Matching Items (4)
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- All Subjects: Combinatorial analysis
- All Subjects: Error Correction
- Creators: Colbourn, Charles
- Creators: Bevilacqua, Vincent Frank
Description
Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both in structure and in the types of objects that they list. More specific types of Gray codes are universal cycles and overlap sequences. Universal cycles are Gray codes on a set of strings of length n in which the first n-1 letters of one object are the same as the last n-1 letters of its predecessor in the listing. Overlap sequences allow this overlap to vary between 1 and n-1. Some of our main contributions to the areas of Gray codes and universal cycles include a new Gray code algorithm for fixed weight m-ary words, and results on the existence of universal cycles for weak orders on [n]. Overlap cycles are a relatively new structure with very few published results. We prove the existence of s-overlap cycles for k-permutations of [n], which has been an open research problem for several years, as well as constructing 1- overlap cycles for Steiner triple and quadruple systems of every order. Also included are various other results of a similar nature covering other structures such as binary strings, m-ary strings, subsets, permutations, weak orders, partitions, and designs. These listing structures lend themselves readily to some classes of combinatorial objects, such as binary n-tuples and m-ary n-tuples. Others require more work to find an appropriate structure, such as k-subsets of an n-set, weak orders, and designs. Still more require a modification in the representation of the objects to fit these structures, such as partitions. Determining when and how we can fit these sets of objects into our three listing structures is the focus of this dissertation.
ContributorsHoran, Victoria E (Author) / Hurlbert, Glenn H. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Colbourn, Charles (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2012
Description
The primary focus of this dissertation lies in extremal combinatorics, in particular intersection theorems in finite set theory. A seminal result in the area is the theorem of Erdos, Ko and Rado which finds the upper bound on the size of an intersecting family of subsets of an n-element set and characterizes the structure of families which attain this upper bound. A major portion of this dissertation focuses on a recent generalization of the Erdos--Ko--Rado theorem which considers intersecting families of independent sets in graphs. An intersection theorem is proved for a large class of graphs, namely chordal graphs which satisfy an additional condition and similar problems are considered for trees, bipartite graphs and other special classes. A similar extension is also formulated for cross-intersecting families and results are proved for chordal graphs and cycles. A well-known generalization of the EKR theorem for k-wise intersecting families due to Frankl is also considered. A stability version of Frankl's theorem is proved, which provides additional structural information about k-wise intersecting families which have size close to the maximum upper bound. A graph-theoretic generalization of Frankl's theorem is also formulated and proved for perfect matching graphs. Finally, a long-standing conjecture of Chvatal regarding structure of maximum intersecting families in hereditary systems is considered. An intersection theorem is proved for hereditary families which have rank 3 using a powerful tool of Erdos and Rado which is called the Sunflower Lemma.
ContributorsKamat, Vikram M (Author) / Hurlbert, Glenn (Thesis advisor) / Colbourn, Charles (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Kierstead, Henry (Committee member) / Arizona State University (Publisher)
Created2011
Description
As robots become more prevalent, the need is growing for efficient yet stable control systems for applications with humans in the loop. As such, it is a challenge for scientists and engineers to develop robust and agile systems that are capable of detecting instability in teleoperated systems. Despite how much research has been done to characterize the spatiotemporal parameters of human arm motions for reaching and gasping, not much has been done to characterize the behavior of human arm motion in response to control errors in a system. The scope of this investigation is to investigate human corrective actions in response to error in an anthropomorphic teleoperated robot limb. Characterizing human corrective actions contributes to the development of control strategies that are capable of mitigating potential instabilities inherent in human-machine control interfaces. Characterization of human corrective actions requires the simulation of a teleoperated anthropomorphic armature and the comparison of a human subject's arm kinematics, in response to error, against the human arm kinematics without error. This was achieved using OpenGL software to simulate a teleoperated robot arm and an NDI motion tracking system to acquire the subject's arm position and orientation. Error was intermittently and programmatically introduced to the virtual robot's joints as the subject attempted to reach for several targets located around the arm. The comparison of error free human arm kinematics to error prone human arm kinematics revealed an addition of a bell shaped velocity peak into the human subject's tangential velocity profile. The size, extent, and location of the additional velocity peak depended on target location and join angle error. Some joint angle and target location combinations do not produce an additional peak but simply maintain the end effector velocity at a low value until the target is reached. Additional joint angle error parameters and degrees of freedom are needed to continue this investigation.
ContributorsBevilacqua, Vincent Frank (Author) / Artemiadis, Panagiotis (Thesis director) / Santello, Marco (Committee member) / Trimble, Steven (Committee member) / Barrett, The Honors College (Contributor) / Mechanical and Aerospace Engineering Program (Contributor)
Created2013-05
Description
Quantum computers provide a promising future, where computationally difficult
problems can be executed exponentially faster than the current classical computers we have in use today. While there is tremendous research and development in the creation of quantum computers, there is a fundamental challenge that exists in the quantum world. Due to the fragility of the quantum world, error correction methods have originated since 1995 to tackle the giant problem. Since the birth of the idea that these powerful computers can crunch and process numbers beyond the limit of the current computers, there exist several mathematical error correcting codes that could potentially give the required stability in the fragile and fault tolerant quantum world. While there has been a multitude of possible solutions, there is no one single error correcting code that is the key to solving the problem. Almost every solution presented has shared with it a limiting factor or an issue that prevents it from becoming the breakthrough that is desperately needed.
This paper gives an introductory knowledge of what is the quantum world and why there is a need for error correcting topologies. Finally, it introduces one recent topology that could be added to the list of possible solutions to this central problem. Rather than focusing on the mathematical frameworks, the paper introduces the main concepts so that most readers even outside the major field of computer science can understand what the main problem is and how this topology attempts to solve it.
problems can be executed exponentially faster than the current classical computers we have in use today. While there is tremendous research and development in the creation of quantum computers, there is a fundamental challenge that exists in the quantum world. Due to the fragility of the quantum world, error correction methods have originated since 1995 to tackle the giant problem. Since the birth of the idea that these powerful computers can crunch and process numbers beyond the limit of the current computers, there exist several mathematical error correcting codes that could potentially give the required stability in the fragile and fault tolerant quantum world. While there has been a multitude of possible solutions, there is no one single error correcting code that is the key to solving the problem. Almost every solution presented has shared with it a limiting factor or an issue that prevents it from becoming the breakthrough that is desperately needed.
This paper gives an introductory knowledge of what is the quantum world and why there is a need for error correcting topologies. Finally, it introduces one recent topology that could be added to the list of possible solutions to this central problem. Rather than focusing on the mathematical frameworks, the paper introduces the main concepts so that most readers even outside the major field of computer science can understand what the main problem is and how this topology attempts to solve it.
ContributorsAhmed, Umer (Author) / Colbourn, Charles (Thesis director) / Zhao, Ming (Committee member) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05