Roper-Suffridge算子上的偏差定理下界估计的简单证明

Liczberski-Starkov first found a lower bound for ||D(f)|| near the origin, where is the Roper-Suffridge operator on the unit ball Bn in Cn and F is a normalized convex function on the unit disk. Later, Liczberski-Starkov and Hamada-Kohr proved the lower bound holds on the whole unit ball using a complex computation. Here we provide a rather short and easy proof for the lower bound. Similarly, when F is a normalized starlike function on the unit disk, a lower bound of ||D(f)|| is obtained again.     

P-Bloch函数的偏差定理和单叶半径

In this paper, we establish distortion theorems for both normalized p-Bloch functions with branch points and normalized locally univalent p-Bloch functions defined on the unit disk, respectively. These distortion theorems give lower bounds on |f′(z)| and ■f′(z). As applications of these distortion theorems, the lower bounds of the radius of the largest schlicht disk on these Bloch functions are given, respectively. Notice that when p = 1, our results reduce to that of Liu and Minda.     

Unsteady viscous flow over a rotating stretchable disk with deceleration

In this work, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated. The three dimensional Navier–Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method. Mathematically, two solution branches are found for the similarity equations. The lower solution branch may not be physically feasible due to a negative velocity in the circumferential direction. For the physically feasible solution branch, namely the upper solution branch, the fluid behavior is greatly influenced by the disk stretching parameter and the unsteadiness parameter. With disk stretching, the disk can be friction free in both the radial and the circumferential directions, depending on the values of the controlling parameters. The results provide an exact solution to the whole unsteady NS equations with new nonlinear phenomena and multiple solution branches.     

Two-layer film flow over a rotating disk

Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.     

Buckling localization in a non‐elastic rotating disk

Buckling localization of a rotating disk made of elastic‐perfectly plastic material is investigated using stress‐rate formulation of the stability boundary‐value problem. The phenomenon of plastic buckling localization and its analogy with elastic buckling localization is discussed. For a thin rotating disk, it is shown that buckling develops at a speed lower than one at which the disk passes to fully plastic state, or in other words, before the limit load has been attained. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Amortized analysis of some disk scheduling algorithms: SSTF,SCAN, andN-StepSCAN

The amortized analysis is a useful tool for analyzing the time-complexity of performing a sequence of operations. The disk scheduling problem involves a sequence of requests in general. In this paper, the performances of representative disk scheduling algorithms,SSTF, SCAN, andN-StepSCAN, are analyzed in the amortized sense. A lower bound of the amortized complexity for the disk scheduling problem is also derived. According to our analysis,SCAN is not only better thanSSTF andN-StepSCAN, but also an optimal algorithm. Various authors have studied the disk scheduling problem based on some probability models and concluded that the most acceptable performance is obtained fromSCAN. Our result therefore supports their conclusion.This research was supported by the National Science Council, Taiwan R. O. C. under contract: NSC80-0408-E009-11.     

Randomized on-line algorithms and lower bounds for computing large independent sets in disk graphs

We study the on-line version of the maximum independent set problem, for the case of disk graphs which are graphs resulting from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lower bounds for deterministic on-line independent set algorithms and present new upper and lower bounds.     

Computation of a rotating flow past a permeable bed

Summary The rotating flow of a viscous incompressible fluid between two disks is studied when there is a porous layer on the lower disk. The motion relative to a rotating frame is caused by a differential rotation of the disks. Generalised Darcy's law represents the flow. The numerical solution is obtained using a shooting method.     

Uniform densities of regular sequences in the unit disk

The upper and lower uniform densities of some regular sequences are computed. These densities are used to determine sequences of sampling and interpolation for Bergman spaces of the unit disk.

     

Lagrange Interpolation for the Disk Algebra: The Worst Case

We consider Lagrange interpolation polynomials for functions in the disk algebra with nodes on the boundary of the unit disk. In case that the closure of the set of nodes does not cover the boundary of the unit disk we prove that there exists a residual set of functions in the disk algebra, such that the Lagrange interpolation polynomials of each of these functions form a dense subset of the space of all holomorphic functions defined on the unit disk.     

Symmetrization of condensers and inequalities for functions multivalent in a disk

Using the circular symmetrization of sets and condensers on Riemann surfaces, we establish new inequalities for multivalent functions with conditions on the critical values of the functions or on the coverings of concentric circles. Two-point distortion theorems, an inequality for the initial coefficients, and a lower bound for the modulus of functions (of diverse classes) p-valent in a disk are proved.     

Finite element analysis of thermoelastic field in a rotating FGM circular disk

This study focuses on the finite element analysis of thermoelastic field in a thin circular functionally graded material (FGM) disk subjected to a thermal load and an inertia force due to rotation of the disk. Due to symmetry, the FGM disk is assumed to have exponential variation of material properties in radial direction only. As a result of nonuniform coefficient of thermal expansion (CTE) and nonuniform temperature distribution, the disk experiences an incompatible eigenstrain which is taken into account. Based on the two dimensional thermoelastic theories, the axisymmetric problem is formulated in terms of a second order ordinary differential equation which is solved by finite element method. Some numerical results of thermoelastic field are presented and discussed for an Al₂O₃/Al FGM disk. The analysis of the numerical results reveals that the thermoelastic field in an FGM disk is significantly influenced by temperature distribution profile, radial thickness of the disk, angular speed of the disk, and the inner and outer surface temperature difference, and can be controlled by controlling these parameters.     

Steady motion of a rotating stratified fluid bounded by porous disks

The motion of a stratified fluid between two parallel infinite porous disks rotating about a vertical axis with slightly different angular velocities has been investigated. The closed form solutions are presented either when the temperature of the disks are prescribed or when the heat flow from the upper to the lower disk is prescribed. The effects of porosity on the flow field have been discussed.     

Cobordism of disk knots

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots D ⁿ⁻² ↪ D ⁿ . Any two disk knots are cobordant if the cobordisms are not required to fix the boundary sphere knots, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of J. Levine on the cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also ask which algebraic cobordism classes can be realized given a fixed boundary knot and provide a complete classification when the boundary knot has no 2-torsion in its middle-dimensional Alexander module. In the course of this classification, we establish a close connection between the Blanchfield pairing of a disk knot and the Farber-Levine torsion pairing of its boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfield pairing will determine the boundary Farber-Levine pairing). In addition, we study the dependence of disk knot Seifert matrices on choices of Seifert surface, demonstrating that all such Seifert matrices are rationally S-equivalent, but not necessarily integrally S-equivalent.     

Monte Carlo Microflow Simulation for the Gas Film Lubrication in Modern Magnetic Disk Storage

A two‐dimensional unsteady problem of gas flow in an extremely narrow channel with an inclined upper wall and moving lower one is studied by the DSMC method. This is a model of gas film lubrication which occurs in modern magnetic disk storage, that is now under development. Far from the magnetic head the flow produced by the disk motion could be described by solution of the Rayleigh problem. Space and time distributions of the pressure on the upper wall as well as density and average velocity inside and outside of the channel were obtained. They show that as a result of the flow slowing‐down by the front wall of the magnetic head the region with an increased density is formed there. At the same time marked non‐homogeneity of gas velocity before the inlet of the channel is observed.     

Modelling and control of the motion of a disk rolling on a spherical dome

This work deals with the modelling and control of the motion of a disk rolling without slipping on a rigid spherical dome. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. First, a mathematical model of the motion of the disk rolling on the dome is derived. Then, by using a kind of an inverse control transformation, a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the spherical dome.     

Boundedness properties of univalent functions with positive coefficients

The present contribution investigates sharp lower bounds of the real parts of certain types of rational functions defined in terms of functions which are analytic and univalent in the open unit disk, and which also involve the familiar fractional derivative operator. Some worthwhile consequences of the main results are also pointed out.     

NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

We present NC-approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of λ-precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for λ-precision unit disk graphs many more graph problems have efficient approximation schemes.Our NC-approximation schemes can also be extended to obtain efficient NC-approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann, and Widmayer. The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.     

Numerical analysis of the electrodynamic characteristics of some radiators for mirror antennas

We investigate the fields produced by electrodynamic models of mirror-antenna radiators with reflectors in the shape of a disk or a disk with a scattering projection. The analysis is based on a numerical method which reduces the problem to integrodifferential equations and then solves them by computer. We consider the amplitude and phase structure of the field of an ideally conducting disk excited by a rectangular slit and an elementary oscillator perpendicular to the disk axis. We also consider the phase structure of the field of a reflector in the shape of a disk with a conical projection excited by an elementary oscillator.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 207–211, 1985.     

Quasiconformal extensions to space of Weierstrass-Enneper lifts

The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the Weierstrass-Enneper lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions to space. The extension is defined through the family of best Möbius approximations to the lift applied to a bundle of euclidean circles orthogonal to the disk. Extension of the planar harmonic map is also obtained subject to additional assumptions on the dilatation. The hypotheses involve bounds on a generalized Schwarzian derivative for harmonic mappings in terms of the hyperbolic metric of the disk and the Gaussian curvature of the minimal surface. Hyperbolic convexity plays a crucial role.