Second order supersymmetry transformations which involve a pair of complex conjugate factorization energies and lead to real non-singular potentials are analyzed. The generation of complex potentials with real spectra is also studied. The theory is applied to the free particle, one-soliton well and one-dimensional harmonic oscillator.

In a survey paper [1], Agarwal and Ding summarized the advances achieved in the study of A-harmonic equations. Some recent results about A-harmonic equations can also be found in [2, 3, 5, 6]. The purpose of this note is to establish some estimates about Green’s operator and the Hodge codifferential operator d∗, which will enrich the existing literature in the field of A-harmonic equations. Let...

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means subordination, analytic parts which are reciprocal starlike (or convex) functions. Further, discuss geometric properties classes, such as integral expression, coefficient estimation, distortion theorem, Jacobian growth estimates, and covering theorem.

Let X be a Hilbert-space valued martingale and Y a real-valued supermartingale which are orthogonal and with Y diierentially subordinate to X. Then where the constant (2) is Catalan's constant whose approximate value is 0:915965594. The constant K is B. Davis' constant in the Kolmogorov's weak-type inequality for conjugate harmonic functions in the unit disk. The inequality is sharp.

A novel approach to computing time-harmonic solutions of Maxwell's equations by time-domain simulations is presented. The method, EM-WaveHoltz, results in a positive definite system which makes it amenable iterative solution with the conjugate gradient method or GMRES. Theoretical guaranteeing convergence away from resonances Numerical examples illustrating properties EM-WaveHoltz are given.

We consider known expressions for the eigenvalue of Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in N=4 super Yang-Mills theory as a real valued function two variables. define new functions complex conjugate variables that have definite complexity analogous to weight nested harmonic sums. argue those span general space BFKL at any order perturbation theory.

A second-generation pulse-burst laser system for high-speed flow diagnostics is described in detail. The laser can produce a burst of high-energy pulses (of the order of hundreds of millijoules per pulse) with individual pulse durations of less than 10 ns and pulse separations as short as 1 micros. A key improvement is the addition of a phase-conjugate mirror, which effectively isolates the hig...