Generalized Cantor Expansions

There are many ways to represent a number, commonly known as base expansions. The most frequently used base is ten, which is the basis for our decimal number system. However a more uncommon way to represent a number is the so called Cantor expansion of the number. This system uses factorials rather than numbers to powers as the basis for the system, and it can be shown that this produces a unique expansion for every natural number. However, if you view factorials as products, then it becomes natural to ask what happens if you use other types of products as bases. This paper explores that question and shows there are an uncountably infinite number of bases which can be used to represent the natural, and real numbers uniquely. By using these new and interesting types of bases, it becomes possible to formulate bases in which all rational numbers have a terminating expansion

PHOTOPLETHYSMOGRAPHIC WAVEFORM ANALYSIS DURING LOWER BODY NEGATIVE PRESSURE SIMULATED HYPOVOLEMIA AS A TOOL TO DISTINGUISH REGIONAL DIFFERENCES IN MICROVASCULAR BLOOD FLOW REGULATION.

The purpose of this investigation was to explore modulation of the photoplethsymographic (PPG) waveform in the setting of simulated hypovolemia as a tool to distinguish regional differences in regulation of the microvasculature. The primary goal was to glean useful physiological and clinical information as it pertains to these regional differences in regulation of microvascular blood flow. This entailed examining the cardiovascular, autonomic nervous, and respiratory systems interplay in the functional hemodynamics of regulation of microvascular blood flow to both central (ear, forehead) and peripheral (finger) sites. We monitored ten healthy volunteers (both men and women age 24-37 ) non-invasively with central and peripheral photoplethysmographs and laser Doppler flowmeters during Lower Body Negative Pressure (LBNP). Waveform amplitude, width, and oscillatory changes were characterized using waveform analysis software (Chart, ADInstruments). Data were analyzed with the Wilcoxon Signed Ranks Test, paired t-tests, and linear regression. Finger PPG amplitude decreased by 34.6 ± 17.6% (p = 0.009) between baseline and the highest tolerated LBNP. In contrast, forehead amplitude changed by only 2.4 ± 16.0% (p=NS). Forehead and finger PPG width decreased by 48.4% and 32.7%, respectively. Linear regression analysis of the forehead and finger PPG waveform widths as functions of time generated slopes of -1.113 (R = -0.727) and -0.591 (R = -0.666), respectively. A 150% increase in amplitude density of the ear PPG waveform was noted within the range encompassing the respiratory frequency (0.19-0.3Hz) (p=0.021) attributable to changes in stroke volume. We also noted autonomic modulation of the ear PPG signal in a different frequency band (0.12 0.18 Hz). The data indicate that during a hypovolemic challenge, healthy volunteers had a relative sparing of central cutaneous blood flow when compared to a peripheral site as indicated by observable and quantifiable changes in the PPG waveform. These results are the first documentation of a local vasodilatation at the level of the terminal arterioles of the forehead that may be attributable to recently documented cholinergic mechanisms on the microvasculature

Cometary Escape in the Restricted Circular Planar Three Body Problem

The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. This principle is utilized along with Aubry-Mather theory to construct regions of instability for a certain three body problem, given by a Hamiltonian system of two degrees of freedom. In principle, these methods can be applied to construct instability regions for a variety of Hamiltonian systems with $2$ degrees of freedom. The Hamiltonian model considered in this thesis describes the dynamics of a Sun-Jupiter-Comet system and under some simplifying assumptions, the existence of instabilities for the orbit of the comet is shown. In particular it is shown that a comet which starts close to an orbit in the shape of an ellipse of eccentricity $e=0.66$ can increase in eccentricity to beyond $e=1$. Furthermore, there exist ejection orbits for the comet. Such orbits are initially well within the range of our solar system. This might give an indication of why most objects rotating around the Sun in our solar system have relatively low eccentricity. Several new theoretical tools are introduced in this thesis as well. The most notable is a checkable sufficient condition to verify that an exact area preserving map is an exact area preserving twist map in a certain coordinate system. This coordinate system is constructed by ``spreading the cumulative twist'' which arises from the long term dynamics of system. Many of the results of the thesis are `computer assisted' and utilize recent advances in rigorous numerical integration. It is through the application of these advances in computing that it has become possible to state deep results for realistic solar systems. This has been the dream of many since humans first observed the stars so long ago

Trauma and acute care surgeons report prescribing less opioids over time.

IntroductionConfronted with the opioid epidemic, surgeons must play a larger role to reduce risk of opioid abuse while managing acute pain. Having a better understanding of the beliefs and practices of trauma and acute care surgeons regarding discharge pain management may offer potential targets for interventions beyond fixed legal mandates.MethodsAn Institutional Review Board-approved electronic survey was sent to trauma and acute care surgeons who are members of the American Association for the Surgery of Trauma, and trauma and acute care surgeons and nurse practitioners at a Level 1 trauma center in February 2018. The survey included four case-based scenarios and questions about discharge prescription practices and beliefs.ResultsOf 66 respondents, most (88.1%) were at academic institutions. Mean number of opioid tablets prescribed was 20-30 (range 5-90), with the fewest tablets prescribed for elective laparoscopic cholecystectomy and the most for rib fractures. Few prescribed both opioid and non-opioid medications (22.4% to 31.4 %). Most would not change the number/strength of medications (69.2%), dose (53.9%), or number of tablets of opioids (83.1%) prescribed if patients used opioids regularly prior to their operation. The most common factors that made providers more likely to prescribe opioids were high inpatient opioid use (32.4%), history of opioid use/abuse (24.5%), and if the patient lives far from the hospital (12.9%). Most providers in practice >5 years reported a decrease in opioids (71.9%) prescribed at discharge.ConclusionTrauma and acute care surgeons and nurse practitioners reported decreasing the number/amount of opioids prescribed over time. Patients with high opioid use in the hospital, history of opioid use/abuse, or who live far from the provider may be prescribed more opioids at discharge.Level of evidenceLevel IV

Investigation of Age Related Differences in the Rewiring of P2-Olfactory Receptor Neurons

Olfactory receptor neurons (ORNs) maintain the ability to regenerate. These neurons reside in the olfactory epithelium and project axons that connect to the olfactory bulbs. Despite the diffuse distribution of ORNs in the olfactory epithelium, they converge at discrete glomeruli in the olfactory bulb. In the P2 IRES tau-lacZ mouse, the P2 ORN subtype has been previously mapped to two glomeruli, using X-gal staining. To determine if age affects ORN regeneration, left olfactory nerve transections were performed on P2 mice from immature (five-weeks old) and mature (1 6-weeks old) groups. Following recovery, the olfactory bulbs were processed to observe ORN regeneration. A significant difference was seen in the number and mapping of full P2 glomeruli between lesioned and control olfactory bulbs, but not between the age groups. This suggests that age differences between the two groups in this study were not large enough to affect the regeneration of P2 ORNs

Analysis of the Stochastic Stability and Asymptotically Stationary Statistics for a Class of Nonlinear Attitude Estimation Algorithms

Attitude estimation algorithms are critical components of satellite control systems, aircraft autopilots, and other applications. Attitude estimation systems perform their task by fusing attitude and gyroscope measurements; however, such measurements are typically corrupted by random noise and gyroscopes may have significant bias. Variations of the extended Kalman filter are commonly used, but this technique relies on instantaneous linearization of the underlying nonlinear dynamics and global stability cannot be guaranteed. Nonlinear attitude observers with guaranteed global stability have been derived and experimentally demonstrated, but only for the deterministic setting where no stochastic effects are present. The first part of this thesis extends a deterministic nonlinear attitude estimator by introducing additional dynamics that allow learning variations of gyro bias as a function of operating temperature, a common source of bias variation in rate gyro readings. The remainder of the thesis formally addresses the problem of stochastic stability and asymptotic performance for this family of estimators when the measurements contain random noise. Analysis tools from stochastic differential equation theory and stochastic Lyapunov analysis are used together to demonstrate convergence of the filter states to a stationary distribution, and to bound the associated steady-state statistics as a function of filter gains and sensor parameters. In many cases these bounds are conservative, but solutions have been found for the associated stationary Fokker-Planck PDEs for two cases. When only the gyro measurement contains noise, the attitude estimation errors are shown to converge to a bipolar Bingham distribution. When the gyro measurement is further assumed to have constant bias, the estimation errors are shown to converge to a joint bipolar Bingham and multivariate Gaussian distribution. Knowledge of the stationary distributions allow for exact computation of steady-state statistics. Further, the analysis suggests a method for modeling a continuous quaternion noise process with specified statistics on SO(3); this model is used for analyzing estimator performance when both the gyro and the attitude measurements contain noise. Bounds and exact predictions for the different noise models are validated using a high fidelity numerical integration method for nonlinear stochastic differential equations

Analysis of the Stochastic Stability and Asymptotically Stationary Statistics for a Class of Nonlinear Attitude Estimation Algorithms

Attitude estimation algorithms are critical components of satellite control systems, aircraft autopilots, and other applications. Attitude estimation systems perform their task by fusing attitude and gyroscope measurements; however, such measurements are typically corrupted by random noise and gyroscopes may have significant bias. Variations of the extended Kalman filter are commonly used, but this technique relies on instantaneous linearization of the underlying nonlinear dynamics and global stability cannot be guaranteed. Nonlinear attitude observers with guaranteed global stability have been derived and experimentally demonstrated, but only for the deterministic setting where no stochastic effects are present. The first part of this thesis extends a deterministic nonlinear attitude estimator by introducing additional dynamics that allow learning variations of gyro bias as a function of operating temperature, a common source of bias variation in rate gyro readings. The remainder of the thesis formally addresses the problem of stochastic stability and asymptotic performance for this family of estimators when the measurements contain random noise. Analysis tools from stochastic differential equation theory and stochastic Lyapunov analysis are used together to demonstrate convergence of the filter states to a stationary distribution, and to bound the associated steady-state statistics as a function of filter gains and sensor parameters. In many cases these bounds are conservative, but solutions have been found for the associated stationary Fokker-Planck PDEs for two cases. When only the gyro measurement contains noise, the attitude estimation errors are shown to converge to a bipolar Bingham distribution. When the gyro measurement is further assumed to have constant bias, the estimation errors are shown to converge to a joint bipolar Bingham and multivariate Gaussian distribution. Knowledge of the stationary distributions allow for exact computation of steady-state statistics. Further, the analysis suggests a method for modeling a continuous quaternion noise process with specified statistics on SO(3); this model is used for analyzing estimator performance when both the gyro and the attitude measurements contain noise. Bounds and exact predictions for the different noise models are validated using a high fidelity numerical integration method for nonlinear stochastic differential equations

Prospective observational study of point-of-care creatinine in trauma.

Background:Patients with trauma are at risk for renal dysfunction from hypovolemia or urological injury. In austere environments, creatinine values are not available to guide resuscitation. A new portable device, the Stat Sensor Point-of-care (POC) Whole Blood Creatinine Analyzer, provides accurate results in <30 s and requires minimal training. This device has not been evaluated in trauma despite the theoretical benefit it provides. The purpose of this study is to determine the clinical impact of the POC device in trauma. Methods:40 patients with trauma were enrolled in a prospective observational study. One drop of blood was used for creatinine determination on the Statsensor POC device. POC creatinine results were compared to the laboratory. Turnaround time (TAT) for POC and laboratory methods was calculated as well as time elapsed to CT scan if applicable. Results:Patients (n=40) were enrolled between December 2014 and March 2015. POC creatinine values were similar to laboratory methods with a mean bias of 0.075±0.27 (p=0.08). Mean analytical TATs for the POC measurements were significantly faster than the laboratory method (11.6±10.0 min vs 78.1±27.9 min, n=40, p<0.0001). Mean elapsed time before arrival at the CT scanner was 52.9±34.2 min. Conclusions:The POC device reported similar creatinine values to the laboratory and provided significantly faster results. POC creatinine testing is a promising development for trauma practice in austere environments and workup of a subset of stable patients with trauma. Further study is warranted to determine clinical impact, both in hospital-based trauma and austere environments

Range and Intensity Image-Based Terrain and Vehicle Relative Pose Estimation System

A navigation system includes an image acquisition device for acquiring a range image of a target vehicle, at least one processor, a memory including a target vehicle model and computer readable program code, where the processor and the computer readable program code are configured to cause the navigation system to convert the range image to a point cloud having three dimensions, compute a transform from the target vehicle model to the point cloud, and use the transform to estimate the target vehicle's attitude and position for capturing the target vehicle

Destruction of invariant curves in the restricted circular planar three body problem using comparison of action

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