FORECASTING MINERAL SUPPLY FROM MINERAL RENT DATA

An optimal control model of exhaustible resources is used to clarify the long run relationship between mineral rent and depletion cost at the industry level. A standard first order condition of the time rate of change of rents is reformulated to reveal that rent data may be used to help forecast the rise in extraction costs resulting from resource depletion. This application of the theory of exhaustible resources is illustrated using historical mineral industry rent and extraction cost data. A forecast of U.S. coal extraction costs, following the method proposed in this paper, suggests that future rates of extraction cost increases will be similar to rates experienced in the past.     

Numerical Modeling of Solitons in a Low Birefringent Optical Fiber

A numerical investigation, based on the split-step Fourier transform algorithm of all optical switching of solitons in a low birefringent optical fiber is presented. The numerical algorithm is described in detail. This revised version was published online in August 2006 with corrections to the Cover Date.     

C 1 Quintic Splines on Type-4 Tetrahedral Partitions

Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type-4) partition and associated trivariate C ¹ quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is also shown how the spaces can be used to solve certain Hermite and Lagrange interpolation problems.     

Approximating weak Chebyshev subspaces by Chebyshev subspaces

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties.     

On the evolution of islands

Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked cells. Let ξ _k denote the random number of islands afterk cells have been marked. We give explicit expressions for moments of products of ξ _k ’s and for moments of products of 1/ξ _k ’s. These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4.     

3,4‐[2,2‐Bis(methoxyethoxymethoxymethyl)propylenedithio]‐3′,4′‐(ethylenedithio)tetrathiafulvalene: a spiro‐substituted BEDT–TTF analogue

The crystal structure of the title compound [systematic name: 2‐(1,3‐dithiolo[4,5‐b][1,4]dithiin‐2‐ylidene)‐6,6‐bis(methoxyethoxymethoxymethyl)‐1,3‐dithiolo[4,5‐b][1,4]dithiepine], C₂₁H₃₀O₆S₈, a spiro‐substituted BEDT–TTF analogue [BEDT–TTF is bis(ethylenedithio)tetrathiafulvalene], has a strongly bent heterocyclic framework. The seven‐membered ring adopts a pseudo‐chair conformation with notably widened ring bond angles, especially at the methylene C atoms [119.49 (11) and 117.60 (11)°]. The axial side chain adopts an extended conformation, but the equatorial side chain curls back on itself and the O atom nearest the ring system is involved in three short contacts to H atoms (2.45–2.53 Å). The molecules pack in centrosymmetrically related pairs, which are isolated from each other by columns of the polyether side chains. This study emphasizes the ease of distortion of the neutral bis(propylenedithio)tetrathiafulvalene ring structure, and how the need to accommodate side chains can easily override the tendency of these donor systems to form stacks in the crystalline state.     

Information and asymptotic efficiency of the case-cohort sampling design in Cox's regression model

Efficiencies of the maximum pseudolikelihood estimator and a number of related estimators for the case-cohort sampling design in the proportional hazards regression model are studied. The asymptotic information and lower bound for estimating the parametric regression parameter are calculated based on the effective score, which is obtained by determining the component of the parametric score orthogonal to the space generated by the infinite-dimensional nuisance parameter. The asymptotic distributions of the maximum pseudolikelihood and related estimators in an i.i.d. setting show that these estimators do not achieve the computed asymptotic lower bound. Simple guidelines are provided to determine in which instances such estimators are close enough to efficient for practical purposes.