There has been burgeoning interest in the computation of partial differential equations on curves and surfaces. Models involving partial differential equations on surfaces arise in many areas including material science, bio-physics, fluid mechanics and image processing. For example, we refer to [10, 27, 29] for applications of the Allen–Cahn and Cahn–Hilliard equations to phase ordering and separation on surfaces. Models for thin fluid films on surfaces have been developed in [21, 24]. For image processing and geometry applications we mention geodesic flow of curves on surfaces and active contours for segmentation on surfaces, [6, 22, 23, 28].
The work in this paper is concerned with an approach to the formulation and approximation of parabolic equations on a prescribed stationary n-dimensional surface ? in Rn+1 (n = 1, 2) using an implicit representation of the surface. The surface is just one level set of a prescribed function ? and the partial differential equation and its solution are extended to a domain ? ? R n+1 containing the surface. A general framework for formulating partial differential equations on implicit surfaces was proposed by the authors of [3]. They considered time dependent second order linear and nonlinear diffusion equations in the context of finite difference approximations on rectangular grids independent of the surfaces. In [20, 19] the authors presented finite difference methods for fourth order parabolic equations on implicit surfaces. A finite element approximation of elliptic equations on implicit surfaces is presented in [5].
This document introduces the Cocoa application environment using the Java language and teaches you how to leverage Apple’s development tools to build robust, object-oriented applications. Cocoa provides the best way to build modern, multimedia-rich, object-oriented applications for consumers and enterprise customers alike. This document assumes you are familiar with Java programming but does not assume you have previous experience with Cocoa or Xcode Tools.
This document is intended for Java programmers interested in developing Cocoa applications. Keep in mind, however, that Java is not Cocoa’s native language. To develop Cocoa applications that you intend to release to end users, you must use Objective-C. No Java interfaces for new Cocoa features will be added to Mac OS X versions after 10.4. Therefore, features added to Cocoa in subsequent versions ac OS X will not be available to Cocoa applications developed using Java.
Many natural resources and energy sources are commodities. According to the Webster’s New Universal Unabridged Dictionary (1996 [43]), a commodity is “any unprocessed or partially processed good, as grains, fruits, and vegetables, or precious metals.” Coal, oil, and natural gas are additional examples. For a storable commodity, the economic interpretation of storage is the amount of commodity carried over to the next period from the current period; that is, storage from the prior period plus the difference between the commodity production and consumption in the current period (Williams and Wright 1991 [45]). Professional commodity storers, hereafter referred to as merchants, trade this surplus in wholesale markets that, for basic commodities, resemble a situation of perfect competition; that is, they are characterized by many small players who behave as price takers. For example, such a setting is the natural gas market at Henry Hub, Louisiana, the delivery location of the New York Mercantile Exchange (NYMEX) natural gas futures contract.
Merchants need access to storage facilities to support their commodity trading activities. They may own such facilities themselves, or hold contracts on their capacity. In this paper, a storage asset refers to the facility where a commodity can be physically stored, or a contractual agreement that entitles its owner to usage of a portion of such a facility. These assets feature two distinctive characteristics. On the operational side, while the storage technology may take many forms (from conventional warehouses and oil tanks to underground depleted reservoirs, aquifers, and salt domes used to store natural gas), minimum/maximum inventory levels (space) and injection/withdrawal capacity limits are ubiquitous. On the financial side, commodity prices are notoriously variable and volatile (Seppi 2002 [34]), and storage assets give their managers (merchants) the real option (Trigeorgis 1996 [40]) to buy the commodity at one point in time, store it, and sell it at a later point in time to exploit price variability and volatility.
