In this paper we derive a formula for divided differences of composite functions of several variables with respect to rectangular grids of points. Letting the points coalesce yields a chain rule for partial derivatives of multivariate functions. Math Subject Classification: 41A05, 65D05, 26A06

Abstract In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio...

We interpret divided power structures on the homotopy groups of simplicial commutative rings as having a counterpart in divided power structures on chain complexes coming from a non-standard symmetric monoidal structure.

By a result of H. Cartan (cf. 5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of Cartan operations in the context of simplicial algebras over an operad.

The concept of comprehensive triangular decomposition (CTD) was first introduced by Chen et al. in their CASC’2007 paper and could be viewed as an analogue of comprehensive Gröbner systems for parametric polynomial systems. The first complete algorithm for computing CTD was also proposed in that paper and implemented in the RegularChains library in Maple. Following our previous work on generic ...

Leibniz superalgebras with nilindex $n + m$ and characteristic sequence $(n-1, 1 \ | m)$ divided into four parametric classes that contain a set of non-isomorphic superalgebras. In this paper, we give complete classification solvable whose nilradical is nilpotent superalgebra m)$. We obtain condition for the value parameters such which they have extension. Moreover, Lie maximal given.

In an effort to develop a robust and efficient front end for a chirped-pulse parametric amplification chain, we demonstrate a broadband difference-frequency converter driven by a monolithic femtosecond Yb-doped-fiber amplifier and emitting carrier-envelope-offset-free pulses with the energy of tens of nanojoules tunable in the wavelength range from 1200 nm to beyond 2 mum. Next to providing the...

A controller design method is proposed to control quadrotors with six degrees of freedom. The vehicle system is divided into four subsystems: the longitudinal, lateral, yaw, and height subsystems. A linear and decoupled nominal model is obtained for each subsystem, while coupling and nonlinear dynamics, parametric perturbations, and external disturbances are considered as uncertainties. For eac...

We demonstrate a two-crystal mid-infrared dual-wavelength optical parametric oscillator, synchronously pumped by a high power femtosecond Yb:fiber laser. The singly-resonant ring cavity, containing two periodically poled lithium niobate crystals, is capable of generating two synchronized idler wavelengths, independently tunable over 30 THz in the 2.9 - 4.2 μm wavelength region, due to the casca...