Let f be a transcendental meromorphic function of order less than one. The authors prove that the exact difference Δf =(z+1)-f(z) has infinitely many fixed points, if a ∈ ℂ and ∞ are Borel exceptional values (or Nevanlinna deficiency values) of f. These results extend the related results obtained by Chen and Shon.

Theory of systems on homogeneous time scales unifies theories of continuous-time and discretetime systems. The characterizations of external dynamic equivalence known for continuous-time and discrete-time systems are extended here to systems on time scales. Under assumption of uniform observability, it is shown that two analytic control systems with output are externally dynamically equivalent ...

A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on l(Z≥0), leading to a proof of Favard’s theorem stating that polynomials satisfying a three-term recurrence relation are orthogonal polynomials. We discuss the link...

In this paper, continuous genetic algorithm is introduced as an efficient solver for systems of second-order boundary value problems where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. The solution methodology is based on representing each derivative in the system of differential equations by its finite di...

We derive explicit formulas for the eigenvalues and eigenvectors of the Discrete Laplacian on a rectangular grid for the standard finite difference and finite element methods in 1D, 2D, and 3D. Periodic, Dirichlet, Neumann, and mixed boundary conditions are all considered. We show how the higher dimensional operators can be written as sums of tensor products of one dimensional operators, and th...

Although applied economists often estimate interaction terms to infer how the effect of one independent variable on the dependent variable depends on the magnitude of another independent variable, most researchers misinterpret the coefficient in nonlinear models. The magnitude of the interaction effect does not equal the coefficient on the interaction term, can be of opposite sign, and is cond...

with Dirichlet boundary conditions, acting on l2(N,C2), resp. L2([0,∞),C2), where c > 0 represents the speed of light, m ≥ 0 the mass of a particle, I2 is the 2× 2 identity matrix and V is a bounded real potential. In the discrete case D is the finite difference operator defined by (Dφ)(n) = φ(n+1)−φ(n), with adjoint (Dφ)(n) = φ(n − 1) − φ(n), and in the continuous case D = D = −i d dx . Model ...

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy-Kovalevskaya extension theorem for discre...

We study maps preserving the Heisenberg commutation relation ab − ba = 1. We find a one-parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative. It involves a non-local “coordinate” operator while the dual “derivative” is just the Jackson finite-difference operator. Substitution of this realization into any differential operator in...

In this paper, using a generalized Jacobi-Dunkl translation operator, we prove a generalization of Titchmarsh’s theorem for functions in the k-JacobiDunkl-Lipschitz class defined by the finite differences of order k ∈ N∗ and Sobolev spaces associated with the Jacobi-Dunkl operator.