Sampling theorems and bases in a Hilbert space

A unified approach to sampling theorems for (wide sense) stationary random processes rests upon Hilbert space concepts. New results in sampling theory are obtained along the following lines: recovery of the process [chi](t) from nonperiodic samples, or when any finite number of samples are deleted; conditions for obtaining [chi](t) when only the past is sampled; a criterion for restoring [chi](t) from a finite number of consecutive samples; and a minimum mean square error estimate of [chi](t) based on any (possibly nonperiodic) set of samples.In each case, the proofs apply not only to the recovery of [chi](t), but are extended to show that (almost) arbitrary linear operations on [chi](t) can be reproduced by linear combinations of the samples. Further generality is attained by use of the spectral distribution function F([middle dot]) of [chi](t), without assuming F([middle dot]) absolutely continuous.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32353/1/0000424.pd

The operator theory of the pseudo-inverse II. Unbounded operators with arbitrary range

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32075/1/0000119.pd

On the statistics of random pulse processes

Statistics are obtained for pulse trains in which the pulse shapes as well as the time base are random. The general expression derived for the mean and spectral density of the pulse train require neither independence of intervals between time base points nor independence of the pulses. The spectral density appears as an infinite series that can be summed to closed form in many applications (e.g., pulse duration modulation with skipped and jittered samples). If the time base is a Poisson point process and the pulse shapes are independent, stronger results become available; we are then able to calculate joint characteristic functions for the pulse process, thus providing a more complete statistical description. Examples are given, illustrating use of the above results for pulse duration modulation (with arbitrary pulse shapes) and telephone traffic.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33662/1/0000172.pd

On stationarity conditions for a certain periodic random process

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32356/1/0000428.pd

On two discrete-time system stability concepts and supermartingales

A random discrete-time system {xn}, N = 0, 1, 2, ... is called stochastically stable if for every [epsilon] > 0 there exists a [lambda] > 0 such that the probability P[(supn || xn ||) > [epsilon]] P[|| x0 || > [lambda]] V([middle dot]) satisfies the supermartingale definition on {V(xn)} in a neighborhood of the origin; earlier proofs of stochastic stability require additional restrictions. A criterion for xn --> 0 almost surely is developed. It consists of a global inequality on {U(xn)} stronger than the supermartingale defining inequality, but applied to a U([middle dot]) that need not be a Lyapunov function. The existence of such a U([middle dot]) is exhibited for a stochastically unstable nontrivial stochastic system. This indicates that our criterion for xn --> 0 is "tight," and that the two stability concepts studied are substantially distinct.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33783/1/0000037.pd

Random sampling of random processes: Stationary point processes

This is the first of a series of papers treating randomly sampled random processes. Spectral analysis of the resulting samples pre-supposes knowledge of the statistics of tn, the random point process whose variates represent the sampling times. We introduce a class of stationary point processes, whose stationarity (as characterized by any of several equivalent criteria) leads to wide-sense stationary sampling trains when applied to wide-sense stationary processes. Of greatest importance are the nth forward [backward] recurrence times (distances from t to the nth point thereafter [preceding]), whose distribution functions prove more useful to the computation of covariances than interval statistics, and which possess remarkable properties that facilitate the analysis.The moments of the number of points in an interval are evaluated by weighted sums of recurrence time distribution functions, the moments being finite if and only if the associated sum converges. If the first moment is finite, these distribution functions are absolutely continuous, and obey some convexity relations. Certain formulas relate recurrence statistics to interval length statistics, and conversely; further, the latter are also suitable for a direct evaluation of moments of points in intervals.Our point process requires neither independent nor identically distributed interval lengths. It embraces most of the common sampling schemes (e.g., periodic, Poisson, jittered), as well as some new models. Of particular interest are point processes obtained from others by a random deletion of points (skip processes), as for instance a jittered cyclically periodic process with (random or systematic) skipping. Computation of the statistics for several point processes yields new results of interest not only for their own sake, but also of use for spectral analyses appearing in other papers of this series.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33420/1/0000822.pd

The operator theory of the pseudo-inverse I. Bounded operators

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32074/1/0000118.pd

Optimal policies for controlled Markov chains with a constraint

The time average reward for a discrete-time controlled Markov process subject to a time-average cost constraint is maximized over the class of al causal policies. Each epoch, a reward depending on the state and action, is earned, and a similarly constituted cost is assessed; the time average of the former is maximized, subject to a hard limit on the time average of the latter. It is assumed that the state space is finite, and the action space compact metric. An accessibility hypothesis makes it possible to utilize a Lagrange multiplier formulation involving the dynamic programming equation, thus reducing the optimization problem to an unconstrained optimization parametrized by the multiplier. The parametrized dynamic programming equation possesses compactness and convergence properties that lead to the following: If the constraint can be satisfied by any causal policy, the supremum over time-average rewards respective to all causal policies is attained by either a simple or a mixed policy; the latter is equivalent to choosing independently at each epoch between two specified simple policies by the throw of a biased coin.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25498/1/0000039.pd

Reference Ranges for the Clinical Laboratory Derived from a Rural Population in Kericho, Kenya

The conduct of Phase I/II HIV vaccine trials internationally necessitates the development of region-specific clinical reference ranges for trial enrolment and participant monitoring. A population based cohort of adults in Kericho, Kenya, a potential vaccine trial site, allowed development of clinical laboratory reference ranges. Lymphocyte immunophenotyping was performed on 1293 HIV seronegative study participants. Hematology and clinical chemistry were performed on up to 1541 cohort enrollees. The ratio of males to females was 1.9∶1. Means, medians and 95% reference ranges were calculated and compared with those from other nations. The median CD4+ T cell count for the group was 810 cells/µl. There were significant gender differences for both red and white blood cell parameters. Kenyan subjects had lower median hemoglobin concentrations (9.5 g/dL; range 6.7–11.1) and neutrophil counts (1850 cells/µl; range 914–4715) compared to North Americans. Kenyan clinical chemistry reference ranges were comparable to those from the USA, with the exception of the upper limits for bilirubin and blood urea nitrogen, which were 2.3-fold higher and 1.5-fold lower, respectively. This study is the first to assess clinical reference ranges for a highland community in Kenya and highlights the need to define clinical laboratory ranges from the national community not only for clinical research but also care and treatment

Behavioral and Immune Responses to Infection Require Gαq- RhoA Signaling in C. elegans

Following pathogen infection the hosts' nervous and immune systems react with coordinated responses to the danger. A key question is how the neuronal and immune responses to pathogens are coordinated, are there common signaling pathways used by both responses? Using C. elegans we show that infection by pathogenic strains of M. nematophilum, but not exposure to avirulent strains, triggers behavioral and immune responses both of which require a conserved Gαq-RhoGEF Trio-Rho signaling pathway. Upon infection signaling by the Gαq pathway within cholinergic motorneurons is necessary and sufficient to increase release of the neurotransmitter acetylcholine and increase locomotion rates and these behavioral changes result in C. elegans leaving lawns of M. nematophilum. In the immune response to infection signaling by the Gαq pathway within rectal epithelial cells is necessary and sufficient to cause changes in cell morphology resulting in tail swelling that limits the infection. These Gαq mediated behavioral and immune responses to infection are separate, act in a cell autonomous fashion and activation of this pathway in the appropriate cells can trigger these responses in the absence of infection. Within the rectal epithelium the Gαq signaling pathway cooperates with a Ras signaling pathway to activate a Raf-ERK-MAPK pathway to trigger the cell morphology changes, whereas in motorneurons Gαq signaling triggers behavioral responses independent of Ras signaling. Thus, a conserved Gαq pathway cooperates with cell specific factors in the nervous and immune systems to produce appropriate responses to pathogen. Thus, our data suggests that ligands for Gq coupled receptors are likely to be part of the signals generated in response to M. nematophilum infection