The Experts below are selected from a list of 103044 Experts worldwide ranked by ideXlab platform
Evgeniy Zorin - One of the best experts on this subject based on the ideXlab platform.
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explicit bounds for rational points near planar curves and metric diophantine approximation
Advances in Mathematics, 2010Co-Authors: Victor Beresnevich, Evgeniy ZorinAbstract:Abstract The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C 3 non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C 1 (lowest possible). This way we broaden the notion of non-degeneracy in a Natural Direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.
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explicit bounds for rational points near planar curves and metric diophantine approximation
arXiv: Number Theory, 2010Co-Authors: Victor Beresnevich, Evgeniy ZorinAbstract:The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to $C^1$ (lowest possible). This way we broaden the notion of non-degeneracy in a Natural Direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of non-degeneracy.
Victor Beresnevich - One of the best experts on this subject based on the ideXlab platform.
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explicit bounds for rational points near planar curves and metric diophantine approximation
Advances in Mathematics, 2010Co-Authors: Victor Beresnevich, Evgeniy ZorinAbstract:Abstract The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C 3 non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C 1 (lowest possible). This way we broaden the notion of non-degeneracy in a Natural Direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.
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explicit bounds for rational points near planar curves and metric diophantine approximation
arXiv: Number Theory, 2010Co-Authors: Victor Beresnevich, Evgeniy ZorinAbstract:The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to $C^1$ (lowest possible). This way we broaden the notion of non-degeneracy in a Natural Direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of non-degeneracy.
Insup Lee - One of the best experts on this subject based on the ideXlab platform.
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hierarchical scheduling framework for virtual clustering of multiprocessors
Euromicro Conference on Real-Time Systems, 2008Co-Authors: Insik Shin, Arvind Easwaran, Insup LeeAbstract:Scheduling of sporadic task systems on multiprocessor platforms is an area which has received much attention in the recent past. It is widely believed that finding an optimal scheduler is hard, and therefore most studies have focused on developing algorithms with good utilization bounds. These algorithms can be broadly classified into two categories: partitioned scheduling in which tasks are statically assigned to individual processors, and globalscheduling in which each task is allowed to execute on any processor in the platform. In this paper we consider a third, more general, approach called cluster-based scheduling. In this approach each task is statically assigned to a processor cluster, tasks in each cluster areglobally scheduled among themselves, and clusters in turn are scheduled on the multiprocessor platform. We develop techniques to support such cluster-based scheduling algorithms, and also consider properties that minimize processor utilization of individual clusters. Since neither partitioned nor global strategies dominate over the other, cluster-based scheduling is a Natural Direction for research towards achieving improved utilization bounds.
Akshayaram Srinivasan - One of the best experts on this subject based on the ideXlab platform.
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revisiting the cryptographic hardness of finding a nash equilibrium
International Cryptology Conference, 2016Co-Authors: Sanjam Garg, Omkant Pandey, Akshayaram SrinivasanAbstract:The exact hardness of computing a Nash equilibrium is a fundamental open question in algorithmic game theory. This problem is complete for the complexity class PPAD. It is well known that problems in PPAD cannot be $$\mathrm {NP}$$ -complete unless $$\mathrm {NP}=\mathrm {coNP}$$ . Therefore, a Natural Direction is to reduce the hardness of PPAD to the hardness of problems used in cryptography. Bitansky, Paneth, and Rosen [FOCS 2015] prove the hardness of PPAD assuming the existence of quasi-polynomially hard indistinguishability obfuscation and sub-exponentially hard one-way functions. This leaves open the possibility of basing PPAD hardness on simpler, polynomially hard, computational assumptions. We make further progress in this Direction and reduce PPAD hardness directly to polynomially hard assumptions. Our first result proves hardness of PPAD assuming the existence of polynomially hard indistinguishability obfuscation $$i\mathcal {O}$$ and one-way permutations. While this improves upon Bitansky et al.'s work, it does not give us a reduction to simpler, polynomially hard computational assumption because constructions of $$i\mathcal {O}$$ inherently seems to require assumptions with sub-exponential hardness. In contrast, public key functional encryption is a much simpler primitive and does not suffer from this drawback. Our second result shows that $$\mathsf{PPAD}$$ hardness can be based on polynomially hard compact public key functional encryption and one-way permutations. Our results further demonstrate the power of polynomially hard compact public key functional encryption which is believed to be weaker than indistinguishability obfuscation. Our techniques are general and we expect them to have various applications.
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revisiting the cryptographic hardness of finding a nash equilibrium
2016Co-Authors: Sanjam Garg, Omkant Pandey, Akshayaram SrinivasanAbstract:The exact hardness of computing a Nash equilibrium is a fundamental open question in algorithmic game theory. This problem is complete for the complexity class PPAD. It is well known that problems in PPAD cannot be NP-complete unless NP = coNP. Therefore, a Natural Direction is to reduce the hardness of PPAD to the hardness of problems used in cryptography. Bitansky, Paneth, and Rosen [FOCS 2015] prove the hardness of PPAD assuming the existence of quasi-polynomially hard indistinguishability obfuscation and sub-exponentially hard one-way functions. This leaves open the possibility of basing PPAD hardness on simpler, polynomially hard, computational assumptions. We make further progress in this Direction and reduce PPAD hardness directly to polynomially hard assumptions. Our first result proves hardness of PPAD assuming the existence of polynomially hard indistinguishability obfuscation (iO) and one-way permutations. While this improves upon Bitansky et al.’s work, it does not give us a reduction to simpler, polynomially hard computational assumption because constructions of iO inherently seems to require assumptions with sub-exponential hardness. In contrast, public key functional encryption is a much simpler primitive and does not suffer from this drawback. Our second result shows that PPAD hardness can be based on polynomially hard compact public key functional encryption and one-way permutations. Our results further demonstrate the power of polynomially hard compact public key functional encryption which is believed to be weaker than indistinguishability obfuscation. Our techniques are general and we expect them to have various applications. ∗University of California, Berkeley, sanjamg@berkeley.edu †Stony Brook University, omkant@gmail.com ‡University of California, Berkeley, akshayaram@berkeley.edu
Insik Shin - One of the best experts on this subject based on the ideXlab platform.
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hierarchical scheduling framework for virtual clustering of multiprocessors
Euromicro Conference on Real-Time Systems, 2008Co-Authors: Insik Shin, Arvind Easwaran, Insup LeeAbstract:Scheduling of sporadic task systems on multiprocessor platforms is an area which has received much attention in the recent past. It is widely believed that finding an optimal scheduler is hard, and therefore most studies have focused on developing algorithms with good utilization bounds. These algorithms can be broadly classified into two categories: partitioned scheduling in which tasks are statically assigned to individual processors, and globalscheduling in which each task is allowed to execute on any processor in the platform. In this paper we consider a third, more general, approach called cluster-based scheduling. In this approach each task is statically assigned to a processor cluster, tasks in each cluster areglobally scheduled among themselves, and clusters in turn are scheduled on the multiprocessor platform. We develop techniques to support such cluster-based scheduling algorithms, and also consider properties that minimize processor utilization of individual clusters. Since neither partitioned nor global strategies dominate over the other, cluster-based scheduling is a Natural Direction for research towards achieving improved utilization bounds.