We examine the problem of the existence and calculation of Hermitian solutions P of a linear matrix inequality corresponding to the spectral factorization of a proper rational spectral density. In particular, zeros are permitted on the finite imaginary axis or at infinity. As part of our construction of solutions, we show explicitly that each such imaginary-axis eigenvalue, either finite or inf...

A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor’...

most of the theologians and muslim philosophers, at least in accordance with tradition and narrative, believe in bodily resurrection as a necessary faith; but they have different conceptualizations of the “body of resurrection” such as: elemental body, heavenly body, intermediary body and imaginary body. the presentation of these conceptualizations will help us to find guidelines for more accur...

It is unusual for an irreducible polynomial to have a root with rational real part or with rational imaginary part. Of course, such polynomials exist: one can simply take the minimal polynomial of, say, 1+ i √ 2 or √ 2+ i. The same applies to polynomials having a root of rational modulus. But it turns out to be of interest to characterize these three kinds of polynomials. We therefore define ou...

In [4] it was shown for a suitable power n of a pair of units u, v of the quaternions algebras over the ring of integers of imaginary rational extensions A = H(o Q √ −d ) that the group generated by u, v is a free group in the unit group of A when d ≡ 7 (mod 8) is a positive square free integer. We extend this result for any imaginary rational extension and show that the group generated by u, v...