The theory of stationary point processes.
Axiomatic formulation for stationary point processes interpreted as ordered sequences of points randomly located on real line, noting relation to set theory
SET THEORY↗
SEARCH · Search NASA
Results for “Set theory”
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
TOPOLOGICALLY CONVEX SETS AND FIXED POINT THEORY
Topologically convex sets and fixed point theory - mapping
TOPOLOGY↗
Path integral suppression of badly behaved causal sets
Abstract Causal set theory is a discrete model of spacetime that retains a notion of causal structure. We understand how to construct causal sets that approximate a given spacetime, but most causal sets are not at all manifold-like, and must be dynamically excluded if something like our Universe is to emerge from the theory. Here we show that the most common of these ‘bad’ causal sets, the Kleitman–Rothschild orders, are strongly suppressed in the gravitational path integral, and we provide evidence that a large class of other ‘bad’ causal sets are similarly suppressed. It thus becomes plausible that continuum behavior could emerge naturally from causal set quantum theory.
Astronomy & Astrophysics↗
The PDF4LHC21 combination of global PDF fits for the LHC Run III
A precise knowledge of the quark and gluon structure of the proton, encoded by the parton distribution functions (PDFs), is of paramount importance for the interpretation of high-energy processes at present and future lepton–hadron and hadron–hadron colliders. Motivated by recent progress in the PDF determinations carried out by the CT, MSHT, and NNPDF groups, we present an updated combination of global PDF fits: PDF4LHC21. It is based on the Monte Carlo combination of the CT18, MSHT20, and NNPDF3.1 sets followed by either its Hessian reduction or its replica compression. Extensive benchmark studies are carried out in order to disentangle the origin of the differences between the three global PDF sets. In particular, dedicated fits based on almost identical theory settings and input datasets are performed by the three groups, highlighting the role played by the respective fitting methodologies. We compare the new PDF4LHC21 combination with its predecessor, PDF4LHC15, demonstrating their good overall consistency and a modest reduction of PDF uncertainties for key LHC processes such as electroweak gauge boson production and Higgs boson production in gluon fusion. We study the phenomenological implications of PDF4LHC21 for a representative selection of inclusive, fiducial, and differential cross sections at the LHC. The PDF4LHC21 combination is made available via the LHAPDF library and provides a robust, user-friendly, and efficient method to estimate the PDF uncertainties associated to theoretical calculations for the upcoming Run III of the LHC and beyond.
quantum chromodynamics↗
Intelligent System Development Using a Rough Sets Methodology
The purpose of this research was to examine the potential of the rough sets technique for developing intelligent models of complex systems from limited information. Rough sets a simple but promising technology to extract easily understood rules from data. The rough set methodology has been shown to perform well when used with a large set of exemplars, but its performance with sparse data sets is less certain. The difficulty is that rules will be developed based on just a few examples, each of which might have a large amount of noise associated with them. The question then becomes, what is the probability of a useful rule being developed from such limited information? One nice feature of rough sets is that in unusual situations, the technique can give an answer of 'I don't know'. That is, if a case arises that is different from the cases the rough set rules were developed on, the methodology can recognize this and alert human operators of it. It can also be trained to do this when the desired action is unknown because conflicting examples apply to the same set of inputs. This summer's project was to look at combining rough set theory with statistical theory to develop confidence limits in rules developed by rough sets. Often it is important not to make a certain type of mistake (e.g., false positives or false negatives), so the rules must be biased toward preventing a catastrophic error, rather than giving the most likely course of action. A method to determine the best course of action in the light of such constraints was examined. The resulting technique was tested with files containing electrical power line 'signatures' from the space shuttle and with decompression sickness data.
Anderson, Gray T.↗
Optimal Scheduling of a Multi-User, Multi-Tasking Automatic Robotic Telescope
An algorithm is introduced to generate the optimal schedule of observations for a multi-user, multi-tasking automatic robotic telescope. The schedule is optimal m that it will take the highest possible quality data, perform the largest number of tasks, and fairly serve multiple users in terms of data quality, allocated telescope time, and observation priorities. In addition, the algorithm addresses problems due to overloading and underloading the telescope. The algorithm, which employs dynamic-programming-based decision theory, is analytically proven to produce the optimal schedule using set theory. Simulations conducted using actual astronomical data are presented which show significant improvement over existing scheduling techniques. Once completed the new scheduling algorithm will be tested on the Tennessee State University/Harvard-Smithsonian Center for Astrophysics 32-inch automatic Photoelectric Telescope at Fairborn Observatory, Arizona.
Tantaris, R. N.↗
Causal sets and an emerging continuum
Abstract Causal set theory offers a simple and elegant picture of discrete physics. But the vast majority of causal sets look nothing at all like continuum spacetimes, and must be excluded in some way to obtain a realistic theory. I describe recent results showing that almost all non-manifoldlike causal sets are, in fact, very strongly suppressed in the gravitational path integral. This does not quite demonstrate the emergence of a continuum—we do not yet understand the remaining unsuppressed causal sets well enough—but it is a significant step in that direction.
Astronomy & Astrophysics↗
Scattering Amplitudes and QFT Insight (Final Scientific Report)
The calculation of scattering amplitudes provides an invariant window into the physical dynamic content of relativistic quantum field theory. Yet with traditional methods, for empirically relevant theories, these calculations scale with a factorial complexity in external particles and precision. If we aspire to collapse the theoretical uncertainty obscuring new physics hidden at all scales from the microscopic probed at high energy colliders to the largest effective field theory in the universe governing the evolution of large scale structure, this complexity challenge necessitates new ideas and methods in calculation. Novel approaches like the color-kinematics duality and the associated double-copy construction have drastically simplified the situation, relating both gauge and gravity theory predictions to a much smaller kernel of invariant kinematic data. This project supporting research by the PI's Amplitudes and Insights group at Northwestern University looked to push insight deep into both the IR and the UV by establishing how novel structures must constrain the predictions of counterterms in both gauge and gravity theories, setting the groundwork to exploring the high energy behavior of particular gravity theories via constituent gauge-theory calculations, and confronting fundamental challenges at the interface between QFT amplitudes analysis and next generation gravitational wave science as well as well as inflationary and large scale structure cosmology.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
North American Fuzzy Logic Processing Society (NAFIPS 1992), volume 1
This document contains papers presented at the NAFIPS '92 North American Fuzzy Information Processing Society Conference. More than 75 papers were presented at this Conference, which was sponsored by NAFIPS in cooperation with NASA, the Instituto Tecnologico de Morelia, the Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP), the Instituto Tecnologico de Estudios Superiores de Monterrey (ITESM), the International Fuzzy Systems Association (IFSA), the Japan Society for Fuzzy Theory and Systems, and the Microelectronics and Computer Technology Corporation (MCC). The fuzzy set theory has led to a large number of diverse applications. Recently, interesting applications have been developed which involve the integration of fuzzy systems with adaptive processes such as neural networks and genetic algorithms. NAFIPS '92 was directed toward the advancement, commercialization, and engineering development of these technologies.
Villarreal, James A.↗
North American Fuzzy Logic Processing Society (NAFIPS 1992), volume 2
This document contains papers presented at the NAFIPS '92 North American Fuzzy Information Processing Society Conference. More than 75 papers were presented at this Conference, which was sponsored by NAFIPS in cooperation with NASA, the Instituto Tecnologico de Morelia, the Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP), the Instituto Tecnologico de Estudios Superiores de Monterrey (ITESM), the International Fuzzy Systems Association (IFSA), the Japan Society for Fuzzy Theory and Systems, and the Microelectronics and Computer Technology Corporation (MCC). The fuzzy set theory has led to a large number of diverse applications. Recently, interesting applications have been developed which involve the integration of fuzzy systems with adaptive processes such a neural networks and genetic algorithms. NAFIPS '92 was directed toward the advancement, commercialization, and engineering development of these technologies.
Villarreal, James A.↗
Two Methods for Efficient Solution of the Hitting-Set Problem
A paper addresses much of the same subject matter as that of Fast Algorithms for Model-Based Diagnosis (NPO-30582), which appears elsewhere in this issue of NASA Tech Briefs. However, in the paper, the emphasis is more on the hitting-set problem (also known as the transversal problem), which is well known among experts in combinatorics. The authors primary interest in the hitting-set problem lies in its connection to the diagnosis problem: it is a theorem of model-based diagnosis that in the set-theory representation of the components of a system, the minimal diagnoses of a system are the minimal hitting sets of the system. In the paper, the hitting-set problem (and, hence, the diagnosis problem) is translated from a combinatorial to a computational problem by mapping it onto the Boolean satisfiability and integer- programming problems. The paper goes on to describe developments nearly identical to those summarized in the cited companion NASA Tech Briefs article, including the utilization of Boolean-satisfiability and integer- programming techniques to reduce the computation time and/or memory needed to solve the hitting-set problem.
Vatan, Farrokh↗
Deriving Laws from Ordering Relations
It took much effort in the early days of non-Euclidean geometry to break away from the mindset that all spaces are flat and that two distinct parallel lines do not cross. Up to that point, all that was known was Euclidean geometry, and it was difficult to imagine anything else. We have suffered a similar handicap brought on by the enormous relevance of Boolean algebra to the problems of our age-logic and set theory. Previously, I demonstrated that the algebra of questions is not Boolean, but rather is described by the free distributive algebra. To get to this stage took much effort, as many obstacles-most self-placed-had to be overcome. As Boolean algebras were all I had ever known, it was almost impossible for me to imagine working with an algebra where elements do not have complements. With this realization, it became very clear that the sum and product rules of probability theory at the most basic level had absolutely nothing to do with the Boolean algebra of logical statements. Instead, a measure of degree of inclusion can be invented for many different partially ordered sets, and the sum and product rules fall out of the associativity and distributivity of the algebra. To reinforce this very important idea, this paper will go over how these constructions are made, while focusing on the underlying assumptions. I will derive the sum and product rules for a distributive lattice in general and demonstrate how this leads to probability theory on the Boolean lattice and is related to the calculus of quantum mechanical amplitudes on the partially ordered set of experimental setups. I will also discuss the rules that can be derived from modular lattices and their relevance to the cross-ratio of projective geometry.
Knuth, Kevin H.↗
An Image Processing Algorithm Based On FMAT
Information deleted in ways minimizing adverse effects on reconstructed images. New grey-scale generalization of medial axis transformation (MAT), called FMAT (short for Fuzzy MAT) proposed. Formulated by making natural extension to fuzzy-set theory of all definitions and conditions (e.g., characteristic function of disk, subset condition of disk, and redundancy checking) used in defining MAT of crisp set. Does not need image to have any kind of priori segmentation, and allows medial axis (and skeleton) to be fuzzy subset of input image. Resulting FMAT (consisting of maximal fuzzy disks) capable of reconstructing exactly original image.
Wang, Lui↗
Category Theory
Functions started with an implicit definition - the dependency of some quantity on others. Functions were coined by Leibniz in 1673, were given the f(x) notation by Euler in 1734, were formalized in the language of set theory from around 1847 to 1939. Functions have since been further abstract, often capture and represent structure.
Alan Hylton↗
MFANS 2024 - Formally Proving Characteristics of Cyber-Physical Systems
Cyber-physical systems (CPS) are engineered systems that rely on the smooth integration of computational algorithms and physical elements. This integration presents new challenges for verifying that systems will behave as expected. The goal of this presentation is to present current challenges and potential solutions for the formal verification of cyber-physical systems. For cyber systems, formal methods refer to systematically rigorous mathematical techniques employed in the specification, development, analysis, and verification of both software and hardware systems. Recent advancements in computer science have yielded sophisticated tools specifically designed to address challenges associated with formal methods in complex systems. These tools leverage various foundational concepts such as logic, formal languages, program semantics, type systems, type theory, and automata theory. A notable achievement in the application of formal methods is the seL4 microkernel, claimed to be the first general-purpose operating-system kernel to be verified. Its proof implies the absence of bugs and guarantees that the kernel meets specifications. For physical systems, dynamic and control theory has a history of using rigorous analytic techniques to prove functional correctness. Lyapunov, optimal, classical, modern, and robust control theories all provide rigorous mathematical methods both to analyze system performance and to design controller that can be guaranteed to meet certain objectives. Recent computational techniques like level set theory and reachability analysis provide assertions that a system's state will avoid unsafe regions. Even though success has been independently achieved for cyber systems and physical systems, the integration of such systems creates new challenges. In particular, there is an obvious discrepancy between finite-state machines and infinite-state systems, resulting in different approaches for modeling and analyzing these system. While it is possible to simulate hybrid systems, this provides only a demonstration of a performance and not proof. For hybrid systems, current formal methods and system analysis approaches typically require a workarounds to work on hybrid systems like CPS. This paper will outline the state of the art and limits of current practice for formally verifying CPS and will identify possible research directions that require attention.
97 MATHEMATICS AND COMPUTING↗
Multi-layered reasoning by means of conceptual fuzzy sets
The real world consists of a very large number of instances of events and continuous numeric values. On the other hand, people represent and process their knowledge in terms of abstracted concepts derived from generalization of these instances and numeric values. Logic based paradigms for knowledge representation use symbolic processing both for concept representation and inference. Their underlying assumption is that a concept can be defined precisely. However, as this assumption hardly holds for natural concepts, it follows that symbolic processing cannot deal with such concepts. Thus symbolic processing has essential problems from a practical point of view of applications in the real world. In contrast, fuzzy set theory can be viewed as a stronger and more practical notation than formal, logic based theories because it supports both symbolic processing and numeric processing, connecting the logic based world and the real world. In this paper, we propose multi-layered reasoning by using conceptual fuzzy sets (CFS). The general characteristics of CFS are discussed along with upper layer supervision and context dependent processing.
Takagi, Tomohiro↗
Forced nutations of the earth: Influence of inner core dynamics. I - Theory. II - Numerical results and comparisons. III - Very long interferometry data analysis
A treatment is presented of the nutation problem for an oceanless, elastic, spheroidally stratified earth, with the dynamical role of the inner core explicitly included in the formulation. Solving the enlarged system of equations shows that a new almost diurnal eigenfrequency emerges. A rough estimate places it not far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Tables are provided that exhibit the sensitivities of various relevant quantities, the eigenfrequencies and the coefficients which appear in the resonance expansion, as well as the nutation amplitudes at important tidal frequencies, to possible errors in the earth parameters which enter the theory set forth. Finally, the analysis of 798 VLBI experiments performed between July 1980 and February 1989 and the determination from this analysis of corrections to selected coefficients in the International Astronomical Union 1980 theory of the nutations of the earth are discussed.
Mathews, P. M.↗
Application of fuzzy theories to formulation of multi-objective design problems
Much of the decision making in real world takes place in an environment in which the goals, the constraints, and the consequences of possible actions are not known precisely. In order to deal with imprecision quantitatively, the tools of fuzzy set theory can by used. This paper demonstrates the effectiveness of fuzzy theories in the formulation and solution of two types of helicopter design problems involving multiple objectives. The first problem deals with the determination of optimal flight parameters to accomplish a specified mission in the presence of three competing objectives. The second problem addresses the optimal design of the main rotor of a helicopter involving eight objective functions. A method of solving these multi-objective problems using nonlinear programming techniques is presented. Results obtained using fuzzy formulation are compared with those obtained using crisp optimization techniques. The outlined procedures are expected to be useful in situations where doubt arises about the exactness of permissible values, degree of credibility, and correctness of statements and judgements.
Dhingra, A. K.↗