The Old is Made New

This panel will consider old and new technologies, materials and processes of iron casting, mold making and foundry practice in the light of contemporary attitudes on environmental impacts. Are some of the old methods and practices of iron casting more environmentally “green” than contemporary practice

Transfer Entropy as a Log-likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic chi-squared distribution is established for the transfer entropy estimator. The result generalises the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense

Multi-Site Application of the Geomechanical Approach for Natural Fracture Exploration

In order to predict the nature and distribution of natural fracturing, Advanced Resources Inc. (ARI) incorporated concepts of rock mechanics, geologic history, and local geology into a geomechanical approach for natural fracture prediction within mildly deformed, tight (low-permeability) gas reservoirs. Under the auspices of this project, ARI utilized and refined this approach in tight gas reservoir characterization and exploratory activities in three basins: the Piceance, Wind River and the Anadarko. The primary focus of this report is the knowledge gained on natural fractural prediction along with practical applications for enhancing gas recovery and commerciality. Of importance to tight formation gas production are two broad categories of natural fractures: (1) shear related natural fractures and (2) extensional (opening mode) natural fractures. While arising from different origins this natural fracture type differentiation based on morphology is sometimes inter related. Predicting fracture distribution successfully is largely a function of collecting and understanding the available relevant data in conjunction with a methodology appropriate to the fracture origin. Initially ARI envisioned the geomechanical approach to natural fracture prediction as the use of elastic rock mechanics methods to project the nature and distribution of natural fracturing within mildly deformed, tight (low permeability) gas reservoirs. Technical issues and inconsistencies during the project prompted re-evaluation of these initial assumptions. ARI's philosophy for the geomechanical tools was one of heuristic development through field site testing and iterative enhancements to make it a better tool. The technology and underlying concepts were refined considerably during the course of the project. As with any new tool, there was a substantial learning curve. Through a heuristic approach, addressing these discoveries with additional software and concepts resulted in a stronger set of geomechanical tools. Thus, the outcome of this project is a set of predictive tools with broad applicability across low permeability gas basins where natural fractures play an important role in reservoir permeability. Potential uses for these learnings and tools range from rank exploration to field-development portfolio management. Early incorporation of the permeability development concepts presented here can improve basin assessment and direct focus to the high potential areas within basins. Insight into production variability inherent in tight naturally fractured reservoirs leads to improved wellbore evaluation and reduces the incidence of premature exits from high potential plays. A significant conclusion of this project is that natural fractures, while often an important, overlooked aspect of reservoir geology, represent only one aspect of the overall reservoir fabric. A balanced perspective encompassing all aspects of reservoir geology will have the greatest impact on exploration and development in the low permeability gas setting

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment

A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) is usually not attributed a wave function but only a reduced density matrix $\rho_1$. Nevertheless, there is a precise way of attributing to it a random wave function $\psi_1$, called its conditional wave function, whose probability distribution $\mu_1$ depends on the entangled wave function $\psi\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about $\mu_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of $\mathscr{H}_2$, most entangled states $\psi$ with given reduced density matrix $\rho_1$ are such that $\mu_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP(\rho_1)$. We also show that, for most entangled states $\psi$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+\delta E]$) and most orthonormal bases of $\mathscr{H}_2$, $\mu_1$ is close to $GAP(\mathrm{tr}_2 \rho_{mc})$ with $\rho_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then $\mu_1$ is close to $GAP(\rho_\beta)$ with $\rho_\beta$ the canonical density matrix on $\mathscr{H}_1$ at inverse temperature $\beta=\beta(E)$. This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that $GAP$ measures describe the thermal equilibrium distribution of the wave function.Comment: 27 pages LaTeX, no figures; v2 major revision with simpler proof

Identifying Oil Exploration Leads using Intergrated Remote Sensing and Seismic Data Analysis, Lake Sakakawea, Fort Berthold Indian Reservation, Willistion Basin

The Fort Berthold Indian Reservation, inhabited by the Arikara, Mandan and Hidatsa Tribes (now united to form the Three Affiliated Tribes) covers a total area of 1530 mi{sup 2} (980,000 acres). The Reservation is located approximately 15 miles east of the depocenter of the Williston basin, and to the southeast of a major structural feature and petroleum producing province, the Nesson anticline. Several published studies document the widespread existence of mature source rocks, favorable reservoir/caprock combinations, and production throughout the Reservation and surrounding areas indicating high potential for undiscovered oil and gas resources. This technical assessment was performed to better define the oil exploration opportunity, and stimulate exploration and development activities for the benefit of the Tribes. The need for this assessment is underscored by the fact that, despite its considerable potential, there is currently no meaningful production on the Reservation, and only 2% of it is currently leased. Of particular interest (and the focus of this study) is the area under the Lake Sakakawea (formed as result of the Garrison Dam). This 'reservoir taking' area, which has never been drilled, encompasses an area of 150,000 acres, and represents the largest contiguous acreage block under control of the Tribes. Furthermore, these lands are Tribal (non-allotted), hence leasing requirements are relatively simple. The opportunity for exploration success insofar as identifying potential leads under the lake is high. According to the Bureau of Land Management, there have been 591 tests for oil and gas on or immediately adjacent to the Reservation, resulting in a total of 392 producing wells and 179 plugged and abandoned wells, for a success ratio of 69%. Based on statistical probability alone, the opportunity for success is high

Local Versus Global Thermal States: Correlations and the Existence of Local Temperatures

We consider a quantum system consisting of a regular chain of elementary subsystems with nearest neighbor interactions and assume that the total system is in a canonical state with temperature $T$. We analyze under what condition the state factors into a product of canonical density matrices with respect to groups of $n$ subsystems each, and when these groups have the same temperature $T$. While in classical mechanics the validity of this procedure only depends on the size of the groups $n$, in quantum mechanics the minimum group size $n_{min}$ also depends on the temperature $T$! As examples, we apply our analysis to a harmonic chain and different types of Ising spin chains. We discuss various features that show up due to the characteristics of the models considered. For the harmonic chain, which successfully describes thermal properties of insulating solids, our approach gives a first quantitative estimate of the minimal length scale on which temperature can exist: This length scale is found to be constant for temperatures above the Debye temperature and proportional to $T^{-3}$ below.Comment: 12 pages, 5 figures, discussion of results extended, accepted for publication in Phys. Rev.

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Realization of low-loss mirrors with sub-nanometer flatness for future gravitational wave detectors

The second generation of gravitational wave detectors will aim at improving by an order of magnitude their sensitivity versus the present ones (LIGO and VIRGO). These detectors are based on long-baseline Michelson interferometer with high finesse Fabry-Perot cavity in the arms and have strong requirements on the mirrors quality. These large low-loss mirrors (340 mm in diameter, 200 mm thick) must have a near perfect flatness. The coating process shall not add surface figure Zernike terms higher than second order with amplitude >0.5 nm over the central 160 mm diameter. The limits for absorption and scattering losses are respectively 0.5 and 5 ppm. For each cavity the maximum loss budget due to the surface figure error should be smaller than 50 ppm. Moreover the transmission matching between the two inputs mirrors must be better than 99%. We describe the different configurations that were explored in order to respect all these requirements. Coatings are done using IBS. The two first configurations based on a single rotation motion combined or not with uniformity masks allow to obtain coating thickness uniformity around 0.2 % rms on 160 mm diameter. But this is not sufficient to meet all the specifications. A planetary motion completed by masking technique has been studied. With simulated values the loss cavity is below 20 ppm, better than the requirements. First experimental results obtained with the planetary system will be presented

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

Use of approximations of Hamilton-Jacobi-Bellman inequality for solving periodic optimization problems

We show that necessary and sufficient conditions of optimality in periodic optimization problems can be stated in terms of a solution of the corresponding HJB inequality, the latter being equivalent to a max-min type variational problem considered on the space of continuously differentiable functions. We approximate the latter with a maximin problem on a finite dimensional subspace of the space of continuously differentiable functions and show that a solution of this problem (existing under natural controllability conditions) can be used for construction of near optimal controls. We illustrate the construction with a numerical example.Comment: 29 pages, 2 figure