Algebraic treatment of the Bateman Hamiltonian

We apply the algebraic method to the Bateman Hamiltonian and obtain its natural frequencies and ladder operators from the adjoint or regular matrix representation of that operator. Present analysis shows that the eigenfunctions compatible with the complex eigenvalues obtained earlier by other authors are not square integrable. In addition to this, the ladder operators annihilate an infinite number of such eigenfunctions.

A Note on Simulating Predecessor-Successor Relationships in Critical Path Models

For any n entities, we exhaust all possible ordered relationships, from rank (or the highest number of connections in a linear chain, comparable to matrix rank) 0 to (n - 1). As an example, we use spreadsheets with the “RAND” function to simulate the case of n = 8 with the order-length = 3, as from a total of 10000possibilities by the number of combinations of selecting 2 (a pair of predecessor-successor) out of 5 (= card{A, B, C, D} + 1) matching- destinations followed by an exponentiation of 4 (= 8 – card{A, B, C, D}). Since the essence of this paper is about ordered structures of networks, our findings here may serve multi-disciplinary interests, in particular, that of the critical path method (CPM) in operations with management. In this connection, we have also included, toward the end of this exposition, a linear algebraic treatment that renders a deterministic mathematical programming for optimal predecessor-successor network structures.

An algebraic treatment of the Askey biorthogonal polynomials on the unit circle

A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak {J}$ .

Algebraic treatment of the Pais–Uhlenbeck oscillator and its PT-variant

The algebraic method enables one to study the properties of the spectrum of a quadratic Hamiltonian through the mathematical properties of a matrix representation called regular or adjoint. This matrix exhibits exceptional points where it becomes defective and can be written in canonical Jordan form. It is shown that any quadratic function of K coordinates and K momenta leads to a 2K differential equation for those dynamical variables. We illustrate all these features of the algebraic method by means of the Pais–Uhlenbeck oscillator and its PT-variant.