The Central Limit Theorem and the Estimation of the Concentration of Measure for Fractional Brownian Motion

The principal result of Chapter 1 is a new, direct and elementary proof of the general Central Limit Theorem (CLT). Two important stepping-stones are, first, a new, similarly direct and elementary proof of the CLT for Bernoulli random variables defined on [0,1]; this was initially proved by Bernoulli in the 1700\u27s. The second important stepping-stone is a new result for Bernstein polynomials of continuous functions. Bernstein polynomials are a fundamental object of mathematical analysis. It is well known that Bernstein polynomials of a continuous function on intervals $[0,b_{n}]$ when $n$ tends to infinity return the value of the function for an appropriate rate of $b_{n}$, but uniform convergence is sacrificed. Nothing was known for the symmetric interval $[-b_{n},b_{n}]$. We have proven that for these intervals the limit does not recover the function but rather its integral with respect to Gaussian measure. The extension to our direct proof of the of the general CLT involves a new and surprising connection between the CLT and the Haar basis on [0, 1]: the i.i.d. sequence of random variable is transformed to a sequence defined on [0,1] and the random variables in the transformed sequence are then expanded with respect to the Haar basis.Our work on the estimation of the concentration of measure for fractional Brownian motion requires finding the intersections of ellipsoidal and spherical shells for Gaussian measure in $\mathbb{R}^{N}.$ Gaussian measure is concentrated on a small shell of a sphere of radius the square root of N. We want to determine how large this shell must be to include the majority of the Gaussian measure. This result determines the rate of convergence of averages of squares for fractional Brownian increments. It requires understanding the spectrum of the covariance operator as a function of dimension N and the Hurst index. To help understand the spectrum, we compute the exact rate of the largest eigenvalue of this operator

Carbon deposition in a Bosch process using a cobalt and nickel catalyst

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1980.MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.Bibliography: leaves 190-192.by James Edwin Garmirian.Ph.D

Impaired Distal Thermoregulation in Diabetes and Diabetic Polyneuropathy

Objective: To determine how thermoregulation of the feet is affected by diabetes and diabetic polyneuropathy in both wakefulness and sleep. Research Design and Methods: Normal subjects, diabetic subjects without neuropathy, diabetic subjects with small-fiber diabetic polyneuropathy, and those with advanced diabetic polyneuropathy were categorized based on neurological examination, nerve conduction studies, and quantitative sensory testing. Subjects underwent foot temperature monitoring using an iButton device attached to the foot and a second iButton for recording of ambient temperature. Socks and footwear were standardized, and subjects maintained an activity diary. Data were collected over a 32-h period and analyzed. Results: A total of 39 normal subjects, 28 patients with diabetes but without diabetic polyneuropathy, 14 patients with isolated small-fiber diabetic polyneuropathy, and 27 patients with more advanced diabetic polyneuropathy participated. No consistent differences in foot temperature regulation between the four groups were identified during wakefulness. During sleep, however, multiple metrics revealed significant abnormalities in the diabetic patients. These included reduced mean foot temperature (P < 0.001), reduced maximal temperature (P < 0.001), increased rate of cooling (P < 0.001), as well as increased frequency of variation (P = 0.005), supporting that patients with diabetic polyneuropathy and even those with only diabetes but no diabetic polyneuropathy have impaired nocturnal thermoregulation. Conclusions: Nocturnal foot thermoregulation is impaired in patients with diabetes and diabetic polyneuropathy. Because neurons are highly temperature sensitive and because foot warming is part of the normal biology of sleep onset and maintenance, these findings suggest new potentially treatable mechanisms of diabetes-associated nocturnal pain and sleep disturbance

Biomarkers in motor neuron disease: A state of the art review

Motor neuron disease can be viewed as an umbrella term describing a heterogeneous group of conditions, all of which are relentlessly progressive and ultimately fatal. The average life expectancy is 2 years, but with a broad range of months to decades. Biomarker research deepens disease understanding through exploration of pathophysiological mechanisms which, in turn, highlights targets for novel therapies. It also allows differentiation of the disease population into sub-groups, which serves two general purposes: (a) provides clinicians with information to better guide their patients in terms of disease progression, and (b) guides clinical trial design so that an intervention may be shown to be effective if population variation is controlled for. Biomarkers also have the potential to provide monitoring during clinical trials to ensure target engagement. This review highlights biomarkers that have emerged from the fields of systemic measurements including biochemistry (blood, cerebrospinal fluid, and urine analysis); imaging and electrophysiology, and gives examples of how a combinatorial approach may yield the best results. We emphasize the importance of systematic sample collection and analysis, and the need to correlate biomarker findings with detailed phenotype and genotype data

An Asymptotic Preserving Discrete Velocity Method based on Exponential Bhatnagar-Gross-Krook Integration

The BGK model of the Boltzmann equation allows for efficient flow simulations, especially in the transition regime between continuum and high rarefaction. However, ensuring efficient performances for multiscale flows, in which the Knudsen number varies by several orders of magnitude, is never straightforward. Discrete velocity methods as well as particle-based solvers can each reveal advantageous in different conditions, but not without compromises in specific regimes. This article presents a second-order asymptotic preserving discrete velocity method to solve the BGK equation, with the particularity of thoroughly maintaining positivity when operations are conducted on the distribution function. With this procedure based on exponential differencing, it is therefore also possible to construct an adapted version of this second-order method using the stochastic particle approach, as previously presented. The deterministic variant is detailed here and its performances are evaluated on several test cases. Combined to the probabilistic solver and with the possibility of a future coupling, our exponential differencing DVM provides a robust toolbox that would be useful in efficiently simulating multiscale gas phenomena