Estimation of the parameters in exponential decontamination models
Parameter estimation in exponential decontamination models
PROBABILITY THEORY↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Minimum weight structural design with probability-of-failure constraint.
Probability of failure concept and minimum weight structural design, discussing mathematical programming problem
Ghista, D. N.↗
On the atmosphere as an optical communication channel.
Mathematical model for atmosphere as optical communication channel, noting error probability dependence on amount of turbulence and diversity
Hoversten, E. V.↗
Potential functions in mathematical pattern recognition
Mathematical pattern recognition, using potential functions superposition methods for reconstructing probability density from samples
Meisel, W. S.↗
Deriving Laws from Ordering Relations
The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra. These rules are known as the sum rule, the product rule and Bayes Theorem, and the measure resulting from this generalization is probability. In this paper, I will describe how Cox s technique can be further generalized to include other algebras and hence other problems in science and mathematics. The result is a methodology that can be used to generalize an algebra to a calculus by relying on consistency with order theory to derive the laws of the calculus. My goals are to clear up the mysteries as to why the same basic structure found in probability theory appears in other contexts, to better understand the foundations of probability theory, and to extend these ideas to other areas by developing new mathematics and new physics. The relevance of this methodology will be demonstrated using examples from probability theory, number theory, geometry, information theory, and quantum mechanics.
Knuth, Kevin H.↗
A mathematical model of hardware and system testing
Hardware and system testing mathematical model assuming stochastic design and manufacturing defects detection and nondetection probability equations analogous to reliability function
Lind, M. M.↗
The semi-Markov unreliability range evaluator program
The SURE program is a design/validation tool for ultrareliable computer system architectures. The system uses simple algebraic formulas to compute accurate upper and lower bounds for the death state probabilities of a large class of semi-Markov models. The mathematical formulas used in the program were derived from a mathematical theorem proven by Allan White under contract to NASA Langley Research Center. This mathematical theorem is discussed along with the user interface to the SURE program.
Butler, R. W.↗
Space Mathematics: A Resource for Secondary School Teachers
A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.
Kastner, Bernice↗
Semi-Markov Unreliability Range Evaluator
Semi-Markov Unreliability Range Evaluator, SURE, computer program is software tool for analysis of reconfigurable, fault-tolerant systems. Traditional reliability analyses based on aggregates of fault-handling and fault-occurrence models. SURE provides efficient means for calculating accurate upper and lower bounds for probabilities of death states for large class of semi-Markov mathematical models, and not merely those reduced to critical-pair architectures.
Butler, Ricky W.↗
The Investigation Of Carbon Contamination And Sputtering Effects Of Xenon Ion Thrusters
The Electro-Physics Branch of the NASA Glenn Research Center investigates the effect of atomic oxygen, environmental durability of high performance power materials and surfaces, and low earth orbit. One of its current projects involves the analysis of ion thrusters. Ion thrusters are devices that initiate a beam of ions to a target area. The type of ion thruster that I have been working with this Summer of 2004 emits positively charged Xenon (Xe(+)) atoms through two grids, the screen grid and the accelerator grid, after it enters an ionization chamber. Insulators are used to mechanically hold and separate these two grids. A propellant isolator, an instrument that closely resembles insulators, is placed in front of the ionization chamber. Both the insulator and isolator are made with a ceramic compound and filled with insulating beads. The main difference between the two devices is that the propellant isolator allows gas to flow through, in this case, the gas is Xe(+) and the insulators do not. In order to avoid carbon deposits and other contaminating chemicals to settle on the insulators and propellant isolator, a metal shadow shield is placed around them. These shadow shields function as a protectant and can be shaped in numerous configurations. Part of my job responsibility this summer is to investigate the effectiveness of different shadow shields that are utilized on three different ion engines: the NSTAR (NASA Solar Electric Propulsion Technology Application Readiness), JIMO (Jupiter Icy Moons Orbiter), and NEXIS (Nuclear Electric Xenon Ion System). Using calculus and other mathematical tactics, I was asked to find the total flux of carbon contamination that was able to pass the protectant shadow shield. I familiarized myself with the software program, MathCad2004, to help perform some mathematical computations such as complex integration. Another method of studying the probability of contamination is by experimental simulation. After attaining the precise parameters of the actual shadow shields, I created replicas of three types of shadow shielding to be used to undergo testing. It will be placed in a machine that produces carbon atoms at a high temperature of 200 C. or beam is aimed at a targeted material. As a result of this collision, atoms and other particles are ejected out of the target surface. Another part of my internship consisted of research on sputter ejection, or the angle distribution of sputtered material. This research entailed finding the past results of sputter ejection investigation as well as creating another type of mock simulation. Other minor projects include calculating the path of Xe(+) gas through the insulating beads of the isolators and assisting my mentor in collecting data for his paper for the Joint Propulsion Conference & Exhibit to be held July 11-14,2004 in Fort Lauderdale, Florida.
Prak, Moline K.↗
Model-Based Method for Sensor Validation
Fault detection, diagnosis, and prognosis are essential tasks in the operation of autonomous spacecraft, instruments, and in situ platforms. One of NASA s key mission requirements is robust state estimation. Sensing, using a wide range of sensors and sensor fusion approaches, plays a central role in robust state estimation, and there is a need to diagnose sensor failure as well as component failure. Sensor validation can be considered to be part of the larger effort of improving reliability and safety. The standard methods for solving the sensor validation problem are based on probabilistic analysis of the system, from which the method based on Bayesian networks is most popular. Therefore, these methods can only predict the most probable faulty sensors, which are subject to the initial probabilities defined for the failures. The method developed in this work is based on a model-based approach and provides the faulty sensors (if any), which can be logically inferred from the model of the system and the sensor readings (observations). The method is also more suitable for the systems when it is hard, or even impossible, to find the probability functions of the system. The method starts by a new mathematical description of the problem and develops a very efficient and systematic algorithm for its solution. The method builds on the concepts of analytical redundant relations (ARRs).
Vatan, Farrokh↗
Safety Assessment of Conformance Monitoring for Situational Awareness in UTM Operations
This report presents a systematic approach to evaluating the safety benefit of utilizing strategic deconfliction and Conformance Monitoring for Situational Awareness to manage the traffic that is comprised of both conforming aircraft and contingent aircraft. First, we developed a Monte-Carlo-based simulation platform that generates a range of nominal flight paths, a set of flight paths associated with contingent aircraft, encounters between conforming and contingent aircraft, and simulates aircraft's reported positions within conformance bounds. Next, we derived the mathematical equations for quantifying the level of safety, represented by the probability of mid-air collision per flight hour, using the flight data from the simulation platform. Finally, we compared the safety benefit of CMSA services in scenarios generated at different traffic densities and rates of contingent aircraft. This study suggests that strategic deconfliction and CMSA services can collaboratively handle traffic mixed with conforming and contingent aircraft more safely than strategic deconfliction alone.
CMSA↗
Self-Supervised Dynamical Systems
Some progress has been made in a continuing effort to develop mathematical models of the behaviors of multi-agent systems known in biology, economics, and sociology (e.g., systems ranging from single or a few biomolecules to many interacting higher organisms). Living systems can be characterized by nonlinear evolution of probability distributions over different possible choices of the next steps in their motions. One of the main challenges in mathematical modeling of living systems is to distinguish between random walks of purely physical origin (for instance, Brownian motions) and those of biological origin. Following a line of reasoning from prior research, it has been assumed, in the present development, that a biological random walk can be represented by a nonlinear mathematical model that represents coupled mental and motor dynamics incorporating the psychological concept of reflection or self-image. The nonlinear dynamics impart the lifelike ability to behave in ways and to exhibit patterns that depart from thermodynamic equilibrium. Reflection or self-image has traditionally been recognized as a basic element of intelligence. The nonlinear mathematical models of the present development are denoted self-supervised dynamical systems. They include (1) equations of classical dynamics, including random components caused by uncertainties in initial conditions and by Langevin forces, coupled with (2) the corresponding Liouville or Fokker-Planck equations that describe the evolutions of probability densities that represent the uncertainties. The coupling is effected by fictitious information-based forces, denoted supervising forces, composed of probability densities and functionals thereof. The equations of classical mechanics represent motor dynamics that is, dynamics in the traditional sense, signifying Newton s equations of motion. The evolution of the probability densities represents mental dynamics or self-image. Then the interaction between the physical and metal aspects of a monad is implemented by feedback from mental to motor dynamics, as represented by the aforementioned fictitious forces. This feedback is what makes the evolution of probability densities nonlinear. The deviation from linear evolution can be characterized, in a sense, as an expression of free will. It has been demonstrated that probability densities can approach prescribed attractors while exhibiting such patterns as shock waves, solitons, and chaos in probability space. The concept of self-supervised dynamical systems has been considered for application to diverse phenomena, including information-based neural networks, cooperation, competition, deception, games, and control of chaos. In addition, a formal similarity between the mathematical structures of self-supervised dynamical systems and of quantum-mechanical systems has been investigated.
Zak, Michail↗
Stochastic Error Cancellation in Analog Quantum Simulation
Analog quantum simulation is a promising path towards solving classically intractable problems in many-body physics on near-term quantum devices. However, the presence of noise limits the size of the system and the length of time that can be simulated. In our work, we consider an error model in which the actual Hamiltonian of the simulator differs from the target Hamiltonian we want to simulate by small local perturbations, which are assumed to be random and unbiased. We analyze the error accumulated in observables in this setting and show that, due to stochastic error cancellation, with high probability the error scales as the square root of the number of qubits instead of linearly. We explore the concentration phenomenon of this error as well as its implications for local observables in the thermodynamic limit. Moreover, we show that stochastic error cancellation also manifests in the fidelity between the target state at the end of time-evolution and the actual state we obtain in the presence of noise. This indicates that, to reach a certain fidelity, more noise can be tolerated than implied by the worst-case bound if the noise comes from many statistically independent sources.
Analog quantum simulation↗
Structural Reliability Analysis and Optimization: Use of Approximations
This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different approximations including the higher-order reliability methods (HORM) for representing the failure surface. This report is divided into several parts to emphasize different segments of the structural reliability analysis and design. Broadly, it consists of mathematical foundations, methods and applications. Chapter I discusses the fundamental definitions of the probability theory, which are mostly available in standard text books. Probability density function descriptions relevant to this work are addressed. In Chapter 2, the concept and utility of function approximation are discussed for a general application in engineering analysis. Various forms of function representations and the latest developments in nonlinear adaptive approximations are presented with comparison studies. Research work accomplished in reliability analysis is presented in Chapter 3. First, the definition of safety index and most probable point of failure are introduced. Efficient ways of computing the safety index with a fewer number of iterations is emphasized. In chapter 4, the probability of failure prediction is presented using first-order, second-order and higher-order methods. System reliability methods are discussed in chapter 5. Chapter 6 presents optimization techniques for the modification and redistribution of structural sizes for improving the structural reliability. The report also contains several appendices on probability parameters.
Grandhi, Ramana V.↗
Computing the Instantaneous Collision Probability between Satellites using Characteristic Function Inversion
The probability that two satellites overlap in space at a specified instant of time is called their instantaneous collision probability. Assuming Gaussian uncertainties and spherical satellites, this probability is the integral of a Gaussian distribution over a sphere. This paper shows how to compute the probability using an established numerical procedure called characteristic function inversion. The collision probability in the short-term encounter scenario is also evaluated with this approach, where the instant at which the probability is computed is the time of closest approach between the objects. Python and R code is provided to evaluate the probability in practice. Overall, the approach has been established for over fifty years, is implemented in existing software, does not rely on analytical approximations, and can be used to evaluate two and three dimensional collision probabilities.
79 ASTRONOMY AND ASTROPHYSICS↗
Cost reduction potential of the DSN data base
The cost of the DSN data base can be reduced by computerizing and unifying the current multiplicity of separate manual, computer, and hybrid data bases. Savings would accrue from eliminating all manual system costs, increasing efficiency in data base implementation and maintenance-and-operation, and increasing data accuracy. By applying a simple mathematical savings model to current data base costs, this study estimates the probable range of net ten-year savings. The minimum net savings, under the assumptions of the study, are calculated as $7.5 million.
Mckenzie, M.↗
Dispositional logic
The applicability of conventional mathematical analysis (based on the combination of two-valued logic and probability theory) to problems in which human judgment, perception, or emotions play significant roles is considered theoretically. It is shown that dispositional logic, a branch of fuzzy logic, has particular relevance to the common-sense reasoning typical of human decision-making. The concepts of dispositionality and usuality are defined analytically, and a dispositional conjunctive rule and dispositional modus ponens are derived.
Le Balleur, J. C.↗