for the existence of a state feedback controller, which guarantees that all solutions of the closed-loop system converge exponentially (with a pre-specified convergence rate) within a ball whose radius is minimized. The obtained condition is given in terms of matrix inequalities with one parameter needing to be tuned, which can be solved by using a one-dimensional search method with Matlab’s LMI Toolbox to minimize the radius of the ball. Two numerical examples are given to illustrate the superiority of the proposed method.     

On intersections of abelian and nilpotent subgroups in finite groups. I

We consider Pontryagin’s generalized nonstationary example with identical dynamic and inertial capabilities of the players under phase constraints on the evader’s states. The boundary of the phase constraints is not a “death line” for the evader. The set of admissible controls is a ball centered at the origin, and the terminal sets are the origin. We obtain sufficient conditions for a multiple capture of one evader by a group of pursuers in the case when some functions corresponding to the initial data and to the parameters of the game are recurrent.