Publications (8)5.6 Total impact
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ABSTRACT: Let K be a cyclic number field of prime degree ℓ. Heilbronn showed that for a given ℓ there are only finitely many such fields that are normEuclidean. In the case of ℓ = 2 all such normEuclidean fields have been identified, but for ℓ ≠ 2, little else is known. We give the first upper bounds on the discriminants of such fields when ℓ > 2. Our methods lead to a simple algorithm which allows one to generate a list of candidate normEuclidean fields up to a given discriminant, and we provide some computational results.  [Show abstract] [Hide abstract]
ABSTRACT: Assuming the Generalized Riemann Hypothesis (GRH), we show that the normEuclidean Galois cubic fields are exactly those with discriminant $\Delta=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for $\zeta_K(s)$. If $38(\ell1)^2(\log f)^6\log\log f<f$, then $K$ is not normEuclidean. 
Article: NormEuclidean Galois fields
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ABSTRACT: Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are normEuclidean. In the case of $\ell=2$ all such normEuclidean fields have been identified, but for $\ell\neq 2$, little else is known. We give the first upper bounds on the discriminants of such fields when $\ell>2$. Our methods lead to a simple algorithm which allows one to generate a list of candidate normEuclidean fields up to a given discriminant, and we provide some computational results.  [Show abstract] [Hide abstract]
ABSTRACT: We give an explicit version of a result due to D. Burgess. Let $\chi$ be a nonprincipal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $\chi$ takes on a particular value is less than $\left\{\frac{\pi e\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.  [Show abstract] [Hide abstract]
ABSTRACT: Let $\chi$ be a nonprincipal Dirichlet character modulo a prime $p$. Let $q_1<q_2$ denote the two smallest prime nonresidues of $\chi$. We give explicit upper bounds on $q_2$ that improve upon all known results. We also provide a good upper estimate on the product $q_1 q_2$ which has an upcoming application to the study of normEuclidean Galois fields.  [Show abstract] [Hide abstract]
ABSTRACT: This paper describes the problem of feedback control for stabilization of the plasma vertical instability in a tokamak. Such controllers are typically designed based on a model that assumes the plasma mass m is identically zero while in reality the mass is small but positive. The assumption that m is zero can lead to a controller that appears to be stabilizing according to the massless analysis but in fact can increase the instability of the physical system.In this work, we consider a general class of controllers, which contains as a special case the type of controller most commonly used in operating tokamaks to stabilize the vertical instability, a proportionalderivative controller. Suppose C is a controller in this class which stabilizes the vertical instability with plasma mass assumed to be zero. We give easytocheck necessary and sufficient conditions for C to also stabilize the physical system, in which the plasma actually has a small mass. We allow for the possibility that the tokamak could have both superconducting and resistive conductors.The practical implications of the results presented provide substantial insight into some longstanding issues regarding feedback stabilization of the vertical instability with PD controllers and also provide a rigorous foundation for the common practice of designing controllers assuming m=0. For controllers that operate only on the plasma vertical position, we settle the question: when are m=0 models predictive of actual plasma behavior? 
Conference Paper: Conditions for a massless plasma analysis to predict stabilization of the tokamak plasma vertical instability
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ABSTRACT: This paper describes the problem of feedback control for stabilization of the plasma vertical instability in a tokamak. Such controllers are typically designed based on a model that assumes the plasma mass m is identically zero. However, the assumption of m = 0 can lead to a controller C that appears to be stabilizing according to the massless analysis but in fact will increase the instability of the physical system. In this work, we consider the most commonly used type of controller, a proportionalderivative controller. Suppose C is a PD controller which stabilizes the vertical instability with plasma mass assumed to be zero. We give easytocheck necessary and sufficient conditions for C to also stabilize the physical system, in which the plasma actually has a small mass.  [Show abstract] [Hide abstract]
ABSTRACT: A formula for the sum of any positiveintegral power of the first N positive integers was published by Johann Faulhaber in the 1600s. In this paper, we generalize Faulhaber's formula to nonintegral complex powers with real part greater than 1.
Publication Stats
8  Citations  
5.60  Total Impact Points  
Top Journals
Institutions

20112012

Oregon State University
 Department of Mathematics
Corvallis, Oregon, United States


20072010

University of California, San Diego
 Department of Mathematics
San Diego, California, United States
