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Let A=a_(ij) be a matrix with positive coefficients and lambda_0 be the positive eigenvalue in the Frobenius theorem, then the n-1 eigenvalues lambda_j!=lambda_0 satisfy the ...

The eigenvalues lambda satisfying P(lambda)=0, where P(lambda) is the characteristic polynomial, lie in the unions of the disks |z|<=1 |z+b_1|<=sum_(j=1)^n|b_j|.

Given a sequence {a_k}_(k=1)^n, a partial sum of the first N terms is given by S_N=sum_(k=1)^Na_k.

F(x,s) = sum_(m=1)^(infty)(e^(2piimx))/(m^s) (1) = psi_s(e^(2piix)), (2) where psi_s(x) is the polygamma function.

If all elements a_(ij) of an irreducible matrix A are nonnegative, then R=minM_lambda is an eigenvalue of A and all the eigenvalues of A lie on the disk |z|<=R, where, if ...

If mu=(mu_1,mu_2,...,mu_n) is an arbitrary set of positive numbers, then all eigenvalues lambda of the n×n matrix a=a_(ij) lie on the disk |z|<=m_mu, where ...

f(z)=k/((cz+d)^r)f((az+b)/(cz+d)) where I[z]>0.

Let {y^k} be a set of orthonormal vectors with k=1, 2, ..., K, such that the inner product (y^k,y^k)=1. Then set x=sum_(k=1)^Ku_ky^k (1) so that for any square matrix A for ...

A quadratic form Q(x) is said to be positive semidefinite if it is never <0. However, unlike a positive definite quadratic form, there may exist a x!=0 such that the form is ...

Let alpha be a nonzero rational number alpha=+/-p_1^(alpha_1)p_2^(alpha_2)...p_L^(alpha_L), where p_1, ..., p_L are distinct primes, alpha_l in Z and alpha_l!=0. Then ...