In the article, the noted Darboux’s formula of functions with single variable is generalized to that of functions of two independent variables with integral remainder, some important special cases of Darboux’s formula of functions with two variables are obtained, and some estimates of the integral remainders and Darboux’s expansion of the function ln(x + y) are given. These results generalize A...

Methods Clinical study which enrolled 108 cases, from 3 to 16 years of age who was diagnosed as BJH previously by Beighton Criteria. The children separately examined by two qualified pediatric rheumatologists and the Beighton score for hypermobility was calculated again; eight children were excluded from the study during re-exams and for remainders (100 cases) examinations were performed based ...

For a prime p and a matrix A ∈ Zn×n, write A as A = p(A quo p)+ (A rem p) where the remainder and quotient operations are applied element-wise. Write the p-adic expansion of A as A = A[0] + pA[1] + p2A[2] + · · · where each A[i] ∈ Zn×n has entries between [0, p − 1]. Upper bounds are proven for the Z-ranks of A rem p, and A quo p. Also, upper bounds are proven for the Z/pZ-rank of A[i] for all ...

A bstract We present the two-loop leading colour QCD helicity amplitudes for process pp ? W ( l? ) ? + j . implement a complete reduction of amplitudes, including leptonic decay -boson, using finite field arithmetic, and extract analytic remainders recently identified basis special functions. Simplified expressions are obtained after considering permutations rational kinematic parametrisation m...

Based on the blossoming theory, in this work we develop a new method for deriving Hermite-Padé approximants of certain hypergeometric series. Its general principle consists building identities generalising Hermite identity exponentials, and then applying their blossomed versions to appropriate tuples simultaneously produce explicit expressions integral representations corresponding remainders. ...

We consider two types of Cheney–Sharma operators for functions defined on a triangle with all straight sides. construct their product and Boolean sum, we study interpolation properties the orders accuracy give different expressions corresponding remainders, highlighting symmetry between methods. also some illustrative numerical examples.