For transformation optics, comprising two or three cylindrical lenses, incoming standard Hermite Gaussian (SHG) modes are converted to twisted Hermite-Gaussian beams with complex argument if the rotational angle differs fromp/4.We show that these twisted modes can be expressed as a superposition of a set of elegant Hermite-Gaussian (EHG) modes with complex arguments multiplied by a Gaussian function. Expressions for the free propagation of these modes, is then derived for the first time. Examples are given for the far-field distributions of converted modes (using new expression for twisted Hermite-Gaussian with complex arguments).

In this paper, we propose to extend the hierarchical bivariate Hermite Interpolant to the spherical case. Let T be an arbitrary spherical triangle of the unit sphere S and let u be a function defined over the triangle T. For kϵN, we consider a Hermite spherical Interpolant problem Hk defined by some data scheme Dk (u) and which admits a unique solution pk in the space Bnk (T) of homogeneous Bernstein-Bezier polynomials of degree nk=2 k (resp. nk=2k+1) defined on T. We discuss the case when the data scheme Dr(u) are nested, i.e., Dr-1 (u) ÌDr(u) for all 1£r£k. This, give a recursive formulae to compute the polynomialpk. Moreover, this decomposition give a new basis for the space Bnk (T), which are the hierarchical structure. The method is illustrated by a simple numerical example.

In this paper, one numerical method is presented for numerical approximation of linear constrained optimal control problems with quadratic performance index. The method with variable coefficients is based on Hermite polynomials. The properties of Hermite polynomials with the operational matrices of derivative are used to reduce optimal control problems to the solution of linear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, and pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix polynomials are underlined.

In this paper, a new inequality for generalized convex functions which is related to the left side of generalized Hermite-Hadamard type inequality is obtained. Some applications for some generalized special means are also given.

The main aim of this article is to present error approximations for Hermite-Hadamard inequalities, which are connected with $_{\omega_{1}}\mathfrak{q}$-fractional calculus operators and are based on $\mathrm{(\rho, m)}$-convex functions. The investigation revolves around our primary results, which are established using a specific identity. Moreover, we demonstrate that the results  from earlier literature can be derived as special cases from our more comprehensive findings.