The purpose of this paper is to establish sev eral identities containing Gaussian binomial coefficient. These results generali ze several Mercier's results. Key words:Gaussian binomial coefficient; identity; the ...The purpose of this paper is to establish sev eral identities containing Gaussian binomial coefficient. These results generali ze several Mercier's results. Key words:Gaussian binomial coefficient; identity; the functio n A(T,T 2,...,T n)culating eige nvalues of auto-correlation matrix of the physical control force of actuators. T he optimization algorithm calculating the optimal actuator placement is then put forward via the minimization of an energy criterion, which is chosen as the con trol index. Numerical examples show the effectiveness of the proposed method.展开更多
We prove some 3-adic congruences for binomial sums,which were conjectured by Zhi-Wei Sun.For example,for any integer m≡1（mod 3）and any positive integer n,we have v3（1/n ∑n-1X k=0 1/mk 2k/ k〉））≥min{3（n）,3（...We prove some 3-adic congruences for binomial sums,which were conjectured by Zhi-Wei Sun.For example,for any integer m≡1（mod 3）and any positive integer n,we have v3（1/n ∑n-1X k=0 1/mk 2k/ k〉））≥min{3（n）,3（m-1）-1},where 3（n）denotes the 3-adic order of n.In our proofs,we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.展开更多
文摘The purpose of this paper is to establish sev eral identities containing Gaussian binomial coefficient. These results generali ze several Mercier's results. Key words:Gaussian binomial coefficient; identity; the functio n A(T,T 2,...,T n)culating eige nvalues of auto-correlation matrix of the physical control force of actuators. T he optimization algorithm calculating the optimal actuator placement is then put forward via the minimization of an energy criterion, which is chosen as the con trol index. Numerical examples show the effectiveness of the proposed method.
基金supported by National Natural Science Foundation of China (Grant Nos. 11271185,11171140 and 11226277)the Initial Founding of Scientific Research for the Introduction of Talents of Nanjing Institute of Technology,China (Grant No. YKJ201115)
文摘We prove some 3-adic congruences for binomial sums,which were conjectured by Zhi-Wei Sun.For example,for any integer m≡1（mod 3）and any positive integer n,we have v3（1/n ∑n-1X k=0 1/mk 2k/ k〉））≥min{3（n）,3（m-1）-1},where 3（n）denotes the 3-adic order of n.In our proofs,we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.
