The classical algorithms require order n ~ operations to compute the first n terms in the reversion of a power series or the composition of two series, and order nelog n operations if the fast Founer transform is used for power series multiplication In this paper we show that the composition and reversion problems are equivalent (up to constant factors), and we give algorithms which require only order (n log n) ~/2 operations In many cases of practical importance only order n log n operations are required, these include certain special functions of power series and power series solution of certain differential equations Applications to root-finding methods which use inverse interpolation and to queueing theory are described, some results on multivariate power series are stated, and several open questions are mentioned.

The importance of pushing the performance envelope of disk drives continues to grow, not just in the server market but also in numerous consumer electronics products. One of the most fundamental factors impacting disk drive design is the heat dissipation and its effect on drive reliability, since high temperatures can cause off-track errors, or even head crashes. Until now, drive manufacturers have continued to meet the 40 % annual growth target of the internal data rates (IDR) by increasing RPMs, and shrinking platter sizes, both of which have counter-acting effects on the heat dissipation within a drive. As this paper will show, we are getting to a point where it is becoming very difficult to stay on this roadmap. This paper presents an integrated disk drive model that captures the close relationships between capacity, performance and thermal characteristics over time. Using this model, we quantify the drop off in IDR growth rates over the next decade if we are to adhere to the thermal envelope of drive design. We present two mechanisms for buying back some of this IDR loss with Dynamic Thermal Management (DTM). The first DTM technique exploits any available thermal slack, between what the drive was intended to support and the currently lower operating temperature, to ramp up the RPM. The second DTM technique assumes that the drive is only designed for average case behavior, thus allowing higher RPMs than the thermal envelope, and employs dynamic throttling of disk drive activities to remain within this envelope.

This paper describes the first infrastructure for integrated studies of the performance and thermal behavior of storage systems. Using microbenchmarks running on this infrastructure, we first gain insight into how I/O characteristics can affect the temperature of disk drives. We use this analysis to identify the most promising, yet simple, “knobs ” for temperature optimization of high speed disks, which can be implemented on existing disks. We then analyze the thermal profiles of real workloads that use such disk drives in their storage systems, pointing out which knobs are most useful for dynamic thermal management when pushing the performance envelope.

To account for surface relaxation in ultra-thin films, we consider the simplest one-dimensional discrete chain with harmonic interactions of up to second nearest neighbors. We assume that the springs, describing interactions of the nearest neighbors (NN) and next to nearest neighbors (NNN) have incompatible reference lengths, which introduce a hyper-pre-stress and results in a formation of the exponential surface boundary layers. For a finite body loaded by a system of (double) forces at the boundary, we explicitly find the displacement field and compute the energies of the inhomogeneous stressed and reference configurations. We then obtain a simple expression for the hyper-pre-stress related contribution to the surface energy and show an unusual scaling of the total energy with the film thickness. For ultra-thin films we report an anomalous stiffness increase due to the overlapping of the surface boundary layers. Implications of the micro level hyper-pre-stress in fracture mechanics and in the theory of non-Bravais lattices are also discussed.

The aim of this paper is the study of the kernel and acceleration properties of sequence transformations of the form T n = L(S n =D n)=L(1=D n) , where (S n) is the sequence for which we want to compute the limit, (D n) is an error estimate and L is a linear difference operator. We will obtain those properties for different classes of operators L and we will give a procedure for constructing, for a given class of sequences, an operator for which the corresponding transformation accelerates that class.

We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrödinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of in-phase or anti-phase excited nodes, separated by an arbitrary sequence of empty nodes. By using stability analysis, we prove that the discrete solitons are all unstable near the anti-continuum limit, except for the solitons, which consist of alternating anti-phase excited nodes. We classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons. 1

The importance of pushing the performance envelope of disk drives continues to grow in the enterprise storage market. One of the most fundamental factors impacting disk drive design is heat dissipation, since it directly affects drive reliability. Until now, drive manufacturers have continued to meet the 40 % annual growth target of the internal data-rates (IDR) by increasing RPMs and shrinking platter sizes, both of which have counteracting effects on the heat dissipation within a drive. In this article, we shall show that we are getting to a point where it is going to be very difficult to stay on this roadmap. We first present detailed models that capture the close relationships between capacity, performance, and thermal characteristics over time. Using these models, we quantify the drop-off in IDR growth rates over the next decade if we are to adhere to the thermal design envelope. We motivate the need for continued improvements in IDR by showing that the response times of real workloads can be improved by 30–60 % with a 10K increase in the RPM for disks used in their respective storage systems. We then present two dynamic thermal management (DTM) techniques that can be used to buy back some of this IDR loss. The first DTM technique exploits the thermal slack between what the drive was intended to support and the currently lower operating temperature to ramp up the RPM. The second DTM technique assumes that the drive is

. In the present paper we give new formulas for a general solution of the linear difference equation of finite order with constant complex coefficients without necessity of solving the characteristic equation Introduction. In this paper we deal with the following difference equation of order m: (1) xn+m = m X r=1 a r xn+m\Gammar with constant complex coefficients a 1 ; : : : ; am . Our Theorem gives a simple formula for the general solution depending only on the coefficients a 1 ; : : : ; am . We do not have to solve the characteristic equation as it is usually done (cf for instance [1], [2]) and, in general, it is often impossible to find the exact solutions of it. To formulate our Theorem we adopt the following convention: (\Gamma1)! \Delta 0 = 1 : Theorem. Let x 0 ; : : : ; xm\Gamma1 be arbitrary complex numbers, let h 1 ; : : : ; hm be nonnegative integers. The general solution of equation (1) is of the form (2): xn = m\Gamma1 X l=0 " X 1h1+\Delta\Delta\Delta+mh m=n\Gammal...

Thermal-aware design of disk-drives is important because high temperatures can cause reliability problems. Dynamic thermal management (DTM) techniques have been proposed to operate the disk at the average case temperature, rather than at the worst case by modulating the activities to avoid thermal emergencies caused by unexpected events, such as fan-breaks, increased inlet air temperature, etc. A delay-based approach to adjust the disk seek activities is one such DTM solution for disk-drives. Even if such a DTM approach could overcome thermal emergencies without stopping disk activity, it suffers from long delays when servicing the requests. In this paper, we investigate the possibility of using a multispeed disk-drive (called dynamic rotations per minute (DRPM)), which dynamically modulates the rotational speed of the platter for implementing the DTM technique. Using a detailed performance and thermal simulator of a storage system, we evaluate two possible DTM policies—time-based and watermark-based—with a DRPM disk-drive and observe that dynamic RPM modulation is effective in avoiding thermal emergencies. However, we find that the time taken to transition between different rotational speeds of the disk is critical for the effectiveness of this DTM technique. �DOI: 10.1115/1.2993152�

We study discrete vortices in the anti-continuum limit of the discrete two-dimensional nonlinear Schrödinger (NLS) equations. The discrete vortices in the anti-continuum limit represent a finite set of excited nodes on a closed discrete contour with a non-zero topological charge. Using the Lyapunov–Schmidt reductions, we find sufficient conditions for continuation and termination of the discrete vortices for a small coupling constant in the discrete NLS lattice. An example of a closed discrete contour is considered that includes the vortex cell (also known as the off-site vortex). We classify the symmetric and asymmetric discrete vortices that bifurcate from the anti-continuum limit. We predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete vortices. 1