2018 — 2021 |
Ford, Kevin |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Primes, Divisors, and Permutations @ University of Illinois At Urbana-Champaign
Questions about properties of positive integers, especially the way in which integers factor and the distribution of prime numbers, have fascinated people for thousands of years and have recently found applications in computer science, information security and signal processing. Goals of the research are to enhance our understanding of the distribution of prime numbers, properties of divisors of integers, the distribution of random permutations. An equally important objective is to more fully understand the deeper connections between these seemingly dissimilar objects, that is, primes and permutations, thus building bridges between several areas of mathematics such as number theory, combinatorics, probability, and group theory.
One of the projects aims to develop a new methods, based on probabilistic reasoning, for establishing the existence of long strings of consecutive composite values in sparse sequences of integers, in order to gain a better understanding of the gaps between prime values of the sequence. Of particular interest are the sequences of values of a given polynomial. Another project will show that there exist extremely large discrepancies, in a precise quantitative sense, in the distribution of primes in arithmetic progressions to a given modulus when many residue classes are viewed together. This will also involve the creation of new tools for dealing with large deviations of multivariate distributions. The third project is dedicated toward a better understanding of the the concentration of divisors of integers and the concentration of fixed sets of random permutations. The research will improve existing results about the distribution of the maximal concentration of divisors in dyadic intervals, and maximal number of fixed sets of any size, and obtain sharp bounds on the largest interval which contains k divisors of a typical integer, for any quantity k. A common probabilistic model for divisors and permutations will guide the investigations.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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0.948 |
2018 |
Ford, Kevin Ray |
R21Activity Code Description: To encourage the development of new research activities in categorical program areas. (Support generally is restricted in level of support and in time.) |
Real-Time Optimized Biofeedback Utilizing Sport Techniques (Robust)
? DESCRIPTION (provided by applicant): Real-time Optimized Biofeedback Utilizing Sport Techniques (ROBUST) represents an innovative new approach to reduce traumatic anterior cruciate ligament (ACL) injuries. Over the last four decades, these debilitating injuries have occurred at a 2 to 10-fold greater rate in female compared to male athletes with the highest prevalence occurring between the ages of 16-18 years. As a consequence, there is a large population of females that endure significant pain, functional limitations and knee osteoarthritis (OA) as early as 5 years after the initial unintentional injury. To reduce the burden of OA, The National Public Health Agenda for Osteoarthritis recommends both expanding and refining evidence-based prevention of ACL injury. There currently is a gap in knowledge regarding how to maximize the effectiveness of injury prevention training in young female athletes. The long-term goal is to reduce ACL injuries in young female athletes. The objective of this application is to increase the efficacy of biofeedback training to reduce the risk of ACL injury. This proposal tests the central hypothesis that biofeedback methodology is needed to maximize the effectiveness of neuromuscular prophylactic interventions. The rationale supporting this proposal is that once the proposed research is completed, health professionals will be more successful at preventing devastating ACL injuries through properly optimized and targeted biofeedback training for young at-risk females. Specific Aim 1 will identify the most optimal, focused approach for biofeedback in adolescent females at high risk for ACL injury. A six-week randomized, pre/post-testing design will be used to identify biofeedback training effects. Specific Aim 2 will determine the effects of hip strategy on retention of decreased knee abduction load with focused biofeedback. A six-month follow-up design will be used to test retention of real-time biofeedback intervention. This research is innovative because it represents a new and substantive departure from the status quo by recognizing the need to optimize the application of biofeedback training. The work will contribute clinically relevant data in support of a future more robust clinical trial. The proposed research will be significant because it will lead to reduced rates of ACL injury in young females. Reduction of female injury rates to equal that of males would allow females annually to continue the health benefits of sports participation and avoid the long- term complications of osteoarthritis, which occurs with a 10 to 100-fold greater incidence in ACL-injured than in uninjured athletes.
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0.958 |