S.W. Colomb proposed the conjecture——there exists a positive integer q_0 such that every non-zero element in GF(q) can be written as a sum of two primitive roots if q>q_0.Such a q_0 has been obtained, but quite l...S.W. Colomb proposed the conjecture——there exists a positive integer q_0 such that every non-zero element in GF(q) can be written as a sum of two primitive roots if q>q_0.Such a q_0 has been obtained, but quite large. In this paper we consider whether or not every non-zero element in GF(q) can be written as a sum of two primitive roots for any prime power q=p^n. We will prove that the answer of this question is in the affirmative for one of the following cases: (i) q>6.62×10, and q≠300690391, (ii) n>1 and q≠2~2.If prime power q<10500, the answer of this question is in the negative for q=2, 3, 4, 5,7, 11, 13, 19, 31, 43, 61 and in the affirmative for the others.展开更多
Let q be a power of a prime, F_q a finite field with q elements, b a fixed primitive root of F_q and e a given divisor of q-1. Then the cyclotomic number (h, k )_e of order e in F_q is defined as the number of ordered...Let q be a power of a prime, F_q a finite field with q elements, b a fixed primitive root of F_q and e a given divisor of q-1. Then the cyclotomic number (h, k )_e of order e in F_q is defined as the number of ordered pairs (s, t )展开更多
最近,Takashi Agoh对于素数p≡1(mod4)给出了计算二次域Q(p^(1/2))的类数h的一个公式,此公式仅依赖于Q(p^(1/2))的基本单位ε,素数p以及数a=1+sum from k=1 to(p-1)/2((-1)~kN_K).孙琦教授对奇素数p,得到N_k的若干性质和计算N_2,N_3,N_...最近,Takashi Agoh对于素数p≡1(mod4)给出了计算二次域Q(p^(1/2))的类数h的一个公式,此公式仅依赖于Q(p^(1/2))的基本单位ε,素数p以及数a=1+sum from k=1 to(p-1)/2((-1)~kN_K).孙琦教授对奇素数p,得到N_k的若干性质和计算N_2,N_3,N_4的公式.本文对奇素数p,得到N_k的若干新性质和N_5,N_6的计算公式,展开更多
In this paper, neural networks are used to approximately solve the finite-horizon constrained input H-infinity state feedback control problem. The method is based on solving a related Hamilton-Jacobi-Isaacs equation o...In this paper, neural networks are used to approximately solve the finite-horizon constrained input H-infinity state feedback control problem. The method is based on solving a related Hamilton-Jacobi-Isaacs equation of the corresponding finite-horizon zero-sum game. The game value function is approximated by a neural network with time- varying weights. It is shown that the neural network approximation converges uniformly to the game-value function and the resulting almost optimal constrained feedback controller provides closed-loop stability and bounded L2 gain. The result is an almost optimal H-infinity feedback controller with time-varying coefficients that is solved a priori off-line. The effectiveness of the method is shown on the Rotational/Translational Actuator benchmark nonlinear control problem.展开更多
文摘S.W. Colomb proposed the conjecture——there exists a positive integer q_0 such that every non-zero element in GF(q) can be written as a sum of two primitive roots if q>q_0.Such a q_0 has been obtained, but quite large. In this paper we consider whether or not every non-zero element in GF(q) can be written as a sum of two primitive roots for any prime power q=p^n. We will prove that the answer of this question is in the affirmative for one of the following cases: (i) q>6.62×10, and q≠300690391, (ii) n>1 and q≠2~2.If prime power q<10500, the answer of this question is in the negative for q=2, 3, 4, 5,7, 11, 13, 19, 31, 43, 61 and in the affirmative for the others.
文摘Let q be a power of a prime, F_q a finite field with q elements, b a fixed primitive root of F_q and e a given divisor of q-1. Then the cyclotomic number (h, k )_e of order e in F_q is defined as the number of ordered pairs (s, t )
文摘最近,Takashi Agoh对于素数p≡1(mod4)给出了计算二次域Q(p^(1/2))的类数h的一个公式,此公式仅依赖于Q(p^(1/2))的基本单位ε,素数p以及数a=1+sum from k=1 to(p-1)/2((-1)~kN_K).孙琦教授对奇素数p,得到N_k的若干性质和计算N_2,N_3,N_4的公式.本文对奇素数p,得到N_k的若干新性质和N_5,N_6的计算公式,
基金This work was supported by the National Science Foundation (ECS-0501451)Army Research Office (W91NF-05-1-0314).
文摘In this paper, neural networks are used to approximately solve the finite-horizon constrained input H-infinity state feedback control problem. The method is based on solving a related Hamilton-Jacobi-Isaacs equation of the corresponding finite-horizon zero-sum game. The game value function is approximated by a neural network with time- varying weights. It is shown that the neural network approximation converges uniformly to the game-value function and the resulting almost optimal constrained feedback controller provides closed-loop stability and bounded L2 gain. The result is an almost optimal H-infinity feedback controller with time-varying coefficients that is solved a priori off-line. The effectiveness of the method is shown on the Rotational/Translational Actuator benchmark nonlinear control problem.
