Fourth-order Matrices Research Articles

Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices \(\mathfrak{P}_ +\) . Since elements of \(\mathfrak{P}_ +\) are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, \(\mathfrak{P}_ + \in \mathbb{R}^{15}\) . We define \(\mathfrak{P}_ +\) explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

Efficient optimal controller for nuclear power plants

A new scheme for the optimal control of nuclear power plants is proposed. It differs from previous applications of optimal control theory to nuclear reactors in that here it is possible to select the weighting matrices in the quadratic cost functional so that desired pole placement, and subsequent transient response, is achieved. The desired weighting matrix and corresponding optimal control law are found sequentially. From the computational point of view the proposed design algorithm is very efficient since the dynamical system is aggregated at each stage of the sequential process to a first- or a second-order system. Thus, almost the entire computation involves second- or fourth-order matrices.

Lyapunov stability and sign definiteness of a quadratic form in a cone

The use of the second Lyapunov method in many problems in the theory of stability of motion leads to the problem of sign definiteness of a quadratic form whose variables are defined in a convex polyhedral cone C ⊂R n. A method of obtaining the necessary and sufficient conditions is given for this problem. The conditions imposed on the elements of the third- and fourth-order matrices are given. The problem of asymptotic stability of a system with resonance /1/ is solved as an example. A number of problems of the theory of the stability of motion require that the sign definiteness of the quadratic form be established, with conditions written in the form of linear inequalities. Usually, the conditions are those of non-negativity /1–3/, and the more general conditions can be reduced to them. The problem of sign definiteness of a quadratic form under the conditions of non-negativeness was considered for an arbitrary number of variables in /4/. However, the results obtained there can be reduced to the problem of the compatibility of systems of inequalities and pose well-known difficulties when used to solve specific problems. The problem of the sign definiteness of a quadratic form in a convex cone (in general, infinitely-sided) belonging to a Hubert space is considered in /5/, and the necessary and sufficient conditions are obtained, but in the finite-dimensional case discussed below the above result is the same as that obtained in /4/.

Energy Levels in Ca XV

A discussion of the structure of term spectra in isoelectronic sequences. shows that spectral terms fall naturally into certain groups, which we call “ complexes ”. A complex is completely specified by the set of principal quantum numbers ( n ) and the parity p , and contains one or more entire configurations. The interval between terms that belong to the same complex varies asymptotically as the nuclear charge Z for large values of Z , while the interval between terms that belong to complexes with different sets of principal quantum numbers ( n ) varies asymptotically as Z2 . The electrostatic coupling between configurations whose terms belong to different complexes tends to vanish with increasing Z ; but the coupling between configurations whose terms belong to the same complex approaches constancy. This circumstance greatly simplifies the discussion of configuration mixing in highly ionized atoms. In Ca XV the configurations |$({1s}^{2})\,({2s}^{2})\,({2p}^{2})\,\text{and}\,({1s}^{2})\,({2p}^{4})$| are strongly coupled, but the complex |$({1}^{2}{2}^{4})$| is weakly coupled with higher complexes. On the approximation of zero coupling between |$({1}^{2}{2}^{4})$| and higher complexes, the energy levels of |$({1}^{2}{2}^{4})$| are the eigenvalues of a tenth-order matrix, which reduces to two fourth-order matrices and a second-order matrix. We have evaluated the matrix elements partly by extrapolation along the isoelectronic sequence and partly by theoretical considerations. The predicted wavelengths for the transitions |${2p}^{2}\,^{3}{P}_{2\rightarrow 1},\,^3{P}_{1\rightarrow 0}$| are 5456 A and 5650 A respectively. These agree with the observed wave-lengths of the coronal lines λλ 5445, 5694 to within the uncertainty of the calculation, thus confirming the identifications of Edlén and Waldmeier.

On positive stochastic matrices with real characteristic roots

Suleῐmanova (3) describes a geometrical technique* for discussing the conditions under which a given set of n + 1 numbers can be a set of characteristic roots of a positive stochastic matrix of order n + 1. Suleῐmanova states that, using this technique, it is possible to prove that a set of n + 1 real numbers 1, λ1, λ2, …, λn, where | λi | < 1 for i = 1, 2, …, n, is a set of characteristic roots of a positive stochastic matrix provided that the sum of the moduli of the negative numbers of the set is less than unity. No indication of the method to be employed is given in (3), and it would seem that the problem is a difficult one. In the present paper I give a full discussion of how Suleῐmanova's technique may be applied to solve the problem for third-order matrices, and simply an indication of the solution for fourth-order matrices; and in, the latter case I point out that Suleῐmanova's result may be replaced by a stronger one. For second-order matrices the problem is almost trivial. As a preliminary I prove two general theorems.